Doubts and equilibria

Abstract

In real life strategic interactions decision-makers are likely to entertain doubts about the degree of optimality of their play. To capture this feature of real choice-making, we present a model based on the doubts felt by an agent about how well is playing a game. The doubts are coupled with (and mutually reinforced by) imperfect discrimination capacity, which we model by means of similarity relations. These cognitive features, together with an adaptive learning process guiding agents’ choice behavior leads to doubt-based selection dynamic systems. We introduce the concept of Mixed Strategy Doubt Equilibrium and study its relationship with the Nash equilibrium concept.

Appendices

Appendix

A Satisficing procedural preferences based on similarity judgements

It is safe to say that doubts are closely related to imperfect discrimination capacity. Thus, we will assume that agents observe the expected payoff and the popularity attached to their current pure strategy with some noise. We model this imperfection by means of an extension of Rubinstein type of similarity relations (see Rubinstein 1988), which we call correlated similarity relations (see Aizpurua et al. 1993; Uriarte 1999).¹⁰ Then the agents would build a procedural preference relation compatible with those similarity relations, as in Rubinstein (1988).

We proceed as follows, first we shall build the correlated similarity relations and second, we show how the agent may proceed to build his preference and decide about the pure strategies.

To be more specific, let (π _ki , f _ki ) be the vector of expected payoff-proportion of agents of player population k attached to strategy i ∈ S _k at time t with f _ki  ∈ (0,1).

I Correlated Similarities on ${\Pi _{ki}}$ and ${F_{ki}}$

The doubt function serves to build correlated similarity relations on both Π _ki and F _ki . Let (π _ki ,f _ki ) and $(\overline{ \pi }_{ki},\overline{f}_{ki})$ be two vectors in Π _ki ×F _ki , with $\overline{f}_{ki}$, f _ki ∈ (0,1).

• (a) On the space of expected payoffs, Π _ki

The doubt function d _ki defines correlated similarities of the difference-type as follows: given f _ki we say that $\overline{\pi }_{ki}$ is similar to π _ki , (formally written as $\overline{\pi }_{ki}S\Pi \lbrack f_{ki}]\pi _{ki}$ ), if and only if $\left\vert \overline{\pi } _{ki}-\pi _{ki}\right\vert \leqq$ d _ki (f _ki ), where $\left\vert.\right\vert$ stands for absolute value. Thus, there is one similarity relation on Π _ki , for each $\overline{f}_{ki}\in (0,1)$.

Then the similarity interval of π _ki , given f _ki is:

$$ \lbrack \pi _{ki}-d_{ki}(f_{ki}),\pi _{ki}+d_{ki}(f_{ki})] $$

Note that d _ki (f _ki ), the doubt level felt by ∑ agent ki given the proportion f _ki , becomes the threshold level in the definition of this type of similarity relation. If f _ki increases, the threshold, d _ki (f _ki ), decreases and so the similarity intervals of π _ki shrink (giving rise to the vertical cone-shaped form in Fig. 1). This means that when f _ki increases, the discrimination capacity on the space of expected payoffs to strategy i increases (probably because the accumulated experience with strategy i has increased due to the increased number of agents from population k currently playing strategy i). When f _ki is such that π _ki  − d _ki (f _ki ) ≤ m and π _ki  + d _ki (f _ki ) ≥ M, the whole set Π _ki = [m,N] is similar to π _ki and when f _ki  = 1 only π _ki is similar to itself. This variations in perception induces a vertical wedge type form, as it can be seen in Fig. 1.

Fig. 1

Preference-indifference relation compatible with correlated similarities. Relative to (π _ki , f _ki ), U = U _α ∪ U _β ∪ U _δ denotes the upper-contour (or aspiration) set, L = L _α ∪ L _β ∪ L _δ the lower contour set and the darker area is the indifference set

Notice that for a Cartesian skeptical agent, the similarity interval is

$$ \lbrack \pi _{ki}-d_{ki}(f_{ki}),\pi _{ki}+d_{ki}(f_{ki})]=[m,M] $$

That is, since in this case d _ki  ∈ D _M , then d _ki (f _ki ) > M for all f _ki  ∈ (0,1), thus m is similar to M, and hence the similarity on Π _ki is degenerate (see Rubinstein 1988).

• (b) On the strategy frequency space, F _ki

The doubt function d _ki defines correlated similarity relations of the ratio-type by means of the continuous function λ _ki :Π _ki →R, which is defined as follows: for a given f _ki  ∈ (0,1) and d _ki ,

$$ \lambda _{ki}(\pi _{ki})=\frac{\pi _{ki}}{\pi _{ki}-d_{ki}(f_{ki})} $$

The properties of λ _ki are the following:

• (i) Given f _ki and d _ki , if π _ki increases, λ _ki (π _ki ) decreases continuously and thus, the similarity interval shrinks. This means that when the expected payoffs at stake increase, the discrimination efforts on the frequency space, F _ki , increases (generating a kind of horizontal wedge type form, as it is shown in Fig. 1)

• (ii) Keeping the function d _ki , and π _ki constant, if the frequency f _ki increases, then λ _ki (π _ki ) decreases and so the similarity intervals of the higher frequency shrink.

Note that:

• 1. For the herding agents: d _ki  ∈ D _m and the function λ _ki  > 1 is then used to define on F _ki correlated similarity relations of the ratio-type whose similarity interval for that f _ki is:

  $$ \lbrack f_{ki}/\lambda _{ki}(.)\text{, \ }f_{ki}.\lambda _{ki}(.)] $$

• 2. For the skeptical agents: d _ki  ∈ D _M , then λ _ki  < 0 will define a degenerate similarity relation (see Rubinstein 1988). Thus, when doubts are of a skeptical nature, the similarity relations on both Π _ki and F _ki are degenerate; that is, 0 is similar to 1. Formally, on F _ki , given a f _ki  ∈ (0,1), the correlated similarity relation SF _ki [π _ki , f _ki ] will induce the following similarity intervals for f _ki :

  $$ \lbrack f_{ki}/\lambda _{ki}(.)\text{, \ }f_{ki}/\lambda _{ki}(.)]=[0,1] $$

  The size of this degenerate similarity interval does not change with π _ki ; it remains constant for any value of π _ki .

