1 Introduction

The growing interest in evolutionary games with continuous strategy spaces is primarily because of the fact that many applications in economics are modelled as evolutionary games with continuous strategy spaces. Some of the important applications include oligopoly games, bargaining games, harvest pre-emptive game, and war of attrition. In these games, the strategy space is typically a compact subset of an Euclidean space. Considering the vast literature on evolutionary games with finite strategy spaces (see, e.g. [9, 16, 19, 20, 24, 28]), a natural approach is to approximate the strategy space by finite sets.

Oechssler and Riedel [21] have already considered such an approach for the Harvest pre-emption game. It is observed (see Sect. 5.4 in [21]) that the limit of the ESS of the finite approximation is not ESS of the harvest pre-emption game. The reason for such a negative result is the infinite dimensional nature of the space of all mixed strategy spaces.

Evolutionary games with continuous strategy spaces were first studied by Bomze and Pötscher through what they called as “generalized” mixed strategy games [5]. There is a vast literature relating evolutionary stability and dynamic stability under various dynamics for games with finite strategy spaces (see, e.g. [6, 9, 17, 24, 28]). However, this is not the case with games with continuous strategy spaces. In the works [3–5, 10, 21, 22], the relation is explored for replicator dynamics. For the literature concerning other dynamics, see [8, 13, 15, 18].

Replicator dynamics is one of the most important dynamics. In this article, we study the relation between evolutionary stability and dynamic stability under replicator dynamics for the polymorphic population states. In [5, 21, 22], such a connection has been studied for monomorphic population states.

The population states in evolutionary games with continuous strategy spaces are nothing but probability measures on its strategy space. We can define the static stability concepts of evolutionarily stable strategy (ESS) and uninvadability in these games. We can also define the replicator dynamics for these games which capture the evolution of the population over time. Certain other stability concepts can also be defined, but they depend on the notion of “closeness” of the population states.

The closeness of the population states can be made precise using various metrics. In this article, we study the evolutionary dynamics and related stability results under the metric defined by the variational or strong norm, which gives rise to the strong topology. In particular, we can define the static stability concepts of strong uninvadability [3, 4] and strong unbeatability [6] along with the dynamic stability concepts such as Lyapunov stability and strongly attracting population states [5, 21]. We can also define the concepts of evolutionary robustness [22] and weakly attracting population states [21] when we consider the metric associated with the weak topology.

Evolutionary games with continuous strategy spaces with the underlying topology as the strong topology are studied in [21]. The authors connected the static and dynamic stabilities for a population state where all the individuals in the entire population play one and the same pure strategy x. Such a population state is called a monomorphic population state, and it is represented by the Dirac measure \(\delta _x\). They proved that an uninvadable monomorphic population state is Lyapunov stable. Moreover, if the initial population state is close to this monomorphic population state in the strong sense and the payoff function is continuous, then the monomorphic population state is weakly attracting. Some results regarding the stability of dimorphic population states have been discussed in [14].

In [22], Oechssler and Riedel proved that for a doubly symmetric game with continuous payoff function and compact strategy space, an evolutionary robust population state is Lyapunov stable when the underlying topology is the weak topology. They also prove the asymptotic stability of monomorphic evolutionary robust strategies under various assumptions. Cressman has also studied evolutionary games with continuous strategy games in [10]. One of the main results he proves in this paper is regarding population states which have a finite support, which are also known as polymorphic population states. He proved that a dimorphic (or polymorphic) neighbourhood superior population state is neighbourhood attracting. Furthermore, this paper also contains a motivating example (Example 1c) of a neighbourhood attracting dimorphic population state. In [11] (Sect. 6), Cressman and Hofbauer give a class of examples having globally superior dimorphic population states.

Bomze has proved a couple of results regarding the asymptotic stability of population states in [3]. In the first result [3, Theorem 2], one of the condition that he imposes is that the set of all population states has to be compact under the given topology. When the underlying topology is strong, this becomes a very strong imposition. We can weaken the conditions of this theorem considerably when we talk about the stability of polymorphic population states. In fact, in this article, we prove that for the Lyapunov stability of the polymorphic population state, the unbeatability condition is sufficient.

Another result that Bomze gives [3, Theorem 3], depends on a first-order condition for strong uninvadability of a population state. This condition may not be satisfied always when the population state is strongly uninvadable. We present here an example (Example 2 in Sect. 3) where this first-order condition is not satisfied even though the population state is strongly uninvadable. In this article, we also prove that in the case of polymorphic population states, uninvadability is a sufficient condition for asymptotic stability which is our main result (Theorem 7 in Sect. 3). Note that such a result is not expected in general, if the underlying topology is weak because weak neighbourhoods of polymorphic states may contain states whose support is far away from that of the polymorphism, as illustrated in [10] (Example 3).

In all the results mentioned above, to obtain dynamic stability for a population state, say P, the initial population state, say Q(0), for the replicator dynamics is taken from a small neighbourhood of P. Moreover, Q(0) is chosen such that its support contains the support of P. This is a necessary condition to study stability with respect to replicator dynamics since the replicator dynamics can only increase or decrease the frequency of the strategies which already exists at the start of the dynamics. Thus, in general, population state P will be not be stable with respect to its complete neighbourhood.

