The evolutionary stability of optimism, pessimism, and complete ignorance

Abstract

We seek an evolutionary explanation for why in some situations humans maintain either optimistic or pessimistic attitudes toward uncertainty and are ignorant to relevant aspects of their environment. Players in strategic games face Knightian uncertainty about opponents’ actions and maximize individually their Choquet expected utility with respect to neo-additive capacities (Chateauneuf et al. 2007) allowing for both an optimistic or pessimistic attitude toward uncertainty as well as ignorance to strategic dependencies. An optimist (resp. pessimist) overweighs good (resp. bad) outcomes. A complete ignorant never reacts to opponents’ changes of actions. We focus on sub- and supermodular aggregative games and provide monotone comparative statics w.r.t. optimism/pessimism. With qualifications, we show that in finite populations, optimistic (resp. pessimistic) complete ignorance is evolutionary stable and yields a strategic advantage in submodular (resp. supermodular) games with aggregate externalities. Moreover, this evolutionary stable preference leads to Walrasian behavior in these classes of games.

Proofs

1.1 Proof of Lemma 1

If $A_i$ is a subcomplete sublattice of a complete lattice for all players $i \in N$, then it is a complete lattice. The direct product $A_{-i} = \times _{j \in N {\setminus } \{i\}} A_j$ of complete lattices is a complete lattice. By the Frink–Birkhoff theorem, a lattice is complete if and only if it is compact in its interval topology (Topkis 1998, pp. 29). Since $\pi _i$ is continuous in $a_{-i}$ on $A_{-i}$ for every $a_i$, it follows from Weierstrass’ theorem that it attains minima and maxima w.r.t. $a_{-i} \in A_{-i}$ for every $a_i \in A_i$. Thus, both $\arg \max _{a_{-i} \in A_{-i}} \pi _i(a_i, a_{-i})$ and $\arg \min _{a_{-i} \in A_{-i}} \pi _i(a_i, a_{-i})$ are non-empty for every $a_i \in A_i$. Since $A_{-i}$ is a complete lattice, it has a greatest and least element in $A_{-i}$ which are $sup_{A_{-i}} A_{-i}$ and $\inf _{A_{-i}} A_{-i}$, respectively. The remainder of the proof now follows directly from the definitions of positive and negative externalities, respectively. $\square$

1.2 Proof of Lemma 2

We prove a more general ordinal version of Lemma 2 that might be interesting on its own right. To this end, we introduce some definitions. A real-valued function f on a partially ordered set $X \times T$ satisfies the (dual) single-crossing property in (x, t) on $X \times T$ if for all $x'' \rhd x'$ and $t'' \rhd t'$:

$$\begin{aligned} f(x'', t') \ge f(x', t')&\Rightarrow (\Leftarrow )&f(x'', t'') \ge f(x', t''), \end{aligned}$$

(10)

$$\begin{aligned} f(x'', t')> f(x', t')&\Rightarrow (\Leftarrow )&f(x'', t'') > f(x', t''). \end{aligned}$$

(11)

We say that f satisfies the strict (dual) single-crossing property in (x, t) on $X \times T$ if for all $x'' \rhd x'$ and $t'' \rhd t'$:

$$\begin{aligned} f(x'', t') \ge (>) f(x', t')&\Rightarrow (\Leftarrow )&f(x'', t'') > (\ge ) f(x', t''). \end{aligned}$$

(12)

It is straight-forward to verify that increasing (decreasing) differences implies the (dual) single-crossing property but not vice versa. The same holds for the strict versions.

Lemma 5

Let $G = \langle N, (A_i), (\pi _i) \rangle$ be an aggregative strategic game with $A_i$ being a chain for each $i \in N$, and let $\mathbf {a}^{\circ }$ be a symmetric ATS profile and $\mathbf {a}^*$ be a symmetric Nash equilibrium of G. If the aggregative game G is such that $\pi _i$ satisfies:

• (i) the dual single-crossing property in $(a_i, \aleph ^n(\mathbf {a}))$ on $A_i \times X$ and G has strict negative (resp. positive) aggregate externalities, or

• (ii) the strict dual single-crossing property in $(a_i, \aleph ^n(\mathbf {a}))$ on $A_i \times X$ and G has negative (resp. positive) aggregate externalities,

then $\mathbf {a}^{\circ } \trianglerighteq \mathbf {a}^*$ (resp. $\mathbf {a}^{\circ } \unlhd \mathbf {a}^*$).