II Procedural Preference on ${\Pi _{ki}\times F_{ki}}$

Based on a model developed in Uriarte (1999), we show now how the above two correlated similarity relations build a (non-complete and non-transitive) preference-indifference relation defined on the space of expected payoffs and frequencies, Π _ki ×F _ki , attached to pure strategy i ∈ S _k . Let us assume that each agent ki compares pairs of alternatives in Π _ki ×F _ki with the aid of a pair of correlated similarity relations to decide which of the two is preferred. The agent may define a procedural preference $\succsim _{ki}$on Π _ki ×F _ki by means of the pair of correlated similarities and know his aspiration set U at each t (which we identify with the upper contour set of the vector (π _ki ,f _ki ) at t, U = U _α  ∪ U _β  ∪ U _δ ; see Fig. 1). That is, given a pair of vectors $(\overline{\pi }_{ki}, \overline{f}_{ki})$ and (π _ki ,f _ki ) in Π _ki ×F _ki , the vector $(\overline{\pi }_{ki},\overline{f}_{ki})$ will be declared to be preferred to (π _ki ,f _ki ), i.e. $(\overline{\pi }_{ki},\overline{f} _{ki})\succ _{ki}$ (π _ki ,f _ki ), whenever the agent ki perceives that one of the following three conditions is met. (Note that since $( \overline{\pi }_{ki},\overline{f}_{ki})$ is to be preferred, the conditional similarity relation SΠ on Π _ki given $\overline{f}_{ki}$ and the conditional similarity relation SF on F _ki given $\overline{\pi } _{ki}$ and $\overline{f}_{ki}$ are to be used):

Condition α

$\overline{\pi }_{ki}>\pi _{ki}$, and no $\overline{\pi }_{ki}S\Pi \lbrack \overline{f}_{ki}]\pi _{ki}$; while $\overline{f}_{ki}SF[\overline{\pi }_{ki},\overline{f}_{ki}]f_{ki}.$

In words, $\overline{\pi }_{ki}$ is bigger than π _ki and, given $\overline{f}_{ki}$, $\overline{\pi }_{ki}$ is perceived to be not similar to π _ki ; while, $\overline{f}_{ki}$ is perceived to be similar to f _ki . U _α in Fig. 1 is the area implied by this condition.

Condition β

$\overline{f}_{ki}>f_{ki}$ and no $\overline{ f}_{ki}SF[\overline{\pi }_{ki},\overline{f}_{ki}]f_{ki};$while $\overline{ \pi }_{ki}S\Pi \lbrack \overline{f}_{ki}]\pi _{ki}.$

In words, $\overline{f}_{ki}$ is bigger than f _ki and, given $\overline{ \pi }_{ki}$ and $\overline{f}_{ki},$ $\overline{f}_{ki}$ is perceived to be not similar to f _ki ; while, given $\overline{f}_{ki},$ $\overline{\pi }_{ki}$ is perceived to be similar to π _ki .U _β in Fig. 1 is the area implied by this condition.

Condition δ

$\overline{\pi }_{ki}>\pi _{ki}$ and no $\overline{\pi }_{ki}S\Pi _{ki}[\overline{f}_{ki}]\pi _{ki}$; $\overline{f} _{ki}>f_{ki}$ and no $\overline{f}_{ki}SF[\overline{\pi }_{ki},\overline{f}_{ki}]f_{ki}$.

That is, vector $(\overline{\pi }_{ki},\overline{f}_{ki})$ is strictly bigger than (π _ki ,f _ki ) and no similarity is perceived in both instances. U _δ in Fig. 1 is the area implied by this condition.

Indifference

Whenever both expected payoffs and strategy proportions are perceived to be similar, then the two vectors will be declared indifferent; i.e. when $\overline{\pi } _{ki}S\Pi \lbrack \overline{f}_{ki}]\pi _{ki}$, $\pi _{ki}S\Pi \lbrack f_{ki}]\overline{\pi }_{ki}$, $\ \overline{f}_{ki}SF[\overline{\pi }_{ki}, \overline{f}_{ki}]f_{ki}$ and $f_{ki}SF[\pi _{ki},f_{ki}]\overline{f}_{ki}$, then $(\overline{\pi }_{ki},\overline{f}_{ki})\sim _{ki}$ (π _ki ,f _ki ).

When none of these four situations takes place, then the two vectors would be non-comparable (see Fig. 1).

The distance to the aspiration set U depends on how thick the indifference set of (π _ki , f _ki ) is. We assume here that agents are preference-satisficers; that is, they choose a strategy to reduce the distance from (π _ki , f _ki ) to U. The Herding agent can achieve this by reducing doubts by means of playing more popular strategies and/or increasing expected payoffs. The smaller (greater) that distance the more satisfied (dissatisfied) the ki-agent will be with his current strategy. It can be seen that the properties (i) and (ii) of λ _ki capture the changes in the thickness of the indifference sets. Hence, the λ _ki function can be thought of as an index of how dissatisfied the ki-agent is with his current strategy. Notice that a doubt-less agent’s indifference classes will consists of almost singletons: $\sim \lbrack (\pi _{ki}$, f _ki )] ≅ (π _ki , f _ki ) conveying the idea that with almost no doubts about the goodness of the current strategy, the doubt-less agent feels very satisfied and, very likely, will not switch to a different strategy.

The Skeptical agent will have indifference sets that will cover the entire choice space because the similarity intervals on both the Π _ki and F _ki spaces are degenerate. Thus, we will say that the Skeptical agent’s preference relation is degenerate. Thus, for tis type of agent any pair of expected payoffs will be similar, as well as any pair of strategy frequencies. Hence, in terms of preferences, the agent will not perceive real differences between any two different vectors in Π _ki ×F _ki and he will declare to be indifferent among them. Thus, having the thickest indifference sets that are possible, the upper contour sets (i.e. agents’s aspiration set) will appear to be unreachable and the Skeptical agent will feel highly dissatisfied.

B The index of dissatisfied agents

Given the expected payoffs and the frequencies attached to each of the pure strategies of population k, we propose the index of dissatisfied agents with pure strategy i to be represented by the agent ki ^′s dissatisfaction level relative to the total dissatisfaction level of population k (or, the agent $ki {\acute{}}$s threshold divided by player population $k {\acute{}}$s threshold on F _ki ):

$$ \alpha _{ki}=\alpha _{ki}(\pi _{ki},f_{ki},\pi _{kj},f_{kj})=\frac{\lambda _{ki}}{\Sigma _{_{i=1}}^{m_{k}}\lambda _{ki}}=\frac{\lambda _{ki}}{\lambda _{k}}\text{, }i\neq j $$

To avoid the use of different doubt parameters, we will only assume that either they are all herding agents or skeptical ones; no mixed populations of doubtful agents are allowed. Further, we assume that the herding doubts agents do perceive the changes in payoffs and frequencies. But we cannot assume the same for the skeptical agents. This is so because, as said above, the skeptical agent has similarity intervals that are degenerate and, as a consequence, in terms of preferences, he does not distinguish between any two different vectors in Π _ki ×F _ki . Therefore, the stimulus intensity received by this agent from any vector (π _ki , f _ki ) would be the same and hence the response probability is the same for each strategy. Furthermore, since a Cartesian skeptical agent is endowed with universal doubts, he will always be dissatisfied and continuously experimenting with every available strategy, no matter the level of payoffs and popularity attached to each strategy. For this reason, we may say that this type of agents react in a “non-standard” way to the changes in the expected payoffs and strategy frequencies.