In [26], van Veelen and Spreij proved that when a population state P is asymptotically stable with respect to a complete (strong) neighbourhood under the replicator dynamics, then P should be a polymorphic population state. The results we prove in this article establish the converse of the result by van Veelen and Spreij.

There are many references studying polymorphic population models in various contexts; see, e.g. [1, 7, 12, 25]. In [25], the use of game theory is shown to be more effective in explaining the behaviour of males and females by studying their strategies simultaneously. Broom et al. [7] study evolutionary stability of various population mixtures in the context of bird populations. Haplotypes are considered as pure strategies in [12], and evolutionary stability of certain state/sets is illustrated. Furthermore, a review on game theory of public goods in one-shot social dilemmas without assortment is given in [1] wherein the evolution of cooperation in N-person games is discussed.

The rest of the article is structured as follows. Section 2 gives the preliminary notations, definitions, and results to study continuous strategy evolutionary games with the underlying topology as the strong topology. Section 3 is divided into two parts. In the first part, we discuss some properties of polymorphic population states. In the second part, we discuss the stability of polymorphic population states. We prove that an uninvadable polymorphic population state is asymptotically stable, whereas for Lyapunov stability, it is enough if the population state is unbeatable. Concluding remarks are given in Sect. 4. An “Appendix” is devoted to study the Lyapunov stability for differential equations in infinite dimensional spaces.

2 Preliminaries and Problem Description

Let S be a Polish space (i.e. complete separable metric space) with the associated metric d. We consider a symmetric two player game \(G = (S, u)\). Here, \(u : S \times S \rightarrow \mathbb {R}\) represents the payoff function, which is bounded and measurable. Recall that in a symmetric game, if a player chooses \(z \in S\) and the other player chooses \(w \in S\), then the player choosing z will get a payoff u(z, w).

Let \(\mathscr {B}\) denote the Borel sigma-algebra on S, i.e. the sigma-algebra generated by all open sets in S. Following the tradition of evolutionary game theory, a population state of the game G is defined to be a probability measure, Q, on the measurable space \((S,\mathscr {B})\). The set of all population states is denoted by \(\Delta \). The average payoff to a population P playing against a population Q is given by

$$\begin{aligned} E(P,Q) := \int _S \int _S u(z,w) ~Q(\mathrm{{d}}w)~P(\mathrm{{d}}z). \end{aligned}$$

We recall a few definitions from evolutionary game theory.

Definition 1

[5, 19] A population state P is called an evolutionarily stable strategy if for every “mutation” \(Q \ne P\), there is an invasion barrier \(\epsilon (Q) > 0\), such that, for all \(0 < \eta \le \epsilon (Q)\),

$$\begin{aligned} E(P,(1-\eta )P+\eta Q) > E(Q,(1-\eta )P+\eta Q). \end{aligned}$$
(1)

Definition 2

[27] A population state P is called uninvadable if, in Definition 1, \(\epsilon (Q)\) can be chosen independent of \(Q \in \Delta \), \(Q \not = P\).

Note that we can rewrite the condition (1) in the ESS definition as

$$\begin{aligned} E(P,R) > E(R,R), \end{aligned}$$

where \(R = (1-\eta )P+\eta Q\) for all \(0 < \eta \le \epsilon (Q)\). A neighbourhood of P can be completely characterized by R, with \(\eta \) sufficiently small, when the set of pure strategies is finite; but not when the set of pure strategies is infinite. In games with infinite strategy set, the neighbourhoods of the population state P can be determined using various topologies. In this article, we consider the topology generated by the variational (or strong) norm, i.e. the variational (or strong) topology. The variational norm of a probability measure P is given by

$$\begin{aligned} \Vert P \Vert = 2~ \sup _{B \in \mathscr {B}} |P(B)|. \end{aligned}$$

Thus, the distance between two probability measures P and Q is given by

$$\begin{aligned} \Vert P-Q \Vert = 2~ \sup _{B \in \mathscr {B}} |P(B)-Q(B)|. \end{aligned}$$

We next define strong uninvadability and strong unbeatability.

Definition 3

[3, 4] A population state P is called strongly uninvadable if there is an \(\epsilon > 0\) such that for all population states \(R \ne P\) with \(\Vert R - P \Vert \le \epsilon \), we have

$$\begin{aligned} E(P,R) > E(R,R). \end{aligned}$$

Definition 4

[6] A population state P is called strongly unbeatable if there is an \(\epsilon > 0\) such that for all population states \(R \ne P\) with \(\Vert R-P\Vert \le \epsilon \), we have

$$\begin{aligned} E(P,R) \ge E(R,R). \end{aligned}$$

It can be easily seen that a strongly uninvadable state is uninvadable and an uninvadable state is an ESS [4].

