# Towards the assignments for $1^{1}D_{2}$ and $1^{3}D_{2}$ meson nonets

Xue-Chao Feng 1 fxchao<EMAIL_ADDRESS>Ke-Wei Wei 2
<EMAIL_ADDRESS>Jie Wu 1<EMAIL_ADDRESS>Xue-Zhen Zhai 1<EMAIL_ADDRESS>Shi-Zhuo Wang 1<EMAIL_ADDRESS>1 Department of Technology and Physics,
Zhengzhou University of Light Industry, 450002 Zhengzhou, China
2 School of Science, Henan University of Engineering, 451191 Zhengzhou, China

###### Abstract

In this work, we investigate the mass spectrum of $1^{1}D_{2}$ and
$1^{3}D_{2}$ meson nonets in the framework of the meson mass matrix and Regge
phenomenology. The results are compared with the values from different
phenomenological models and may be useful for the assignment of the
$1^{1}D_{2}$ and $1^{3}D_{2}$ meson nonets in the future.

11.55.Jy, 12.40.Yx, 14.40.Be

##  1 Introduction

Quantum Chromodynamics (QCD) was established half a century ago as a theory of
strong interactions. In the past few years, QCD has achieved great success and
continued to provide satisfactory explanations for many puzzling experimental
results. However, we also need to face the fact objectively that there is
still a long way to obtain a comprehensive understanding of strong
interactions. One of the problems to be solved is that perturbative QCD is not
applicable at low energy, and it is difficult to produce accurate results from
non-perturbative calculations [1]. In this case, it is necessary to build
models that can reveal the most important features of QCD, e.g., quenched
lattice QCD [2, 3], the Dyson-Schwinger formalism [4], constituent quark
models [5], and light cone QCD [6, 7]. In addition, since mesons are an ideal
laboratory for studying strong interactions in the strongly coupled non-
perturbative regime, the study of the meson spectrum is of great significance
for better understanding the dynamics of strong interactions [8, 9, 10, 11].

In this work, we will discuss the assignment of the $1^{1}D_{2}$ and
$1^{3}D_{2}$ meson nonets. The masses and widths of the $1^{1}D_{2}$ and
$1^{3}D_{2}$ meson nonets are listed in Table 1. Here, the values are taken
from PDG [12].

Table 1. The masses and widths of $1^{3}D_{2}$ and $1^{1}D_{2}$ meson nonets
(in units of MeV). $\rho_{2}(???)$, $\omega_{2}(???)$, and $\phi_{2}(???)$
remain to be found in experiments.  State ($1^{1}D_{2}$) mass width State
($1^{3}D_{2}$) mass width $\pi_{2}(1670)$ $1670.6^{+2.9}_{-1.2}$
$258^{+8}_{-9}$ $\rho_{2}(???)$ $\eta_{2}(1645)$ $1617\pm 5$ $181\pm 11$
$\omega_{2}(???)$ $\eta_{2}(1870)$ $1842\pm 8$ $225\pm 14$ $\phi_{2}(???)$
$K_{2}(1770)^{{\dagger}}$ $1773\pm 8$ $186\pm 14$ $K_{2}(1820)^{{\dagger}}$
$1819\pm 12$ $264\pm 34$

For the $1^{1}D_{2}$ meson nonet, the states $\pi_{2}(1670)$, $\eta_{2}(1645)$
and $\eta_{2}(1870)$ are well established. The state $\pi_{2}(1670)$ was
observed in the reaction $\pi^{+}p\rightarrow p\pi^{+}\pi^{+}\pi^{-}$ half a
century ago [13]. In 2002, the E852 Collaboration performed a partial-wave
analysis of the reaction $\pi^{-}p\rightarrow\pi^{+}\pi^{-}\pi^{-}p$ and the
$\pi_{2}(1670)$ was confirmed [14]. In 2005, three isovector $2^{-+}$ states
$\pi_{2}(1670)$, $\pi_{2}(1880)$ and $\pi_{2}(1970)$ were seen in the
$\omega\rho^{-}$ decay channel [15]. In Ref. [16], the state $\pi_{2}(1670)$
was measured with mass $1642^{+12}_{-1}$ MeV and width $311^{+12}_{-23}$ MeV.
For the isoscalar, the Crystal Barrel Collaboration observed $\eta_{2}(1645)$,
and $\eta_{2}(1870)$ in the reaction
$p\bar{p}\rightarrow\eta\pi^{0}\pi^{0}\pi^{0}$ [17]. Subsequently, the
$\eta_{2}(1645)$ and $\eta_{2}(1870)$ were studied by the WA102 Collaboration
in the $a_{2}(1320)\pi$, $f_{2}(1270)\pi$, $a_{0}(1320)\pi$ channels [18, 19].
For the kaon sector, the assignment of $K_{2}(1770)$ remains interesting. The
$K_{2}(1770)$ was first reported in the reaction $K^{-}p\rightarrow
pK^{-}\pi^{+}\pi^{-}$ and $K^{-}p\rightarrow p\bar{K}^{0}\pi^{-}\pi^{0}$ [20].
Moreover, the evidence for two $J^{P}=2^{-}$ strange mesons was presented in
the reaction $K^{-}p\rightarrow K^{-}\pi^{+}\pi^{-}\pi^{0}p$ [21]. Unlike the
$\pi_{2}(1670)$, $\eta_{2}(1645)$ and $\eta_{2}(1870)$, the kaon has non-
diagonal C-parity, the spin-singlet ($1^{1}D_{2}$) and spin-triplet
($1^{3}D_{2}$) can mix to produce the physical states $K_{2}(1770)$ and
$K_{2}(1820)$ [22]. The cognition of the $1^{3}D_{2}$ meson nonet is even
lower than that of the $1^{1}D_{2}$ meson nonet. Since $1^{3}D_{2}$ meson has
the same spin angular momentum and orbital momentum as $1^{1}D_{2}$ and
$1^{3}D_{1}$ meson, the mass of the corresponding states may have a large
overlap. Moreover, the large decay widths also make it very difficult to find
these states. Till now, only the kaon $K_{2}(1820)$ is listed in Table 1, the
other members are still not established in the experiments. Therefore, both
the theoretical and experimental analysis of the assignment for $1^{1}D_{2}$
and $1^{3}D_{2}$ meson nonets is required.

In this work, the mass spectrum of $1^{1}D_{2}$ and $1^{3}D_{2}$ meson nonets
is investigated in the framework of the meson mass mixing matrix and Regge
phenomenology. This work is organized as follows. The mass mixing matrix is
described in section 2. The mass relations are derived from Regge
phenomenology in section 3. The summary is given in section 4.

##  2 Mass mixing matrix of isoscalar states

In the quark model, mesons exist as $q\bar{q^{\prime}}$ bound states of quark
$q$ and antiquark $\bar{q^{\prime}}$ (the flavors of $q$ and
$\bar{q^{\prime}}$ may be different). The $q\bar{q^{\prime}}$ bound states can
be classified based on the representations of SU(3) flavor group. In general,
states with the same $J^{P}$ and additive quantum numbers can mix, so the bare
isoscalar states can mix and form two physical states [23, 24, 25].

In the $n\bar{n}=(u\bar{u}+d\bar{d})/\sqrt{2}$ and $s\bar{s}$ basis, the mass
matrix describing the mix of the physical isoscalar states can be expressed as

$M^{2}=\left(\begin{array}[]{cc}M^{2}_{n\bar{n}}+2A_{nn}&\sqrt{2}A_{ns}\\\
\sqrt{2}A_{ns}&M^{2}_{s\bar{s}}+A_{ss}\end{array}\right)$ (1)

with

$A_{nn}=\frac{\zeta}{m_{n}m_{n}},\quad A_{ns}=\frac{\zeta}{m_{n}m_{s}},\quad
A_{ss}=\frac{\zeta}{m_{s}m_{s}},$ (2)

where $M_{n\bar{n}}$ and $M_{s\bar{s}}$ are the masses of states $n\bar{n}$
and $s\bar{s}$, respectively; $A_{nn}$, $A_{ns}$ and $A_{ss}$ are the mixing
parameters, which describe the $q\bar{q}\leftrightarrow
q^{\prime}\bar{q^{\prime}}$ transition amplitudes [26, 27]; $\zeta$ is an
SU(3)-invariant phenomenological parameter, $m_{n}$, $m_{s}$ are the masses
constituent quarks ($n$ denotes the up or down quark).

For a given meson nonet, the physical isoscalar states $\Phi$ and
$\Phi^{\prime}$ are the eigenstates of the mass matrix $M^{2}$

$UM^{2}U^{\dagger}=\left(\begin{array}[]{cc}M^{2}_{\Phi}&0\\\
0&M^{2}_{\Phi^{\prime}}\end{array}\right),$ (3)

where $U$ is the unitary matrix, $M^{2}_{\Phi}$ and $M^{2}_{\Phi^{\prime}}$
are masses of states $\Phi$ and $\Phi^{\prime}$, respectively.

From the realtions (1), (2) and (3), one has

$M^{2}_{n\bar{n}}+\frac{2\zeta}{m^{2}_{n}}+M^{2}_{s\bar{s}}+\frac{\zeta}{m^{2}_{s}}=M^{2}_{\Phi}+M^{2}_{\Phi^{\prime}},$
(4)

$\left(M^{2}_{n\bar{n}}+\frac{2\zeta}{m^{2}_{n}}\right)\left(M^{2}_{s\bar{s}}+\frac{\xi}{m^{2}_{s}}\right)-\frac{2\xi}{m^{2}_{n}m^{2}_{s}}=M^{2}_{\Phi}M^{2}_{\Phi^{\prime}}.$
(5)

The relations (4) and (5) are the trace and determinant of relation (3),
respectively.

Applying the relations (4) and (5) for the $1^{1}D_{2}$ and $1^{1}P_{1}$ meson
nonets, we obtain the masses $M_{s\bar{s}({1^{1}D_{2}})}=1849.0\pm 8.1$ MeV or
$1609.1\pm 5.7$ MeV and $M_{s\bar{s}({1^{1}P_{1}})}=1423.9\pm 8.7$ MeV or
$1156.4\pm 8.1$ MeV. The masses $M_{s\bar{s}({1^{1}D_{2}})}=1609.1\pm 5.7$ MeV
and $M_{s\bar{s}({1^{1}P_{1}})}=1156.4\pm 8.1$ MeV should be discarded since
they are smaller than the masses of the isovector states $\pi_{2}(1670)$ and
$b_{1}(1235)$. In this work, we use the quark masses $m_{n}=0.29$ GeV,
$m_{s}=0.46$ GeV as extracted from meson spectroscopy [28, 29].

