Fixation Probabilities of Strategies for Trimatrix Games and Their Applications to Triadic Conflict

Abstract

This study extends existing formulae of the fixation probabilities of strategies for symmetric games and bimatrix games in finite populations and derives a counterpart for trimatrix games. This allows us to describe the stochastic evolutionary game dynamics when three players are assigned different roles and therefore are not interchangeable. Following previous studies, we also derived two types of stochastic stability conditions based on the obtained fixation probabilities; “strong stochastic stability,” which requires that for any initial frequencies of strategies, the fixation probability of a combination of specific strategies is higher than that under neutrality and those of any other combinations are lower than neutrality; and “stochastic stability,” which only requires that the fixation probability of a specific strategy combination be higher than that under neutrality for any initial frequencies of strategies. Thus, for the former, we obtain a clear correspondence with bimatrix games, but not necessarily for the latter. The results of applying our findings to triadic conflicts (the Impartial person and mediator game and the Fish in troubled waters game), the volunteer’s dilemma, and coordination games are also reported.

Appendices

Appendices

A1 Impact of population size ratios

In Sect. 3.1.1, assuming $M=N=L$, we derived the condition for the strategy triplet $(\alpha _1, \beta _1, \gamma _1)$ to fix from the initial condition $(i.j,k) = (M/2, N/2, L/2)$ more likely than under neutrality, Ineq. (36), which is equivalent to

$$\begin{aligned} \begin{aligned} 7a_{111}&+2a_{112}+2a_{121}+a_{122}+7b_{111}+2b_{112}+2b_{211}+b_{212}+7c_{111}+2c_{211}\\&\quad +2c_{121}+c_{221} > 7a_{211}+2a_{212}+2a_{221}+a_{222}+ 7b_{121}+2b_{122}\\&\quad +2b_{221}+b_{222}+7c_{112}+2c_{212}+2c_{122}+c_{222}. \end{aligned} \end{aligned}$$

(41)

In this section, while maintaining the assumption of $(i,j,k)=(M/2, N/2, L/2)$, we examine how population size ratios change the coefficient for each term in Ineq. (41), by further assuming that $M=N$ and $L=\lambda M$ $(0<\lambda <\infty )$, which allows us to know the impact of a change in the size of one population. This assumption could be reasonable when two of the three players play symmetric roles, as Players 1 and 2 in the Impartial person and mediator game, and a snipe and a clam in the Fish in troubled waters game.

By inserting $M=N, L=\lambda M$, and $(i,j,k)=(M/2, M/2, \lambda M/2)$ into Eq. (32), we found that the coefficient for each term in the counterpart of Ineq. (41) is as follows:

$$\begin{aligned} W_{a_{111}}= & {} 3\lambda ^2+8\lambda +3; \, W_{a_{112}}= 3\lambda ^2+\lambda ; \, W_{a_{121}}= \lambda ^2+2\lambda +1; \, W_{a_{122}}= \lambda ^2+\lambda ; \,\qquad \end{aligned}$$

(42)

$$\begin{aligned} W_{c_{111}}= & {} 8\lambda ^3+5\lambda ^2+\lambda ; \, W_{c_{211}}= 3\lambda ^2+\lambda ; \, \text {and} \, W_{c_{221}}= \lambda ^2+\lambda , \end{aligned}$$

(43)

where $W_{\circ }$ indicates the weight (or coefficient) on payoff $\circ$. Further, $a_{211}$, $b_{111}$, and $b_{121}$ are assigned the same coefficient as $a_{111}$; $a_{212}$, $b_{112}$, and $b_{122}$ are assigned the same coefficient as $a_{112}$; $a_{221}$, $b_{211}$, and $b_{221}$ are assigned the same coefficient as $a_{121}$; $a_{222}$, $b_{212}$, and $b_{222}$ are assigned the same coefficient as $a_{122}$; $c_{112}$ is assigned the same coefficient as $c_{111}$; $c_{212}$, $c_{121}$, and $c_{122}$ are assigned the same coefficient as $c_{211}$; $c_{221}$ is assigned the same coefficient as $c_{222}$. Note that to obtain Ineq. (41), both sides should be divided by two as well as inserting $\lambda =1$ into the coefficients obtained above.

