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ScienceDirect J. Differential Equations ••• (••••) •••–••• www.elsevier.com/locate/jde

Evolutionary dynamics on measurable strategy spaces: Asymmetric games ✩ Saul Mendoza-Palacios ∗ , Onésimo Hernández-Lerma Mathematics Department, CINVESTAV-IPN, A. Postal 14-740, México D.F. 07000, Mexico Received 15 January 2015; revised 11 June 2015

Abstract The theory of evolutionary dynamics in asymmetric games has been mainly studied for games with a finite strategy space. In this paper we introduce an evolutionary dynamics model for asymmetric games where the strategy sets are measurable spaces (separable metric spaces). Under this hypothesis the replicator dynamics is in a Banach Space. We specify conditions under which the replicator dynamics has a solution. Furthermore, under suitable assumptions, a critical point of the system is stable. Finally, an example illustrates our results. © 2015 Elsevier Inc. All rights reserved. MSC: 91A22; 91A10; 34G20; 34A34 Keywords: Asymmetric evolutionary games; Evolutionary games; Population games; Replicator dynamics; Space of finite signed measures; Ordinary differential equations in measure spaces

1. Introduction Evolutionary games form a class of noncooperative games in which the interaction of the strategies is studied from two different approaches, static and dynamic. The static approach captures evolutionary ideas through defining and studying equilibrium concepts. The dynamic approach, on the other hand, studies the interactions of the strategies as a dynamical system, ✩

This research was partially supported by CONACYT Grant 221291.

* Corresponding author.

E-mail address: [email protected] (S. Mendoza-Palacios). http://dx.doi.org/10.1016/j.jde.2015.07.005 0022-0396/© 2015 Elsevier Inc. All rights reserved.

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which is determined by a system of differential equations. In this paper we are interested in the dynamic approach with a specific dynamical system known as the replicator dynamics. An evolutionary game is said to be symmetric if there are two players only and, furthermore, they have the same sets of strategies and the same payoff functions. This type of games models interactions of the strategies of a single population. In this paper we address asymmetric evolutionary games, also known as multipopulation games, in which there is a finite set of players (or populations) each of which has a different set of strategies and different payoff functions. In our model, the strategy set of each player (or population) is a measurable space and consequently the dynamical system lives in a Banach space. We establish conditions for the existence of solutions to the replicator dynamics, and also conditions that ensure the stability of the system. This allows us to generalize the usual asymmetric evolutionary game, in which the strategy set of each player is finite. Symmetric models can be seen, of course, as a particular case of our proposed model. Evolutionary game dynamics with strategies in metric spaces has been developed for the symmetric case, that is, when we have two players with the same set of strategies and the same payoff functions, and the system of differential equations is reduced to a single equation (see Section 3.2 below). Conditions for the existence of solutions to the replicator equation in the symmetric case are given by several authors including Bomze [6], Oechssler and Riedel [23], and more generally (including dynamics different from the replicator equation) by Cleveland and Ackleh [9]. Similarly, conditions for dynamic stability in these models have been developed with respect to different topologies, see e.g. Bomze [5], Oechssler and Riedel [23] and [24], Eshel and Sansone [13], Veelen and Spreij [32], Cressman and Hofbauer [12]. Asymmetric evolutionary game dynamics (or of several populations) has been developed for games where the strategy set of each player is finite, as in Balkenborg and Schlag [2], Ritzberger and Weibull [26], Samuelson and Zhang [28], Selten [31]. Nevertheless, there are well-known cases where the sets of strategies are metric spaces, such as oligopoly models and Nash bargaining games (Cressman [11]). In evolutionary games, instead of assuming rationality of players and perfect information of the game, we assume that each player (or population) has an evolutionary dynamics in the use of their strategies, which is modeled by a certain dynamical system. With our proposed model we can introduce an evolutionary dynamics in asymmetric games where the strategy sets are Borel spaces (that is, Borel subsets of complete and separable metric spaces). This includes, for example, the models where the strategy sets are finite or games with strategies in metric spaces such as some models in oligopoly theory, international trade theory, war of attrition, and public goods, among others. It should be noted that extending symmetric evolutionary games to the asymmetric case is not evident. Bomze and Potscher [7] suggest an approach (similar to Selten’s [31] for the case with finite strategy sets) in which asymmetric games are reduced to the symmetric case; for details see Section 3.3 below. This approach, however, has some disadvantages. For instance, the relationship between Nash equilibria and the replicator dynamics is unclear. It is also unclear how to extend stability concepts and results to the asymmetric case. In contrast, with our proposed model is it easy to see the relationship between a Nash equilibrium and the replicator dynamics (see Section 5) and, in addition, stability concepts have a natural construction from the symmetric case to the asymmetric situation (see Section 6). The main goal of this paper is to introduce a theoretical foundation for asymmetric evolutionary games. Hence, in particular, we would like to know which results for symmetric games can be extended to the asymmetric case. This can be done, in principle, in many different ways

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because for evolutionary games there are several interesting dynamics, for instance, the imitation dynamics, the monotone-selection dynamics, the best-response dynamics, the Brown–von Neumann–Nash dynamics, and so forth (see, for instance, Hofbauer and Sigmund [18,19], Sandholm [30]). Here, however, we selected the replicator dynamics partly because it is the most studied dynamics for games with strategies in metric spaces, and partly because it has many interesting properties, as can be seen in Cressman [10], Hofbauer and Weibull [20], and many other references. In particular, with the replicator dynamics it is not difficult to construct a proof of the existence of Nash equilibria and, moreover, when the strategy sets Ai are finite, we can give a geometric characterization of the set of Nash equilibria; see Harsanyi [15], Hofbauer and Sigmund [18], Ritzberger [25]. The paper is organized as follows. Section 2 presents notation and technical requirements. Section 3 describes the asymmetric evolutionary game and the replicator equation. Section 4 establishes conditions for the existence of a solution to the system of differential equations (replicator dynamics), and gives some characterizations of the solution. Section 5 establishes a relationship between the Nash equilibrium concept and the replicator dynamics. Section 6 introduces conditions to establish the stability of the system. Section 7 proposes an example to illustrate our results. We conclude in Section 8 with some general comments on possible extensions. Appendix A, contains the proof of a technical fact. 2. Technical preliminaries 2.1. Signed measure spaces Consider a separable metric space X and its Borel σ -algebra B(X). Let M(X) be the Banach space of finite signed measures μ on B(X) endowed with the total variation norm μ := sup f (s)μ(ds) = |μ|(X), f ≤1

(1)

X

where |μ| = μ+ + μ− denotes the total variation of μ, and μ+ , μ− stand for the positive and the negative parts of μ, respectively. The supremum in (1) is taken over functions in the Banach space B(X) of real-valued bounded measurable functions on X, with the supremum norm f := sup |f (x)|.

(2)

x∈X

The following definition introduces some convergence concepts in M(X). Definition 2.1. A sequence of measures μn ∈ M(X) is said to be setwise convergent if there exists μ ∈ M(X) such that ∀g ∈ B(X) :

g(s)μn (ds) =

lim

n→∞ X

g(s)μ(ds). X

(3)

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If this convergence is uniform, that is, lim sup g(s)μn (ds) − g(s)μ(ds) = lim μn − μ = 0, n→∞ g≤1 n→∞ X

(4)

X

then it is called strong convergence or convergence in the total variation norm. Consider the subset CB (X) ⊂ B(X) of all real-valued continuous and bounded functions on X. A sequence of measures μn ∈ M(X) is said to be weakly convergent if there exists μ ∈ M(X) such that (3) holds over all CB (X). If M(X) is the space of probability measures on X, then sometimes we say that μn converges in distribution to μ. 2.2. Product spaces Consider two separable metric spaces X and Y with their Borel σ -algebras B(X) and B(Y ). We denote by σ [X ×Y ] the σ -algebra on X ×Y generated by the Cartesian product B(X) ×B(Y ). For μ ∈ M(X) and ν ∈ M(Y ), we denote their product by μ × ν ∈ M(X × Y ). Proposition 2.2. For μ ∈ M(X) and ν ∈ M(Y ), it holds that μ × ν ≤ μν.