We now consider the evolution of the population over time using the replicator dynamics [21, 22]. To this end, we note that the success (or lack of success) of a strategy \(z \in S\) against a strategy \(w \in S\) is given by

$$\begin{aligned} \sigma (z,w) := u(z,w) - u(w,w). \end{aligned}$$

The average success (or lack of success) of a strategy \(z \in S\) against a population \(Q \in \Delta \) is given by

$$\begin{aligned} \sigma (z,Q) := \int _S u(z,w) ~ Q(\mathrm{{d}}w) - \int _S \int _S u(\bar{z},\bar{w})~ Q(d\bar{w})~ Q(d\bar{z}) = E(\delta _z,Q) - E(Q,Q), \end{aligned}$$

where the Dirac measure \(\delta _z\) represents a monomorphic population state.

The replicator dynamics is derived based on the idea that the relative increment in the frequency of strategies in a set \(B\in \mathscr {B}\) is given by the average success of strategies in B. That is, for every \(B \in \mathscr {B}\),

$$\begin{aligned} Q'(t)(B) = \frac{\mathrm{{d}}Q(t)}{\mathrm{{d}}t}(B) = \int _B \sigma (z,Q(t))~Q(t)(\mathrm{{d}}z) \end{aligned}$$
(2)

where Q(t) denotes the population state at time t.

The replicator dynamics Eq. (2) can be also written as

$$\begin{aligned} Q'(t) = F(Q(t)), \end{aligned}$$
(3)

where for every \(B \in \mathscr {B}\), \(F(Q(t))(B) = \int _B \sigma (z,Q(t))~Q(t)(\mathrm{{d}}z)\); that is, F(Q(t)) is the signed measure whose Radon–Nikodym derivative \(\dfrac{dF(Q(t))}{\mathrm{{d}}Q(t)}\), w.r.t. Q(t) is \(\sigma (\cdot ,Q(t))\).

Since the payoff function u is bounded and measurable, it follows that the replicator dynamics is well posed [21, Theorem 2] which in turn assures the existence of a unique solution to the replicator dynamics (2) with the initial condition Q(0).

We can now introduce a few dynamic stability definitions for population states. Let P be a rest point of the replicator dynamics, i.e. \(F(P) = 0.\)

Definition 5

Rest point P is called Lyapunov stable if for all \(\epsilon > 0\), there exists an \(\eta > 0\) such that,

$$\begin{aligned} ||Q(0)-P||< \eta ~~ \Rightarrow ~ ||Q(t)-P|| < \epsilon \,\, \text{ for } \text{ all } t > 0. \end{aligned}$$

Definition 6

P is called strongly attracting if there exists an \(\eta > 0\) such that Q(t) converges to P strongly as \(t \rightarrow \infty \), whenever \(||Q(0) - P|| < \eta \).

Definition 7

P is called asymptotically stable if P is Lyapunov stable and strongly attracting.

One of the main interest in studying games with continuous strategy spaces is to establish conditions under which the population states will be dynamically stable. We recall here some of the existing results in this direction.

Oechssler and Riedel [21] provide with sufficient conditions for a monomorphic population state \(Q^* = \delta _x\) to be Lyapunov stable and “weakly attracting”. More precisely, they prove the following result.

Theorem 1

[21] If \(Q^*=\delta _x\) is an uninvadable, monomorphic population state, then \(Q^*\) is Lyapunov stable. Moreover, if u is continuous then \(Q^*\) is weakly attracting, in the sense that the trajectory w.r.t the replicator dynamics converges to \(Q^*\) weakly when the initial population state is from a small (strong) neighbourhood of \(Q^*\).

Bomze, in [3], proved couple of results regarding the asymptotic stability of population states under very strong assumptions. The first theorem that he gives is as follows.

Theorem 2

[3] Suppose that \(\Delta \) is relatively \(\tau \)-compact, where \(\tau \) is a topology on the L-space, \(\mathscr {L}\), of \((S, \mathscr {B}, \Delta )\) such that the map \(Q \mapsto \Vert Q\Vert \) from \(\mathscr {L}\) to \(\mathbb {R}\) is lower semi-continuous. If \(P \in \Delta \) is strongly uninvadable, and if the map \(Q \mapsto E(P,Q)-E(Q,Q)\) on \(\Delta \) is \(\tau \)-continuous, then every replicator dynamics trajectory Q(t), \(t\ge 0\), starting in

$$\begin{aligned} \mathscr {U}_P = \left\{ Q \in \Delta : P \ll Q~~ \text{ and }~ \int _S \ln \left( \frac{\mathrm{{d}}P}{\mathrm{{d}}Q} \right) ~\mathrm{{d}}P < \delta \right\} \end{aligned}$$

converges to P as \(t \rightarrow \infty \), with respect to \(\tau \), provided that \(\delta > 0\) is small enough.

The \(\tau \)-compactness condition in the above theorem is a very strong condition when \(\tau \) is taken to be the strong topology. Bomze gives another result regarding the asymptotic stability of the population state P with \(\tau \) as the strong topology under the condition of the following theorem.

Let \(\mathscr {M}\) be the linear span of \(\Delta \), with variational norm, and \(\mathscr {F}\) be the space of all bounded measurable functions with the norm \(\Vert F\Vert _\infty = \sup _{z\in S}|F(z)|\). Also, for \(F_Q \in \mathscr {F}\), \(F_Q(z)\) denotes the mean payoff to \(z \in S\) against \(Q \in \Delta \).

