Evolution of Risk-Taking Behaviour and Status Preferences in Anti-coordination Games

Abstract

This paper analyses how risk-taking behaviour and preferences over consumption rank can emerge when individuals have an incentive to coordinate their actions. Using an evolutionary game theory framework, it is shown that when ex-ante homogeneous individuals face a strategic interaction where they benefit from choosing distinct actions, i.e. an anti-coordination game, stable types must be willing to accept risky gambles over consumption to differentiate themselves. This is the case despite an assumed concavity of the objective function, which makes any gamble costly in expectation. Consumption differences act as a form of costly communication that allows for coordination. More specifically, it is shown that when individuals have access to any fair consumption lottery, there exists a neutrally stable equilibrium where individuals choose a risky lottery and condition their action in the anti-coordination game on relative consumption. Furthermore, the evolutionarily optimal risk-taking behaviour can be induced by preferences over consumption rank. This suggests status preferences might have evolved and are salient in settings where miscoordination is particularly detrimental.

Appendices

Proofs

Proof of Lemma 1

Define the expected fitness payoff (in the stage game) of a player i with endowment c choosing action h with probability $\phi _i$ against an opponent j choosing h with probability $\phi _j$ as:

$$\begin{aligned} \Pi (\phi _i, \phi _j | c) :=\phi _i \, (1-\phi _j) \, f(c,{{\overline{s}}})\! + \!(1-\phi _i) \, \phi _j \, f(c,{\underline{s}}) \!+\! (1-\phi _i-\phi _j+2\phi _i\phi _j)f(c,0). \end{aligned}$$

Suppose now, to the contrary of the claim, that a type $\tau$ is stable against all $\tau ' \in {\hat{\mathop {\mathrm {{\mathscr {T}}}}\limits }}$ and there exists a subset of non-identical message combinations ${\hat{M}}\times {\hat{M}}' \subset M\times M$ such that for all $(c,c') \in {\hat{M}}\times {\hat{M}}'$, we have $\phi _{\tau }(c|c') \in (0,1)$ and $\lambda _L(c -c_0) \cdot \lambda _L(c'-c_0)>0$, meaning $\tau$ randomizes between actions for a set of messages with positive measure (given lottery L). Then, either both actions yield the same expected fitness, i.e. $\Pi (1, \phi _{\tau }(c'|c)|c) = \Pi (0, \phi _{\tau }(c'|c)|c)$, or there is a strict best response, meaning $\Pi (1, \phi _{\tau }(c'|c)|c) \ne \Pi (0, \phi _{\tau }(c'|c)|c)$. Note that different cases can apply to different subsets of ${\hat{M}}\times {\hat{M}}'$.

Suppose first the latter is the case and there is a strict best response for a (sub-)set of message combinations $\hat{{\underline{M}}}\times \hat{{\underline{M}}}' \subseteq {\hat{M}}\times {\hat{M}}'$. We can construct a type $\tau '$ that has identical preferences over lotteries as $\tau$, meaning that for any configuration $(\tau , \tau ', \epsilon )$, we have $L_{\tau }^{\epsilon } = L_{\tau '}^{\epsilon }$, with $L_{\tau }^{\epsilon }$ the lottery choice of a type $\tau$ given the configuration. Furthermore, let the strategy of a type $\tau '$ in the stage game be as follows:

$$\begin{aligned} \phi _{\tau '}(c'|c) = {\left\{ \begin{array}{ll} \arg \max _{x \in \{0,1\}} \Pi (x,\phi _{\tau }(c'|c)|c), &{} \text {if} \; (c,c') \in \hat{{\underline{M}}} \times \hat{{\underline{M}}}'\\ \phi _{\tau }(c'|c), &{} \text {otherwise},\\ \end{array}\right. } \end{aligned}$$

meaning $\tau '$ best-responds in the stage game for all message combinations in $\hat{{\underline{M}}}\times \hat{{\underline{M}}}'$, and reacts identical to $\tau$ otherwise. It follows that since $\lambda _{L}(\hat{{\underline{M}}}) \lambda _{L}(\hat{{\underline{M}}}') > 0$, $\lim _{\epsilon \rightarrow 0} V_{\tau '}(\tau , \tau ', \epsilon ) - V_{\tau }(\tau , \tau ', \epsilon ) > 0,$ which contradicts $\tau$ being stable.

