population game.
1.4 Coordination Games
A symmetric coordination game^ C, satisfies the following Definition:
D efinition ^/Cm isja coordination game if for all i e A, ira > TTji Vj ^ %. Definition 6 states that payoff is maximized when both players choose the s
action. As a result, any probability distribution that assigns probabihty one to only one action constitutes a strict Nash-equihbrium of Cm- For exactly the same rea son, any probabihty distribution that assigns probabihty one to (i,z) is a symmetric correlated equihbrium of Cm- While in general N{Cm) Ç N{Cm) in the same way as
"^{Cm) G T(Cm), by focusing on pure strategy equihbria the same relation holds with the equality sign, as we explicitly state in the next Remark:
R em ark 3 For a coordination game Cm L
= N ’ iCm) 4"'(Cm) = «'’(Cm),
We now turn to the associated population game, F = (H, C, fi). As far as the latter is concerned, the results of Remark 1 and 2 are of straight-forward application. If
equilibria of the associated population game are those configurations of play where the frequencies with which available actions are chosen in the population reproduce any of the Nash-equihbria of the underlying game. However, if the population is finite and/or matching is local, the relevant set of equihbria to be looked at is that of the symmetric correlated equihbria of the underlying game, and these might not reproduce any Nash- equihbrium of the underlying game. For example, in a uniform population matching model for a 2-2 coordination game we know that the only equihbria involve all players adopting the same action. This is no longer the case if the matching technology is arbitrarily specified, in that players might hold, in equihbrium, different ’’views of the world” and on the basis of the latter, choose different actions. As a result, both action can coexist in equihbrium^.
We conclude this Section with a general remark on the latter point, where we address the following question. We know from Remark 2 that, if matching is uniform, the set of equihbria of the population game can be identified, in the hmit, with the set of symmetric Nash equihbria of the underlying game. Clearly the remark holds also if we require players to be adopting a strict best-reply, in which case in the hmit we would only observe frequencies that reproduce degenerate equihbria. Given that the identification of the latter set is particularly simple for an underlying coordination game, we wonder whether by introducing further assumptions on the payoff structure, we can obtain the same result for a finite population.
To clarify, consider F = (Q, C2,Hu)' Assume in an equihbrium of F there is at least player w adopting the first action. Then it must be that, for w:
Can there be a player s adopting, in equilibrium action 2? The answer is no: s would in fact adopt action 2 in equihbrium only if:
but this cannot be the case, because by adding ——- (tth — 7T2i — 7Ti2 + 7T22) (> 0
pS2 — 1
under Assumption 6) to the first equihbrium condition we contradict the second. Hence Assumption 6 is necessary and sufficient to rule out equihbria where both action are represented in the population. This is no longer the case for a generic m-by-m coordination game, as shown in the Remark that foUows.
R e m a rk 4 Given T = (Çl,Cm, A^(r) 3 and M{T) D N^{Cm)- I f for all j ^ l ^ (ttü - TTii) > {iTij - TTy), then Ad(F) = if^{Cm) and M{T) = N^(Cm)- P ro o f. The 3 is obvious. For the Ç in the second part of the statement, it is
sufficient to prove that, given an equihbrium 6, if og(w) = %, then ag{s) = % for ah s G f2. If w chooses i, then:
+ g - Try) > 0 V Zf i
By adding ■ — tth — Tr^ + tt^) > 0 we get, for all actions j ^ i : pS2 — 1
+ § i
j
S
r
°
which rules out optimality for any action j t
The above Remark shows that, if the population is finite, a further assumption on the underlying payoffs (besides the one that defines a coordination game) allows to characterize equihbria of the population game exclusively in terms of strict equihbria
of the underlying game. The condition requires that the comparative advantage of action i over action I be increasing in the probability with which a player’s opponent adopts action i. To see this, let q be any probabihty distribution over A such th at Qj {'^ij “ TTij) > 0 VZ ^ 2. The latter function quantifies the relative advantage
of action i over any action I ^ i. Its partial derivative with respect to % is given by (TTit—TTii) — (TTij—TTij) for any j and is strictly increasing in % if the above condition holds. In other words, if action i is the best-reply to ç, then this is also true for any q such that % > % and qj < qj Vj ^ i. If the population is finite and matching is uniform, one player who adopts action z as his best reply faces a proportion of other z-players. As it is clear, any other player wih observe at least the same proportion of i.
In the Chapter that foUows we provide a general introduction to the learning processes that constitute the focus of this work.