# The evolution of forecasting
for decision making in dynamic environments

Andrew R. Tilman  Department of Biology, University of Pennsylvania,
Philadelphia, PA, USA  Vítor V. Vasconcelos  Erol Akçay  Department of
Biology, University of Pennsylvania, Philadelphia, PA, USA  Joshua B. Plotkin
Department of Biology, University of Pennsylvania, Philadelphia, PA, USA

###### Abstract

Economic and evolutionary theories of strategic behavior make similar
predictions in a fixed environment, even though one approach assumes perfect
rationality and the other assumes no cognition whatsoever. But predictions
differ in a dynamic environment. Here we explore a middle ground between
economic rationality and evolutionary myopia. Starting from a population of
purely myopic agents, whose actions feed back on the environment to produce
cyclical dynamics, we study whether evolution can favor the emergence of new
type that forms forecasts about the future environment. We show that
forecasting types have an evolutionary advantage, even when they pay a cost to
produce forecasts. Forecasting types can invade but not overtake the
population, resulting in a stable coexistence with myopic types. Moreover,
forecasters provide a public good by reducing the amplitude of environmental
oscillations and thereby increasing the mean fitness for forecasting and
myopic types alike. We interpret our results as a natural evolution of
cognitive styles, and we discuss applications to problems of decision-making
with environmental feedback in both human and non-human systems.

## Introduction

Economic and evolutionary theory both address the problem of decision making
in strategic settings. But economic and evolutionary analyses make vastly
different assumptions about what information is available to individuals, and
how individuals use that information. In a classical economical analysis,
equilibrium strategies are identified such that fully informed and perfectly
rational agents have no incentive to unilaterally deviate – that is, a Nash
equillibria (Nash et al.,, 1950). On the other hand, evolutionary game theory
does not require individuals who know much, if anything, about the game they
are playing (Maynard Smith,, 1982), but who rather make strategic decisions
through a myopic search process. Such agents that lack cognition altogether
will eventually reach an equilibrium, called an evolutionarily stable strategy
(ESS) (Maynard Smith,, 1982). Remarkably, in static environments, these two
concepts are tightly coupled: all ESS’s are Nash equilibria, and all strict
Nash equilibria are ESS’s.

The remarkable concordance between the long-term behavior of fully rational
individuals versus the behavior of simple myopic agents does not carry over
when strategic interactions take place in a changing environment, or when the
strategies themselves generate environmental feedback. In a changing
environment, a strategy that is favorable today may be detrimental tomorrow.
In this setting, there is a wide gulf between the behaviors predicted by
traditional game theory versus those that arise dynamically in an evolving
population of myopic agents.

Here, we explore a middle ground between myopic agents and fully rational
agents – that is, between evolutionary and economic theories of decision
making. The goal of our study is not to determine which models of decision-
making or strategic behavior are best. Rather, we take the myopic setting as a
starting point and we ask when evolution will favor the emergence of
individuals who use more information about their environment and the game they
are playing when making strategic decisions. That is, we study how more
sophisticated modes of decision-making can arise in a population. In
particular, we study the emergence of decision making by individuals who
forecast the future state of the environment and account for future payoffs
based on their forecasts. Such individuals are not strictly myopic, but also
not fully rational and perfectly informed. We study both the emergence of
forecasting types as well as the resulting effects on strategic and
environmental dynamics.

The question of myopic decision-making versus decision-making using forecasts
is relevant not only to humans but also to a diversity of non-human organisms
that live in variable environments. Flowering plants are most successful when
they time their blossoming to align with favorable environmental conditions,
such as temperature, pollinator abundance, and density of competitors.
Accounting for environmental cues, including environmental change, may help
plants to achieve optimal timing of flowering (Vermeulen,, 2015). Migratory
species face a similar decision about when to leave one habitat and head for
another (Johansson and Jonzén,, 2012); and some form of environmental forecast
is beneficial and may be promoted by natural selection.

Environmental change can also be caused endogenously, by the decisions of
individuals. This is readily apparent in socio-ecological systems, e.g. when
resource exploitation decisions affect resource abundances. It can also be
found in predator-prey systems. For example, female Dusky Warblers modify
their nest site choice in response to changes in predator abundance
(Forstmeier and Weiss,, 2004). These behavioral responses may in turn alter
the abundance of their predators, producing a feedback on the payoffs for
different nest site strategies. In this setting as well, selection might
promote traits that allow individuals to account for the changing environment,
even at a cognitive cost, when choosing between alternative nesting
strategies.

Decision-making in the context of environmental feedbacks is described by an
emerging theory of “eco-evolutionary” games (Weitz et al.,, 2016; Tilman et
al.,, 2020; Wang et al.,, 2020; Wang and Fu,, 2020; Lin and Weitz,, 2019;
Barfuss et al.,, 2020). Eco-evolutionary games arise when the strategies
adopted by individuals influence the state of the environment, and the
environment in turn influences game payoffs, generating feedback. Even the
simplest of such systems can generate persistent oscillations with an ever
changing environment (Weitz et al.,, 2016; Tilman et al.,, 2020). When the
environmental quality is high, a strategy with high environmental impact does
well; but this leads to environmental collapse and the subsequent dominance of
a low-impact strategy, followed by environmental recovery – producing a cycle
of strategic and environmental oscillations. These types of linked
oscillations arise in myriad systems, including systems of great importance to
human welfare. Climate change is the result decades of individual, corporate,
and national decisions that have applied strong forcing to our climate system,
such that an overshoot of the 2°C warming target is now likely (Pachauri et
al.,, 2014). This overshoot is characteristic of the environmental
oscillations seen in the theory of eco-evolutionary games (Menard et al.,,
2020; Weitz et al.,, 2016; Tilman et al.,, 2020). The slow timescale at which
the environment responds to our strategic behaviors sets the stage for this
overshoot. Similar dynamics occur in fisheries, where harvester-driven
collapse is ubiquitous (Pauly et al.,, 1998; Essington et al.,, 2015), which
leads to reduced harvesting and eventual recovery of fish stocks (Hutchings,,
2000; Worm et al.,, 2009). Understanding how modes of decision-making can
aggravate or alleviate these human-driven environmental changes has profound
practical implications. How might human behavior and decision-making produce
stable outcomes, as opposed to an unstable environmental or wildly oscillating
environment?

To address these questions, here we study the emergence of decision-making
that uses environmental forecasting in the setting of eco-evolutionary games.
Whereas a population composed entirely of myopic individuals can experience
persistent cycles, we show that one composed entirely of forecasters can
produce a stable equilibrium. Moreover, the average fitness in the population
of forecasters can be higher than it would be in the myopic population,
despite the fact that we assume forecasters pay a fixed cost, to model the
cognitive or economic burden of making environmental forecasts. We show that
forecasting types can invade a myopic population, but their invasion is self-
limiting. Once forecasters stabilize the environment, then forecasting no
longer has value (the future will be just like the present) and then the
myopic individuals outperform the forecasters by avoiding the cost of
forecasting. As a result, forecasters tend to increase when rare, partly
mitigating environmental cycles, but they seldom overtake a population
altogether.