Theorem 3

[3] Let \(P \in \Delta \) be a rest point and assume that the map \(Q \mapsto F_Q\) from \(\Delta \) to \(\mathscr {F}\) is Fréchet differentiable at \(Q=P\) in the sense that there is a continuous linear map \(DF_P : \mathscr {M} \rightarrow \mathscr {F}\) such that for all \(\eta > 0\) there is a \(\rho > 0 \) fulfilling

$$\begin{aligned}&\Vert F_Q - F_P - DF_P(Q - P)\Vert _\infty \le \eta \Vert Q-P\Vert \\&\quad \text{ whenever }~~~\Vert Q-P\Vert < \rho ~~\text{ and }~ Q \in \Delta . \end{aligned}$$

P is strongly uninvadable if there is a constant \(c > 0\) such that

$$\begin{aligned} \int _S DF_P(Q-P)~d(Q-P) \le -c ~\Vert Q-P\Vert ^2~~\text{ for } \text{ all } Q\in \Delta . \end{aligned}$$
(4)

Note that games satisfying this last condition are known as negative definite games. Negative definite games possess many interesting properties and they have been studied extensively in the literature [8, 18, 24].

The above theorem gives a first-order condition for P to be strongly uninvadable. The next one gives another set of conditions for asymptotic stability of a population state P.

Theorem 4

[3] Under the assumptions of Theorem 3, every replicator dynamics trajectory Q(t), \(t\ge 0\), starting in \(\mathscr {U}_P\) (as in Theorem 2) satisfies \(\Vert Q(t) - P\Vert \rightarrow 0\) as \(t \rightarrow \infty \).

The first-order condition for strong uninvadability of P, given in Theorem 3 is not a necessary condition, as illustrated by Example 2 in the next section. We observe that the conditions for stability can be weakened when we are dealing with polymorphic population states. In the next section, we focus on the polymorphic population states and provide with conditions for their stability with the underlying topology as the strong topology.

3 Stability of Polymorphic Population States

In this section, we will first study some properties of polymorphic population states, and then, we will move on to the stability of these population states. As the name suggests, polymorphic population states have a finite support. Moreover, we can view them as convex combinations of monomorphic population states.

3.1 Properties of Polymorphic Population States

We begin by characterizing rest points of the replicator dynamics (3).

Lemma 1

A population state P is a rest point of the replicator dynamics (3) if and only if \(\int _S u(z, w) P(\mathrm{{d}}w)\) is constant a.s. z(P).

Proof

Clearly, P is a rest point of the replicator dynamics if and only if for all \(B \in \mathscr {B}\),

$$\begin{aligned} F(P)(B) =\int _B \sigma (z,P)~ P(\mathrm{{d}}z) = 0. \end{aligned}$$

This is equivalent to

This implies and is implied by

$$\begin{aligned} E(\delta _{z},P) = E(P,P) ~~~~~\text{ a.s. } z(P). \end{aligned}$$

From this, it follows that P is a rest point of the replicator dynamics if and only if, \( \int _S u(z, w) P(\mathrm{{d}}w)\) is independent of z a.s.(P). \(\square \)

In the case of the polymorphic population state given by

$$\begin{aligned} P^* = \alpha _1 \delta _{x_1} + \alpha _2 \delta _{x_2} + \ldots + \alpha _k \delta _{x_k}, \end{aligned}$$
(5)

where \(x_1, x_2, \ldots , x_k\) are distinct points in S and the sum of the positive numbers \(\alpha _1, \alpha _2, \ldots , \alpha _k\) is 1, this lemma reduces to the following corollary.

Corollary 1

Let \(P^*\) be a polymorphic population state given by (5). Then, \(P^*\) is a rest point of the replicator dynamics if and only if the sum \( \sum \limits _{j=1}^k \alpha _j u(x_i,x_j)\) is independent of i.

Proof

The proof follows since the support of \(P^*\) is \(\{x_1, x_2, \ldots , x_k \}\). \(\square \)

We illustrate the above corollary using the following example.

Example 1

Let \(S = [0,1]\) and the payoff function be defined by

$$\begin{aligned} u(z,w) = \left\{ \begin{array}{cl} w &{}\quad \text {if}\quad z < w\\ z-w &{}\quad \text {if}\quad z \ge w \end{array} \right. \end{aligned}$$

Consider the polymorphic population state

$$\begin{aligned} P^* = \alpha _1 \delta _{x_1} + \alpha _2 \delta _{x_2} + \alpha _3 \delta _{x_3} = \frac{1}{3} \delta _0 + \frac{1}{3} \delta _{1/2} + \frac{1}{3} \delta _1. \end{aligned}$$

Then, for \(i=1\),

$$\begin{aligned} \sum _{j=1}^k \alpha _j u(x_1,x_j)&= \frac{1}{3} \sum _{j=1}^3 u(0,x_j) = \frac{1}{3} \left\{ u(0,0) + u(0,1/2) + u(0,1) \right\} \\&= \frac{1}{3} \left\{ 0 + \frac{1}{2} + 1 \right\} = \frac{1}{2}. \end{aligned}$$

Similarly, for \(i=2\) and \(i=3\), we get the sum \(\sum _{j=1}^k \alpha _j u(x_i,x_j)\) as 1 / 2. Thus, by Corollary 1, \(P^*\) is a rest point of the replicator dynamics.