Now, suppose instead for all $(c,c') \in {\hat{M}}\times {\hat{M}}'$, we have $\Pi (1, \phi _{\tau }(c'|c)|c) = \Pi (0, \phi _{\tau }(c'|c)|c)$, meaning actions h and d yield the same payoff in expectation. The anti-coordination nature of the stage game then implies that $f(c,{\underline{s}}) > \Pi (0, \phi _{\tau }(c'|c)|c) = \Pi (1, \phi _{\tau }(c'|c)|c)$. Let $\Pi _{\tau }(c) :=\Pi (0, \phi _{\tau }(c'|c)|c)$. Now, consider a type $\tau '$ with $\phi _{\tau '}(c|c') = 1$ and $\phi _{\tau '}(c'|c) = 0$ for all $(c,c') \in {\hat{M}}\times {\hat{M}}'$, and $\phi _{\tau '} = \phi _{\tau }$ otherwise. Given some $(c,c') \in {\hat{M}}\times {\hat{M}}'$, a type $\tau '$ receives an expected fitness payoff of $\Pi _{\tau '}(c) \ge \epsilon \cdot f(c,{\underline{s}}) + (1-\epsilon ) \Pi _{\tau }(c) > \Pi _{\tau }(c)$, noting that in an interaction with another $\tau '$ they receive at least $f(c,{\underline{s}})$, and if interacting with a type $\tau$, they obtain $\Pi _{\tau }(c)$ in expectation. Since the set ${\hat{M}}\times {\hat{M}}'$ is assumed to have positive measure and payoffs are equal for message combinations outside of the set, this implies as in the previous case that $\lim _{\epsilon \rightarrow 0} V_{\tau '}(\tau , \tau ', \epsilon ) - V_{\tau }(\tau , \tau ', \epsilon ) > 0$, contradicting the stability of $\tau$. $\square$

Proof of Lemma 2

Sufficiency: Let $L \in \mathop {\mathrm {{\mathscr {L}}}}\limits$ be the lottery chosen by a type $\tau$ and a type $\tau '$. Let $\phi _{\tau }(c'|c) \in \{0,1\}$ and $\phi _{\tau }(c|c') = 1- \phi _{\tau }(c'|c)$ for almost all $c, c' \in M$. Suppose there is a set ${\hat{M}} \times \hat{M'} \subset M \times M$ with positive measure (according to L) such that $\phi _{\tau '}(c|c') \in (0,1)$. For any c, this yields an expected fitness payoff in the stage game for a type $\tau '$ of:

$$\begin{aligned} \begin{aligned} \Pi _{\tau '}^{\epsilon }(c)&:=\epsilon \cdot \bigl [ \phi _{\tau '}(c|c') \bigl (1-\phi _{\tau '}(c'|c)\bigr ) f(c,{{\overline{s}}}) + \bigl (1-\phi _{\tau '}(c|c')\bigr ) \phi _{\tau '}(c'|c) f(c,{\underline{s}}) \\&\quad +\bigl (\phi _{\tau '}(c|c')\phi _{\tau '}(c'|c) + (1-\phi _{\tau '}(c|c'))(1-\phi _{\tau '}(c'|c)) \bigr )f(c,0) \bigr ]\\&\quad + (1-\epsilon ) \cdot \bigl [ \phi _{\tau '}(c|c') \bigl (1-\phi _{\tau }(c'|c)\bigr ) f(c,{{\overline{s}}}) + \bigl (1-\phi _{\tau '}(c|c')\bigr ) \phi _{\tau }(c'|c) f(c,{\underline{s}})\\&\quad \bigl (\phi _{\tau '}(c|c')\phi _{\tau }(c'|c) + (1-\phi _{\tau '}(c|c'))(1-\phi _{\tau }(c'|c)) \bigr )f(c,0) \bigr ]. \end{aligned} \end{aligned}$$

