Continuous and Global Stability in Innovative ... - Semantic Scholar

Continuous and Global Stability in Innovative Evolutionary Dynamics Josef Hofbauer Department of Mathematics University of Vienna

J¨org Oechssler Department of Economics University of Bonn

Frank Riedel Department of Economics Stanford University February 26, 2003 Abstract Innovation plays a central role in the development of modern economies, as does the regret of those who have missed the opportunity to try a successful new strategy. In contrast to purely biological environments, where new strategies emerge mainly by random mutation, human societies tend to exhibit more deliberate, although possibly imperfect inventions of new strategies. In this paper, we study such a class of evolutionary dynamics for the case of continuous strategy spaces, the Brown–von Neumann–Nash dynamics. For two general classes of games, we establish global convergence to Nash equilibria that are continuously stable or evolutionarily robust. Journal of Economic Literature Classification Numbers: C70, 72. Key words:

1

Introduction

Innovation plays a central role in the development of modern economies, as does the regret of those who have missed the opportunity to try a successful new strategy. In contrast to purely biological environments, where new strategies emerge mainly by random mutation, human societies tend to exhibit more deliberate, although possibly imperfect inventions or discoveries of new strategies. Therefore, it is important for the understanding of the evolution of behavior in human societies to allow for innovation and regret–based decision rules. In this paper, we study such a class of evolutionary dynamics for the case of continuous strategy spaces. In the finite strategy case, these Brown–von Neumann–Nash dynamics (BNN) have recently been suggested as a plausible boundedly rational learning process in games (see e.g. Skyrms [24], Swinkels [26], Weibull [31], Berger and Hofbauer [3], Hofbauer [15], and Sandholm [20]). Selection dynamics like the replicator dynamics from evolutionary biology (see e.g. [27]) do not allow to introduce new strategies into the population. A strategy may be absolutely perfect but if it has not been around at the very beginning or happens to die out once, it can never be used in the future. For the case of human strategic interaction this seems too restrictive. In contrast, the BNN dynamics satisfy the property of “inventiveness” (Weibull, [31]), or equivalently, “noncomplacency” (Sandholm [20]). Namely, if there are any (used or unused) strategies that (would) perform better than the current population average, at least one of them must increase in frequency. In particular, new strategies can enter if they yield better than average payoffs. Inventive dynamics can be motivated by a search story. Suppose players do not know the payoff function. But each round they get information (through a “newspaper search”, say) on the payoffs some other strategies would have against the current population profile and they choose the best among those strategies. This implies on the one hand that better strategies are adopted more frequently because they are more likely to be the best in the sample. On the other hand, it is clear that the overall best strategy in not necessarily chosen as it may fail to appear in the random sample. Thus, the resulting process is not of the best–reply variety. Furthermore, it seems plausible to assume that players have some inertia and are reluctant to switch to a new strategy. Why would a player not immediately adopt a newly found strategy with seemingly higher expected payoff than the current strategy? An answer is given by Skyrms [24, p. 30] in terms of a “rational deliberation”: The player “...would not on pain of incoherence if she believes that she is in an informational feedback situation and if she assigns any positive probability at all to the possibility that informational feedback may lead her 1