Now that we have established the condition for the polymorphic population state \(P^*\) to be a rest point of the replicator dynamics, we move on to characterizing small neighbourhoods of \(P^*\) with respect to the variational topology.

Consider population states P and Q from \(\Delta \). Then, by Lebesgue decomposition, we can decompose Q in terms of the Borel measures \(Q_1\) and \(Q_2\) such that

$$\begin{aligned} Q = Q_1 + Q_2 \end{aligned}$$

where \(Q_1\) is absolutely continuous with respect to P and \(Q_2\) is singular with respect to P. Now

$$\begin{aligned} \Vert Q - P \Vert = 2 \sup _{B \in \mathscr {B}} | Q_1(B) + Q_2(B) - P(B) | \ge 2 |Q_1 (A) - P(A)| \end{aligned}$$

for every Borel set \(A \subseteq \text {Support}{(P)}\).

We can similarly decompose the population state Q, by taking \(P = P^*\), the polymorphic population state. Now since the support of \(P^* = \{x_1, x_2, \ldots , x_k\}\), note that whenever the support of Q is a strict subset of the support of \(P^*\), form the above inequality, we obtain

$$\begin{aligned} \Vert Q - P^* \Vert \ge 2~ \inf ~ \{ \alpha _j : x_j \not \in \text {Support}{(Q)} \}. \end{aligned}$$

Let \( 0< \epsilon < 2~\inf ~\{ \alpha _j : j = 1, 2, \ldots , k \}\). From the above, it follows that for every Q in the \(\epsilon \)-neighbourhood of \(P^*\), the support of \(Q_1\) must be equal to the support of \(P^*\).

In conclusion, every population state Q sufficiently close to \(P^*\) will be of the form

$$\begin{aligned} Q = \sum _{j=1}^k \beta _j \delta _{x_j} + \beta _{k+1} R~; ~~~~\sum _{j=1}^{k+1} \beta _j = 1 \end{aligned}$$
(6)

where the support of \(R \in \Delta \) is disjoint from the support of \(P^*\).

One useful consequence of this fact is the following lemma, whose proof is a straight forward application of the representation (6).

Lemma 2

Let \(P^*\) be the polymorphic state given by (5). Then, for sufficiently small \(\epsilon \), \(P^*\) is absolutely continuous with respect to Q, for every Q in \(\epsilon \)-neighbourhood of \(P^*\).

Not surprisingly, this lemma fails when \(P^*\) is infinitely supported. In fact, the Lebesgue measure on [0, 1] provides a counter example.

Another consequence of the Eq. (6) is the following lemma which gives bounds for the variational distance between population states in a small neighbourhood of \(P^*\).

Lemma 3

Let \(\epsilon > 0\) be small enough such that all population states in the neighbourhood \(\Omega (\epsilon ) := \{Q \in \Delta : \Vert Q-P^*\Vert < \epsilon \}\) are of the form (6). If

$$\begin{aligned} Q_1&= \sum _{j=1}^k \beta _j \delta _{x_j} + \beta _{k+1} R_1~; ~\sum _{j=1}^{k+1} \beta _j = 1,\\ Q_2&= \sum _{j=1}^k \gamma _j \delta _{x_j} + \gamma _{k+1} R_2~; ~~\sum _{j=1}^{k+1} \gamma _j = 1 \end{aligned}$$

are population states in \(\Omega (\epsilon )\) then we have,

$$\begin{aligned} 2 \max _{1 \le j \le k} \left\{ |\beta _j - \gamma _j| \right\} \le \Vert Q_1-Q_2\Vert \le 2 \max \left\{ \sum _{j=1}^k |\beta _j - \gamma _j|~, ~2 \left( 1 - \sum _{j=1}^k \beta _j \right) \right\} . \end{aligned}$$

Proof

The proof follows from straight forward calculations using the definition of the variational distance. \(\square \)

From this lemma, we can write the bounds for variational distance of \(P^*\) and a population state Q in its neighbourhood, with the form given in (6), as

$$\begin{aligned} \max _{1 \le j \le k} |\alpha _j - \beta _j | \le \frac{1}{2}\Vert Q - P^*\Vert \le \ \max \left\{ \sum _{j=1}^k |\alpha _j - \beta _j|, 2 \left( 1 - \sum _{j=1}^k \beta _j \right) \right\} . \end{aligned}$$
(7)

Remark 1

The above lemma and its application not only gives us lower and upper bounds for the variational distance but it also proves that \(\Vert Q-P^*\Vert \rightarrow 0\) if and only if \(|\beta _j - \alpha _j| \rightarrow 0\) for every \(j = 1, 2, \ldots , k\). Thus, to prove convergence of a sequence of population states to \(P^*\), it is enough to prove the convergence of the weights on each of \(x_j\)’s.