Apart from the mixing of isoscalar states, for the kaons of $1^{1}D_{2}$ and
$1^{3}D_{2}$, the spin-singlet and spin-triplet can mix to form physical
states $K_{2}(1770)$ and $K_{2}(1820)$. The mixing can be expressed as

$K_{2}(1773)=K_{2}\left({1}^{1}D_{2}\right)\cos\theta_{K}+K_{2}\left({1}^{3}D_{2}\right)\sin\theta_{K},$
(6)
$K_{2}(1820)=-K_{2}\left({1}^{1}D_{2}\right)\sin\theta_{K}+K_{2}\left({1}^{3}D_{2}\right)\cos\theta_{K},$
(7)

where $\theta_{K}$ is the ($1^{1}D_{2}$)-($1^{3}D_{2}$) mixing angle. The same
analysis process is used for axial-vector mesons [30]. The masses of
$K_{2}(1^{1}D_{2})$ and $K_{2}(1^{3}D_{2})$ can be related with
$M_{K_{2}(1773)}$ and $M_{K_{2}(1820)}$ by the following relation,

$\left(\begin{array}[]{cc}\cos\theta_{K}&\sin\theta_{K}\\\
-\sin\theta_{K}&\cos\theta_{K}\end{array}\right)\left(\begin{array}[]{cc}M^{2}_{K_{2}(1^{1}D_{2})}&\xi\\\
\xi&M^{2}_{K_{2}(1^{1}D_{2})}\end{array}\right)\left(\begin{array}[]{cc}\cos\theta_{K}&-\sin\theta_{K}\\\
\sin\theta_{K}&\cos\theta_{K}\end{array}\right)$
$=\left(\begin{array}[]{cc}M^{2}_{K_{2}(1773)}&0\\\
0&M^{2}_{K_{2}(1820)}\end{array}\right),$ (8)

where $\xi$ represents a parameter describing the $K_{2}(1^{1}D_{2})$ and
$K_{2}(1^{3}D_{2})$ mixing. From the relations (6), (7), and (8), the mixing
angle $\theta_{K}$ and the mass relations between $M_{K_{2}(1^{1}D_{2})}$ and
$M_{K_{2}(1^{3}D_{2})}$ are obtained, i.e.,

$M_{K_{2}\left({1}^{1}D_{2}\right)}^{2}=M_{K_{2}(1773)}^{2}\cos^{2}\theta_{K}+M_{K_{2}(1820)}^{2}\sin^{2}\theta_{K},$
(9)
$M_{K_{2}\left({1}^{3}D_{2}\right)}^{2}=M_{K_{2}(1773)}^{2}\sin^{2}\theta_{K}+M_{K_{2}(1820)}^{2}\cos^{2}\theta_{K},$
(10)

$\cos\left(2\theta_{K}\right)=\frac{M_{K_{2}\left({1}^{1}D_{2}\right)}^{2}-M_{K_{2}\left({1}^{3}D_{2}\right)}^{2}}{M_{K_{2}(1773)}^{2}-M_{K_{2}(1820)}^{2}},$
(11)
$M_{K_{2}\left({1}^{1}D_{2}\right)}^{2}+M_{K_{2}\left({1}^{3}D_{2}\right)}^{2}=M_{K_{2}(1773)}^{2}+M_{K_{2}(1820)}^{2}.$
(12)

As described in the previous section, based on the mass mixing matrix of
isoscalar states and the mixing of the spin-singlet and spin-triplet, we build
a bridge that connects the masses of meson nonet members. In the subsequent
analysis, we will determine the masses of the unknown meson states by applying
relations (4), (5), and (12) to the known meson masses.

##  3 Regge phenomenology and mass relations

In this section, we use Regge theory for the assignment of meson multiplets.
The Regge theory, which connects the high energy behavior of the scattering
amplitude with singularities in the complex angular momentum plane of the
partial wave amplitudes, was developed in the 1960s [31]. Recently, Regge
theory has attracted renewed attention in that it can be used to predict the
meson masses or to determine the quantum numbers of newly observed states in
experiments [9, 28, 32, 33, 34, 35]. Regge theory indicates that mesons are
associated with Regge poles, which move in the complex angular momentum plane
as a function of energy. The plots of Regge trajectories of hadrons in the
$(J,M^{2})$ plane are usually called Chew-Frautschi plots, where $J$ and $M$
are the total spins and the masses of the hadrons, respectively. According to
the Chew-Frautschi conjecture, the poles fall onto linear trajectories in the
$(J,M^{2})$ plane,

$J=\alpha_{n\bar{n}N}(0)+\alpha^{\prime}_{n\bar{n}N}M^{2}_{n\bar{n}N},$ (13)
$J=\alpha_{n\bar{s}N}(0)+\alpha^{\prime}_{n\bar{s}N}M^{2}_{n\bar{s}N},$ (14)
$J=\alpha_{s\bar{s}N}(0)+\alpha^{\prime}_{s\bar{s}N}M^{2}_{s\bar{s}N},$ (15)

where $N$ is the radial quantum number. The $\alpha^{\prime}$ and $\alpha$ are
the slope and intercept of the Regge trajectory, respectively. In this work,
the intercept and slope can be expressed as

$\alpha_{n\bar{n}N}(0)+\alpha_{s\bar{s}N}(0)=2\alpha_{n\bar{s}N}(0),$ (16)
$\frac{1}{\alpha^{\prime}_{n\bar{n}N}}+\frac{1}{\alpha^{\prime}_{s\bar{s}N}}=\frac{2}{\alpha^{\prime}_{n\bar{s}N}}.$
(17)

The intercept relation was derived from the dual-resonance model [36], and is
satisfied in two-dimensional QCD [37], the dual-analytic model [38], the quark
bremsstrahlung model [39]. The slope relation (17) was obtained in the
framework of topological expansion and the $q\bar{q}$-string picture of
hadrons [40].

From relations (13)-(17), one has

$M^{2}_{n\bar{n}N}\alpha^{\prime}_{n\bar{n}N}+M^{2}_{s\bar{s}N}\alpha^{\prime}_{s\bar{s}N}=M^{2}_{n\bar{s}N}\alpha^{\prime}_{n\bar{s}N}.$
(18)

Based on the assumption that the slopes of parity partner trajectories
coincide and are independent of charge conjugation [9, 41], that is to say,

$\left\\{\begin{array}[]{c}\alpha^{\prime}_{n\bar{n}}(1^{1}P_{1})=\alpha^{\prime}_{n\bar{n}}(1^{1}D_{2})=\alpha^{\prime}_{n\bar{n}}(1^{3}D_{2})\\\
\\\
\alpha^{\prime}_{n\bar{s}}(1^{1}P_{1})=\alpha^{\prime}_{n\bar{s}}(1^{1}D_{2})=\alpha^{\prime}_{n\bar{s}}(1^{3}D_{2})\\\
\\\
\alpha^{\prime}_{s\bar{s}}(1^{1}P_{1})=\alpha^{\prime}_{s\bar{s}}(1^{1}D_{2})=\alpha^{\prime}_{s\bar{s}}(1^{3}D_{2})\\\
\end{array}\right..$ (19)

we have the following relations, which are related to meson multiplets with
the same regge slopes by eliminating the slopes,

$\frac{4M^{2}_{n\bar{s}({1^{1}D_{2}})}M^{2}_{n\bar{n}({1^{1}P_{1}})}-4M^{2}_{n\bar{n}({1^{1}D_{2}})}M^{2}_{n\bar{s}({1^{1}P_{1}})}}{M^{2}_{n\bar{n}({1^{1}D_{2}})}M^{2}_{s\bar{s}({1^{1}P_{1}})}-M^{2}_{s\bar{s}({1^{1}D_{2}})}M^{2}_{n\bar{n}({1^{1}P_{1}})}}$
$=\frac{M^{2}_{n\bar{s}({1^{1}D_{2}})}(M^{2}_{n\bar{n}({1^{1}P_{1}})}-M^{2}_{s\bar{s}({1^{1}P_{1}})})-M^{2}_{n\bar{s}({1^{1}P_{1}})}(M^{2}_{n\bar{n}({1^{1}D_{2}})}-M^{2}_{s\bar{s}({1^{1}D_{2}})})}{M^{2}_{n\bar{s}({1^{1}D_{2}})}M^{2}_{s\bar{s}({1^{1}P_{1}})}-M^{2}_{s\bar{s}({1^{1}D_{2}})}M^{2}_{n\bar{s}({1^{1}P_{1}})}},$
(20)

$\frac{4M^{2}_{n\bar{s}({1^{3}D_{2}})}M^{2}_{n\bar{n}({1^{1}P_{1}})}-4M^{2}_{n\bar{n}({1^{3}D_{2}})}M^{2}_{n\bar{s}({1^{1}P_{1}})}}{M^{2}_{n\bar{n}({1^{3}D_{2}})}M^{2}_{s\bar{s}({1^{1}P_{1}})}-M^{2}_{s\bar{s}({1^{3}D_{2}})}M^{2}_{n\bar{n}({1^{1}P_{1}})}}$
$=\frac{M^{2}_{n\bar{s}({1^{3}D_{2}})}(M^{2}_{n\bar{n}({1^{1}P_{1}})}-M^{2}_{s\bar{s}({1^{1}P_{1}})})-M^{2}_{n\bar{s}({1^{1}P_{1}})}(M^{2}_{n\bar{n}({1^{3}D_{2}})}-M^{2}_{s\bar{s}({1^{3}D_{2}})})}{M^{2}_{n\bar{s}({1^{3}D_{2}})}M^{2}_{s\bar{s}({1^{1}P_{1}})}-M^{2}_{s\bar{s}({1^{3}D_{2}})}M^{2}_{n\bar{s}({1^{1}P_{1}})}}.$
(21)

Inserting the masses of states into relations (12), (20), (21), with the aid
of
$M^{2}_{s\bar{s}({1^{3}D_{2}})}=2M^{2}_{n\bar{s}({1^{3}D_{2}})}-M^{2}_{n\bar{n}({1^{3}D_{2}})}$
[24, 42], we obtain the values of $M_{n\bar{s}({1^{1}D_{2}})}$,
$M_{n\bar{s}({1^{3}D_{2}})}$, $M_{n\bar{n}({1^{3}D_{2}})}$,
$M_{s\bar{s}({1^{3}D_{2}})}$ and list the results in Table 3 and 4. In this
work, we take

$M^{2}_{n\bar{s}({1^{1}P_{1}})}=\frac{1}{2}M^{2}_{K_{1}(1270)}+\frac{1}{2}M^{2}_{K_{1}(1400)},$
(22)

the $K_{1A}$ and $K_{1B}$ are nearly equal $45^{\circ}$ mixtures of the
$K_{1}(1270)$ and $K_{1}(1400)$ [43].

