On the other hand, if the game is not self-signaling, both the efficient and the inefficient equilibrium outcomes are stable.

Evolutionary Equilibrium Selection with Location Choices

Kenichi Amaya

Abstract

This paper studies evolutionary equilibrium selection in 2 × 2 symmet- ric coordination games. There is a large population of players who are matched to play the game. In contrast to the standard literature in evo- lutionary game theory in which the matching rule is exogenously given, this paper consider the case where the players have some control over how they are matched. Assume there are multiple locations. Each player se- lects one location prior to playing the coordination game, and is matched to another player who selects the same location. Adjustment of location choice is assumed to be infinitely faster than the adjustment of actions of the coordination game. We obtain an efficient-equilibrium-selection result if the underlying coordination game satisfies the self-signaling condition.

On the other hand, if the game is not self-signaling, both the efficient and the inefficient equilibrium outcomes are stable.

1 Introduction

There are many interesting economic and social problems that can be analyzed as coordination games. Since coordination games have multiple equilibria, the question of equilibrium selection has attracted much interest.

There has been a substantial literature in evolutionary game theory which discusses this issue. The literature in stochastic evolution has demonstrated that the risk dominant equilibirum is selected (e.g., Kandori, Mailath and Rob (1993), Ellison (1993), Young (1993)). Another strand of literature in evolutionary game theory argues that if players can communicate each other before playing the game, then the evolutionary forc leads to Pareto efficient equilibria (e.g., Robson (1990), W¨ arneryd (1991), Matsui (1991), and Kim and Sobel (1995)).

In the evolutionary models in the existing literature, there is a large pop- ulation of players and each player is matched with another player to play a

Vol. , No. ・ , March , −

certain game, according to some exogenous rule. Most of the literature assumes uniform random matching. One exception is the local interaction model by El- lison (1993), who studies the case where the players are located on a circle and are matched to play the game only with their neighbours. In either case, the matching rule is exogenously given and the players has no control over it.

This paper considers an evolutionary model in which the players have some control over with whom to play the game. In our model, there are multiple locations. Each player selects one location prior to playing the coordination game, and is matched to another player who selects the same location.

This is a fairly simple model and may be interpreted as a reduced form representation of more complicated procedure. For example, suppose that a pair of players is matched to play the game only upon mutual agreement, and the players can send a cheap talk message before matching. If the players choose to be paired to another player who sends the same messege, the outcome will be the same as in the location choice model.

A player’s decision includes two components: which location to go and which strategy of the coordination game to take. In evolutionary models, the players gradually adjust their decisions over time. Generally speaking, the speed of adjustment can be different between the two components. In this paper, we consider the case where the adjustment of location choice is much faster than the adjustment of strategy of the coordination game.

For example, suppose the players are firms and the strategies in the coor- dination games are the firms’ product characteristics. Since an adjustment of product characteristics requires certain investments, it occurs only occasionally.

On the other hand, the firm can visit different marketplaces every day to find a business partners.

Of course, we can think of the cases where the adjustment of strategy of the coordination game is faster than the location choice. However, the analysis of such cases is left as future research.

We analyze 2 × 2 symmetric coordination games with two symmetric pure

strategy Nash equilibria that are Pareto ranked. We show that the equilibrium

selection result depends on the structure of the underlying coordination game,

i.e., whether the game is self-signaling or not.

a b

a 2 , 2 r, q b q, r 1 , 1 Figure 1.

A 2 × 2 symmetric coordination game is said to be self-signaling if given a player’s own strategy, she is better off if the opponent chooses the same strategy.

Consider the game of Figure 1. Assume q < 2 and r < 1, so that there are two pure strategy Nash equilibria ( a, a ) and ( b, b ).

The game is self-signaling if q < 1. In this case, if a player is going to play b , she is better off if the opponent plays b as well rather than a . Also, if a she is going to play a , she is better off if the opponent plays a as well rather than b .

On the other hand, the game is not self-signaling if 1 < q < 2. In this case, if a player is going to play b , she is better off if the opponent plays a rather than b .