We also show that forecasters and myopic agents can stably persist together,
and that this coexistence generates both public and private benefits. First,
forecasting generates public benefit because even a small sub-population of
forecasters serves to reduce the magnitude of environmental variability, which
increases the average fitness of both forecasting and myopic types when they
coexist. Second, forecasting also generates private benefits because
forecasters anticipate environmental change and are better able to deploy the
right strategies at the right times. The private benefits allow the
evolutionary emergence of forecasters, who then provide public benefits to
all.

Finally, we interpret our results on the emergence of forecasters as the first
step in the transition between purely myopic agents and fully rational
individuals with sophisticated cognition – that is, as a middle ground between
evolutionary and economic theories of strategic decision making, where both
cognitive types coexist in the population. We conclude by discussing the
implications of our results in both human and non-human contexts, where
strategic decisions are linked with the state of the environment.

## Model

A broad class of eco-evolutionary games produce cyclic dynamics (Weitz et
al.,, 2016; Tilman et al.,, 2020). These cycles are driven by the strategic
decisions of myopic agents. Taking this setting as a starting point, we model
the emergence of decision makers who use environmental forecasting in a
resident population of myopic agents. Myopic individuals update their strategy
following standard replicator dynamics. They switch between strategies based
on the current payoffs they experience. Their choices feedback to alter the
state of the environment. But myopic individuals do not know anything about
the underlying environment; they attend only to the strategies that others
employ and the resulting payoffs they receive.

In contrast to myopic types, forecasting types know and collect more
information when making their strategic choices. Forecasting individuals make
forecasts of the future states of the environment, and they account for how
the changing environment will influence their future payoffs, discounting the
expected future payoffs in combination with present value of each alternative
strategy. We also model this with replicator dynamics.

To model the emergence of forecasting, we must also consider the process by
which forecasting and myopic types compete. We assume that the act of
forecasting carries a cognitive or economic cost. While forecasters may
believe that their current strategy will yield large payoffs in the future,
this does not give them an advantage over myopic types in the present. Thus,
we assume that forecasting and myopic types compete with each other based
solely on their instantaneous fitness.

With these assumptions the deck is stacked against forecasters. How could a
population of forecasters emerge when each forecaster is at an inherent
fitness disadvantage and must compete with myopic types based only on
instantaneous payoffs? We find that, nonetheless, environmental forecasting
allows individuals to anticipate when near-term environmental changes will
cause a different strategy to be favored, leading to instantaneous advantages
that allow forecasters to invade.

### Eco-evolutionary games

We assume that agents are engaged in a two-strategy eco-evolutionary game,
building on the framework developed by Tilman et al., (2020). The two
strategies are called the “low impact" $L$ and “high impact" $H$ alternatives,
to denote the magnitude of their effects on the environmental state. When all
individuals follow the low-impact strategy, the environmental state tends
towards its highest value. But when the high-impact strategy dominates in the
population, the state of the environment declines towards its lowest possible
value. The environmental state is described by a normalized quantity, $n$,
that is bounded between zero and one. The payoffs for the eco-evolutionary
game are assumed to be linear in the state of the environment, $n$, and in the
frequencies of the low- and high-impact strategies. Therefore, the game can be
represented by an environmentally dependent payoff matrix:

$\Pi(n)=(1-n)\begin{bmatrix}R_{0}&S_{0}\\\
T_{0}&P_{0}\end{bmatrix}+n\begin{bmatrix}R_{1}&S_{1}\\\
T_{1}&P_{1}\end{bmatrix},$ (1)

where the payoff to the low-impact strategy corresponds to the first row of
the matrix, and the payoff to the high-impact strategy corresponds to the
second row of the matrix.

Allowing for both forecasting and myopic types of agents, there are in total
four kinds of individuals in the population: forecasting or myopic types that
follow either strategy $L$ or strategy $H$. We use $z_{L}^{m}$ to denote the
frequency of $L$-strategy myopic individuals, $z_{H}^{m}$ the frequency of
$H$-strategy myopic individuals, $z_{L}^{f}$ the frequency of $L$-strategy
forecasting individuals, and $z_{H}^{f}$ the frequency of $H$-strategy
forecasting individuals. The overall frequency of strategy $L$ in the whole
population is then $z_{L}=z_{L}^{f}+z_{L}^{m}$. And the overall frequency of
forecasting types is $z^{f}=z_{L}^{f}+z_{H}^{f}$.

The immediate payoffs to each strategy are

$\displaystyle\pi_{L}(n,z_{L})=(1-n)(R_{0}z_{L}+S_{0}(1-z_{L}))+n(R_{1}z_{L}+S_{1}(1-z_{L})),$
(2)
$\displaystyle\pi_{H}(n,z_{L})=(1-n)(T_{0}z_{L}+P_{0}(1-z_{L}))+n(T_{1}z_{L}+P_{1}(1-z_{L})).$
(3)

Myopic types consider only these immediate payoffs when updating their
strategies. In contrast, forecasting types also account for their expectations
of the future. These expectations integrate both environmental forecasting and
payoff discounting.

### Forecasting and Discounting

When forecasting the future state of the environment, forecasting types form
expectations about future payoffs. We assume that forecasters observe the
current rate of environmental change and use linear extrapolation to project
future environmental states,

$\hat{n}(t)=n+t\dot{n}$ (4)

where $\hat{n}(t)$ is the forecasted state of the environment $t$ units of
time into the future, $n$ is the current state of the environment, and
$\dot{n}$ is the current rate of change of the environment. This linear
forecast will be accurate in the short term, but it becomes increasingly
unreliable over the long term. Forecasts are continuously updated, so the
accuracy of near-term predictions will remain high throughout the process.

A forecasting type must place some value on their future payoffs for their
forecast to have any effect on their behavior. We assume that forecasting
individuals put greater weight on their near-term payoffs than long-term
payoffs. We model this through a normalized discount function of the form

$\omega(t)=re^{-rt}$ (5)

where $\omega(t)$ is the weight given to time $t$ into the future and $r$ is
the discount rate. This discounting function is normalized so that regardless
of $r$, $\omega(t)$ integrates to $1$. Thus, increasing $r$ shifts an agents
preferences toward the near term without changing the total weight that they
give to the expected payoffs from the game being played.

Figure 1: We study decision making and the emergence of cognitive complexity
in a dynamic environment. The game payoff matrix $\Pi$ for two alternative
strategies depends on the state of the environment, $n(t)$, which changes over
time in response to individuals’ actions. A myopic type individual chooses
between two alternative strategies based only on the instantaneous payoff of
each strategy $s$ in current environment, $\Pi_{s}(n(0))$. Whereas a more
complex type forms a forecast about the future environmental states,
$\hat{n}(t)$ using linear extrapolation, and makes decisions that account for
expected future payoffs, discounting the future relative to the present.
Starting from a population of purely myopic agents we show that evolution can
favor the emergence of forecasting types, even when they pay a cost to produce
forecasts.