Apart from the ground meson, the radial excitations can be estimated in the
framework of Regge phenomenology. Based on the assumption that the ground and
the radial excitation have the same slopes [34], we obtain the following from
the relations (13), (14), and (15),

$M^{2}_{n\overline{n}N}\alpha^{\prime}_{n\overline{n}1}-M^{2}_{n\bar{n}1}\alpha^{\prime}_{n\overline{n}1}=\alpha_{n\bar{n}1}(0)-\alpha_{n\bar{n}N}(0),$
(23)
$M^{2}_{s\overline{s}N}\alpha^{\prime}_{s\overline{s}1}-M^{2}_{s\bar{s}1}\alpha^{\prime}_{s\overline{s}1}=\alpha_{s\bar{s}1}(0)-\alpha_{s\bar{s}N}(0),$
(24)

$M^{2}_{n\overline{s}N}\alpha^{\prime}_{n\overline{s}1}-M^{2}_{n\bar{s}1}\alpha^{\prime}_{n\overline{s}1}=\alpha_{n\bar{s}1}(0)-\alpha_{n\bar{s}N}(0).$
(25)

In Refs. [44, 45], Filipponi et al. indicate that the values of
$\alpha_{n\bar{n}1}(0)-\alpha_{n\bar{n}N}(0)$,
$\alpha_{n\bar{s}1}(0)-\alpha_{n\bar{s}N}(0)$, and
$\alpha_{s\bar{s}1}(0)-\alpha_{s\bar{s}N}(0)$ depend on the constituent quark
masses through the combination $m_{i}+m_{j}$ ($m_{i}$ and $m_{j}$ are the
constituent masses of quark and antiquark). In this case, a factor
$f_{i\overline{j}}(m_{i}+m_{j})$ is introduced into relations (23)-(25) [32,
46]. Then these relations are expressed as

$M^{2}_{n\bar{n}N}=M^{2}_{n\bar{n}1}+\frac{(N-1)}{\alpha^{\prime}_{n\bar{n}}}(1+f_{n\bar{n}}(m_{n}+m_{n})),$
(26)

$M^{2}_{n\bar{s}N}=M^{2}_{n\bar{s}1}+\frac{(N-1)}{\alpha^{\prime}_{n\bar{s}}}(1+f_{n\bar{s}}(m_{n}+m_{s})),$
(27)

$M^{2}_{s\bar{s}N}=M^{2}_{s\bar{s}1}+\frac{(N-1)}{\alpha^{\prime}_{s\bar{s}}}(1+f_{s\bar{s}}(m_{s}+m_{s})).$
(28)

In this work, the parameters used as input are taken from our previous work
(Table 2) [46].

Table 2. The slopes (GeV-2) and the parameters $f_{nn}$ ,$f_{ns}$ and $f_{ss}$
(GeV-1) of relations (19), (20), and (21).  parameters
$\alpha^{\prime}_{n\bar{n}}$ $\alpha^{\prime}_{n\bar{s}}$
$\alpha^{\prime}_{s\bar{s}}$ $f_{nn}$ $f_{ns}$ $f_{ss}$ value $0.7218$
$0.6613$ $0.76902$ $0.3556$ $0.1376$ $0.2219$

The results are shown in Table 3, 4 and Figure 1, 2.

Table 3. The radial excitation masses of the $1^{1}D_{2}$ multiplet in this
work are compared with the other predictions (in units of MeV). This work (1)
and (2) are the solutions of relation (20). The mass used as input for our
calculation is shown in boldface.  $1^{1}D_{2}$ $N$ This work(1) This work(2)
[47] [48] [49] [50] [51] 1 $1670.6^{+2.9}_{-1.2}$ 1643 1680 $M_{n\bar{n}}$ 2
$2111.9{\pm 2.3}$ 1960 2130 3 $2476.1{\pm 2.0}$ 2216 1 $1761.7\pm 5.6$
$1767.4\pm 5.9$ 1709 1778 1780 $M_{n\bar{s}}$ 2 $2188.7\pm 4.6$ $2193.3\pm
4.8$ 2066 2121 2230 3 $2545.0\pm 3.8$ $2549.0\pm 4.1$ 2380 1 $1849.0\pm 8.1$
1825 1893 1909 1890 $M_{s\bar{s}}$ 2 $2262.4\pm 6.7$ 2282 2336 2321 3
$2611.1\pm 5.8$ 2685 2723 2662

Table 4. The radial excitation masses of the $1^{3}D_{2}$ multiplet in this
work are compared with the other predictions (in units of MeV). This work (1)
and (2) are the solutions of relation (21).  $1^{3}D_{2}$ $N$ This work(1)
This work(2) [47] [48] [49] [50] [51] 1 $1687.7\pm 15.5$ $1692.3\pm 15.6$ 1661
1700 $M_{n\bar{n}}$ 2 $2125.9\pm 12.3$ $2129.6\pm 12.4$ 1983 2150 3 $2488.1\pm
10.6$ $2491.2\pm 10.6$ 2241 1 $1824.8\pm 9.6$ $1829.7\pm 9.8$ 1824 1789 1810
$M_{n\bar{s}}$ 2 $2236.8\pm 7.9$ $2243.8\pm 8.0$ 2163 2131 2260 3 $2589.1\pm
6.8$ $2592.6\pm 7.0$ 2388 1 $1952.4\pm 13.5$ $1957.5\pm 13.6$ 1840 1904 1910
$M_{s\bar{s}}$ 2 $2347.7\pm 11.3$ $2351.9\pm 11.4$ 2297 2348 3 $2685.3\pm 9.9$
$2689.0\pm 10.1$ 2701 2734

Figure 1:  The radial excitation of $N^{1}D_{2}$ meson nonet.

Figure 2:  The radial excitation of $N^{1}D_{2}$ meson nonet.

##  4 Conclusion

In this work, we investigate the mass spectrum of $1^{1}D_{2}$ and
$1^{3}D_{2}$ meson nonets in the framework of the meson mass matrix and Regge
phenomenology. The mass of $1^{1}D_{2}(s\bar{s})$ is determined to be $1849.0$
MeV, which agrees with the state $\eta_{2}(1870)$ with a mass of $1842\pm 8$
MeV in PDG. The consistency of masses also indicates that $\eta_{2}(1870)$ is
the candidate for $1^{1}D_{2}$ meson nonet with a small mixing with $n\bar{n}$
component, the corresponding results need to be further confirmed in the
experiment. Apart from the ground meson, the masses of radial excitations of
$1^{1}D_{2}$ are obtained. There are five isovector states with
$J^{PC}=2^{-+}$ ($\pi_{2}(1670)$, $\pi_{2}(1880)$, $\pi_{2}(2050)$,
$\pi_{2}(2100)$, $\pi_{2}(2285)$) are listed in PDG. The $\pi_{2}(2100)$ is
omitted from the summary table, the states $\pi_{2}(2050)$ and $\pi_{2}(2285)$
are classified as “further states”. In this work, the isovector of
$2^{1}D_{2}$ mass is determined to be $2111.9$ MeV, which is consistent with
the mass of $\pi_{2}(2100)$ ($2090\pm 29$ MeV). We suggest that
$\pi_{2}(2100)$ would be the plausible candidate of $2^{1}D_{2}$ meson state.
For the $1^{3}D_{2}$ meson nonet, only the kaon $K_{2}(1820)$ is found, and it
will mix with $K_{2}(1770)$ state. In PDG, the four states $\rho_{2}(1940)$,
$\omega_{2}(1975)$, $\omega_{2}(2195)$, $\rho_{2}(2225)$ with $J^{PC}=2^{--}$
are interpreted as “further states”. In this work, we find that there is still
a large deviation between our results and the masses of these states. Till
now, since the experimental information of $1^{3}D_{2}$ meson nonet is not
enough, it is necessary to conduct a systematic and complete study for these
states. The results may be useful for the discovery of unobserved states in
the experiments (BESIII, CMD-3, and COMPASS).

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# More can be better: An analysis of single-mutant fixation probability
functions under $2\times 2$ games

Diogo L<EMAIL_ADDRESS>City, University of London,
Northampton Square, London, EC1V 0HB, UK Mark<EMAIL_ADDRESS>City, University of London, Northampton Square, London, EC1V 0HB, UK

###### Abstract

Evolutionary game theory has proved to be a powerful tool to probe the self-
organisation of collective behaviour by considering frequency-dependent
fitness in evolutionary processes. It has shown that the stability of a
strategy depends not only on the payoffs received after each encounter but
also on the population’s size. Here, we study $2\times 2$ games in well-mixed
finite populations by analysing the fixation probabilities of single mutants
as functions of population size. We proved that 9 out of the 24 possible games
always lead to monotonically decreasing functions, similarly to fixed fitness
scenarios. However, fixation functions showed increasing regions under 12
distinct anti-coordination, coordination, and dominance games. Perhaps
counter-intuitively, this establishes that single-mutant strategies often
benefit from being in larger populations. Fixation functions that increase
from a global minimum to a positive asymptotic value are pervasive but may
have been easily concealed by the weak selection limit. We obtained sufficient
conditions to observe fixation increasing for small populations and three
distinct ways this can occur. Finally, we describe fixation functions with
increasing regions bounded by two extremes under intermediate population
sizes. We associate their occurrence with transitions from having one global
extreme to other shapes.

##  1 Introduction

Evolutionary game theory has proved to be an essential framework in the
context of studying the self-organisation of collective behaviour such as
signalling [1, 2, 3], cooperation [4, 5, 6], or aggressive territorial
behaviour [7, 8]. The fitness of individuals in a population was first
considered to depend on the frequency of their types in evolutionary models of
infinite populations, under which either a static analysis of Evolutionarily
Stable Strategies (ESS) [7] or finding attractors of the replicator equation
[9, 10] indicated stable solutions of the evolutionary dynamics.

Only later in [11], the original Moran process [12] was adapted to incorporate
frequency-dependent fitness, thus initiating the study of evolutionary games
on finite populations. Soon afterwards, it was shown that the stability of
strategies depends not only on the payoffs received by individuals but on the
size of the considered finite population as well [13]. For strong selection,
however, the evolutionary outcomes were shown to correspond to the classical
infinite population ones, first under the frequency-dependent Moran process
[14], and then under general fitness mappings [15].

These models introduced a new concept of evolutionary stability $ESS_{N}$
adapted from the definition used under infinite populations [7]. An
evolutionarily stable strategy became one on which selection opposes both
invasion and fixation by other strategies [16, 11]. These two requirements
respectively provide equilibrium and stability conditions [8] of a given
strategy. Selection opposes invasion when no single mutant holds a higher
fitness than the residents in an otherwise pure population. Selection opposes
fixation when no single mutant is able to fixate in the population with a
probability above neutral fixation [11]. It was shown that of the $16$
possible combinations of selection favouring or opposing the invasion and
fixation of strategists $B$ by strategists $A$ and vice-versa, only $8$ of
those are possible evolutionary scenarios under the frequency-dependent Moran
process [13].