In a symmetric coordination game, if a player knows what the opponent intends to play, he wants to play the same strategy. If the game is self-signaling, a player has a right incentive to reveal her intention, given the opponent best responds to it. If the game is not self-signaling, a player has an incentive to convince the opponent that she intends to play something different.

For general 2 × 2 symmetric coordination games, if we call the efficient equi- librium strategy a and the inefficient equilibrium strategy b , then the game is self-signaling if and only if ( b, b ) gives a higher payoff than ( b, a ) to the row player. ¹

This paper shows that whether we obtain the equilibrium selection result depends on whether the game is self-signaling or not. If the game is self-signaling, only the Pareto efficient Nash equilibrium is stable. On the other hand, if the game is not self-signaling, both the efficient and the inefficient equilibrium are stable. Therefore, an efficiet equilibrium selection result is obtained if and only if the underlying game is self-signaling.

1

For more discussion on the self-signaling condition, see a survey of the cheap talk literature

by Farrell and Rabin (1996).

The logic behind the result is similar to that in the literature of uniform matching with pre-play communication. In those literature, the inefficient equi- librium outcome is destabilized because of the following “secret handshake” ar- gument. In the game of Figure 1, suppose initially all the players are playing b and a small population of mutants enters here. The mutants send a new message that is not used by the incumbents, and plays a if and only if the opponent also sends this new message. They play b otherwise. Then, ( b, b ) is played when two incumbents are matched and when an incumbent and a mutant are matched.

However, ( a, a ) is played when two mutants are matched. Therefore, the mu- tants receive a higher payoff on average than the incumbents and invade the population. The key here is that the mutants can separate themselves from the incumbents through their messages and coordinate on the efficient equilibrium only among themselves.

In our model, suppose initially all the players are playing b (they are called type b ). and a small population of mutants who plays a (type a ) enters here. In short run, each player’s type is fixed, and she can choose the location given her type. If the game is self-signaling, a type a player is happier if she is matched with another type a , and a type b player is happier if he is matched with another type b . Therefore, after the adjustment of location choices, different types are separated to different locations. Then, in the example of Fig. 1, All type a players’ payoff is 2 and all type b players’ payoff is 1. Therefore, the initial population is not stable against the injection of type a .

However, if the game is not self-signaling, a type b player prefers to be matched with type a . Under the short run adjustment of location choices, type b incumbents always try to pool with type a mutants. Therefore, type a cannot receive a higher payoff than type b and eventually vanishes out.

We analyze a model where the action in the coordination game (or types) and the location choice evolve at different speeds. We assume the evolution of location choice is infinitely faster than the evolution of types in the following sense. For each fixed type distribution, we look for stable outcomes in the adjustment of location choices and examine how well each type performs in the stable outcomes. We model the evolution of the type distribution in the way

4

that the type earning a higher payoff in the stable outcomes thrives better.

This is the same approach as the literature in evolution of preferences (e.g. Ely and Yilankaya (2001)), in which the players’ preferences evolve slowly, while they frequently adjust their behavior in games taking the current preferences as exogenously fixed.

The rest of the paper is organized as follows. Section 2 describes the stage game to be repeated. Section 3 considers the short run evolution of location choices under a fixed type distribution. Section 4 studies the two-speed evolution model which includes the adjustment of actions in the coordination games and contains our main results.

2 Location Choice Game

2.1 Base Game

The base game is a 2 × 2 symmetric coordination game. The set of pure strategies is {a, b} and π ij , i, j = a, b is the payoff when a player plays i and the opponent plays j . Thus the payoff matrix is given by Figure 2.

a b

a π aa , π aa π ab , π ba

b π ba , π ab π bb , π bb

Figure 2.

Assume π aa > π ba , π bb > π ab , and π aa > π bb . By the first two inequalities, the game has two pure strategy Nash equilibria, ( a, a ) and ( b, b ). The last inequality says the former equilibrium is Pareto efficient. Also assume there is no tie in payoffs. The game is said to be self-signaling if π bb > π ba . The strategies in the base game are called actions

2.2 Location Choice Game

There is a continuous population of players. Time is continuous and at each

moment the players play the following two stage game. In stage 1, each player

chooses one location. The set of available locations is countable and finite and is denoted by L = { 1 , 2 , · · · , N } . A typical element of L is denoted by l . In stage 2, each player is randomly matched to another player who has chosen the same location and play the base game described above.