Before proceeding further to study the stability of polymorphic population states, we present an example which shows that the first-order condition in Theorem 3 is not necessary to guarantee that \(P^*\) is strongly uninvadable.

Example 2

Let \(S = [-1,1]\) and the payoff function be defined as

$$\begin{aligned} u(z,w) = 2-zw~~\text{ for } \text{ all } z,w \in S. \end{aligned}$$

The polymorphic state \(P^* = \alpha \delta _{-1} + (1 - \alpha ) \delta _1\) with \(\alpha = 1/2\) is a rest point of the replicator dynamics. Now consider a population state Q from an arbitrarily small strong neighbourhood of \(P^*\) (as given in Lemma 2). Then, Q will be of the form

$$\begin{aligned} Q = \beta \delta _{-1} + \gamma \delta _1 + (1-\beta -\gamma )R, \end{aligned}$$

where \(R\in \Delta \) such that \(R(\{-1, 1\}) = 0\) and \(0< \beta +\gamma \le 1\).

Note that \(E(\delta _z,P^*) = 2 \) for all \(z \in S\) which implies that

$$\begin{aligned} E(P^*,P^*) = E(P^*,Q) = E(Q,P^*) = 2. \end{aligned}$$

By definition of u,

$$\begin{aligned} E(Q,Q) = 2 - \left( \gamma - \beta + (1-\beta -\gamma ) \mu \right) ^2 \end{aligned}$$

where \(\mu = \int _S z R(\mathrm{{d}}z)\).

Therefore, \(P^*\) is strongly uninvadable since \(E(P^*,Q) - E(Q,Q) > 0\) for every \(Q\not = P^* \) in a strong neighbourhood of \(P^*\).

However, we can show that the condition (4) is not true. In fact, the map \(F_Q = \int _S u(\cdot ,w)~Q(\mathrm{{d}}w)\) is Frèchet differentiable and \(DF_{P^*}(Q-P^*) = F_{Q-P^*}\). Hence,

$$\begin{aligned} \int _S DF_{P^*}(Q-P^*)~d(Q-P^*) = E(Q,Q) - E(P^*,Q) - E(Q,P^*) + E(P^*,P^*). \end{aligned}$$

Taking \(Q = \frac{1}{2} (\delta _{-1/2} + \delta _{1/2})\) we note that

$$\begin{aligned} \int _S DF_{P^*}(Q-P^*)~d(Q-P^*) = 0. \end{aligned}$$

Thus, the game is not negative definite game.

3.2 Stability of \(P^*\)

We are now ready to discuss the stability of polymorphic population states. First, we recall the following result from [26, Proposition 13]. Since this result forms the background for our work, we provide a proof which is slightly different from [26] .

Theorem 5

[26, Proposition 13] Every asymptotically stable rest point of the replicator dynamics in variational distance is finitely supported.

Proof

From [4, Lemma 2], we have

$$\begin{aligned} \text {Support}{(Q(t))} = \text {Support}{(Q)} \end{aligned}$$

where \(Q(\cdot )\) is the trajectory of the replicator dynamics (2) with initial condition \(Q(0) = Q\). If Q(t) converges to P strongly, then, by Portmanteau theorem [2, Theorem 2.1], we must have

$$\begin{aligned} \text {Support}{(P)} \subseteq \text {Support}{(Q)}. \end{aligned}$$
(8)

To prove the theorem, we exhibit a probability measure Q in any arbitrary neighbourhood of P contradicting (8), provided P is not finitely supported.

If P is not finitely supported, then for each \(\epsilon > 0\), we can find a set C such that \(0< P(C) < \epsilon \) [26, Lemma 15]. Choose Q which is defined by

$$\begin{aligned} Q(B) = \frac{1}{ 1 - P(C) } P(B {\setminus } C), ~~\text{ for } \text{ all }~ B \in \mathscr {B}. \end{aligned}$$

Now, it is easy to verify that \(\Vert P - Q \Vert < \epsilon \), giving the required contradiction. \(\square \)

Remark 2

In fact, the above proof also proves the result in the case of weak convergence. See Proposition 14 in [26].

Let \(P^*\) be a rest point of the replicator dynamics where \(P^*\) is as in (5). Let Q(0) be a population state in a small neighbourhood of \(P^*\) as in Lemma 2. Hence,

$$\begin{aligned} Q(0) = \sum _{j=1}^k \beta _j \delta _{x_j} + \beta _{k+1} R(0)~; ~~~~\sum _{j=1}^{k+1} \beta _j = 1 \end{aligned}$$
(9)

where \(R(0) \in \Delta \) with \(R(0)(\{x_1, x_2, \ldots , x_k\}) = 0\).

Consider the solution \(Q(\cdot )\) of the replicator dynamics Eq. (2) starting from Q(0). Since the support of Q(0) and Q(t) is the same, \(Q(t)(\{x_j\}) > 0\) for all \(j = 1,2, \ldots , k\).

