The decrease in a person's generosity to others was less obvious in Japanese than in Germans. This pattern was similar to that in Strombach et al.'s (2014) comparison of Chinese and Germans. It is also congruent with the findings of Buchan et al. (2006), who showed that the effect of group membership was weaker in Chinese, Japanese, and Koreans than in North Americans in terms of trustworthiness. This result suggests that although Germans are more generous to others in gains than Japanese overall, the cultural difference in generosity is more pronounced among closer friends than among distant others. Thus, whereas Germans might behave generously to specific others to whom they think they can pay a cost (i.e., forego a reward) to maintain friendships with others, the generosity of Japanese might be moderate but relatively fair to acquaintances.

Whereas Japanese discounted social gain more than social loss, which is consistent with Takahashi's (2013) finding, no difference in social discounting between gain and loss was found among Germans. The Japanese pattern is consistent with the sign effect in delay discounting whereby people are more likely to behave impulsively in gains than in losses. In contrast, as the German participants in this study might have sufficiently high other-regarding motives, place great weight on others' outcomes, and likely dismiss their own outcome regardless of the frames, a difference between gain and loss might not appear. As this research provided the first evidence on cultural similarities and differences in the social discounting of losses and the sign effect, future work should examine the validity of the current findings in different cultures.

To understand cultural differences in social discounting, it would be informative to explore what socioecological factors influence costly generosity to others. A person's generosity to others based on sacrificing his or her reward might be more useful in a mobile environment where interpersonal relationships consist of relatively weak ties and showing one's goodness and trustworthiness through one's prosocial and generous behaviors are crucial to form and maintain relationships, compared to in a stable environment, where interpersonal relationships are fixed and taken for granted. Moreover, it will also be important to look at whether and to what extent socioecological factors moderate the asymmetry between the social discounting of gains and losses. Residential mobility (Oishi, 2010) and relational mobility (Schug et al., 2010) might be useful for explaining the cultural differences in social discounting.

The present research has several limitations. First, although it followed the manipulation of social distance used in previous research (Jones and Rachlin, 2006; Rachlin and Jones, 2008) and assumed that the representations of social distance do not vary across cultures, we did not verify whether this assumption was correct. Thus, the present research cannot deny the possibility that the relationship type the participants imagined corresponding to a given social distance might differ across cultures, or that the cultural difference in the imagined relationship type might influence the cultural difference in social discounting. Given that in social discounting, a person's generosity increases when his or her partners are members of mates and genetic kinships (Hackman et al., 2015), the current finding that Germans are more generous than Japanese to others might be because Germans are more likely than Japanese to imagine members of mates and genetic kinships as close friends. Moreover, even if the imagined relationship type does not differ across cultures, perceived emotional closeness to a person at a given social distance might differ between cultures. Hackman et al. (2015) demonstrated that independently of the effect of the relationship type, emotional closeness increases a person's generosity. This result suggests a possibility that compared to Japanese, Germans perceived greater emotional closeness to others, particularly to closer friends, and that cultural difference led to the Germans' greater generosity. Future work should focus on these issues. Second, we used hypothetical gains and losses in this study, although Locey et al. (2011) demonstrated that there is no difference in social discounting between real and hypothetical rewards. Another limitation is that we did not consider how the amount of the gains and losses influences social discounting. Rachlin and Jones (2008, Experiment 2) indicated that people discount another person's reward more as the amount increases. The possibility that the cultural effect in social discounting might change as a function of the amount of gains could be tested in future work.

Despite these limitations, we believe that the present research showing the usefulness of the hyperbolic with exponent model cross-culturally and the mediating role of harmony-seeking in the

cultural differences in social discounting in gains contributes to our understanding of the mechanism of individuals' generosity against the backdrop of the characteristics of sociocultural environments. We hope that future research addresses the generalizability of our findings in divergent cultural contexts.

#### AUTHOR CONTRIBUTIONS

KI designed the research and wrote the paper. KI and CE performed the research and analyzed the data.

#### REFERENCES

#### FUNDING

The research was supported by Grant-in-Aid for Scientific Research 26380843 and 15KK0120 from the Ministry of Education, Culture, Sports, Science and Technology of Japan.

#### ACKNOWLEDGMENTS

We thank Yuki Takahashi for her support in carrying out this work.

**Conflict of Interest Statement:** The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Copyright © 2018 Ishii and Eisen. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

## The Magnitude and "Peanuts" Effects: Searching Implications

Salvador Cruz Rambaud and Ana M. Sánchez Pérez\*

Departamento de Economía y Empresa, Universidad de Almería, Almería, Spain

The framework of this paper is the field of decision-making processes in which people face the choice between probabilistic and dated rewards. Traditionally, the preferences for probabilistic outcomes have been analyzed by the Expected Utility (EU) model whilst the preferences for dated rewards have been studied by the Discounted Utility (DU) model. Nevertheless, recent empirical findings have revealed the existence of several anomalies or paradoxes in both contexts. Specifically, EU and DU models exhibit an anomaly affecting the amount of the reward, viz the "peanuts" and the magnitude effects, respectively, which seem to go in opposite directions. The aim of this paper is to analyze both effects jointly in a wide setting involving choices subject to risk and over a period of time, and thereby identify and consider the implications of one anomaly on the other.

Keywords: intertemporal choice, discounted utility model, expected utility model, magnitude effect, "peanuts" effect

#### Edited by:

Sergio Ortobelli Lozza, University of Bergamo, Italy

#### Reviewed by:

Alessandro Staino, University of Calabria, Italy Simon Grima, University of Malta, Malta

\*Correspondence: Ana M. Sánchez Pérez<EMAIL_ADDRESS>
#### Specialty section:

This article was submitted to Mathematical Finance, a section of the journal Frontiers in Applied Mathematics and Statistics

> Received: 05 May 2018 Accepted: 18 July 2018 Published: 17 August 2018

#### Citation:

Cruz Rambaud S and Sánchez Pérez AM (2018) The Magnitude and "Peanuts" Effects: Searching Implications. Front. Appl. Math. Stat. 4:36. doi: 10.3389/fams.2018.00036

### 1. INTRODUCTION

This manuscript deals with the interaction between magnitude effect and "peanuts" effect, a term used by Prelec and Loewenstein [1], Weber and Chapman [2], and Haisley et al. [3] to describe the effect of a decreasing risk-aversion with decreasing monetary rewards. This is of particular importance since these two effects move in opposite directions, and the analysis of their interactions can be useful to validate any decision-making model which takes into account the elements of time and risk. The first part of the paper considers some implications of peanuts effect, magnitude effect, subendurance, and their reverse versions. The second part of the paper focuses on discount functions V given by the product of a utility function u and a function g depending on p and t, and goes on to analyze the implications which the magnitude effect and peanuts effect have on the utility function u.

Individual decision-making has been studied within disciplines which range from economics to psychology, passing through areas such as neuroscience [4]. However, these studies have always been carried out using two principal models:

1. The Discounted Utility (DU) model which is employed to assess a stream of rewards with different maturities [5]. In this way, individuals try to maximize their discounted payoff or "utility" which is given by:

$$U\_0 = \sum\_{t=0}^{T} \delta^t u\_t,\tag{1}$$

where U0 is the present value of the sequence, ut is the utility obtained from the outcome available at instant t (t = 1, 2, . . . , T) and δ represents the discount factor (which is lower than or equal to 1).

2. The Expected Utility (EU) theory which is employed to model those decision processes involving risky choices [6], that is to say, when the amounts have been specified in terms of probability. In the same way, individuals try to maximize their expected utility:

$$U\_0 = \sum\_{k=0}^{n} p\_k u\_k,\tag{2}$$

where U0 is the present value of the experience, uk the utility obtained from the k-th experience and pk the probability associated with this experience (where obviously the n associated probabilities amount to 1: Pn k = 0 pk = 1).

The DU and EU models have a similar structure given that they are based on the same theoretical principles, since alternatives are assessed by taking into account the sum of their utilities [7]. Moreover, DU and EU are general models with the power to predict, and their high level of acceptance is explained by their simplicity and the fact that they are based on the traditional systems of calculation of the present and actuarial values, respectively.

Nowadays, the DU and EU models are the standard theories of rational choice over time and under risk, respectively, in many social and behavioral sciences [8]. However, it is well known that models are simplified representations of reality, and so this is a limitation when trying to describe the actual behavior of people [9]. In effect, several anomalies have recently been detected which must be taken into account when analyzing a real situation.

Despite their similarity, decisions involving intertemporal choices and uncertainty have traditionally been studied in different research areas, given that delayed and risky rewards do not require the same treatment. Some cautious attempts to integrate DU and EU models, such as the Discounted Expected Utility (DEU) [10, 11], have been made in order to analyze individual behavior in decisions involving time delay and risk [11, 12]. In the same vein, in psychology, it is usual to interpret time in a probabilistic way [13], or to translate risks into delays when facing risky choices [14].

Other scholars [1, 7, 15] have studied the analogies and the anomalies observed in DU and EU models by arguing that they start from certain fundamental psychological properties of multidimensional prospect valuation. In the following paragraph, some anomalies of the DU (intertemporal choices) and EU (risky choices) models are jointly presented:

In spite of the fact that the magnitude and the peanuts effects are embedded in different frameworks, intuitively they seem to move in opposite ways because "the peanuts effect seems to reveal decreasing sensitivity to payoffs at larger stakes, while the magnitude effect seems to reveal increasing sensitivity to payoffs at large stakes" [11]. In effect, whilst the magnitude effect occurs in the DU model, the peanuts effect makes sense in the EU model. For this reason, in this paper we will follow a joint model in which time and risk preferences are integrated, in the same line as Schneider [11] and Baucells and Heukamp [17].

In this paper, we will focus on the study of the magnitude effect and its relation with the peanuts effect both depending on the amounts of reward. The importance of this research is obvious: "Prelec and Loewenstein could not explain both effects, and this challenge has remained unresolved over the subsequent twenty-five years, posing an apparent impossibility result that no common approach to modeling risk and time preferences can capture both of these basic behaviors" [11].

The DU and EU models propose that relative preference between two options is consistent, even if their amounts of reward are increased by a constant factor [18, 19]. However, some authors [20–23] have demonstrated that in decisions with delayed rewards, the preference increases as its amount increases. This magnitude effect is based on the premise that the patience of individuals is directly related to the reward amount and that individuals are more patient for large rewards than for those of smaller amounts, leading to an increased preference for the larger later outcomes [24, 25]. Following Schneider [11], the magnitude effect may be represented as follows:

$$(\mathbf{x}, \mathbf{p}, \mathbf{s}) \sim (\mathbf{y}, \mathbf{p}, t) \text{ implies } (k\mathbf{x}, \mathbf{p}, \mathbf{s}) \prec (k\mathbf{y}, \mathbf{p}, t), \tag{3}$$

where x < y, s < t and k > 1, where p represents the probability of occurrence which is considered constant in both alternatives. This paradox explains how large outcomes are discounted at a lower rate than smaller ones, the discount rate being a decreasing function of the size of the reward [7].

On the other hand, in choices involving risk, the amount of the reward has the opposite effect on the decision-making process. As pointed out by some scholars [26, 27], the peanuts effect has not yet been addressed as thoroughly as the magnitude effect. Only some recent studies [11] have mathematically analyzed this effect. It may be defined as follows (x < y):

$$(\mathbf{x}, p, t) \sim (\mathbf{y}, q, t) \text{ implies } \ (k\mathbf{x}, p, t) \succ (k\mathbf{y}, q, t), \tag{4}$$

the probabilities of occurrence being p and q, with p > q, and k > 1. Another definition of the peanuts effect was provided by Schneider and Day [28], as follows. Consider a lottery f :(x, p; 0, 1 − p), with x > 0. Let c: = {f , E(f)}. Then, the peanuts effect holds if f ≻c E(f), for sufficiently small x, and E(f) ≻c f , for sufficiently large x, for every p ∈ (0, 1).

Weber and Chapman [2] revealed that "a utility function cannot account for the decrease in the size of the peanuts effect for smaller probabilities". In the same way, Schneider and Day [28] proved that the peanuts effect cannot hold for Cumulative Prospect Theory (CPT) or any non-choice-set-dependent EU model. Finally, Leland and Schneider [8] define the indifference relationship SS∼btLL in the framework of the so-called Salience Weighted Utility over Presentations (SWUP) model in the following way:

$$
\mu(\mathfrak{x}, \mathfrak{y})[\mathfrak{u}(\mathfrak{y}) - \mathfrak{u}(\mathfrak{x})] \frac{\delta^s + \delta^t}{2} = \pi(\mathfrak{s}, t)(\delta^s - \delta^t) \frac{\mathfrak{u}(\mathfrak{y}) + \mathfrak{u}(\mathfrak{x})}{2}
$$

and the preference relationship SS′≺btLL′ by the following inequality:

$$
\mu(\mathfrak{x}, \mathfrak{y})[\mathfrak{u}(\mathfrak{y}) - \mathfrak{u}(\mathfrak{x})] \frac{\delta^s + \delta^t}{2} > \pi(\mathfrak{s}, t)(\delta^s - \delta^t) \frac{\mathfrak{u}(\mathfrak{y}) + \mathfrak{u}(\mathfrak{x})}{2},
$$

where µ(x, y) and π(s, t) are the so-called "salience functions". In this way, they characterize the magnitude and the peanuts effects by imposing some conditions on the utility and the salience functions.