### Eco-evolutionary dynamics

We model strategic and environmental change based on pairwise interactions
among the four possible kinds of individuals in the population. We assume that
the act of forecasting is inherently costly.

For strategy change among forecasting types, discounting and forecasting are
integrated into their assessment of the expected payoffs to the low- and high-
impact strategies. This leads to the expected payoff of the low-impact
strategy,

$f_{L}=\int_{0}^{\infty}\omega(t)\pi_{L}(\hat{n}(t),z_{L})\mathrm{d}t$ (6)

which can be solved exactly as

$\displaystyle f_{L}(z_{L},n,\dot{n})$
$\displaystyle=\pi_{L}(n,z_{L})+\frac{\dot{n}}{r}\frac{\partial\pi_{L}}{\partial
n}(n,z_{L}).$ (7)

An equivalent expression holds for the high-impact strategy.

For myopic individuals the immediate payoff difference drives strategy
dynamics, that is

$\pi_{L}(n,z_{L})-\pi_{H}(n,z_{L}).$ (8)

For forecasting individuals, the payoff difference between the strategies is

$f_{L}-f_{H}=\pi_{L}(n,z_{L})-\pi_{H}(n,z_{L})+\frac{\dot{n}}{r}\frac{\partial}{\partial
n}\left[\pi_{L}(n,z_{L})-\pi_{H}(n,z_{L})\right],$ (9)

where a small value of the discount rate $r$ means that agents put a larger
weight on future payoffs. Equation 9 shows how forecasting and discounting
alter the perception of fitness differences between strategies for
forecasters, relative to myopic types. In particular, the first term in
Equation 9 is the immediate difference in payoff between the $L$ and $H$
strategies – which is the quantity that alone determines myopic decision-
making – whereas the second term reflects how forecasting changes the
decision-making process. Note that in the special case that the environment is
not changing, that is $\dot{n}=0$, then the payoff difference perceived by
forecasters equals the payoff difference perceived by myopic types– which
makes sense because, in this case, the forecasted environment is constant and
so forecasting and myopic types make the same strategic decisions.

Now we develop dynamic equations that describe how the frequencies of all four
kinds of individuals change in the population. First, we consider $L$-strategy
myopic individuals. These individuals can emulate $H$-strategy myopic
individuals, or they can choose to adopt either strategy of forecasting
individuals. This leads to three terms in the rate of change of $z_{L}^{m}$,
following mass-action:

$\dot{z}_{L}^{m}=z_{L}^{m}z_{H}^{m}\left(\pi_{L}-\pi_{H}\right)+\epsilon_{2}z_{L}^{m}z_{L}^{f}C+\epsilon_{2}z_{L}^{m}z_{H}^{f}\left(\pi_{L}-\pi_{H}+C\right).$
(10)

The first term corresponds to replicator dynamics among myopic individuals.
The second term corresponds to the flux of $L$-strategy forecasters into the
myopic $L$-strategists caused by the cost of forecasting: all $L$-strategists
receive the same immediate payoff from the game at any point in time, but
those who are forecasters pay a fixed cost $C$ for the forecasting ability.
Herein lies the dilemma of forecasting.

The last term represents the flux of $H$-strategy forecasters to $L$-strategy
myopic individuals, driven both by the (immediate) payoff advantage of playing
strategy $L$ $\left(\pi_{L}-\pi_{H}\right)$ and by the cost of forecasting
($C$). The timescale of strategy switching is likely to be faster than the
timescale of type switching. The parameter $\epsilon_{2}$ controls the
relative timescale of type switching as compared to strategy switching.

Next, we consider $L$-strategy forecasting individuals. The dynamics of this
sub-population of types follow

$\dot{z}_{L}^{f}=z_{L}^{f}z_{H}^{f}(f_{L}-f_{H})-\epsilon_{2}z_{L}^{f}z_{L}^{m}C+\epsilon_{2}z_{L}^{f}z_{H}^{m}\left(\pi_{L}-\pi_{H}-C\right).$
(11)

The first term reflects that forecasters switch between strategies $L$ and $H$
based on their perception of the net-present value of each strategy, given
forecasting and discounting. The second term reflects that forecasters are
never favored over myopic types within strategy $L$ or $H$, because they pay a
cost to forecast, but the rate of switching types (myopic or forecasting) may
be slower than the rate of switching strategies ($L$ or $H$) if
$\epsilon_{2}<1$. The last term shows that $L$-strategy forecasters transition
to $H$-strategy myopic individuals according to the immediate payoff
difference between the strategies and the cost of forecasting.

We can construct the other equations, for $\dot{z}_{H}^{f}$ and
$\dot{z}_{H}^{m}$ in a similar manner, leading to a dynamical system governed
by four equations for the four types/strategies, plus one equation for the
linked environmental dynamic:

$\displaystyle\dot{z}_{L}^{m}$
$\displaystyle=z_{L}^{m}z_{H}^{m}(\pi_{L}-\pi_{H})+\epsilon_{2}z_{L}^{m}z_{L}^{f}C+\epsilon_{2}z_{L}^{m}z_{H}^{f}(\pi_{L}-\pi_{H}+C)$
(12) $\displaystyle\dot{z}_{H}^{m}$
$\displaystyle=-z_{H}^{m}z_{L}^{m}(\pi_{L}-\pi_{H})+\epsilon_{2}z_{H}^{m}z_{L}^{f}(-\pi_{L}+\pi_{H}+C)+\epsilon_{2}z_{H}^{m}z_{H}^{f}C$
(13) $\displaystyle\dot{z}_{L}^{f}$
$\displaystyle=z_{L}^{f}z_{H}^{f}(f_{L}-f_{H})-\epsilon_{2}z_{L}^{f}z_{L}^{m}C+\epsilon_{2}z_{L}^{f}z_{H}^{m}(\pi_{L}-\pi_{H}-C)$
(14) $\displaystyle\dot{z}_{H}^{f}$
$\displaystyle=-z_{H}^{f}z_{L}^{f}(f_{L}-f_{H})+\epsilon_{2}z_{H}^{f}z_{L}^{m}(-\pi_{L}+\pi_{H}-C)-\epsilon_{2}z_{H}^{f}z_{H}^{m}C$
(15) $\displaystyle\dot{n}_{~{}}$
$\displaystyle=\epsilon_{1}(z_{L}^{f}+z_{L}^{m}-n).$ (16)

The equation for the environmental dynamic describes a decaying environmental
variable (Tilman et al.,, 2020), which can correspond to, for example,
pollution levels. In this case, the high- and low-impact strategies generate
emissions of the resource. Tilman et al., (2020) show that systems with either
decaying or renewing intrinsic environmental dynamics (corresponding, for
example, a harvested population) generate qualitatively similar eco-
evolutionary dynamics. The timescale of environmental dynamics can differ from
the timescales of strategy and type dynamics. Here, $\epsilon_{1}$ is the
relative timescale of environmental dynamics, compared to strategy dynamics.
In total, there are three timescales in the model. Without loss of generality,
we consider relative timescales, with strategy dynamics as the reference
point. Thus there are only two timescale parameters in the model.