We refer to this two stage game, the base game preceded by a location choice stage, as the location choice game.

3 Types of the players and short run games

The same location choice game is repeated over time and the players gradually adjust their behavior over time as in usual evolutionary models. The players’

behavior includes two elements: the locations to choose in stage 1 and the actions of the base game to be chosen in stage 2.

We assume that players receive opportunities to revise locations infinitely more frequently than opportunities to revise actions.

The type θ of a player describes the action she takes in the second stage.

The set of possible types is Θ ≡ {a, b} .

In this section we consider the evolutionary dynamics of location choice under fixed type distribution. Suppose the type distribution x = ( x a , x b ) is fixed, where x θ , θ = a, b is the fraction of type θ players in the whole population. Let ΔΘ be the set of probability distribution over Δ.

3.1 Notation

For θ ∈ {a, b} , let p θ = ( p θ (1) , p θ (2) , · · · , p θ ( N )) be the distribution of location choices of type θ players, where p θ ( l ), l ∈ L denotes the fraction of type θ players who chooses location l among all type θ players. We have

l∈L p θ ( l ) = 1 for θ ∈ {a, b} . Let r ( l ) = x a p a ( l ) + x b p b ( l ) be the fraction of players of all types who chooses location l , and let ¯ L be the set of locations chosen by players of positive measure, i.e.,

L ¯ ≡ {l ∈ L|r ( l ) > 0 }.

For l ∈ L ¯ , let q θ ( l ) be the fraction of type θ players among those who chooses location l , i.e.,

q θ ( l ) = x θ p θ ( l ) r ( l ) . For θ ∈ a, b , let

L θ ≡ {l ∈ L|q ¯ θ ( l ) ≥ q θ ( l ) ∀l ∈ L} ¯

be the set of locations with the highest fraction of type θ players.

3.2 Dynamics

Time is continuous. Each player receives opportunities to revise her location with Poisson rate λ .

A player with a revision opportunity chooses the location that yields her the highest expected payoff. We refer to such a location as best location. If a type θ player chooses location l , she will be matched to a type a player with probability q a ( l ) and to a type b player with probability q b ( l ), and thus her expected payoff is

q a ( l ) π θa + q b ( l ) π θb .

Therefore, if π θa > π θb , then the set of best locations for a type θ is L a , and if π θb > π θa , then the set of best locations for a type θ is L b ,

Assume that if there are multiple best locations for a player and she has been in one of the best locations before revision, then she keeps choosing the same location as before. If she has not been in a best location before, she picks one of the best locations randomly according to a certain probability distribution.

When the set of best responses is L k ( k = a, b ), this distribution is denoted by ψ k ( l ), where

l∈L

ψ k ( l ) = 1 for k = a, b .

The assumption on the payoff structure of the base game implies that the set of best locations for type a players is L a . The set of best locations for type b players is L b if the game is self-signaling, and L a otherwise.

A special case is when q a ( l ) = x a and q b ( l ) = x b for all l ∈ L ¯ . The proportion

of each type of players is equivalent across all location inhabited by positive

measure of players. The proportion should be equal to the proportion of the types in the whole population. In this case, L a = L b = ¯ L and thus all players with a revision opportunity keep choosing the same location as before.

From the formulation above, we can derive the dynamics of p θ as follows.

Dynamics of type a

If q θ ( l ) = x θ for all θ ∈ {a, b} and l ∈ L ¯ , p ˙ a ( l ) = 0 ∀l ∈ L.

If q θ ( l ) = q θ ( l ) for some θ ∈ {a, b} and l, l ∈ L ¯ , p ˙ a ( l ) =

−λp a ( l ) if l / ∈ L a

λψ a ( l )

l

∈L /

p a ( l ) if l ∈ L a . Dynamics of type b (if the base game is self-signaling)

If q θ ( l ) = x θ for all θ ∈ {a, b} and l ∈ L ¯ , p ˙ b ( l ) = 0 ∀l ∈ L.