As indicated in the previous paragraphs, the magnitude and peanuts effects seem to move in opposite directions, and can be explained by two a priori psychological principles:

The influence of negative emotion is likely to be greater in risky choice than in intertemporal choice because the possibility of gaining a reward smaller than expected is implicit in risky choices. The presented studies confirm that, from a psychological point of view, probabilistic discounting is not identical to that which underlies temporal discounting.

The methodology employed in this paper is the interaction between time and risk preferences. Its main contribution is the way by which time and risk preferences interact given certain assumptions. As a result, a wide variety of mathematical relationships arise by using the concepts of regularity and subendurance, and an important representation of the discount function V(x, p, t) as u(x)g(p, t).

After describing the state of the art concerning the mathematical treatment in the existing literature of both the magnitude and the peanuts effects, this paper is organized as follows. In section 2, the possible relationships between the magnitude and the peanuts effects are analyzed by using a very general definition of discount function involving time delay and risk, and the presence or absence of the so-called subendurance. In section 3, the general expression of the discount function V(x, p, t) is restricted to the functional form u(x)g(p, t), where u is a utility function. Under these circumstances, some significant relationships can be obtained between the reverse magnitude and the peanuts effects in the presence of a regular discount function. Finally, section 4 summarizes and concludes.

#### 2. FRAMEWORK AND GENERAL RESULTS

#### 2.1. Preliminaries

It will prove useful to begin with some definitions [17].

**Definition 1.** Consider the set M = X × P × T, where X = [0, +∞), P = [0, 1], and T = [0, +∞). A discount function is a continuous real-valued function V(x, p, t) defined on M which is strictly increasing in the first and second components, and strictly decreasing in the third.

However, Baucells and Heukamp [17] require that V tends to zero whenever xpe−t tends to zero, but in this paper this restriction will be relaxed, and we will assume that V converges to zero only when x → 0 or p → 0, by allowing that

$$\lim\_{t \to +\infty} V(\mathbf{x}, p, t) := L(\mathbf{x}, p) \ge 0.1$$

In the first case, V is said to be regular whilst, if L(x, p) > 0, V is said to be singular. The paper by Baucells and Heukamp [17] implicitly includes the regularity of V. In some further results we will require this condition but in others this condition will not be necessary. The following definitions have been obtained from Schneider [11].

**Definition 2.** The peanuts effect (resp. reverse peanuts effect) is said to hold if, for every 0 < x < y, p > q and k > 1, (x, p, t) ∼ (y, q, t) implies (kx, p, t) ≻ (ky, q, t) (resp. (kx, p, t) ≺ (ky, q, t)).

**Definition 3.** The magnitude effect (resp. reverse magnitude effect) is said to hold if, for every 0 < x < y, s < t and k > 1, (x, p,s) ∼ (y, p, t) implies (kx, p,s) ≺ (ky, p, t) (resp. (kx, p,s) ≻ (ky, p, t)).

The following definition reflects the idea that the larger the reward, the more subjects are willing to wait in exchange for improved probabilities [17].

**Definition 4.** Subendurance (resp. reverse subendurance) is said to hold if, for every 0 < x < y, s < t and p < q, (y, p,s) ∼ (y, q, t) implies (x, p,s) ≻ (x, q, t) (resp. (x, p,s) ≺ (x, q, t)).

#### 2.2. General Results

**Lemma 1.** Given an x ∈ X, let Vx : P×T → R be the real function defined by Vx(p, t):= V(x, p, t). If V is regular, then, for every (p, t) ∈ P×T and every q > p, there exists a 1 = 1(x, p, q, t) > 0 such that Vx(q, t + 1) = Vx(p, t).

Proof. Given an x ∈ X, for every (p, t) ∈ P×T and every q > p, let us consider the following real-valued function:

$$V\_{\mathbf{x},q}: T \to \mathbb{R}$$

defined as:

$$V\_{x,q}(r) := V\_x(q,r).$$

Taking into account the definition of V, the inequality Vx,q(t) > Vx(p, t) holds. Moreover, as V is regular, when r → +∞, Vx,q(r) → 0. Therefore, there exists a r0, large enough, such that Vx,q(r0) ≤ Vx(p, t) < Vx,q(t). As V is continuous and decreasing in t, by the Intermediate Value Theorem there exists a value 1 > 0 such that Vx,q(t + 1) = Vx(p, t), from where Vx(q, t + 1) = Vx(p, t).

**Corollary 1.** Let Vp : X × T → R be the real function defined by Vp(x, t):= V(x, p, t). If V is regular, then, for every (x, t) ∈ X × T and every y > x, there exists a 1 = 1(x, y, p, t) > 0 such that Vp(y, t + 1) = Vp(x, t).

Proof. The proof is analogous to that of Lemma 1 because the amount x and the probability p play similar rôles.

Observe that the next two results do not require the condition of regularity of V.

**Lemma 2.** Let Vt : X × P → R be the real function defined by Vt(x, p):= V(x, p, t). For every (x, p) ∈ X × P and every q > p, there exists a 0 < k = k(x, p, q, t) < 1 such that Vt(kx, q) = Vt(x, p).

Proof. Given a t ∈ T, for every (x, p) ∈ X × P and every q > p, let us consider the following real-valued function:

$$V\_{q,t}: X \to \mathbb{R}$$

defined as:

$$V\_{q,t}(z) := V\_t(z,q).$$

Taking into account the definition of V, the inequality Vq,t(x) > Vt(x, p) holds. Moreover, as Vq,t(0) = 0, then Vq,t(0) ≤ Vt(x, p) < Vq,t(x). As V is continuous and increasing in x, by the Intermediate Value Theorem there exists a value 0 < k = k(x, p, q, t) < 1 such that Vq,t(kx) = Vt(x, p), from where Vt(kx, q) = Vt(x, p).

**Corollary 2.** Let Vt : X × P → R be the real function defined by Vt(x, p):= V(x, p, t). For every (x, p) ∈ X × P and every y > x, there exists a 0 < k = k(x, p, q, t) < 1 such that Vt(y, kp) = Vt(x, p).

Proof. The proof is analogous to that of Lemma 1 because the amount x and the probability p play similar rôles.

**Observation 1.** It can be shown that the peanuts effect holds if and only if, for every 0 < x < y, p > q and 0 < h < 1, (x, p, t) ∼ (y, q, t) implies (hx, p, t) ≺ (hy, q, t). In effect, assume that (x, p, t) ∼ (y, q, t), with 0 < x < y and p > q. If, for a given 0 < h < 1, (hx, p, t) (hy, q, t), then Vp,t(0) ≤ V(hy, q, t) ≤ Vp,t(hx). As Vp,t is continuous and increasing, by the Intermediate Value Theorem, there would be a h0 ≤ h such that (h0x, p, t) ∼ (hy, q, t). As 1/h0 > 1, by hypothesis,

$$\left(\frac{1}{h\_0}h\_0\chi, p, t\right) \succ \left(\frac{1}{h\_0}h\mathcal{Y}, q, t\right) \succeq (\mathcal{Y}, q, t),$$

which is a contradiction. Therefore, (hx, p, t) ≺ (hy, q, t). The proof of the reciprocal implication is analogous.

**Observation 2.** Analogously to Observation 1, it can be shown that the magnitude effect holds if and only if, for every 0 < x < y, s < t and 0 < h < 1, (x, p,s) ∼ (y, p, t) implies (hx, p,s) ≻ (hy, p, t).

**Observation 3.** Finally, observe that it can be shown that subendurance holds if and only if, for every 0 < x < y, s < t and p < q, (x, p,s) ∼ (x, q, t) implies (y, p,s) ≺ (y, q, t). The proof is analogous to that of observations 1 and 2.

**Theorem 1.** The magnitude and the peanuts effects imply subendurance.

Proof. In effect, assume that the magnitude and the peanuts effects hold. In order to show that subendurance is satisfied, we are going to start from the following indifference relation:

$$(\wp, p, s) \sim (\wp, q, t),\tag{5}$$

where s < t, and p < q. By Lemma 2, there is a "trade-off " between y and q, in the way that the amount y can be increased until z, in exchange for diminishing the probability up to the value p, keeping the former indifference between the involved rewards, that is to say, such that:

$$(\wp, p, s) \sim (z, p, t),\tag{6}$$

where z > y. Therefore, as the magnitude effect holds, one has:

$$(k\wp, p, s) \prec (kz, p, t),\tag{7}$$

with k > 1. On the other hand, we can apply transitivity to equivalences (5) and (6), and so derive that:

$$(\wp, q, t) \sim (z, p, t). \tag{8}$$

By applying now the peanuts effect to the former equivalence and for the same value of k, one has:

$$(k\circ, q, t) \succ (kz, p, t). \tag{9}$$

Finally, the transitivity applied again to preferences (7) and (9) results in:

$$(ky, p, s) \prec (ky, q, t),\tag{10}$$

from where subendurance holds (see Observation 3).

**Theorem 2.** The following statements hold1 :

1The additional condition of the discount function being regular will be taken into account when a trade-off between reward amount (or probability) and time is necessary in order to apply Lemma 1 and Corollary 1.

Proof. The proof is analogous to that of Theorem 1.

All the implications included in theorems 1 and 2 can be schematized in **Figure 1**2 . Observe that:

#### 3. SEARCHING PARTICULAR IMPLICATIONS

#### 3.1. Introducing a Discount Function

Assume that the discount function has the functional form V(x, p, t):= u(x)g(p, t), where u is a utility function and g(p, t) is a non-negative continuous function defined in P × [0, t0) (t0 can even be +∞, in which case [0, t0) = T) satisfying the following conditions:

1. g(p, t) > 0.

The discount function is said to be [33]

p ∈ P (see **Figure 3**).

3. With bounded domain if t0 ∈ R.

#### 3.2. Particular Results

According to Baucells and Heukamp [17], this model is not compatible with subendurance and reverse subendurance. In these conditions, we can put forward the following propositions.

**Proposition 1.** A sufficient condition for the magnitude effect (resp. reverse magnitude effect) is that, for every 0 < x < y, s < t, p ≥ q and k > 1, (x, p,s) ∼ (y, q, t) implies (kx, p,s) ≺ (ky, q, t) (resp. (kx, p,s) ≻ (ky, q, t)). If V is regular, this condition is also necessary.

Proof. Obviously, the condition is sufficient (it suffices to take p = q). Let us see the necessity. In effect, assume that the magnitude effect holds. In order to show that the aforementioned condition holds, we are going to start from the following indifference relation:

$$(\mathbf{x}, \mathbf{p}, \mathbf{s}) \sim (\mathbf{y}, \mathbf{q}, \mathbf{t}),\tag{11}$$

where 0 < x < y, s < t and p ≥ q. From the former equivalence, it can be deduced that

$$
u(x)\emptyset(p,s) = \mu(\wp)\emptyset(q,t). \tag{12}$$

By Lemma 1, the next step is to propose a "trade-off " between q and t, in the way that the instant t can be delayed until t ′ , in exchange for increasing the probability up to p, keeping the indifference between the involved rewards, that is to say, such that:

$$(\mathfrak{x}, \mathfrak{p}, s) \sim (\mathfrak{y}, \mathfrak{p}, t').$$

In effect, it suffices to define (taking into account that g(p, ·) is continuous and decreasing)

$$t' := \mathcal{g}(\boldsymbol{\rho}, \cdot)^{-1} \left[ \frac{\boldsymbol{u}(\boldsymbol{\alpha})}{\boldsymbol{u}(\boldsymbol{\gamma})} \mathcal{g}(\boldsymbol{\rho}, \boldsymbol{s}) \right] > \boldsymbol{t}. \tag{13}$$

The existence of such t ′ is guaranteed because V is regular. Therefore, as the magnitude effect holds, one has:

$$(k\chi, p, s) \prec (k\chi, p, t'),\tag{14}$$

for every k > 1.

Finally, observe that, from the former equalities and inequalities, the following inequality is satisfied:

$$\frac{\mu(k\boldsymbol{\alpha})}{\mu(k\boldsymbol{\gamma})} < \frac{\operatorname{g}(\boldsymbol{p},t')}{\operatorname{g}(\boldsymbol{p},s)} = \frac{\mu(\boldsymbol{\alpha})}{\mu(\boldsymbol{\gamma})} = \frac{\operatorname{g}(\boldsymbol{q},t)}{\operatorname{g}(\boldsymbol{p},s)},$$

which obviously derives in:

$$(k\mathbf{x}, \mathbf{p}, \mathbf{s}) \prec (ky, q, t)$$

and so the condition is necessary. The reasoning for the reverse magnitude effect is analogous.

**Proposition 2.** The peanuts effect (resp. reverse peanuts effect) holds if and only if, for every 0 < x < y, s ≤ t, p > q and k > 1, (x, p,s) ∼ (y, q, t) implies (kx, p,s) ≻ (ky, q, t) (resp. (kx, p,s) ≺ (ky, q, t)).