Based on these coefficients, let us consider the question of which elements of the payoff trimatrix are more important, and to what extent, for the strategy triplet $(\alpha _1,\beta _1,\gamma _1)$ to be more likely to fix than neutrality. Note that because of the symmetry, the results for $\alpha$-strategists hold true for $\beta$-strategists.

First, we perform a pairwise comparison of the payoffs that can be obtained by the same strategist. For an $\alpha _1$-strategist,

• (1) $a_{111}$ vs. $a_{112}$: (i) $W_{a_{111}}>W_{a_{112}}$ for any $\lambda$; (ii) $\lim \limits _{\lambda \rightarrow \infty }\dfrac{W_{a_{111}}}{W_{a_{112}}} = 1$; (iii) $\lim \limits _{\lambda \rightarrow 0^{+}}\dfrac{W_{a_{111}}}{W_{a_{112}}} = \infty$

• (2) $a_{111}$ vs. $a_{121}$: (i) $W_{a_{111}}>W_{a_{121}}$ for any $\lambda$; (ii) $\lim \limits _{\lambda \rightarrow \infty }\dfrac{W_{a_{111}}}{W_{a_{121}}} = 3$; (iii) $\lim \limits _{\lambda \rightarrow 0^{+}}\dfrac{W_{a_{111}}}{W_{a_{121}}} = \infty$

• (3) $a_{111}$ vs. $a_{122}$: (i) $W_{a_{111}}>W_{a_{122}}$ for any $\lambda$; (ii) $\lim \limits _{\lambda \rightarrow \infty }\dfrac{W_{a_{111}}}{W_{a_{122}}} = 3$; (iii) $\lim \limits _{\lambda \rightarrow 0^{+}}\dfrac{W_{a_{111}}}{W_{a_{122}}} = \infty .$

The above results show that the most weighted payoff is that obtained when players adopt the three strategies constituting the focal triplet $(\alpha _1,\beta _1,\gamma _1)$, $a_{111}$.

The weight to an $\alpha _1$-strategist’s payoff obtained from the deviation of $\gamma _1$-strategy to $\gamma _2$-strategy from the focal triplet (i.e., $a_{112}$) gets closer to that to $a_{111}$ as $\lambda$ increases ((1)–(ii)). This means that the choice of players in a larger group is equally important, whether it is a strategy that constitutes the focal triplet or a strategy that deviates from it.

The weight to the payoff obtained from the deviation of $\beta _1$-strategy to $\beta _2$-strategy from the focal triplet (i.e., $a_{121}$) also approaches that to $a_{111}$ as $\lambda$ increases, but the weight to $a_{111}$ is at least three times greater ((2)–(ii)). This means that an increase in the size of one population (i.e., population $\gamma$) also changes the impact of the choices of players who belong to a population of the same size (i.e., population $\beta$) as a population to which a focal player belongs (i.e., population $\alpha$), but its weight is not as great as that led by the larger population.

Further, the weight to the payoff obtained from the deviation of both $\beta _1$-strategy and $\gamma _1$-strategy (i.e., $W_{a_{122}}=\lambda ^2+\lambda$) is the same in the order of magnitude as that of the weight to the payoff obtained from the deviation of only $\beta _1$-strategy (i.e., $W_{a_{121}}= \lambda ^2+2\lambda +1$).

For a $\gamma _1$-strategist,

• (4) $c_{111}$ vs. $c_{211}$: (i) $c_{111}>c_{211}$ for any $\lambda$; (ii) $\lim \limits _{\lambda \rightarrow \infty }\dfrac{c_{111}}{c_{211}} = \infty$; (iii) $\lim \limits _{\lambda \rightarrow 0}\dfrac{c_{111}}{c_{211}} = 1.$

• (5) $c_{111}$ vs. $c_{221}$: (i) $c_{111}>c_{221}$ for any $\lambda$; (ii) $\lim \limits _{\lambda \rightarrow \infty }\dfrac{c_{111}}{c_{221}} = \infty$; (iii) $\lim \limits _{\lambda \rightarrow 0}\dfrac{c_{111}}{c_{221}} = 1.$

This result means that the relatively larger a population size, the greater the weight given to the payoff of a player belonging to the population obtained when players belonging to a smaller group adopt strategies that constitute the focal triplet (i.e., $c_{111}$).

Next, we perform the pairwise comparison of the payoffs that can be obtained by different strategists.

