Evolution of preferences in multiple populations

Abstract

We study the evolution of preferences in multi-population settings that allow matches across distinct populations. Each individual has subjective preferences over potential outcomes, and chooses a best response based on his preferences and the information about the opponents’ preferences. Individuals’ realized fitnesses are given by material payoff functions. Following Dekel et al. (Rev Econ Stud 74:685–704, 2007), we assume that individuals observe their opponents’ preferences with probability p. We first derive necessary and sufficient conditions for stability for $p=1$ and $p=0$, and then check the robustness of our results against small perturbations on observability for the case of pure-strategy outcomes.

Appendix A: Proofs of Theorems

Theorem 3.5

Let $(\mu ,b)$ be a stable configuration in $(G,\Gamma _1(\mu ))$. Then for each $\theta \in {{\,\textrm{supp}\,}}\mu$, the outcome $b(\theta )$ is Pareto efficient with respect to $\pi$.

Proof

Suppose that there exists $\bar{\theta }\in {{\,\textrm{supp}\,}}\mu$ such that $b(\bar{\theta })$ is not Pareto efficient, that is, there exists $\sigma \in \prod _{i\in N}\Delta (A_i)$ such that $\pi _i(\sigma )\ge \pi _i\big (b(\bar{\theta })\big )$ for all $i\in N$ and $\pi _j(\sigma ) > \pi _j\big (b(\bar{\theta })\big )$ for some $j\in N$. Let an indifferent mutant profile $\widetilde{\theta }^0 = (\widetilde{\theta }^0_1, \ldots , \widetilde{\theta }^0_n)$ be introduced with its population share vector $\varepsilon = (\varepsilon _1, \ldots , \varepsilon _n)$. Let $\widetilde{b}\in B_1(\widetilde{\mu }^{\varepsilon };b)$ be the played equilibrium satisfying (1) $\widetilde{b}(\widetilde{\theta }^0) = \sigma$; (2) for any proper subset $T\varsubsetneq N$ and any $\theta _{-T}\in {{\,\textrm{supp}\,}}\mu _{-T}$, we have $\widetilde{b}(\widetilde{\theta }^0_T, \theta _{-T}) = b(\bar{\theta }_T, \theta _{-T})$.⁴⁰ Then for every $i\in N$, the difference between the average fitnesses of $\widetilde{\theta }^0_i$ and $\bar{\theta }_i$ is

$$\begin{aligned} \varPi _{\widetilde{\theta }^0_i}(\widetilde{\mu }^{\varepsilon };\widetilde{b}) - \varPi _{\bar{\theta }_i}(\widetilde{\mu }^{\varepsilon };\widetilde{b}) = \widetilde{\mu }^{\varepsilon }_{-i}(\widetilde{\theta }^0_{-i}) \big [ \pi _i(\sigma ) - \pi _i\big (b(\bar{\theta })\big ) \big ]. \end{aligned}$$

Thus, for any vector $\varepsilon \in (0,1)^n$, we have $\varPi _{\widetilde{\theta }^0_i}(\widetilde{\mu }^{\varepsilon };\widetilde{b})\ge \varPi _{\bar{\theta }_i}(\widetilde{\mu }^{\varepsilon };\widetilde{b})$ for every $i\in N$, and $\varPi _{\widetilde{\theta }^0_j}(\widetilde{\mu }^{\varepsilon };\widetilde{b}) > \varPi _{\bar{\theta }_j}(\widetilde{\mu }^{\varepsilon };\widetilde{b})$ for some $j\in N$. This means that the configuration $(\mu ,b)$ is not stable. $\square$

Lemma 3.7

Let $(\mu ,b)$ be a stable configuration in $(G,\Gamma _1(\mu ))$. Then the equality $\pi \big (b(\theta )\big ) = \pi \big (b(\theta ')\big )$ holds for every $\theta$, $\theta '\in {{\,\textrm{supp}\,}}\mu$.

Proof

Let $(\mu ,b)$ be a stable configuration. We claim that for any $j\in N$ and any $\theta '_j$, $\theta ''_j\in {{\,\textrm{supp}\,}}\mu _j$, the equality $\pi \big ( b(\theta '_j, \theta _{-j}) \big ) = \pi \big ( b(\theta ''_j, \theta _{-j}) \big )$ holds for all $\theta _{-j}\in {{\,\textrm{supp}\,}}\mu _{-j}$. To see this, we first suppose that there exist $\theta '_j$, $\theta ''_j\in {{\,\textrm{supp}\,}}\mu _j$ for some $j\in N$ such that $\pi _j\big (b(\theta '_j, \bar{\theta }_{-j})\big )> \pi _j\big (b(\theta ''_j, \bar{\theta }_{-j})\big )$ for some $\bar{\theta }_{-j}\in {{\,\textrm{supp}\,}}\mu _{-j}$. Let $\widetilde{\theta }^0_j\in {({{\,\textrm{supp}\,}}\mu _j)}^{\textsf{c}}$ be an indifferent type entering the jth population. Let the played equilibrium $\widetilde{b}\in B_1(\widetilde{\mu }^{\varepsilon };b)$ satisfy $\widetilde{b}(\widetilde{\theta }^0_j, \bar{\theta }_{-j}) = b(\theta '_j, \bar{\theta }_{-j})$ and $\widetilde{b}(\widetilde{\theta }^0_j, \theta _{-j}) = b(\theta ''_j, \theta _{-j})$ for all $\theta _{-j}\ne \bar{\theta }_{-j}$. Then for an arbitrary population share $\varepsilon$ of $\widetilde{\theta }^0_j$, the difference between the average fitnesses of $\widetilde{\theta }^0_j$ and $\theta ''_j$ is

$$\begin{aligned} \varPi _{\widetilde{\theta }^0_j}(\widetilde{\mu }^{\varepsilon };\widetilde{b}) - \varPi _{\theta ''_j}(\widetilde{\mu }^{\varepsilon };\widetilde{b}) = \mu _{-j}(\bar{\theta }_{-j})\big [ \pi _j\big (b(\theta '_j,\bar{\theta }_{-j})\big ) - \pi _j\big (b(\theta ''_j,\bar{\theta }_{-j})\big ) \big ]> 0, \end{aligned}$$

which means that the configuration $(\mu ,b)$ is not stable. This shows that for any $j\in N$ and any $\theta '_j$, $\theta ''_j\in {{\,\textrm{supp}\,}}\mu _j$, we have

$$\begin{aligned} \pi _j\big (b(\theta '_j, \theta _{-j})\big ) = \pi _j\big (b(\theta ''_j, \theta _{-j})\big ) \end{aligned}$$

(A.1)

for all $\theta _{-j}\in {{\,\textrm{supp}\,}}\mu _{-j}$.

Next suppose that there exist $\theta '_j$, $\theta ''_j\in {{\,\textrm{supp}\,}}\mu _j$ and $\bar{\theta }_{-j}\in {{\,\textrm{supp}\,}}\mu _{-j}$ for some $j\in N$ such that $\pi _k\big (b(\theta '_j, \bar{\theta }_{-j})\big )> \pi _k\big (b(\theta ''_j, \bar{\theta }_{-j})\big )$ for some $k\in N$ with $k\ne j$. Consider suitable mutant types $\widetilde{\theta }^0_j$ and $\widetilde{\theta }_k$ entering the populations j and k, respectively. Let the played focal equilibrium $\widetilde{b}$ satisfy: (1) $\widetilde{b}(\widetilde{\theta }^0_j, \theta _{-j}) = b(\theta ''_j, \theta _{-j})$ for all $\theta _{-j}\in {{\,\textrm{supp}\,}}\mu _{-j}$, and $\widetilde{b}(\widetilde{\theta }_k, \theta _{-k}) = b(\bar{\theta }_k, \theta _{-k})$ for all $\theta _{-k}\in {{\,\textrm{supp}\,}}\mu _{-k}$; (2) $\widetilde{b}(\widetilde{\theta }^0_j, \widetilde{\theta }_k, \bar{\theta }_{-j -k}) = b(\theta '_j, \bar{\theta }_{-j})$ and $\widetilde{b}(\widetilde{\theta }^0_j, \widetilde{\theta }_k, \theta _{-j -k}) = b(\theta ''_j, \bar{\theta }_k, \theta _{-j -k})$ for any $\theta _{-j -k}\ne \bar{\theta }_{-j -k}$. Then by (A.1), the equality $\varPi _{\widetilde{\theta }^0_j}(\widetilde{\mu }^{\varepsilon };\widetilde{b}) = \varPi _{\theta ''_j}(\widetilde{\mu }^{\varepsilon };\widetilde{b})$ is true for any vector $\varepsilon$ of the population shares of the two mutant types. Moreover, our assumptions about $\widetilde{b}$ also lead to $\varPi _{\widetilde{\theta }_k}(\widetilde{\mu }^{\varepsilon };\widetilde{b}) > \varPi _{\bar{\theta }_k}(\widetilde{\mu }^{\varepsilon };\widetilde{b})$, and hence the configuration $(\mu ,b)$ is not stable. The claim follows. Now for any $(\theta _1, \ldots , \theta _n)$, $(\theta '_1, \ldots , \theta '_n)\in {{\,\textrm{supp}\,}}\mu$, by applying the claim at most n times, we obtain