This formulation of the eco-evolutionary game rests on the assumption of mass-
action kinetics, where there must be an encounter between any pair of types in
order for a transition to occur between them. The likelihood of encounters is
proportional to relative abundance, and so encounters among forecasters with
different strategies will be rare when forecasting types are rare overall. We
focus on the case where type switching is slow relative to strategy switching
($\epsilon_{2}<1$). We also assume that transitions between forecasting types
and myopic types are based on the immediate payoff difference, not based on
forecasters expectations of net-present payoffs.

An alternative formulation of competitive dynamics would consider a
hierarchical form of strategy and type imitation, where strategy dynamics
within each type are unaffected by the overall abundance of the type
(forecaster or myopic) and follow standard two-strategy replicator dynamics.
In this formulation, switching between types is based on the mean fitness
difference between forecasting and myopic types. When a myopic individual
becomes a forecaster (and vice versa) they choose a strategy (L or H) in
proportion to the current frequency of each strategy within that type. We
analyze this alternative formulation in Supplementary Information section A,
whereas we focus on the mass-action model across all four strategies/types in
the main text.

## Results

### Myopic types alone

We focus our analysis on eco-evolutionary games in which a population of
myopic individuals generate cyclical dynamics. These parameter regimes have
been identified by prior work on eco-evolutionary games with myopic agents
(Tilman et al.,, 2020). Figure 2 panel (a) shows the temporal dynamics in such
a population of myopic individuals, which approaches a limit cycle where the
state of the environment and the frequencies of high- and low-environmental
impact strategies oscillate. These persistent cycles reduce the long-term
average fitness of the population compared to the fitness that could be
achieved under a stable environment.

### Forecasting types alone

In contrast to a myopic population, a population consisting entirely of
forecasters generates a qualitatively different outcome. Provided forecasters
care sufficiently about the future (i.e., have a sufficiently small discount
rate), then forecasting types can produce a stable outcome for the same eco-
evolutionary game that would exhibit cyclical dynamics in a myopic population.
Figure 2 panel (b) illustrates a case where a population of forecasters
produces a stable mixed equilibrium, with a fixed proportion of high- versus
low-impact strategies and a fixed state of the environment. At this
equilibrium, the average fitness attained by the population of forecasting
types exceeds the average fitness that would arise in a population of myopic
types. In other words, a population of pure forecasters can stabilize the eco-
evolutionay system and thereby increase average public welfare, compared to
myopic decision makers.

### Coexistence of myopic and forecasting types

When forecasting types and myopic compete and interact, the coupled system
does not approach equilibrium as quickly. Figure 2 panel (c) shows the long-
run dynamics of forecasting types and myopic types who can each assume either
a high-impact strategy or a low-impact strategy. Since forecasting is costly,
it is not possible for forecasters to displace the myopic agents altogether –
if they did, the environment would be completely stabilized, in which case the
myopic agents make the same strategic choice as forecasters, while avoiding
the cost of forecasting. As a result, forecasting types and myopic types both
persist, and so do environmental oscillations; but the amplitude of
oscillations is diminished compared to what occurs in a population of myopic
agents alone.

These results show that both cognitive types can coexist and that the presence
of forecasting types, even when they comprise only a small fraction of the
population, can have a significant impact on the dynamics of the system and
the fitness of both cognitive types. Despite the persistence of some
environmental oscillations when forecasting and myopic types co-exist, the
resulting population mean fitness over one environmental cycle is much greater
than when only myopic types are present, and it can even exceed the fitness
when only forecasting types are present.

(a)(b)(c)

Figure 2: (a) Eco-evolutionary game dynamics in a population of myopic types,
who can each assume either the high-impact strategy ($H$) or the low-impact
strategy ($L$). The feedback between strategic choices and the environment
creates persistent cycles, in which the $L$ strategy is advantageous when the
environmental state is high and the $H$ strategy is advantageous when
environmental state is low. (b) If all individuals are forecasting types who
integrate environmental forecasts into their strategy updating, then the state
of the environment and the strategic frequencies reach a stable equilibrium.
(c) Forecasting types can co-exist with myopic types. After a lengthy
transient (see Figure 2), both types co-exist in a stable limit cycle with
reduced magnitudes of environmental and strategic variation compared to a
population of myopic types alone.
$\left(\epsilon_{1}=.3,~{}\epsilon_{2}=.1,~{}r=.15,~{}C=5/1000,~{}R_{0}=5,~{}R_{1}=0,~{}S_{0}=2,~{}S_{1}=0,~{}T_{0}=0,~{}T_{1}=2,~{}P_{0}=0,~{}P_{1}=4\right)$

### Invasion of forecasting types

We have shown that a forecasters can coexist with myopic types and that they
have a large effect on strategic and environmental dynamics. But the question
remains: can the forecasting type invade a population initially composed of
myopic types and reach a substantial frequency? To study the invasion of
forecasting, we initialize simulations with only myopic individuals and let
this system relax to its steady state, which features a stable limit cycle
with large oscillations. We then perturb the system by introducing forecasters
at very low frequency and simulate that invasion process.

For any particular set of parameters, we find that the likelihood of a
successful invasion is greater when the initial frequency of forecasting types
is greater. The success of invasion also depends on the strategy composition
among the invading forecasters as well as the exact timing of the introduction
of forecasters within the environmental cycle.

Nonetheless, given sufficiently slow switching between forecaster and myopic
types ($\epsilon_{2}<<1$), our simulations show that a successful invasion
results in system-level behavior that always approaches the same limit-cycle,
regardless of invasion timing and strategy mix. The long-run dynamics after
forecasters successfully invade and coexist with myopic types are relatively
simple: both types persist in a stable limit cycle with a relatively small
amplitude of oscillation (Figure 2c). But the invasion process itself is quite
intricate.

(a)(b)

Figure 3: (a) Environmental forecasting is assumed to be costly, but
forecasting types can nonetheless invade a resident population of myopic types
and eventually reach co-existence, while reducing the amplitude of
oscillations. As this example show, a successful invasion of forecasters takes
orders of magnitude longer than a single environmental cycle. (b) Dashed lines
indicate the long-run fitness that would be attained by population of purely
myopic or purely forecasting types. Solid lines indicate the average fitness
of forecasting and myopic sub-populations during the invasion process with
both types present. The amplitude of environmental variability is reduced once
forecasters reach appreciable frequency, and this increases the fitness of
forecasting and myopic types alike, eventually producing fitness as high as
when only forecasters are present. Forecasting is thus a public good: it is
individually costly and beneficial for all.
$\left(\epsilon_{1}=.3,~{}\epsilon_{2}=.1,~{}r=.15,~{}C=5/1000,~{}R_{0}=5,~{}R_{1}=0,~{}S_{0}=2,~{}S_{1}=0,~{}T_{0}=0,~{}T_{1}=2,~{}P_{0}=0,~{}P_{1}=4\right)$
The initial frequency of forecasting types is $1/70$.