Studying fixation probabilities thus became an important aspect of
understanding the evolution of collective behaviour. They were studied as
functions of the initial number of mutants [17], and it has been proved that
each of the $8$ possible evolutionary scenarios has associated to it one (and
only one) of three function shapes. In [18, 19], they were studied in relation
to the initial number of mutants and intensity of selection under the pairwise
comparison process. They have also been extensively studied in the context of
structured populations [20, 21, 22, 23].

Fixation probabilities describe the likelihood of changing from one pure state
to another after one mutation occurs, conditional on mutations being rare
enough. Following [24], under games with two or more strategies they can be
used to construct a simplified Markov chain between pure states, and allow the
computation of the long-term stationary distribution over them [25, 26, 27,
28]. The validity of these approximations under rare mutations has been
analysed analytically in [29] and through comparison with simulations in [25,
26]. The concerns raised over the existence of mixed stable states leading to
quasi-stationary distributions [30, 31, 32] have been tackled in this context
by considering higher-orders approximations including those states as
configurations of interest [33].

The dependence of fixation probabilities on population size has been analysed,
for instance through the calculation of their asymptotic limits. This was
initially done by [34], who obtained quantitative limits for some cases, but
only qualitative ones in others, whose error was later estimated by [17].
These results were later expanded for two different limit orders of weak
selection by [35], who also corrected some borderline cases under arbitrary
values of intensity of selection.

Taking into consideration the central importance that single-mutant fixation
probabilities have in the context of evolutionary theory, the present work
analyses them systematically as functions of population size under the
simplest $2\times 2$ games. To do so, we classify games based on the ordering
of their payoff matrix’ entries, leading to a total of $24$ different fixation
orderings. We start by proving in section 3.1 that nine of these always lead
to monotonically decreasing fixation probability functions. We tested the
remaining orderings and concluded that three additional orderings may not have
any non-monotonically decreasing functions. In total, these included the
fixation of $6$ dominated strategies, one dominating strategy, and five
strategies from coordination games.

However, increasing population size may not only change whether mutants fixate
below or above neutrality [11, 13], but can correspond to actual increases of
single-mutant fixation probabilities. We observed diverse fixation functions
with increasing regions under the twelve remaining orderings. These included
all six strategies from anti-coordination games such as the Hawk-
Dove/Snowdrift game [7, 8, 36, 37], the fixation of five dominating strategies
such as defectors in the Prisoner’s Dilemma [4, 5], and the fixation of stag
hunters in the Stag Hunt game [38, 6] (the only exception in coordination
games). In section 3.2, fixation functions that increased from a global
minimum to a positive asymptotic value were explored and found to be
pervasive. In some anti-coordination games (e.g. fixation of doves) this shape
was found every time the payoff matrix led to a positive asymptotic value.
These functions have passed mostly unnoticed in the past with the exception of
[20], where it was briefly noted that they could be observed under the Hawk-
Dove game. We propose that they may have been hidden by the weak selection
limit [11, 18, 19, 39], especially if this limit was considered to be dominant
over the large population one [35].

In section 3.3, we show that it should be possible to see fixation increasing
for the smallest populations $N=2$ under $6$ different orderings: three
dominating, two anti-coordination, and one coordination game strategy
orderings. We find three different ways this can happen: functions might
increase monotonically, or they increase up to a global maximum and then
decrease to a positive asymptotic value or to zero.

Finally, in section 3.4, we explore fixation functions with two extremes:
decreasing for small populations, increasing for intermediate populations, and
decreasing again for larger populations. These were observed both with
positive and zero asymptotic values. The first were observed under the
fixation of two dominating strategies, and the second under anti-coordination
games and the Stag Hunt game. These were all observed for transitions between
functions with one global extreme and other function shapes.

##  2 Model

Let us consider a well-mixed population of individuals who interact pairwise
according to a general $2\times 2$ symmetric game, and call $A$ and $B$ the
strategies at their disposal. After each encounter, they receive a payoff
defined by the payoff matrix from table 1. Individuals using $A$ receive $a$
and $b$ respectively against individuals using $A$ and $B$; while individuals
using $B$ receive $c$ and $d$ respectively against individuals using $A$ and
$B$. We consider all payoff matrix entries to be strictly positive.

| $A$ | $B$
---|---|---
$A$ | a | b
$B$ | c | d
Table 1: Payoff matrix of a $2\times 2$ game.

Each individual in a well-mixed population interacts on average with the same
frequency with all others. In this context, their fitness is simply the
average of the payoffs received over the encounters they have with other
individuals in the population. Considering a population with $N$ individuals,
where $i$ individuals are using strategy $A$ and $N-i$ using $B$, the fitness
of individuals using $A$ and $B$ are, respectively, the following:

$f_{i}^{N}=\frac{a(i-1)+b(N-i)}{N-1},$ (1)
$g_{i}^{N}=\frac{ci+d(N-i-1)}{N-1}.$ (2)

However, selection may depend on factors beyond the studied game, which on
average should represent equal contributions to each individual’s fitness,
irrespective of their strategy. To account for this, we use the fitness
formulation proposed in [11], which includes a parameter representing the
intensity of selection $w\in[0,1]$:

$f_{i}^{N}=1-w+w\frac{a(i-1)+b(N-i)}{N-1},$ (3)
$g_{i}^{N}=1-w+w\frac{ci+d(N-i-1)}{N-1}.$ (4)

Changing $w$ is equivalent to performing a transformation of the original
payoff matrix. However, as this transformation does not change the original
game ordering, we will set $w=1$ for most of our analysis (leading back to
equations 1 and 2), except when intending to study the explicit effects of
intensity of selection.

Selection may act in different ways on the evolutionary process, and hence the
fitness of each individual may be considered on the first (birth) event [11,
13], on the second (death) event [40], on simultaneous events [41], or to
depend exponentially on both individuals’ payoffs [18]. Here we are focusing
on the birth-death process with selection in the birth event, which is
typically described as the frequency-dependent Moran process [11, 13]. This is
a process under which it has been suggested that cooperators perform generally
worse [40, 42] when compared to the results obtained under the others.

During each step of this process, one individual in the population gives birth
proportionally to their fitness and another one dies randomly in the
population. Taking this into consideration, for each evolutionary step, the
probabilities of having the number of individuals $i$ using strategy $A$
increasing by one $\left(P_{i^{+}}^{N}\right)$, decreasing by one
$\left(P_{i^{-}}^{N}\right)$, or remaining the same
$\left(P_{i^{=}}^{N}\right)$ are the following:

$P_{i^{+}}^{N}=\frac{if_{i}^{N}}{if_{i}^{N}+(N-i)g_{i}^{N}}\frac{N-i}{N},$ (5)
$P_{i^{-}}^{N}=\frac{(N-i)g_{i}^{N}}{if_{i}^{N}+(N-i)g_{i}^{N}}\frac{i}{N},$
(6) $P_{i^{=}}^{N}=1-P_{i^{+}}^{N}-P_{i^{-}}^{N}.$ (7)

All the other transitions are impossible by definition of the dynamics. This
stochastic process is defined as a Markov chain, under which there are two
absorbing states: $i=0$ and $i=N$. If the population falls into one of these,
it will stay there for the remaining evolutionary time.

To compute the probability of one single mutant $A$ fixating in the
population, i.e. seeing the system transitioning from $i=1$ to $i=N$, we have
to consider a recursive relation based on equations 5, 6 and 7. Following
[43], this relation leads to a closed-form expression for the fixation
probability

$\rho_{N}=\frac{1}{1+\sum_{j=1}^{N-1}\prod_{k=1}^{j}\gamma_{k}^{N}},$ (8)

where $\gamma_{k}^{N}$ is defined as

$\gamma_{k}^{N}=\frac{P_{k^{-}}^{N}}{P_{k^{+}}^{N}}.$ (9)

Applying equations 5 and 6, the term $\gamma_{k}^{N}$ reduces to the relative
fitness of residents in a population with $k$ mutants and $N$ total
individuals. It will prove useful to express it as an explicit function of the
number of mutants $k$ and the population size $N$ when $w=1$:

$\gamma_{k}^{N}=\frac{g_{k}^{N}}{f_{k}^{N}}=\frac{ck+d(N-k-1)}{a(k-1)+b(N-k)}.$
(10)

##  3 Results

General $2\times 2$ matrix games are often defined by the ordering of the four
payoff matrix entries from table 1 (see e.g. [8, 19]). In the present work, we
have only considered cases where all payoff matrix entries are positive and
distinct pairwise. Even though there are $24$ possible orderings, the pairs
that have $b$ and $c$ swapped as well as $a$ and $d$ correspond to the same
game with the definitions of strategies $A$ and $B$ swapped. Therefore, there
are a total of $12$ independent $2\times 2$ games, each with a pair of
complementary orderings. These pairs are shown together in table 2 and
delimited by horizontal lines. In each ordering, strategy $A$ is considered to
be the mutant and strategy $B$ the resident.

In infinite populations, equilibria under these games are limited to three
scenarios [7, 9, 10, 44]. Games either have one pure evolutionarily stable
state ($ESS$), one inner $ESS$, or two pure $ESS$. These are respectively
named games of dominance, anti-coordination, and coordination, and directly
correspond to three invasion scenarios in stochastic models when $N\to\infty$
[13]. The correspondence between these games and the ordering of their payoffs
is set in the first two columns of table 2. For each pair of dominance
orderings on the table, the first ordering corresponds to having mutant $A$
invading $B$, while the second corresponds to $B$ invading $A$. All other
pairs of complementary game orderings did not follow any particular order.