A type $\tau$ obtains:

$$\begin{aligned} \begin{aligned} \Pi _{\tau }^{\epsilon }(c)&:=\epsilon \cdot \bigl [ \phi _{\tau }(c|c') \bigl (1-\phi _{\tau '}(c'|c)\bigr ) f(c,{{\overline{s}}}) + \bigl (1-\phi _{\tau }(c|c')\bigr ) \phi _{\tau '}(c'|c) f(c,{\underline{s}})\\&\quad \times \bigl (\phi _{\tau }(c|c')\phi _{\tau '}(c'|c) + (1-\phi _{\tau }(c|c'))(1-\phi _{\tau '}(c'|c)) \bigr )f(c,0) \bigr ]\\&\quad + (1-\epsilon ) \cdot \bigl [ \bigl (1-\phi _{\tau }(c'|c)\bigr ) f(c,{{\overline{s}}}) + \phi _{\tau }(c'|c)f(c,{\underline{s}})\bigr ], \end{aligned} \end{aligned}$$

where we made use of the fact that $\phi _{\tau }(c'|c) = 1- \phi _{\tau }(c|c')$. Since $\phi _{\tau '}(c'|c) \in (0,1)$ and $f(c,{\underline{s}}) > f(c,0)$, we have $\lim _{\epsilon \rightarrow 0} \Pi _{\tau }^{\epsilon }(c) > \lim _{\epsilon \rightarrow 0} \Pi _{\tau '}^{\epsilon }(c)$. Note further that this extends to $\phi _{\tau '}(c'|c) \in [0,1]$, which given that strategies are non-identical would imply $\phi _{\tau '}(c'|c) = 1-\phi _{\tau }(c'|c)$, which leads to an allocation of 0 social good in any such pairing. As both types are assumed to choose the same L, this applies to all $c \in {\hat{M}}$ and we can conclude that $\lim _{\epsilon \rightarrow 0} V_{\tau }(\tau , \tau ', \epsilon ) - V_{\tau '}(\tau , \tau ', \epsilon )>0$, as required.

Necessity: This follows directly from the proof of Lemma 1. $\square$

Proof of Lemma 3

Take a population configuration $(\tau , \tau ', \epsilon )$ and suppose the resident types preferences over lotteries are such that they choose the trivial lottery in every equilibrium. Consider a sequence of lotteries $\{L_n\}_{n=1}^{\infty }$ with each $L_n$ such that there is a probability mass of $\frac{1}{2n+1}$ at $-\delta$, and a probability mass of $1-\frac{1}{2n+1}$ distributed uniformly over the interval $[0, \frac{\delta }{n}]$, with $\delta < c_0$. Note that every such $L_n$ is a fair and bounded lottery. It follows from Lemma 2 that if a type $\tau$ is stable, then if there are distinct messages, we have $1- \phi _{\tau '}(c'|c) \in \{0,1\}$ and $\phi _{\tau }(c|c') = 1- \phi _{\tau '}(c'|c)$. Suppose $\tau$ follows such a strategy.