Proof. Obviously, the condition is sufficient (it suffices to take s = t). Let us see the necessity. In effect, assume that the peanuts effect holds. In order to show that the aforementioned condition holds, we are going to start from the following indifference relation:

$$(\propto, p, s) \sim (\psi, q, t),\tag{15}$$

where 0 < x < y, s ≤ t and p > q. From the former equivalence, it can be deduced that

$$
\mu(\mathfrak{x})\mathfrak{g}(\mathfrak{p},\mathfrak{s}) = \mathfrak{u}(\mathfrak{y})\mathfrak{g}(\mathfrak{q},\mathfrak{t}).\tag{16}
$$

2 In order to interpret this chart and the next one correctly, it is necessary to take into account the following criterium: if a cell can be vertically embedded in a contiguous cell, the property indicated in the first (smaller) cell implies the property enclosed in the second (bigger) cell.

FIGURE 1 | Implications between the magnitude and the peanuts effects. Green denotes de presence of the mentioned effect. Yellow denotes de absence of the indicated effect. Red represents the existence of the reverse of the mentioned effect. Source: own elaboration.

The next step is to propose a "trade-off " between q and t, in the way that the probability q can be reduced until q ′ , in exchange for anticipating the availability of the second reward up to s, keeping the indifference between the involved rewards, that is to say, such that:

$$(x, p, s) \sim (y, q', s).$$

In effect, it suffices to define (taking into account that g(·,s) is continuous and decreasing, and g(0,s) = 0)

$$q' := \mathcal{g}(\cdot, s)^{-1} \left[ \frac{\mu(\boldsymbol{x})}{\mu(\boldsymbol{y})} \mathcal{g}(\boldsymbol{p}, s) \right] < q. \tag{17}$$

Therefore, as the peanuts effect holds, one has:

$$(k\chi, p, s) \succ (k\chi, q', s),\tag{18}$$

for every k > 1.

However, observe that, from the former equalities and inequalities, the following inequality is satisfied:

$$\frac{\mu(k\infty)}{\mu(k\circ)} > \frac{\underline{g}(q',s)}{\underline{g}(p,s)} = \frac{\mu(\infty)}{\underline{\mu(\mathcal{Y})}} = \frac{\underline{g}(q,t)}{\underline{g}(p,s)},$$

which obviously derives in:

$$(k\infty, p, s) \succ (ky, q, t)$$

and so the condition is necessary. The reasoning for the reverse peanuts effect is analogous.

The following results are a direct consequence of propositions 1 and 2 in the framework defined at the beginning of section 3.

**Corollary 3.** The peanuts effect implies the reverse magnitude effect. Moreover, if the discount function involved in the intertemporal choice is regular, then the reverse magnitude effect implies the peanuts effect.

**Corollary 4**. If the model involved in the intertemporal choice is the q-exponential discounting, then the reverse magnitude effect is equivalent to the peanuts effect.

Proof. In effect, take into account that the q-exponential (in particular, the exponential, the hyperbolic and the linear discounting) discount function [34] is regular, as required by Corollary 3.

The following proposition provides a characterization of the utility function u involved in the context of the peanuts effect.

**Proposition 3**. The peanuts effect holds if and only if the elasticity of the utility function

$$\epsilon\_{\mathfrak{u}}(z) := z [\ln \mathfrak{u}(z)]'$$

is decreasing.

Proof. Let us assume that the peanuts effect holds and that 0 < x < y. Let p and q (p > q) be two probabilities such that (x, p, t) ∼ (y, q, t). In such a case,

$$
\mu(\mathfrak{x})\mathfrak{g}(\mathfrak{p},t) = \mathfrak{u}(\mathfrak{y})\mathfrak{g}(\mathfrak{q},t).
$$

By the peanuts effect, (kx, p, t) ≻ (ky, q, t) for every k > 1, and so

$$
\mu(k\infty)\lg(p,t) > \mu(ky)\lg(q,t).
$$

By dividing the left and the right-hand sides of the former expressions, one has:

$$\frac{u(k\infty)}{u(\infty)} > \frac{u(k\wp)}{u(\wp)}$$

and so

$$
\ln \mu(k\boldsymbol{\omega}) - \ln \mu(\boldsymbol{\omega}) \succ \ln \mu(k\boldsymbol{\chi}) - \ln \mu(\boldsymbol{\chi}).
$$

As k > 1, we can write k := 1 + h, with h > 0. Therefore,

$$
\ln \mu(\mathbf{x} + h\mathbf{x}) - \ln \mu(\mathbf{x}) \\
> \ln \mu(\mathbf{y} + h\mathbf{y}) - \ln \mu(\mathbf{y}).
$$

Dividing both sides of the former inequality by h and letting h → 0, we obtain:

$$\propto \left. \frac{\operatorname{d} \ln \mathfrak{u}(z)}{\operatorname{d} z} \right|\_{z=x} \geq \mathcal{Y} \left. \frac{\operatorname{d} \ln \mathfrak{u}(z)}{\operatorname{d} z} \right|\_{z=y}.$$

Consequently, the elasticity of the utility function, z[ln u(z)]′ , is decreasing. The converse implication can be easily shown.

This result coincides with that provided by [35].

**Corollary 5**. If the peanuts effect holds then u is ln-concave.

FIGURE 5 | Convexity, and peanuts and magnitude effects Green denotes de presence of the mentioned effect. Yellow denotes de absence of the indicated effect. Red represents the existence of the reverse of the mentioned effect. Source: own elaboration.

Proof. In effect, if the peanuts effect holds, by Proposition 3, z[ln u(z)]′ is decreasing. As z is increasing, then [ln u(z)]′ is decreasing. Therefore, u is ln-concave.

The converse statement is not true. In effect, the utility function u(z) = exp{ √ z} − 1 is ln-concave (see **Figure 4**, line in blue) but

$$[z[\ln \mu(z)]']' = \frac{1}{2} \frac{\sqrt{z} \exp\{\sqrt{z}\}}{\exp\{\sqrt{z}\} - 1}$$

is increasing (see **Figure 4**, line in green).

Analogously, we can show the following statements.

**Proposition 4**. The magnitude effect holds if and only if the elasticity of the utility function is increasing.

**Corollary 6**. If u is ln-convex then the magnitude effect holds.

All the implications between the analyzed effects and the ln-convexity of u can be seen in **Figure 5**.

#### 4. CONCLUSIONS

In behavioral finance, the decision-making processes should be explained through appropriate theoretical models and taking into account the possible anomalies that human behaviors entail. The treatment of these paradoxes can improve the explanatory power of the economic models by involving some suitable tools from the fields of economics and psychology.

Although sometimes time and risky preferences may be considered as analogous concepts, the temporal and probabilistic discount functions are not identical. To illustrate the difference between intertemporal and probabilistic choices, we refer to two anomalies of the discounted and expected utility models: the magnitude and the peanuts effects, respectively.

The magnitude effect occurs in intertemporal choices where the larger reward is usually related to a longer waiting time and

#### REFERENCES

a lower discount rate. Given that these choices occur free of negative feelings such as disappointment, the decision-maker, in search of greater profits, may prefer to wait. However, the peanuts effect occurs in uncertain choices in which the disappointment experienced is directly related to the amount and probability of the result. Specifically, the decision-maker is more prone to make risky decisions when a small amount is involved in the experiment.

This paper has presented a model simultaneously applied to time and risk which could explain both the magnitude effect in choices over time and the peanuts effect in choices under risk. In pursuit of establishing the relationship between both effects, we have obtained some implications in a broad framework by considering the presence or absence of subendurance. In the particular case in which V(x, p, t) adopts the expression u(x)g(p, t), where u is a utility function, the reverse magnitude and the peanuts effects are equivalent when the discount function is regular. Finally, some implications have been deduced involving the ln-convexity of u(x).

#### AUTHOR CONTRIBUTIONS

SC and AS contributed to the design and implementation of the research, and to the writing of the manuscript.

#### FUNDING

The authors gratefully acknowledge financial support from the Spanish Ministry of Economy and Competitiveness [National R&D Project DER2016-76053-R].

#### ACKNOWLEDGMENTS

We are very grateful for the valuable comments and suggestions offered by two referees.

**Conflict of Interest Statement:** The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Copyright © 2018 Cruz Rambaud and Sánchez Pérez. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

## How Are Individual Time Preferences Aggregated in Groups? A Laboratory Experiment on Intertemporal Group Decision-Making

#### Manami Tsuruta1 \* and Keigo Inukai 2

1 Graduate School of Economics, Osaka University, Osaka, Japan, 2 Department of Economics, Meiji Gakuin University, Tokyo, Japan

#### Edited by:

Taiki Takahashi, Hokkaido University, Japan

## Reviewed by:

Barret Pengyuan Shao, Independent Researcher, Washington, DC, United States Fuat Balci, Koç University, Turkey Naoshi Tsuchida, Bank of Japan, Japan

> \*Correspondence: Manami Tsuruta<EMAIL_ADDRESS>
#### Specialty section:

This article was submitted to Mathematical Finance, a section of the journal Frontiers in Applied Mathematics and Statistics

> Received: 20 April 2018 Accepted: 23 August 2018 Published: 02 October 2018

#### Citation:

Tsuruta M and Inukai K (2018) How Are Individual Time Preferences Aggregated in Groups? A Laboratory Experiment on Intertemporal Group Decision-Making. Front. Appl. Math. Stat. 4:43. doi: 10.3389/fams.2018.00043 The study of intertemporal decision-making is an interdisciplinary scientific topic of economics, psychology, and neuroscience. Most of these studies focus on individual intertemporal decisions, but little is known about the relationship between groups and individual time preferences. As a result, we intend to assess the role of group intertemporal decision-making. We experimentally investigate how to aggregate individual time preferences by clarifying who has the most influence on group decisions among heterogeneous group members. We formulate two hypotheses. The first is the multilateral bargaining hypothesis, which is based on the multilateral bargaining model. If people employ this model to reach agreement, the most patient member in a group has the greatest impact on group choices. The second is the median voter hypothesis, which is based on the median voter model. When people employ this model to reach agreement, the median patient member in a group has the greatest impact on group choices. Here, we find that the median patient member in a group has a significant impact on group decisions in an unstructured bargaining situation. This finding suggests that people use the majority voting rule during group intertemporal decision-making. Thus, our findings support the median voter hypothesis. Furthermore, the results of a chat analysis show that this result is partially due to people's conformity with the majority opinion.

Keywords: intertemporal choice, impatience, group decision-making, multilateral bargaining, median voter model, laboratory experiment, chatting, conformity with majority

#### INTRODUCTION

Many essential economic decisions are made by groups such as companies and households. For example, investment plans in companies are decided in meetings with multiple people, and saving plans in households are decided by partners. Environmental decisions are also intertemporal choices. For example, we are faced with a decision between consuming ecological resources today vs. enjoying a rich environment later. These types of decisions are mainly made by the society, such as local communities, governments, and so on.

In general, economists describe a group choice as a summation of individual choices or representative individuals. Do real-life group decisions over time work in this way? Although it is important that we understand how intertemporal group decision are made, there is little empirical evidence on the relationship between group and individual time preferences. Very few studies have used experiments to examine groups' intertemporal choices1 . Yang and Carlsson [2] investigated whether a group consensus related to intertemporal decisionmaking was derived from individual time preferences. In their research, participants were wives and husbands in rural China, where each couple answered intertemporal questions. They found that 11% of the consensus decisions made by couples were more patient than both the wife's and husband's individual choices and that 9% of the consensus decisions were more impatient than the individuals' choices. Interestingly, their result suggested that some proportion of group consensus decisions were not simply a summation of individuals' choices. Nevertheless, they failed to reveal why couples' consensus decisions were beyond the individuals' time preferences. Furthermore, they focused on family financial decisions made by spouses. Thus, we cannot apply their findings to groups' intertemporal choices in general. Yang and Carlsson [2] and Carlsson et al. [3] also investigated who in the family had the most influence on group consensus decisions, identifying that husbands had a greater impact than wives did in joint choices. However, this result could be culture specific and, thus, might not be generalizable to other regions or countries.

The purpose of this study is to investigate the mechanism behind groups' intertemporal decision-making in heterogeneous time preferences by clarifying who has a significant impact on groups' intertemporal choices. As such, we contribute to the literature by investigating the difference between group preferences and individual preferences. Charness and Sutter [4] indicate that groups make more self-interested decisions than individuals. He and Villeval [5] show that group decisions reflect the same level of inequality aversion as individual decisions. Many studies investigated the difference between group and individual risk preferences [6–12]. Although an increasing number of studies are comparing groups' and individuals' decision-making, few studies focus on groups' intertemporal decision-making [2, 3]. Our study is the first to empirically examine the mechanism behind how groups reach intertemporal decisions. Our second contribution to the literature is that we use anonymous experimental protocols to eliminate various unobservable effects. In previous studies [2, 3], the participants are families, who use face-to-face dialogue to make decisions. Here, we gather unrelated participants, who then communicate with each other via text messages on a computer. The third contribution of our study is that our groups contain more than two members (i.e., three members). This makes it possible to generalize the results to real-life group settings. Our fourth contribution is that we analyze the chat messages during group decision-making to establish how the group forms a consensus. The fifth contribution of our study is to show how individual time preferences are aggregated in groups. The mechanism behind time preference aggregation has not been studied previously. In summary, the goal of this study is to determine the processes behind group decisions in an intertemporal context.