• (6) $a_{111}$ vs. $c_{111}$: (i) $c_{111}\gtreqless a_{111}$ for $\lambda \gtreqless 1$; (ii) $\lim \limits _{\lambda \rightarrow \infty }\dfrac{c_{111}}{a_{111}} = \infty$; (iii) $\lim \limits _{\lambda \rightarrow 0}\dfrac{c_{111}}{a_{111}} = 0.$

• (7) $a_{112}$ vs. $c_{211}$: Both are always equal.

• (8) $a_{122}$ vs. $c_{221}$: Both are always equal.

The above result shows that the relatively larger the population to which a player belongs, the more weight is given to his or her payoff obtained from the game with strategists constituting the focal triplet (i.e., $c_{111}$ in (6)).

The impact of deviating from the focal triplet on weights is the same even if the deviation occurs in populations of different sizes $\alpha$ and $\gamma$, given strategy $\beta _1$ ($\beta _2$) is adopted in (7) ((8)).

A2 Proof of Theorem 1

We only find the condition for the strategy triplet $(\alpha _1,\beta _1,\gamma _1)$ to be strongly stochastic stable, without loss of generality.

We basically use the property shown in the proof of Theorem 2 in the main text here as well: that is, because once two of i, j, and k in the left-hand side of Ineq. (35) are fixed, it is a linear function of the remaining one, only focusing on fixation probabilities from initial conditions near the corner of the state space is equivalent to considering those from all initial conditions.

However, to prove Theorem 1, we have to derive a condition in which the fixation probability of each strategy triplet except $(\alpha _1,\beta _1,\gamma _1)$ is lower than that under neutrality. Therefore, subscripts of payoffs must be replaced appropriately to convert Eq. (33), representing the fixation probability of the focal triplet $(\alpha _1,\beta _1,\gamma _1)$, into expressions for fixation probabilities of other invading triplets.

For example, to investigate the condition where the fixation probability of the triplet ($\alpha _2, \beta _1, \gamma _1$) is less than that under neutrality, we should not only substitute $M-i$ for i, N for j, and L for k in Eq. (33), but also swap 1 and 2 in the first subscript of payoffs. After that we set $M=N=L$, and take it to be large enough. Hereafter we take sufficiently large $M=N=L$ in each step without explicitly stating it. The condition for that quantity to be negative when $i=1$ and $i=M-1$ is

$$\begin{aligned} a_{211} < a_{111}. \end{aligned}$$

(44)

Similarly, the counterpart of ($\alpha _1, \beta _2, \gamma _1$) is

$$\begin{aligned} b_{121} < b_{111}, \end{aligned}$$

(45)

and the counterpart of ($\alpha _1, \beta _1, \gamma _2$) is

$$\begin{aligned} c_{112} < c_{111}. \end{aligned}$$

(46)

For ($\alpha _2, \beta _2, \gamma _1$), in addition to swapping 1 and 2 in the first and second subscripts of payoffs in Ineq. (35), by fixing $(i,j,k)=(1,1,L)$,

$$\begin{aligned} a_{221} + b_{221} < a_{121} + b_{211}, \end{aligned}$$

(47)

by fixing $(i,j,k)=(1,N-1,L)$,

$$\begin{aligned} b_{221} < b_{211}, \end{aligned}$$

(48)

by fixing $(i,j,k)=(M-1,1,L)$,

$$\begin{aligned} a_{221} < a_{121}, \end{aligned}$$

(49)

and

by fixing $(i,j,k)=(M-1,M-1,L)$,

$$\begin{aligned} a_{211} + a_{221} + b_{121} + b_{221} < a_{111} + a_{121} + b_{211} + b_{111}, \end{aligned}$$

(50)

are obtained, respectively.

For ($\alpha _2, \beta _1, \gamma _2$), in addition to swapping 1 and 2 in the first and third subscripts of payoffs in Ineq. (35),

by fixing $(i,j,k)=(1,N,1)$,

$$\begin{aligned} a_{212} + c_{212} < a_{112} + c_{211}, \end{aligned}$$

(51)

by fixing $(i,j,k)=(1,N,L-1)$,

$$\begin{aligned} c_{212} < c_{211}, \end{aligned}$$

(52)

by fixing $(i,j,k)=(M-1,N,1)$,

$$\begin{aligned} a_{212} < a_{112}, \end{aligned}$$

(53)

and

by fixing $(i,j,k)=(M-1,N,L-1)$,

$$\begin{aligned} a_{211} + a_{212} + c_{112} + c_{212} < a_{111} + a_{112} + c_{111} + c_{211}, \end{aligned}$$

(54)

are obtained, respectively.