The dynamics of a successful invasion occur at a much slower timescale than
the environmental oscillations (Figure 3a), so that the invasion process takes
many environmental cycles before the forecasters reach an appreciable relative
abundance. Figure 3b shows the dynamics of fitness for forecasting and myopic
types (averaged over one environmental cycle). The dashed lines represent the
long-run average fitness attained by a population composed entirely of myopic
individuals, or by a population of pure forecasters. For these parameters,
when only forecasting types are present, the environments equilibrates to a
fixed point and this leads to a greater long-run fitness. When only myopic
types are present, long-run average fitness is lower, and the persistent
environmental oscillations are substantial. Forecasters can invade under
highly variable environments because forecasting allows them to foresee and
predict the best time to switch strategies ($L$ or $H$). This mitigates
environmental oscillations and leads to a decrease in the magnitude of
environmental variability. The reduction in environmental variability that
forecasters provide acts as a rising tide that lifts all boats: when the
environment is more stable, both forecasters and myopic individuals achieve
payoffs as high as those that are attained when only forecasters are present.
Since forecasting is individually costly and its environmental benefits are
shared by all, forecasting is a public good. Despite serving as a public good,
we see that forecasting can nonetheless emerge via evolution and increase the
fitness of both myopic and forecasting individuals, even when forecasters
comprise only a small portion of the population over the long term ($\sim
15\%$ in the example of Figure 4).

### Successful and unsuccessful invasions

Whether or not forecasters will successfully invade and reach a stable
frequency depends on initial conditions. In the simulations described above,
we assume that when forecasters arise, they have the same mix of strategies as
the resident myopic population. Nonetheless, the timing of the invasion within
the environmental cycle and the initial frequency of invading forecasters
determine whether the invasion will ultimately be successful and lead to
coexistence.

Figure 4a shows invasion points that are evenly spaced in time, within a phase
plane plot. The outer black orbit represents the long-run dynamics of a
population of purely myopic individual, with dynamics proceeding in a counter-
clockwise direction. Whether or not a small frequency of forecasters can
invade depends upon at what time point, within this periodic cycle, the
forecasters are introduced: successful timings are shown as red dots, and
unsuccessful timings as brown dots. Even a successful invasion requires many
environmental cycles to establish a stable frequency of forecasters (Figure
4b).

The inner black orbit in Figure 4a shows the long-run population-level dynamic
after a successful invasion of forecasters. The magnitude of environmental and
strategic variability is decreased substantially, after invasion, which is
reflected by a much smaller orbit in phase space.

(a)(b)

Figure 4: While forecasters can invade and co-exist with myopic types, a
successful invasion is not assured and depends upon the timing of
introduction. (a) In a purely myopic population, eco-evolutionary game
dynamics converge to the outer black orbit and proceed in a counterclockwise
direction. We study the potential invasion of forecasting by introducing a
small sub-population of forecasters at some point along this orbit. We assume
that the forecasting sub-population has the same mix of strategies as the
resident myopic population at the time of introduction. Points on the outer
orbit indicate simulated introduction points that are evenly spaced in time.
The color of the point illustrates whether or not the invasion of forecasters
is successful from this point. The inner black orbit shows the long-run
dynamics of the system, which are independent of the timing of the invasion
provided it is successful. The purple and teal orbits illustrate the
corresponding long-run dynamics among forecasting and myopic sub-populations.
(b) Invasions progress very slowly, so that many environmental cycles pass
before they approach the long-run dynamics illustrated in panel (a). The
frequency of forecasters oscillates rapidly throughout this slow invasion
process, and it may take a long time before it is clear whether the invasion
will succeed or
fail.$\left(\epsilon_{1}=3/10,~{}\epsilon_{2}=1/10,~{}r=5/100,~{}C=5/1000,~{}T_{1}=R_{1}=2,~{}P_{1}-S_{1}=4,~{}S_{0}-P_{0}=1,~{}R_{0}-T_{0}=3\right)$
The initial frequency of forecasting types is $1/550$.

Whether or not forecasters can successfully invade also depends on how much
they value the present versus the future. In general, if forecasters place
greater value on expected future payoffs (that is, a small discount rate $r$),
this tends to increase their probability of successful invasion and
establishment in a myopic resident population, all else equal (SI Figure 3).
Whereas if forecasters primarily value the present (large $r$) then they do
not differ much in their strategic behavior compared to the myopic types,
except they must pay a cost and so they cannot invade (SI Figure 3). And so
placing great value on the future always benefits forecasters, even if they
end up putting more decision-making weight on predictions for the far-future
that may prove false. These results make intuitive sense considering that
forecasters types are continually updating their forecasts based on the
current trajectory of environmental change.

## Discussion

Strategic interactions often play out in changing environments. But if the
environment is always changing, will individuals who account for environmental
dynamics be favored when making strategic decisions? We find that in many
circumstances environmental forecasters, who pay a cost for their forecasting
ability, can indeed invade otherwise myopic populations engaged in eco-
evolutionary games. Although a population of pure forecasters can produce a
completely stable environment, forecasters cannot, in these cases, entirely
over-take a population of myopic agents. Nonetheless, even when forecasters
remain at low abundance, they have a striking effect on the dynamics of the
eco-evolutionary game, greatly mitigating the amplitude of environmental
variation and increasing fitness for all.

Whereas classical bio-economic approaches to decision making assume that
agents have complete information about the dynamics of the environment,
evolutionary approaches typically assume the opposite: individuals update
their strategy based solely on present conditions. Here, we analyze a middle
ground and ask whether agents who act slightly more like those studied in a
bio-economic approaches can emerge in a strictly myopic population. That this
invasion can be successful means that evolution by natural selection, or
myopic imitation, can promote the emergence of more sophisticated cognition. A
surprising and important caveat is that forecasters cannot replace myopic
individuals, instead, both types coexist.

The invasion of forecasters in an eco-evolutionary game acts as a rising tide
that lifts all boats, increasing everyone’s fitness. In our analysis,
forecasting operates like a public good – providing benefits to forecasters
and myopic types alike – and, thus, the invasion of forecasters can be thought
of as a meta social dilemma, where both strategies ($L$ or $H$) and decision-
making types (myopic or forecasting) exhibit features of a social dilemma.
When a myopic population experiences cyclic dynamics, the long run average
payoffs are even lower than what would occur under the tragedy of the commons.
This phenomenon has been referred to as a oscillating tragedy of the commons
(Weitz et al.,, 2016), but given the damaging consequences for individuals’
fitness, the ‘catastrophe of the commons’ may be more apt. The invasion of
forecasters resolves this catastrophe by ushering in greater environmental
stability and increased fitness for forecasting and myopic types alike.