Using this, from the replicator dynamics Eq. (2), we obtain,

$$\begin{aligned} Q^\prime (t)(\{x_j\}) = Q(t)(\{x_j\}) ~\sigma (x_j, Q(t)), ~~~~ Q(0)(\{x_j\}) = \beta _j \end{aligned}$$
(10)

for \(j=1, 2, \ldots , k\).

In the following theorem, we prove the Lyapunov stability of polymorphic population states.

Theorem 6

Let \(P^*\) be the polymorphic population state as in (5). If \(P^*\) is unbeatable then \(P^*\) is Lyapunov stable.

Proof

Let the polymorphic population state \(P^*\) be unbeatable. By arguments similar to that of Lemma 1 in [21], it follows that \(P^*\) is strongly unbeatable. This implies that there exists \(\epsilon > 0\) such that for \(R (\not = P^*)\), with \(\Vert R-P^*\Vert \le \epsilon \),

$$\begin{aligned} E(P^*,R) \ge E(R,R) \end{aligned}$$

Let \(\delta < 2 \min \{\alpha _1, \alpha _2, \ldots , \alpha _k\}\), \(\theta = \min \{\epsilon , \delta \}\) and \(\Omega = \{Q \in \Delta : \Vert Q-P^*\Vert < \theta \}\).

By the definition of \(\theta \), it follows from Lemma 2 that \(P^*\) is absolutely continuous with respect to Q, for every \(Q \in \Omega \). Therefore, for every \(Q \in \Omega \) and \(B \in \mathscr {B}\), we have

$$\begin{aligned} P^*(B) = \int _B \frac{\mathrm{{d}}P^*}{\mathrm{{d}}Q}~\mathrm{{d}}Q. \end{aligned}$$

Putting \(B = \{x_j\}\), we get,

$$\begin{aligned} \frac{\mathrm{{d}}P^*}{\mathrm{{d}}Q}(x_j) = \frac{\alpha _j}{Q(\{x_j\})};~~~~j=1,2,\ldots ,k. \end{aligned}$$
(11)

Define \(V:\Omega \rightarrow \mathbb {R}\) by,

$$\begin{aligned} V(Q) = \int _S \ln \left( \frac{\mathrm{{d}}P^*}{\mathrm{{d}}Q} \right) ~\mathrm{{d}}P^*. \end{aligned}$$
(12)

Since \(P^*\) is polymorphic, using (11), we can rewrite (12) as follows.

$$\begin{aligned} V(Q) = \sum _{j=1}^k \alpha _j ~ \ln \left( \frac{\mathrm{{d}}P^*}{\mathrm{{d}}Q}(x_j) \right) = \sum _{j=1}^k \alpha _j ~ \ln \left( \frac{\alpha _j}{Q(\{x_j\})} \right) . \end{aligned}$$
(13)

Using continuity of the log function, one may show that V is continuous in \(\Omega \). Moreover, \(V(P^*) = 0\) and for \(Q \in \Omega \) such that \(Q \not = P^*\), we have,

$$\begin{aligned} V(Q)&= \sum _{j=1}^k \alpha _j \ln \left( \frac{\alpha _j}{Q(\{x_j\})} \right) \\&= - \sum _{j=1}^k \alpha _j \ln \left( \frac{Q(\{x_j\})}{\alpha _j} \right) \\&> - \sum _{j=1}^k \alpha _j \left( \frac{Q(\{x_j\})}{\alpha _j} - 1 \right) ~~~~(\because ln(z) < z - 1 ~~\text{ for }~~z \not = 1)\\&= 1 - \sum _{j=1}^k Q(\{x_j\})~~~\ge 0. \end{aligned}$$

Thus, \(V(Q) \ge 0\) and the equality holds if and only if \(Q = P^*\); in other words, V is positive definite.

From Pinsker’s inequality (see [23, Eqs. (3.3.6) and (3.3.9)] and [4, Lemma 3]) we see that \(\Vert Q-P^*\Vert ^2 \le V(Q)\) for every \(Q \in \Omega \).

Now, let Q(t) be the trajectory of the replicator dynamics with the initial population state as \(Q \in \Omega \). Then,

Therefore,

$$\begin{aligned} \dot{V}(Q) = - E(P^*,Q) + E(Q,Q). \end{aligned}$$

Since \(P^*\) is unbeatable, and in turn strongly unbeatable, \(\dot{V}(Q) \le 0\) for any \(Q \in \Omega \) which proves that V is non-increasing along replicator dynamics trajectories.

Thus, by Theorem 8 (in the “Appendix”), we can conclude that \(P^*\) is Lyapunov stable. \(\square \)

Our next results establishes the asymptotic stability of polymorphic population states.

Theorem 7

Let \(P^*\) be the polymorphic population state as in (5). If \(P^*\) is uninvadable then \(P^*\) is asymptotically stable.