Proof

Since $\mathbf {a}^{\circ }$ is an ATS and $\mathbf {a}^*$ is a Nash equilibrium of G, we have by definition:

$$\begin{aligned} \pi _i(a_i^{\circ }, \aleph ^n(a_i^{\circ }, a_{-i}^{\circ }))\ge & {} \pi _i(a_i^*, \aleph ^n(a_i^{\circ }, a_{-i}^{\circ })), \end{aligned}$$

(13)

$$\begin{aligned} \pi _i(a_i^*, \aleph ^n(a_i^*, a_{-i}^*))\ge & {} \pi _i(a_i^{\circ }, \aleph ^n(a_i^{\circ }, a_{-i}^*)), \end{aligned}$$

(14)

for all $i \in N$. Consider case (i), and let G have strict negative aggregate externalities and $A_i$ be a chain. Suppose to the contrary that for $i \in N$, we have $a^{\circ }_i \lhd a^*_i$. By the dual single-crossing property of $\pi _i$ in $(a_i, \aleph (\mathbf {a}))$ on $A_i \times X$:

$$\begin{aligned} \pi _i(a_i^*, \aleph ^n(a_i^*, a_{-i}^*))\ge & {} \pi _i(a_i^{\circ }, \aleph ^n(a_i^{\circ }, a_{-i}^*)) \Rightarrow \pi _i(a_i^*, \aleph ^n(a_i^*, a_{-i}^{\circ }))\nonumber \\\ge & {} \pi _i(a_i^{\circ }, \aleph ^n(a_i^{\circ }, a_{-i}^{\circ })). \end{aligned}$$

(15)

Since G has strict negative aggregate externalities:

$$\begin{aligned} \pi _i(a_i^*, \aleph ^n(a_i^*, a_{-i}^{\circ }))\ge & {} \pi _i(a_i^{\circ }, \aleph ^n(a_i^{\circ }, a_{-i}^{\circ })) \Rightarrow \pi _i(a_i^*, \aleph ^n (a_i^{\circ }, a_{-i}^{\circ })) \nonumber \\> & {} \pi _i(a_i^{\circ }, \aleph ^n(a_i^{\circ }, a_{-i}^{\circ })), \end{aligned}$$

(16)

which is a contradiction to inequality (13). The case for strict positive externalities follows analogously.

Now, consider case (ii). Again, suppose to the contrary that, for $i \in N$, we have $a^{\circ }_i \lhd a^*_i$. By the strict dual single-crossing property of $\pi _i$ in $(a_i, \aleph (\mathbf {a}))$ on $A_i \times X$:

$$\begin{aligned} \pi _i(a_i^*, \aleph ^n(a_i^*, a_{-i}^*))\ge & {} \pi _i(a_i^{\circ }, \aleph ^n(a_i^{\circ }, a_{-i}^*)) \Rightarrow \pi _i(a_i^*, \aleph ^n(a_i^*, a_{-i}^{\circ }))\nonumber \\> & {} \pi _i(a_i^{\circ }, \aleph ^n(a_i^{\circ }, a_{-i}^{\circ })). \end{aligned}$$

(17)

Since G has negative aggregate externalities:

$$\begin{aligned} \pi _i(a_i^*, \aleph ^n(a_i^*, a_{-i}^{\circ }))> & {} \pi _i(a_i^{\circ }, \aleph ^n(a_i^{\circ }, a_{-i}^{\circ })) \Rightarrow \pi _i(a_i^*, \aleph ^n (a_i^{\circ }, a_{-i}^{\circ })) \nonumber \\> & {} \pi _i(a_i^{\circ }, \aleph ^n(a_i^{\circ }, a_{-i}^{\circ })), \end{aligned}$$

(18)

which is a contradiction to inequality (13). The case for positive externalities follows analogously. $\square$

Lemma 2 is now an immediate corollary of Lemma 5.

1.3 Proof of Lemma 3

We need to show that if $\pi _i$ is supermodular in $a_i$ on $A_i$ and has increasing (resp. decreasing) differences in $(a_i, a_{-i})$ on $\mathbf {A}$, then $u_i(\alpha _i, \delta _i)$ is supermodular in $a_i$ on $A_i$ and has increasing (resp. decreasing) differences in $(a_i, a_{-i})$ on $\mathbf {A}$ for each $(\alpha _i, \delta _i) \in [0, 1]^2$. If $\pi _i$ is supermodular in $a_i$ on $A_i$ and has increasing (resp. decreasing) differences in $(a_i, a_{-i})$ on $\mathbf {A}$, then for any scalar $\gamma \ge 0$ also $\gamma \pi _i$ is supermodular in $a_i$ on $A_i$ and has increasing (resp. decreasing) differences in $(a_i, a_{-i})$ on $\mathbf {A}$ (Topkis 1998, Lemma 2.6.1. (a)). $\heartsuit _i$ and $\maltese _i$ are both supermodular in $a_i$ on $A_i$ by definition and constant in $a_{-i}$ on $A_{-i}$. Since $u_i$ is for each $(\alpha _i, \delta _i) \in [0, 1]^2$ a sum of supermodular functions in $a_i$ on $A_i$ having increasing (resp. decreasing) differences in $(a_i, a_{-i})$ on $\mathbf {A}$, it is supermodular in $a_i$ on $A_i$ and has increasing (resp. decreasing) differences $(a_i, a_{-i})$ on $\mathbf {A}$ (Topkis 1998, Lemma 2.6.1. (b)). By analogous arguments, the result extends to the strict versions (see Topkis 1998, p. 49). $\square$