(5)

As a consequence, μ × ν is in M(X × Y ). Proof. See Heidergott and Leahu [17], Lemma 4.2.

2

Now consider a finite family of metric spaces {Xi }ni=1 and their σ -algebras B(Xi ), as well as the Banach spaces M(Xi ). For i = 1, . . . , n, let μi ∈ M(Xi ). Consider the elements μ = (μ1 , μ2 , . . . , μn ) in the product space M(X1 ) × M(X2 ) × . . . × M(Xn ) for which μ∞ = (μ1 , . . . , μn )∞ := max μi < ∞. 1≤i≤n

(6)

These elements form a Banach space with · ∞ as a norm. We call it the direct product of the Banach spaces M(Xi ). 2.3. Differentiability Definition 2.3. Let X be a separable metric space. We say that a mapping μ : [0, ∞) → M(X) is strongly differentiable if there exists μ (t) ∈ M(X) such that, for every t > 0, μ(t + ) − μ(t)

lim − μ (t) = 0. →0

(7)

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Note that, by (1), the left-hand side of (7) can be expressed as ⎡ ⎤ 1

⎦ ⎣ lim sup g(x)μ(t + , dx) − g(x)μ(t, dx) − g(x)μ (t, dx) . →0 g≤1 X

X

X

The strong derivative μ (t) ∈ M(X) is also called a Fréchet derivative in the Banach space M(X). If μ : [0, ∞) → M(X) is weakly differentiable in the sense of weak convergence and μ (t) ∈ M(X) is continuous with the norm (1), then μ(t) is strongly differentiable (see Heidergott, Hordijk and Leahu [16], Proposition 1). 3. The model 3.1. Asymmetric evolutionary games Let I := {1, 2, . . . , n} be the set of different species (or players). Each individual of the species i ∈ I can choose a single element ai in a set of characteristics (strategies or actions) Ai , which is a separable metric space. For every i ∈ I and every vector a := (a1 , . . . , an ) in the Cartesian product A := A1 × . . . × An , we write a as (ai , a−i ) where a−i := (a1 , . . . , ai−1 , ai+1 , . . . , an ) is in A−i := A1 × . . . × Ai−1 × Ai+1 × . . . × An . For each i ∈ I , let B(Ai ) be the Borel σ -algebra of Ai , and P(Ai ) the set of probability measures on Ai , also known as the set of mixed strategies. A probability measure μi ∈ P(Ai ) assigns a population distribution over the action set Ai of the species i. Finally, for each species i we assign a payoff function Ji : P(A1 ) × . . . × P(An ) → R that explains the interrelation with the population of other species, and which is defined as Ji (μ1 , . . . , μn ) :=

...

A1

Ui (a1 , . . . , an )μn (dan )...μ1 (da1 ),

(8)

An

where Ui : A1 × . . . × An → R is a given measurable function. For every i ∈ I and every vector μ := (μ1 , . . . , μn ) in P(A1 ) × . . . × P(An ), we sometimes write μ as (μi , μ−i ), where μ−i := (μ1 , . . . , μi−1 , μi+1 , . . . , μn ) is in P(A1 ) × . . . × P(Ai−1 ) × P(Ai+1 ) × . . . × P(An ). If δ{ai } is a probability measure concentrated at ai ∈ Ai , the vector (δ{ai } , μ−i ) is written as (ai , μ−i ), and so Ji (δ{ai } , μ−i ) = Ji (ai , μ−i ).

(9)

We express the right-hand side of (8) as I(μ1 ,...,μn ) Ui :=

...

A1

An

Ui (a1 , . . . , an )μn (dan )...μ1 (da1 ).

(10)

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Hence (9) becomes Ji (ai , μ−i ) =

Ui (ai , a−i )μ−i (da−i )

A−i

= I(μ1 ,...,μi−1 ,μi+1 ,...,μn ) Ui .

(11)

In particular, (8) yields Ji (μi , μ−i ) :=

Ji (ai , μ−i )μi (dai ).

(12)

Ai

In an evolutionary game, the dynamics of the strategies is determined by the solution of a system of differential equations of the form μ i (t) = Fi (μ1 (t), . . . , μn (t))

∀ i ∈ I, t ≥ 0,

(13)

with some initial condition μi (0) = μi,0 for each i ∈ I . The notation μ i (t) represents the Fréchet derivative of μi (t) in the Banach space M(Ai ) (see Definition 2.3). For each i ∈ I , Fi (·) is a mapping Fi : P(A1 ) × . . . × P(An ) → M(Ai ). Let F : P(A1 ) × . . . × P(An ) → M(A1 ) × . . . × M(An ), where F (μ) := (F1 (μ), . . . , Fn (μ)), and consider the vector μ (t) := (μ 1 (t), . . . , μ n (t)). Then the system (13) can be expressed as μ (t) = F (μ(t)),

(14)

and we can see that the system lives in the Cartesian product of signed measures M(A1 ) × . . . × M(An ), which is a Banach space with norm as in (6), i.e. μ∞ = (μ1 , . . . , μn )∞ := max μi . i∈I

(15)

More explicitly, we may write (13) as μ i (t, Ei ) = Fi (μ(t), Ei ) ∀ i ∈ I, Ei ∈ B(Ai ), t ≥ 0,

(16)

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where μ i (t, Ei ) and Fi (μ(t), Ei ) are the signed-measures μ i (t) and Fi (μ(t)) valued at Ei ∈ B(Ai ). We shall be working with a special class of asymmetric evolutionary games which can be described as

I, P(Ai )

i∈I

, Ji (·)

i∈I

, μ i (t) = Fi (μ(t))

i∈I

,

(17)

where i) I = {1, . . . , n} is the finite set of players; ii) for each player i ∈ I we have a set of mixed actions P(Ai ) and a payoff function Ji : P(A1 ) × . . . × P(An ) → R (as in (12)); and iii) the replicator dynamics Fi (μ(t)), where

Fi (μ(t), Ei ) :=

Ji (ai , μ−i (t)) − Ji (μi (t), μ−i (t)) μi (t, dai ).

(18)

Ei

3.2. The symmetric case We can obtain from (17) a symmetric evolutionary game when I := {1, 2} and the sets of actions and payoff functions are the same for both players, i.e., A = A1 = A2 and U (a, b) = U1 (a, b) = U2 (b, a), for all a, b ∈ A. As a consequence, the sets of mixed actions and the expected payoff functions are the same for both players, i.e., P(A) = P(A1 ) = P(A2 ) and J (μ, ν) = J1 (μ, ν) = J2 (ν, μ), for all μ, ν ∈ P(A). This kind of model determines the dynamic interaction of strategies of a unique species through the replicator dynamics ν (t) = F (ν(t)), where F : P(A) → M(A) is given by F (ν(t), E) :=

J (a, ν(t)) − J (ν(t), ν(t)) ν(t, da) ∀E ∈ B(A).

(19)

E

Finally, as in (17), we can describe a symmetric evolutionary games as

I = {1, 2}, P(A), J (·), ν (t) = F (ν(t)) .