We formulate two hypotheses. First, group intertemporal choices are decided based on the multilateral bargaining model; that is, the most patient member has a significant impact on group choices (multilateral bargaining hypothesis). Second, group intertemporal decisions are determined based on the median voter model; that is, the median patient member has a substantial impact on group choices (median voter hypothesis). To test these hypotheses, we conduct laboratory experiments. Here, participants make intertemporal choices individually in individual conditions. In addition, participants make intertemporal choices in a group (three people) through discussion to reach group decisions in group conditions. We analyze the individual discount factors elicited in the individual conditions and the group discount factors elicited in the group conditions. Groups consist of a most patient member, a median patient member, and a least patient member2 . Here, we examine whose discount factor is the closest to the group discount factor and interpret this member as having the greatest impact on a group decision. Consequently, we check the results using a regression analysis, post-estimation analysis, and Bayesian analysis.

Based on the results of the regression and post-estimation analyses, we reject the multilateral bargaining hypothesis. However, rejecting the multilateral bargaining hypothesis and not rejecting the median voter hypothesis does not guarantee support for the median voter hypothesis [13]. To resolve this problem, we conduct a Bayesian analysis and calculate the Bayes factors. The results support the median voter hypothesis. Finally, we examine the text data generated by the participants' computer-based chats to investigate why the median voter hypothesis is supported. Based on this analysis, we find that the third person to express an opinion (i.e., the last person) tends to follow the majority opinion. This phenomenon might lead to the observed result, even though the experimental group setting is a free discussion and follows the unanimity rule.

The rest of the paper is organized as follows. Section Methods describes our experimental design and procedures. Section Hypotheses presents the two hypotheses. Section Results analyzes the results using regression, post-estimation, Bayesian, and chat analyses. Section Conclusion concludes the paper.

#### METHODS

The experiment took place at Osaka University in Japan. We conducted six sessions with a total of 105 student participants, with each session comprising 15 or 18 participants and lasting

1Bixter et al. [1] also studied group and individual time preferences. They investigated whether individual choices after group decisions are affected by group agreement.

2Of course, there are cases where all the members' individual discount factors are same or two of the three members' individual discount factors are same. We explain how we handled these cases in section Regression Analysis and Post Estimation (Regression analysis and post estimation).

for approximately 1 h. All participants gave written informed consent according to the Declaration of Helsinki and the guidelines approved by the ethical committee (Institute of Social and Economic Research Ethical Committee at Osaka University) prior to the experiment. The experiment was computerized using z-Tree [14].

Participants expressed their time preferences by making a series of choices between early and delayed options of different denominations (i.e., using choice titration [15]). For example, participants were asked whether they would prefer JPY 1,750 today or JPY 2,000 1 month later. After making the choice, the amount of the delayed option was changed, which meant participants faced new options. We explained how to obtain the amount of the delayed option to which participants were indifferent between the delayed option and the early option in a simplified manner. For example, person X was indifferent between receiving JPY 1,750 today and JPY 2,100 1 month later. Thus, the amount of the delayed option to which the participant was indifferent to receiving the early option was JPY 2,100. We tested what amount of the delayed option was indifferent to JPY 1,750 by asking the question, "Do you prefer receiving JPY 1,750 today or JPY 2,000 1 month later?" When person X chose JPY 1,750 today, we then asked, "Would you prefer to receive JPY 1,750 today or JPY 2,120 1 month later?" In this way, we obtained an approximate indifference amount for the delayed option, relative to the amount of the early option. We repeated this process four times in order to obtain the indifference amount of the delayed option. All alternatives are shown in **Figure 1**. We calculated the indifference amount of the delayed option as follows. We used two values: "High up," which was the highest amount of the delayed option a participant did not choose; and "Lowdown," which was the lowest amount of the delayed option that a participant did choose. We defined the mean of "High up" and "Lowdown" Highup + Lowdown /2 as the indifference amount of the delayed option, relative to the amount of the early option. All indifference values are shown in the **Supplementary Material**. For example, when a participant reached C in **Figure 1** and (early date, later date) = (today, 1 month), the indifference amount of the delayed option (1 month) relative to the amount (JPY 1,750) of the early option (today) was JPY 2,165. In this case, "High up" was JPY 2,150 and "Lowdown" was JPY 2,180. The participants were not informed of all the available alternatives or the titration rule before the task. The combinations of dates for the early and delayed options were categorized into two types: (early date, later date) = (today, 1 month) and (1 month, 2 months). The order of the two types of dates was determined randomly.

After obtaining the indifference amount of the delayed option relative to the amount of the early option, we calculated the discount factor. In this study, we referred to the discount factors as time preferences. We used a linear utility function: u (x) = x. We assumed the indifference amount of the delayed option relative to the amount of the early option to be xdelayed, and that of the early option to be xearly. Later, we calculated the discount factor δ as xearly = δ t xdelayed i.e., δ = xearly/xdelayed 1 t , for year t. Because there were two types of dates for the two options, (early date, later date) = (today, 1 month) and (1 month, 2 months), the values of t were 1/12 and 1/12, respectively. We referred to δ as the patience. A high value of δ denoted patience and a low value denoted impatience. Thus, the most impatient preference was A, and the most patient preference was P in **Figure 1** because xearly was always 1,750. The impatience preference ranking was A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P.

There were three choice conditions: an individual condition, a different condition ("for another"), and a group condition. All participants were tested for all three conditions. To control the order effect of the choice conditions, we conducted various combinations of all three conditions. In other words, we conducted six sessions. For the individual condition, participants chose alternatives for themselves; for the "for another" condition, participants chose alternatives for other members of the same group on an individual basis. Here, the chosen option was paid to the other members of their group. The payment mechanism is described in the next paragraph. We failed to analyze the "for another" condition here, because it was described in detail by Truruta et al. [16] 3 . For the group condition, three participants per group discussed the options using text messages on a computer. As a result, participants could not identify the other group members' gender, visual aspect, race, and so on. We did this to eliminate unobservable effects on decision-making that would result from a face-to-face discussion. Moreover, this enabled us to analyze the text data for each group discussion: for example, the extent of discussion, who expresses an opinion first, and whether persuasion occurs. Subsequently, we investigated our finding that median voter hypothesis was supported (i.e., the median patient member had the greatest impact on group choices). In the group condition, the amount of the reward shown was per person: for example, "Each member is going to receive JPY 1,750 today or JPY 2,000 one month later." Participants, then, needed to make a group choice through discussion. Hence, the decision rule was unanimity. No time limits were imposed in any of the conditions.

The amount of the reward per person was the same in all three choice conditions. One of the options that the participants chose was selected randomly by the computer, and the selected option was paid to participants. The option that was paid was selected from all group choices a person's group chose under the group condition, all individual choices s/he chose under the individual condition, and all individual choices the other group members (two persons) chose under the "for another" condition. The reward was an Amazon gift card, which was e-mailed to the participant. The advantage of this reward was that we could make the transaction cost the same between the immediate reward and the delayed reward. The average reward was approximately JPY 1,890.

After participants arrived at the laboratory, the instructions were distributed. The instructions4 are included in the **Supplementary Material**. Participants sat in cubicles and

3We do not include the "for another" condition in our analysis because we are not interested in this category in this study. See Tsuruta [16] for a discussion of this condition. Thus, we analyze only the group and individual conditions. 4The instruction is written in Japanese.

impatient preference is A, and the most patient preference is P. The impatience preference ranking is A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P.

chose the alternatives using a computer. After three choice conditions, we informed participants about the reward they were to be paid, and they completed a post-experimental questionnaire that inquired about their demographic data.

### HYPOTHESES

We formulate two hypotheses. The first hypothesis is based on the multilateral bargaining model. According to this hypothesis, the most patient member in a group has the greatest impact on group choices (multilateral bargaining hypothesis). The second hypothesis is based on the median voter model. According to this hypothesis, the median patient member has the greatest impact on group choices (median voter hypothesis).

### Multilateral Bargaining Hypothesis

This subsection describes the multilateral bargaining hypothesis, which is based on the multilateral bargaining model. This hypothesis predicts that the most patient member has the greatest impact on group choices. Our experimental setting is close to multilateral bargaining, in that members of a group negotiate with each other over group choices.

The bilateral bargaining model is well known as a result of Rubinstein's pioneering work [17]. The multilateral bargaining model was developed by Baron and Ferejohn [18] and Banks and Duggan [19] and had been studied mainly in the fields of political science and economics. In the popular model setting of multilateral bargaining [18], a proposer is selected from all members with some probability, who then proposes the allocation of the surplus. Then, the members vote on the proposals (i.e., accept or not accept). Under the majority voting rule, the proposal is implemented and the game ends when the majority of the members vote to accept. Otherwise, the procedure is repeated, including the selection of a proposer [18]. Under the unanimity voting rule, the proposal is implemented and the game ends when all members vote to accept. Otherwise, the procedure is repeated by selecting another proposer [20].

Our experiment differs from these bargaining models, where members decide how to allocate the resource among themselves, because members decide on a common group discount factor (i.e., time preferences) in our experimental setting. Here, we follow Ambrus et al. [6], who studied how individual risk preferences were aggregated in groups and also employed the multilateral bargaining model as a theoretical background. We do so because we also investigate how individual preferences are aggregated in groups. Moreover, applying the multilateral bargaining model is plausible, because people have heterogeneous opinions and make group decisions through discussion in our experimental setting.

The main difference between our hypothesis and that of Ambrus et al. [6] is that we assume heterogeneous individual discount factors, whereas they assume common discount factors among members of a group. Our assumption is closer to how group decisions are made in real-life. Another difference is that we employ the unanimity rule; that is, group members' choices must be the same after the discussion5 .

Several theoretical studies based on the multilateral bargaining model assume heterogeneity of individual discount factors and employ the unanimity rule [21–23]. According to these studies, a more patient member receives a higher payoff and has stronger bargaining power. This is because patient members can reject unfavorable proposals more easily than impatient members. The unfavorable option refers to an objective option (i.e., a lower reward). For example, suppose we have a proposer A, a patient member B (δB = 0.9), and an impatient member C (δC = 0.2), when proposer A proposes that A receives 60, B receives 20, and

5 In their experimental setting, Ambrus et al. [6] employed the unanimity rule, as we did. However, they built their hypothesis using both the unanimity rule and the majority rule.

C receives 20 at time t, the patient member B compares 20 to δB (= 0.9) × (the expected allocation of B at t + 1) to decide whether or not to accept the proposal. Similarly, the impatient member C compares 20 to δC (= 0.2) × (the expected allocation of C at t + 1) to decide whether or not to accept the proposal. Here, the impatient member C's expected payoff at t + 1 is likely to be lower than that of the patient member owing to the difference in the discount factors. When B and C's allocations at t + 1 are both 40, the patient member B rejects the proposal at t (20 < 0.9 × 40), but the impatient member C accepts the proposal at t (20 > 0.2 × 40).

We apply this logic to our experimental setting. Thus, the multilateral bargaining hypothesis states that the most patient member has the greatest impact on group choices.

#### Median Voter Hypothesis

This subsection describes the median voter hypothesis, which is based on the median voter model. This hypothesis predicts that the median patient member has the greatest impact on group choices in our experiment.

The median voter model is well known and is applied in various academic fields [24]. Here, the median voter's choices are selected as the group choices under majority rule if all voters have a single peaked preference [25]. We apply this logic to our experimental setting. Thus, we hypothesize that the median patient member has a significant impact on group choices. According to this hypothesis, we predict that group decisions are conducted under the majority voting rule, not bargaining, even though they can discuss the options freely.

In our experimental setting, the median patient member's final choices are the same as the final group choices when the group members employ the majority voting rule. The explanation is as follows (see **Figure 1**). First, we consider the situation in which all the members' preferences are different. For example, there are three members of the same group, each of whom prefer a different option (D, E, and F). For the first question (JPY 1,750 at an early date or JPY 2,000 at a later date), all three members select the early option. For the second question (JPY 1,750 at an early date or JPY 2,120 at a later date), the person who likes D the best selects the early option, but the other members select the delayed option. Consequently, the group chooses the delayed option because they employ majority voting. For the third question (JPY 1,750 at an early date or JPY 2,060 at a later date), all three members select the early option. For the fourth question (JPY 1,750 at an early date or JPY 2,090 at a later date), the person who likes F the best selects the delayed option, but the other members select the early option. In this case, the group chooses the early option because they employ majority voting. Therefore, the final group choice is E, which is the same as the median patient member's final choice. The results for the other cases are the same. Second, we consider the situation in which two of the members' final choices are the same. The median patient member always belongs to the majority, because there are three group members. For example, if one member prefers C but the other two prefer D, then the median patient member is the person who prefers D. If one member prefers I but two prefer E, then the median patient member is the person who prefers E. Therefore, the median

TABLE 1 | Discount factors for each rank under each combination of dates.

Here, (today vs. 1 month) and (1 month vs. 2 months) refer to the combinations of dates for the early option and the delayed option. Using (today vs. 1 month) as an example, participants choose to receive a small reward today or a larger reward one month later. Each rank refers to the order of the discount factor under the individual condition in the same group. Rank 1 refers to the most patient member (the highest discount factor) in a group, rank 2 refers to the median member (the second highest discount factor), and rank 3 refers to the most impatient member (the third highest discount factor, i.e., the lowest discount factor). Group refers to the discount factor for the group condition.

patient member's final choice is also the final group choice under the majority voting rule, because the median patient member always belongs to the majority. Finally, we consider the situation in which all the members' final choices are the same. Evidently, the median patient member's final choice is the same as the final group choice in this situation.

Consequently, the median voter hypothesis states that the median patient member has the greatest impact on group choices.