Proof

Let the polymorphic population state \(P^*\) be uninvadable. By Lemma 1 in [21], \(P^*\) is strongly uninvadable and hence, there exists \(\epsilon > 0\) such that for all \(R (\not = P^*)\) with \(\Vert R-P^*\Vert \le \epsilon \),

$$\begin{aligned} E(P^*,R) > E(R,R). \end{aligned}$$

We can define \(\Omega \) and the function V as in the proof of Theorem 6 where V is a positive definite continuous function for which

$$\begin{aligned} \dot{V}(Q) = - E(P^*,Q) + E(Q,Q) \end{aligned}$$

for every \(Q \in \Omega \). Since, \(P^*\) is uninvadable, and in turn strongly uninvadable, \(\dot{V}(Q) < 0\) for any \(Q \in \Omega \), \(Q \not = P^*\) which proves that V is strictly decreasing along replicator dynamics trajectories which remain in \(\Omega \).

Now, for any \(0< \epsilon _1 < \theta \), by Theorem 6, there exists \(\delta _1 > 0\) such that every trajectory starting from the open ball centred at \(P^*\) with radius \(\delta _1\) (denoted by \(B(P^*,\delta _1)\)) will remain in \(B(P^*,\frac{\epsilon _1}{2k})\).

Consider the trajectory \(Q(t) = Q(t;Q_0)\) starting from \(Q_0 \in B(P^*,\delta _1)\). For this trajectory Q(t), clearly, there exists a sequence \(t_n \rightarrow \infty \) such that \(Q(t_n)(\{x_j\})\) converges to a limit, say \(\beta _j^*\); \(j = 1,2,\ldots ,k\).

Since \(Q(t_n) \in B(P^*,\frac{\epsilon _1}{2k})\), it follows from (7) that \(|\alpha _j - \beta _j^*| \le \frac{\epsilon _1}{2k}\) for every \(j = 1,2,\ldots ,k\) and hence \(\sum \limits _{j=1}^k |\alpha _j - \beta _j^*| \le \frac{\epsilon _1}{2} < \theta \). In particular, by the definition of \(\theta \) we now have \(\beta _j^* > 0\) for every \(j = 1,2,\ldots ,k\).

This implies that

$$\begin{aligned} V(Q(t_n)) = \sum _{j=1}^k \alpha _j \ln \left( \frac{\alpha _j}{Q(t_n)(\{x_j\})} \right) \end{aligned}$$

converges to

$$\begin{aligned} V(Q^*) = \sum _{j=1}^k \alpha _j \ln \left( \frac{\alpha _j}{\beta _j^*} \right) \end{aligned}$$

for any (fixed) \(Q^* \in \Lambda \subset \Omega \) where

$$\begin{aligned} \Lambda = \left\{ Q \in \Omega ~\mid ~ Q = \sum _{j=1}^k \beta _j^* \delta _{x_j} + \left( 1 - \sum _{j=1}^k \beta _j^* \right) R; ~~R(\{x_1,x_2,\ldots ,x_k\}) = 0 \right\} . \end{aligned}$$

For \(s > 0\), by the replicator dynamics Eq. (10), we know that

$$\begin{aligned} Q(s;Q^*)(\{x_j\})&= \beta _j^* ~ \exp \left( \int _0^s \sigma \left( x_j,Q(t;Q^*) \right) ~\mathrm{{d}}t \right) \nonumber \\&= \beta _j^* ~T(s) \end{aligned}$$
(14)

and

$$\begin{aligned} Q(s;Q(t_n))(\{x_j\})&= Q(t_n)(\{x_j\}) ~ \exp \left( \int _0^s \sigma \left( x_j,Q(t;Q(t_n)) \right) ~\mathrm{{d}}t \right) \nonumber \\&= Q(t_n)(\{x_j\}) ~T_n(s). \end{aligned}$$
(15)

Therefore we have,

$$\begin{aligned}&| V(Q(s;Q^*)) - V(Q(s,Q(t_n))) |\nonumber \\&\quad = \Bigg | \sum _{j=1}^k \alpha _j \ln \left( \frac{\alpha _j}{\beta _j^* ~T(s)}\right) - \sum _{j=1}^k \alpha _j \ln \left( \frac{\alpha _j}{Q(t_n)(\{x_j\}) ~T_n(s)}\right) \Bigg | \nonumber \\&\quad = \sum _{j=1}^k \alpha _j \Bigg | \ln \left( \frac{Q(t_n)(\{x_j\}) ~T_n(s)}{\beta _j^* ~T(s)} \right) \Bigg | \end{aligned}$$
(16)

Since \(\sigma (\cdot ,Q)\) is bounded, it follows that \(\dfrac{T_n(s)}{T(s)} \rightarrow 1\) (uniformly in n) as \(s \downarrow 0\) and hence from (16), we get,

$$\begin{aligned} \lim _{s \downarrow 0 ,~ n \uparrow \infty } | V(Q(s;Q^*)) - V(Q(s,Q(t_n))) | = 0. \end{aligned}$$

Thus, by Theorem 9 (in the “Appendix”), we can conclude that \(P^*\) is asymptotically stable.

\(\square \)

4 Conclusions

In this article, we studied the stability of polymorphic population states in games with continuous strategy spaces. We proved that uninvadability is a sufficient condition for asymptotic stability of a polymorphic population state, whereas unbeatability is enough for the Lyapunov stability. Beyond finitely supported population states, one cannot establish similar stability results unless we weaken the notion of stability. This is an interesting topic for future research in games with continuous strategy spaces.