$$\begin{aligned} \pi \big ( b(\theta _1, \theta _2, \ldots , \theta _n) \big )&= \pi \big ( b(\theta '_1, \theta _2, \ldots , \theta _n) \big )\\&= \cdots = \pi \big ( b(\theta '_1, \theta '_2, \ldots , \theta '_n) \big ), \end{aligned}$$

as desired. $\square$

Theorem 3.8

Let $(\mu ,b)$ be a stable configuration in $(G,\Gamma _1(\mu ))$, and let $\varphi _{\mu ,b}$ be the aggregate outcome of $(\mu ,b)$. Then $(\mu , b)$ is balanced, and for each $i\in N$,

$$\begin{aligned} \varPi _{\bar{\theta }_i}(\mu ;b) = \pi _i\big ( b(\theta ) \big ) = \pi _i(\varphi _{\mu ,b}) \end{aligned}$$

for any $\bar{\theta }_i\in {{\,\textrm{supp}\,}}\mu _i$ and any $\theta \in {{\,\textrm{supp}\,}}\mu$.

Proof

Since $(\mu ,b)$ is stable, by Lemma 3.7, we let $v^* = \pi \big ( b(\theta ) \big )$ for $\theta \in {{\,\textrm{supp}\,}}\mu$. Then for each $i\in N$ and any $\bar{\theta }_i\in {{\,\textrm{supp}\,}}\mu _i$, the equality $\varPi _{\bar{\theta }_i}(\mu ;b) = v^*_i$ is obvious, and thus $(\mu , b)$ is balanced. Finally, for any $i\in N$, we have

$$\begin{aligned} \pi _i(\varphi _{\mu ,b}) = \sum _{a\in A}\varphi _{\mu ,b}(a) \pi _i(a) = \sum _{\theta \in {{\,\textrm{supp}\,}}\mu }\mu (\theta ) \sum _{a\in A} \bigg ( \prod _{s\in N}b_s(\theta )(a_s) \bigg )\pi _i(a) = v^*_i \end{aligned}$$

since $\pi _i\big ( b(\theta ) \big ) = \sum _{a\in A} \big ( \prod _{s\in N}b_s(\theta )(a_s) \big ) \pi _i(a)$ for all $\theta \in {{\,\textrm{supp}\,}}\mu$. $\square$

Theorem 3.9

Let G be a two-player game, and suppose that $(a^*_1,a^*_2)$ is Pareto efficient with respect to $\pi$. If $(a^*_1,a^*_2)$ is a strict Nash equilibrium of G, then $(a^*_1,a^*_2)$ is stable in $(G,\Gamma _1(\mu ))$ for some $\mu \in {\mathcal {M}}(\Theta ^2)$.

Proof

Let $(a^*_1,a^*_2)$ be a strict Nash equilibrium of G, and suppose that it is not a stable strategy profile. We shall show that $(a^*_1,a^*_2)$ is not Pareto efficient with respect to $\pi$. To see this, consider a monomorphic configuration $(\mu , b)$ where each ith population consists of $\theta ^*_i$ for which $a^*_i$ is the strictly dominant strategy. Then $(a^*_1, a^*_2)$ is the aggregate outcome of $(\mu ,b)$, and hence this configuration is not stable under our assumptions on $(a^*_1,a^*_2)$. This means that there exists a mutant sub-profile $\widetilde{\theta }_J$ for some $J\subseteq \{1, 2\}$ such that for every $\bar{\epsilon }\in (0,1)$, these mutants, with some population share vector $\varepsilon \in (0,1)^{|J|}$ satisfying $\Vert \varepsilon \Vert \in (0,\bar{\epsilon })$, can play an equilibrium $\widetilde{b}\in B_1(\widetilde{\mu }^{\varepsilon };b)$ to outperform the incumbents, that is, $\varPi _{\widetilde{\theta }_j}(\widetilde{\mu }^{\varepsilon };\widetilde{b})\ge \varPi _{\theta ^*_j}(\widetilde{\mu }^{\varepsilon };\widetilde{b})$ for all $j\in J$, with strict inequality for some j.

In the case when $|J|=1$, it is clear that mutants have no fitness advantage since $(a^*_1,a^*_2)$ is a strict Nash equilibrium. Let $J = \{1,2\}$, and suppose that $(\widetilde{\theta }_1, \widetilde{\theta }_2)$ is a mutant pair having an evolutionary advantage. For $\varepsilon \in (0,1)^2$ and $\widetilde{b}\in B_1(\widetilde{\mu }^{\varepsilon };b)$, the post-entry average fitnesses of $\theta ^*_i$ and $\widetilde{\theta }_i$ are, respectively,

$$\begin{aligned} \varPi _{\theta ^*_i}(\widetilde{\mu }^{\varepsilon };\widetilde{b}) = (1-\varepsilon _{-i}) \pi _i(a^*_1,a^*_2) + \varepsilon _{-i} \pi _i\big ( a^*_i, \widetilde{b}_{-i}(\theta ^*_i, \widetilde{\theta }_{-i}) \big ) \end{aligned}$$

and

$$\begin{aligned} \varPi _{\widetilde{\theta }_i}(\widetilde{\mu }^{\varepsilon };\widetilde{b}) = (1-\varepsilon _{-i}) \pi _i\big ( \widetilde{b}_i(\widetilde{\theta }_i, \theta ^*_{-i}), a^*_{-i} \big ) + \varepsilon _{-i} \pi _i\big ( \widetilde{b}(\widetilde{\theta }_1,\widetilde{\theta }_2) \big ). \end{aligned}$$

Using the instability assumption on $(a^*_1,a^*_2)$, we gradually reduce $\bar{\epsilon }$ to 0, and then the sequence of the norms of the corresponding population share vectors converges to 0. We can choose a sequence $\{\widetilde{b}^t\}$ from the corresponding focal equilibria such that one of the following three cases occurs. To complete the proof, we will show that $(a^*_1,a^*_2)$ is Pareto dominated in any one of these cases.

Case 1: $\widetilde{b}^t_i(\widetilde{\theta }_i, \theta ^*_{-i}) = a^*_i$ for each i and each t. Since $(\widetilde{\theta }_1, \widetilde{\theta }_2)$ has an evolutionary advantage, it follows that each $\widetilde{b}^t(\widetilde{\theta }_1, \widetilde{\theta }_2)$ Pareto dominates $(a^*_1, a^*_2)$.

Case 2: $\widetilde{b}^t_i(\widetilde{\theta }_i, \theta ^*_{-i})\ne a^*_i$ and $\widetilde{b}^t_{-i}(\theta ^*_i, \widetilde{\theta }_{-i}) = a^*_{-i}$ for fixed i and for every t. Without loss of generality, suppose that $i = 1$. Let $\widetilde{b}^t_1(\widetilde{\theta }_1,\theta ^*_2) = (1-\zeta ^t_1)a^*_1 + \zeta ^t_1\sigma ^t_1$, where $\sigma ^t_1\in \Delta ( A_1{{\setminus}} \{a^*_1\} )$ and $\zeta ^t_1\in (0,1]$ for all t. Since $(\widetilde{\theta }_1, \widetilde{\theta }_2)$ has an evolutionary advantage and $(a^*_1,a^*_2)$ is a strict Nash equilibrium, we have

$$\begin{aligned} \frac{\varepsilon ^t_2}{1-\varepsilon ^t_2}\ge \frac{\zeta ^t_1 [ \pi _1(a^*_1,a^*_2) - \pi _1(\sigma ^t_1,a^*_2) ]}{\pi _1\big ( \widetilde{b}^t(\widetilde{\theta }_1, \widetilde{\theta }_2) \big ) - \pi _1(a^*_1,a^*_2)}> 0 \end{aligned}$$

in which $\pi _1\big ( \widetilde{b}^t(\widetilde{\theta }_1, \widetilde{\theta }_2) \big )> \pi _1(a^*_1,a^*_2)$ and