1.4 Proof of Lemma 4

By Lemma 3, if G is supermodular (resp. submodular), then $G(\varvec{\alpha }, \varvec{\delta })$ is supermodular (resp. submodular) for each $(\varvec{\alpha }, \varvec{\delta }) \in [0, 1]^n \times [0, 1]^n$. Thus, $u_i$ is supermodular in $a_i$ on $A_i$ and has increasing (resp. decreasing) differences in $(a_i, a_{-i})$ on $\mathbf {A}$. Since $u_i$ is supermodular in $a_i$ on $A_i$, $b_i(a_{-i}, \alpha _i, \delta _i)$ is a sublattice of $A_i$ for each $a_{-i} \in A_{-i}$ and $(\alpha _i, \delta _i) \in [0, 1]^2$ by Topkis (1998), Theorem 2.7.1. Since $u_i$ has increasing (resp. decreasing) differences in $(a_i, a_{-i})$ on $\mathbf {A}$, $b_i(a_{-i}, \alpha _i, \delta _i)$ is increasing (resp. decreasing) in $a_{-i}$ on $\{a_{-i} \in A_{-i} : b_i(a_{-i}, \alpha _i, \delta _i) \ne \emptyset \}$ for any $(\alpha _i, \delta _i) \in [0, 1]^2$ by Topkis (1998), Theorem 2.8.1. $\square$

1.5 Proof of Proposition 1

If $\pi _i$ is continuous on $A_{-i}$ for every $a_i \in A_i$, then $u_i$ is well defined (Remark 1). If $\pi _i$ is upper semicontinuous on $A_i$, then $u_i$ is upper semicontinuous on $A_i$, since limits are preserved under algebraic operations. The result follows now from Lemmata 3 and 4 and Zhou’s (1994) generalization of Tarski’s fixed-point theorem. $\square$

Note that, in Proposition 1, we do not claim that the set of equilibria under Knightian uncertainty is a sublattice of $\mathbf {A}$. Thus, if $\mathbf {a} = (a_1, \ldots ,a_n)$ and $\mathbf {a}' = (a_1', \ldots , a_n')$ are both equilibria under Knightian uncertainty for $G(\varvec{\alpha }, \varvec{\delta })$, then $(a_1 \vee a_1', \ldots , a_n \vee a_n')$ and $(a_1 \wedge a_1', \ldots , a_n \wedge a_n')$ may not be equilibria under Knightian uncertainty for the game $G(\varvec{\alpha }, \varvec{\delta })$ (see, for an example, Zhou (1994), p. 299).

Remark 3

In addition to the assumptions of Proposition 1, assume that $n = 2$ and that $A_i$ is a chain for $i \in \{1, 2\}$. Then, for any $(\varvec{\alpha }, \varvec{\delta }) \in [0, 1]^n \times [0, 1]^n$, the set of equilibria under Knightian uncertainty is a subcomplete sublattice and a greatest and least equilibrium exist.

Proof of Remark

This follows from Proposition 1 and a result by Echenique (2003) who observed that the set of Nash equilibria forms a sublattice in two-player games with totally ordered action sets for which each player’s best-response correspondence is increasing in the strong set order. $\square$

1.6 Proof of Proposition 2

Since $\pi _i$ is defined on $A_i$ and the range of sums of opponents actions and continuous in latter, by arguments similar to the ones for proving Remark 1, $G(\varvec{\alpha }, \varvec{\delta })$ is well defined for all $(\varvec{\alpha }, \varvec{\delta }) \in [0, 1]^n \times [0, 1]^n$. Submodularity of G implies by Lemma 3 submodularity of $G(\varvec{\alpha }, \varvec{\delta })$ for all $(\varvec{\alpha }, \varvec{\delta }) \in [0, 1]^n \times [0, 1]^n$. Thus, for all $i \in N$, $u_i(\alpha _i, \delta _i)$ has decreasing differences in $(a_i, \sum _{j \in N {\setminus }\{i\}} a_{j})$. By Lemma 4, $b_i(\alpha _i, \delta _i)$ is decreasing in $\sum _{j \in N {\setminus } \{i\}} a_{j}$. Since $\pi _i$ is continuous in both variables, $u_i(\alpha _i, \delta _i)$ is continuous in $a_i$ and $\sum _{j \in N {\setminus } \{j\}} a_j$, because limits are preserved under algebraic operations. Thus, $b_i(\sum _{j \in N {\setminus } \{i\}} a_{j}, \alpha _i, \delta _i)$ is non-empty for any $\sum _{j \in N {\setminus } \{i\}} a_j$. It also implies that $b_i(\alpha _i, \delta _i)$ is upper hemicontinuous in $\sum _{j \in N {\setminus } \{i\}} a_{j}$. Thus, the conditions are sufficient for a theorem by Kukushkin (1994) by which there exists a Nash equilibrium in pure actions of the game $G(\varvec{\alpha }, \varvec{\delta })$. $\square$

1.7 Proof of Proposition 3

By Remark 2, $b_i(a_{-i}, \alpha _i, 1)$ is constant and, therefore, trivially increasing in $a_{-i}$ on $A_{-i}$ for any $\alpha _i \in [0, 1]$ for all $i \in N$. Hence, the result follows as a special case from the proof of Proposition 1. $\square$