Type of game |  Game |  Ordering |  Shapes of fixation functions $\rho_{N}$ with |  Always decreasing $\rho_{N}$ |  Possibility of $\rho_{3}>\rho_{2}$ |
---|---|---|---|---|---|---
under $N\to\infty$ | positive asymptotic value | null asymptotic value |
Dominance Games (one pure $ESS$) | PD (D) | $b>d>a>c$ | DP DUP |  |  |  |
PD (C) | $c>a>d>b$ |  | D0 | tested |  |
| $a>c>b>d$ | DP DUP UP UDP DUDP |  |  | yes |
| $d>b>c>a$ |  | D0 | tested |  |
| $a>b>c>d$ | DP DUP UP UDP DUDP |  |  | yes |
| $d>c>b>a$ |  | D0 | proven (T1+T2) |  |
| $a>b>d>c$ | DP |  | proven (T1) |  |
| $d>c>a>b$ |  | D0 | proven (T1) |  |
| $b>a>d>c$ | DP DUP |  |  |  |
| $c>d>a>b$ |  | D0 | tested |  |
| $b>a>c>d$ | DP DUP UP |  |  | yes |
| $c>d>b>a$ |  | D0 | proven (T2) |  |
Anti-Coordination Games (one mixed $ESS$) | HD (D) | $c>a>b>d$ | DUP | D0 DUD0 |  |  |
HD (H) | $b>d>c>a$ | DP DUP | D0 DUD0 |  |  |
| $b>c>a>d$ | DUP UP |  |  | yes |
| $c>b>d>a$ |  | D0 DUD0 |  |  |
| $b>c>d>a$ | DUP UP | D0 DUD0 |  | yes |
| $c>b>a>d$ | DUP | D0 DUD0 |  |  |
Coordination Games (two pure $ESS$) | SH (S) | $a>c>d>b$ |  | D0 UD0 DUD0 |  | yes |
SH (H) | $d>b>a>c$ |  | D0 | proven (T1) |  |
| $a>d>b>c$ |  | D0 | proven (T1) |  |
| $d>a>c>b$ |  | D0 | proven (T1) |  |
| $a>d>c>b$ |  | D0 | proven (T1) |  |
| $d>a>b>c$ |  | D0 | proven (T1) |  |

Table 2: Summary of fixation probability results under general $2\times 2$
games. Games can be defined by the ordering of the payoff matrix’s entries.
There is a total of $24$ orderings, corresponding to $12$ independent $2\times
2$ games with distinguishable strategies. Pairs of orderings of the same game
are enclosed between horizontal lines. We split games into three types –
dominance, anti-coordination and coordination games – depending on the
equilibria observed in the asymptotic limit $N\to\infty$ under them. Orderings
corresponding to known games are signalled – Prisoner’s Dilemma (PD), Hawk-
Dove or Snowdrift (HD), Stag Hunt (SH) – and the strategy considered to be
mutant $A$ is indicated in parenthesis. The first/second ordering of each pair
of dominance games corresponds to the fixation of the dominating/dominated
strategy. Under the other two types of games, pairs did not follow any
particular ordering. The shapes of fixation functions $\rho_{N}$ found under
each of the orderings are listed, separated by the ones having positive and
null asymptotic values. The last two columns summarise the results of theorems
from sections 3.1 and 3.3 respectively. Some game orderings were proven to
always have decreasing fixation functions $\rho_{N}$, either by Theorem $1$
(T1) or by Theorem $2$ (T2). One of the orderings resulted from a joint proof
(T1+T2) for its two subsets: $bc\leq ad$ by T1 and $bc\geq ad$ by T2. We did
not include the results for incomplete subsets of other orderings. Some
orderings were observed to always lead to decreasing functions after being
tested systematically. The remaining orderings were found to have non-
decreasing functions, as can be confirmed by the listing of shapes found under
each ordering. Condition $\rho_{3}>\rho_{2}$ was met under the orderings
signalled in the last column when eq. 29 was fulfilled.

However, under finite populations, evolutionary outcomes are defined not only
by whether selection favours or opposes invasion, but also by whether it does
so with fixation [13]. In the following sections, we will explore some of the
possible ways in which fixation probabilities may depend on population size in
$2\times 2$ games. We found that there are at least $8$ shapes that single-
mutant fixation probability functions $\rho_{N}$ can take:

  1. 1.

Always decreasing to a positive value (DP);

  2. 2.

Always decreasing to $0$ (D0);

  3. 3.

Decreasing to a global minimum and then increasing up to a positive value
(DUP);

  4. 4.

Always increasing up to a positive value (UP);

  5. 5.

Increasing up to a global maximum and then decreasing to a positive value
(UDP);

  6. 6.

Increasing up to a global maximum and then decreasing to $0$ (UD0);

  7. 7.

Decreasing to a minimum, then increasing up to a maximum, and finally
decreasing to a positive value (DUDP);

  8. 8.

Decreasing to a minimum, then increasing up to a maximum, and finally
decreasing to $0$ (DUD0).

We observed a higher diversity of fixation functions with increasing regions
mainly under anti-coordination games (e.g. Hawk-Dove/Snowdrift game), the
fixation of dominating strategies (e.g. defectors in the Prisoner’s Dilemma),
and the fixation of stag hunters under the game with the same name (the only
exception in coordination games). A summary of the fixation function shapes
can be found in figure 1. In table 2, we have represented both the fixation
function shapes observed under each game, and how the analytical results
reflected on them. In section 3.1, we prove that some orderings always have
decreasing fixation probability functions, and then explore these functions
(shapes 1 and 2). In section 3.2, we explore functions with one global minimum
(shape 3), state these are pervasive across dominance and anti-coordination
games and suggest that these might have been concealed by the weak selection
limit. In section 3.3, we obtain the conditions under which it is possible to
see fixation probability functions increasing for very small populations and
explore the functions under which this happens (shapes 4, 5 and 6). Finally,
in section 3.4, we study the settings under which fixation probabilities are
seen to have two extremes – a local minimum and a local maximum – before
decreasing either to a positive value (shape 7) or to zero (shape 8) and
associate these with transitions from functions with one global extreme
(minimum or maximum) to other shapes.

(a)

(b)

Figure 1: Summary of shapes of single-mutant fixation probability functions
$\rho_{N}$ under $2\times 2$ games. Figure 1(a) represents curves normalized
by their positive asymptotic values ($\rho_{\infty}=1-d/b$) for
simplification, and figure 1(b) represents curves with zero asymptotic values.
The coding of function shapes describes their successive trends of going up
(U) and down (D), and whether their asymptotic limit is positive (P) or zero
($0$). Results were obtained for the following payoff values: DP
$[80,6,1,1.5]$, UDP $[20,3,1.9,1.1]$, DUDP $[8,1.8,2,0.9]$, UP $[10,3,2,1]$,
DUP $[3,4,2,1]$, D0 $[2,1,3,4]$, UD0 $[13,0.5,3,1]$, DUD0 $[3,2,4,1.235]$.
These payoff values were chosen to get optimal clarity of figures.

###  3.1 Decreasing fixation probability functions

Under the fixed fitness Moran process [12], increasing the size $N$ of a
finite population always leads to a decrease in the probability that a single
mutant has of fixating on the whole population. When frequency-dependent
fitness is introduced [11], despite this not being necessarily true, it is
still observed to happen often. To probe the $2\times 2$ games under which
$\rho_{N+1}>\rho_{N}$ is always true, we obtain Theorems 1 and 2.

######  Theorem 1.

If the payoff matrix entries of a $2\times 2$ game satisfy $b<a$ and $c\leq d$
or $b\geq a$ and $bc\leq ad$, then the fixation probability $\rho_{N}$ of a
single mutant using $A$ decreases monotonically with $N$.

###### Proof.

We try to prove that $\rho_{N}>\rho_{N+1}$ is always true in some games. We
start by comparing the closed-form expressions for the fixation probabilities
recalling equation 8 and focusing on their denominators:

$\begin{split}\rho_{N}>\rho_{N+1}&\Leftrightarrow\frac{1}{1+\sum_{j=1}^{N-1}\prod_{k=1}^{j}\gamma_{k}^{N}}>\frac{1}{1+\sum_{j=1}^{N}\prod_{k=1}^{j}\gamma_{k}^{N+1}}\Leftrightarrow\\\
&\\\
&\Leftrightarrow\sum_{j=1}^{N-1}\prod_{k=1}^{j}\gamma_{k}^{N}<\sum_{j=1}^{N}\prod_{k=1}^{j}\gamma_{k}^{N+1}.\\\
\end{split}$ (11)

Isolating the extra term on the sum on the right hand-side and joining the
remaining two sums of $N-1$ terms, we get the following condition:

$\rho_{N}>\rho_{N+1}\Leftrightarrow\sum_{j=1}^{N-1}\left[\prod_{k=1}^{j}\gamma_{k}^{N}-\prod_{k=1}^{j}\gamma_{k}^{N+1}\right]<\prod_{k=1}^{N}\gamma_{k}^{N+1}.$
(12)

The right hand-side product is positive because all values of
$\gamma_{k}^{N+1}$ are positive as well. Then, if the following more strict
condition is fulfilled

$\sum_{j=1}^{N-1}\left[\prod_{k=1}^{j}\gamma_{k}^{N}-\prod_{k=1}^{j}\gamma_{k}^{N+1}\right]\leq
0,$ (13)

we will necessarily have $\rho_{N}>\rho_{N+1}$.

Any payoff matrix fulfilling $\gamma_{k}^{N}\leq\gamma_{k}^{N+1}\forall
k\in\\{1,...,N-1\\};\forall N\in\\{2,3,...\\}$, satisfies condition 13. Thus,
single-mutant fixation probabilities in such games will necessarily be
decreasing functions of $N$. Under the frequency-dependent Moran Process,
$\gamma_{k}^{N}$ is equivalent to the relative fitness of resident strategy
$B$ as it is seen in equation 10. Therefore,
$\gamma_{k}^{N}\leq\gamma_{k}^{N+1}$ leads to the following condition in that
case:

$\frac{ck+d(N-k-1)}{a(k-1)+b(N-k)}\leq\frac{ck+d(N-k)}{a(k-1)+b(N-k+1)},$ (14)

which after some algebra becomes

$(bc-ad)k\leq(b-a)d.$ (15)

We are interested in the games that satisfy this condition for all values of
$k\in\\{1,...,N-1\\}$ and $N\in\\{2,3,...\\}$, which is equivalent to saying
that

$bc-
ad\leq\inf\left(\left\\{\frac{(b-a)d}{k}:k\in\\{1,...,N-1\\};N\in\\{2,3,...\\}\right\\}\right).\\\
$ (16)

The value of $k$ for which the infimum occurs depends only on whether $(b-a)d$
is negative or positive.

  1. 1.

Case $b<a$:

This case leads to $(b-a)d<0$, and therefore the infimum on equation 16 occurs
for the minimum value that $k$ can assume $k=1$, regardless of the value of
$N$. Thus, the condition becomes $bc-ad\leq(b-a)d$, which simplifies to

$c\leq d.$ (17)

  2. 2.

Case $b\geq a$:

This case leads to $(b-a)d\geq 0$, and therefore the infimum on equation 16
occurs for the largest possible value of $k$, which will be $k=N-1$. Since
$(b-a)d$ is non-negative and $k$ is unbounded, the infimum is $0$, and the
condition becomes

$bc\leq ad.$ (18)

All games contained in these two cases will necessarily have a fixation
probability with a decreasing function on $N$.