#### RESULTS

#### Regression Analysis and Post-estimation Analysis

We classify group members into three ranks, as follows. Rank 1, rank 2, and rank 3 represent the order of the discount factor under the individual condition in the same group6 . Rank 1 refers to the most patient member (the highest discount factor) in a group, rank 2 refers to the median member (the second highest discount factor), and rank 3 refers to the most impatient member (the third highest discount factor: i.e., the lowest discount factor). If two members in the same group have the same value, we handle it as follows. For example, when the value of member A's discount factor is 0.8, that of member B's discount factor is 0.8, and that of member C's discount factor is 0.7, we set rank 1 and rank 2 to 0.8 and rank 3 to 0.7. If all members of the group have the same value, we set all three rank values to the same value.

**Table 1** shows the descriptive statistics for each rank and the group discount factors for each date. **Figure 2** shows the mean discount factors of the (today, 1 month) condition. **Figure 3** shows the mean discount factors of the (1 month, 2 months) condition. In both cases, rank 2 seems to be close to the group discount factor.

6We explain how to calculate discount factor (i.e., time preferences) in section Methods.

FIGURE 2 | Mean discount factors for the group condition and the individual conditions (Rank 1, Rank 2, and Rank 3). The combination of dates for the early option and the delayed option is (today vs. 1 month). Participants choose to receive a smaller reward today or a larger reward one month later. Each rank refers to the order of the discount factor under the individual condition in the same group. Rank 1 refers to the most patient member (the highest discount factor) in a group, rank 2 refers to the median member (the second highest discount factor), and rank 3 refers to the most impatient member (the third highest discount factor, i.e., the lowest discount factor). Group refers to the discount factor for the group condition.

FIGURE 3 | Mean discount factors for the group condition and individual conditions (Rank 1, Rank 2, and Rank 3). The combination of dates for the early option and the delayed option is (1 month vs. 2 months). Participants choose to receive a smaller reward one month later or a larger reward two months later. Each rank refers to the order of the discount factor under the individual condition in the same group. Rank 1 refers to the most patient member (the highest discount factor) in a group, rank 2 refers to the median member (the second highest discount factor), and rank 3 refers to the most impatient member (the third highest discount factor. i.e., the lowest discount factor). Group refers to the discount factor for the group condition.

For a more detailed analysis, we conduct a regression analysis and a post-estimation analysis. These analyses are partially based on the work of Ambrus et al. [6], who also analyze how individual preferences are aggregated at the group level. They investigated risk preferences using a lottery task and selfishness using a gift exchange game. We focus on the model in which the group decision is a linear function of δ (rank i) g i=1,2,3 :

$$
\delta\_{\mathcal{g}}^{group} = constant + \alpha\_1 \delta\_{\mathcal{g}}^{(rank1)} + \alpha\_2 \delta\_{\mathcal{g}}^{(rank2)} + \alpha\_3 \delta\_{\mathcal{g}}^{(rank3)} + \epsilon\_{\mathcal{g}}, \text{(1)}.
$$

where δ group g denotes group g's elicited group discount factor, and δ (rank i) g denotes rank i's elicited individual discount factor in group g. We use δ (rank j) g to refer to the jth highest discount factor among the individuals in group g (in particular, δ (rank 1) g refers to the highest and δ (rank 3) g refers to the lowest discount factors). We interpret the coefficients (i.e., α1, α2, α3) as the influence of each rank on the group discount factors.

Next, we conduct a post-estimation analysis using our two hypotheses (i.e., the multilateral bargaining hypothesis and the median voter hypothesis): see section Hypotheses. Later, we conducted post-estimation analyses, as follows. First, we test whether the most patient member has the greatest impact on group choices by analyzing whether we can reject α2 = α3 = 0. Subsequently, we test this more strictly by analyzing whether we can reject α2 = α3 = 0 and α1 = 1. If these tests are rejected, we can interpret that the multilateral bargaining hypothesis is rejected. Second, we test whether the median patient member has the greatest impact on group choices by analyzing whether we can rejectα1 = α3 = 0. We also test this more strictly by analyzing whether we can reject α1 = α3 = 0 and α2 = 1. If these tests are rejected, we can interpret that the median voter hypothesis is rejected. Third, we test whether the most impatient member has an impact on group choices. Here, we test whether we can reject α1 = α2 = 0, and we test this more strictly by analyzing whether we can reject α1 = α2 = 0 and α3 = 1. If these tests are rejected, we can interpret from the results that the most impatient member does not have the greatest impact on group choices.

**Table 2** shows the results of the regression of the group discount factors on the ordered individual discount factors. The coefficient of the median member's discount factor, α2, is positive and significant for both combinations of dates. **Table 3** shows the results of the post-estimation analyses. Here, we reject that "the most patient member has the greatest impact on group choices," "the most patient member strongly has the greatest impact on group choices," "the most impatient member has the greatest impact on group choices," and "the most impatient member strongly has the greatest impact on group choices" for both combinations of dates at the 1% level. We cannot reject "the median patient member has the greatest impact on group choices" and "the median patient member strongly has the greatest impact on group choices" for both combinations of dates at the 1% level. Therefore, we reject the multilateral bargaining hypothesis. In addition, the most impatient member does not have the greatest impact on group choices.

#### Bayesian Analysis

In this subsection, we conduct a Bayesian analysis. As mentioned in the previous subsection, we reject the multilateral bargaining hypothesis. However, rejecting this hypothesis and not rejecting the median voter hypothesis does not guarantee support for the median voter hypothesis. In other words, not rejecting a hypothesis is not the same as supporting

Robust standard errors in parentheses.

\*, \*\*, \*\*\* denote significance at the 1%, 5%, 10% level.

Here, (today vs. 1 month) and (1 month vs. 2 months) refer to the combinations of dates for the early option and the delayed option. Using (today vs. 1 month) as an example, participants choose to receive a small reward today or a larger reward one month later. Each rank refers to the order of the discount factor under the individual condition in the same group. Rank 1 refers to the most patient member (the highest discount factor) in a group, rank 2 refers to the median member (the second highest discount factor), and rank 3 refers to the most impatient member (the third highest discount factor, i.e., the lowest discount factor). Group refers to the discount factor for the group condition.

TABLE 3 | Results from the post-estimation analysis (p-values).

\*, \*\*, \*\*\* denote significance at the 1%, 5%, 10% level.

In this table reports the post-estimation analysis results from Table 2. Statistical significance means that the hypothesis is rejected. For example, the result of first row and column (0.0000\*\*\*) means that α2 = α3 = 0 is rejected with a p-value of 0.0000.

the hypothesis [13]. To resolve this problem, we conduct a Bayesian analysis and calculate the Bayes factors. Then, we analyze which hypothesis best fits the experimental data.

We first conduct the following four linear regressions using Bayesian methods and, then, calculate the Bayes factors to judge which hypothesis best fits the experimental data.

$$\delta\_{\mathcal{S}}^{group} = constant + b\_1 \delta\_{\mathcal{S}}^{(rank\ 1)} + b\_2 \delta\_{\mathcal{S}}^{(rank\ 2)} + b\_3 \delta\_{\mathcal{S}}^{(rank\ 3)} + \epsilon\_{\mathcal{S}} \tag{2}$$

$$
\delta\_\mathcal{g}^{group} = \ constant + c\_1 \delta\_\mathcal{g}^{(rank\ 1)} + \epsilon\_\mathcal{g} \tag{3}
$$

$$
\delta\_\%^{group} = \text{constant} + c\_2 \delta\_\%^{(rank\ 2)} + \epsilon\_\% \tag{4}
$$

$$
\delta\_\mathcal{g}^{\text{group}} = \text{constant} + \mathfrak{c}\_\mathfrak{d} \mathfrak{f}\_\mathfrak{g}^{\text{rank 3}} + \mathfrak{e}\_\mathfrak{g}, \tag{5}
$$

where δ group g denotes group g's elicited group discount factor, and δ (rank i) g denotes rank i's elicited individual discount factor in group g. We use δ (rank j) g to refer to the jth highest discount factor among the individuals in group g (in particular, δ (rank 1) g refers to the highest and δ (rank 3) g refers to the lowest discount factors). Model (2) includes all the members' discount factors as explanatory variables. Model (3) includes only the most patient member's discount factor (i.e., δ (rank 1) g ) as an explanatory variable. Model (4) includes only the median patient member's discount factor (i.e., δ (rank 2) g ) as an explanatory variable. Model (5) includes only the most impatient member's discount factor (i.e., δ (rank 3) g ). If the median voter hypothesis is supported, Model (4) will fit the experimental data better than Models (3) and (5).

To fit a Bayesian parametric model, we need to specify the likelihood function or the distribution of the data and the prior distributions for all model parameters. In Model (2), the Bayesian linear model has five parameters: four regression coefficients (i.e., a constant, δ (rank 1) , δ (rank 2) , and δ (rank 3) ) and the variance of the data. We assume a normal distribution for the dependent variable (i.e., δ group) and start with a non-informative Jeffreys prior [26] for the parameters. Under the Jeffreys prior, the joint prior distribution of the coefficients and the variance is proportional to the inverse of the variance. In Models (3), (4), and (5), we also assume a normal distribution for the dependent variable (i.e., δ group) and start with a non-informative Jeffreys prior for the three parameters (i.e., a constant, δ (rank R) , and the variance of the data). Here, R is set to 1 in Model (3), 2 in Model (4), and 3 in Model (5).

We can write model (2) as follows:

δ

$$\begin{aligned} \text{group} & \sim N\left(\mathbf{X}\boldsymbol{\beta}, \sigma^2\right) \\ \left(\boldsymbol{\beta}, \sigma^2\right) & \sim 1/\sigma^2, \end{aligned}$$

where X is design matrix, and β = constant, δ (rank 1) , δ (rank 2) , δ (rank 3) ′ , which is a vector

#### TABLE 4 | Results of Bayesian linear regression.

Here, (today vs. 1 month) and (1 month vs. 2 months) refer to the combinations of dates for the early option and the delayed option. Using (today vs. 1 month) as an example, participants choose to receive a small reward today or a larger reward one month later. Each rank refers to the order of the discount factor under the individual condition in the same group. Rank 1 refers to the most patient member (the highest discount factor) in a group, rank 2 refers to the median member (the second highest discount factor), and rank 3 refers to the most impatient member (the third highest discount factor, i.e., the lowest discount factor). Group refers to the discount factor for the group condition. Mean reports the estimates of the posterior means. Std. Dev. reports the estimates of the posterior standard deviations. The credible interval reports the 95% probability that the coefficient is in the described range.

of coefficients. In Models (3), (4), and (5), the vector of coefficients is β = constant, δ (rank R) ′ , where R is 1 in Model (3), 2 in Model (4), and 3 in Model (5). Next, we calculate the Bayes factors. The Bayes factors compute the relative probabilities of how well each model fits the data, as compared with the base model. We report the log Bayesian factors. If a log Bayesian factor is larger than zero, the reference model fits the data better than the base model and vice versa. We calculate two cases, namely, where the base model is Model (2) and the base model is Model (4). We use the Metropolis-Hastings method. All statistical analyses are performed using STATA 14.0.

**Table 4** shows the results of Bayesian linear regression. The mean reports the estimates of posterior means, which are the means of the marginal posterior distributions of the parameters. Std. Dev. reports the estimates of the posterior standard deviations, which are the standard deviations of the marginal posterior distributions. The credible interval reports the 95% probability that the coefficient is in the described range. In the case of Model (2), only rank 2 has a lower bound of the 95% credible interval that is positive for both combinations of dates for the early option and the delayed option. Therefore, we surmise that rank 2 has a strong influence on group choices. **Table 5** shows the log Bayes factors. The first and third columns report the log Bayes factors using Model (2) as a base model. As shown, the value of the log Bayes factor is positive when the reference model is Model (4) for both combinations of dates. Thus, Model (4) fits the data better than Model (2). In other words, the model that includes only the constant and rank 2 is better than the model that includes all of the coefficients. The second and fourth columns report the log Bayes factors using Model (4) as a base model. As shown, the value of the log Bayes factor is negative when the reference model is Model (3) for both combinations of dates. Thus, Model (4) fits the data better than Model (3). In other

#### TABLE 5 | Log Bayes factors.

Here, (today vs. 1 month) and (1 month vs. 2 months) refer to the combinations of dates for the early option and the delayed option. Using (today vs. 1 month) as an example, participants choose to receive a small reward today or a larger reward one month later. Models (2), (3), (4), and (5) are described in the Bayes analysis subsection. If a log Bayesian factor is larger than zero, the reference model fits the data better than the base model and vice versa.

words, the model that includes the constant and rank 2 is better than the model that includes the constant and rank 1. From these results, we conclude that the median voter hypothesis is supported.

#### Chat Analysis

We reject the multilateral bargaining model using a regression analysis and a post-estimation analysis in section Regression Analysis and Post-estimation Analysis. In addition, we discover that the median voter hypothesis is supported using a Bayesian analysis in section Bayesian Analysis. In this subsection, we investigate why the median voter hypothesis is supported by analyzing the text data from the group chats.

Why does the median patient member have a strong impact on group choices, even though the experimental setting uses the unanimity rule? Here, we focus on the order of preference expression, which we label as follows. The person who expresses an opinion first is 1st, the person who expresses an opinion second is 2nd, and the person who expresses an opinion third is 3rd.