$$\begin{aligned} \zeta ^t_1[ \pi _2(a^*_1,a^*_2) - \pi _2(\sigma ^t_1,a^*_2) ]\ge \pi _2(a^*_1,a^*_2) - \pi _2\big ( \widetilde{b}^t(\widetilde{\theta }_1, \widetilde{\theta }_2) \big ) \end{aligned}$$

for every t. If we let $\pi _2(a^*_1,a^*_2)> \pi _2\big ( \widetilde{b}^t(\widetilde{\theta }_1, \widetilde{\theta }_2) \big )$ for all t, then $\pi _2(a^*_1,a^*_2)> \pi _2(\sigma ^t_1,a^*_2)$ for all t; otherwise, $(a^*_1, a^*_2)$ is Pareto dominated by $\widetilde{b}^t(\widetilde{\theta }_1, \widetilde{\theta }_2)$ for some t, as desired. Now, in the case when $\pi _2(a^*_1,a^*_2)> \pi _2\big ( \widetilde{b}^t(\widetilde{\theta }_1, \widetilde{\theta }_2) \big )$ for all t, we define

$$\begin{aligned} \kappa = \min \left\{ \, \frac{\pi _1(a^*_1,a^*_2) - \pi _1(\sigma ^t_1,a^*_2)}{\pi _2(a^*_1,a^*_2) - \pi _2(\sigma ^t_1,a^*_2)} \biggm | \sigma ^t_1\in \Delta ( A_1{\setminus}\{a^*_1\} ), \ t\in {\mathbb {Z}}^{+} \,\right\} , \end{aligned}$$

and we can deduce

$$\begin{aligned} \frac{\varepsilon ^t_2}{1-\varepsilon ^t_2}\ge \frac{\kappa \big [ \pi _2(a^*_1,a^*_2) - \pi _2\big ( \widetilde{b}^t(\widetilde{\theta }_1, \widetilde{\theta }_2) \big ) \big ]}{\pi _1\big ( \widetilde{b}^t(\widetilde{\theta }_1, \widetilde{\theta }_2) \big ) - \pi _1(a^*_1,a^*_2)}> 0 \end{aligned}$$

for all t. Since $\bar{\epsilon }$ converges to 0, we obtain

$$\begin{aligned} \lim _{t\rightarrow \infty } \frac{\pi _2(a^*_1,a^*_2) - \pi _2\big ( \widetilde{b}^t(\widetilde{\theta }_1, \widetilde{\theta }_2) \big )}{\pi _1\big ( \widetilde{b}^t(\widetilde{\theta }_1, \widetilde{\theta }_2) \big ) - \pi _1(a^*_1,a^*_2)} = 0. \end{aligned}$$

If $\pi \big ( \widetilde{b}^t(\widetilde{\theta }_1, \widetilde{\theta }_2) \big )$ does not converge to $\pi (a^*_1,a^*_2)$, then by applying the fact that the payoff region $\pi \big ( \prod _{i=1}^{2}\Delta (A_i) \big )$ is compact, there exists a strategy profile $(\widetilde{\sigma }_1, \widetilde{\sigma }_2)$ such that $\pi (\widetilde{\sigma }_1, \widetilde{\sigma }_2)$ is a limit point of the set $\{\, \pi \big ( \widetilde{b}^t(\widetilde{\theta }_1, \widetilde{\theta }_2) \big ) \mid t\in {\mathbb {Z}}^{+} \,\}$, and it Pareto dominates $(a^*_1,a^*_2)$ in terms of $\pi _1(\widetilde{\sigma }_1, \widetilde{\sigma }_2)> \pi _1(a^*_1,a^*_2)$ and $\pi _2(\widetilde{\sigma }_1, \widetilde{\sigma }_2) = \pi _2(a^*_1,a^*_2)$, as desired. Otherwise, by summing up the above, there exists a sequence $\{ \pi \big ( \widetilde{b}^t(\widetilde{\theta }_1, \widetilde{\theta }_2) \big ) \}$ converging to $\pi (a_1^*, a_2^*)$ with $\pi _1\big ( \widetilde{b}^t(\widetilde{\theta }_1, \widetilde{\theta }_2) \big )> \pi _1(a_1^*, a_2^*)$ and $\pi _2(a_1^*, a_2^*)> \pi _2\big ( \widetilde{b}^t(\widetilde{\theta }_1, \widetilde{\theta }_2) \big )$ for all t; moreover, the curve connecting the sequence has a horizontal tangent line at $\pi (a_1^*, a_2^*)$. If $\pi (a_1^*, a_2^*)$ lies on the Pareto frontier of $\pi \big ( \prod _{i=1}^{2}\Delta (A_i) \big )$, then intuitively it seems that there exists a strictly convex subregion of $\pi \big ( \prod _{i=1}^{2}\Delta (A_i) \big )$ including this sequence, which contradicts the shape of a noncooperative payoff region. Therefore $(a_1^*, a_2^*)$ should not be a Pareto-efficient strategy profile. This can be formally proved using the properties of extreme points of a noncooperative payoff region; see (Tu and Juang 2017).

Case 3: $\widetilde{b}^t_i(\widetilde{\theta }_i, \theta ^*_{-i})\ne a^*_i$ for all i and all t. For each $i\in \{1,2\}$, let $\widetilde{b}^t_i(\widetilde{\theta }_i,\theta ^*_{-i}) = (1-\xi ^t_i)a^*_i + \xi ^t_i\sigma ^t_i$, where $\sigma ^t_i\in \Delta ( A_i{{\setminus}} \{a^*_i\} )$ and $\xi ^t_i\in (0,1]$ for all t. Note that $(a^*_1,a^*_2)$ is a strict Nash equilibrium, that the mutant pair $(\widetilde{\theta }_1, \widetilde{\theta }_2)$ has an evolutionary advantage, and that the corresponding norm $\Vert \varepsilon ^t\Vert$ converges to 0. Thus, by comparing the post-entry average fitnesses of $\theta ^*_i$ and $\widetilde{\theta }_i$, we obtain

$$\begin{aligned} \frac{\varepsilon ^t_{-i}}{1 - \varepsilon ^t_{-i}}\ge \frac{\xi ^t_i[\pi _i(a^*_1, a^*_2) - \pi _i(\sigma ^t_i, a^*_{-i})]}{\pi _i\big ( \widetilde{b}^t(\widetilde{\theta }_1, \widetilde{\theta }_2) \big ) - \pi _i(a^*_1, a^*_2) + \xi ^t_{-i}[\pi _i(a^*_1, a^*_2) - \pi _i(a^*_i, \sigma ^t_{-i})]}> 0 \end{aligned}$$

for all i and all t, and it follows that

$$\begin{aligned} \frac{1}{\xi ^t_i}\big [ \pi _i\big ( \widetilde{b}^t(\widetilde{\theta }_1, \widetilde{\theta }_2) \big ) - \pi _i(a^*_1, a^*_2) \big ]\rightarrow \infty \quad \text {or} \quad \frac{\xi ^t_{-i}}{\xi ^t_i}[\pi _i(a^*_1, a^*_2) - \pi _i(a^*_i,\sigma ^t_{-i})]\rightarrow \infty \end{aligned}$$

for each $i\in \{1,2\}$. We discuss all the possibilities. If

$$\begin{aligned} \frac{1}{\xi ^t_1}\big [ \pi _1\big ( \widetilde{b}^t(\widetilde{\theta }_1, \widetilde{\theta }_2) \big ) - \pi _1(a^*_1, a^*_2) \big ]\rightarrow \infty \quad \text {and} \quad \frac{1}{\xi ^t_2}\big [ \pi _2\big ( \widetilde{b}^t(\widetilde{\theta }_1, \widetilde{\theta }_2) \big ) - \pi _2(a^*_1, a^*_2) \big ]\rightarrow \infty \end{aligned}$$

occur simultaneously, then $(a^*_1, a^*_2)$ is Pareto dominated. Next, because $\xi ^t_2/\xi ^t_1\rightarrow \infty$ and $\xi ^t_1/\xi ^t_2\rightarrow \infty$ cannot occur simultaneously, the remaining possibility is that

$$\begin{aligned} \frac{1}{\xi ^t_i}\big [ \pi _i\big ( \widetilde{b}^t(\widetilde{\theta }_1, \widetilde{\theta }_2) \big ) - \pi _i(a^*_1, a^*_2) \big ]\rightarrow \infty \quad \text {and} \quad \frac{\xi ^t_i}{\xi ^t_{-i}}[\pi _{-i}(a^*_1, a^*_2) - \pi _{-i}(\sigma ^t_i, a^*_{-i})]\rightarrow \infty , \end{aligned}$$

where either $i = 1$ or $i = 2$. In each case, we have $\xi ^t_1\rightarrow 0$ and $\xi ^t_2\rightarrow 0$.