∎

######  Theorem 2.

If the payoff matrix entries of a $2\times 2$ game satisfy $c\geq d\geq b\geq
a$, or $bc\geq ad$, $c+d\geq 2b$ and $c<d$, then the fixation probability
$\rho_{N}$ of a single mutant using $A$ decreases monotonically with $N$.

###### Proof.

As in the proof of Theorem 1, we start by comparing the closed-form
expressions for the fixation probabilities (eq. 11). We then follow an
alternative path, isolating the first term ($j=1$) instead of the last one
($j=N$) in the sum with $N$ elements, and obtain the following relation:

$\rho_{N}>\rho_{N+1}\Leftrightarrow\sum_{j=1}^{N-1}\left[\prod_{k=1}^{j}\gamma_{k}^{N}-\gamma_{1}^{N+1}\prod_{k=1}^{j}\gamma_{k+1}^{N+1}\right]<\gamma_{1}^{N+1}.$
(19)

Following the same reasoning as in the previous proof, because
$\gamma_{1}^{N+1}$ is strictly positive, a game meeting the more strict
condition

$\sum_{j=1}^{N-1}\left[\prod_{k=1}^{j}\gamma_{k}^{N}-\gamma_{1}^{N+1}\prod_{k=1}^{j}\gamma_{k+1}^{N+1}\right]\leq
0,$ (20)

will necessarily lead to $\rho_{N}>\rho_{N+1}$.

By the same reasoning used before, this includes all the payoff matrices that
meet $\gamma_{k}^{N}\leq\gamma_{k+1}^{N+1}$ and $\gamma_{1}^{N+1}\geq 1\forall
k\in\\{1,...,N-1\\};\forall N\in\\{2,3,...\\}$. The second part of the
condition had to be added to assure that the constant coefficient on the right
hand-side product did not make it smaller than the left hand-side product.
Applying the form that $\gamma_{k}^{N}$ assumes under the frequency-dependent
Moran process (equation 10), having $\gamma_{k}^{N}\leq\gamma_{k+1}^{N+1}$ and
$\gamma_{1}^{N+1}\geq 1$ is equivalent to

$\begin{dcases}&a(c-d)\leq(bc-ad)(N-k)\\\ &d-c\leq(d-b)N.\\\ \end{dcases}$
(21)

To fulfill these equations for all $k\in\\{1,...,N-1\\}$ and
$N\in\\{2,3,...\\}$, we get the following condition:

$\begin{dcases}&bc-
ad\geq\sup\left(\left\\{\dfrac{a(c-d)}{N-k}:k\in\\{1,...,N-1\\};N\in\\{2,3,...\\}\right\\}\right)\\\
&d-b\geq\sup\left(\left\\{\dfrac{d-c}{N}:N\in\\{2,3,...\\}\right\\}\right).\\\
\end{dcases}$ (22)

Parallel to the proof of Theorem 1, the suprema in both equations depend only
on the sign of the terms $a(c-d)$ and $d-c$, which are related.

  1. 1.

Case $c<d$:

Since $a(c-d)<0$ and $N$ is unbounded, the supremum on the top equation in 22
is $0$. Since $d-c>0$, the one in the bottom equation is the maximum of that
expression obtained when $N=2$. Therefore, condition 22 is fulfilled if $bc-
ad\geq 0$ and $d-b\geq(d-c)/2$, leading to:

$bc\geq ad\hskip 2.84544pt\text{,}\hskip 8.5359ptc+d\geq 2b.$ (23)

  2. 2.

Case $c\geq d$:

Since $a(c-d)\geq 0$, the supremum in the top of equation 22 occurs for
$k=N-1$. Since $d-c\leq 0$ and $N$ is unbounded, the supremum is simply $0$ in
the bottom one. Therefore, condition 22 turns into $bc-ad\leq a(c-d)$ and
$d-b\geq 0$, which lead to:

$b\geq a\hskip 2.84544pt\text{,}\hskip 8.5359ptd\geq b.$ (24)

Any game meeting these conditions will necessarily have a fixation probability
with a decreasing function on $N$.

∎

Under Theorem 1, there are a total of $7$ orderings of the payoff matrix
values represented in table 2 that always satisfy the obtained condition. The
first condition includes exclusively $6$ complete orderings corresponding to
the fixation of strategies from the fourth dominance game listed in table 2,
and the two last coordination games. The second condition is always satisfied
by ordering $d>b>a>c$ corresponding to the fixation of hare hunters in the
Stag Hunt game, and it is partially met for subsets of other $4$ dominance
game orderings: $b>d>a>c$, $d>b>c>a$, $d>c>b>a$ and $b>a>d>c$.

Under Theorem 2, the first condition in the union defines exclusively the
complete ordering $c>d>b>a$ corresponding to the fixation process of a
dominated strategy. The second condition is only satisfied partially under the
two orderings $d>b>c>a$ and $d>c>b>a$, both of which are already partially
covered by the condition from Theorem 1. Joining Theorems 1 and 2, it can be
easily shown that ordering $d>c>b>a$ ends up covered completely by the union
of both final conditions, thus always holding decreasing fixation functions.

Additionally, if the dependence of the fixation probability on population size
were considered only for large $N$, the second condition in equation 22 would
simply lead to $d\geq b$. Together with the conditions from Theorem 1, the
large $N$ conditions would further cover the complete ordering $d>b>c>a$ as
well.

It is worth noting that both Theorems 1 and 2 were proved by analysing the
transition probability ratios $\gamma_{k}^{N}$ (eq. 9), which under the
frequency-dependent Moran process become the relative fitness of individuals
using B (eq. 10). It can be seen that increasing population size by one
individual impacted fixation probabilities in two ways: 1) added an extra term
to the sum, and 2) changed ratios $\gamma_{k}^{N}$ to $\gamma_{k}^{N+1}$ in
the already existing terms. Contrary to what happens under these orderings,
fixation probability functions might not be monotonically decreasing functions
of $N$ if the sum of all terms in the denominator of equation 8 decreases with
$N$, despite the increase of the number of terms. As will be shown in later
sections 3.2, 3.3, and 3.4, this is possible under cases where adding
individuals to the population decreases the relative fitness $\gamma_{k}^{N}$
enough to compensate for the extra individuals that one single mutant would
have to replace.

####  3.1.1 Games with decreasing functions

Focusing on the games under which fixation probabilities are decreasing
functions of $N$, these might have either a zero or positive asymptotic limit,
depending on the values in the payoff matrix. According to [34], dominance
games have a mutant strategy holding a positive asymptotic value if $a>c$ and
$b>d$, otherwise this value is zero. Looking at the entries in table 2
corresponding to these games, we may conclude that the first game ordering for
every pair (i.e. the dominating strategy) always holds a positive asymptotic
value (DP when decreasing), while the second (i.e. the dominated strategy)
always holds a null asymptotic value (D0 when decreasing).

Decreasing fixation probability functions were observed under all dominance
game orderings, both for dominating and dominated mutant strategies. Three of
the orderings corresponding to the fixation of dominated strategies were
proved to only have decreasing functions, and the remaining three were tested
for a wide choice of values with no non-decreasing counter-examples found. On
the other side, all five dominating mutants which were not proven to hold
decreasing fixation functions were found to have examples of fixation
probability functions which increased for some value of population size, thus
holding alternative shapes.

Regarding anti-coordination games, the borderline case corresponding to
switching from having a mutant strategy fixating positively in the asymptotic
limit to fixating with null probability was obtained in [34] and depends only
on the payoff and intensity of selection values. The original expressions were
redefined in [35] based on $I$, the definite integral of $\ln(\gamma(\alpha))$
evaluated between $0$ and $1$, which under $w=1$ becomes

$I=\ln\left(\dfrac{c^{\frac{c}{c-d}}b^{\frac{b}{a-b}}}{d^{\frac{d}{c-d}}a^{\frac{a}{a-b}}}\right).$
(25)

The function $\gamma(\alpha)$ is the relative fitness $\gamma_{k}^{N}$ (eq.
10) under $N\to\infty$, which depends only on the fraction of mutants
$\alpha=k/N$ instead of $k$ and $N$ independently.

It was shown that the general borderline case is obtained under $I=0$. If the
parameterization of intensity of selection $w$ is considered (see eqs. 3 and
4), and the weak selection limit taken after the large population limit in
exactly this order, condition $I=0$ is equivalent to $a+b=c+d$. Asymptotic
fixation probabilities in that case can be represented as first-order terms in
$w$ under $a+b\geq c+d$, but only as higher-order terms for $a+b<c+d$.

We observed that under $I>0$, i.e. when the asymptotic fixation probability is
null, it is always possible to see decreasing fixation functions (D0) under
these games. Otherwise, under $I<0$, i.e. when the mutant strategy fixates
positively in the asymptotic limit, the fixation of hawks under the Hawk-Dove
game represented the only one of those game orderings which could have
strictly decreasing functions (DP) for some choices of the payoff matrix. The
remaining $4$ anti-coordination orderings that satisfied $I<0$ for some of
their subsets, e.g. fixation of mutant doves under the Hawk-Dove game, were
tested systematically and never observed to decrease monotonically to a
positive asymptotic value. In section 3.2, we propose an explanation for why
fixation function shapes beyond the decreasing one have not been referred to
in most of the previous literature, with the exception of [20].

Under coordination games, asymptotic fixation values are always null
independent of the strategy considered as the mutant, as noted by [34]. Here
we add that five of these orderings (two pairs of complementary orderings and
the one representing hare hunters’ fixation under the Stag Hunt game), were
proven to always have strictly decreasing fixation probabilities (D0).
Regarding the other ordering, which corresponds to stag hunters’ fixation
under the Stag Hunt game, there were striking examples of alternative fixation
probability functions which will be mentioned in sections 3.3 and 3.4.

###  3.2 Fixation probability functions with one minimum

Under some of the explored $2\times 2$ games, there were alternatives to the
way the fixation probability of a single mutant depended on population size
$N$. One common pattern found in these functions across anti-coordination
games and the fixation of dominating strategies started with the occurrence of
a plunge for small population sizes until a global minimum of the fixation
probability was reached under $N_{min}$. This was followed by a steady
increase up to a positive asymptotic value which could be computed following
[34].

This type of dependence was not observed under any of the $6$ coordination
game orderings nor the other $6$ orderings referring to the fixation of
dominated strategies. It was so because all of these fixation probabilities
have null asymptotic values. On the other hand, it was sometimes observed
under the fixation of dominating strategies, such as $b>d>a>c$, $a>c>b>d$,
$a>b>c>d$, $b>a>d>c$, and $b>a>c>d$. This included the fixation of defectors
under the Prisoner’s Dilemma (e.g. under payoffs $[2,4,1.9,2.1]$). The
occurrence of this dependence under this set of games is absent from the
previous literature.