Given the population configuration and lottery choices, the fitness payoff of a type $\tau$ in any equilibrium is bounded above by

$$\begin{aligned} f_{\tau }(\epsilon ) :=(1-\epsilon ) f\bigl (c_0, s_0\bigr ) + \epsilon f\bigl (c_0, {{\overline{s}}}\bigr ), \end{aligned}$$

where $s_0 :=\frac{{{\overline{s}}}{\underline{s}}}{{{\overline{s}}}+{\underline{s}}}$, i.e. the expected interior Nash equilibrium payoff, and we made use of Jensen’s inequality, noting that $f(c_0, s_0) \ge {{\,\mathrm{{\mathbb {E}}}\,}}[f(c_0, S(a_i, a_j)) | p_{\text {NE}}(a_i,a_j)]$ for a f weakly concave in s, with $p_{\text {NE}}(a_i,a_j)$ the Nash equilibrium probability distribution over action combinations.

The expected fitness of a type $\tau '$ choosing a lottery $L^n$ and stage game strategy $\phi _{\tau '} = \phi _{\tau }$ is bounded below by:

$$\begin{aligned} f_{\tau '}^n(\epsilon ) :=\left( 1-\frac{1}{2n}\right) f\bigl (c_0, {\underline{s}}\bigr ) \end{aligned}$$

where we set the fitness of all types with consumption outcomes $c < c_0$ to 0. Note that $s_0 < {\underline{s}}$ and hence $f(c_0,{\underline{s}}) > f(c_0,s_0)$. We can conclude that:

$$\begin{aligned} \lim _{\epsilon \rightarrow 0}\lim _{n \rightarrow \infty } f_{\tau }(\epsilon ) - f_{\tau '}^n(\epsilon ) < 0. \end{aligned}$$

It follows from continuity that we can find $n^*$ and $\epsilon ^*$ such that for every $n>n^*$ and every $\epsilon <\epsilon ^*$, we have

$$\begin{aligned} V_{\tau '}(\tau ,\tau ',\epsilon ) \ge f_{\tau '}^n(\epsilon ) > f_{\tau }(\epsilon ) \ge V_{\tau }(\tau ,\tau ',\epsilon ) \end{aligned}$$

in some equilibrium if $L^n \in \mathop {\mathrm {{\mathscr {L}}}}\limits$. The result follows. $\square$

Proof of Proposition 1

Take a configuration $(\tau , \tau ', \epsilon )$. It follows from Lemma 1 that if (i) is violated, $\tau$ cannot be stable. Suppose now (i) holds and there is an equilibrium in which type $\tau$ chooses a lottery L and $\tau '$ chooses a lottery ${\hat{L}} \ne L$, $\; {\hat{L}} \in \mathop {\mathrm {{\mathscr {L}}}}\limits$, while $\phi _{\tau } = \phi _{\tau '}$. Let ${{\overline{M}}}_L(c_0)$ be the smallest interval that contains $M_L(c_0)$.

Case $c_0 \in M_L(c_0)$: Condition (ii) implies that $\int F_{\tau }(c|G) \lambda _L(c-c_0) dc = F_{\tau }(c_0|G)$. Furthermore, (ii) and (iii) together imply that

$$\begin{aligned} \int F_{\tau }(c|G) \lambda _{{\hat{L}}}(c-c_0) dc \le F_{\tau }(c_0|G) = \int F_{\tau }(c|G) \lambda _{L}(c-c_0) dc. \end{aligned}$$

Any such lottery$\; {\hat{L}}$ yields weakly lower expected fitness than L and hence $V_{\tau }(\tau , \tau ',\epsilon ) \ge V_{\tau '}(\tau , \tau ',\epsilon )$.

Case $c_0 \notin M_L(c_0)$: Take any $c^* \notin M_L(c_0)$ but $c^* \in {{\overline{M}}}_L(c_0)$. As any L must be fair, there exist $c_1, c_2, c_3 \in M_L(c_0)$ such that $c_1< c^*< c_2 < c_3$. Suppose $F_{\tau }(c^*) > \alpha F_{\tau }(c_3) + (1-\alpha ) F_{\tau }(c_1)$, with $\alpha = \frac{c^* - c_1}{c_3 - c_1}$. Note that the right-hand side corresponds to the expected (fitness) outcome of an individual with consumption $c^*$ that chooses a (fair) lottery $L_{\alpha } \in \mathop {\mathrm {{\mathscr {L}}}}\limits$ that yields $c_3- c^*$ with probability $\alpha = \frac{c_3 - c^*}{c_3 - c_1}$ and $c_1$ with $1-\alpha$.