(20)

3.3. Another approximation to asymmetric games Bomze and Pötscher [7] suggest an approach in which asymmetric games are reduced to symmetric ones. They construct a new strategy set A¯ and a new payoff function J : A¯ × A¯ → R. The strategy set A¯ decomposes into mutually disjoint sets Ai , that is A¯ := ∪i∈I Ai , where Ai is the set of strategies of the species i ∈ I . Then any measurable E ⊂ A¯ may be expressed as a union of mutually disjoint sets Ei , that is E = ∪i∈I Ei , where Ei = E ∩ Ai . Then μ(E) = μ(E ) = i i∈I i∈I μi (E)μ(Ai ), where μi (E) := μ(E|Ai ) =

μ(E ∩ Ai ) . μ(Ai )

(21)

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The new payoff function is given by J (μ, ν) =

μ(Ai )Ji (μi , ν−i ),

i∈I

where ν−i := (ν1 , . . . , νi−1 , νi+1 , . . . , νn ) and νj as (21) and Ji (μi , ν−i ) as (8). The replicator dynamics is constructed as in the symmetric case (19), with F (μ(t), E) :=

i∈I

μ(Ai )

Ji (ai , μ−i (t)) − μ(Ai )Ji (μi (t), μ−i (t)) μi (t, dai ).

Ei

4. Existence In this section we introduce conditions for the existence and uniqueness of solutions to the differential system (13). For this purpose we give conditions under which the operator F in (14) is Lipschitz, when this operator is defined as in (18). For each i ∈ I and t ≥ 0, let βi (ai |μ(t)) := Ji (ai , μ−i (t)) − Ji (μi (t), μ−i (t)).

(22)

Hence, by (18), βi (·|μ(t)) is the Radon–Nikodym density of Fi (μ(t)) with respect to μi (t), i.e., Fi (μ(t), Ei ) =

βi (ai |μ(t))μi (t, dai ) ∀Ei ∈ B(Ai ).

(23)

Ei

Remark 4.1. i) We will use the usual notation μ << ν to indicate that μ is absolutely continuous respect to ν (i.e. for every set E ∈ B(A) with ν(E) = 0 we have μ(E) = 0). ii) Let A be a separable metric space with Borel σ -algebra B(A). Suppose that ν, η ∈ M(A) and c1 , c2 ≥ 0, and let μ = c1 η + c2 ν. If there exists a positive measure κ ∈ M(A) such that ν << κ and η << κ, then also μ << κ. Moreover, the Radon–Nikodym densities ϕνκ =

dν dκ

ϕμκ =

dμ = c1 ϕηκ + c2 ϕνκ . dκ

and ϕηκ =

dη , dκ

are such that

Lemma 4.2. Let ν, η, μ, κ and ϕμκ be as in Remark 4.1. Then the total variation norm of μ is given by μ =

|ϕμκ (a)|κ(da). A

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In particular, the distance between the signed measures ν and η is given by ν − η =

|(ϕνκ − ϕηκ )(a)|κ(da). A

The following proposition extends to our context some results by Bomze [6] (Lemma 1), and Oechssler and Riedel [23] (Lemma 3) in the case of symmetric evolutionary games. Theorem 4.3. Suppose that, for each i ∈ I , the function βi (·|μ) in (22) satisfies: i) there exists Ci ≥ 0 such that |βi (ai |μ)| ≤ Ci for each ai ∈ Ai and μ∞ ≤ 2; ii) there is a constant Di > 0, such that sup |βi (ai |η) − βi (ai |ν)| ≤ Di η − ν∞

ai ∈Ai

for each ν, η with η∞ , ν∞ ≤ 2. Then there exists a bounded Lipschitz map G : M(A1 ) × . . . × M(An ) → M(A1 ) × . . . × M(An ), which coincides with F on P(A1 ) × . . . × P(An ). i| Proof. For each i ∈ I and ν, η with η∞ , ν∞ ≤ 2, let μi = |ηi |+|ν . Then μi ≤ 2, ηi << 2 dηi dνi = ϕηi μi and dμ = μi and νi << μi . Whence there exist the Radon–Nikodym densities dμ i i ϕνi μi . Using (23) and Lemma 4.2 we have that

Fi (η) − Fi (ν) =

βi (ai |η)ϕηi μi (ai ) − βi (ai |ν)ϕνi μi (ai ) μi (dai ) Ai

≤

βi (ai |η) − βi (ai |ν)ϕηi μi (ai )μi (dai )

Ai

+ βi (ai |ν)ϕηi μi (ai ) − ϕνi μi (ai )μi (dai ) Ai

≤ βi (ai |η) − βi (ai |ν)|ηi |(dai ) Ai

+ βi (ai |ν)ϕηi μi (ai ) − ϕνi μi (ai )μi (dai ) Ai

≤ 2Di max ηj − νj + Ci ηi − νi j ∈I

≤ Ki η − ν∞ ,

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where Ki := max{2Di , Ci }. Therefore F (η) − F (ν) = max Fi (η) − Fi (ν) ≤ Kη − ν∞ , i∈I

for all η, ν with η∞ , ν∞ ≤ 2, with K := max{Ki : i ∈ I }. Hence, F is Lipschitz continuous on the subset of M(A1 ) × . . . × M(An ) with norm · ∞ ≤ 2. Let us now consider the function G(μ) := (2 − μ∞ )+ F (μ),

(24)

with (2 − μ∞ )+ := max{0, 2 − μ∞ }. It is clear that G(·) is bounded and coincides with F (·) on P(A1 ) × . . . × P(An ). It remains to show that G(·) is Lipschitz. Consider η and ν in M(A1 ) × . . . × M(An ). If η∞ , ν∞ ≥ 2, then G(η) = G(ν) = 0 and there is nothing to prove. Now, if η∞ > 2 ≥ ν∞ , then G(η) − G(ν)∞ = (2 − ν∞ )F (ν)∞ , and F (ν)∞ = max βj (aj |ν)|νj |(daj ) ≤ max Cj νj ≤ Cν∞ j ∈I

j ∈I

Aj

where C = maxj ∈I {Cj }. Hence G(η) − G(ν)∞ ≤ (2 − ν∞ )Cν∞ ≤ 2C(η∞ − ν∞ ) ≤ 2Cη − ν∞ .

(25)

Finally, if η∞ , ν∞ ≤ 2, then G(η) − G(ν)∞ = (2 − η∞ )F (η) − (2 − ν∞ )F (ν)∞ ≤ (2 − η∞ )F (η) − F (ν)∞ + F (ν)∞ |ν∞ − η∞ | ≤ 2Kη − ν∞ + 2Cν − η∞ .

(26)

Using (25) and (26) we see that, for any η, ν ∈ M(A1 ) × . . . × M(An ), we have G(η) − G(ν)∞ ≤ 2(K + C)η − ν∞ .

2

The following proposition is an extension to our asymmetric games of Lemma 4 of Oechssler and Riedel [23] for symmetric games. Proposition 4.4. Let i ∈ I . If the payoff function Ui (·) is bounded, then βi (·|μ) satisfies the conditions i) and ii) of Theorem 4.3.