If q θ ( l ) = q θ ( l ) for some θ ∈ {a, b} and l, l ∈ L ¯ , p ˙ b ( l ) =

−λp b ( l ) if l / ∈ L b

λψ b ( l )

l

∈L /

p b ( l ) if l ∈ L b . Dynamics of type b (if the base game is not self-signaling)

If q θ ( l ) = x θ for all θ ∈ {a, b} and l ∈ L ¯ , p ˙ b ( l ) = 0 ∀l ∈ L.

If q θ ( l ) = q θ ( l ) for some θ ∈ {a, b} and l, l ∈ L ¯ ,

p ˙ b ( l ) =

−λp b ( l ) if l / ∈ L a

λψ a ( l )

l

∈L /

p b ( l ) if l ∈ L a .

3.3 Stability

We define the stability of the evolutionary dynamics under fixed type distribu-

tion.

Let p = ( p a , p b ) be the distribution vector, or state and P be the set of all possible states. Define the distance between two states p, p ∈ P by

d ( p, p ) ≡ max

θ,l |p θ ( l ) − p θ ( l ) |.

For a nonempty closed set of states A ⊂ P and a state p , define the distance between P and p by

d ( A, p ) ≡ min

p∈A d ( p, p ) .

For a nonempty closed set of states A ⊂ P , define the -neighborhood of A by N ( A ) ≡ {p ∈ P |d ( A, p ) < }.

We say a dynamic process of p converges to A if the distance d ( A, p ) converges to zero.

Definition 1 A nonempty closed set of states A ⊂ P is stable under type dis- tribution x = ( x a , x b ) if there exists > 0 such that every dynamic process of p starting in the -neighborhood of A converges to A , and no proper subset of A meets this property.

In the rest of this subsection, we solve for the stable set of states. In 3.3.1, we consider the case where only one type is present. In 3.3.2, we consider the case where both types are present and the base game is self-signaling. In 3.3.3, we consider the case where both types are present and the base game is not self-signaling.

3.3.1 One type

Consider the case where ( x a , x b ) = (1 , 0) or ( x a , x b ) = (0 , 1).

Proposition 1 If x a = 1 or x b = 1, then the set of all possible states P is the unique stable set.

Proof:

We only prove the case of x a = 1. The same argument applies for the case of

x b = 1.

Since every dynamic process stays in P for ever, it is obvious that there exists > 0 such that every dynamic process of p starting in the -neighborhood of P converges to P .

Next we prove no proper subset of P satisfies the property above. Since only type a players are present, q a ( l ) = 1 for all l ∈ L ¯ . Thus, ¯ L = L a and all players are choosing the best location. In any p ∈ P , no player with a revision opportunity switches the location. Therefore, if we take any proper subset A of P and state p in the -neighborhood of A , the dynamic process starting from p does not converge to A .

3.3.2 Two types, Self-signaling Suppose x a > 0 and x b > 0.

Let P ^SS ⊂ P be the set of states such that for all l ∈ L , one of the following three conditions holds:

1. p a ( l ) > 0 and p b ( l ) = 0 2. p a ( l ) = 0 and p b ( l ) > 0 3. p a ( l ) = p b ( l ) = 0

In such states, every location is eiher inhabited only by type a , inhabited only by type b , or empty. Therefore, all players are matched to a player of the same type. Accordingly, type a players receive payoff π aa and type b players receive payoff π bb .

Proposition 2 If x a , x b > 0 and the base game is self-signaling, P ^SS is the unique stable set.

Proof:

To prove the proposition, we prove the following three statement: (i) There exists > 0 such that every dynamic process of p starting in the -neighborhood of P ^SS converges to P ^SS (ii) No proper subset of P ^SS meets the property above.

(iii) No set which contains a state p / ∈ P ^SS is stable.

First, we prove (ii). Let P 1 be a nonemply closed proper subset of P ^SS . Then for any > 0 we can find a state p’ such that p ∈ P ^SS ∪ N ( P 1 ). Since p ∈ P ^SS , all type a players are in L a and all type b players are in L b , i.e., all players are in their best location. Therefore, no player changes the location upon a revision opportunity. Thus the dynamic process stays at p for ever and does not converge to P 1 . This implies P 1 is not stable.