For ($\alpha _1, \beta _2, \gamma _2$), in addition to swapping 1 and 2 in the second and third subscripts of payoffs in Ineq. (35),

by fixing $(i,j,k)=(M,1,1)$,

$$\begin{aligned} b_{122} + c_{122} < b_{112} + c_{121}, \end{aligned}$$

(55)

by fixing $(i,j,k)=(M,1,L-1)$,

$$\begin{aligned} c_{122} < c_{121}, \end{aligned}$$

(56)

by fixing $(i,j,k)=(M,N-1,1)$,

$$\begin{aligned} b_{122} < b_{112}, \end{aligned}$$

(57)

and

by fixing $(i,j,k)=(M,N-1,L-1)$,

$$\begin{aligned} b_{121} + b_{122} + c_{112} + c_{122} < b_{111} + b_{112} + c_{111} + c_{121}, \end{aligned}$$

(58)

are obtained, respectively.

For ($\alpha _2, \beta _2, \gamma _2$), in addition to swapping 1 and 2 in the first, second, and third subscripts of payoffs in Ineq. (35),

by fixing $(i,j,k)=(1,1,1)$,

$$\begin{aligned} a_{222} + b_{222} + c_{222} < a_{122} + b_{212} + c_{221}, \end{aligned}$$

(59)

by fixing $(i,j,k)=(M-1,1,1)$,

$$\begin{aligned} a_{222} < a_{122}, \end{aligned}$$

(60)

by fixing $(i,j,k)=(1,N-1,1)$,

$$\begin{aligned} b_{222} < b_{212}, \end{aligned}$$

(61)

by fixing $(i,j,k)=(1,1,L-1)$,

$$\begin{aligned} c_{222} < c_{221}, \end{aligned}$$

(62)

by fixing $(i,j,k)=(M-1,N-1,1)$,

$$\begin{aligned} a_{222} + b_{222} < a_{122} + b_{212}, \end{aligned}$$

(63)

by fixing $(i,j,k)=(M-1,1,L-1)$,

$$\begin{aligned} a_{222} + c_{222} < a_{122} + c_{221}, \end{aligned}$$

(64)

by fixing $(i,j,k)=(1,N-1,L-1)$,

$$\begin{aligned} b_{222} + c_{222} < b_{212} + c_{221}, \end{aligned}$$

(65)

and

by fixing $(i,j,k)=(M-1,N-1,L-1)$,

$$\begin{aligned} \begin{aligned} 2a_{222}&+ 2a_{211} + a_{212} + a_{221} + 2b_{222} + 2b_{121} + b_{122} + b_{221} + 2c_{222} \\&+ 2c_{112} + c_{122} + c_{212} < 2a_{111} + 2a_{122} + a_{112} + a_{121} \\&+ 2b_{212} + 2b_{111} + b_{112} + b_{211} + 2c_{221} + 2c_{111} + c_{121} + c_{211}, \end{aligned} \end{aligned}$$

(66)

are obtained, respectively.

Inequalities (48) and (49) entail Ineq. (47). Inequalities (52) and (53) entail Ineq. (51). Inequalities (56) and (57) entail Ineq. (55). Inequalities (60) and (61) entail Ineq. (63). Inequalities (60) and (62) entail Ineq. (64). Inequalities (61) and (62) entail Ineq. (65). Inequalities (44), (45), (48), and (49) entail Ineq. (50). Inequalities (44), (46), (52), and (53) entail Ineq. (54). Inequalities (45), (46), (56), and (57) entail Ineq. (58). All the conditions that were not entailed above entail Ineq. (66). Therefore, the condition for a strategy triplet $(\alpha _1, \beta _1, \gamma _1)$ to be strongly stable is all Ineqs. (44), (45), (46), (48), (49), (52), (53), (56), (57), (60), (61), and (62) to be satisfied, which indicates that strategies $\alpha _1, \beta _1,$ and $\gamma _1$ constitute the dominant strategy equilibrium. $\square$