The properties of α _ki follow naturally from those of λ _ki ; hence, the properties of α _ki for the herding agent ki are the following:

• 1. The proportion of dissatisfied agents with their current pure strategy i ∈ S _k will decrease if expected payoffs to strategy i ∈ S _k , π _ki , increase.

  $$ \frac{\partial \alpha _{ki}}{\partial \pi _{ki}}=\frac{\frac{\partial \lambda _{ki}}{\partial \pi _{ki}}\lambda _{k}-\frac{\partial \lambda _{ki}}{ \partial \pi _{ki}}\lambda _{ki}}{\lambda _{k}^{2}}=\frac{\frac{ -d_{ki}(f_{ki})}{\left( \pi _{ki}-d_{ki}(f_{ki})\right) ^{2}}\left( \lambda _{k}-\lambda _{ki}\right) }{\lambda _{k}^{2}}<0 $$

• 2. The proportion of dissatisfied agents with their current pure strategy i ∈ S _k should increase if expected payoffs to strategy j ∈ S _k , π _kj , increase.

  $$ \frac{\partial \alpha _{ki}}{\partial \pi _{kj}}=\frac{-\frac{\partial \lambda _{kj}}{\partial \pi _{kj}}\lambda _{ki}}{\lambda _{k}^{2}}=\frac{ \frac{-d_{kj}(f_{kj})}{\left( \pi _{kj}-d_{kj}(f_{kj})\right) ^{2}}\left( -\lambda _{ki}\right) }{\lambda _{k}^{2}}>0 $$

• 3. If agents ki’s doubts decrease, because the popularity of strategy i , f _ki , has increased, the proportion of dissatisfied should decrease too.

  $$ \frac{\partial \alpha _{ki}}{\partial f_{ki}}=\frac{\frac{\partial \lambda _{ki}}{\partial f_{ki}}\lambda _{k}-\frac{\partial \lambda _{ki}}{\partial f_{ki}}\lambda _{ki}}{\lambda _{k}^{2}}=\frac{\frac{\pi _{ki}\frac{\partial d_{ki}(f_{ki})}{\partial f_{ki}}}{\left( \pi _{ki}-d_{ki}(f_{ki})\right) ^{2} }\left( \lambda _{k}-\lambda _{ki}\right) }{\lambda _{k}^{2}}<0\ $$

• 4. If the popularity of strategy j ∈ S _k , f _kj , increases, the proportion of dissatisfied agents with their current pure strategy i ∈ S _k should increase.

  $$ \frac{\partial \alpha _{ki}}{\partial f_{kj}}=\frac{-\frac{\partial \lambda _{kj}}{\partial f_{kj}}\lambda _{ki}}{\lambda _{k}^{2}}=\frac{\frac{\pi _{kj} \frac{\partial d_{kj}(f_{kj})}{\partial f_{kj}}}{\left( \pi _{kj}-d_{kj}(f_{kj})\right) ^{2}}\left( -\lambda _{ki}\right) }{\lambda _{k}^{2}}>0 $$

C Proofs of propositions

Let

	(y) L	R
(x) U	a 11, b 11	a 12, b 12
D	a 21, b 21	a 22, b 22

denote the 2×2 constant-sum game G, and $I^{\ast }\equiv \left[ \left( x^{\ast },1-x^{\ast }\right),\left( y^{\ast },1-y^{\ast }\right) \right]$, with x ^∗ > 0 and y ^∗ > 0, the Mixed strategy Nash Equilibrium of G. To get this equilibrium, we may assume, without loss of generality, that a ₁₁ > a ₂₁, b ₁₁ < b ₁₂, a ₁₂ < a ₂₂, and b ₂₂ < b ₂₁. Recall that payoffs are normalized so that they take values on [m,M]. To avoid the use of four different doubt parameters, we shall assume that the four doubt functions are the same: d _D  = d _U  = d _R  = d _L  = d. The doubt-based selection dynamics (for definition (a) of λ _ki ) are represented by the following system:

$$\begin{array}{rll} \overset{\cdot }{x} &=&\frac{x\left( 1-x\right) }{\pi _{U}\left( \pi _{D}-d_{D}\right) +\pi _{D}\left( \pi _{U}-d_{U}\right) }\left( \pi _{U}d_{D}-\pi _{D}d_{U}\right) \nonumber\\ &=&\frac{x\left( 1-x\right) }{\pi _{U}\left( \pi _{D}-d_{D}\right) +\pi _{D}\left( \pi _{U}-d_{U}\right)}\nonumber\\ &&\times\left( (a_{11}y+a_{12}(1-y))d_{D}\left( 1-x\right) -(a_{21}y+a_{22}(1-y))d_{U}\left( x\right) \right) \nonumber\\ &\equiv &G_{1}(x,y)F_{1}(x,y) \end{array}$$

(C.1)

$$\begin{array}{rll} \overset{\cdot }{y} &=&\frac{y\left( 1-y\right) }{\pi _{L}\left( \pi _{R}-d_{R}\right) +\pi _{R}\left( \pi _{L}-d_{L}\right) }\left( \pi _{L}d_{R}-\pi _{R}d_{L}\right) \nonumber\\ &=&\frac{y\left( 1-y\right) }{\pi _{L}\left( \pi _{R}-d_{R}\right) +\pi _{R}\left( \pi _{L}-d_{L}\right)} \nonumber\\ &&\times\left( (b_{11}x+b_{21}(1-x))d_{R}\left( 1-y\right) -(b_{12}x+b_{22}(1-x))d_{L}\left( y\right) \right) \nonumber\\ &\equiv &G_{2}(x,y)F_{2}(x,y) \end{array}$$

(C.2)