When people express an opinion, this opinion is considered to be closest to their own preferences7 . They can discuss whether their opinions differ from those of the other group members. Here, we refer to the choices in the individual condition as a member's own preferences. However, the proportion (the number expressing the opinion closest to own preference) ÷ (the number of all expressions) is only 80%8 . In other words, 20% of those expressing an opinion fail to express their individual conditions. Are the proportions of each order of preference expression the same? **Table 6** shows the results9 . The data in **Table 6** show the sum of the two combinations of dates (i.e., {Early date, later date} = {today, 1 month} and {1 month, 2 months}). The proportion of expressions closer to the own preferences declines from 1st to 3rd (i.e., 84.05% in 1st, 79.77% in 2nd, 73.93% in 3rd). The proportions in each order of preference expressions are statistically different [χ 2 (2) = 8.0602, p = 0.018]. Multiple comparisons show that the proportion between 1st and 3rd is significantly different (p = 0.014, Bonferroni test), and the proportion between 2nd and 3rd is not significantly different (p = 0.306, Bonferroni test). Thus, those who express their own opinion third fail to express own preferences less often than those who do so first.

Why do people who express 3rd fail to express their own preferences? There are two situations in which "3rd" express their own opinion. First, the first person and the second person express the same opinion. Second, the first person and the second person express different opinions10. If the first person and the second person express the same opinion, the third person's preference differs from theirs, but the third person follows the first two. We call this "dishonesty due to conformity with majority." For example, 1st chooses the early option and 2nd chooses the early option. Consequently, although 3rd prefers the delayed option, s/he chooses the early option. We call this "conformity with majority." In the chat data, the total number of dishonest choices by 3rd is 67, as shown in **Table 6**. Out of these, the number of "dishonesty due to conformity with the majority" is 63. Therefore, there might be a tendency for people to follow the majority, even though they have their own opinions.

TABLE 6 | Table of whether the first expressions in the group chats are dishonest.

The values show the sums of the two combinations of dates (i.e., {Early date, later date} = {today, 1 month} and {1 month, 2 months}). The order of preference expression is as follows. The person who expresses an opinion first is 1st, the person who expresses an opinion second is 2nd, and the person who expresses an opinion third is 3rd. Honest means that the first expression in the group chat is the same as that of the individual condition. Dishonest means that the first expression in the group chat is not the same as that of the individual condition. We exclude cases where no group chat occurred and where the order of preference expression is ambiguous.

We guess that this tendency affected our main findings that the median patient member has the greatest impact on group choices. The "dishonesty due to conformity with majority" works to the median member's advantage, as follows. For example, suppose we have two options, A and B, and three members (x, y, z) who have single-peaked preferences; in this situation, there are four patterns of members' preferences (i.e., {x, y, z} = {A, A, A}, {A, A, B}, {A, B, B}, {B, B, B}). Taking {x, y, z} = {A, A, B} as an example, the number of combinations of the order of preference expressions and own opinions is six (i.e., {1st, 2nd, 3rd} = {Ax, Ay, Bz}, {Ay, Ax, Bz}, {Bz, Ax, Ay}, {Bz, Ay, Ax}, {Ax, Bz, Ay}, and {Ay, Bz, Ax}). Here, Ax indicates that person x chooses option A. It is possible that "dishonesty due to conformity with the majority" occurs when {1st, 2nd, 3rd} = {Ax, Ay, Bz} and {Ay, Ax, Bz}. When the third person z fails to express his/her own opinion B, but instead expresses A to follow the majority, "dishonesty due to conformity with the majority" occurs. Consequently, all three members choose option A. Thus, alternative A (i.e., median person y's preference) is selected as the group choice. We reach the same conclusion for {x, y, z} = {A, B, B}. These cases might increase the significance of the median patient member's effect on group choices.

As discussed above, we guess that the median patient members have a significant impact on group choices, partially because there are many cases of "dishonesty due to conformity with the majority." Naturally, there might be other reasons why median patient members have an effect on group choices. This is left for future research.

#### CONCLUSION

This study investigates how individual intertemporal preferences are aggregated in groups through deliberation, by clarifying who has a significant effect on group choices. We formulated two hypotheses. First, the multilateral bargaining hypothesis

7For example, the person who prefers D in **Figure 1** is considered to choose the early option for the first question (JPY 1,750 at an early date or JPY 2,000 at a later date).

8We check whether people's opinions are the same as those of the individual conditions. We do not count opinions that change during the discussion and count only the first expression in each question. There is one expression per person in each question. As mentioned in the Method section, there are four expressions for one combination of dates for the early option and the delayed option, there are two combinations of dates (i.e., {today, 1 month} and {1 month, 2 months}), and there are 105 participants. Therefore, the number of all first expressions is 840 (4 × 2 × 105 = 840). Out of these, the number of first expressions not close to the own preferences is 168. Therefore, (the number choosing an option NOT closer to own preference) ÷ (the number of all expressions) = 168/840 = 0.2.

9We exclude cases where no chat occurred or where the order of preference expression is ambiguous in **Table 6**. Therefore, the final sample is 771.

10As mentioned in the Method section, participants always face a choice between two options.

is based on the multilateral bargaining model, which predicts that the most patient member in a group has the greatest impact on group choices. Second, the median voter hypothesis is based on the median voter model, which predicts that the median patient member in a group has the greatest impact on group choices. We found that the median patient member has a substantial impact on group choices; that is, the median voter hypothesis is supported. Moreover, we examined the text data from the group chats to investigate why the median voter hypothesis is supported. According to the chat analysis, people who express their own opinion third (i.e., last) tend to follow the majority opinion. This may be one reason for our result (i.e., the median member has the greatest impact on group choices), even though the experimental setting is a free discussion and uses the unanimity rule.

Our results indicate that a median patient member in a group has the greatest impact on group intertemporal choices, even while using a free discussion and the unanimity rule. When workers make investment decisions under these conditions in a meeting, the median patient opinion might be accepted. This finding is surprising because many economists assume that the most patient member has the strongest bargaining power in group choices.

In our experiment, the groups are small and contain an odd number of members (i.e., three). Thus, we cannot generalize our results for an even number of members or for large groups. For an even number of members, we cannot determine whether the more patient median member or the less patient median member has a greater impact. Furthermore, median patient's power may decrease in large groups when compared with small groups. Median member becomes a pivotal member in group decision less often in a large group than in a small group. Thus, the answer to how much impact which median member has in a large group is unknown. Resolving these issues is left to future research.

#### REFERENCES

#### DATA AVAILABILITY STATEMENT

The raw data supporting the conclusions of this manuscript will be made available by the authors, without undue reservation, to any qualified researcher.

#### AUTHOR CONTRIBUTIONS

MT developed the study concept and the experimental paradigm and conducted the experiment. MT performed the data analysis and interpretation under the supervision of KI. MT and KI drafted the manuscript. Both authors contributed to the discussion section of the manuscript and approved the work for publication.

#### FUNDING

This work was supported by the Japan Society for the Promotion of Science Grant-in-Aid for Scientific Research JP17H04780 JP 15K13007 for KI and 15H05728 for Yoshiyasu Ono, the Top-Setting Program to Advance Cutting-Edge Humanities and Social Sciences Research, and the Joint Usage/Research Center at ISER, Osaka University.

#### ACKNOWLEDGMENTS

We thank the faculty and staff of the Center for Behavioral Economics at ISER, Osaka University, who kindly allowed us to use their laboratory resources.

#### SUPPLEMENTARY MATERIAL

The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/fams. 2018.00043/full#supplementary-material

**Conflict of Interest Statement:** The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Copyright © 2018 Tsuruta and Inukai. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

## A Testing Method of Probability Weighting Functions From an Axiomatic Perspective

Kazuhisa Takemura1,2 \* and Hajime Murakami 2

*1 Institute for Decision Research, Waseda University, Tokyo, Japan, 2 Department of Psychology, Waseda University, Tokyo, Japan*

This study presents a testing approach to examine various models of probability weighting functions that are considered nonlinear functions of probability in behavioral decision theory, such as prospect theory. Although there are several empirical psychometric tests to examine probability weighting functions, there is no concrete method to examine these functions' axiomatic properties. We propose axiomatic properties and a testing method to examine the generalized hyperbolic logarithmic model, power model, and exponential power model of the probability weighting functions, and provide an illustrative example of the testing method.

#### Edited by:

*Taiki Takahashi, Hokkaido University, Japan*

#### Reviewed by:

*Zari Rachev, Texas Tech University, United States G. Charles-Cadogan, University of Leicester, United Kingdom*

> \*Correspondence: *Kazuhisa Takemura<EMAIL_ADDRESS>
#### Specialty section:

*This article was submitted to Quantitative Psychology and Measurement, a section of the journal Frontiers in Applied Mathematics and Statistics*

> Received: *20 April 2018* Accepted: *21 September 2018* Published: *11 October 2018*

#### Citation:

*Takemura K and Murakami H (2018) A Testing Method of Probability Weighting Functions From an Axiomatic Perspective. Front. Appl. Math. Stat. 4:48. doi: 10.3389/fams.2018.00048* Keywords: axiomatic approach, decision under risk, hyperbolic logarithmic discounting, probability weighting function, time discounting

#### INTRODUCTION

A probability weighting function W(p) is a nonlinear function of an objective probability p, where p is determined primarily from the frequentist view. Recently, they have received substantial empirical and theoretical attention [1–3]. They are used in many fields, such as behavioral decision theory, behavioral economics and neuroscience [4].

Several psychometric models have been proposed to represent probability weighting functions (e.g., [1–3, 5, 6]). Some proposed probability weighting models derive from time discounting models [2, 3, 7]. Rachin et al. [7] derived the model from the original hyperbolic function. Takahashi [2] used a q-exponential time discount function [8] to derive Prelec's [6] probability weighting function and an exponential power model. Takemura and Murakami [3] used a more direct assumption of time discounting to derive the hyperbolic logarithmic function model and the generalized hyperbolic logarithmic model as probability weighting functions.

Takemura and Murakami [3] used a generalized hyperbolic time discounting model that assumes both Fechner's [9] psychophysical law of time and a geometric distribution of trials. From this, they derived hyperbolic logarithmic type models. They were then able to examine the generalized hyperbolic model in the context of an axiomatic system. They used Gonzalez and Wu's [10] procedure to estimate the function parameters. To investigate goodness of fit, they computed both the Akaike Information Criterion (AIC) and Bayes Information Criterion (BIC). The results indicated that the generalized hyperbolic logarithmic mode originally proposed by Prelec [6] fitted better than median time discounting models, the one-parameter Prelec model, and the Tversky and Kahneman model.

Takemura and Murakami Axiomatic Testing of Probability Weighting Functions

Takemura and Murakami [3] made a key contribution by supporting, both theoretically and empirically, a possible psychological interpretation of a probability weighting function in the context of time discounting. However, the empirical method used in the study was a psychometric nonlinear regression study. Although there are several empirical psychometric tests available to examine probability weighting functions, there is no concrete method to examine the axiomatic properties of the probability weighting functions. Prelec [6] had already proposed the axiomatic properties for some weighting functions. However, no concrete axiomatic properties distinguished the individual models he proposed, and no testing method was suggested. Based on their axiomatic considerations, we propose axiomatic properties and a testing method to examine the generalized hyperbolic logarithmic model, power model, and exponential power model of the probability weighting functions, and provide an illustrative example of the testing method.

#### AXIOMATIC SYSTEM OF GENERALIZED HYPERBOLIC LOGARITHMIC MODEL, EXPONENTIAL POWER MODEL, AND POWER MODEL FOR PROBABILITY WEIGHTING FUNCTIONS

Counterexamples, such as the Allais paradox [11] and the Ellsberg paradox [12], have been identified in earlier studies. These paradoxes are interpreted as deviations from the independence axiom. Recently, they have been explained using theory systems. More specifically, these systems include the nonlinear utility theory [13–15]—which does not require this independence axiom—as well as the generalized expected utility theory [16]. Prospect theory [5, 17] integrates knowledge and past findings in nonlinear utility theory (or generalized expected utility theory) and behavioral decision-making theory.

In prospect theory, we assume a non-additive probability function, where a non-additive probability is a set function π : 2 Ω → [0, 1] from an aggregate of subsets of a nonempty set Ω to a closed interval [0, 1]. The non-additive probability function is a set function satisfying both a boundedness condition (π (φ) = 0, π (Ω) = 1) and a monotonicity condition [if A ⊆ B, then π (A) ≤ π(B), where A, B are subsets of Ω]. A non-additive probability does not necessarily satisfy additivity conditions. Prelec [6] showed psychometric functions of non-additive probability (probability weighting functions) and axiomatic properties of the probability weighting functions based on prospect theory.

Based on the theoretical work of Prelec [6], we show axiomatic properties of the generalized hyperbolic logarithmic model, exponential power model, and power model for probability weighting functions.

For the set A of probability distributions P, Q, . . . on X = [x −, x +], where x − < 0 < x +, let < be a preference relation. Prospects are considered distributions with finite support. Then, we assume the following axioms [6].

**W1.

**W2. Strict stochastic dominance**: P > Q if both P 6= Q and P is stochastically dominants over Q.

**W3. Certainty equivalent condition:** For every P, ∃ x such that (x) ∼ P.

**W4. Continuity in probabilities:** If (y, p) > (x) where 0 < p < 1, then ∃ q, r such that q < p < r, (y, q) > (x), and (y,r) > (x).