Now, consider a lottery $L_{\beta }$ that yields $c_3 - c_2$ with probability $\beta = \frac{c_2 - c_1}{c_3 - c_1}$ and $c_1 - c_2$ with $1-\beta$. Clearly, $L_{\beta }$ is also fair and bounded and hence in $\mathop {\mathrm {{\mathscr {L}}}}\limits$. It follows that

$$\begin{aligned} \int F(c|G)\lambda _{L_{\beta }}(c-c_2) dc = \beta F_{\tau }(c_3|G) + (1-\beta ) F_{\tau }(c_1|G) = F_{\tau }(c_2|G) \end{aligned}$$

where the last equality follows from (ii), noting that $M_{L_{\beta }}(c_2) = \{c_1, c_3\} \subset M_L(c_0)$. Consider another lottery $L_{\gamma }$ that yields $c_3 - c_2$ with probability $\gamma = \frac{c_3 - c_2}{c_3-c^*}$ and $c^* - c_2$ with $1-\gamma$. This is again a fair and bounded lottery. By construction:

$$\begin{aligned} \gamma F_{\tau }(c_3|G) + (1-\gamma ) F_{\tau }(c^*|G) >\gamma F_{\tau }(c_3|G) + (1-\gamma )\bigl [\alpha F_{\tau }(c_3|G) + (1-\alpha ) F_{\tau }(c_1|G)\bigr ] \end{aligned}$$

But then

$$\begin{aligned} \begin{aligned} \gamma F_{\tau }(c_3) + (1-\gamma ) F_{\tau }(c^*|G) >&\; \bigl (\gamma + (1-\gamma )\alpha \bigr ) F_{\tau }(c_3|G) + (1-\gamma )(1-\alpha ) F_{\tau }(c_1|G)\bigr )\\ =&\; \beta F_{\tau }(c_3|G) + (1-\beta ) F_{\tau }(c_1|G), \end{aligned} \end{aligned}$$

This contradicts (iii). It follows that for any $c^*$ in ${{\overline{M}}}_{L}(c_0)$, we have $F_{\tau }(c^*|G) \le \int F_{\tau }(c|G) \lambda _{L^*}(c-c^*) dc$, for any $L^* \in \mathop {\mathrm {{\mathscr {L}}}}\limits$ such that $M_{L^*}(c^*) \subseteq M_L(c_0)$. Furthermore, this holds with equality for any $c^* \in M_L(c_0)$. It then follows that $\int F(c|G) \lambda _{L}(c-c_0) dc \ge \int F_{\tau }(c|G) \lambda _{L'}(c-c_0) dc$ for any $L' \in \mathop {\mathrm {{\mathscr {L}}}}\limits$ with $M_{L'}(c_0) \subseteq M_{L}(c_0)$. It follows directly from (iii) that this inequality also holds for an arbitrary ${\hat{L}} \in \mathop {\mathrm {{\mathscr {L}}}}\limits$.

The final step is to show that a departure from a strategy $\phi _{\tau }$ that efficiently correlates actions cannot lead to higher expected fitness if the lottery choice of the resident satisfies (ii) and (iii). If $\phi _{\tau '}$ differs from $\phi _{\tau }$ for a measurable set of message combinations in an equilibrium with an aggregate distribution G, then as $\epsilon \rightarrow 0$, $F_{\tau '}(c|G)$ approaches

$$\begin{aligned} F_{\tau '}(c|G) \!=\! \int \Bigl ( \phi _{\tau '}(x|c)(1-\phi _{\tau }(c|x))\cdot f(c,{{\overline{s}}}) + (1-\phi _{\tau '}(x|c))(\phi _{\tau }(c|x))\cdot f(c,{\underline{s}}) \Bigr ) dG(x). \end{aligned}$$