Without loss of generality, let $i = 1$. It is enough to consider the sequence $\{ \widetilde{b}^t \}$ satisfying $\pi _1\big ( \widetilde{b}^t(\widetilde{\theta }_1, \widetilde{\theta }_2) \big )> \pi _1(a^*_1, a^*_2)$. If the inequality $\pi _2\big ( \widetilde{b}^t(\widetilde{\theta }_1, \widetilde{\theta }_2) \big )\ge \pi _2(a^*_1, a^*_2)$ also holds for some t, then $(a^*_1, a^*_2)$ is Pareto dominated by some $\widetilde{b}^t(\widetilde{\theta }_1, \widetilde{\theta }_2)$, as desired. Hence we only have to suppose that $\pi _2(a^*_1, a^*_2)> \pi _2\big ( \widetilde{b}^t(\widetilde{\theta }_1, \widetilde{\theta }_2) \big )$ for all t. From comparing the post-entry average fitness of $\theta ^*_2$ with that of $\widetilde{\theta }_2$, we know that for each t,

$$\begin{aligned} \pi _2\big ( \widetilde{b}^t(\widetilde{\theta }_1, \widetilde{\theta }_2) \big ) - \pi _2(a^*_1, a^*_2) + \xi ^t_1[\pi _2(a^*_1, a^*_2) - \pi _2(\sigma ^t_1, a^*_2)]> 0. \end{aligned}$$

Then

$$\begin{aligned} \frac{\xi ^t_1[\pi _2(a^*_1, a^*_2) - \pi _2(\sigma ^t_1, a^*_2)]}{\pi _1\big ( \widetilde{b}^t(\widetilde{\theta }_1, \widetilde{\theta }_2) \big ) - \pi _1(a^*_1, a^*_2)}> \frac{\pi _2(a^*_1, a^*_2) - \pi _2\big ( \widetilde{b}^t(\widetilde{\theta }_1, \widetilde{\theta }_2) \big )}{\pi _1\big ( \widetilde{b}^t(\widetilde{\theta }_1, \widetilde{\theta }_2) \big ) - \pi _1(a^*_1, a^*_2)}> 0 \end{aligned}$$

for all t. Since we let $i = 1$ and the term $\pi _2(a^*_1, a^*_2) - \pi _2(\sigma ^t_1, a^*_2)$ is bounded, we conclude that

$$\begin{aligned} \lim _{t\rightarrow \infty } \frac{\pi _2(a^*_1,a^*_2) - \pi _2\big ( \widetilde{b}^t(\widetilde{\theta }_1, \widetilde{\theta }_2) \big )}{\pi _1\big ( \widetilde{b}^t(\widetilde{\theta }_1, \widetilde{\theta }_2) \big ) - \pi _1(a^*_1,a^*_2)} = 0. \end{aligned}$$

Then, as discussed in Case 2 of this proof, the strategy profile $(a^*_1, a^*_2)$ cannot be Pareto efficient. $\square$

Theorem 3.14

Let $(a^*_1, \ldots , a^*_n)$ be a strict union Nash equilibrium in an n-player game G. Then $(a^*_1, \ldots , a^*_n)$ is stable in $(G,\Gamma _1(\mu ))$ for some $\mu \in {\mathcal {M}}(\Theta ^n)$.

Proof

Let $a^* = (a^*_1, \ldots , a^*_n)$ be a strict union Nash equilibrium of G, and let $(\mu , b)$ be a monomorphic configuration where each ith population consists of $\theta _i^*$ for which $a^*_i$ is the strictly dominant strategy, so that $a^*$ is the aggregate outcome of $(\mu , b)$. For any nonempty subset J of N, consider a mutant sub-profile $\widetilde{\theta }_J$ entering with its population share vector $\varepsilon$. To verify stability, we shall show that there exists a uniform invasion barrier $\bar{\epsilon }\in (0,1)$ such that for every $\varepsilon \in (0,1)^{|J|}$ with $\Vert \varepsilon \Vert \in (0,\bar{\epsilon })$, the inequality

$$\begin{aligned} \sum _{i\in J} \varepsilon _i \Big ( \varPi _{\theta _i^*}(\widetilde{\mu }^{\varepsilon };\widetilde{b}) - \varPi _{\widetilde{\theta }_i}(\widetilde{\mu }^{\varepsilon };\widetilde{b}) \Big ) \ge 0 \end{aligned}$$

holds for all $\widetilde{b}\in B_1(\widetilde{\mu }^{\varepsilon };b)$. By direct calculation,

$$\begin{aligned}{} & {} \sum _{i\in J} \varepsilon _i \Big ( \varPi _{\theta _i^*}(\widetilde{\mu }^{\varepsilon };\widetilde{b}) - \varPi _{\widetilde{\theta }_i}(\widetilde{\mu }^{\varepsilon };\widetilde{b}) \Big ) =\sum _{i\in J} \varepsilon _i \prod _{j\in J_{-i}}(1 - \varepsilon _j) \big [ \pi _i(a^*) - \pi _i\big ( \widetilde{b}(\widetilde{\theta }_i, \theta ^*_{-i}) \big ) \big ]\nonumber \\{} & {} \quad + \sum _{i\in J} \sum _{\begin{array}{c} |T|=1\\ T\subseteq J_{-i} \end{array}} \varepsilon _i \varepsilon _T \prod _{j\in J_{-i}{\setminus}T}(1 - \varepsilon _j) \big [ \pi _i\big ( \widetilde{b}(\theta ^*_i, \widetilde{\theta }_T, \theta ^*_{-i -T}) \big ) - \pi _i\big ( \widetilde{b}(\widetilde{\theta }_i, \widetilde{\theta }_T, \theta ^*_{-i -T}) \big ) \big ] \nonumber \\{} & {} \quad + \ldots + \sum _{i\in J} \varepsilon _i \varepsilon _{J_{-i}} \big [ \pi _i\big ( \widetilde{b}(\theta ^*_i, \widetilde{\theta }_{J_{-i}}, \theta ^*_{-J}) \big ) - \pi _i\big ( \widetilde{b}(\widetilde{\theta }_i, \widetilde{\theta }_{J_{-i}}, \theta ^*_{-J}) \big ) \big ], \end{aligned}$$

(A.2)

where $\varepsilon _T$ denotes $\prod _{j\in T}\varepsilon _j$, and it continues to be used.

On the right-hand side of (A.2), for any $i\in J$ and any $T\subseteq J_{-i}$, write the difference between the two fitnesses as

$$\begin{aligned}{} & {} \pi _i\big ( \widetilde{b}(\theta ^*_i, \widetilde{\theta }_T, \theta ^*_{-i -T}) \big ) - \pi _i\big ( \widetilde{b}(\widetilde{\theta }_i, \widetilde{\theta }_T, \theta ^*_{-i -T}) \big )\\{} & {} \quad = \big [ \pi _i\big ( \widetilde{b}(\theta ^*_i, \widetilde{\theta }_T, \theta ^*_{-i -T}) \big ) - \pi _i(a^*) \big ] + \big [ \pi _i(a^*) - \pi _i\big ( \widetilde{b}(\widetilde{\theta }_i, \widetilde{\theta }_T, \theta ^*_{-i -T}) \big ) \big ]. \end{aligned}$$

Then classify the sum in (A.2) according to the number of mutants occurring in those differences between $\pi _i(a^*)$ and $\pi _i\big ( \widetilde{b}(\widetilde{\theta }_H, \theta ^*_{-H}) \big )$, where $i\in J$ and $H\subseteq J$. In this way, any classified part with k mutants in matched tuples is either of the form

$$\begin{aligned}{} & {} \sum _{i\in J} \sum _{\begin{array}{c} |T|=k-1\\ T\subseteq J_{-i} \end{array}} \varepsilon _i \varepsilon _T \prod _{j\in J_{-i}{\setminus}T}(1 - \varepsilon _j) \big [ \pi _i(a^*) - \pi _i\big ( \widetilde{b}(\widetilde{\theta }_i, \widetilde{\theta }_T, \theta ^*_{-i -T}) \big ) \big ]\nonumber \\{} & {} \quad +\sum _{i\in J} \sum _{\begin{array}{c} |T|=k\\ T\subseteq J_{-i} \end{array}} \varepsilon _i \varepsilon _T \prod _{j\in J_{-i}{\setminus}T}(1 - \varepsilon _j) \big [ \pi _i\big ( \widetilde{b}(\theta ^*_i, \widetilde{\theta }_T, \theta ^*_{-i -T}) \big ) - \pi _i(a^*) \big ] \end{aligned}$$

(A.3)

where $1\le k\le |J| - 1$, or of the form $\sum _{i\in J} \varepsilon _J \big [ \pi _i(a^*) - \pi _i\big ( \widetilde{b}(\widetilde{\theta }_J, \theta ^*_{-J}) \big ) \big ]$ whenever $k = |J|$. First, it is clear that

$$\begin{aligned} \sum _{i\in J} \varepsilon _J \big [ \pi _i(a^*) - \pi _i\big ( \widetilde{b}(\widetilde{\theta }_J, \theta ^*_{-J}) \big ) \big ] \ge 0 \end{aligned}$$

for any $\widetilde{b}\in B_1(\widetilde{\mu }^{\varepsilon };b)$, since $a^*$ is a strict union Nash equilibrium of G.