ultimately to a different decision.” To be specific let us assume that players are more likely to adopt a new strategy the greater the payoff difference between the new strategy and the current population average. This careful, conservative rule is reminiscent of the proportional imitation rule of Schlag [22] who shows that it has certain optimality properties in the framework of imitation in a simple decision problem. It is also closely related to the regret–matching rule of Hart and Mas–Colell [13] where better strategies are also adopted with a probability that is proportional to the apparent gains the new strategy yields, only that gains are calculated using the regret one has for not playing the new strategy from the very beginning as compared to the actual payoff one obtained. [3] and [15] derive several useful results relating to the (asymptotic) stability of Nash equilibria with respect to the BNN dynamics in finite normal form games. The current paper seeks to extend those results to the case of more general strategy spaces. When studying the stability properties of BNN dynamics there appear some important differences between the case of a finite number of pure strategies (as studied by [3] and [15]) and the continuous strategy case studied here. Probably most important is the fact that strict equilibria are not necessarily (Lyapunov) stable in the continuous case.1 Interestingly, static stability concepts originally developed for the replicator dynamics (like a “continously stable strategy” (CSS) [9] and Evolutionary Robustness [19]) become relevant for stability with respect to the BNN dynamics. We establish global dynamic stability for CSS in two classes of games. We introduce the notion of CSS games: these are given by symmetric payoff funtions whose best reply function has a slope less than 1 everywhere. In particular, this class of games includes submodular games or games with strategic complements. For these games, we show that a unique Nash equilibrium in pure strategies exist; this Nash equilibrium is also a CSS and a global attractor under BNN. The second class of games we study is the class of negative definite games. This includes zero–sum games and the war of attrition, among others. For these games, every Nash equilibrium is globally neutrally evolutionarily robust. The BNN dynamics converge to the set of Nash equilibria and a strict Nash equilibrium, if it exists, is a global attractor. The next section sets up the formal background for this model. Section 3 introduces the continuous BNN dynamics and shows that they are well–defined and induce a weakly continuous flow. Section 4 reviews the relevant stability and equilibrium concepts for continuous evolutionary game theory. Section 5 introduces and analyzes CSS games and 1

This is also the case for other dynamics like the replicator dynamics, see e.g. Oechssler and Riedel, [18] and [19].

2

Section 6 treats the class of negative definite games. An appendix collects some proofs.

2

Preliminaries

We consider symmetric two–player games with a (pure) strategy set S. Let A be a σ–algebra on S and µ be a finite measure on (S, A). Let f : S × S → R be a bounded measurable function, where f (x, y) is the payoff for player 1 when he plays x and player 2 plays y. (Our main example is a compact interval S ⊂ R with the Lebesgue measure. Most of our results require S to be a compact metric space with the Borel σ–algebra B.) A population is identified with the aggregate play of its members and is described by a probability measure P on the measure space (S, A). We denote by ∆ = M1 (S, A) the set of all populations (probability measures or mixed strategies) on S. The average payoff of a population P against a population Q is defined as the bilinear extension of f to Me × Me , i.e. Z Z (1) f (x, y)dQ(y)dP (x) . E(P, Q) = S

S

Let σ(x, P ) := E(δx , P ) − E(P, P ) denote the difference between the payoff of strategy x ∈ S (identified with the Dirac measure δx on x) and the average population payoff. The excess payoff of pure strategy x when matched against population P is defined as σ+ (x, P ) := max(σ(x, P ), 0).

3

The BNN dynamics

In the dynamics we want to study, the number of players who adopt a given strategy x changes according to the current excess payoff of that strategy. This can be interpreted in two ways as we have indicated above. On the one hand, one might think of a deliberate, yet imperfect process of discovery in which new strategies are found and adopted with a probability that is proportional to its excess payoff. In particular, strategies that are not a better reply to the current population than the population itself are never adopted. On the other hand, the excess payoff can be interpreted as the regret players have of not

3

having used a certain strategy. By using the excess payoff function as a random device, they choose new strategies. These reflections lead to the following kind of dynamics, generalizing the Brown–von Neumann–Nash dynamics to the case of continuous strategy sets S. For the measure space (S, A, µ), BNN is given by a differential equation on ∆ Z Z ˙ σ+ (x, P )dµ(x) − P (A) σ+ (s, P )dµ(s). (2) P (A) = R

A

S

The total excess J(P ) := S σ+ (s, P )dµ(s) vanishes if σ(x, P ) ≤ 0 or E(δx , P ) ≤ E(P, P ) for µ–a.a. x ∈ S, in particular, if P is a Nash equilibrium. A nice property of the BNN dynamics is that for continuous f (and thus, in particular, in the finite strategy case) the rest points of the dynamic coincide with the Nash equilibria.2 Proposition 3.1 Suppose f is continuous. Then P is a rest point of the BNN dynamics if and only if (P, P ) is a Nash equilibrium. Proof : If P is a best reply to itself, then σ+ (x, P ) = 0 for all x, and stationarity follows. Let P ∗ be a stationary point of BNN, that is Z σ+ (x, P ∗ )dx = P ∗ (A)J(P ∗ ) (3) A

for all Borel sets A. We distinguish two cases, J(P ∗ ) = 0 and J(P ∗ ) > 0. Case 1: J(P ∗ ) = 0. In this case, for Lebesgue almost every x, we have σ+ (x, P ∗ ) = 0.