Using the same argument as in the proof of Lemma 2, we can conclude that $\lim _{\epsilon \rightarrow 0} F_{\tau '}(c|G) - F_{\tau }(c|G) \le 0$ for all $c \in M_G$, with the inequality strict for any c where $\phi _{\tau '}(c'|c)$ differs from $\phi _{\tau }(c'|c)$ for a set of messages with positive measure according to G. Combining this with the previous argument, we can further conclude that for any ${\hat{L}} \in \mathop {\mathrm {{\mathscr {L}}}}\limits$, as $\epsilon \rightarrow 0$, we have:

$$\begin{aligned} \int F_{\tau '}(c|G) \lambda _{{\hat{L}}}(c-c_0) dc < \int F_{\tau }(c|G) \lambda _{{\hat{L}}}(c-c_0) dc \le \int F_{\tau }(c|G) \lambda _{L}(c-c_0) dc \end{aligned}$$

with L the lottery choice of $\tau$ in equilibrium. Accordingly, $\lim _{\epsilon \rightarrow 0} V_{\tau '}(\tau ,\tau ',\epsilon ) - V_{\tau }(\tau ,\tau ',\epsilon ) < 0$. The result follows. $\square$

Proof of Proposition 2

Let $v: {\mathop {\mathrm {{\mathbb {R}}}}\limits }_+ \times [0,1] \rightarrow {\mathop {\mathrm {{\mathbb {R}}}}\limits }_+$ be a Bernoulli utility function over consumption and rank defined as follows:

$$\begin{aligned} v(c,r) :=r \cdot f(c,{{\overline{s}}}) + (1-r) \cdot f(c,{\underline{s}}). \end{aligned}$$

(5)

Suppose preferences over lotteries in $\mathop {\mathrm {{\mathscr {L}}}}\limits$, given an aggregate consumption distribution G and a consumption endowment c, can be represented as follows:

$$\begin{aligned} V(L \; | \; c, G) = \int v\bigl (c+x,r_G(c+x)\bigr ) \lambda _{L}(x-c) dx, \end{aligned}$$

(6)

where $r_G(x)$ captures the measure of individuals with consumption below x (or a convex combination between the measure of individuals strictly and weakly below x in the case of a mass point at x), defined as

$$\begin{aligned} r_G(x) :={\left\{ \begin{array}{ll} G(x), &{} \text {if } \lambda _G(x) = 0\\ \alpha {\underline{G}}(x) + (1-\alpha ) G(x), &{} \text {otherwise}, \end{array}\right. } \end{aligned}$$

for some $\alpha \in (0,1)$ and ${\underline{G}}(x)$ the left-limit of G(x). $V(L \; | \; c, G)$ amounts to the expected utility from consumption and consumption rank outcomes under lottery L, given G and c. It follows from the proof of Lemma 3 that a lottery choice from the set of all fair and bounded lotteries that maximizes (6) cannot result in a mass point at any $c>0$ for a pure population in any equilibrium.

Consider the following strategy in the stage game:

$$\begin{aligned} \phi (c'|c) = {\left\{ \begin{array}{ll} 1, &{} \text {if } c > c',\\ 0, &{} \text {if } c < c',\\ \frac{f(c,{{\overline{s}}})- f(c,0)}{f(c,{{\overline{s}}}) + f(c,{\underline{s}}) - 2 f(c,0)}, &{} \text {otherwise}. \end{array}\right. } \end{aligned}$$

There exist preferences in the stage game, conditional on c and $c'$, that admit this as a unique equilibrium strategy (i.e. Table 5). It follows from Lemma 2 that a type with such preferences is evolutionarily stable against any other type that has the same choice behaviour over lotteries. The remaining question is whether V leads to equilibrium choices over lotteries that are neutrally stable.