Next, for a given nonempty proper subset S of J, the sum of the terms involving differences between $\pi _i(a^*)$ and $\pi _i\big ( \widetilde{b}(\widetilde{\theta }_S, \theta ^*_{-S}) \big )$, $i\in J$, in (A.3) can also be written as

$$\begin{aligned}{} & {} \sum _{i\in S} \varepsilon _S \prod _{j\in J{\setminus}S}(1 - \varepsilon _j) \big [ \pi _i(a^*) - \pi _i\big ( \widetilde{b}(\widetilde{\theta }_S, \theta ^*_{-S}) \big ) \big ]\nonumber \\{} & {} \quad + \sum _{i\in J{\setminus}S} \varepsilon _i \varepsilon _S \prod _{j\in J_{-i}{\setminus}S}(1 - \varepsilon _j) \big [ \pi _i\big ( \widetilde{b}(\widetilde{\theta }_S, \theta ^*_{-S}) \big ) - \pi _i(a^*) \big ]. \end{aligned}$$

(A.4)

We claim that there exists a uniform invasion barrier $\bar{\epsilon }_S$ such that the sum in (A.4) is greater than or equal to 0 for any $\varepsilon$ with $\Vert \varepsilon \Vert \in (0,\bar{\epsilon }_S)$ and for any $\widetilde{b}\in B_1(\widetilde{\mu }^{\varepsilon };b)$. To show it, suppose that for every $j\in S$, $\widetilde{b}_j(\widetilde{\theta }_S,\theta ^*_{-S}) = (1-\gamma _j)a^*_j + \gamma _j\sigma _j$, where $\sigma _j\in \Delta ( A_j{{\setminus}} \{a^*_j\} )$ and $\gamma _j\in [0,1]$. Then the term in the expansion of the fitness $\pi _i\big ( \widetilde{b}(\widetilde{\theta }_S, \theta ^*_{-S}) \big )$ of each player i can be represented as follows:

$$\begin{aligned} \prod _{j\in S} c_j(\gamma _j, \alpha _j) \pi _i\Big ( \big (x_j(\gamma _j, \alpha _j)\big )_{j\in S}, a^*_{-S} \Big ), \end{aligned}$$

where the index $\alpha _j$ takes the value 0 or 1; for any given $\alpha _j$, the coefficient function $c_j$ is defined by letting $c_j(\gamma _j, \alpha _j)$ be $(1 - \gamma _j)^{\alpha _j} \gamma _j^{1 - \alpha _j}$ if $0< \gamma _j< 1$ and be 1 otherwise; the strategy function $x_j$ is defined by

$$\begin{aligned} x_j(\gamma _j, \alpha _j) = {\left\{ \begin{array}{ll} a^*_j &{} \text {if}\, \gamma _j = 0,\\ \sigma _j &{} \text {if}\, \gamma _j = 1,\\ \sigma _j &{} \text {if}\, \gamma _j\in (0, 1)\, \text {and}\, \alpha _j = 0,\\ a^*_j &{} \text {if}\, \gamma _j\in (0, 1)\, \text {and}\, \alpha _j = 1. \end{array}\right. } \end{aligned}$$

Thus, for any one term in the expansion of the difference $\pi _i(a^*) - \pi _i\big ( \widetilde{b}(\widetilde{\theta }_S, \theta ^*_{-S}) \big )$, we can write it as

$$\begin{aligned} \prod _{j\in S} c_j(\gamma _j, \alpha _j) \Big [ \pi _i(a^*) - \pi _i\Big ( \big (x_j(\gamma _j, \alpha _j)\big )_{j\in S}, a^*_{-S} \Big ) \Big ] \end{aligned}$$

for some $(\alpha _j)_{j\in S}\in \{0, 1\}^{|S|}$. Similarly for the expansion of the difference $\pi _i\big ( \widetilde{b}(\widetilde{\theta }_S, \theta ^*_{-S}) \big ) - \pi _i(a^*)$. Collecting the terms that have the same coefficient $\prod _{j\in S} c_j(\gamma _j, \alpha _j)$ in (A.4), we obtain the sum

$$\begin{aligned}{} & {} \varepsilon _S \prod _{j\in J{\setminus}S}(1 - \varepsilon _j) \prod _{j\in S} c_j(\gamma _j, \alpha _j) \Bigg \{ \sum _{i\in S} \Big [ \pi _i(a^*) - \pi _i\Big ( \big (x_j(\gamma _j, \alpha _j)\big )_{j\in S}, a^*_{-S} \Big ) \Big ]\nonumber \\{} & {} \quad + \sum _{i\in J{\setminus}S} \Big ( \frac{\varepsilon _i}{1 - \varepsilon _i} \Big ) \Big [ \pi _i\Big ( \big (x_j(\gamma _j, \alpha _j)\big )_{j\in S}, a^*_{-S} \Big ) - \pi _i(a^*) \Big ] \Bigg \}. \end{aligned}$$

(A.5)

Of course, the sum in (A.5) is equal to 0 if $x_j(\gamma _j, \alpha _j) = a^*_j$ for all $j\in S$. Otherwise, since $a^*$ is a strict union Nash equilibrium of G, there exists $m > 0$ such that for every $\sigma _j$ satisfying $\sigma _j(a^*_j) = 0$,

$$\begin{aligned} \sum _{i\in S} \Big [ \pi _i(a^*) - \pi _i\Big ( \big (x_j(\gamma _j, \alpha _j)\big )_{j\in S}, a^*_{-S} \Big ) \Big ]\ge m. \end{aligned}$$

Furthermore, the difference $\pi _i\Big ( \big (x_j(\gamma _j, \alpha _j)\big )_{j\in S}, a^*_{-S} \Big ) - \pi _i(a^*)$ is bounded for every $i\in J{\setminus}S$, and we know that $\{0, 1\}^{|S|}$ is a finite set. Therefore, we can find a uniform invasion barrier $\bar{\epsilon }_S$ such that for any $\varepsilon$ with $\Vert \varepsilon \Vert \in (0,\bar{\epsilon }_S)$ and any $\widetilde{b}\in B_1(\widetilde{\mu }^{\varepsilon };b)$, the sum in (A.4) is greater than or equal to 0. This completes the proof since the number of subsets of J is finite.⁴¹$\square$

Lemma 4.1

Let $(\mu ,s)$ b e a balanced configuration in $(G,\Gamma _0(\mu ))$ with the aggregate outcome x. Then

$$\begin{aligned} \varPi _{\theta _i}(\mu ;s) = \pi _i(x) \end{aligned}$$

for every $i\in N$ and every $\theta _i\in {{\,\textrm{supp}\,}}\mu _i$.

Proof

Since $(\mu ,s)$ is a balanced configuration, we have that for each $i\in N$ and for any fixed $\theta _i\in {{\,\textrm{supp}\,}}\mu _i$,

$$\begin{aligned} \pi _i\big (s_i(\theta _i),x_{-i}\big ) = \varPi _{\theta _i}(\mu ;s) = \varPi _{\theta '_i}(\mu ;s) = \pi _i\big (s_i(\theta '_i),x_{-i}\big ) \end{aligned}$$

for every $\theta '_i\in {{\,\textrm{supp}\,}}\mu _i$. This implies that

$$\begin{aligned} \pi _i(x)=\sum _{\theta '_i\in {{\,\textrm{supp}\,}}\mu _i}\mu _i(\theta '_i)\pi _i\big (s_i(\theta '_i),x_{-i}\big ) =\pi _i\big (s_i(\theta _i),x_{-i}\big )=\varPi _{\theta _i}(\mu ;s), \end{aligned}$$

and the proof is complete. $\square$

Theorem 4.5

If a configuration $(\mu ,s)$ is stable in $(G,\Gamma _0(\mu ))$, then the aggregate outcome of $(\mu ,s)$ is a Nash equilibrium of G.