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Proof. Suppose that μ∞ ≤ 2 and let i ∈ I . Since Ui (·) is bounded, there exists Ci > 0 such that |Ui (a)| ≤ Ci for all a ∈ A. Then, by Proposition 2.2, |βi (ai |μ)| = Ui (ai , a−i )μ−i (da−i ) − Ui (a)μ(da) A−i A ≤ Ui (ai , a−i )μ−i (da−i ) + Ui (a)μ(da) A A−i ≤ Ci μ1 × . . . × μi−1 × μi+1 ... × μn + Ci μ1 × . . . × μn ≤ 2n−1 Ci + 2n Ci . Letting Ci := Ci (2n−1 + 2n ), the condition i) follows. To prove the condition ii) in Theorem 4.3, note that for any η and ν with η∞ , ν∞ ≤ 2, using the notation in (10), and substracting and adding terms, we obtain, for every i ∈ I , Ui (a)η(da) − Ui (a)ν(da) A

A

≤ |I(η1 ,η2 ,...,ηn ) Ui − I(ν1 ,η2 ,...,ηn ) Ui | + |I(ν1 ,η2 ,η3 ,...,ηn ) Ui − I(ν1 ,ν2 ,η3 ,...,ηn ) Ui | + ... + |I(ν1 ,...,νn−2 ,ηn−1 ,ηn ) Ui − I(ν1 ,...,νn−2 ,νn−1 ,ηn ) Ui | + |I(ν1 ,...,νn−1 ,ηn ) Ui − I(ν1 ,...,νn−1 ,νn ) Ui | ≤ Ui η2 × .... × ηn η1 − ν1 + Ui ν1 × η3 × . . . × ηn η2 − ν2 + ... + Ui ν1 × . . . × νn−2 × ηn ηn−1 − νn−1 + Ui ν1 × .... × νn−1 ηn − νn ≤ 2n−1 Ui max ηj − νj . j ∈I

(27)

Similarly, for every i ∈ I , Ui (a)η−i (da−i ) ≤ 2n−2 Ui max ηj − νj . Ui (a)ν−i (da−i ) − j =i A−i A−i

(28)

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Then by (27) and (28) |βi (ai |η) − βi (ai |ν)| = Ui (a)η−i (da−i ) − Ui (a)η(da) A−i A −

Ui (a)ν−i (da−i ) +

A−i

A

Ui (a)ν(da)

Ui (a)ν−i (da−i ) ≤ Ui (a)η−i (da−i ) − A−i A−i + Ui (a)ν(da) − Ui (a)η(da) A

A

≤ 2 Ui max ηj − νj . n

j ∈I

To conclude, the latter inequality yields sup |βi (ai |η) − βi (ai |ν)| ≤ Di η − ν∞ ,

ai ∈Ai

with Di = 2n Ui .

2

By Theorem 4.3, the differential equation μ (t) = G(μ(t)),

(29)

with G as in (24) has a unique solution in the space M(A1 ) × . . . × M(An ). If μ(t) is a solution to (29) and μ(t) ∈ P(A1 ) × . . . × P(An ) ∀t ≥ 0, then μ(t) is also a solution of the differential equation (14), and it is unique since F (·) is Lipschitz in the open ball V2 (0) = {μ ∈ M(A1 ) × . . . × M(An ) : μ∞ < 2}. Let μ(·) be a solution of (29) (or (14)). We say that a set C ⊂ M(A1 ) × . . . × M(An ) is an invariant set for (29) (or (14)), if μ(t) is in C for all t > 0 when μ(0) is in C. The following proposition ensures that the set P(A1 ) × . . . × P(A2 ) is an invariant set for (29). Therefore the replicator dynamics has a solution.

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13

Theorem 4.5. If μ(t) is a solution to (29), with initial condition μ(0) in P(A1 ) × . . . × P(An ), then μ(t) remains in P(A1 ) × . . . × P(An ) for all t > 0. Moreover, μ(t) is also the unique solution to the replicator dynamics (14) with F (·) as in (18). Proof. First, note that dμi (t, Ei ) = μ (t, Ei ) dt

∀i ∈ I, Ei ∈ B(Ai ), t ≥ 0.

(30)

Indeed, dμi (t, Ei )

(t, E ) − μ i i dt μi (t + , Ei ) − μi (t, Ei )

= lim − μi (t, Ei ) →0 ⎡ ⎤ 1 ⎥ ⎢ = lim ⎣ 1Ei μi (t + , dai ) − 1Ei μi (t, dai )⎦ − 1Ei μ i (t, dai ) →0 Ai Ai Ai μi (t + ) − μi (t) − μ i (t) ≤ lim = 0. →0 Now, if μ(t) is a solution to (29), then by (30) and (24), for each i ∈ I , Ei ∈ B(Ai ) and t ≥ 0, we have dμi (t, Ei ) = (2 − μ(t)∞ )+ dt

Ji (ai , μ−i (t))μi (t, dai ) − Ji (μi (t), μ−i (t))μi (t, Ei ) . Ei

(31) In particular, for every i ∈ I , dμi (t, Ai ) = (2 − μ(t)∞ )+ [1 − μi (t, Ai )]Ji (μi (t), μ−i (t)). dt

(32)

We can express (32) as a system of differential equations in Rn , say dμi (t, Ai ) = fi (t, μi (t, Ai )) dt

for i = 1, . . . , n,

where we can see the vector [fi (t, μi (t, Ai ))]i∈I as a function f : [0, ∞) × Rn → Rn with f (t, μ1 (t, A1 ), . . . , μn (t, An )) = [fi (t, μi (t, Ai ))]i∈I . The system (32) has a critical point if μi (t, Ai ) = 1 for i = 1, . . . , n. Then if μi (0, Ai ) = 1, we have that μi (t, Ai ) = 1 for all t ≥ 0 and i ∈ I . Hence the set

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B := {μ ∈ M1 × . . . × Mn : μi (Ai ) = 1 ∀i ∈ I } , is an invariant set for (29). Moreover, if Ei ∈ B(Ai ), t ≥ 0 and μi (t , Ei ) = 0, then by (31), μi (t, Ei ) = 0 for all t ≥ t . In particular for each Ei ∈ B(Ai ) and i ∈ I , |μi (t, Ei ) − μi (s, Ei )| ≤ μi (t) − μi (s)

∀t, s ≥ 0.

(33)

Since for each i in I the map t → μi (t) is continuous, then by (33) so is the map t → μi (t, Ei ) for each Ei ∈ B(Ai ). Therefore, if μi (0, Ei ) ≥ 0, then we have μi (t, Ei ) ≥ 0 for all t > 0 and Ei ∈ B(Ai ). It follows that, P(A1 ) × . . . × P(An ) ⊂ B is an invariant set for the system of differential equations (29). Finally, if μ(t) is a solution to (29) and μ(0) is in P(A1 ) × . . . × P(An ), then μ(t) is a solution to (14), and since F is Lipschitz for all μ with μ∞ ≤ 2, this solution is unique. 2 Theorem 4.6. Suppose that the conditions i) and ii) of Theorem 4.3 are satisfied. If μ(t) is a solution to (14) with the initial condition μ(0) in P(A1 ) × . . . × P(A1 ), then: i) for every i ∈ I and t > 0, if μi is in P(Ai ), then μi (0) << μi (t) and μi (t) << μi (0), with Radon–Nikodym density t dμi (t) (ai ) = e 0 βi (ai |μ(s))ds . dμi (0)

(34)

ii) In particular, for every i ∈ I and t > 0, if νi is a probability measure satisfying that νi << μi (t) whenever νi << μi (0), then dνi dνi log (ai ) = log (ai ) − dμi (t) dμi (0)

t βi (ai |μ(s))ds.

(35)

0

Proof. The following proof is an adaptation of Ritzberger [25] (Lemma 2) and Bomze [6] (Lemma 2). Let μ(t) be the solution to (14), with μ(0) ∈ P(Ai ) and ϕi (t, ai ) := e

t 0

βi (ai |μ(s))ds

≥0

∀i ∈ I.