1.8 Proof of Proposition 4

For an equilibrium under Knightian uncertainty to exist for $(\varvec{\alpha }, \varvec{1})$, $u_i(\alpha _i, 1)$ must be well defined. If $\pi _i$ is strictly concave in $a_i$, then for any $\alpha _i \in [0, 1]$, $u_i(\alpha _i, 1)$ is strictly concave in $a_i$, since it is a weighted sum of strictly concave functions. Strict concavity of $u_i(a_i, 1)$ is sufficient for $b_i(a_{-i}, \alpha _i, 1)$ being unique for all $a_{-i} \in A_{-i}$. By Remark 2, if $\delta _i = 1$, then $b_i(a_{-i}, \alpha _i, 1)$ is constant for all $a_{-i} \in A_{-i}$. Hence, if there exists an equilibrium under Knightian uncertainty with $\delta _i = 1$ for all $i \in N$, then it must be unique with each player choosing her strictly dominant action. $\square$

1.9 Proof of Proposition 5

Proposition 5 is a direct corollary of the following two lemmata:

Lemma 6

If $\pi _i$ has:

• increasing differences in $(a_i, a_{-i})$ on $\mathbf {A}$ and positive (resp. negative) externalities, or

• decreasing differences in $(a_i, a_{-i})$ on $\mathbf {A}$ and negative (resp. positive) externalities,

then $u_i$ has increasing (resp. decreasing) differences in $(a_i, \alpha _i)$ on $A_i \times [0, 1]$ for all $a_{-i} \in A_{-i}$ and $\delta _i \in [0, 1]$. The result extends to the strict versions.

Proof of Lemma

If $\pi _i$ has decreasing (increasing) differences in $(a_i, a_{-i})$ on $\mathbf {A}$, then for all $a_i'' \rhd a_i'$:

$$\begin{aligned} \pi _i(a_i'', \sup _{\mathbf {A}} A_{-i}) - \pi _i(a_i', \sup _{\mathbf {A}} A_{-i}) \le (\ge ) \pi _i(a_i'', \inf _{\mathbf {A}} A_{-i}) - \pi _i(a_i', \inf _{\mathbf {A}} A_{-i}). \end{aligned}$$

By Remark 1, it follows that if $\pi _i$ has [decreasing differences in $(a_i, a_{-i})$ on $\mathbf {A}$ and negative (resp. positive) externalities] or [increasing differences in $(a_i, a_{-i})$ on $\mathbf {A}$ and positive (resp. negative) externalities], then for all $a_i'' \rhd a_i'$:

$$\begin{aligned} \maltese _i(a_i'') - \maltese _i(a_i') \le (\ge ) \heartsuit _i(a_i'') - \heartsuit _i(a_i'). \end{aligned}$$

(19)

Let $\alpha _i', \alpha _i'' \in [0, 1]$ with $\alpha _i'' \ge \alpha _i'$. It follows from the last inequality that:

$$\begin{aligned}&\alpha ''_i [(\heartsuit _i(a_i'') - \heartsuit _i(a_i')) - (\maltese _i(a_i'') - \maltese _i(a_i'))] \ge (\le ) \nonumber \\&\quad \alpha _i' [(\heartsuit _i(a_i'') - \heartsuit _i(a_i')) - (\maltese _i(a_i'') - \maltese _i(a_i'))]. \end{aligned}$$

(20)

This is equivalent to:

$$\begin{aligned}&\alpha ''_i [\heartsuit _i(a_i'') - \heartsuit _i(a_i')] - \alpha ''_i [\maltese _i(a_i'') - \maltese _i(a_i')] \ge (\le ) \nonumber \\&\quad \alpha _i' [\heartsuit _i(a_i'') - \heartsuit _i(a_i')] - \alpha _i'[\maltese _i(a_i'') - \maltese _i(a_i')] \end{aligned}$$

(21)

$$\begin{aligned}&\alpha ''_i [\heartsuit _i(a_i'') - \heartsuit _i(a_i')] + [\maltese _i(a_i'') - \maltese _i(a_i')] - \alpha ''_i [\maltese _i(a_i'') - \maltese _i(a_i')] \ge (\le ) \nonumber \\&\quad \alpha _i' [\heartsuit _i(a_i'') - \heartsuit _i(a_i')] + [\maltese _i(a_i'') - \maltese _i(a_i')] - \alpha _i'[\maltese _i(a_i'') - \maltese _i(a_i')] \end{aligned}$$

(22)

$$\begin{aligned}&\alpha ''_i [\heartsuit _i(a_i'') - \heartsuit _i(a_i')] + (1 - \alpha ''_i) [\maltese _i(a_i'') - \maltese _i(a_i')] \ge (\le ) \nonumber \\&\quad \alpha _i' [\heartsuit _i(a_i'') - \heartsuit _i(a_i')] + (1 - \alpha _i') [\maltese _i(a_i'') - \maltese _i(a_i')]. \end{aligned}$$

(23)

Consider any $\delta _i \in [0, 1]$. Then, the previous inequality implies:

$$\begin{aligned}&\alpha ''_i \delta _i [\heartsuit _i(a_i'') - \heartsuit _i(a_i')] + (1 - \alpha ''_i) \delta _i [\maltese _i(a_i'') - \maltese _i(a_i')] \ge (\le ) \nonumber \\&\quad \alpha _i' \delta _i [\heartsuit _i(a_i'') - \heartsuit _i(a_i')] + (1 - \alpha _i') \delta _i [\maltese _i(a_i'') - \maltese _i(a_i')], \end{aligned}$$