Next we prove statements (i) and (iii). To do so, it suffices to show that any dynamic process starting from an initial state such that q a ( l ) = q a ( l ) for some l, l ∈ L ¯ converges to P ^SS .

Consider such a process. First, notice that if a location l does not belong to L a at time t , then l / ∈ L a for all t > t . Therefore, if l / ∈ L a for some t , then lim _t→∞ p a ( l ) = 0. Therefore, lim _t→∞ p a ( l ) > 0 implies l ∈ L a for all t, which in turn implies l / ∈ L b for all t, thus lim _t→∞ p b ( l ) = 0. Likewise, lim _t→∞ p b ( l ) > 0 implies lim _t→∞ p a ( l ) = 0. Thus, any location l satisfies one of the following.

1. lim _t→∞ p a ( l ) > 0 and lim _t→∞ p b ( l ) = 0 2. lim _t→∞ p a ( l ) = 0 and lim _t→∞ p b ( l ) > 0 3. lim _t→∞ p a ( l ) = lim _t→∞ p b ( l ) = 0 Therefore, the process converges to P ^SS .

3.3.3 Two types, Not self-signaling Suppose x a > 0 and x b > 0.

Let P ^NSS ⊂ P be the set of states such that for all l ∈ L , one of the following two conditions holds:

1. p a ( l ) = p b ( l ) > 0 (and thus q θ ( l ) = x θ for θ = a, b ) 2. p a ( l ) = p b ( l ) = 0

In such states, in all locations with positive measure of players, the ratio of type

a and type b are same and equal to the fixed total population ratio. Therefore, all

players are matched to a type a player with probability x a and to a type b player

with probability x b . Accordingly, type a players receive payoff x a π aa + x b π ab and type b players receive payoff x a π ba + x b π bb .

Proposition 3 If x a , x b > 0 and the base game is not self-signaling, P ^NSS is the unique stable set.

Proof:

To prove the proposition, we prove the following three statement: (i) There exists > 0 such that every dynamic process of p starting in the -neighborhood of P ^NSS converges to P ^NSS (ii) No proper subset of P ^NSS meets the property above. (iii) No set which contains a state p / ∈ P ^NSS is stable.

First, we prove (ii). Let P 1 be a nonemply closed proper subset of P ^NSS . Then for any > 0 we can find a state p’ such that p ∈ P ^NSS ∪ N ( P 1 ). Since p ∈ P ^NSS , all players are in L a = L b = ¯ L , i.e., all players are in their best location. Therefore, no player changes the location upon a revision opportunity.

Thus the dynamic process stays at p for ever and does not converge to P 1 . This implies P 1 is not stable.

Next we prove statements (i) and (iii). To do so, it suffices to show that any dynamic process starting from an initial state such that q a ( l ) = q a ( l ) for some l, l ∈ L ¯ converges to P ^NSS . This claim may be restated as the following lemma.

Lemma 1 Consider any dynamic process starting from an initial state such that q a ( l ) = q a ( l ) for some l, l ∈ L ¯ . For any > 0, there exists no l ∈ L such that for all t > 0 there exists some t > t such that |q a ( l ) − x a | > and r ( l ) > .

If the lemma holds, for any location l , either q a ( l ) converges to x a or r ( l ) = x a p a ( l ) + x b p b ( l ) converges to zero.

Now we prove the lemma. First, for some l , suppose q a ( l ) < x a for some t . This implies l / ∈ L a at t and forever after. Therefore, lim _t→∞ r ( l ) = 0. This proves the lemma where |q a ( l ) − x a | > is replaced by q a ( l ) − x a < − .

Next we prove the lemma where |q a ( l ) − x a | > is replaced by q a ( l ) − x a > .

Suppose, to the contrary, that for some > 0, there exists l ∈ L such that for

any t > 0 there exists some t > t such that |q a ( l ) − x a | > and r ( l ) > . For

such t , since

l

∈L

r ( l ) q a ( l ) = x a

and thus

l

∈L

r ( l )( q a ( l ) − x a ) = 0 , it must hold that

l

∈L\{l}

r ( l )( q a ( l ) − x a ) < − ² and thus

1 N − 1

l

∈L\{l}

r ( l )( x a − q a ( l )) > 1 N − 1 · ² .