Rationality is a bedrock assumption that underlies much of micro-economic
theory. However, whether rationality applies human decision makers has long
been questioned. Bounded rationality is an alternative model that acknowledges
the limits of human cognition (Simon,, 1955). Similarly, prospect theory
accounts for decision-making heuristics that violate classical notions of
rationality and alter behavior (Tversky and Kahneman,, 1974). These advances
were central the formation of behavioral economics, which explores how
economic, social and cultural factors effect decision-making. More recent work
has considered the potential for multiple modes of decision-making to coexist
in a population or within a single individual (Kahneman,, 2011; Rand et al.,,
2017). Models starting from an evolutionary perspective have integrated
foresight, where agents consider how changes in their behavior may alter the
choices of others in the future (Perry and Gavrilets,, 2020; Perry et al.,,
2018). However, little work has explicitly analyzed the consequences of
feedback among modes of decision-making, dynamics of behavior, and the
environmental setting, with a few notable exceptions (Adamson and Hilker,,
2020; Austrup,, 2020).

While we have studied the evolution of a new cognitive style that forms
beliefs about future environmental states, a closely related phenomenon called
theory of mind falls outside the scope of our analysis. Theory of mind
describes the ability to conceptualize the way in which others make decisions.
Future work could study forecasting of others’ strategy dynamics, in addition
to or in lieu of environmental forecasting. We expect that strategic
forecasting may have qualitatively distinct population-level impacts than
environmental forecasting, and it may be favored in different settings.
Additionally, here we have analyzed a well-mixed model where all individuals
of all types and strategies interact; in some contexts a hierarchical model
might be more realistic. In the SI, we describe a hierarchical model where
forecasting types and myopic types form two distinct sub-populations that
compete with each other at the population level. The hierarchical model
produces dynamics of greater complexity than the well-mixed model presented in
the main text.

Rather than predicting a steady advance in the sophistication of cognitive
styles, our analysis suggests that a more likely outcome is the coexistence of
multiple modes of decision making. This finding aligns with results from
psychology that indicate most people have automatic modes of decision making
as well as more deliberative, and mentally taxing, modes (Kahneman,, 2011;
Rand et al.,, 2017). Our finding of coexistence between forecasting and myopic
types provides theoretical support for behavioral-economic hypotheses about
how decisions are actually made. Our results help explain why automatic
decision making persists, and they describe a context in which higher-level
cognition can be favored evolutionarily while also bringing public benefits to
all cognitive types.

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## The evolution of forecasting
for decision making in dynamic environments
Supplementary Information

In the main text, we describe and analyze an eco-evolutionary model to explore
the emergence of forecasting types in dynamically variable environments. The
model describes competition among forecasting and myopic types based on mass
action kinetics. In other words, the populations of all four states of agents
(high- or low-impact strategies and myopic or forecasting types) are well
mixed, and interactions occur proportional to frequency. An alternative
formulation of the model could consider a hierarchical structure, where
forecasting and myopic sub-populations update their strategies within their
type, and the sub-populations of cognitive types compete at the group level.

Here, we present the alternative, hierarchical formulation of the model, and
we illustrate some of the complex dynamics that this alternative formulation
can generate.

Next, we perform a change of variables so that the hierarchical and mass
action formulations of the model can be directly compared.

Finally, we present additional findings for the mass-action model. We
illustrate the importance of the discount rate for invasion success, and find
that the more weight the future has in decision making, the more likely
forecasters are to invade. Finally, we show that forecasting does not entirely
resolve the tragedy of the commons: optimal population-level fitness is not
achieved under dominance by forecasting types.

##  Appendix A Hierarchical model of forecaster emergence

In this section, we consider an alternative eco-evolutionary dynamic, where
there is a hierarchical structure to the system. Forecasters interact only
with forecasters, and myopic individuals interact only with myopic individuals
for determining strategy dynamics. Then, forecasting and myopic populations
compete, based on present payoffs. This hierarchical dynamic leads to a
different system of equations governing system dynamics, resulting in some
qualitatively distinct outcomes.

We can write the dynamics of this hierarchical system as

$\displaystyle\dot{y}_{L}^{m}$
$\displaystyle=y_{L}^{m}(1-y_{L}^{m})]\left(\pi_{L}-\pi_{H}\right)$ (17)
$\displaystyle\dot{y}_{L}^{f}$
$\displaystyle=y_{L}^{f}(1-y_{L}^{f})\left(f_{L}-f_{H}\right)$ (18)
$\displaystyle\dot{z}^{f}$
$\displaystyle=\epsilon_{2}z^{f}(1-z^{f})\left[(y_{L}^{f}-y_{L}^{m})\left(\pi_{L}-\pi_{H}\right)-C\right]$
(19) $\displaystyle\dot{n}$
$\displaystyle=\epsilon_{1}\left[y_{L}^{f}z^{f}+y_{L}^{m}(1-z^{f})-n\right].$
(20)

Notice that $\dot{n}$ also appears in the dynamical equation for the frequency
of strategy $L$ among forecasters, this is because forecasters use the current
rate of change of the environment to forecast likely future environmental
states. Further, notice that $C$ is the cost of forecasting. We assume that
both myopic and forecasting agents update their strategies on the same
timescale, and that $\epsilon_{2}$ and $\epsilon_{1}$ are the relative
timescales (as compared to strategy dynamics) of switching between forecasting
and myopic, and environmental dynamics, respectively. In sum, we have a four-
dimensional system with each variable bound between 0 and 1.

#### Equation for $\dot{y}^{m}_{L}$

Myopic agents update their strategy according to standard replicator dynamics.
The payoff difference between the strategies controls the dynamics of
switching.

#### Equation for $\dot{y}^{f}_{L}$

Forecasting agents update their strategy based on the present state of the
systems as well as based on how the changes in the environment will change
future payoffs. Taking into account this future modifies the equation for
selection in this case, with forecasted payoffs utilized for strategy
dynamics.

#### Equation for $\dot{z}^{f}$

While we assumed that forecasters update their their strategy according to
predicted future outcomes, however, forecaster must nonetheless compete with
myopic agents in the present. Thus the frequency of forecasters evolves in
response to the present payoffs of myopic and forecasting types. The payoff to
a myopic agent is

$y_{L}^{m}\pi_{L}+(1-y_{L}^{m})\pi_{H},$ (21)

and the payoff to forecasting agents is

$y_{L}^{f}\pi_{L}+(1-y_{L}^{f})\pi_{H}-C,$ (22)

Where $\pi_{L}$ and $\pi_{H}$ are the payoffs of strategies 1 and 2. With
these, we can write the payoff difference between forecasting agents and
myopic agents as

$(y_{L}^{f}-y_{L}^{m})\pi_{L}+(1-y_{L}^{f}-1+y_{L}^{m})\pi_{H}-C=(y_{L}^{f}-y_{L}^{m})(\pi_{L}-\pi_{H})-C.$
(23)

Intuitively, when strategy $L$ yields higher payoffs than strategy $H$ given
the state of the system, forecasters will have higher payoffs than myopic
agents if the frequency of strategy $L$ among forecasters, $y_{L}^{f}$, is
greater than among myopic agents, $y_{L}^{m}$. Further, since forecasting is
costly, the benefits of forecasting must outweigh the cost, $C$, for
forecasters to increase in frequency.