By continuity of f (and hence of σ+ (x, P ∗ )), it follows that σ+ (x, P ∗ ) = 0 holds true for all x ∈ S. This is equivalent to E(δx , P ∗ ) ≤ E(P ∗ , P ∗ ), and it follows that P ∗ is a best reply to itself. Case 2:J(P ∗ ) > 0. Since P ∗ is a stationary point of BNN, we get from (3) that P ∗ has a density p∗ with respect to Lebesgue measure and p∗ (x) = 2

σ+ (x, P ∗ ) J(P ∗ )

Sandholm [21] calls this property “Nash stationarity”.

4

for P ∗ -almost every x. For every x with p∗ (x) > 0, we have thus σ+ (x, P ∗ ) > 0,

or E(δx , P ∗ ) > E(P ∗ , P ∗ ). By integrating, we get ∗

∗

E(P , P ) =

Z

E(δx , P ∗ )p∗ (x)dx > E(P ∗ , P ∗ ),

x:p∗ (x)>0

a contradiction. Hence, we cannot have J(P ∗ ) > 0 for a stationary point P ∗ . This concludes the proof. 2 If J(P ) > 0 then the relative excess for a subset A ⊂ S is denoted by RP (A) := R 1 σ (x, P )dµ(x) and defines a measure on (S, A), absolutely continuous with reJ(P ) A + 1 spect to µ, with density function r(x) = J(P σ (x, P ). ) + Then (2) can be rewritten as P˙ (A) = J(P )(RP (A) − P (A)).

(4)

Hence, under the BNN dynamics, a population P moves toward its relative excess measure RP , and the speed of motion slows down proportionally to the total excess. For later reference note that by construction of RP we have that E(RP , P ) ≥ E(P, P ) for all P .

3.1

Solutions of the BNN dynamics

Theorem 3.2 For each P = P (0) ∈ ∆ there is a unique solution P (t) ∈ ∆ of the ordinary differential equation (2) for t ∈ [0, ∞[.

3.2

Weak Continuity of the Flow

Given that a unique solution to the BNN dynamics exists, we can define the semiflow B : ∆ × [0, ∞[→ ∆, 5

where B(P, t) = P (t) denotes the population at time t when the BNN dynamics start in P = P (0). In the following, we will use the metric k·kBL on ∆ which metrizes the weak topology (cf. [23, p. 352]). Let S be a separable metric space and B be the Borel σ–algebra on S. For a Lipschitz continuous, bounded function g : S → R, let |g(x) − g(y)| (5) kgkBL := sup |g(x)| + sup . |x − y| x∈S x6=y Abusing notation slightly, we define the dual norm k·kBL on Me (S) via šZ › kQkBL = sup gdQ ,

(6)

where the supremum is taken over all bounded, Lipschitz continuous functions g with kgkBL ≤ 1. Theorem 3.3 Let S be a separable metric space and f satisfy the boundedness and Lipschitz conditions (22) and (23). Then the semiflow B is continuous with respect to the weak topology of measures and the Euclidean topology on [0, ∞[; more precisely: whenever kP n − P kBL → 0 and tn → t, then kB (P n , tn ) − B (P, t)kBL → 0. The proof can be found in the appendix.

4

Stability Concepts

In the case of finite strategy sets S it is straightforward to show that strict Nash equilibria are asymptotically stable with respect to BNN dynamics (see e.g. Berger [3]). The simple example below shows that this is not the case anymore for general S. As it turns out, concepts originally developed for the continuous version of the replicator dynamics in evolutionary biology like CSS [9] and Evolutionary Robustness [19] become relevant for the BNN dynamics as well. The classical definition of an evolutionary stable strategy (ESS) (Maynard Smith [16]) requires that for all mutant populations R there exists an invasion barrier ε such that the original population P does better against the mixed population (1 − η)P + ηR than R does for all η ≤ ε. In this definition some invasion barrier exists for each R. Eshel and Motro [9] introduced the following definition. Definition 4.1 (CSS) A strategy u is a continuously stable strategy (CSS) if (1) it is a ESS and (2) there exists an ε > 0 such that for all v with |v − u| < ε there exists an η > 0 such that for all x with |v − x| < η f (v, x) > f (x, x) if and only if |v − u| < |x − u|. 6

As shown by Eshel [10] if f is twice differentiable, a necessary condition for an ESS u to be a CSS is that fxx (u, u) + fxy (u, u) ≤ 0.