In addition, let μ˜ i (t, Ei ) :=

ϕi (t, ai )μi (0, dai )

∀Ei ∈ B(Ai ),

Ei

and, by (23), Fi (μ˜ i (t), Ei ) =

βi (ai |μ(t))μ˜ i (t, dai ). Ei

(36)

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15

We will prove that μ˜ (t) − F (μ(t)) ˜ ∞ = 0, where μ˜ (t) = (μ˜ 1 (t), . . . , μ˜ n (t)) and F (μ(t)) ˜ = (F1 (μ(t)), ˜ . . . , Fn (μ(t)). ˜ Let i ∈ I and fix t > 0. Then μ˜ i (t) − Fi (μ(t)) ˜ 1 g(ai )[ϕi (t + h, ai ) − ϕi (t, ai )]μ(0, dai ) = lim sup h→0 g≤1 h Ai − g(ai )βi (ai |μ(t))ϕ(t, ai )μi (0, dai ) Ai 1 ≤ lim [ϕi (t + h, ai ) − ϕi (t, ai )] − βi (ai |μ(t))ϕi (t, ai ) μi (0, dai ) h→0 h Ai

which, by (36), t+h βi (ai |μ(s))ds t −1 0 βi (ai |μ(s))ds e t ≤ sup e − βi (ai |μ(t)) μi (0, dai ) lim h→0 h ai ∈Ai Ai

t+h e t βi (ai |μ(s))ds − 1 tCi ≤ sup e lim − βi (ai |μ(t)) μi (0, dai ) = 0, h→0 h ai ∈Ai Ai

where the latter equality follows from the conditions i) and ii) of Theorem 4.3 together with the dominated convergence theorem. To conclude μ˜ (t) − F (μ(t)) ˜ ∞=0

∀t > 0.

By the uniqueness in Corollary 1.7, page 72 of Lang [22] we thus get (34), and therefore μi (t) << μi (0) ∀i ∈ I. By the condition ii) of Theorem 4.3, for each i ∈ I and t > 0, there exists Ci ≥ 0 such that t −tCi ≤ 0 βi (ai |μ(s)) ≤ tCi . Therefore 0 < e−tCi ≤ e

t 0

βi (ai |μ(s))

≤ etCi .

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Hence, by (34),

e−tCi μi (0, dai ) ≤

Ei

t e 0 βi (ai |μ(s)) μi (0, dai ) = μi (t, Ei ); Ei

thus μi (0) << μi (t). The assertion ii) follows from i) and an application of the chain rule for Radon–Nikodym densities (see Bartle [3], Chapter 8). 2 5. Nash equilibrium and the replicator equation In this section we consider an asymmetric evolutionary game as in (17). We wish to study the relation between a Nash equilibrium of a normal form game and the replicator equation (see Theorem 5.4 below). Also we introduce the concept of strongly uninvadable profile (Definition 5.5), and its relation with -equilibrium(Definition 5.1). A normal form game (also known as a game in strategic form) can be described as

:= I, P(Ai )

i∈I

, Ji (·)

i∈I

,

(37)

where i) I = {1, 2, . . . n} is the set of players, ii) for each player i ∈ I we specify a set of actions (or strategies) P(Ai ) and a payoff function Ji : P(A1 ) × . . . × P(An ) → R. Definition 5.1. Let be a normal form game. A vector μ∗ in P(A1 ) × . . . × P(An ) is called -equilibrium ( > 0) if for all i ∈ I Ji (μ∗i , μ∗−i ) ≥ Ji (μi , μ∗−i ) −

∀μi ∈ P(Ai ).

If the inequality is true when = 0, then μ∗ is called a Nash equilibrium. The following proposition states an important fact about probability measures on separable metric spaces. Proposition 5.2. Let A be a separable metric space and μ ∈ P(A). Then there is a unique closed set S ⊂ A (called the support of μ) such that μ(A − S) = 0 and μ(O ∩ S) > 0 for every open set O for which O ∩ S = φ. Proof. See Royden [27], page 408.

2

Lemma 5.3. Supposes that μ∗ = (μ∗1 , . . . , μ∗n ) is a Nash equilibrium of , and let Si be the support of μ∗i for some i ∈ I . Then Ji (ai , μ∗−i ) = Ji (μ∗ , μ∗−i ) for all ai ∈ Si , i.e., Ji (μ∗i , μ∗−i ) = Ji (ai , μ∗−i ) μ∗i -a.s.

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17

Proof. Using Proposition 5.2, the proof is similar to the case when the strategy sets are finite (see, e.g., Webb [33]). 2 The following theorem gives an important property, namely the relation between a Nash equilibrium of a normal form game and the replicator equation. Theorem 5.4. Suppose that μ∗ = (μ∗1 , . . . , μ∗n ) is a Nash equilibrium of . Then μ∗ is a critical point of (14), i.e., F (μ∗ ) = 0, when F (·) is described by the replicator dynamics (18). Proof. First note that any vector of Dirac measures δa = (δa1 , . . . , δan ) (sometimes called a profile of pure strategies) is a critical point of (14), since for every Ei ∈ B(Ai ) and i ∈ I : Fi (δa , Ei ) =

) − Ji (δa , δa ) δa (dai ) = 0. Ji (ai , δa−i i −i i

Ei

Then if μ∗ is a pure Nash equilibrium, i.e., μ∗ = δa ∗ , the theorem holds. Suppose now that the Nash equilibrium μ∗ is not pure and let Si∗ be the support of μ∗i for some i ∈ I . By Lemma 5.3, for all ai ∈ Si∗ Ji (ai , μ∗−i ) = Ji (μ∗i , μ∗−i ). Therefore, for any Ei ∈ B(Ai ), Fi (μ , Ei ) = Ji (ai , μ∗−i ) − J (μ∗i , μ∗−i ) μ∗i (da) ∗

Ei

=

Ji (ai , μ∗−i ) − J (μ∗i , μ∗−i ) μ∗i (da) = 0.

2

Ei ∩Si∗

The following definition is an extended version of strongly uninvadables strategies of symmetric games (for details see Bomze [6]). Definition 5.5. A vector μ∗ ∈ P(A1 ) × P(A2 ) × . . . × P(An ) is called a strong uninvadable profile (SUP) in a set C if μ∗ is in C and the followings holds. There exists > 0 such that for any μ ∈ C with μ − μ∗ ∞ < , and every i ∈ I , Ji (μ∗i , μ−i ) > Ji (μi , μ−i ) if μi = μ∗i . In particular if C = P(A1 ) × P(A2 ) × . . . × P(An ), μ∗ is simply called strong uninvadable profile (SUP). In both cases, we call the global invasion barrier. Lemma 5.6. Let μ, ν ∈ P1 (A) × · · · × P(An ) and δ > 0. Then there exists α in (0, 1) such that γ − μ∞ ≤ δ if γ = αν + (1 − α)μ. Proof. Let 0 < α <

δ ν−μ∞ .

Then

γ − μ∞ = αν + (1 − α)μ − μ∞ = αν − μ∞ < δ.

2

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18

As usual, the open neighborhood with center μ∗ and radius ε > 0 is defined as Vε (μ∗ ) := {μ ∈ P(A1 ) × . . . × P(An ) : μ − μ∗ ∞ < ε}.

(38)

The following theorem gives the relation between an -equilibrium (or Nash equilibrium) and strong uninvadable profiles. Theorem 5.7. Suppose that the payoff function Ui (·) in (8) is bounded for all i ∈ I . Let μ∗ be a SUP in a set C with global invasion barrier 1 > 0. If the set C ∩ V1 (μ∗ ) has a convex and nonempty interior, then μ∗ is an 2 -equilibrium of , where 2 (·) > 0 is a function of 1 . Moreover, if μ∗ is a SUP, then μ∗ is a Nash equilibrium and the boundedness hypothesis is not required. Proof. Suppose that μ∗ is not an 2 -equilibrium of for any 2 > 0. Then for 2 > 0, there exists i ∈ I and ν ∈ P(A1 ) × · · · × P(An ) such that Ji (νi , μ∗−i ) − 2 > J (μ∗i , μ∗−i ).