**W5. Simple continuity:** Let the set of all k nonpositive and (n − k) nonnegative rank-ordered n-tuples from X be S(k, n), where 0 ≤ k ≤ n. If the preference relation induced on each set S(k, n) is continuous for any probability vector (p1, p2, · · · , pn), then there is simple continuity.

**W6. Tradeoff consistency:** Consider a prospect (x, pi; **x**−**i** , **p**−**i**) with outcome c of rank i singled out and the set R(k, n, p) of all sign-order and rank-order compatible prospects with a p-chance of a negative outcome. Assume there are not eight prospects, (x, pi; **a**−**i** , **p**−**i**), (y, pi; **b**−**i** , **p**−**i**), (x ′ , pi; **a**−**i** , **p**−**i**), (y ′ , pi; **b**−**i** , **p**−**i**), (x ′ , qj; **c**−**j** , **q**−**j**), (y ′ , qj; **d**−**j** , **q**−**j**), (x, qj; **c**−**j** , **q**−**j**), and (y, qj; **d**−**j** , **q**−**j**), such that the first and second groups of four belong to the same sign-order and rank-order compatible set, and

$$
\begin{array}{rcl}
\left(\boldsymbol{\chi}, \ p\_{i}; \ \mathbf{a}\_{-i}, \mathbf{p}\_{-i}\right) & \mathrel{\mathop{\rm b}}{\mathop{\rm b}}\left(\boldsymbol{\chi}, \ p\_{i}; \ \mathbf{b}\_{-i}, \mathbf{p}\_{-i}\right), \\
\left(\boldsymbol{\chi}', \ p\_{i}; \ \mathbf{a}\_{-i}, \mathbf{p}\_{-i}\right) & \mathrel{\mathop{\rm b}}{\mathop{\rm b}}\left(\boldsymbol{\chi}', \ p\_{i}; \ \mathbf{b}\_{-i}, \mathbf{p}\_{-i}\right), \\
\left(\boldsymbol{\chi}', \ q\_{j}; \ \mathbf{c}\_{-j}, \mathbf{q}\_{-j}\right) & \mathrel{\mathop{\rm b}}{\mathop{\rm b}}\left(\boldsymbol{\chi}', \ q\_{j}; \ \mathbf{d}\_{-j}, \mathbf{q}\_{-j}\right), \\
\left(\boldsymbol{\chi}, \ q\_{j}; \ \mathbf{c}\_{-j}, \mathbf{q}\_{-j}\right) & \mathrel{\mathop{\rm b}}{\mathop{\rm b}}\left(\boldsymbol{\chi}, \ q\_{j}; \ \mathbf{d}\_{-j}, \mathbf{q}\_{-j}\right).
\end{array}
$$

Then, tradeoff consistency holds.

The following assumptions are as described by Prelec [6].

**Assumption 1:** < satisfies axioms W1-W6, which support a sign-dependent and rank-dependent representation with a continuous and strictly increasing ratio scale v(x), as well as a strictly increasing unique w −(p), w +(p) that is continuous on (0, 1), and satisfies w +(0) = w −(0) = 0, w +(1) = w −(1) = 1.

**Assumption 2:** There is a separable representation of the restriction of < to simple prospects, with v(x), w − p , and w +(p) satisfying the Assumption 1 conditions.

**Definition 1 Conditional invariance [6]:** < has **conditional invariance** if the following holds for any outcomes x, y, x ′ , y ′ ∈ X, probabilities q, p,r,s ∈ [0, 1], and conditional probability λ, 0 < λ < 1:

If (x, p) ∼ (y, q) and (x, r) ∼ (y,s), then (x ′ , λp)∼(y′ ,λq) implies (x ′ , λr)∼ (y ′ , λs) or (x ′ , λr)∼(y ′ ,λs).

**Definition 2 Projection invariance [6]:** < has **projection invariance** if the following holds for any outcomes x, y ∈ X, probabilities q, p,r,s ∈ [0, 1], and conditional probability λ, 0 < λ < 1:

If (x, p) ∼ (y, q) and x, rp ∼ y,sq , then (x, r 2p) ∼ (y,s 2 q).

#### **Proposition 1: The generalized hyperbolic logarithmic model proposition**

Let < be a preference relation on R + where either Assumption 1 or 2 holds, conditional invariance (Definition 1) does not hold, and projection invariance (Definition 2) holds. Then, the probability weighting function W(p) is a hyperbolic logarithm,

$$W(p) = (1 - k \log p)^{\beta},$$

where p is probability (0
 0.

#### **Proof**

The proof of Proposition 1 is trivial and derived from a combination of Propositions 4 and 5 in the original theoretical work by Prelec [6]. Prelec [6] found that if < is a preference relation on R + where either Assumption 1 or 2 holds and conditional invariance (Definition 1) holds, then the weighting function (0<p) is either an exponential-power function or a power function (Proposition 4). Prelec [6] also found that if < is a preference relation on R + where either Assumption 1 or 2 holds and projection invariance (Definition 2) holds, then the weighting function (0<p) is either a hyperbolic logarithm or a power function (Proposition 5). Therefore, if < is a preference relation on R + satisfying Assumption 1 or 2, conditional invariance (Definition 1) does not hold, and projection invariance (Definition 2) holds, then the probability weighting function W(p) is a hyperbolic logarithmic function.

#### **Proposition 2: Proposition of the exponential power model**

If < is a preference relation on R + satisfying Assumption 1 or 2, conditional invariance (Definition 1) holds, and projection invariance (Definition 2) does not hold, then the probability weighting function W(p) is an exponential power function such as

$$W(p) = \exp\left\{-k\left(1 - p^{\beta}\right)\right\},$$

where p is probability (p>0), and k and β are positive constants, k, β > 0.

#### **Proof**

The proof of Proposition 2 is trivial and also derived from a combination of Propositions 4 and 5 in the original theoretical work by Prelec [6]. As in the same inference of Proposition 1, if < is a preference relation on R + satisfying Assumption 1 or 2, conditional invariance (Definition 1) holds and projection invariance (Definition 2) does not hold, then the probability weighting function W(p) should be an exponential power function.

#### **Proposition 3: Proposition of the power model**

If < is a preference relation on R + satisfying Assumption 1 or 2, conditional invariance (Definition 1), and projection invariance (Definition 2), then the probability weighting function W(p) is an exponential power

$$W\left(p\right) = p^{\beta},$$

where p is probability, and β is a positive constant, β > 0.

**Proof**

The proof of for this proposition is trivial and also derived from a combination of Propositions 4 and 5 in the original theoretical work by Prelec [6]. As in the same inference of Proposition 1, if < is a preference relation on R + satisfying Assumption 1 or 2, conditional invariance (Definition 1) holds and projection invariance (Definition 2) holds, then the probability weighting function W(p) is a power function.

#### A TESTING METHOD TO EXAMINE AXIOMATIC PROPERTIES OF GENERALIZED HYPERBOLIC LOGARITHMIC MODEL, EXPONENTIAL POWER MODEL, AND POWER MODEL FOR PROBABILITY WEIGHTING

We propose a testing method to examine the generalized hyperbolic logarithmic model, power model, and exponential power model of the probability weighting functions, and provide an illustrative example of the testing method. First, we present the testing method using verification tasks of projection invariance and conditional invariance. We then give an example verifying the reliability and axioms and showing the goodness of fit of the models.

**Figure 1** illustrates the testing experimental process, which had participants choose one option from two gambles. To assess reliability, projection invariance and conditional invariance should be examined at least twice. Additionally, trials are done at least 30 times to stabilize the responses.

#### The Projection Invariance Verification Process

Experimental screens and the task processes in the projection invariance verification process are shown in **Figure 2**. The task presented to the participants was to choose one option from two gambles as shown in **Table 1**. The participants were instructed to choose a preferred option from the experimenter.

The verification process of projection invariance is presented in **Table 1**. In the verification of projection invariance in **Table 1**, y = 10,000 yen, p = 100%, q = 50%, and s = 50% are given. In addition, **Table 1** presents values of the responses by participants shown in bold typeface.

TABLE 1 | Verification process of projection invariance.

*The values given in responses by the participants are shown in bold typeface. y* = *10,000 yen, p* = *100%, q* = *50%, s* = *50%.*

The verification projection invariance tasks comprise three steps. To explain the verification process using the example presented in **Table 1**, in Step 1, x in the alternative A equivalent to the alternative B (to obtain 10,000 yen with 50%) is estimated from a pair comparison of the alternative A and alternative B in **Figure 2**. Next, in Step 2, r in the alternative A equivalent to the alternative B (to obtain 10,000 yen with 25%) is estimated. For x of the alternative A in Step 2, the x obtained in Step 1 is used. Finally, in Step 3, using x and r obtained in Step 1 and Step 2, the alternative A (to get 5,000 yen with 25%) is made. Then yˆ is estimated (to obtain yˆ yen with 12.5%). Here, when yˆ obtained in Step 3 is 10,000 yen, which is the same as y, projection invariance is regarded as satisfied. Additionally, because y = 10,000 yen is given in Step 3, to estimate yˆ, 20,000 yen and 0 yen, respectively, the maximum value and the minimum value of yˆ were presented to the participants. Then they were asked to do the pair comparison, as shown in **Figure 2**.

Steps 1, 2, and 3 respectively comprise 9 trials, 9 trials, and 12 trials. The stimulation sequences used to verify projection invariance are shown in **Table 2**. Nine sequences of stimulation were prepared. Furthermore, because y is fixed at 10,000 yen in Step 3, when a participant gives a response to satisfy the axioms, the participant might continue to give the same response and is likely to change a response due to fluctuation of the psychological process. Therefore, with three sequences of dummy stimulation added to the nine sequences, Step 3 has 12 trials in all.

#### Verification Process of Conditional Invariance

Experimental screens and task processes were prepared in an identical form to that used for projection invariance in the verification process of conditional invariance. The participants were also instructed to choose a preferred option from two gambles, as shown in **Table 3**. The verification process of conditional invariance is presented in **Table 3**. For verification of the conditional invariance in **Table 3**, y = 20,000 yen, y ′ = 10,000 yen, p = 100%, q = 50%, s = 10%, and λ = 50% are given. Additionally, the values given in responses by the participants are shown in bold typeface in **Table 3**.

The verification tasks of conditional invariance comprise four steps. To explain the verification process using the example in **Table 3**, in Step 1, x of alternative A equivalent to alternative B (to get 20,000 yen with 50%) is estimated through a pair comparison between alternatives A and B as shown in **Figure 2**. Next, in Step 2, r of the alternative A equivalent to the alternative B (to get 20,000 yen with 10%) is estimated. For x of alternative A in Step 2, the x obtained in Step 1 is used. In Step 3, x ′ of the alternative A (to obtain x ′ yen with 50%) equivalent to the alternative B (to get 10,000 yen with 25%) is estimated. In Step 4, using r and x ′ obtained in Step 2 and Step 3, respectively, alternative A (to get 5,000 yen with 10%) is made. Then, yˆ ′ of alternative B (to get yˆ ′ yen with 5%) is estimated. Here, when yˆ ′ obtained in Step TABLE 3 | Verification process of conditional invariance.

*The values given in responses by the participants are shown in bold typeface. y* = *20,000 yen, y*′ = *10,000 yen, p* = *100%, q* = *50%, s* = *10%,* λ = *50%.*

TABLE 4 | Stimulus sequences of conditional invariance (%).

4 is 10,000 yen, which is the same value as y ′ , the conditional invariance is regarded as satisfied.

Steps 1, 2, and 3 are composed respectively of 9 trials, 9 trials, and 12 trials. The stimulus sequences used for verification of conditional invariance are presented in **Table 4**. Nine sequences of stimuli were prepared. In Step 4, although three dummy stimulus sequences were also prepared for the same reason as those for projection invariance, three stimuli from the sequences were randomly provided twice because of the experimental program's errors. As a result, Step 4 had 9 sequences plus 3 trials, i.e., 12 trials in total.

#### AN EXAMPLE OF THE TESTING METHOD

#### Participants

The participants were 14 undergraduate students (eleven female and three male) studying psychology at Waseda University aged between 21 and 25 years old. They were paid 1500 Japanese yen (about 15 dollars) to participate in a 1.5-h test. This study has ethical approval from the Academic Research Ethical Review Committee, Waseda University concerning Guidelines Regarding Academic Research Ethics, Waseda University. Participants provided written informed consent.

#### Materials and Procedure

We asked the participants to select their preferred option from two alternatives while watching the screen shown TABLE 5 | Reliability of projection invariance and conditional invariance (final step).

*Intraclass coefficients that are 0.6 or greater presented in bold typeface.*

in **Figure 2**. The verification procedure of projection invariance and conditional invariance were as described above.

#### Examination of Reliability

A participant was asked to work on the tasks twice in a row to assess the reliability. Intraclass correlation coefficients of projection invariance and conditional invariance by the participant was calculated. The intraclass correlation coefficients calculated from the first and second answers in the final step are shown in **Table 5**. Those calculated in all steps are shown in **Table 6**, with intraclass correlation coefficients that are 0.6 or greater presented in bold typeface. The medians of intraclass correlation coefficients calculated in the final step were 0.216 (maximum value, 0.871; minimum value, −0.324) in projection invariance and −0.131 (maximum value, 0.668; minimum value, −0.384) in conditional invariance. However, the medians of intraclass correlation coefficients calculated in all steps was 0.897 (maximum value, 0.992; minimum value, 0.401) in projection invariance and 0.765 (maximum value, 0.968; minimum value, 0.476) in conditional invariance.