(7)

Condition (7) is sufficient if the weak inequality is replaced by a strict one. A similar concept was introduced by Apaloo [1]. Definition 4.2 (NIS) A strategy u is a neighborhood invader strategy (NIS) if there exists an ε > 0 such that for all x with |x − u| < ε f (u, x) > f (x, x). In the twice differentiable case a necessary condition for u to be a NIS is that (see Oechssler and Riedel [19]) fxx (u, u) + 2fxy (u, u) ≤ 0.

(8)

Again, condition (8) is sufficient if the weak inequality is replaced by a strict one. The following condition was introduced by Oechssler and Riedel [19] and is stronger than both, CSS and NIS. Definition 4.3 (ER) A population P ∗ ∈ ∆(S) is evolutionarily robust (ER) if there exists ε > 0 such that for all Q 6= P ∗ with ρ(Q, P ∗ ) < ε we have E(P ∗ , Q) > E(Q, Q).

(9)

When (9) holds with weak inequality but for all Q ∈ ∆, we say that P is globally neutrally evolutionarily robust. When there is strict inequality for all Q = 6 P , P is called globally evolutionarily robust. The following example provides an interesting case study and shows that strict Nash equilibria need not be stable under BNN dynamics. Example. Let S be an interval around 0 and f (x, y) = −x2 + axy be a typical linear–quadratic game. 0 is a strict NE for every a. For a < 2, this equilibrium satisfies CSS (by the strict version of (7)). For a = 2 every pure strategy is a strict NE; for a > 2 it looks ’unstable’.

7

Let Pi =

R

S

xi P (dx) denote the ith moment of P . Then Z E(δx , P ) = (−x2 + axy)P (dy) = −x2 + axP1

(10)

S

Z Z

(−x2 + axy)P (dx)P (dy) S S “2 ’Z Z 2 xP (dx) = − x P (dx) + a

E(P, P ) =

(11)

S

S

= −P2 + aP12 Therefore

σ+ (x, P ) = [x(aP1 − x) − aP12 + P2 ]+

(12)

and the density r(x, P ) are cutoffs of a quadratic function in x. Both the maximizer and the mean are at x = a2 P1 . Hence the mean value of P changes under (2) according to a (13) P˙ 1 = J(P )( − 1)P1 . 2 Hence for a < 2, P1 (t) → 0, whereas for a = 2, P1 (t) ≡ P1 (0), and for a > 2, P1 (t) moves away from 0. Now restrict to measures P with mean value P1 = 0. Then (12) reduces to σ+ (x, P ) = 3/2 [−x2 + P2 ]+ . A simple calculation shows that J(P ) = 34 P2 and that the variance of the corresponding relative excess density r equals P2 /5. Hence P2 16 5/2 P˙2 = J(P )( − P2 ) = − P2 5 15

(14)

shows that the variance of P (t) goes to 0 and hence P (t) converges weakly to the point measure δ0 . In particular, δ0 is the global attractor for probability measures with mean 0. For a < 2, (13) implies this for all probability measures. A different way to analyze this game is by observing that it is a potential game: It is strategically equivalent to the common interest game with payoff function f (x, y) = −x2 + axy − y 2 = f (y, x). Then f (x, x) = (a − 2)x2 and E(P, P ) = −2P2 + aP12 = −2V ar(P ) + (a − 2)P12 . This function is a Lyapunov function, increasing along solutions of (2). For a < 2, P = δ0 is the unique maximizer of E, and hence globally asymptotically stable. For a > 2, δ0 is a minimum of f (x, x), and hence unstable. This argument is made precise and generalized in the following section.