Proof of Proposition 2

• 1. We must first show that a Mixed Strategy Nash Equilibrium (MSNE) converges to a Mixed Strategy Doubt-Full Equilibrium (MSDFE) as δ converges to zero in the class of doubt functions $D^{1-\delta }\subset D_{M}$. An interior rest point of C.1–C.2, (i.e. a MSDE), satisfies:

  $$\begin{array}{lll} \left( a_{11}y+a_{12}\left( 1-y\right) \right) d_{D}\left( 1-x\right) -\left( a_{21}y+a_{22}\left( 1-y\right) \right) d_{U}\left( x\right) &=&0 \\ \left( b_{11}x+b_{21}\left( 1-x\right) \right) d_{R}\left( 1-y\right) -\left( b_{12}x+b_{22}\left( 1-x\right) \right) d_{L}\left( y\right) &=&0 \end{array}$$

  Then, if $d_{i}\in D^{1-\delta }$ for i ∈ {U,D,L,R},

  $$ \lim_{\delta \rightarrow 0}\frac{d_{U}\left( x\right) }{d_{D}\left( 1-x\right) }=\lim_{\delta \rightarrow 0}\frac{d_{L}\left( y\right) }{ d_{R}\left( 1-y\right) }=1\text{, for all }(x,y)\in \left( 0,1\right) \times \left( 0,1\right) $$

  Now suppose that we are in the MSNE, $(x^{\ast },y^{\ast })\in \left( 0,1\right) \times \left( 0,1\right)$, of G and that $d_{i}\in D^{1-\delta }$. Then, the strategies available to each player get the same expected payoff; that is $a_{11}y^{\ast }+a_{12}\left( 1-y^{\ast }\right) =a_{21}y^{\ast }+a_{22}\left( 1-y^{\ast }\right)$ and $b_{11}x^{\ast }+b_{21}\left( 1-x^{\ast }\right) =b_{12}x^{\ast }+b_{22}\left( 1-x^{\ast }\right)$. Thus,

  $$\begin{array}{lll} &&{\kern-6pt} \lim\limits_{\delta \rightarrow 0}\frac{\big(a_{11}y^{\ast }+a_{12}\left( 1-y^{\ast }\right)\!\big) d_{D}\left( 1-x^{\ast }\right) }{\big(a_{21}y^{\ast }+a_{22}\left( 1-y^{\ast }\right)\!\big) d_{U}\left( x^{\ast }\right)} \\ &&{\kern4pt} =\lim\limits_{\delta \rightarrow 0}\frac{\big( b_{11}x^{\ast }+b_{21}\left( 1-x^{\ast }\right)\! \big) d_{R}\left( 1-y^{\ast }\right) }{ \big( b_{12}x^{\ast }+b_{22}\left( 1-x^{\ast }\right)\! \big) d_{L}\left( y^{\ast }\right) }=1 \end{array}$$

  This, plus continuity, establishes the result.

• 2. We show that for all $(x^{\prime },y^{\prime })\in \left( 0,1\right) \times \left( 0,1\right)$, there exists a sequence of functions d ^δ ∈ D ^δ and a δ ^′ low enough that the rest point of C.1–C.2 cannot be any $C\neq \left[ \left( x^{\prime },1-x^{\prime }\right),\left( y^{\prime },1-y^{\prime }\right) \right]$ for any δ ≤ δ ^′ and then the result follows.

An interior rest point of C.1–C.2 must satisfy:

$$\begin{array}{lll} \left( a_{11}y+a_{12}\left( 1-y\right) \right) d_{D}\left( 1-x\right) -\left( a_{21}y+a_{22}\left( 1-y\right) \right) d_{U}\left( x\right) &=&0 \\ \left( b_{11}x+b_{21}\left( 1-x\right) \right) d_{R}\left( 1-y\right) -\left( b_{12}x+b_{22}\left( 1-x\right) \right) d_{L}\left( y\right) &=&0 \end{array}$$

which implies that

$$\begin{array}{lll} \left( a_{11}y+a_{12}\left( 1-y\right) \right) \frac{d_{D}\left( 1-x\right) }{d_{U}\left( x\right) }-\left( a_{21}y+a_{22}\left( 1-y\right) \right) &=&0 \\ \left( b_{11}x+b_{21}\left( 1-x\right) \right) \frac{d_{R}\left( 1-y\right) }{d_{L}\left( y\right) }-\left( b_{12}x+b_{22}\left( 1-x\right) \right) &=&0 \end{array}$$

Let first x ^′ ≤ 1/2. We construct the doubt functions d _U (x) and d _D (1 − x) in D _m as follows:

$$ d_{ki}^{x^{\prime }}(f_{ki})=\left\{\begin{array}{c} m\delta \left( 1-f_{ki}\right) {\kern62pt}\text{if}{\kern3pt} f_{ki}\leq x^{\prime } \\ m\delta \left( 1-f_{ki}\right) \dfrac{\left( 1-f_{ki}\right) ^{1/\delta } \text{ }}{\left( 1-x^{\prime }\right) ^{1/\delta }}{\kern6pt} \text{ if } f_{ki}>x^{\prime } \end{array} \right. $$

where ki ∈ {U,D} and δ > 0. Note that as δ approaches 0, the graph of $d_{ki}^{x^{\prime }}$ function approaches the horizontal axis and the agent is said to be in a doubt-less mode.

Now, for x > x ^′

$$ \frac{d_{D}\left( 1-x\right) }{d_{U}\left( x\right) }=\frac{mx\text{ }}{ m\left( 1-x\right) \frac{\left( 1-x\right) ^{1/\delta }\text{ }}{\left( 1-x^{\prime }\right) ^{1/\delta }}}=\frac{x}{1-x}\left( \frac{1-x^{\prime }}{ 1-x}\right) ^{1/\delta } $$

Since 1 − x ^′ > 1 − x we can make$\big(\frac{1-x^{\prime }}{1-x}\big) ^{1/\delta }$ as big as we want by choosing a sufficiently small δ. Then

$$ \frac{x}{1-x}\left( \frac{1-x^{\prime }}{1-x}\right) ^{1/\delta }>\frac{ \left( a_{21}y+a_{22}\left( 1-y\right) \right) }{\left( a_{11}y+a_{12}\left( 1-y\right) \right) } $$

Hence,

$$ \left( a_{11}y+a_{12}\left( 1-y\right) \right) \frac{x}{1-x}\left( \frac{ 1-x^{\prime }}{1-x}\right) ^{1/\delta }-\left( a_{21}y+a_{22}\left( 1-y\right) \right) >0 $$