#### Equation for $\dot{n}$

This is a straightforward modification of the decaying resource environmental
feedback from Tilman et al., (2020) with the frequency of strategy $L$ written
in terms of our variables.

Comparing the equations for the pairwise interaction (mass action) model with
the equations that arise from the hierarchical model provides insight into how
the two model structures diverge, and why. In both models, the dynamical
equations that govern the frequency of forecasters, $z^{f}$, are identical. In
both models forecasting and myopic types compete based on present fitness.
This does not imply that the invasion processes withing the two models are
equivalent, however.

In the pairwise interaction model, when forecasters are rare, they mostly
interact with myopic individuals, this limits the utility of forecasting,
since the forecast is only utilized when interacting with another forecaster.
This accounts for the existence of the $z^{f}(f_{L}-f_{H})$ term in forecaster
strategy dynamics under pairwise interaction; when forecasters are rare,
$z^{f}<<1$, $f_{L}-f_{H}$ has a minimal impact on the strategy dynamics of
forecasters.

Under the hierarchical model, in contrast, there is no effect of $z^{f}$ on
strategy dynamics for forecasting or myopic types. This is because there are
assumed to be two well mixed sub populations where the rate of interaction
(and thus strategy dynamics) does not depend on $z^{f}$.

Qualitatively, the two models produce distinct invasion dynamics. As shown,
invasion under the pairwise interaction model occurs with a long transient,
where the frequency of forecasting stays very low. In the long run, a
successful invasion by forecasters leads to a new limit cycle.

Under the hierarchical model, forecasters can increase in frequency much more
rapidly, since their strategy dynamics are not slowed when they are rare. SI
Figure 1 shows that the hierarchical model produces more complex long run
dynamics. For a single set of parameters, a range of long run behaviors are
possible. It appears as though there may exist a stable manifold upon which a
continuum of neutrally stable orbits occur. This finding, however, is without
proof.

\begin{overpic}[width=195.12767pt]{FIGS/SIFig1a.png} \end{overpic} (a)
Invasion at $t=105$

\begin{overpic}[width=195.12767pt]{FIGS/SIFig1b.png} \end{overpic} (b)
Invasion at $t=100$

\begin{overpic}[width=195.12767pt]{FIGS/SIFig1c.png} \end{overpic} (c)
Invasion at $t=95$

SI Figure 1: The three panels show that the hierarchical model produces
different long-run dynamics given different system states when forecasting
types are introduced. In these examples, forecasting types match the frequency
of strategy $L$ and $H$ among the myopic types at the time of invasion. The
outer light grey loop in the panels is the limit cycle reached by a population
of myopic individuals. The dashed lines represent population-level and within
type (forecasting or myopic) strategy-environment dynamics during the invasion
process. Light red and light blue loops are the long run dynamics of myopic
and forecasting types, respectively. Notice that the post-invasion long run
dynamics vary depending on the state of the system at the time of invasion.
Panels (a) and (b) show subtly different long-orbits post invasion. Panel (c)
shows a case where long-run dynamics do not settle into a simple periodic
orbit, but rather exhibit periods of environmental (and strategy) stability
followed by periods of environmental (and strategy) variability.
$\left(\epsilon_{1}=3/10,~{}\epsilon_{2}=1,~{}r=15/100,~{}C=1/100,~{}T_{1}-R_{1}=2,~{}P_{1}-S_{1}=4,~{}S_{0}-P_{0}=1,~{}R_{0}-T_{0}=3\right)$
The initial frequency of forecasting types is $1/100$.

##  Appendix B Relationship between hierarchical and mass-action models

We have described the hierarchical model with a different coordinate system
than we used to describe the mass action model in the main text. Here, we
perform a change of coordinates on the mass-action model so that we can
directly compare the two models, and consider some possible sources of
divergence between their results. First, recall the system of equations
analyzed in the main text, given by

$\displaystyle\dot{z}_{L}^{m}$
$\displaystyle=z_{L}^{m}z_{H}^{m}(\pi_{L}-\pi_{H})+\epsilon_{2}z_{L}^{m}z_{L}^{f}C+\epsilon_{2}z_{L}^{m}z_{H}^{f}(\pi_{L}-\pi_{H}+C)$
$\displaystyle\dot{z}_{H}^{m}$
$\displaystyle=-z_{H}^{m}z_{L}^{m}(\pi_{L}-\pi_{H})+\epsilon_{2}z_{H}^{m}z_{L}^{f}(-\pi_{L}+\pi_{H}+C)+\epsilon_{2}z_{H}^{m}z_{H}^{f}C$
$\displaystyle\dot{z}_{L}^{f}$
$\displaystyle=z_{L}^{f}z_{H}^{f}(f_{L}-f_{H})-\epsilon_{2}z_{L}^{f}z_{L}^{m}C+\epsilon_{2}z_{L}^{f}z_{H}^{m}(\pi_{L}-\pi_{H}-C)$
$\displaystyle\dot{z}_{H}^{f}$
$\displaystyle=-z_{H}^{f}z_{L}^{f}(f_{L}-f_{H})+\epsilon_{2}z_{H}^{f}z_{L}^{m}(-\pi_{L}+\pi_{H}-C)-\epsilon_{2}z_{H}^{f}z_{H}^{m}C$
$\displaystyle\dot{n}_{~{}}$
$\displaystyle=\epsilon_{1}(z_{L}^{f}+z_{L}^{m}-n).$

where $\dot{z}_{L}^{m}+\dot{z}_{H}^{m}+\dot{z}_{L}^{f}+\dot{z}_{H}^{f}=1$,
denoting the population-wide frequencies of the four strategic types, and
$n\in[0,1]$ is the state of the environment.

With the following transformations, we can perform a change of coordinates to
express the mass-action model in the same terms as the hierarchical model.