8

5

Global Stability I: CSS Games

In this subsection, let us study the following subclass of potential games. Definition 5.1 Let S = [a, b] be a closed real interval. A function f : S × S → < is called a CSS game, if it is strictly concave in its first variable, twice continuously differentiable on ]a, b[×]a, b[, symmetric, and satisfies fxx (x, y) + fxy (x, y) < 0

(15)

for all x, y ∈]a, b[. For a CSS game f , the best reply function r(y) = arg max f (x, y) x∈S

is well defined and, as is well known, its derivative is given by r0 (y) = −

fxy (r(y), y) , (a < y < b) . fxx

In terms of the best reply function, condition (15) reduces to r0 (y) < 1.

(16)

By the existence theorem, a CSS game has symmetric pure Nash equilibria. Remark 5.2 It is perhaps not so well known that CSS games have no mixed equilibria. Here is why. For a mixed strategy Q, the function Z f (x, y)Q(dy) F (x) = S

is strictly concave since f is strictly concave. Hence, it has a unique maximum x∗ . Therefore, a mixed strategy cannot be a best reply. The CSS condition implies even more: Proposition 5.3 Every CSS game has a unique symmetric Nash equilibrium (x∗ , x∗ ) in pure strategies. x∗ is also CSS. 9

Proof : By the existence theorem, we can pick a pure strategy x∗ that is a best reply to itself, r(x∗ ) = x∗ . Take some y > x∗ . The best reply condition (16) implies that Z y Z y Z y 0 ∗ ∗ 0 ∗ r (u)du < x + 1du = y. r(y) = r(x ) + r (u)du = x + x∗

x∗

x∗

Thus, y is not a best reply to itself. Similarly, we get for y < x∗ , Z Z x∗ Z y ∗ ∗ ∗ 0 0 r(y) = r(x ) − r (u)du = x − r (u)du > x + x∗

y

y

1du = y.

x∗

We conclude that there is at most one symmetric Nash equilibrium in pure strategies. If x∗ is in the interior of S, the CSS condition (15) yields immediately that x∗ is CSS. 2 Remark 5.4 Under the CSS assumption, a CSS exists, but it can happen that no NIS or ER exists, e.g. for f (x, y) = −x2 + 32 xy − y 2 . Proposition 5.5 BNN converges globally to the unique CSS x∗ . Proof : For symmetric paoff functions f , average payoff E(P, P ) is a Lyapunov function, as the following calculation shows: d E(P (t), P (t)) = 2E(P 0 (t), P (t)) dt = 2E(rP − P, P )J(P ) ≥ 0. and the BNN dynamics converge to a maximum of average payoff. A maximum P ∗ of average payoff satisfies ∗ E(rP − P ∗ , P ∗ )J(P ∗ ) = 0, and is thus a stationary point of BNN. By Proposition 3.1, P ∗ is a Nash equilibrium. Therefore, P = δx∗ . 2 Example 5.6 Consider the two firm Bertrand game with heterogeneous goods. The demand function is given by d(p1 , p2 ) = 1 − p1 + γp2 with γ < 2. The payoff function is f (p1 , p2 ) = p1 d(p1 , p2 ). 1 This game is a CSS game with the unique symmetric Nash equilibrium p1∗ = p∗2 = 2−γ . CSS dynamics converge globally to this equilibrium. For γ close to 2, the equilibrium is not NIS, and therefore not asymptotically stable with respect to replicator dynamics.