Now, for x < x ^′

$$ \frac{d_{D}\left( 1-x\right) }{d_{U}\left( x\right) }=\frac{\text{ }mx\left( \frac{x}{1-x^{\prime }}\right) ^{1/\delta }}{m\left( 1-x\right) }=\frac{x}{ 1-x}\left( \frac{x}{1-x^{\prime }}\right) ^{1/\delta } $$

since x < x ^′ ≤ 1/2, we have that 1 − x ^′ > x so we can make$\left( \frac{x}{1-x^{\prime }}\right) ^{1/\delta }$ as small as we want by choosing a sufficiently small δ. Then

$$ \frac{x}{1-x}\left( \frac{x}{1-x^{\prime }}\right) ^{1/\delta }<\frac{\left( a_{21}y+a_{22}\left( 1-y\right) \right) }{\left( a_{11}y+a_{12}\left( 1-y\right) \right) } $$

Hence

$$ \left( a_{11}y+a_{12}\left( 1-y\right) \right) \frac{x}{1-x}\left( \frac{x}{ 1-x^{\prime }}\right) ^{1/\delta }-\left( a_{21}y+a_{22}\left( 1-y\right) \right) <0 $$

When x ^′ > 1/2 let d _U (x) and d _D (1 − x) in D as follows:

$$ d_{ki}^{x^{\prime }}(f_{ki})=\left\{ \begin{array}{c} m\delta \left( 1-f_{ki}\right) {\kern58pt}\text{if}{\kern3pt} f_{ki}\leq x^{\prime } \\[3pt] m\delta \left( 1-f_{ki}\right) \dfrac{\left( 1-f_{ki}\right) ^{1/\delta } \text{ }}{x^{\prime 1/\delta }}\text{ if }f_{ki}>x^{\prime } \end{array} \right. $$

where ki ∈ {U,D} and δ > 0. Now, for x > x ^′

$$ \frac{d_{D}\left( 1-x\right) }{d_{U}\left( x\right) }=\frac{mx\text{ }}{ m\left( 1-x\right) \frac{\left( 1-x\right) ^{1/\delta }\text{ }}{x^{\prime 1/\delta }}}=\frac{x}{1-x}\left( \frac{x^{\prime }}{1-x}\right) ^{1/\delta } $$

Since x > x ^′ > 1/2, 1 − x < 1/2 we can make$\big(\frac{x^{\prime }}{ 1-x}\big) ^{1/\delta }$ as big as we want by choosing a sufficiently small δ. Then

$$ \frac{x}{1-x}\left( \frac{x^{\prime }}{1-x}\right) ^{1/\delta }>\frac{\left( a_{21}y+a_{22}\left( 1-y\right) \right) }{\left( a_{11}y+a_{12}\left( 1-y\right) \right) } $$

Hence,

$$ \left( a_{11}y+a_{12}\left( 1-y\right) \right) \frac{x}{1-x}\left( \frac{ x^{\prime }}{1-x}\right) ^{1/\delta }-\left( a_{21}y+a_{22}\left( 1-y\right) \right) >0 $$

For x < x ^′

$$ \frac{d_{D}\left( 1-x\right) }{d_{U}\left( x\right) }=\frac{\text{ }mx\left( \frac{x}{x^{\prime }}\right) ^{1/\delta }}{m\left( 1-x\right) }=\frac{x}{1-x} \left( \frac{x}{x^{\prime }}\right) ^{1/\delta } $$

Since x < x ^′, we can make$\left( \frac{x}{x^{\prime }}\right) ^{1/\delta }$ as small as we want by choosing a sufficiently small δ. Then

$$ \frac{x}{1-x}\left( \frac{x}{x^{\prime }}\right) ^{1/\delta }<\frac{\left( a_{21}y+a_{22}\left( 1-y\right) \right) }{\left( a_{11}y+a_{12}\left( 1-y\right) \right) } $$

Hence

$$ \left( a_{11}y+a_{12}\left( 1-y\right) \right) \frac{x}{1-x}\left( \frac{x}{ x^{\prime }}\right) ^{1/\delta }-\left( a_{21}y+a_{22}\left( 1-y\right) \right) <0 $$

The argument for y is analogous. □

Proof of Proposition 3

Let $I^{\ast }\equiv \left[ \left( x^{\ast },1-x^{\ast }\right),\left( y^{\ast },1-y^{\ast }\right) \right] \in (0,1)\times (0,1)$ be an interior Mixed Strategy Nash Equilibrium (MSNE) of G. In this equilibrium, expected payoffs are equalized across strategies; that is, π _U  = π _D and π _L  = π _R . From Proposition 2, we also know that an MSNE is a Mixed Strategy Doubt-Full equilibrium (MSDFE); that is, $\pi _{U}d_{D}(1-x^{\ast })=\pi _{D}d_{U}(x^{\ast })$ and $\pi _{L}d_{R}(1-y^{\ast })=\pi _{R}d_{L}(y^{\ast })$. Hence, an interior MSNE is a stationary state of the system C.1–C.2 if all agents are Cartesian Skeptical. Thus, $F_{1}(x^{\ast },y^{\ast })=0$ and $F_{2}(x^{\ast },y^{\ast })=0$, where

$$\begin{array}{lll} F_{1}(x,y) &=&\pi _{U}d_{D}(1-x)-\pi _{D}d_{U}(x) \\ &=&(a_{11}y+a_{12}(1-y))d_{D}\left( 1-x\right) -(a_{21}y+a_{22}(1-y))d_{U}\left( x\right) \\ F_{2}(x,y) &=&\pi _{L}d_{R}(1-y)-\pi _{R}d_{L}(y) \\ &=&(b_{11}x+b_{21}(1-x))d_{R}\left( 1-y\right) -(b_{12}x+b_{22}(1-x))d_{L}\left( y\right) \end{array}$$

and

$$\begin{array}{lll} \frac{\partial F_{1}(x,y)}{\partial x} &=&\pi _{U}\frac{\partial d_{D}(1-x)}{ \partial x}-\pi _{D}\frac{\partial d_{U}(x)}{\partial x} \\ \frac{\partial F_{1}(x,y)}{\partial y} &=&\left( a_{11}-a_{12}\right) d_{D}(1-x)+\left( a_{22}-a_{21}\right) d_{U}(x) \\ \frac{\partial F_{2}(x,y)}{\partial x} &=&\left( b_{11}-b_{21}\right) d_{R}(1-y)+\left( b_{22}-b_{12}\right) d_{L}(y) \\ \frac{\partial F_{2}(x,y)}{\partial y} &=&\pi _{L}\frac{\partial d_{R}(1-y)}{ \partial y}-\pi _{R}\frac{\partial d_{L}(y)}{\partial y} \end{array}$$