(24)

which in turn implies:

$$\begin{aligned}&\alpha ''_i \delta _i [\heartsuit _i(a_i'') - \heartsuit _i(a_i')] + (1 - \alpha ''_i) \delta _i [\maltese _i(a_i'') - \maltese _i(a_i')] \nonumber \\&\quad + (1 - \delta _i) [\pi _i(a_i'', a_{-i}) - \pi _i(a_i', a_{-i})] \ge (\le ) \nonumber \\&\quad \alpha _i' \delta _i [\heartsuit _i(a_i'') - \heartsuit _i(a_i')] + (1 - \alpha _i') \delta _i [\maltese _i(a_i'') - \maltese _i(a_i')] \nonumber \\&\quad +(1 - \delta _i) [\pi _i(a_i'', a_{-i}) - \pi _i(a_i', a_{-i})] \end{aligned}$$

(25)

$$\begin{aligned}&\delta _i [\alpha ''_i \heartsuit _i(a_i'') + (1 - \alpha ''_i) \maltese _i(a_i'')] + (1 - \delta _i) \pi _i(a_i'', a_{-i}) \nonumber \\&\quad - \delta _i [\alpha _i'' \heartsuit _i(a_i') + (1 - \alpha _i'') \maltese _i(a_i')] - (1 - \delta _i) \pi _i(a_i', a_{-i}) \ge (\le ) \nonumber \\&\quad \delta _i [\alpha '_i \heartsuit _i(a_i'') + (1 - \alpha '_i) \maltese _i(a_i'')] + (1 - \delta _i) \pi _i(a_i'', a_{-i}) \nonumber \\&\quad - \delta _i [\alpha _i' \heartsuit _i(a_i') + (1 - \alpha _i') \maltese _i(a_i')] - (1 - \delta _i) \pi _i(a_i', a_{-i}). \end{aligned}$$

(26)

Hence, we have:

$$\begin{aligned}&u_i(a_i'', a_{-i}, \alpha _i'', \delta _i) - u_i(a_i', a_{-i}, \alpha _i'', \delta _i) \ge (\le ) u_i(a_i'', a_{-i}, \alpha _i', \delta _i)\nonumber \\&\quad - u_i(a_i', a_{-i}, \alpha _i', \delta _i). \end{aligned}$$

(27)

The proof holds analogously for strict versions. $\square$

Lemma 7

(Monotone optimal selections) If $u_i(a_i, a_{-i}, \alpha _i, \delta _i)$ is supermodular in $a_i$ on $A_{i}$ and has increasing (resp. decreasing) differences in $(a_i, \alpha _i)$ on $A_i \times [0, 1]$ for each $a_{-i} \in A_{-i}$ and $\delta _i \in [0, 1]$, then $b_i(a_{-i}, \alpha _i, \delta _i)$ is increasing (resp. decreasing) in $\alpha _i$ on $\{\alpha _i \in [0, 1] : b_i(a_{-i}, \alpha _i, \delta _i) \ne \emptyset \}$ for $\delta _i \in [0, 1]$. If, in addition, $u_i(a_i, a_{-i}, \alpha _i, \delta _i)$ has strictly increasing (resp. decreasing) differences in $(a_i, \alpha _i)$ on $A_i \times [0, 1]$ for each $a_{-i} \in A_{-i}$ and $\delta _i \in [0, 1]$, $\alpha _i'' > \alpha _i'$ in [0, 1], and for any $a_{-i} \in A_{-i}$, $a'_i \in b_i(a_{-i}, \alpha _i', \delta _i)$ and $a''_i \in b_i(a_{-i}, \alpha ''_i, \delta _i)$, then $a_i' \unlhd (\trianglerighteq ) a_i''$. In this case, if one picks any $a_i(\alpha _i)$ in $b_i(a_{-i}, \alpha _i, \delta _i)$ for each $\alpha _i$ with $b_i(a_{-i}, \alpha _i, \delta _i)$ non-empty, then $a_i(\alpha _i)$ is increasing (resp. decreasing) in $\alpha _i$ on $\{\alpha _i \in [0, 1] : b_i(a_{-i}, \alpha _i, \delta _i) \ne \emptyset \}$.

Proof of Lemma

Pick any $\alpha _i''$ and $\alpha _i'$ in [0, 1] with $\alpha _i'' > \alpha _i'$, and for any $(a_{-i}, \delta _i) \in A_{-i} \times [0, 1]$, $a'_i \in b_i(a_{-i}, \alpha _i', \delta _i)$ and $a''_i \in b_i(a_{-i}, \alpha _i'', \delta _i)$.

First, consider strictly increasing differences of $u_i$ in $(a_i, \alpha _i)$, and suppose that it is not true that $a_i' \unlhd a_i''$. Then, $a_i'' \lhd a_i' \vee a_i''$ and so using the hypothesis that $u_i(a_i, a_{-i}, \alpha _i, \delta _i)$ is for any $(a_{-i}, \delta _i) \in A_{-i} \times [0, 1]$ supermodular in $a_i$ and has strictly increasing differences in $(a_i, \alpha _i)$:

$$\begin{aligned} 0\le & {} u_i(a_i', a_{-i}, \alpha _i', \delta _i) - u_i(a_i' \wedge a_i'', a_{-i}, \alpha _i', \delta _i) \nonumber \\\le & {} u_i(a_i' \vee a_i'', a_{-i}, \alpha _i', \delta _i) - u_i(a_i'', a_{-i}, \alpha _i', \delta _i) \nonumber \\< & {} u_i(a_i' \vee a_i'', a_{-i}, \alpha _i'', \delta _i) - u_i(a_i'', a_{-i}, \alpha _i'', \delta _i) \le 0, \end{aligned}$$

(28)

which yields a contradiction.