(39)

By hypothesis, C ∩ V1 (μ∗ ) has a convex and nonempty interior. Hence, by Lemma 5.6, there exist α1 , α2 ∈ [0, 1] such that η − μ∗ < 1 , where η ∈ C and η := (1 − α1 )μ∗ + α1 [(1 − α2 )ν + α2 κ] for some κ in the interior of C ∩ V1 (μ∗ ). Since μ∗ is a SUP in the set C, Ji (μ∗i , η−i ) > Ji (ηi , η−i ), which implies (see Appendix A). (1 − α2 )(1 − α1 )n−1 Ji (μ∗i , μ∗−i ) + [(1 − α2 )α1 ]n−1 Ji (μ∗i , ν−i ) + [α2 α1 ]n−1 Ji (μ∗i , κ−i )]

> (1 − α2 )(1 − α1 )n−1 Ji (νi , μ∗−i ) − α2 (1 − α1 )n−1 Ji (μ∗i , μ∗−i ) − Ji (κi , μ∗−i ) + O(α1 ). Let 2∗ =

α2 1−α2

(40)

L1 , where L = 2n−1 max Ui . i∈I

By (27) |Ji (μ∗i , μ∗−i ) − Ji (κi , μ∗−i )| < L1 ≤ 2

1 − α2 α2

∀ 2 ≥ 2∗ .

Then (1 − α2 )(1 − α1 )n−1 Ji (μ∗i , μ∗−i ) + [(1 − α2 )α1 ]n−1 Ji (μ∗i , ν−i ) + [α2 α1 ]n−1 Ji (μ∗i , κ−i ) > (1 − α2 )(1 − α1 )n−1 [Ji (νi , μ∗−i ) − 2 ] + O(α1 ).

(41)

If (39) is true, there exists α1 in (0, 1) sufficiently close to 0, such that the equation (41) is violated. So we have that μ∗ is an 2 -equilibrium (for 2 ≥ 2∗ ). Now, suppose that μ∗ is a SUP and not a Nash equilibrium of . Then there exists i ∈ I and ν ∈ P(A1 ) × · · · × P(An ) such that (39) is true with 2 = 0. By the Lemma 5.6 there exist α ∈ [0, 1] such that η − μ∗ < 1 where η = (1 − α)μ∗ + αν. Since μ∗ is a SUP, Ji (μ∗i , η−i ) > Ji (ηi , η−i ). Then (see Appendix A)

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(1 − α)n−1 Ji (μ∗i , μ∗−i ) + (α)n−1 [Ji (μ∗i , ν−i )]

> (1 − α)n−1 Ji (νi , μ∗−i )] + O(α).

19

(42)

If μ∗ is not a Nash equilibrium, then for α in (0, 1) sufficiently small (42) is violated. So we have that μ∗ is a Nash equilibrium. 2 6. Stability of the system In this section we are interested in the stability of the differential system (14) (see Definition 6.1). To this end, we establish that uninvadable profiles (Definition 5.5) have some type of stability. Definition 6.1. Let μ∗ be a critical point of (14), i.e., F (μ∗ ) = 0. i) μ∗ is called Lyapunov stable if for every > 0 there exists δ > 0 such that if μ(0) − μ∗ ∞ < δ, then μ(t) − μ∗ ∞ < for all t > 0. ii) μ∗ is called weakly attracting if it is Lyapunov stable and, in addition, there exists δ > 0 such that if μ(0) − μ∗ ∞ < δ, then as t → ∞, μi (t) → μ∗i weakly for all i ∈ I . The following proposition is an extension to asymmetric evolutionary games of Theorem 3 in Oechssler and Riedel [23]. Theorem 6.2. Suppose that the conditions i) and ii) of Theorem 4.3 hold. Let δa ∗ = (δa1∗ , . . . , δan∗ ) be a vector of Dirac measures, and C an invariant set for the differential equation (14). If δa ∗ is a SUP in the set C, then there exists > 0 such that the set C ∩ V (δa ∗ ), is invariant for (14). Moreover, suppose that for all i in I , the map μ → βi (ai∗ |μ) is weakly continuous and the set of strategies Ai is a compact set. If C is a closed set and μ(0) is in C ∩ V (δa ∗ ), then as t → ∞, μ(t) → δa ∗ in distribution. Proof. First note that the vector of Dirac measures δa ∗ = (δa1∗ , . . . , δan∗ ) is a critical point of (14) (see the proof of Theorem 5.4). Then if μ(0) = δa ∗ , we have that μ(t) = δa ∗ for all t > 0 and the theorem holds. Since δa ∗ is a SUP in the set C, there exists > 0 such that for every μ ∈ C with μ − δa ∗ ∞ < and every i ∈ I , Ji (δai∗ , μ−i ) > Ji (μi , μ−i ) if μi = δai∗ . Suppose that μ(0) = δa ∗ and that μ(0) is in C ∩ V (δa ∗ ). By (23), for each i ∈ I and t ≥ 0, μ i (t, {ai∗ }) =

1{a ∗ } (ai )β(ai |μ(t))μi (t, dai ) = β(ai∗ |μ(t))μi (t, {ai∗ }).

Ai

Assume that for each i in I , μ i (0, {ai∗ }) = β(ai∗ |μ(0))μi (0, {ai∗ }) > 0,

(43)

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and define ti,0 := inf{t ≥ 0 : μ i (t, {ai∗ }) = 0}.

(44)

For each i in I , the function βi (ai∗ |μ(t)) is Lipschitz in μ(t), and μ(t) is continuous in t ; hence the map t → βi (ai∗ |μ(t)) is continuous. Also μi (t, {ai∗ }) is continuous in t . Then by (43) the map t → μ i (t, {ai∗ }) is continuous. So for each i ∈ I the set {t ≥ 0 : μ i (t, {ai∗ }) = 0} is closed and μ i (ti,0 , {ai∗ }) = 0. By (44), for any i in I μ i (s, {ai∗ }) = βi (ai∗ |μ(s))μi (s, {ai∗ }) > 0 ∀ 0 ≤ s < t0 ,

(45)

where t0 := min{t1,0 , . . . , tn,0 }. As a consequence of (45) we obtain μi (s, {ai∗ }) > μi (0, {ai∗ }) > 0 ∀ 0 ≤ s < t0 , i ∈ I.

(46)

Note that for any μi ∈ P(Ai ) μi − δai∗ = 2(1 − μi ({ai∗ }))

∀ i ∈ I.

(47)

If μ(0) − δa ∗ ∞ < , then by (46) and (47) we have μ(s) − δa ∗ ∞ <

∀ 0 ≤ s < t0 .

By continuity of μ(t) and (46) we obtain μi (t0 , {ai∗ }) ≥ μi (0, {ai∗ }) > 0

∀ 0 ≤ s < t0 , i ∈ I,

(48)

and by (47) and (48) μi (t0 ) − δai∗ ∞ ≤ μ(0) − δa ∗ ∞ <

∀ i ∈ I.