10

6

Global Stability II: Negatively definite games

Let us assume in the following that the payoff function is (semi)negative definite in the sense that (17) for all P, Q E (P − Q, P − Q) ≤ 0 For the corresponding property in discrete games, see vanDamme or Hofbauer. Examples. Every zero-sum game satisfies Assumption 17. Proof : By definition of a zero-sum game, f (x, y) + f (y, x) = 0 for all x, y ∈ S. This implies that E(P, Q) + E(Q, P ) = 0, and in particular E(P, P ) = 0. Therefore, E(P − Q, P − Q) = 0. 2 Linear quadratic games like f (x, y) = −x2 + axy satisfy Assumption 17 iff a ≤ 0 as one can easily check. An immediate consequence of assumption 17 is Lemma 6.1 1)Under Assumption (17), the set of Nash equilibria is convex. Every Nash equilibrium N is globally neutrally evolutionarily robust. If there exists a strict equilibrium, this equilibrium is unique and globally evolutionarily robust (and thus ESS and NIS). 2) If Assumption 17 holds true with a strict inequality, there exists a (unique) globally evolutionarily robust equilibrium. Proof : Let N be a Nash equilibrium and Q 6= N a population. Then E (N − Q, Q) = E (N − Q, Q − N ) + E (N − Q, N ) ≥ 0 because the first summand is nonnegative by Assumption (17) and the second term is nonnegative by definition of a Nash equilibrium. This shows that every Nash equilibrium is globally evolutionarily robust. For a strict Nash equilibrium, the second summand is strictly positive, hence E (N − Q, Q) > 0. This in turn implies that there is no other Nash equilbrium because E(Q, Q) < E(N, Q) for all Q 6= N . Suppose that N1 and N2 are Nash equilibria. Set N3 = λN1 + (1 − λ)N2 for some λ ∈ (0, 1). We want to show that N3 is a Nash equilibrium. Fill that in! 2 We may now proceed to study stability properties of Nash equilibria. 11

Theorem 6.2 Under Assumption (17), the function Z 1 H(P ) = σ+ (x, P )2 dx 2 S is a Lyapunov function. Every trajectory of BNN converges to the set of Nash equilibria. In particular, a strict Nash equilibria is globally asymptotically stable. Proof : Let us first determine the gradient of σ(x, P ) with respect to P at some point Q. We have ∇σ (x, P ) (Q) = E (δx , Q) − E (P, Q) − E (Q, P ) . From this, we obtain via the chain rule Z d H(Pt ) = σ+ (x, P ) ∇σ (x, P ) (P˙t )dx dt ZS h  ‘   ‘i ‘ ˙ ˙ ˙ = σ+ (x, P ) E δx , Pt − E P, Pt − E Pt , P dx S ‘ ‘ ‘‘     P ˙ ˙ ˙ = J (P ) E r , Pt − E P, Pt − E Pt , P , where we have used the definition of the relative excess measure rP . By definition of the dynamics P˙t , we proceed to € €  €  €  d H(Pt ) = J (P )2 E rP , rP − P − E P, rP − P − E rP − P, P dt € € €   = J (P )2 E rP − P, rP − P − E rP − P, P .  € The first term in brackets, E rP − P, rP − P ≤ 0 by Assumption (17), and the second €  term, E rP − P, P ≥ 0 by definition of the relative excess measure. We thus obtain d H(Pt ) ≤ 0 dt

and the inequality is strict whenever Pt is not a stationary point.

12

2

A A.1

Appendix Solutions of BNN

Since ∆ is not a vector space, we work with the linear span of ∆, that is the space Me (S, A) of all signed measures. Recall that ν is a signed measure on (S, A) if there are two finite measures µ1 and µ2 such that for all sets A ∈ B, ν(A) = µ1 (A) − µ2 (A). The variational norm on Me (S, A) is given by ŒZ Œ Œ Œ kµk = sup ŒŒ g dµŒŒ , g where the sup is taken over all measurable functions g : S → R bounded by 1, sups∈S |g(s)| ≤ 1. Endowed with the variational norm, Me is a Banach space. The strategy for proving the Theorem is the following. Denote by Z Z F (Q) := σ+ (x, Q)dx − Q(·) σ+ (x, Q)dx ·

S

the right–hand side of the BNN–dynamics. Since F is neither bounded nor globally Lipschitz continuous on Me , we construct in the following lemma an auxiliary function F˜ which has these properties and coincides with F on ∆. In particular, we show that F˜ satisfies a global Lipschitz condition    e ˜ ˜ ∃K > 0 s.t. ∀µ, ν ∈ M , F (µ) − F (ν) ≤ K kµ − νk .