Proof

Let x be the aggregate outcome of a stable configuration $(\mu ,s)$, and suppose that x is not a Nash equilibrium. Then there exist $k\in N$ and $a_k\in A_k$ such that $\pi _k(a_k,x_{-k}) > \pi _k(x)$. We have seen in the proof of Lemma 4.1 that a balanced configuration implies that $\pi _i(x) = \pi _i\big (s_i(\theta _i), x_{-i}\big )$ for all $i\in N$ and all $\theta _i\in {{\,\textrm{supp}\,}}\mu _i$, and thus we obtain

$$\begin{aligned} \pi _k(a_k, x_{-k}) > \pi _k\big (s_k(\theta _k), x_{-k}\big ) \end{aligned}$$

for all $\theta _k\in {{\,\textrm{supp}\,}}\mu _k$. Consider a mutant type $\widetilde{\theta }_k\in {({{\,\textrm{supp}\,}}\mu _k)}^{\textsf{c}}$ appearing in the kth population with its population share $\varepsilon$, for which $a_k$ is the strictly dominant strategy. Then for any post-entry equilibrium $\widetilde{s}$, we get $\widetilde{s}_k(\widetilde{\theta }_k) = a_k$, and the average fitnesses of $\widetilde{\theta }_k$ and any $\theta _k\in {{\,\textrm{supp}\,}}\mu _k$ are, respectively,

$$\begin{aligned} \varPi _{\widetilde{\theta }_k}(\widetilde{\mu }^{\varepsilon };\widetilde{s}) = \pi _k(a_k,\widetilde{x}_{-k}) \quad \text {and} \quad \varPi _{\theta _k}(\widetilde{\mu }^{\varepsilon };\widetilde{s}) = \pi _k\big ( \widetilde{s}_k(\theta _k),\widetilde{x}_{-k} \big ), \end{aligned}$$

where $\widetilde{x} = x(\widetilde{\mu }^{\varepsilon },\widetilde{s})$. Since $(\mu ,s)$ is a stable configuration and $\pi _k$ is a continuous function on $\prod _{i\in N}\Delta (A_i)$, it follows that as long as $\bar{\eta }$ and $\varepsilon$ are sufficiently small, the inequality

$$\begin{aligned} \pi _k(a_k,\widetilde{x}_{-k})> \pi _k \big ( \widetilde{s}_k(\theta _k),\widetilde{x}_{-k} \big ) \end{aligned}$$

holds for all $\widetilde{s}\in B_0^{\bar{\eta }}(\widetilde{\mu }^{\varepsilon };s)$ and all $\theta _k\in {{\,\textrm{supp}\,}}\mu _k$ (see the discussion after Definition 4.3). Thus we have arrived at a contradiction. $\square$

Theorem 4.6

If $\sigma ^*$ is a strictly perfect equilibrium of G, then it is stable in $(G,\Gamma _0(\mu ))$ for some $\mu \in {\mathcal {M}}(\Theta ^n)$.

Proof

Let $\sigma ^*$ be supported by a monomorphic configuration $(\mu , s)$ with materialist preferences. Then $s(\theta ) = \sigma ^*$ for $\theta \in {{\,\textrm{supp}\,}}\mu$. Suppose that $(\mu , s)$ is not a stable configuration under no observability. Then, since each population consists of materialist preferences having a fitness advantage, there exist a mutant sub-profile $\widetilde{\theta }_J$ for some $J\subseteq N$ and a positive number $\bar{\eta }$ for which we can choose a sequence $\{ \varepsilon ^t \}$ with $\Vert \varepsilon ^t \Vert$ converging to 0 such that the corresponding nearby set $B_0^{\bar{\eta }}(\widetilde{\mu }^{\varepsilon ^t};s) = \varnothing$ for all t. This means that for any t and any $\widetilde{s}^t\in B_0(\widetilde{\mu }^{\varepsilon ^t})$, we have $d(\widetilde{s}^t(\theta ), s(\theta ))> \bar{\eta }$ for $\theta \in {{\,\textrm{supp}\,}}\mu$. Then for every t, we can conclude that there exist $\widetilde{x}^t_J\in \prod _{j\in J}\Delta (A_j)$ and a mutant sub-profile $\widetilde{\theta }^t_J$ defined by always adopting $\widetilde{x}^t_J$ with its population share vector $\varepsilon ^t$ for which the corresponding nearby set within $\bar{\eta }$ is still empty. This would imply that for any $\widetilde{s}^t\in B_0(\widetilde{\mu }^{\widetilde{x}^t_J})$, where $\widetilde{\mu }^{\widetilde{x}^t_J}$ denotes the type distribution after the entry of $\widetilde{\theta }^t_J$, there exist $i\in N$ and $a_i\in {{\,\textrm{supp}\,}}\sigma ^*_i$ such that

$$\begin{aligned} \pi _i(a_i, \widehat{y}^t_{-i}) < \pi _i\big (\widetilde{s}^t_i(\theta _i), \widehat{y}^t_{-i} \big ), \end{aligned}$$

(A.6)

where $\widehat{y}^t$ is defined by $\widehat{y}^t_i = (1 - \varepsilon ^t_i)\widetilde{s}^t_i(\theta _i) + \varepsilon ^t_i \widetilde{x}^t_i$ if $i\in J$, and $\widehat{y}^t_i = \widetilde{s}^t_i(\theta _i)$ if $i\in N{{\setminus}} J$.

Next, for each t and each $i\in N$, define $\eta ^t_i:A_i\rightarrow {\mathbb {R}}$ by

$$\begin{aligned} \eta ^t_i(a_i) = {\left\{ \begin{array}{ll} \varepsilon ^t_i \widetilde{x}^t_i(a_i) &{} \text {if}\, i\in J\, \text {and}\, \widetilde{x}^t_i(a_i)> 0,\\ 0 &{} \text {otherwise.} \end{array}\right. } \end{aligned}$$

Let $\eta ^t = (\eta ^t_1, \ldots , \eta ^t_n)$ be an error function for all players. Suppose that $\sigma ^*$ is a strictly perfect equilibrium of G, and consider the sequence $\{ G(\eta ^t) \}$ of the perturbed games. Then it is also true that there exists a sequence $\{ \sigma ^t \}$, where each $\sigma ^t$ is a Nash equilibrium of $G(\eta ^t)$, such that $\sigma ^t$ converges to $\sigma ^*$ as t tends to infinity. Therefore for sufficiently large t and for each $i\in N$, we have

$$\begin{aligned} \pi _i(a_i, \sigma ^t_{-i})\ge \pi _i(b_i, \sigma ^t_{-i}) \end{aligned}$$

(A.7)

for all $a_i\in {{\,\textrm{supp}\,}}\sigma ^*_i$ and $b_i\in A_i$. Note that for any Nash equilibrium $\sigma ^t$ of the perturbed game $G(\eta ^t)$, there exists $\widetilde{s}^t\in B_0(\widetilde{\mu }^{\widetilde{x}^t_J})$ such that $\sigma ^t_i = (1 - \varepsilon ^t_i)\widetilde{s}^t_i(\theta _i) + \varepsilon ^t_i \widetilde{x}^t_i$ if $i\in J$, and $\sigma ^t_i = \widetilde{s}^t_i(\theta _i)$ if $i\in N{{\setminus}} J$. This implies that (A.7) is contradictory to (A.6), as desired. $\square$

Theorem 4.8

If $\sigma ^*$ is a quasi-strict equilibrium of G, then it is stable in $(G,\Gamma _0(\mu ))$ for some $\mu \in {\mathcal {M}}(\Theta ^n)$.

Proof

Let $\sigma ^*$ be a quasi-strict equilibrium of $G = (N, A, \pi )$. Then ${{\,\textrm{supp}\,}}\sigma ^*_i$ is the set of all pure best replies of player i against $\sigma ^*_{-i}$. Define the game $\bar{G} = (N, A, \bar{\pi })$ by

$$\begin{aligned} \bar{\pi }_i(a) = {\left\{ \begin{array}{ll} \pi _i(a) &{} \text {if}\, a\in {{\,\textrm{supp}\,}}\sigma ^*,\\ l_i &{} \text {otherwise} \end{array}\right. } \end{aligned}$$

for all $i\in N$ and all $a\in A$, where ${{\,\textrm{supp}\,}}\sigma ^* = \prod _{i\in N} {{\,\textrm{supp}\,}}\sigma _i^*$ and $l_i$ is a constant strictly smaller than $\pi _i(a)$ for all $a\in A$. Then it is obvious that $\sigma ^*$ is a strictly perfect equilibrium of $\bar{G}$.