(49)

Since C is an invariant set, by (49) we see that μ(t0 ) ∈ C ∩ V (δa ∗ ) and so βi (ai∗ |μ(t0 )) > 0 because δa ∗ is a SUP in the set C. Then by (48) μ i (t0 , {ai∗ }) = βi (ai∗ |μ(t0 ))μi (t0 , {ai∗ }) > 0

∀i ∈ I,

so t → μi (t, {ai∗ }) is increasing for each i in I and, moreover, μ(t) ∈ C ∩ V (δa ∗ )

∀ t ≥ 0.

(50)

By hypothesis, Ai is compact for each i ∈ I , so P(Ai ) is compact in the weak topology (see page 186, Corollory 5.7.6, Bobrowski [4]) for all i ∈ I . Then C ∩ P(A1 ) × . . . × P(An ) is compact in the product topology. On the other hand, δa ∗ is a SUP in the set C and, by (50), βi (ai∗ |μ(t)) > 0 for all t > 0 and i in I . Moreover, by Theorem 4.6, μi (t, {ai∗ }) = μi (0, {ai∗ })e

t 0

βi (ai∗ |μ(s))ds

≤ 1 ∀ i ∈ I, t ≥ 0;

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21

hence lim βi (ai∗ |μ(t)) = 0 ∀ i ∈ I.

t→∞

Finally, let v = (v1 , . . . , vn ) ∈ C ∩ P(A1 ) × . . . × P(An ) be an accumulation point of the trajectory μ(t) = (μ1 (t), . . . , μn (t)). By (50) the distance from v to δa ∗ is at most . Since δa ∗ is a SUP in C and the map μ → βi (ai∗ |μ) is weakly continuous, if v is such that βi (ai∗ |v) = Ji (ai∗ , v−i ) − Ji (vi , v−i ) = 0 ∀ i ∈ I, yields that δa ∗ = v, which proves that μi (t) → δai∗ in distribution for all i in I .

2

If the vector δa ∗ in the Theorem 6.2 is a SUP, then we obtain the following corollary, taking C = P(A1 ) × . . . × P(An ). Corollary 6.3. Suppose that the conditions i) and ii) of Theorem 4.3 hold. Let δa ∗ = (δa1∗ , . . . , δan∗ ) be a vector of Dirac measures, and suppose that it is a SUP. Then δa ∗ is Lyapunov stable for the replicator dynamics. Moreover, if the map μ → βi (ai∗ |μ) is weakly continuous and the set of strategies Ai is compact for all i ∈ I , then δa ∗ is weakly attracting. Remark 6.4. Note that if for each i in I the payoff function Ui (·) in (8) is continuous, then the map μ → βi (ai∗ |μ) is weakly continuous. This fact is of relevance because many games satisfy that Ui (·) in (8) is continuous. 7. Example In this section we consider games in which we have two players with the following payoff functions: U1 (x, y) = −a1 x 2 − b1 xy + c1 x + d1 y, U2 (x, y) = −a2 y 2 − b2 yx + c2 y + d2 x, with a1 , a2 , b1 , b2 , c1 , c2 > 0 and d1 , d2 any real numbers. Let A1 = [0, M1 ] and A2 = [0, M2 ] for M1 , M2 > 0 and large enough, be the strategy sets. This class of games could represent a Cournot duopoly or models of international trade with linear demand and linear cost (see Bagwell and Wolinsky [1]). It can also represent some models of public good games (see Ewald [14]). If (2a2 c1 − b1 c2 ), (2a1 c2 − b2 c1 ), (4a1 a2 − b1 b2 ) are all positive, then we have an interior Nash equilibrium (x ∗ , y ∗ ) =

2a2 c1 − b1 c2 2a1 c2 − b2 c1 , . 4a1 a2 − b1 b2 4a1 a2 − b1 b2

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S. Mendoza-Palacios, O. Hernández-Lerma / J. Differential Equations ••• (••••) •••–•••

Let C1 := (μ, ν) ∈ P(A1 ) × P(A2 ) : μ(x ∗ , M1 ] = ν(y ∗ , M2 ] = 0 , C2 := (μ, ν) ∈ P(A1 ) × P(A2 ) : μ[0, x ∗ ) = ν[0, y ∗ ) = 0 , and C = C1 ∪ C2 . The set C is invariant for the replicator dynamics (14) and (δx ∗ , δy ∗ ) is in C. On the other hand, let x¯ μ :=

xμ(dx),

A1

y¯ μ :=

yμ(dy). A2

If (μ, ν) is in C1 , then by Jensen’s inequality J1 (δx ∗ , ν) =

U1 (x ∗ , y)ν(dy) = U1 (x ∗ , y¯ ν ) > U1 (x¯ μ , y¯ ν ) ≥ J1 (μ, ν)

A2

J2 (μ, δy ∗ ) =

U2 (x, y ∗ )μ(dx) = U2 (x¯ μ , y ∗ ) > U2 (x¯ μ , y¯ ν ) ≥ J2 (μ, ν).

A1

This is also true if (μ, ν) is in C2 . Hence, for any > 0, the vector (δx ∗ , δy ∗ ) is a SUP in the set C. Therefore, by Theorem 6.2, for > 0 the set C ∩ V (δa ∗ ) is invariant for (14). Moreover, since for every i in I , the payoff functions Ui (·) are continuous and the sets of strategies Ai are compact sets, we conclude by Theorem 6.2 and Remark 6.4 that if μ(0) ∈ C ∩ V (δa ∗ ), then μ(t) → δa ∗ in distribution. 8. Comments In this paper, we introduced a model of asymmetric evolutionary games with strategies on measurable spaces. The model can be reduced, of course, to the particular case of evolutionary games with finite strategy sets. We established conditions under which the replicator dynamics has a solution and we also characterized that solution (Theorem 4.6). Then stability conditions were established, and finally we gave an example which may be applicable in oligopoly models, theory of international trade, and public good models. There are many questions, however, that remain open. For instance, in symmetric evolutionary games with continuous strategy spaces, there are stability conditions with different metrics and topologies. Are these conditions satisfied in the asymmetric case? On the other hand, normal form games with continuous strategies can be approximated by games with discrete strategies. Hence, it would be interesting to investigate if the replicator dynamics with continuous strategies in the asymmetric case can be approximated, in some sense, by games with discrete strategies. Another important issue on the stability analysis for evolutionary game theory, is the relation between the evolutionary dynamics and potential games (see Sandholm [29] and Cheung [8]), as well as the convergence of learning processes. (For instance, see Hopkins [21] for the positive definite adaptive dynamics.)

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S. Mendoza-Palacios, O. Hernández-Lerma / J. Differential Equations ••• (••••) •••–•••

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Appendix A We prove the inequality (40) under the hypothesis of Theorem 5.7. Let η1 = (1 − α1 )μ, η2 = α1 (1 − α2 )ν, η3 = α1 α2 κ, thus η := η1 + η2 + η3 . Also note that, for every i in I , ηi1 = (1 − α1 )μi , ηi2 = α1 (1 − α2 )νi and ηi3 = α1 α2 κi . Then η = (η1 , .., ηn ) = (η11 + η12 + η13 , . . . , ηn1 + ηn2 + ηn3 ). Since μ∗ is a SUP in the set C, J1 (μ∗1 , η−1 ) > J1 (η1 , η−1 ). Then using the notation in (10) we have the following implications I(μ1 ,η2 ,η3 ,...,ηn ) U1 > I(η1 ,η2 ,η3 ,...,ηn ) U1 ⇒ I(μ1 ,η1 ,η3 ,...,ηn ) U1 + I(μ1 ,η2 ,η3 ,...,ηn ) U1 + I(μ1 ,η3 ,η3 ,...,ηn ) U1 2