Standard arguments (see e.g. Zeidler [32, Corollary 3.9]) then imply that the ordinary differential equation ˙ Q(t) = F˜ (Q(t)), Q(0) = P ˙ has a unique solution (Q(t)). Finally, since Q(t)(S) = 0, Q(t) never leaves ∆, which implies that (Q(t)) also solves differential equation (2) on ∆. Lemma A.1 Suppose f is bounded, then there exists a bounded, Lipschitz continuous function F˜ : Me → Me , which coincides with F on ∆, F˜ (P ) = F (P ), ∀P ∈ ∆. Proof : We define F˜ as F˜ (Q) = (2 − kQk)+ F (Q) .

F˜ is zero for kQk ≥ 2, bounded and coincides with F on ∆ because probability measures have norm 1. It remains to show that Q 7→ F (Q) is Lipschitz for kQk ≤ 2. 13

The estimates |E(δx , Q)| ≤ kf k∞ kQk ,

|E(P, Q)| ≤ kf k∞ kP k kQk

(18)

imply that for each x ∈ S, the functions Q 7→ σ(x, Q) and hence also Q 7→ σ+ (x, Q) are Lipschitz (for kQk ≤ 2) with a Lipschitz constant L independent of x. Then the R map Q 7→ Fˆ (Q) with Fˆ (Q)(A) = A σ+ (x, Q)dµ(x) from Me into itself is Lipschitz with Lipschitz constant Lµ(S). In particular, also Q 7→ J(Q) : Me → R is Lipschitz. Hence F (Q) is Lipschitz in Q. 2 Note that Gronwall’s lemma implies kP (t) − Q(t)k ≤ eLt kP (0) − Q(0)k

(19)

and hence continuous dependence of solutions on initial conditions for finite time.

A.2

Weak Continuity of the Flow

Proof : We prove below that we have |σ (x, P ) − σ (x, Q)| ≤ L kP − QkBL

(20)

for some constant L > 0 and all populations P, Q and all strategies x. The same Lipschitz estimate holds true when we pass to the positive part, so we have |σ+ (x, P ) − σ+ (x, Q)| ≤ L kP − QkBL .

(21)

This implies that the right hand side of (2) is Lipschitz in the norm (6). The claim follows then from general theorems on existence, uniqueness and Lipschitz continuity, i.e., (19) for the BL-norm, of solutions in the dual Banach space BL(S)’, and hence in its forward invariant compact subset ∆. It remains to prove the claim (20). By boundedness and Lipschitz continuity of f , there exist constants L0 , L1 > 0 such that for all strategies x, y, x0 , y 0 |f (x, y)| ≤ L0

q 0 0 |f (x, y) − f (x , y )| ≤ L1 kx − x0 k2 + ky − y 0 k2 .

(22) (23)

For x = x0 , the latter inequality yields |f (x, y) − f (x, y 0 )| ≤ L1 ky − y 0 k . 14

(24)

Let R be a population. Define the function Z g(y) = f (x, y)R(dx). S

Since R is a probability measure, (22) carries over to g, |g(y)| ≤ L0 , and so does (24), |g(y) − g(y 0 )| ≤ L1 ky − y 0 k . Hence, g is a bounded and Lipschitz continuous function with kgkBL ≤ L0 + L1 . We thus obtain ŒZ Œ Œ Œ Œ |E(R, P − Q)| = Œ g(y)(P − Q)(dy)ŒŒ S

≤ kgkBL kP − QkBL

≤ (L0 + L1 ) kP − QkBL .

(25)

By a symmetric argument, we also have |E(P − Q, R)| ≤ (L0 + L1 ) kP − QkBL .

(26)

Now, to prove our claim, note that |σ(x, P ) − σ(x, Q)| ≤ |E(δx , P − Q)| + |E(Q, Q) − E(P, P )| ≤ |E(δx , P − Q)| + |E(Q − P, Q) + E(P, Q − P )| ≤ |E(δx , P − Q)| + |E(Q − P, Q)| + |E(P, Q − P )| Applying (25) for R = δx and R = P, as well as (26) for R = Q, we finally obtain |σ(x, P ) − σ(x, Q)| ≤ 3(L0 + L1 ) kP − QkBL , and the proof is complete.

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