Nonetheless, it was under anti-coordination games that we found this behaviour
to be pervasive. This was present under all $5$ anti-coordination game
orderings which allowed positive asymptotic values. As already noted in
section 3.1, $4$ out of these $5$ anti-coordination game orderings didn’t
accommodate decreasing functions of population size, suggesting that the
existence of increasing regions is the norm rather than the exception under
these games.

Despite being pervasive under these games, which have been thoroughly studied
in the past [7, 36, 37, 8, 45, 46], this effect did not receive much
attention, with the exception of [20]. A significant amount of approaches to
games in finite populations have considered the weak selection limit [18, 39,
35] and show that when selection tends to zero fast enough (when compared to
the increase of population size), we should expect fixation probability
functions to always decrease asymptotically. Motivated by this, we have looked
at the impact of decreasing intensity of selection on the fixation probability
functions, represented in figure 2. For a fixed choice of payoffs $[a,b,c,d]$,
decreasing the intensity of selection $w$ pushes the population size from
which fixation starts to increase $N_{min}$ to larger values. In fact, they
are approximately inversely proportional, $N_{min}(w)\sim 1/w$ (see appendix
A).

Figure 2:  Single-mutant fixation probability functions and the effect of weak
selection on moving global minima under shape DUP to higher population sizes.
The fixation probabilities shown were obtained for the fixation of doves under
the Hawk-Dove game (HD) with $[5.5,5,6,3]$.

These results suggest that even if we consider an arbitrarily small value of
the intensity of selection (i.e. weak selection limit), we may still see
fixation probabilities increase with population size for large enough
populations $N>N_{min}(w)$ (i.e. in the large population limit). This should
not contradict the results obtained in [35], since in the scenario we are
describing the large population limit should dominate over the weak selection
one. Additionally, because turning points $N_{min}(w)$ become very large under
weak selection, fixation probabilities could become less representative of
what happens under anti-coordination games in finite populations. Even though
pure states are absorbing ones, when mutations are considered the population
might spend most of the evolutionary time in particular transient states [34,
33], which are called quasi-stationary states in those cases [32, 47, 30, 31].
In appendix A, we show that average conditional fixation times increase
polynomially with $N_{min}(w)$ as they do with $N$ under neutral fixation,
instead of increasing exponentially as is predicted for anti-coordination
games under a fixed intensity of selection [34]. These results highlight why
it is essential to beware of the impact of the order in which the weak
selection limit $w\to 0$ and the large population limit $N\to\infty$ are
considered under evolutionary dynamics, as it has been clearly stated in [35].

###  3.3 Fixation probability functions increasing under small populations

Under a restricted number of games of all three types (dominance, anti-
coordination, coordination), fixation probabilities were observed to increase
with population size $N$ for small populations. The particular case where
$\rho_{3}>\rho_{2}$ holds is an extreme one where this may happen. However,
the condition represented by that is simple enough to allow an analytical
approach.

######  Theorem 3.

If the payoff matrix entries of a $2\times 2$ game satisfy $c>d$ and
$a+b>2c(c+d)/(c-d)$, then $\rho_{3}>\rho_{2}$.

###### Proof.

The fixation probability values under these population sizes are the
following:

$\rho_{2}=\frac{1}{1+\gamma_{1}^{2}},$ (26)
$\rho_{3}=\frac{1}{1+\gamma_{1}^{3}+\gamma_{1}^{3}\cdot\gamma_{2}^{3}}.$ (27)

These equations lead to the equivalence

$\rho_{3}>\rho_{2}\Leftrightarrow\gamma_{1}^{3}+\gamma_{1}^{3}\cdot\gamma_{2}^{3}<\gamma_{1}^{2}\Leftrightarrow\gamma_{1}^{3}\cdot\left(1+\gamma_{2}^{3}\right)<\gamma_{1}^{2}.$
(28)

Applying the definition of $\gamma_{k}^{N}$ under the frequency-dependent
Moran process (equation 10) on equation 28, we get that the following is a
sufficient condition:

$a+b>\lambda\hskip 2.84544pt\text{and}\hskip 2.84544ptc>d,$ (29)

where we have used $\lambda$, defined as

$\lambda=2\hskip 5.69054pt\frac{c\hskip 2.84526pt(c+d)}{c-d}.$ (30)

∎

This condition is only possible under $6$ out of the $24$ possible payoff
orderings. There are twelve orderings where $c>d$ but only in six of them it
is possible to have $a+b>\lambda$. These are the ones where $c>d$ and either
$a$ or $b$ is the largest entry in the payoff matrix. These orderings are
itemised in the last column of table 2.

Condition 28 highlights the conflict of having one extra individual in a
population. One more resident leads to mutants having another individual to
replace. Thus the presence of an extra term on the left hand-side of equation
28. In order to observe fixation probabilities increasing from population size
$N=2$ to $N=3$, the relative fitness of residents needs to be much lower for
higher proportions of them in the population.

There was a total of three shapes of fixation probability functions found that
met $\rho_{3}>\rho_{2}$. Fixation probabilities might increase monotonically
up to a positive asymptotic value (UP), they might increase up to a maximum
value and then decrease down to a positive value (UDP), or they might increase
up to a maximum and then decrease to zero (UD0). These three shapes are
represented in the summary presented in figure 1.

Three of the orderings that satisfy this condition correspond to the fixation
of dominating strategies. This means that the fixation probability’s
asymptotic value under all of these cases is positive. Under the two orderings
$a>c>b>d$ and $a>b>c>d$ both UP and UDP were found depending on the particular
parameter choices, while under $b>a>c>d$ only UP was found.

Under the two anti-coordination game orderings which are able to satisfy these
conditions, the only shape found from these three was UP. Systematic checks
suggest that equation $I>0$ is either not satisfied at all, or at least at the
same time as equation 29 under those two orderings. This means that when
fixation probabilities increase from $N=2$ to $N=3$ under those two orderings,
we should only observe functions with positive asymptotic fixation
probabilities (i.e. UD0 should never be observed there).

Fixation probability functions with that shape – having one maximum and then
tending to zero (UD0) – were found only for the fixation of stag hunters under
the Stag Hunt game. This is the case exhibited in figure 1(b). As suggested by
the tests mentioned in the previous paragraphs, there was no other ordering
where UD0 was observed, therefore establishing the particularity of this game
and justifying further interest in its study.

###  3.4 Fixation probability functions with two extremes

Under orderings corresponding to games of all three types, fixation
probability functions $\rho_{N}$ were sometimes observed to decrease with
population size $N$ for small populations, have an increasing region for
intermediate population sizes and finally decrease again under large
populations. These increasing regions were necessarily delimited by two local
extremes, one minimum and one maximum.

Under anti-coordination games, these regions were observed only when
asymptotic values were null (DUD0) and occurred when the choice payoffs led to
a positive value of $I$ very close to zero (see equation 25). If $I>0$ then a
mutant using $A$ has a null probability of fixation in the limit $N\to\infty$.
However, close to the border $I=0$ and for small enough populations, they may
fixate similarly as if they were on the negative side. In the particular case
in figure 3, $N=200$ is enough for $I\leq 10^{-3}$. The occurrence of two
extremes and the increasing regions to which they lead present as transitional
features between the decreasing shape D0 (seen under a high enough positive
$I$), and the asymptotically positive DUP (seen under a negative $I$).
Additionally, as we get further closer to $I\to 0^{+}$, we see the positive
values of $\rho_{N}$ breaking down to zero only for larger and larger
population sizes.

Figure 3:  Single-mutant fixation probability functions transitioning from
having shape DUP ($I=0$) to D0 ($I=\\{5\times 10^{-3},10^{-2}\\}$) through
DUD0 ($I=\\{10^{-4}$, $5\times 10^{-4},10^{-3}\\}$). Fixation probabilities
were obtained for games with payoff parameters $[5.5,5,6,d]$ and population
size $N$. The values of $d$ were calculated from each chosen value of $I$ (eq.
25). Their choice maintains the game under ordering $c>a>b>d$, i.e. fixation
of doves under Hawk-Dove game (HD), and reproduces the approach of limit $I\to
0^{+}$. The approximate values of $d$ are, in order, $d=4.530$ ($I=0$),
$d=4.531$, $d=4.535$, $d=4.540$, $d=4.581$, and $d=4.631$.

This shape was also observed in the context of the fixation of stag hunters in
the Stag Hunt game when function shapes transitioned between D0 and UD0, i.e.
between being always decreasing and having a global maximum (instead of a
global minimum as in the previous case). This is shown in figure 4.

Figure 4: Single-mutant fixation probability functions transitioning from
having shape UD0 ($c=18$) to D0 ($c=45$) through DUD0 ($c=\\{25,35\\}$).
Fixation probabilities were obtained for games with payoff parameters
$[50,1,c,2]$ and population size $N$. The values of $c$ were chosen as to
always being under ordering $a>c>d>b$ and allowing condition
$\rho_{3}>\rho_{2}$ (eq. 29) to be met only for the lowest value $c=18$.

A parallel fixation probability function having an intermediate population
size increasing region between two extremes, but with a positive asymptotic
value (DUDP) was observed under the fixation of two dominating strategies.
This can be seen both in the summarising figure 1(a) and in figure 5. This
shape was observed under orderings $a>c>b>d$ and $a>b>c>d$, as a transition
between functions with one global minimum (DUP) and functions which start
increasing up to a global maximum and then decrease to a positive value (UDP).
Those are the only two orderings that may satisfy $\rho_{3}>\rho_{2}$ by
taking the form of UDP.

Figure 5: Single-mutant fixation probability functions transitioning from
having shape UDP ($a=12$) to DUP ($a=4$) through DUDP ($a=\\{10,8,6\\}$).
Fixation probabilities were obtained for games with payoff parameters
$[a,2,1.8,1]$ and population size $N$. The values of $a$ were chosen as to
always being under ordering $a>b>c>d$ and allowing condition
$\rho_{3}>\rho_{2}$ (eq. 29) to be met only for the highest value $a=12$.

These results suggest that the shapes DUDP and DUD0 having two extremes
delimiting an increasing intermediate population region occur for transitions
between fixation functions with one global extreme, either a maximum or a
minimum, and other function shapes. Functions with one global extreme are
characterized by having different trends (increasing/decreasing) for small and
large population sizes. If incrementally changing the payoff matrix values
switches the increasing trend to a decreasing one, an increasing region may
emerge under intermediate populations.