Second, consider strictly decreasing differences of $u_i$ in $(a_i, \alpha _i)$, and suppose that it is not true that $a_i' \trianglerighteq a_i''$. Then, $a_i' \rhd a_i' \vee a_i''$ and so using the hypothesis that $u_i(a_i, a_{-i}, \alpha _i, \delta _i)$ is for any $(a_{-i}, \delta _i) \in A_{-i} \times [0, 1]$ supermodular in $a_i$ and has strictly decreasing differences in $(a_i, \alpha _i)$:

$$\begin{aligned} 0\le & {} u_i(a_i'', a_{-i}, \alpha _i'', \delta _i) - u_i(a_i' \wedge a_i'', a_{-i}, \alpha _i'', \delta _i) \nonumber \\\le & {} u_i(a_i' \vee a_i'', a_{-i}, \alpha _i'', \delta _i) - u_i(a_i'', a_{-i}, \alpha _i'', \delta _i) \nonumber \\< & {} u_i(a_i' \vee a_i'', a_{-i}, \alpha _i', \delta _i) - u_i(a_i', a_{-i}, \alpha _i', \delta _i) \le 0, \end{aligned}$$

(29)

which yields a contradiction. $\square$

The first part of the proof of Lemma 7 is analogous to (Topkis 1998, Theorem 2.8.4.).

1.10 Proof of Proposition 7

Let G be a submodular game with negative externalities and $\varvec{\delta } = \varvec{1}$ (i.e., unit vector), and consider the greatest equilibrium under Knightian uncertainty $\bar{\mathbf {a}}(\varvec{\alpha })$ when the profile of degrees of optimism is $\varvec{\alpha }$. Let $\varvec{\alpha }' \ge \varvec{\alpha }$. Suppose now to the contrary that the greatest equilibrium under Knightian uncertainty $\bar{\mathbf {a}}(\varvec{\alpha }')$ is smaller or incomparable to $\bar{\mathbf {a}}(\varvec{\alpha })$. In both cases, there exists a player i whose equilibrium actions satisfy $\bar{a}_i(\varvec{\alpha }')$ is strictly smaller or incomparable to $\bar{a}_i(\varvec{\alpha })$. Note that $\bar{a}_i(\varvec{\alpha }') \in b_i(\bar{a}_{-i}(\varvec{\alpha }'), \alpha _i', 1)$ and $\bar{a}_i(\varvec{\alpha }) \in b_i(\bar{a}_{-i}(\varvec{\alpha }), \alpha _i, 1)$. Since $\delta _i = 1$, $b_i$ is constant in $a_{-i}$ by Remark 2. Hence, $b_i(\bar{a}_{-i}(\varvec{\alpha }), \alpha _i, 1) \ne b_i(\bar{a}_{-i}(\varvec{\alpha }'), \alpha _i', 1)$ only if $\varvec{\alpha } \ne \varvec{\alpha }'$. By Proposition 5, $b_i(\bar{a}_{-i}(\varvec{\alpha }), \alpha _i, 1)$ is increasing in $\alpha _i$, i.e., $\bar{a}_i(\varvec{\alpha }') \vee \bar{a}_i(\varvec{\alpha }) \in b_i(a_{-i}(\varvec{\alpha }'), \alpha _i', 1)$. Let $\tilde{a}_i(\varvec{\alpha }') = \bar{a}_i(\varvec{\alpha }') \vee \bar{a}_i(\varvec{\alpha })$. Then, $\tilde{a}_i(\varvec{\alpha }') \trianglerighteq \bar{a}_i(\varvec{\alpha }')$. We distinguish two cases: if $\tilde{a}_i(\varvec{\alpha '}) = \bar{a}_i(\varvec{\alpha '})$, then $\bar{a}_i(\varvec{\alpha '}) \trianglerighteq \bar{a}_i(\varvec{\alpha })$, a contradiction. Otherwise, if $\tilde{a}_i(\varvec{\alpha '}) \rhd \bar{a}_i(\varvec{\alpha '})$, then there is a contradiction to $\bar{a}_i(\varvec{\alpha '})$ being a component of the largest equilibrium under Knightian uncertainty with $\varvec{\alpha }'$ and $\varvec{\delta } = \varvec{1}$. An analogous arguments holds for positive externalities and for the least equilibrium under Knightian uncertainty. $\square$

1.11 Proof of Proposition 8

(i) and (ii): First, by assumption, a symmetric ATS exists for G. We denote it by $\mathbf {a}^{\circ }$. It follows from Assumption 2 that there exists a preference $t^{\circ } = (\delta ^{\circ }, \alpha ^{\circ })$ with $\delta ^{\circ } = 1$ (and some $\alpha ^{\circ }$), such that there is a symmetric EKU with $\mathbf {a}^*(\mathbf {t}^{\circ }) = \mathbf {a}^{\circ }$.