On the other hand, the Jacobian J(x,y) of the dynamic system C.1–C.2 evaluated at the steady state (x ^∗,y ^∗) is:

$$ J(x^{\ast },y^{\ast })=\left[ \begin{array}{cc} G_{1}(x^{\ast },y^{\ast })\left. \displaystyle\frac{\partial F_{1}(x,y)}{\partial x} \right\vert _{I^{\ast }} & G_{1}(x^{\ast },y^{\ast })\left. \displaystyle\frac{\partial F_{1}(x,y)}{\partial y}\right\vert _{I^{\ast }} \\[3pt] G_{2}(x^{\ast },y^{\ast })\left. \displaystyle\frac{\partial F_{2}(x,y)}{\partial x} \right\vert _{I^{\ast }} & G_{2}(x^{\ast },y^{\ast })\left. \displaystyle\frac{\partial F_{2}(x,y)}{\partial y}\right\vert _{I^{\ast }} \end{array} \right] $$

In an MSNE, π _U  = π _D , π _L  = π _R . If, on the other hand, agents are playing in a doubt-full mode, (that is, $d_{i}\in D^{1-\delta }$ for i ∈ {U,D,L,R} with $\lim_{\delta \rightarrow \delta ^{\ast }}d_{U}\left( x\right) =\lim_{\delta \rightarrow \delta ^{\ast }}d_{D}\left( 1-x\right) =\lim_{\delta \rightarrow \delta ^{\ast }}d_{L}\left( y\right) =\lim_{\delta \rightarrow \delta ^{\ast }}d_{R}\left( 1-y\right)$ and being nearly 1, for all $(x,y)\in \left( 0,1\right) \times \left( 0,1\right)$; δ ^∗ > 0 but nearly zero, as in Proposition 2). Then, writing $d_{i}\left( .\right) =1$, we would also have π _U d _D  = π _D d _U and π _L d _R  = π _R d _L .

Hence, in an MSNE as an MSDFE:

$$\begin{array}{lll} G_{1}(x^{\ast },y^{\ast }) &=&\frac{x^{\ast }(1-x^{\ast })}{2\pi _{U}\pi _{D}-\pi _{U}d_{D}(1-x^{\ast })-\pi _{D}d_{U}(x^{\ast })} \\ &=&\frac{x^{\ast }(1-x^{\ast })}{\pi _{U}\big(2\pi _{U}-d_{D}(1-x^{\ast })-d_{U}(x^{\ast })\big)} \\ &=&\frac{x^{\ast }(1-x^{\ast })}{2\pi _{U}(\pi _{U}-1)} \\ G_{2}(x^{\ast },y^{\ast }) &=&\frac{y^{\ast }(1-y^{\ast })}{2\pi _{L}(\pi _{L}-1)} \end{array}$$

Thus, the elements of the Jacobian matrix are the following:

$$\begin{array}{rll} j_{11} &=&G_{1}(x^{\ast },y^{\ast })\left. \frac{\partial F_{1}(x,y)}{ \partial x}\right\vert _{I^{\ast }} \\ &=&\frac{x^{\ast }(1-x^{\ast })}{2\pi _{U}(\pi _{U}-1)}\left( \pi _{U}\frac{ \partial d_{D}(1-x)}{\partial x}-\pi _{U}\frac{\partial d_{U}(x)}{\partial x} \right) _{I^{\ast }} \\ &=&\frac{x^{\ast }(1-x^{\ast })}{2(\pi _{U}-1)}\left( \frac{\partial d_{D}(1-x)}{\partial x}-\frac{\partial d_{U}(x)}{\partial x}\right) _{I^{\ast }} \end{array}$$

$$\begin{array}{rll} j_{12} &=&G_{1}(x^{\ast },y^{\ast })\left. \frac{\partial F_{1}(x,y)}{ \partial y}\right\vert _{I^{\ast }} \\ &=&\frac{x^{\ast }(1-x^{\ast })}{2\pi _{U}(\pi _{U}-1)}\big(\left( a_{11}-a_{12}\right) d_{D}(1-x^{\ast })+\left( a_{22}-a_{21}\right) d_{U}(x^{\ast })\big) \end{array}$$

$$\begin{array}{rll} j_{21} &=&G_{2}(x^{\ast },y^{\ast })\left. \frac{\partial F_{2}(x,y)}{ \partial y}\right\vert _{I^{\ast }} \\ &=&\frac{y^{\ast }(1-y^{\ast })}{2\pi _{L}(\pi _{L}-1)}\big(\left( b_{11}-b_{21}\right) d_{R}(1-y^{\ast })+\left( b_{22}-b_{12}\right) d_{L}(y^{\ast })\big) \end{array}$$

$$\begin{array}{rll} j_{22} &=&G_{2}(x^{\ast },y^{\ast })\left. \frac{\partial F_{2}(x,y)}{ \partial y}\right\vert _{I^{\ast }} \\ &=&\frac{y^{\ast }(1-y^{\ast })}{2(\pi _{L}-1)}\left( \frac{\partial d_{R}(1-y)}{\partial y}-\frac{\partial d_{L}(y)}{\partial y}\right) _{I^{\ast }} \end{array}$$

Recall that the real part of the eigenvalues of J(x ^∗,y ^∗) only depends on the sum of the diagonal terms (the trace of the matrix):

$$\begin{array}{rll} \text{Trace of }J(x^{\ast },y^{\ast }) &=&G_{1}(x^{\ast },y^{\ast })\left. \frac{\partial F_{1}(x,y)}{\partial x}\right\vert _{I^{\ast }}+G_{2}(x^{\ast },y^{\ast })\left. \frac{\partial F_{2}(x,y)}{\partial y}\right\vert _{I^{\ast }} \\[6pt] &=&\frac{x^{\ast }(1-x^{\ast })}{2(\pi _{U}-1)}\left( \frac{\partial d_{D}(1-x)}{\partial x}-\frac{\partial d_{U}(x)}{\partial x}\right) _{I^{\ast }} \\[6pt] &&+\,\frac{y^{\ast }(1-y^{\ast })}{2(\pi _{L}-1)}\left( \frac{\partial d_{R}(1-y)}{\partial y}-\frac{\partial d_{L}(y)}{\partial y}\right) _{I^{\ast }} \end{array}$$