##  4 Discussion

Understanding fixation processes and the probability of their occurrence is
essential to the analysis of evolutionary games on finite populations. In this
context, [13] observed that even if individuals are interacting via the same
game, the size $N$ of the population where they are included could lead to
different evolutionarily stable strategies $ESS_{N}$ as defined in [11]. This
is an emerging feature of evolutionary processes under frequency-dependent
fitness since stability under the fixed fitness Moran process never depends on
population size. Here, we provided an extensive analysis of the dependence of
single-mutant fixation probabilities on population size, focusing on the
simplest $2\times 2$ games in well-mixed populations.

We have proved that under nine game orderings, fixation probabilities should
be strictly decreasing functions. Under three other game orderings, extensive
tests were performed without finding any other form of dependence beyond this
one. Altogether, these games included the fixation of all six dominated
strategies, one dominating strategy, and almost all strategies under
coordination game orderings (five out of six).

However, a lot of interesting dependencies on population size $N$ arise in the
remaining twelve game orderings, under which single-mutant fixation functions
were observed to have intervals of population sizes for which they increased
with $N$. As counter-intuitive as this may seem, populations having extra
individuals whom a single mutant would have to replace during fixation might
actually increase that mutant’s chances of fixating.

Under large populations, this happened when the mutant’s relative fitness was
higher under lower proportions of its type in the population, i.e. when
mutants did relatively better among residents than they did among their own
type: $d/b<c/a$. In this limit, the selection dynamics are completely
characterized by these two ratios, as it was already noted in [13]. Increasing
population size guaranteed that both mutants and residents interacted more
frequently with residents in the initial steps of the fixation process.
Therefore, the increase in the number of resident individuals to be replaced
by one mutant was offset by the higher replacement probabilities.

Surprisingly, fixation functions were observed to increase under cases where
mutants did relatively worse among residents than among themselves for large
populations. This was observed under $3$ game orderings, where fixation
probabilities increased for small populations but eventually decreased to
infinity. It was also noted in [13] that payoffs $b$ and $c$ are the only
relevant entries of the payoff matrix under $N=2$. Thus, under small
populations, increasing population size means increasing the maximum frequency
of individuals of the same type that are found in the population before
fixation occurs (e.g. $0\%$ for $N=2$, $50\%$ for $N=3$, $80\%$ for $N=6$).
Increasing population size thus increases the frequency of interactions
between equal types. Having high enough $a>\lambda-b$ and low enough $d<c$
allowed for fixation probabilities to increase for the smallest population
sizes, even when $d/b>c/a$.

The confluence of all these effects in the $12$ remaining orderings resulted
in a high diversity of shapes of single-mutant fixation probability functions,
from functions with one minimum (DUP), to always increasing functions (UP),
initially increasing but asymptotically decreasing ones (UDP and UD0), and
functions with two extremes (DUDP and DUD0). We tried to understand under
which orderings, and why, these function shapes emerged.

Anti-coordination games, such as Hawk-Dove [7, 8] (also called Snowdrift [36,
37]) were shown to hold a wide variety of fixation probability function
shapes. One of the most striking observations was the pervasiveness of
functions with one minimum, a shape already noted in [20] under the fixation
of dove strategy. In four orderings, this was the only shape observed when
asymptotic values were positive, even for arbitrarily low values of intensity
of selection. While it was shown in [35] that fixation probabilities decrease
with population size when the weak selection limit is dominant over the large
population limit, our results suggest that if the considered limit order is
the reverse, weak selection does not necessarily erase these non-decreasing
shapes. Additionally, when approaching the limit where fixation functions
change from being asymptotically null to having this shape with a global
minimum, all anti-coordination game orderings showed functions with two
extremes. Finally, two of these orderings also showed monotonically increasing
shapes when $b$ was sufficiently large, i.e. when mutants benefited the most
from interacting with residents.

The six game orderings associated with the fixation of dominating strategies
[13] (also named unbeatable [48, 11]), such as defectors under the Prisoner’s
Dilemma [4, 5], were observed to hold monotonically decreasing functions, with
one complete ordering and subsets of some of the remaining being comprehended
in the conditions of Theorems 1 and 2. However, they were also shown to have a
wide variety of other function shapes. Functions with one global minimum were
observed under five out of the six orderings. Under three of these orderings,
we observed monotonically increasing functions. Two of them additionally
showed alternative functions keeping an initial increase but decreasing for
larger population sizes, and also functions with two extremes when
transitioning between those with a global minimum and the previous shape. On
the other side, the fixation of dominated strategies, such as cooperation
under the same games, was proved or tested to systematically hold
monotonically decreasing functions.

In the context of coordination games, we have proved that under five out of
the six possible orderings, fixation functions were always monotonically
decreasing. However, the fixation of stag hunters (the reward-dominant
strategy under the Stag Hunt game [38, 6]) was observed to have exceptional
results in this context. If the reward for cooperation $a$ is large enough, we
observe initially increasing functions that then tend to zero thus holding a
maximum. This shows that there is an optimal population size for the fixation
of a single stag hunter in those cases. Functions with two extremes were
observed under these games, when transitioning between the monotonically
decreasing functions and the ones with a global maximum, leading to fixation
probabilities holding similar values for a wide range of population sizes.

There are numerous relevant borderline cases where finite populations might
hold relevant differences. One example leading to particularly different
results in finite populations is the transition through $I=0$ (see [35]),
which in section 3.4 was shown to lead to functions with two extremes.
However, the orderings were explored assuming the entries in the payoff matrix
to be different pairwisely. This ignores potentially interesting equal-payoff
cases, such as the one with $a=c$ and $b=d$ under which there is neutral
selection for infinitely large populations but not for finite ones, leading to
size-dependent equilibria [13].

We have verified that considering the evolutionary dynamics rising from the
pairwise comparison [18] can lead to all eight fixation function shapes
reported here. Additionally, considering multiplayer social dilemmas in well-
mixed populations would allow transition probability ratios $\gamma_{k}^{N}$
to have a non-monotonic dependence on $k$ [49, 50], which could lead to the
emergence of new fixation probability functions of population size. Finally,
fixation probabilities have been studied in the context of structured
populations as well, often holding distinctive dependencies on population size
[20, 21, 22, 23]. Expanding our analysis to structured populations could hold
new effects and lead to a more general theory of fixation probability
functions.

##  Appendix A The Effect of the Weak Selection Limit

In section 3.2, it was observed that anti-coordination games held fixation
probabilities which increased for population sizes above a turning point
$N_{min}(w)$ (see figure 2). Figure 6(a) suggests that turning points are
inversely proportional to the intensity of selection $N_{min}(w)\sim 1/w$,
which means that they would become increasingly large under the weak selection
limit. Under this limit, it could be argued that the increase in the
population sizes for which fixation probabilities start to increase would lead
to a loss of significance of fixation processes due to most of the
evolutionary time being spent in transient/mixed states [34, 33], also called
quasi-stationary states [32, 47, 30, 31].

(a)

(b)

Figure 6: Fixation process under game with payoff values $[5.5,5,6,3]$,
corresponding to dove fixation in the Hawk-Dove game (HD). Figure 6(a)
exhibits the turning point $N_{min}$ as a function of the intensity of
selection $w$, showing that they have an approximately inversely proportional
relation. In figure 6(b), the average conditional fixation times $t_{N}$ were
shown for (1) the Hawk-Dove game with the maximum intensity of selection
($w=1$ and independent $N$), (2) the Hawk-Dove game at the turning points of
figure 6(a) ($w\in(0,1)$ and $N=N_{min}(w)$), (3) the neutral fixation
scenario ($w=0$ and independent $N$), and (4) the simple (symmetric) random
walk.

The average conditional fixation time is the average number of discrete steps
that the population takes to go from state $i=1$ to state $i=N$ (i.e. the
fixation of a single mutant) conditional on it happening:

$t_{N}=\dfrac{\sum_{t=0}^{\infty}t\phi_{N}(t)}{\sum_{t=0}^{\infty}\phi_{N}(t)},$
(31)

where $\phi_{N}(t)$ is the probability that the population gets from state
$i=1$ to state $i=N$ after $t$ discrete steps. It should be clear from this
definition that $\sum_{t=0}^{\infty}\phi_{N}(t)=\rho_{N}$. Following [34], we
can obtain a recursive relation leading to the following expression (see [14,
15] for alternative expressions):

$t_{N}=\sum_{n=1}^{N-1}\dfrac{s_{0,n-1}s_{n,N-1}}{P_{n^{+}}^{N}q_{n}s_{0,N-1}},\hskip
12.0pts_{n,m}=\sum_{k=n}^{m}q_{k},\hskip
12.0ptq_{n}=\prod_{j=1}^{n}\gamma_{j}^{N}.$ (32)

We are interested in comparing the average time obtained in the turning points
$N_{min}(w)$ of figures 2 and 6(a) with three different scenarios: the simple
(symmetric) random walk, the neutral fixation case and the same game for
maximum intensity of selection $w=1$.

The simple (symmetric) random walk is obtained by considering the stochastic
process under which the transition between a state $i$ and one of its two
neighbours is one-half: $P_{i^{+}}=P_{i^{-}}=1/2$. The average conditional
fixation time under this process is

$t_{N}=\dfrac{1}{3}(N^{2}-1).$ (33)

On the other hand, under the frequency-dependent Moran process the transition
probabilities $P_{i^{+}}$ and $P_{i^{-}}$ do not add up to $1$. This is so
because at each time step there is a probability that the population will
remain in the same state $P_{i^{=}}$. Thus, when we are under neutral
selection ($w=0$), equation 32 leads to a different equation than 33, which is
three times slower than the simple random walk

$t_{N}=N(N-1).$ (34)

By analysing fixation processes under $2\times 2$ games, [34] concluded that
anti-coordination games have average conditional fixation times that grow
exponentially with population size for asymptotically large populations. The
base of exponential growth $\tau$ can be calculated from the payoff matrix.
The average conditional fixation time thus becomes

$t_{N}\sim\tau(a,b,c,d)^{N},$ (35)

where $\tau(a,b,c,d)>1$ for any choice of $a$, $b$, $c$, and $d$ leading to an
anti-coordination game with $I<0$.

In figure 6(b), we observe that as intensity of selection decreases, and
turning points increase accordingly $N_{min}(w)\to\infty$, fixation times
under this anti-coordination game keep on being of the order of the neutral
ones $t_{N}\sim N^{2}$ instead of increasing exponentially with $N$. As noted
in [35], considering limits $N\to\infty$ and $w\to 0$ in different orders may
lead to different fixation outcomes. Here we present a case where it is needed
to clarify that relation in order to understand what happens asymptotically.

## Acknowledgements

This project has received funding from the European Union’s Horizon 2020
research and innovation programme under the Marie Skłodowska-Curie grant
agreement No 955708.

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