This implies that there exists at least one location l such that q a ( l ) < x a and r ( l )( x a − q a ( l )) > 1

N − 1 · ² . Since x a − q a ( l ) is at most x a , it must be

r ( l ) > 1 x a · 1

N − 1 · ² .

This implies that for any t > 0 there exists some t > t such that r ( l ) >

x 1

· _N−1 ¹ · ² . This condradits to the argument above that if q a ( l ) < x a for some t then lim _t→∞ r ( l ) = 0.

4 Long-run stability

4.1 Definition of long-run stability

Now we move to the analysis of the two-speed evolution model, where the ad- justment of location choices is infinitely faster than the evolution of types.

For a type distribution x = ( x a , x b ), a state p = ( p a , p b ) and type θ ∈ {a, b} , let

U θ ( x, p ) =

l∈ L ¯

p θ ( l ) {q a ( l ) π θa + q b ( l ) π θb }

be the average payoff of type θ players under type distribution x = ( x a , x b ) and state p = ( p a , p b ). With this notation, now we define the solution concept.

Definition 2 A pair ( x, A ) of type distribution x ∈ ΔΘ and a nonempty closed set of states A ⊂ P is long run stable if the following conditions hold.

1. A is stable under x .

2. For all x ∈ ΔΘ with x = x , there exists ¯ ∈ (0 , 1) such that ∀ ∈ (0 , ),

θ∈Θ

x θ U θ ((1 − ) x + x , p ) >

θ∈Θ

x _θ U θ ((1 − ) x + x , p ) (1) for all p such that p is an element of a stable set under (1 − ) x + x . The idea is essentially the same as the solution concept in Amaya (2006).

The first condition simply states the short run stability of the set of states, taking the type distribution as given. The second condition concerns the stability in the evolution of type distribution. It borrows the backbones from the traditional stability concepts in evolutionary game theory. To illustrate the idea, consider the stability of a type distribution where all players are of the same type θ . Suppose that initially all the players are of type θ . Now inject a small population share of type θ individuals ( θ = θ ). The players immediately adjust to play an element of a stable set of the short run game under the ex-post type distribution (1 − ) θ + θ . The initial type distribution is not invaded by an injection of θ if, for sufficiently small , type θ does not receive a higher payoff than type θ in any element of a stable set under the ex-post type distribution. The initial type distribution is stable if it is not invaded by any small population of mutants, where the mutants may be a mixture of multiple types. The formal definition generalizes this idea and allows both the incumbent population and the mutant population to be a mixture of multiple types.

In what follows, when we say “type distribution x is long run stable”, we mean that there exists A ∈ P such that ( x, A ) is long run stable.

4.2 Results

It is convenient to show the following lemma before giving our results.

Lemma 2 1. Suppose the type distribution x satisfies ( x a , x b ) = (1 , 0). Then ( x, P ) is long run stable if and only if there exists ¯ ∈ (0 , 1) such that

∀ ∈ (0 , )

U a ((1 − , ) , p ) > U b ((1 − , ) , p )

for all p such that p is an element of a stable set under the type distribution (1 − , ).

2. Suppose the type distribution x satisfies ( x a , x b ) = (0 , 1). Then ( x, P ) is long run stable if and only if there exists ¯ ∈ (0 , 1) such that ∀ ∈ (0 , )

U a (( , 1 − ) , p ) > U b (( , 1 − ) , p )

for all p such that p is an element of a stable set under the type distribution ( , 1 − ).

3. Suppose the type distribution x satisfies x a > 0 and x b > 0.

Now we state our main results.

4.2.1 Self-signaling

Proposition 4 Suppose the base game is self-signaling. Then, ( x, A ) is long run stable if and only if x = ( x a , x b ) = (1 , 0) and A = P

This proposition states that the unique long run stable type distribution is where all players are choosing action a . Although the base game has two pure strategy Nash equilibrium ( a, a ) and ( b, b ), the latter is not long run stable. Thus we obtain the equilibrium selection result.