Since the expected values $\pi _{U}=a_{11}y^{\ast }+a_{12}(1-y^{\ast })$ and $\pi _{L}=b_{11}x^{\ast }+b_{21}(1-x^{\ast })$ are smaller than 1, both $\frac{x^{\ast }(1-x^{\ast })}{2(\pi _{U}-1)}$ and $\frac{y^{\ast }(1-y^{\ast })}{2(\pi _{L}-1)}$ are negative. The sign of $\left( \frac{ \partial d_{D}(1-x)}{\partial x}-\frac{\partial d_{U}(x)}{\partial x}\right) _{I^{\ast }}$ and $\left( \frac{\partial d_{R}(1-y)}{\partial y}-\frac{ \partial d_{L}(y)}{\partial y}\right) _{I^{\ast }}$ is clearly positive (that is, the signs of the derivatives of d _D (1 − x) and d _R (1 − y) with respect to x and y, respectively, are positive and those of d _U (x) and d _L (y) are negative). Thus, j ₁₁ < 0 and j ₂₂ < 0 and so the sign of the trace is negative

$$ sign\text{ }\left[ G_{1}(x^{\ast },y^{\ast })\left. \frac{\partial F_{1}(x,y) }{\partial x}\right\vert _{I^{\ast }}+G_{2}(x^{\ast },y^{\ast })\left. \frac{ \partial F_{2}(x,y)}{\partial y}\right\vert _{I^{\ast }}\right] <0 $$

Without loss of generality, we may assume, for an interior equilibrium, that a ₁₁ > a ₂₁, b ₁₁ < b ₁₂, a ₁₂ < a ₂₂, and b ₂₂ < b ₂₁. Then it can be seen that the sign of j ₂₁×j ₁₂ is negative, when the agents are playing in the absent or doubt-full mode:

$$\begin{array}{rll} j_{21}\times j_{12}&=&\left( \frac{y^{\ast }(1-y^{\ast })}{2\pi _{L}(\pi _{L}-1)}((b_{11}-b_{12})+(b_{22}-b_{21})\right) \nonumber\\ &&\times \left( \frac{x^{\ast }(1-x^{\ast })}{2\pi _{U}(\pi _{U}-1)}((a_{11}-a_{21})+(a_{22}-a_{12})) \right) <0 \end{array}$$

Thus, the determinant associated to J(x ^∗,y ^∗), Det J(x ^∗,y ^∗) = j ₁₁× j ₂₂ − j ₂₁×j ₁₂, has a positive sign. Therefore, when every agent is Cartesian skeptical, the MSNE, $I^{\ast }\equiv \left[ \left( x^{\ast },1-x^{\ast }\right),\left( y^{\ast },1-y^{\ast }\right) \right]$, is a sink and therefore is an asymptotically stable equilibrium.□

Proof of Proposition 4

Using the same procedure as in Proposition 3, we can easily prove that, under the doubt-less mode, the MSDLE $[\left( 1/2,1/2\right),$ $\left( 1/2,1/2\right)]$ is a source. Now, to see the trajectory of initial points different from $\left[ \left( 1/2,1/2\right),\left( 1/2,1/2\right) \right]$, we might use the doubt function constructed for the proof of part 2 of Proposition 2.

Note that the denominators of C.1–C.2 are positive in the doubt-less mode of play. Hence the sign of $\overset{\cdot }{x}$ and $\overset{\cdot }{ y}$ depend on the sign of $\left( \pi _{U}d_{D}-\pi _{D}d_{U}\right)$ and $\left( \pi _{L}d_{R}-\pi _{R}d_{L}\right)$, respectively. Now we can proceed as in the proof of part 2 of Proposition 2.

Let first x ^′ ≤ 1/2. We construct the doubt functions d _U (x) and d _D (1 − x) in D _m as follows:

$$ d_{ki}^{x^{\prime }}(f_{ki})=\left\{ \begin{array}{ll} m\delta \left( 1-f_{ki}\right) &\text{if} f_{ki}\leq x^{\prime} \\ m\delta \left( 1-f_{ki}\right) \dfrac{\left( 1-f_{ki}\right) ^{1/\delta } \text{ }}{\left( 1-x^{\prime }\right) ^{1/\delta }} &\text{if} f_{ki}>x^{\prime} \end{array} \right. $$

This means that if x > x ^′

$$ sign\big[ \overset{\cdot }{x}\big] =sign\left[ \left( \pi _{U}-\pi _{D} \frac{\left( 1-x\right) ^{1/\delta }\text{ }}{\left( 1-x^{\prime }\right) ^{1/\delta }}\right) \right] $$

Then there is a $\delta ^{^{\prime }}$ low enough such that for all $0<\delta \leq \delta ^{^{\prime }}$, $\left( 1-x\right) ^{1/\delta }/(1-x^{\prime}) ^{1/\delta}$ is sufficiently small so that $sign\big[ \overset{\cdot }{x}\big] >0$ and hence if x(0) > x ^′, then lim _t→ ∞ x(t) = 1.

If on the other hand x < x ^′

$$ sign\big[ \overset{\cdot }{x}\big] =sign\left[ \left( \pi _{U}\frac{ x^{1/\delta }\text{ }}{\left( 1-x^{\prime }\right) ^{1/\delta }}-\pi _{D}\right) \right] $$

Since x < x ^′ ≤ 1/2, we have that 1 − x ^′ > x so there is a $\delta ^{^{\prime }}$ low enough such that for all $0<\delta \leq \delta ^{^{\prime }}$, x ^1/δ/( 1 − x ^′) ^1/δ is sufficiently small so that $sign\big[ \overset{\cdot }{x}\big] <0$ and hence if x(0) < x ^′, then lim _t→ ∞ x(t) = 0.

When x ^′ > 1/2, we let d _U (x) and d _D (1 − x) in D as follows:

$$ d_{ki}^{x^{\prime }}(f_{ki})=\left\{ \begin{array}{ll} m\delta \left( 1-f_{ki}\right) & \text{if}\, f_{ki}\leq x^{\prime } \\ m\delta \left( 1-f_{ki}\right) \displaystyle\frac{\left( 1-f_{ki}\right) ^{1/\delta } \text{ }}{x^{\prime 1/\delta }} & \text{if}\, f_{ki}>x^{\prime } \end{array} \right. $$

This means that if x > x ^′

$$ sign\big[ \overset{\cdot }{x}\big] =sign\left[ \left( \pi _{U}-\pi _{D} \frac{\left( 1-x\right) ^{1/\delta }\text{ }}{x^{\prime 1/\delta }}\right) \right] $$

Since x > x ^′ > 1/2, 1 − x < 1/2, there is a $\delta ^{^{\prime }}$ low enough such that for all $0<\delta \leq \delta ^{^{\prime }}$, we can make $(x^{\prime})^{1/\delta}/\left( 1-x\right) ^{1/\delta }$ is sufficiently big so that $sign\big[\overset{\cdot }{x}\big] >0$ and hence if x(0) > x ^′, then lim _t→ ∞ x(t) = 1.