$\displaystyle y_{L}^{m}$
$\displaystyle=\frac{z_{L}^{m}}{z_{L}^{m}+z_{H}^{m}}$ $\displaystyle
y_{L}^{f}$ $\displaystyle=\frac{z_{L}^{f}}{z_{L}^{f}+z_{H}^{f}}$
$\displaystyle z^{f}$ $\displaystyle=z_{L}^{f}+z_{H}^{f}$ $\displaystyle n$
$\displaystyle=n.$

With this change of coordinates we can write the dynamical system as

$\displaystyle\dot{y}_{L}^{m}$ $\displaystyle=\frac{\partial
y_{L}^{m}}{\partial z_{L}^{m}}\dot{z}_{L}^{m}+\frac{\partial
y_{L}^{m}}{\partial z_{H}^{m}}\dot{z}_{H}^{m}$ $\displaystyle\dot{y}_{L}^{f}$
$\displaystyle=\frac{\partial y_{L}^{f}}{\partial
z_{L}^{f}}\dot{z}_{L}^{f}+\frac{\partial y_{L}^{f}}{\partial
z_{H}^{f}}\dot{z}_{H}^{f}$ $\displaystyle\dot{z}^{f}$
$\displaystyle=\dot{z}_{L}^{f}+\dot{z}_{H}^{f}$ $\displaystyle\dot{n}$
$\displaystyle=\dot{n}.$

Through collection of terms and substitution of variables, we can write the
system in our new coordinate system as

$\displaystyle\dot{y}_{L}^{m}$
$\displaystyle=y_{L}^{m}\left(1-y_{L}^{m}\right)\left[(1-z^{f})(\pi_{L}-\pi_{H})+\epsilon_{2}z^{f}(\pi_{L}-\pi_{H})\right]$
(24) $\displaystyle\dot{y}_{L}^{f}$
$\displaystyle=y_{L}^{f}\left(1-y_{L}^{f}\right)\left[z^{f}(f_{L}-f_{H})+\epsilon_{2}(1-z^{f})(\pi_{L}-\pi_{H})\right]$
(25) $\displaystyle\dot{z}^{f}$
$\displaystyle=\epsilon_{2}z^{f}(1-z^{f})\left[(y_{L}^{f}-y_{L}^{m})(\pi_{L}-\pi_{H})-C\right]$
(26) $\displaystyle\dot{n}_{~{}}$
$\displaystyle=\epsilon_{1}\left[y_{L}^{f}z^{f}+y_{L}^{m}(1-z^{f})-n\right].$
(27)

This coordinate systems highlights a few important properties about the
system, as constructed. First, the $z^{f}$ equation shows how forecasting can
be favored at the population level. Forecasting types increase in frequency
when the forecasting sub-population has a higher frequency of the favored
strategy. This is achieved when forecasters foresee that the optimal strategy
will soon change, and adopt that strategy earlier and more rapidly than myopic
types. Nonetheless, myopic individuals always are at an advantage, as they
avoid paying the cost $C$, of having the ability to forecast.

Second, notice that the forecasting terms, $f_{L}$ and $f_{H}$, only appear in
the forecaster strategy equations, and when the prevalence of forecasting is
low ($z_{L}<<1$), little weight is given to these forecasts. Instead,
forecasters updating is dominated by the switching of myopic types to
forecasters, but this switching is slow and cannot overcome the cost of
forecasting, $C$. Simulations indicate that forecasters need a critical mass
to invade, indicating bi-stability. For $\epsilon_{2}=1$, inspection of the
dynamical equations near $z^{f}=0$ seems to confirm this, forecasting types
have the same dynamics as myopic types, but pay a fixed cost $C$. While
forecasting has minimal effect when the frequency of forecasting types is low,
we nonetheless have cases where forecasting types can invade.

Third, the strategy dynamic of myopic types is impacted relatively less by
changes in the abundance of myopic individuals ($1-z^{f}$). Within- and cross-
type switching is governed by the same process, compensating somewhat for
$\epsilon_{2}$’s slowing of strategy dynamics.

##  Appendix C Additional mass-action model results

We describe the oscillatory dynamics that occurs under a population composed
entirely of myopic types as a ‘catastrophe of the commons’. This is because
the oscillations that occur lead to further declines in average fitness or
profit for the population that would be expected given a standard tragedy of
the commons. Forecasting can mitigate or eliminate these oscillations, but
when forecasting is costly, environmental stability cannot be attained when
forecasting and myopic types compete. Nonetheless, the fitness gains of
forecasting are nevertheless largely realized.

However, this does not imply that forecasting types entirely resolve the
tragedy of the commons. SI Figure 2 shows that the optimal fitness that could
be attained by a population – by exogenously fixing the frequency of the two
strategic types – exceeds the population fitness under forecasting types
alone, or under the coexistence of forecasting and myopic types. Therefore,
forecasting can resolve a ‘catastrophe of the commons’ but not the tragedy of
the commons. Additional mechanisms are required to resolve problems of
cooperation.

A key parameter associated with forecasting types is their discount rate. The
discount rate determines how much weight forecasters place on the future in
decision making, and so it alters their effective time horizon. SI Figure 3
shows that as the discount rate decreases, and forecasting types care more
about the future, their invasion success rate increases. Lower discount rates
are also associated with a small decrease in the long-run frequency of
forecasting types after invasion. In contrast to Adamson and Hilker, (2020),
we do not find that caring too much about the distant future can backfire and
lead to the reemergence of oscillatory dynamics. This is likely because
forecasting types update their projections and assessments continuously.
Incorrect predictions about the distant future need to not be adhered to by
forecasting types. As environmental trends reverse, so do forecasters
predictions.

(a)(b)

SI Figure 2: (a) Forecasting types can invade a resident population of myopic
types, leading to coexistence while reducing the amplitude of oscillations. As
this example show, a successful invasion of forecasters takes orders of
magnitude longer than a single environmental cycle. (b) Dashed lines indicate
the long-run fitness that would be attained by populations of purely myopic
types, purely forecasting types, or the maximum level of sustained fitness
that could be attained at the population level. Solid lines indicate the
average fitness of forecasting and myopic sub-populations during the invasion
process with both types present. The dynamics of forecasting and myopic types
is a social dilemma, forecasting types create public benefits for both
forecasting and myopic types. Remarkably, the possible fitness gains of
forecasting are attained even in the face of this dilemma. Forecasting
mitigates but does not resolve the tragedy of the commons. The optimal long-
run population average fitness is greater than the fitness that a population
of forecasting types enjoy.
$\left(\epsilon_{1}=.3,~{}\epsilon_{2}=.1,~{}r=.15,~{}C=.005,~{}R_{0}=8,~{}R_{1}=0,~{}S_{0}=2,~{}S_{1}=0,~{}T_{0}=0,~{}T_{1}=2,~{}P_{0}=0,~{}P_{1}=4\right)$
The initial frequency of forecasting types is $1/70$.  SI Figure 3: All else
being equal, the lower the discount rate, $r$, the more likely it is that
forecasting can invade. This implies that valuing the future more always
benefits forecasters, even if they end up putting more decision-making weight
on predictions for the far-future that may prove false. However, there is a
non-monotonic relationship between the long-run frequency of forecasters and
the discount rate that they employ.
$\left(\epsilon_{1}=3/10,\epsilon_{2}=1/10,C=5/1000,T_{1}-R_{1}=2,S_{0}-P_{0}=1,R_{0}-T_{0}=3,P_{1}-S_{1}=4\right)$
Invasion strategy matched resident. The initial frequency of forecasting types
is $1/100$.