Proof:

First we show that ((1 , 0) , P ) is long run stable. Proposition 1 implies that condition 1 in the definition of long run stability is satisfied. To show condition 2 is satisfied, consider a mutation x = (1 −β, β ) where 0 < β ≤ 1 and ∈ (0 , 1).

Then, the ex post type distribution is

(1 − ) x + x = (1 − β, β ) .

From proposition 2, the unique stable set under (1 − ) x + x is P ^SS , and for any p ∈ P ^SS ,

U a ((1 − ) x + x , p ) = π aa

U b ((1 − ) x + x , p ) = π bb .

Therefore, (1) holds because the LHS is equal to π aa , the RHS is equal to (1 − β ) π aa + βπ bb and π aa > π bb . This implies condition 2 is indeed satisfied.

Next we show that no type distribution x = (1 , 0) is long run stable. Let x = (1 − α, α ) with 0 < α ≤ 1. It suffices to show that there exists some x for which (1) does not hold for any ∈ (0 , 1). Let x = (0 , 1). Then, the ex post type distribution is

(1 − ) x + x = (1 − (1 − ) β, (1 − ) β ) .

From proposition 2, the unique stable set under (1 − ) x + x is P ^SS , and for any p ∈ P ^SS ,

U a ((1 − ) x + x , p ) = π aa

U b ((1 − ) x + x , p ) = π bb .

The LHS of (1) is (1 − β ) π aa + βπ bb and the RHS is π aa . Since π aa > π bb and α > 0, (1) does not hold.

4.2.2 Not self-signaling

Proposition 5 Suppose the base game is not self-signaling. Then, ( x, A ) is long run stable if and only if (i) x = ( x a , x b ) = (1 , 0) and A = P or (ii) x = ( x a , x b ) = (0 , 1) and A = P

This proposition states that there are two long run stable type distribution.

In one of them, all players are choosing action a , and in the other, all players are

choosing action b . For each of the two pure surategy Nash equilibrium, there is

a corresponding long run stable type distribution. Therefore, we do not obtain

an equilibrium selection result.

Proof:

First, we show that ((1 , 0) , P ) is long run stable. Proposition 1 implies that condition 1 in the definition of long run stability is satisfied. To show condition 2 is satisfied, consider a mutation x = (1 −β, β ) where 0 < β ≤ 1 and ∈ (0 , 1).

Then, the ex post type distribution is

(1 − ) x + x = (1 − β, β ) .

From proposition 3, the unique stable set under (1 − ) x + x is P ^NSS , and for any p ∈ P ^NSS ,

U a ((1 − ) x + x , p ) = (1 − β ) π aa + βπ ab

U b ((1 − ) x + x , p ) = (1 − β ) π ba + βπ bb . Since π aa > π ba ,

U a ((1 − ) x + x , p ) > U b ((1 − ) x + x , p ) (2) for sufficiently small . The LHS of (1) is U a ((1 − ) x + x , p ) and the RHS is (1 − β ) U a ((1 − ) x + x , p ) + βU b ((1 − ) x + x , p ). Since β > 0, (2) implies the LHS is indeed larger.

Second, we show that ((0 , 1) , P ) is long run stable. Proposition 1 implies that condition 1 in the definition of long run stability is satisfied. To show condition 2 is satisfied, consider a mutation x = ( β, 1 − β ) where 0 < β ≤ 1 and ∈ (0 , 1).

Then, the ex post type distribution is

(1 − ) x + x = ( β, 1 − β ) .

From proposition 3, the unique stable set under (1 − ) x + x is P ^NSS , and for any p ∈ P ^NSS ,

U a ((1 − ) x + x , p ) = βπ aa + (1 − β ) π ab

U b ((1 − ) x + x , p ) = βπ ba + (1 − β ) π bb . Since π bb > π ab ,

U b ((1 − ) x + x , p ) > U a ((1 − ) x + x , p ) (3)

for sufficiently small . The LHS of (1) is U b ((1 − ) x + x , p ) and the RHS is βU a ((1 − ) x + x , p ) + (1 − β ) U b ((1 − ) x + x , p ). Since β > 0, (3) implies the LHS is indeed larger.