If on the other hand x < x ^′

$$ sign\big[ \overset{\cdot }{x}\big] =sign\left[ \left( \pi _{U}\frac{ x^{1/\delta }\text{ }}{x^{\prime 1/\delta }}-\pi _{D}\right) \right] $$

Since x < x ^′, there is a $\delta ^{^{\prime }}$ low enough such that for all $0<\delta \leq \delta ^{^{\prime }}$, x ^1/δ/x ^′1/δ is sufficiently small so that $sign\big[ \overset{\cdot }{x} \big] <0$ and hence if x(0) <  x ^′, then lim _t→ ∞ x(t)  =  0. The argument for y is analogous.□

D Constant doubt-based selection dynamics

The individual choice model that we are going to use in this section is derived from a choice procedure introduced by Aizpurua et al. (1993) in the space of simple lotteries. We consider now the case when the level of doubts felt is constant, for any value of f _ki  ∈ F _ki . This means that society has no influence upon the doubt level of the agents. Formally,

$$ d_{ki}(f_{ki})=\epsilon _{k}\in (0,1) $$

We assume that the constant level of doubts ε _k felt by agent ki induces threshold levels in both expected payoffs and strategy frequencies and that these threshold levels are described by means of similarity relations.

By means of the doubt function that we define a similarity relation on Π _ki  = (0,1] and correlated similarity relations on F _ki  = [0,1]. Suppose that (π _ki ,f _ki ) is the vector of expected payoff-strategy proportion attached to strategy i at time t.

The similarity relation on Π _ki , denoted SΠ _ki , is assumed to be of the difference type and it is defined as follows

$$ \pi _{ki}S\Pi _{ki}\pi _{ki}^{\prime }\Leftrightarrow \left\vert \pi _{ki}-\pi _{ki}^{\prime }\right\vert \leq \epsilon _{k} $$

On F _ki , we define now the correlated similarity relations as follows. First, for all π _ki  > ε _k  > 0 we build the function ϕ _ki :Π _ki →(1, ∞ ] as follows,

$$ \phi _{ki}(\pi _{ki})=\frac{\pi _{ki}}{\pi _{ki}-\epsilon _{k}}>1 $$

Then, we can establish the following similarity relation (of the ratio-type) between f _ki and other frequencies in F _ki , such as $f_{ki}^{\prime }$, given π _ki .

$$ f_{ki}SF_{ki}(\pi _{ki})f_{ki}^{\prime }\Leftrightarrow \frac{1}{\phi _{ki}(\pi _{ki})}\leqq \frac{f_{ki}}{f_{ki}^{\prime }}\leqq \phi _{ki}(\pi _{ki}) $$

We call SF _ki (π _ki ) a correlated similarity relation because the similarity on F _ki depends on the level of expected payoff π _ki at period t. For values of $\pi _{ki}\leqq \epsilon _{k}$ the function ϕ _ki is not defined and we assume that in that case that SF _ki (π _ki ) is the degenerate similarity relation (see Rubinstein 1988).

Remark 2

The threshold level in the frequency space is inversely related to expected payoffs: $\frac{\partial \phi _{ki}(\pi _{ki})}{\partial \pi _{ki}}<0$. This means that as the expected payoffs at stake increases, the discrimination on the frequency space F _ki increases.

We proceed as in the previous case with decreasing doubts; for simplicity we shall write ϕ _ki instead of ϕ _ki (π _ki ). Let the ratio

$$ \frac{\phi _{ki}}{\Sigma _{_{i=1}}^{m_{k}}\phi _{ki}}=\frac{\phi _{ki}}{\phi _{k}} $$

denote the proportion of ki strategists who feel dissatisfied with strategy i. Note that, everything equal, this function increases with ϕ _ki . Hence, an increase in ϕ _ki , due to a decrease in the expected payoffs π _ki , will increase the proportion of dissatisfied ki strategists.

Proceeding as in the case of decreasing doubts, we obtain.

$$\label{eq6} f_{ki}(t+\tau )=f_{ki}(t)-\tau \frac{\phi _{ki}}{\phi _{k}}f_{ki}+\tau \frac{ \overline{\phi }{_{k}}}{\phi _{k}}f_{ki}. $$

(6)

As τ→0, Eq. 6 becomes

$$\label{eq7} \overset{\bullet }{f_{ki}}=f_{ki}\left[ \frac{\overline{\phi }{_{k}-\phi _{ki}}}{{\phi _{k}}}\right] $$

(7)

Proposition 5

• 1. If for all player position $k\in K=\left\{ 1,2,...,n\right\}$ , the strategy set S _k consists of two elements, i.e. if m _k  = 2, then Eq.  7 is just the standard Replicator Dynamics (RD) multiplied by a positive function (i.e. is aggregate monotonic).

• 2. If m _k  > 2, then we obtain a selection dynamics that approximates the RD, but preserves only the positive sign of the RD (i.e. is weakly payoff positive).

Proof

• (a) Let S _k  = $\left\{ 1,2\right\}$ be player population k’s strategy set. Without loss of generality, let us refer to the dynamics of strategy 1. Then, by Eq. 7, we have

  $$\begin{array}{rll} \overset{\bullet }{f_{k1}} &=&f_{k1}{\overline{\phi _{k}}-\phi _{k1}}{\phi _{k}}\\ &=&\frac{\epsilon _{k}}{D(f)}f_{ki}[\pi _{ki}-\overline{\pi }_{k}] \notag \end{array}$$

  where D(f) ≡ π _k1(π _k2 − ε _k ) + π _k2(π _k1 − ε _k ) > 0. Hence, the growth rates $\frac{\overset{\bullet }{f_{ki}}}{f_{ki}}$ equal payoff differences $[\pi _{ki}-\overline{\pi }_{k}]$ multiplied by a (Lipschitz) continuous, positive function $\frac{\epsilon _{k}}{D(f)}$. This concludes the proof. (Note that, given ε _k , a payoff difference $[\pi _{ki}-\overline{\pi }_{k}]$ will have stronger dynamic effect if D(f) is low than if it is high; if ε _k decreases, the dynamic effect of $[\pi _{ki}-\overline{\pi }_{k}]$ decreases).

• (b) Easy.□