2

2

> I(η1 ,η1 η3 ,...,ηn ) U1 + I(η1 ,η2 ,η3 ,...,ηn ) U1 + I(η1 ,η3 ,η3 ,...,ηn ) U1 1

2

1

2

1

2

+ I(η2 ,η1 ,η3 ,...,ηn ) U1 + I(η2 ,η2 ,η3 ,...,ηn ) U1 + I(η2 ,η3 ,η3 ,...,ηn ) U1 1

2

1

2

2

1

+ I(η3 ,η1 ,η3 ,...,ηn ) U1 + I(η3 ,η2 ,η3 ,...,ηn ) U1 + I(η3 ,η3 ,η3 ,...,ηn ) U1 1

2

1

2

1

2

⇒ I(μ1 ,η1 ,η3 ,...,ηn ) U1 + I(μ1 ,η2 ,η3 ,...,ηn ) U1 + I(μ1 ,η3 ,η3 ,...,ηn ) U1 2 2 2 > (1 − α1 ) I(μ1 ,η1 ,η3 ,...,ηn ) U1 + I(μ1 ,η2 ,η3 ,...,ηn ) U1 + I(μ1 ,η3 ,η3 ,...,ηn ) U1 2 2 2 + α1 (1 − α2 ) I(ν1 ,η1 ,η3 ,...,ηn ) U1 + I(ν1 ,η2 ,η3 ,...,ηn ) U1 + I(ν1 ,η3 ,η3 ,...,ηn ) U1 2 2 2 + α1 α2 I(κ1 ,η1 ,η3 ,...,ηn ) U1 + I(κ1 ,η2 ,η3 ,...,ηn ) U1 + I(κ1 ,η3 ,η3 ,...,ηn ) U1 2

2

2

⇒ I(μ1 ,η1 ,η3 ,...,ηn ) U1 + I(μ1 ,η2 ,η3 ,...,ηn ) U1 + I(μ1 ,η3 ,η3 ,...,ηn ) U1 2 2 2 > (1 − α2 ) I(ν1 ,η1 ,η3 ,...,ηn ) U1 + I(ν1 ,η2 ,η3 ,...,ηn ) U1 + I(ν1 ,η3 ,η3 ,...,ηn ) U1 2 2 2 + α2 I(κ1 ,η1 ,η3 ,...,ηn ) U1 + I(κ1 ,η2 ,η3 ,...,ηn ) U1 + I(κ1 ,η3 ,η3 ,...,ηn ) U1 2

2

2

⇒ I(μ1 ,η1 ,η1 ,...,η1 ) U1 + I(μ1 ,η1 ,η2 ,...,η1 ) U1 + . . . + I(μ1 ,η1 ,η3 ,...,η3 ) U1 + . . . 2

3

n

2

3

n

2

3

n

+ I(μ1 ,η2 ,η1 ,...,η1 ) U1 + . . . + I(μ1 ,η2 ,η2 ,...,η2 ) U1 + . . . + I(μ1 ,η2 ,η3 ,...,η3 ) U1 + . . . 2

n

3

2

n

3

2

+ I(μ1 ,η3 ,η1 ,...,η1 ) U1 + . . . + I(μ1 ,η3 ,η3 ,...,η3 ) U1 n n 2 3 2 3 > (1 − α2 ) I(ν1 ,η1 ,η1 ,...,η1 ) U1 + . . . + I(ν1 ,η1 ,η3 ,...,η3 ) U1 2

3

n

2

3

3

3

n

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S. Mendoza-Palacios, O. Hernández-Lerma / J. Differential Equations ••• (••••) •••–•••

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+ I(ν1 ,η2 ,η3 ,...,ηn ) U1 + I(ν1 ,η3 ,η3 ,...,ηn ) U1 2 2 + α2 I(κ1 ,η1 ,η1 ,...,η1 ) U1 + . . . + I(κ1 ,η1 ,η3 ,...,η3 ) U1 n n 2 3 2 3 + I(κ1 ,η2 ,η3 ,...,ηn ) U1 + I(κ1 ,η3 ,η3 ,...,ηn ) U1 2

2

⇒ I(μ1 ,η1 ,η1 ,...,η1 ) U1 + I(μ1 ,η2 ,η2 ,...,η2 ) U1 + I(μ1 ,η3 ,η3 ,...,η3 ) U1 2

3

n

2

3

n

2

3

n

> (1 − α2 )I(ν1 ,η1 ,η1 ,...,η1 ) U1 + α2 I(κ1 ,η1 ,η1 ,...,η1 ) U1 + O(α1 ) 2

3

n

2

3

n

⇒ (1 − α1 )n−1 I(μ1 ,μ2 ,μ3 ,...,μn ) U1 + α2n−1 (1 − α1 )n−1 I(μ1 ,ν2 ,ν3 ,...,νn ) U1 + α2n−1 α1n−1 I(μ1 ,κ2 ,κ3 ,...,κn3 ) U1 > (1 − α2 )(1 − α1 )n−1 I(ν1 ,μ2 ,μ3 ,...,μn ) U1 + α2 (1 − α1 )n−1 I(κ1 ,μ2 ,μ3 ,...,μn ) U1 + O(α1 ) ⇒ (1 − α2 )(1 − α1 )n−1 I(μ1 ,μ2 ,μ3 ,...,μn ) U1 + α2n−1 (1 − α1 )n−1 I(μ1 ,ν2 ,ν3 ,...,νn ) U1 + α2n−1 α1n−1 I(μ1 ,κ2 ,κ3 ,...,κn3 ) U1 > (1 − α2 )(1 − α1 )n−1 I(ν1 ,μ2 ,μ3 ,...,μn ) U1 − α2 (1 − α1 )n−1 I(μ1 ,μ2 ,μ3 ,...,μn ) U1 − I(κ1 ,μ2 ,μ3 ,...,μn ) U1 + O(α1 ), and (40) follows. The inequality (42) is obtained similarly. References [1] K. Bagwell, A. Wolinsky, Game theory and industrial organization, Handb. Game Theory Econ. Appl. 3 (2002) 1851–1895. [2] D. Balkenborg, K.H. Schlag, On the evolutionary selection of sets of Nash equilibria, J. Econom. Theory 133 (1) (2007) 295–315. [3] R.G. Bartle, The Elements of Integration and Lebesgue Measure, John Wiley & Sons, 1995. [4] A. Bobrowski, Functional Analysis for Probability and Stochastic Processes: An Introduction, Cambridge University Press, 2005. [5] I.M. Bomze, Dynamical aspects of evolutionary stability, Monatsh. Math. 110 (3–4) (1990) 189–206. [6] I.M. Bomze, Cross entropy minimization in uninvadable states of complex populations, J. Math. Biol. 30 (1) (1991) 73–87. [7] I.M. Bomze, B.M. Pötscher, Game Theoretical Foundations of Evolutionary Stability, Lecture Notes in Economics and Mathematical Systems, vol. 324, Springer-Verlag, 1989. [8] M.-W. Cheung, Pairwise comparison dynamics for games with continuous strategy space, J. Econom. Theory 153 (2014) 344–375. [9] J. Cleveland, A.S. Ackleh, Evolutionary game theory on measure spaces: well-posedness, Nonlinear Anal. Real World Appl. 14 (1) (2013) 785–797. [10] R. Cressman, Local stability of smooth selection dynamics for normal form games, Math. Social Sci. 34 (1) (1997) 1–19. [11] R. Cressman, Continuously stable strategies, neighborhood superiority and two-player games with continuous strategy space, Internat. J. Game Theory 38 (2) (2009) 221–247.

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S. Mendoza-Palacios, O. Hernández-Lerma / J. Differential Equations ••• (••••) •••–•••

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