Third, we show no type distribution x = ( α, 1 − α ), α ∈ [ α ^∗ , 1) is stable, where

α ^∗ = π bb − π ab

( π aa − π ba ) + ( π bb − π ab ) . It is convenient to check here that for any real number y ,

yπ aa + (1 − y ) πab > ( < ) yπ ba + (1 − y ) π bb iff y > ( < ) α ^∗ . (4) It suffices to show that there exists some x for which (1) does not hold for any ∈ (0 , 1). Let x = (1 , 0). Then, the ex post type distribution is

(1 − ) x + x = ((1 − ) α + , (1 − α )(1 − )) .

From proposition 3, the unique stable set under (1 − ) x + x is P ^NSS , and for any p ∈ P ^NSS ,

U a ((1 − ) x + x , p ) = ((1 − ) α + ) π aa + ((1 − α )(1 − )) π ab

U b ((1 − ) x + x , p ) = ((1 − ) α + ) π ba + ((1 − α )(1 − )) π bb . Since (1 − ) α + > α ^∗ , (4) implies

U a ((1 − ) x + x , p ) > U b ((1 − ) x + x , p ) . (5) The LHS of (1) is αU a ((1 − ) x + x , p ) + (1 − α ) U b ((1 − ) x + x , p ) and the RHS is U a ((1 − ) x + x , p ). Since α < 1, (5) implies (1) does not hold.

Finally, we show no type distribution x = ( α, 1 − α ), α ∈ (0 , α ^∗ ) is stable.

It suffices to show that there exists some x for which (1) does not hold for any ∈ (0 , 1). Let x = (0 , 1). Then, the ex post type distribution is

(1 − ) x + x = ((1 − ) α, 1 − (1 − ) α ) .

From proposition 3, the unique stable set under (1 − ) x + x is P ^NSS , and for any p ∈ P ^NSS ,

U a ((1 − ) x + x , p ) = (1 − ) απ aa + (1 − (1 − )) π ab

U b ((1 − ) x + x , p ) = (1 − ) απ ba + (1 − (1 − )) π bb .

Since (1 − ) α < α ^∗ , (4) implies

U b ((1 − ) x + x , p ) > U a ((1 − ) x + x , p ) . (6) The LHS of (1) is αU a ((1 − ) x + x , p ) + (1 − α ) U b ((1 − ) x + x , p ) and the RHS is U b ((1 − ) x + x , p ). Since α > 0, (6) implies (1) does not hold.

References

[1] Amaya, K. (2006). “Two-Speed Evolution with Pre-Play Communication and Limited Flexibility,” Review of Economic Dynamics, 9, 310-325.

[2] Ellison, G. (1993). “Learning, Local Interaction, and Coordination,” Econo- metrica, 61, 1047-1071.

[3] Ely, J., and Yilankaya, O. (2001). “Nash Equilibrium and the Evolution of Preferences,” Journal of Economic Theory, 97, 255-272.

[4] Farrell, J., and Rabin, M. (1996). “Cheap Talk,” Journal of Economic Per- spectives, 10, 103-118.

[5] Kandori, M., Mailath, G., and Rob, R. (1993) “Learning, Mutation, and Long-Run Equilibria in Games,” Econometrica, 51, 29-56.

[6] Kim, Y.-G., and Sobel, J. (1995). “An Evolutionary Approach to Pre-Play Communication,” Econometrica, 63, 1181-1193.

[7] Matsui, A. (1992). ”Best Response Dynamics and Socially Stable Strate- gies,” Journal of Economic Theory, 57, 343-362.

[8] Robson, A. J. (1990). “Efficiency in Evolutionary Games: Darwin, Nash and the Secret Handshake,” Journal of Theoretical Biology, 144, 379-396.

[9] W¨ arneryd, K. (1991). “Evolutionary Stability in Unanimity Games with Cheap Talk,” Economics Letters, 36, 375-378.

[10] Young, P. (1993). “The evolution of conventions,” Econometrica, 61, 57-84.