Consider a monomorphic configuration $(\mu , s)$ consisting of $\theta _1$, ..., $\theta _n$ with the preferences $\bar{\pi }_1$, ..., $\bar{\pi }_n$, respectively, for which $s(\theta ) = \sigma ^*$. Since $\sigma ^*$ is strictly perfect in $\bar{G}$, we see that for any entry of $\widetilde{\theta }_J$ with $\varepsilon \in (0,1)^{|J|}$ and for any $\eta > 0$, as the claim in the proof of Theorem 4.6, there exists $\bar{\epsilon }\in (0, 1)$ such that $B_0^{\eta }(\widetilde{\mu }^{\varepsilon };s)$ is nonempty for all $\varepsilon$ with $\Vert \varepsilon \Vert \in (0, \bar{\epsilon })$. For $\widetilde{s}\in B_0^{\bar{\eta }}(\widetilde{\mu }^{\varepsilon };s)$, where $\bar{\eta }\in [0, \eta )$, define $\widetilde{x}\in \prod _{i\in N}\Delta (A_i)$ by $\widetilde{x}_i = (1 - \varepsilon _i)\widetilde{s}_i(\theta _i) + \varepsilon _i \widetilde{s}_i(\widetilde{\theta }_i)$ if $i\in J$, and $\widetilde{x}_i = \widetilde{s}_i(\theta _i)$ if $i\in N{{\setminus}} J$. Then $\widetilde{x}$ can be made arbitrarily close to $\sigma ^*$ by making $\bar{\eta }$ and $\varepsilon$ sufficiently small. Since $\sigma ^*$ is a quasi-strict equilibrium of G, we can conclude that if there exists $b_j\in A_j{\setminus}{{\,\textrm{supp}\,}}\sigma ^*_j$ for $j\in N$, then when $\bar{\eta }$ and $\varepsilon$ are small enough,

$$\begin{aligned} \pi _j(b_j, \widetilde{x}_{-j}) < \pi _j(a_j, \widetilde{x}_{-j}) \end{aligned}$$

for all $a_j\in {{\,\textrm{supp}\,}}\sigma ^*_j$. Furthermore, since ${{\,\textrm{supp}\,}}\widetilde{s}_i(\theta _i)\subseteq {{\,\textrm{supp}\,}}\sigma ^*_i$ here for any $i\in N$, it follows that $\pi _j(b_j, \widetilde{x}_{-j})< \pi _i(\widetilde{s}_j(\theta _j), \widetilde{x}_{-j})$ as long as $\bar{\eta }$ and $\varepsilon$ are small enough. Therefore, if ${{\,\textrm{supp}\,}}\widetilde{s}_i(\widetilde{\theta }_i)\nsubseteq {{\,\textrm{supp}\,}}\sigma ^*_i$ for some $i\in J$, the rare mutants would fail to invade for all suitable nearby equilibria.

Next, suppose that ${{\,\textrm{supp}\,}}\widetilde{s}_i(\widetilde{\theta }_i)\subseteq {{\,\textrm{supp}\,}}\sigma ^*_i$ for all $i\in J$. Then the game we have to consider is $\hat{G} = (N, {{\,\textrm{supp}\,}}\sigma ^*, \hat{\pi })$, where $\hat{\pi }$ is the restriction of $\pi$ to ${{\,\textrm{supp}\,}}\sigma ^*$. Now, the incumbents can be viewed as individuals endowed with materialist preferences with respect to $\hat{\pi }$, and therefore such invading mutants cannot outperform them. $\square$

Theorem 5.4

Let $a^*$ be a pure-strategy profile in G. If for any $\bar{p}\in (0,1)$ there exists $p\in (\bar{p},1)$ such that $a^*$ is stable for the degree p, then it is weakly Pareto efficient with respect to $\pi$.

Proof

Let $p\in (0, 1)$ be a degree of observability, and suppose that $a^*$ is not weakly Pareto efficient with respect to $\pi$. Let $(\mu ,b,s)$ be a configuration with the aggregate outcome $a^*$. Then $b(\theta )=a^*$ and $s(\theta )=a^*$ for all $\theta \in {{\,\textrm{supp}\,}}\mu$. To prove the theorem, it suffices to show that there exists $\bar{p}\in (0,1)$ such that the configuration $(\mu ,b,s)$ is not stable for any $p\in (\bar{p},1)$. Because $a^*$ is not weakly Pareto efficient, there exists $\sigma \in \prod _{i\in N}\Delta (A_i)$ such that $\pi _i(\sigma )>\pi _i(a^*)$ for all $i\in N$. Consider an indifferent mutant profile $\widetilde{\theta }^0 = (\widetilde{\theta }^0_1, \ldots , \widetilde{\theta }^0_n)$ entering with its population share vector $\varepsilon = (\varepsilon _1, \ldots , \varepsilon _n)$. Suppose that for $(\widetilde{b}, \widetilde{s})\in B_p(\widetilde{\mu }^{\varepsilon })$, the mutants’ strategies satisfy: (1) $\widetilde{b}(\widetilde{\theta }^0)=\sigma$ and $\widetilde{s}(\widetilde{\theta }^0)=a^*$; (2) for any nonempty proper subset T of N and any $\theta _{-T}\in {{\,\textrm{supp}\,}}\mu _{-T}$, the equality $\widetilde{b}_j(\widetilde{\theta }^0_T, \theta _{-T}) = a^*_j$ holds for all $j\in T$. Then the focal set $B_p(\widetilde{\mu }^{\varepsilon };b,s)$ is nonempty for an arbitrary population share vector $\varepsilon \in (0,1)^n$.

Let $(\widetilde{b}, \widetilde{s})\in B_p(\widetilde{\mu }^{\varepsilon };b,s)$ be played with $\widetilde{b}(\widetilde{\theta }^0_T, \theta _{-T}) = a^*$ for any $T\varsubsetneq N$ and any $\theta _{-T}\in {{\,\textrm{supp}\,}}\mu _{-T}$. Then for each $i\in N$, the difference $\varPi _{\widetilde{\theta }^0_i}(\widetilde{\mu }^{\varepsilon };\widetilde{b}, \widetilde{s}) - \varPi _{\theta _i}(\widetilde{\mu }^{\varepsilon };\widetilde{b}, \widetilde{s})$ in average fitnesses between the mutant $\widetilde{\theta }^0_i$ and an incumbent $\theta _i$ is

$$\begin{aligned} \widetilde{\mu }^{\varepsilon }_{-i}(\widetilde{\theta }^0_{-i}) \bigg ( p^n [\pi _i(\sigma ) - \pi _i(a^*)] + \sum _{\varnothing \ne T\subseteq N} p^{n-|T|}(1-p)^{|T|} [\pi _i(a^*_T,\sigma _{-T}) - \pi _i(a^*)] \bigg ). \end{aligned}$$

Since $\pi _i(\sigma ) > \pi _i(a^*)$ for all $i\in N$ and the game G is finite, there exists $\bar{p}\in (0,1)$ such that for any $p\in (\bar{p},1)$, the inequality $\varPi _{\widetilde{\theta }^0_i}(\widetilde{\mu }^{\varepsilon };\widetilde{b}, \widetilde{s}) > \varPi _{\theta _i}(\widetilde{\mu }^{\varepsilon };\widetilde{b}, \widetilde{s})$ holds for all $i\in N$ and all $\theta _i\in {{\,\textrm{supp}\,}}\mu _i$, regardless of the population shares of the mutants. $\square$

Theorem 5.5

Let $a^*$ be a pure-strategy profile in G. If for any $\bar{p}\in (0,1)$ there exists $p\in (0,\bar{p})$ such that $a^*$ is stable for the degree p, then it is a Nash equilibrium of G.

Proof

Suppose that $p\in (0,1)$, and that $a^*$ is not a Nash equilibrium of G. Let $(\mu ,b,s)$ be a configuration with the aggregate outcome $a^*$. Then $b(\theta ) = a^*$ and $s(\theta ) = a^*$ for all $\theta \in {{\,\textrm{supp}\,}}\mu$. To prove the theorem, it suffices to show that there exists $\bar{p}\in (0,1)$ such that the configuration $(\mu ,b,s)$ is not stable for any $p\in (0, \bar{p})$. Since $a^*$ is not a Nash equilibrium, there exists a strategy $a_k\in A_k$ for some $k\in N$ such that $\pi _k(a_k,a_{-k}^*)> \pi _k(a^*)$. Consider a mutant type $\widetilde{\theta }_k$ entering the kth population with its population share $\varepsilon$, for which $a_k$ is the strictly dominant strategy. Let $p\in (0,1)$ be given. If there exists $\eta > 0$ such that for any $\bar{\epsilon }\in (0, 1)$, the nearby set $B^{\eta }_p(\widetilde{\mu }^{\varepsilon };b,s)$ is empty for some $\varepsilon \in (0, \bar{\epsilon })$, then $(\mu ,b,s)$ is not stable for the degree p, as desired. Now, for any $\eta > 0$, we let $B^{\eta }_p(\widetilde{\mu }^{\varepsilon };b,s)$ be nonempty for all sufficiently small $\varepsilon > 0$.