Given Assumptions 1 and 2, v is continuous and strictly concave in c. This means conditions of Proposition 1 (i) of Ray and Robson [34] are satisfied. This result implies the existence of a unique equilibrium distribution over messages G in a pure population with a corresponding lottery choice $L \in \mathop {\mathrm {{\mathscr {L}}}}\limits$ that is among the maximizers of (5). Furthermore, in any equilibrium, $v(c,r_G(c))$ is linear over the support of G and strictly concave elsewhere. By construction, for a continuous G,

$$\begin{aligned} \begin{aligned} F_{\tau }(c|G) =&\int \phi _{\tau }(x|c) f(x,{{\overline{s}}}) dG(x) + \int \bigl (1-\phi _{\tau }(x|c)\bigr ) f(x,{\underline{s}}) dG(x)\\ =&r_G(c) f(c,{{\overline{s}}}) + (1-r_G(c)) f(c,{\underline{s}}). \end{aligned} \end{aligned}$$

(7)

Hence, in any equilibrium, $F_{\tau }(c|G)$ is linear in c over the support of G and strictly concave elsewhere. Conditions (i)-(iii) of Proposition 2 are satisfied. Such preferences are neutrally stable. $\square$

Additional Result: Necessity of Linearity

The following result shows that stable preferences must be such that they induce a lottery choice and stage game strategy in equilibrium that render $F_{\tau }(c|G)$ linear and strictly increasing over the support of the consumption distribution G. This implies that condition (ii) of Proposition 1 is a necessary condition. And furthermore, while roles in the stage game do not have to be hierarchical in the sense that higher consumption yields a higher allocation of s, expected fitness does need to be strictly increasing in consumption (given the expected allocation of s for each consumption level).

Result 1

A type $\tau \in \mathop {\mathrm {{\mathscr {T}}}}\limits$ is neutrally stable only if for any equilibrium of the pure population with a consumption distribution G, $\; F_{\tau }(c|G)$ is strictly increasing and linear in c, for almost all c in the support of G.

Proof

Let $L_{\tau }$ be the lottery choice corresponding to such a G. Suppose not and there is an interval $I \subset {{\,\textrm{supp}\,}}(L_{\tau })$ such that $F_{\tau }(c|G)$ is weakly decreasing in m over I and strictly increasing elsewhere. Note that for a stability, $L_{\tau }$ needs to be continuous. Then for almost all $c \in \mathop {\mathrm {{\mathbb {R}}}}\limits _+$, $F_{\tau }(c|G)$ is bounded below by $f(c,{\underline{s}})$, which is strictly concave in c. This implies that $F_{\tau }(c|G)$ can only be weakly decreasing over a bounded interval. Let ${{\overline{c}}} \equiv \sup I$, which by the previous argument must exist. Then for almost all $c \in I$, any fair gamble $(1-\alpha ) (c-\epsilon ) + \alpha {{\overline{c}}}+\epsilon$ must deliver expected fitness strictly greater $F_{\tau }(c|G)$ for an arbitrarily small $\epsilon$ and the $\alpha \in (0,1)$ that makes the lottery fair. Let ${\hat{L}}$ be such a lottery.

Then, there exists a mutant type $\tau '$, with preferences such that ${\hat{V}}({\hat{L}}|c,G) > {\hat{V}}(L|c,G)$, for all $L \in \mathop {\mathrm {{\mathscr {L}}}}\limits$, with $L \ne {\hat{L}}$. It follows that for such a $\tau '$, we have $\lim _{\epsilon \rightarrow 0} V_{\tau '}(\tau , \tau ', \epsilon ) - V_{\tau }(\tau , \tau ', \epsilon ) > 0,$ which contradicts $\tau$ being stable. It can be easily verified that the same is true if $F_{\tau }(c|G)$ is increasing but strictly convex over an interval I. $\square$