As **Table 6** shows, the intraclass correlation coefficients are all 0.4 or greater for projection invariance and conditional invariance.

#### Verified Results of Projection Invariance and Conditional Invariance

**Table 7** presents the numbers of sequences in which participants satisfied the axioms of projection invariance and conditional invariance. Because nine sequences were used to verify the axioms, when participants judge in accordance with the axioms in five sequences or more, the numbers are presented in bold typeface.

*Intraclass coefficients that are 0.6 or greater presented in bold typeface.*

TABLE 7 | Number of sequences satisfying the axiom (out of 9 sequences).

*Participants judged in accordance with the axioms in five sequences or more, the numbers are presented in bold typeface.*

#### Examination of Goodness of Fit of the Model

Free parameters, such as β and k, of the probability weighting function by the participant were estimated by the same experiment as in Gonzalez and Wu [10]. **Table 8** presents a list of examined models. In addition, **Table 9** shows models which are the fittest according to AIC. Twelve participants had the best fit with the hyperbolic logarithmic model. One participant had the best fit with the exponential power function. One participant had the best fit with the power function.

#### Relation Between Axioms and Goodness of Fit of Models

The correspondence between axioms and models is presented in **Table 10**. Relations between satisfied axioms and models TABLE 8 | List of models.

TABLE 9 | Models' AICs and the model with the smallest AIC by participant.

*The smallest AIC by participant presented in bold typeface.*

TABLE 10 | Correspondence between axioms and models.

with goodness of fit are shown in **Table 11**. Because nine sequences were used to verify axioms, if axioms were satisfied in five and more sequences, then they are regarded as satisfied. The first participant satisfied projection invariance alone. The third participant satisfied conditional invariance. The other participants satisfied neither projection invariance nor conditional invariance. Results show that there is not a certain correspondence with normal quantitative psychometric methods that used the nonlinear regression method and the model fitting examination by AIC indicator.

#### CONCLUSION AND DISCUSSIONS

This study aimed to present a testing approach used to examine the generalized hyperbolic logarithmic model, power model, and exponential power model of the probability weighting functions that are considered nonlinear functions of probability in behavioral decision theory, for example, in prospect theory [5, 6]. Although many empirical psychometric tests are used to

TABLE 11 | Relations between satisfied axioms and models with goodness of fit.

examine the probability weighting functions, there is no concrete method to examine the axiomatic properties of the probability weighting functions. Therefore, we propose axiomatic properties based on Prelec's [6] theory and a testing method to examine the generalized hyperbolic logarithmic model, power model, and exponential power model of the probability weighting functions, and provide an illustrative example of the testing method.

According to this result of the example experiment, the axiomatic properties of the probability weighting functions did not correspond to the psychometric fitting result of probability weighting functions. A similar result occurs in the additive conjoint systems in judgment and decision making. For example, empirical evaluations of double cancelation for the conjunctive measurement rejected the double cancelation axiom [18, 19]. However, psychometric studies have also indicated that the linear additive model fitted better [20]. There are some contradictions between psychometric studies and axiomatic studies. This case is the same as previous research. Further research is needed to identify why the discrepancies occur.

Luce and Steingrimsson [21] examined the Thomsen condition and the conjoint commutativity axiom, which they showed were equivalent. They also found that brightness and binaural loudness were supporting factors of conjoint commutativity. We must consider the reason for the unclear

#### REFERENCES

correspondence between the axiomatic testing and psychometric testing. One possibility is that the assumptions of the prospect theory did not hold in this experiment. Another is that the essential conditions, such as conditional invariance and projection invariance, did not hold in the experiment. Further research could investigate these possibilities.

In our study, the number of participants was limited and the participants were all trained psychology students. However, our sample size matches those in previous studies [5, 10], so we do not consider this to invalidate the results. Nevertheless, larger sample sizes in future experiments would be beneficial in examining the psychometric model of probability weighting functions.

Although we proposed an axiomatic testing method of Prelec's [6] probability weighting function, there other ways to interpret probability weighting, such as from the perspective of rational dynamic asset pricing theory. Rachev et al. [22] explained the main concepts of prospect theory and probability weighting functions within the framework of rational dynamic asset pricing theory. They derived a modified Prelec weighting function and introduced a new parametric class for weighting probability functions. We did not examine the theoretical notions proposed by Rachev et al. [22]. Further theoretical examinations are needed to seek an adequate probability weighting function.

#### AUTHOR CONTRIBUTIONS

KT and HM developed the theoretical formalism, performed the analytic calculations and performed data analysis. Both authors contributed to the final version of the manuscript. KT supervised the project.

#### FUNDING

This study was supported by a Grant-in-Aid for Scientific Research (A), No. 24243061 and No. 16H02050 from The Ministry of Education, Culture, Sports, Science and Technology, Japan.

#### ACKNOWLEDGMENTS

We sincerely thank Yuki Tamari, Takashi Ideno, Takayuki Sakagami, Yutaka Nakamura, Yiyun Shou and the referees of this journal for their helpful comments.

**Conflict of Interest Statement:** The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Copyright © 2018 Takemura and Murakami. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

## Ecological Validity of Impulsive Choice: Consequences of Profitability-Based Short-Sighted Evaluation in the Producer-Scrounger Resource Competition

#### Yukiko Ogura1 , Hidetoshi Amita2 and Toshiya Matsushima3 \*

*1 Department of Social Psychology, Graduate School of Humanities and Sociology, The University of Tokyo, Tokyo, Japan, 2 Laboratory of Sensorimotor Research, National Eye Institute, National Institute of Health, Bethesda, MD, United States, 3 Department of Biology, Faculty of Science, Hokkaido University, Sapporo, Japan*

Results of intertemporal choice paradigm have been accounted for mostly by psychological terms such as temporal discounting of subjective value. Inability to wait for delayed gratification (choice impulsiveness, as opposed to self-control) is often taken to represent violated rationality. If viewed from foraging ecology, however, such impulsiveness can be accountable as adaptive adjustments to requirements in nature. First, under the circumstance where foragers stochastically encounter food items, the optimal diet-menu model suggests that each option must be evaluated by profitability (*e*/*h*), which is the ratio of energetic gain (*e*) per handling time (*h*), a short-sighted currency. As h includes the delay, profitability will be hyperbolically lower for long-delay food. Second, because of the resource competition between producing and scrounging foragers, profitability of the producer's gain will critically depend on the scrounger's behaviors. We first constructed an analytical model. The model predicted that the profitability of small and short-delay food option (*SS*) can be higher than that of the large and long-delay alternative (*LL*), depending on the duration in which the producer can monopolize a food patch (finder's share). Next, we conducted numerical simulations on the assumption of variable food amount in each patch with realistic set of behavioral parameters. Although non-linearity of profitability function largely reduced profitability for variable amount of food, *SS* still can have a higher profitability than *LL* when the finder's share is small. Because *SS* is consumed more quickly, it is more resistant against scrounging than *LL*. In good accordance, foraging domestic chicks form a synchronized flock and show socially-facilitated investment of effort. If raised in competition, chicks develop a higher degree of choice impulsiveness.

Keywords: foraging theory, social foraging, profitability, competition, social facilitation

#### Edited by:

*Taiki Takahashi, Hokkaido University, Japan*

#### Reviewed by:

*Marcelo N. Kuperman, Bariloche Atomic Centre, Argentina Gergely Zachar, Semmelweis University, Hungary Wataru Toyokawa, University of St Andrews, United Kingdom*

\*Correspondence:

*Toshiya Matsushima<EMAIL_ADDRESS>
#### Specialty section:

*This article was submitted to Quantitative Psychology and Measurement, a section of the journal Frontiers in Applied Mathematics and Statistics*

> Received: *03 July 2018* Accepted: *04 October 2018* Published: *30 October 2018*

#### Citation:

*Ogura Y, Amita H and Matsushima T (2018) Ecological Validity of Impulsive Choice: Consequences of Profitability-Based Short-Sighted Evaluation in the Producer-Scrounger Resource Competition. Front. Appl. Math. Stat. 4:49. doi: 10.3389/fams.2018.00049*

### INTRODUCTION

Spencer introduced the idea of "survival of the fittest" in "The Principles of Biology" (1864), and Darwin adopted this term in his 5th edition of "The Origin of Species" (1869). A naïve biological thinking might therefore be that only optimal individuals have been selected for survival. If it was, we might find a monochromatic world in which only a few fittest phenotypes predominate. Conversely, nature is full of diverse organisms with distinct behavioral phenotypes, even within a sympatric group of animals of the same species. We must consider some adaptive processes that make animals appropriately deviate from the optimality. Deviations due to social inter-individual interaction could be one of such critical processes. In this report, we focus on social foraging behavior as it allows us to theoretically and empirically make quantitative examinations on issues of optimality. Through introducing ecological theories of foraging, we argue that social facilitation and enhanced choice impulsiveness could be described as adaptive deviations.

#### OPTIMAL FORAGING, PROFITABILITY, AND INTERTEMPORAL CHOICES

#### Optimal Diet Menu Model and Impulsive Choices

Classical theories usually assume that optimal foragers maximize the long-term averaged gain rate. Foragers explore food (searching time, abbreviated as Ts) and exploit it (consuming or handling time, Th), so that the gain rate R is given for Ef (net energy gain) as a substitution variable for fitness [1].

$$\mathcal{R} = \frac{E\_f}{T\_s + T\_h}$$

As initially formulated as Holling's disc equation [2] and subsequently by Charnov [3] in his diet menu model, optimal foragers must maximize R. Assuming the stochastic nature of food resources, Charnov reached a somewhat paradoxical conclusion that the R-maximizer must make decisions (i.e., action choice between attacking or passing-over) based on short-sighted estimation of profitability e/h, where e represents energetic gain and h handling time for the encountered food item. Namely, they attack the food if the expected profitability exceeds the lost opportunity, that is the potential food gained by passing over (or giving up) the encountered food item. We should consider (1) prospective (expected) profitability of the food and (2) highly uncertain lost-opportunity, the latter of which may be approximated by the average gain rate they have had (**Figure 1A** left).

As profitability is the product of gain (e) and proximity (1/h), the foraging choice may be isomorphic to the intertemporal choice (ITC) paradigm widely adopted to study impulsive/selfcontrol issues (Ainslie-Rachalin theory [4]; **Figure 1A** right). Here, a set of two options {SS, LL} is given as:

$$\text{LSS} = \left(e\_1, h\_1^{-1}\right) \text{ } \text{LL} = \left(e\_2, h\_2^{-1}\right), \text{where } \left(e\_1 < e\_2\right) \cap \left(h\_1^{-1} > h\_2^{-1}\right).$$

Based on behaviors in the ITC task [5–7], we formulated simple pico-economics for chicks in terms of profitability-based behaviors [8]. Briefly, as chicks feed on grain particles with small e, they must make h small by adopting highly impulsive choices, so that a certain level of profitability (e/h) is achieved. Choice impulsiveness does not necessarily represent internal anomalies in the machineries, but can be an adaptive trait with external (ecological) validity. An extreme case of such foraging economics is found in star-nosed mole rats [9], where the handling time was as short as 120 ms for the tiny food particles they eat. Hyperbolic discounting of future rewards is assumed to be a corollary to the profitability rule, simply because h includes the delay time for food. However, the zeroone rule (another principal prediction derived from the menu model [1]) is not met in many empirical studies including chicks. Instead, matching to the relative profitability has been the norm in most cases. Applicability of the menu model is thus limited.

### Optimal Patch Use Model (Marginal Value Theorem) and Decision of Disengagements

Charnov proposed another important idea, the optimal patchuse model, which is based on marginal value theorem [10] (**Figure 1B** left). He assumes a single forager that sequentially visits a series of food patches. The model is characterized by (1) unevenly distributed food items in patches, and (2) resource depletion by the forager's consumption, as the convex curve of cumulative gain illustrates. The optimal forager must disengage from the patch at a point where gain rate is maximized (indicated by slope of the dashed red line in **Figure 1B** left) by action selection. If staying put in the patch, the forager will find the next food item in a short period of time. If disengaging, it must invest considerable time to find the new patch, which is more beneficial than the food item available in the old patch. Here again, the choice of actions might be translated to the intertemporal choice of options (**Figure 1B** right).

The possible commonality between the patch-use behavior and the ITC task has been challenged both theoretically and empirically [11–18]. So far, the foraging behavior cannot be translated to the ITC task, or these are simply not compatible. Actually, animals are less patient in the ITC task compared with their behaviors in natural foraging situations, and the temporal discounting measured in ITC task does not fit well with the decision to disengage. The underlying decision mechanism is also distinct, as pharmacological treatment using selective serotonin reuptake inhibitor (SSRI, fluvoxamine) suppressed impulsive choices but delayed the disengagement, contrary to the prediction based on the commonality [19]. Furthermore, the impulsivity measure can drastically change depending on how subjects are informed of the relevant parameters, such as the time after the food option was consumed (so-called postreward delay) [7, 20]. Interspecies comparisons suggest a clear dependency on ecological factors such as diet preference [21, 22], but these factors do not uniquely characterize impulsivity of each species.

#### THE ITC PARADIGM IN SOCIAL FORAGING SITUATIONS

#### Producer-Scrounger Resource Conflict