Second, we show that there exists a $t^{\circ }$ with an EKU of G, $\mathbf {a}^*(\mathbf {t}^{\circ }) = \mathbf {a}^{\circ }$, such that $t^{\circ }$ is optimistic. It is sufficient to show that for $\delta ^{\circ } = 1$, there exists $\alpha ^{\circ } \in [0, 1]$ s.t. $\alpha ^{\circ } \ge \alpha ^*$, where $\alpha ^* = \max \{\alpha ' \in [0, 1] : \mathbf {a}^*(\varvec{\alpha }', (1, \ldots , 1)) \in \mathcal {E}^{sym}(G(\varvec{\alpha }', (1, \ldots , 1)) \cap \mathcal {E}^{sym}(G) \}$. Suppose to the contrary that for all $\alpha$ with $\mathbf {a}^*(\varvec{\alpha }) = \mathbf {a}^{\circ }$, we have $\alpha < \alpha ^*$. By Proposition 7, $\mathbf {a}^*(\varvec{\alpha }) \unlhd \mathbf {a}^*(\varvec{\alpha }^*)$, $\mathbf {a}^*(\varvec{\alpha }) \in \mathcal {E}(G(\varvec{\alpha }, (\varvec{1})))$. By Lemma 2, $\mathbf {a}^{\circ } \trianglerighteq \mathbf {a}^*$. Hence, we must have $\mathbf {a}^*(\varvec{\alpha }) = \mathbf {a}^*(\varvec{\alpha }^*)$. But then, set $\alpha = \alpha ^*$, a contradiction.

Third, we show that $t^{\circ }$ is a globally stable fESP. In particular, we show that if the EKU $\mathbf {a}^*(\mathbf {t}^{\circ })$ is a symmetric ATS, then $t^{\circ }$ is a globally stable fESP in G. Denote by $\mathbf {t}' := ({\mathop {\overbrace{t', \ldots , t'}}\limits ^{m}}, t^{\circ }, \ldots , t^{\circ })$ for some $m \in \{1, \ldots , n - 1\}$ and $t' \in T_G$. Recall that we denote by j a mutant (playing $t'$) and by i a non-mutant (playing $t^{\circ }$). By the definition of ATS:

$$\begin{aligned} \pi (a^*_i(\mathbf {t}^{\circ }), \aleph ^n(\mathbf {a}^*(\mathbf {t}^{\circ }))) \ge \pi (a^*_j(\mathbf {t}'), \aleph ^n(\mathbf {a}^*(\mathbf {t}^{\circ }))) \end{aligned}$$

(30)

for all $t' \in T_G$. If $a^*_i(\mathbf {t}^{\circ }) \rhd (\lhd ) a^*_j(\mathbf {t}')$ for all mutants j, then $\mathbf {a}^*(\mathbf {t}^{\circ }) \rhd (\lhd ) \mathbf {a}^*(\mathbf {t}')$. This implies by decreasing differences:

$$\begin{aligned} \pi (a^*_i(\mathbf {t}^{\circ }), \aleph ^n(\mathbf {a}^*(\mathbf {t}'))) \ge \pi (a^*_j(\mathbf {t}'), \aleph ^n(\mathbf {a}^*(\mathbf {t}'))). \end{aligned}$$

(31)

By Remark 2, all non-mutants have constant best-response selections. Therefore, we can select $\mathbf {a}^*(\mathbf {t}') \in \mathcal {E}^{sym}(G(\mathbf {t}'))$, such that $a^*_i(\mathbf {t}^{\circ }) = a^*_i(\mathbf {t}')$ for all non-mutants i. Hence:

$$\begin{aligned} \pi (a^*_i(\mathbf {t}'), \aleph ^n(\mathbf {a}^*(\mathbf {t}'))) \ge \pi (a^*_j(\mathbf {t}'), \aleph ^n(\mathbf {a}^*(\mathbf {t}'))). \end{aligned}$$

(32)

Since this holds for all $m \in \{1, \ldots , n - 1\}$ and all $t' \in T_G$, we have that $t^{\circ }$ is a GfESP in G. The case of positive externalities follows analogously.

(iii): We note that if for all players $\pi$ is strictly concave in the player’s own action $a_i$ on A for all $a_{-i}$ on $A_{-i}$, then so is $u_i$, since it is a sum of strictly concave functions, each term multiplied by positive scalar, and because of aggregate externalities, the worst and best-case actions of the opponents do not depend on the player’s own action. Hence, for all $i \in N$, each $a_{-i}$ and each $t_i$, we have that $b(a_{-i}, t_i)$ is a singleton. By Remark 2 that $b(a_{-i}, \alpha _i)$ is a constant function on $A_{-i}$ for each $\alpha _i$. Therefore, for all non-mutants $a^*_i(\mathbf {t}^{\circ }) = a^*_i(\mathbf {t}')$ for all $\mathbf {a}^*(\mathbf {t}') \in \mathcal {E}^{sym}(G(\mathbf {t}'))$. Hence, $t^{\circ }$ is a robust fESP. This completes the proof of the proposition. $\square$

1.12 Proof of Proposition 9

Lemma 8

Suppose that the strategic game $G = \langle N, A, \pi \rangle$ has positive (resp. negative) externalities, and let $\bar{\mathbf {a}}$ and $\underline{\mathbf {a}}$ be the greatest and least combination of actions in $\mathbf {A}$. If $\underline{\mathbf {a}}$ (resp. $\bar{\mathbf {a}}$) is a Nash equilibrium of G, then $\underline{a}$ (resp. $\bar{a})$ is a finite population evolutionary stable strategy in G.