For $(\widetilde{b}, \widetilde{s})\in B^{\eta }_p(\widetilde{\mu }^{\varepsilon };b,s)$, the post-entry average fitness of an individual $\widehat{\theta }_k$ in the kth population is

$$\begin{aligned}{} & {} \varPi _{\widehat{\theta }_k}(\widetilde{\mu }^{\varepsilon };\widetilde{b}, \widetilde{s}) = (1-p)^n \sum _{\theta '_{-k}\in {{\,\textrm{supp}\,}}\mu _{-k}} \mu _{-k}(\theta '_{-k}) \pi _k\big ( \widetilde{s}(\widehat{\theta }_k, \theta '_{-k}) \big )\\{} & {} \quad + p \sum _{\theta '_{-k}\in {{\,\textrm{supp}\,}}\mu _{-k}} \mu _{-k}(\theta '_{-k}) \sum _{T\varsubsetneq N} p^{n-1-|T|}(1-p)^{|T|} \pi _k\big ( \widetilde{s}(\widehat{\theta }_k,\theta '_{-k})_T, \widetilde{b}(\widehat{\theta }_k,\theta '_{-k})_{-T} \big ), \end{aligned}$$

where we note that there are no entrants except the mutants in the kth population. Because $\pi _k(a_k,a_{-k}^*)> \pi _k(a^*)$ and $\pi _k$ is continuous on $\prod _{i\in N}\Delta (A_i)$, there exist $m> 0$ and $\bar{\eta }> 0$ such that for any $(\widetilde{b}, \widetilde{s})\in B^{\bar{\eta }}_p(\widetilde{\mu }^{\varepsilon };b,s)$,

$$\begin{aligned} \pi _k\big ( \widetilde{s}(\widetilde{\theta }_k, \theta '_{-k}) \big ) - \pi _k\big ( \widetilde{s}(\theta _k, \theta '_{-k}) \big ) = \pi _k\big ( a_k, \widetilde{s}_{-k}(\theta '_{-k}) \big ) - \pi _k\big ( \widetilde{s}(\theta _k, \theta '_{-k}) \big )\ge m \end{aligned}$$

for all $\theta _k\in {{\,\textrm{supp}\,}}\mu _k$ and all $\theta '_{-k}\in {{\,\textrm{supp}\,}}\mu _{-k}$. Thus there exist $\bar{p}\in (0,1)$ and $\bar{\eta }> 0$ such that for any $p\in (0, \bar{p})$ and any $(\widetilde{b}, \widetilde{s})\in B^{\bar{\eta }}_p(\widetilde{\mu }^{\varepsilon };b,s)$, the inequality $\varPi _{\widetilde{\theta }_k}(\widetilde{\mu }^{\varepsilon }; \widetilde{b}, \widetilde{s})> \varPi _{\theta _k}(\widetilde{\mu }^{\varepsilon }; \widetilde{b}, \widetilde{s})$ holds for all $\theta _k\in {{\,\textrm{supp}\,}}\mu _k$. $\square$

Theorem 5.6

Let G be a two-player game, and suppose that $(a^*_1,a^*_2)$ is Pareto efficient with respect to $\pi$. If $(a^*_1,a^*_2)$ is a strict Nash equilibrium of G, then $(a^*_1,a^*_2)$ is stable for all $p\in (0,1)$.

Proof

Let $(a^*_1,a^*_2)$ be a strict Nash equilibrium of G, and suppose that it is not stable for some $p\in (0, 1)$. We will show that $(a^*_1,a^*_2)$ is not Pareto efficient with respect to $\pi$. For each $i\in \{1,2\}$, consider the ith population consisting of the only type $\theta ^*_i$ for which $a^*_i$ is the strictly dominant strategy, and denote this configuration by $(\mu , b, s)$. Then the aggregate outcome of $(\mu ,b,s)$ is $(a^*_1,a^*_2)$, and hence this configuration $(\mu , b, s)$ is not stable for some $p\in (0, 1)$. This means that there exists some chance for a mutant sub-profile $\widetilde{\theta }_J$ to gain an evolutionary advantage over the incumbents in an imperfectly observable environment. Besides, it is obvious that after any entry, the focal set must be nonempty.

In the case when $|J| = 1$, it is obvious that no mutant can have a fitness advantage since $(a^*_1,a^*_2)$ is a strict Nash equilibrium. In the case when $J = \{1, 2\}$, it must be true that for every $\bar{\epsilon }\in (0,1)$, there exist a population share vector $\varepsilon \in (0, 1)^2$ with $\Vert \varepsilon \Vert \in (0,\bar{\epsilon })$ and an equilibrium pair $(\widetilde{b}, \widetilde{s})\in B_p(\widetilde{\mu }^{\varepsilon };b,s)$ such that $\varPi _{\widetilde{\theta }_i}(\widetilde{\mu }^{\varepsilon };\widetilde{b}, \widetilde{s})\ge \varPi _{\theta ^*_i}(\widetilde{\mu }^{\varepsilon };\widetilde{b}, \widetilde{s})$ for all $i\in \{1,2\}$, with strict inequality for some i. To complete the proof, we have to show that $(a^*_1, a^*_2)$ is Pareto dominated in any situation where $(\widetilde{\theta }_1, \widetilde{\theta }_2)$ has an evolutionary advantage.

For $(\widetilde{b}, \widetilde{s})\in B_p(\widetilde{\mu }^{\varepsilon };b,s)$ and $i\in \{1,2\}$, define $\widetilde{z}_i:{{\,\textrm{supp}\,}}\widetilde{\mu }^{\varepsilon }\rightarrow \Delta (A_i)$ by $\widetilde{z}_i(\widehat{\theta }_1, \widehat{\theta }_2) = p\widetilde{b}_{i}(\widehat{\theta }_1, \widehat{\theta }_2) + (1-p)\widetilde{s}_i(\widehat{\theta }_i)$. Then the post-entry average fitness of $\theta ^*_i\in {{\,\textrm{supp}\,}}\mu _i$ is

$$\begin{aligned} \varPi _{\theta ^*_i}(\widetilde{\mu }^{\varepsilon };\widetilde{b}, \widetilde{s}) = (1-\varepsilon _{-i})\pi _i(a_1^*,a_2^*) + \varepsilon _{-i} \pi _i\big ( a_i^*, \widetilde{z}_{-i}(\theta ^*_i, \widetilde{\theta }_{-i}) \big ). \end{aligned}$$

Similarly, the post-entry average fitness of $\widetilde{\theta }_i\in {({{\,\textrm{supp}\,}}\mu _j)}^{\textsf{c}}$ is

$$\begin{aligned} \varPi _{\widetilde{\theta }_i}(\widetilde{\mu }^{\varepsilon };\widetilde{b}, \widetilde{s}) = (1-\varepsilon _{-i}) \pi _i\big ( \widetilde{z}_i(\widetilde{\theta }_i, \theta ^*_{-i}), a_{-i}^* \big ) + \varepsilon _{-i} \pi _i\big ( \widetilde{z}_1(\widetilde{\theta }_1, \widetilde{\theta }_2), \widetilde{z}_2(\widetilde{\theta }_1, \widetilde{\theta }_2) \big ). \end{aligned}$$

With this kind of representation, the two actions $\widetilde{z}_i(\widetilde{\theta }_i, \theta ^*_{-i})$ and $\widetilde{z}_i(\widetilde{\theta }_1, \widetilde{\theta }_2)$ are correlated through $\widetilde{s}_i(\widetilde{\theta }_i)$. Even so, the remaining part of the proof is analogous to the proof of Theorem 3.9. $\square$

Theorem 5.7

Let $(a^*_1, \ldots , a^*_n)$ be a strict union Nash equilibrium in an n-player game G. Then $(a^*_1, \ldots , a^*_n)$ is stable for all $p\in (0,1)$.

Proof

As Theorem 5.6 being proved by a modification of the proof of Theorem 3.9, the proof of this theorem is similar to that of Theorem 3.14 with $\widetilde{b}$ replaced by $\widetilde{z}$, where each function $\widetilde{z}_i:{{\,\textrm{supp}\,}}\widetilde{\mu }^{\varepsilon }\rightarrow \Delta (A_i)$ is defined by $\widetilde{z}_i(\widehat{\theta }_i, \widehat{\theta }_{-i}) = p\widetilde{b}_{i}(\widehat{\theta }_i, \widehat{\theta }_{-i}) + (1-p)\widetilde{s}_i(\widehat{\theta }_i)$. $\square$