Proof of Lemma

If G has positive (resp. negative) externalities, let $a := \underline{a}$ (resp. $a := \bar{a}$). Since $\mathbf {a} = (a, \ldots , a)$ is a Nash equilibrium of G:

$$\begin{aligned} \pi (a, \ldots , a) \ge \pi (a', a, \ldots , a) \text{ for } \text{ all } a' \in A. \end{aligned}$$

(33)

We need to show that:

$$\begin{aligned} \pi (a, a', a, \ldots , a) \ge \pi (a', a, \ldots , a) \text{ for } \text{ all } a' \in A. \end{aligned}$$

(34)

Given both inequalities, it is sufficient to show:

$$\begin{aligned} \pi (a, a', a, \ldots , a) \ge \pi (a, \ldots , a) \text{ for } \text{ all } a' \in A. \end{aligned}$$

(35)

However, last inequality follows immediately from positive (resp. negative) externalities and the fact that $a: = \underline{a}$ (resp. $a: = \bar{a}$). $\square$

Proof of Proposition 9

(i) and (ii): Suppose that the game has positive (resp. negative) aggregate externalities and consider the lowest (resp. highest) symmetric profile of actions $\underline{\mathbf {a}}$ (resp. $\bar{\mathbf {a}}$). By Assumption 3, this profile of actions is a Nash equilibrium of the game G.

If G has positive (resp. negative) externalities, $a := \underline{a}$ (resp. $a := \bar{a}$) is by Lemma 8 a finite population evolutionary stable strategy in G.

We claim that a is an ATS. Since G is an aggregative game, we have by definition of fESS:

$$\begin{aligned} \pi (a, \aleph ^n(a', a, \ldots , a)) \ge \pi (a', \aleph ^n(a', a, \ldots , a)) \text{ for } \text{ all } a' \in A. \end{aligned}$$

(36)

By increasing differences, fESS implies ATS:

$$\begin{aligned} \pi (a, \aleph ^n(a, \ldots , a)) \ge \pi (a', \aleph ^n(a, \ldots , a)) \text{ for } \text{ all } a' \in A. \end{aligned}$$

(37)

By Assumption 2, there exists a preference t with $\delta = 1$, such that a symmetric EKU satisfies $\mathbf {a}^*(\mathbf {t}) = \mathbf {a}$. Consider the preference $t = (\delta , \alpha )$ with $\delta = 1$ and $\alpha = 0$. From Proposition 6, it follows that a symmetric EKU with the symmetric profile of preferences $\mathbf {t} = (t, \ldots , t)$ satisfies $\mathbf {a}^*(\mathbf {t}) = \mathbf {a}$.

Inequality (36) implies:

$$\begin{aligned} \pi (a^*_i(\mathbf {t}), \aleph ^n(\mathbf {a}^*(t'_j, t_{-j}))) \ge \pi (a^*_j(t'_j, t_{-j}), \aleph ^n(\mathbf {a}^*(t'_j, t_{-j}))) \text{ for } \text{ all } t' \in T_G. \end{aligned}$$

(38)

By Remark 2, all non-mutants have constant best-response selections. Therefore, we can select $\mathbf {a}^*(t'_j, t_{-j}) \in \mathcal {E}^{sym}(G(t'_j, t_{-j}))$, such that $a^*_i(\mathbf {t}) = a^*_i(t'_j, t_{-j})$ for all non-mutants with t and any mutant with any $t' \in T_G$. Hence:

$$\begin{aligned} \pi (a^*_i(t'_j, t_{-j}), \aleph ^n(\mathbf {a}^*(t'_j, t_{-j}))) \ge \pi (a^*_j(t'_j, t_{-j}), \aleph ^n(\mathbf {a}^*(t'_j, t_{-j}))) \text{ for } \text{ all } t' \in T, \end{aligned}$$

(39)

i.e., t is a fESP in the game G.

(iii) We note that if $\pi$ is strictly quasiconcave in the player’s own action $a_i$ on A for all $a_{-i}$ on $A_{-i}$, then $u_i(1, 0) = \maltese$ is strictly quasiconcave in the player’s own action $a_i$ on A. Hence, $b(a_{-i}, (1, 0))$ is a singleton for each $a_{-i}$. By Remark 2, we have that $b(a_{-i}, (1, 0))$ is a constant function on $A_{-i}$. Hence, $a^*_i(\mathbf {t}) = a^*_i(t'_j, t_{-j})$ for all equilibria $\mathbf {a}^*(t'_j, t_{-j}) \in \mathcal {E}^{sym}(G(t'_j, t_{-j}))$. This completes the proof of the proposition. $\square$