Mixed Strategy: Symmetric Games and Symmetric Equilibria

The Battling Brewers: Miller and Budweiser

This is an example of symmetric games and symmetric equilibria.  Although we don't have specific numbers for the payoffs in the table, they do have an ordinal ordering: d>b>0>a.

Tough Concede
Budweiser Tough a, a d, 0
Concede 0, d b, b

The best response function for the players can be summarized as

Bud's best response to Miller Miller's best response to Bud
bBud(TMiller) = Concede

bMiller(CBud) = Tough

bBud(CMiller) = Tough bMiller(TBud) = Concede

From the best response table we can see that there are two Nash equilibria. Notice that we could switch the names of the two players and not be able to distinguish the outcome of the new game. These are also pure strategy equilibria.  If the players could coordinate their strategies they might find it in their combined interest to both play Concede and earn a total of 2b.  This would be the case if 2b>d.

Another possibility is that there is a mixed strategy equilibrium.  Suppose that Miller plays Tough with probability p and Concede with probability (1-p).  The consequence for Bud when Miller plays this mixed strategy is

Miller plays mixed strategy (T,p)

Bud plays Tough Bud plays Concede
Bud's payoff ap + d(1-p) b(1-p)

If Bud is indifferent between the two pure strategies (Why?), then they must have the same payoff for him. So then

ap + d(1-p) = b(1-p)

Solving for p we see that in response to Miller's mixed strategy Bud is indifferent to the two pure strategies when Miller has chosen

p = (d-b)/(d-b-a)

Indeed, when Miller uses this probability then Bud could also play a mixed strategy with the same probability!  When the two players use identical mixed strategies their payoffs are equal.  We need only compute the payoff for one of them because of the symmetry in the game.  In doing the calculations remember that the expected payoff is the probability weighted average of the pure strategy payoffs.

E(PayoffBud) = ((d-b)/(d-b-a))(ap + d(1-p) + (1-(d-b)/(d-b-a))(b(1-p))

Substitute in again for p and cancel common terms and you will find that Bud's expected payoff is

E(PayoffBud) = (-ab)/(d-b-a)

When Miller and Bud play the same mixed strategy the game results in a third equilibrium, a mixed strategy Nash equilibrium.

This illustrative Miller v. Budweiser game is known as a symmetric game. When each player has the same strategy in a Nash equilibrium then that solution to the game is a symmetric equilibrium.

In this Miller v. Bud game the pure strategy equilibria are not symmetric, but the mixed strategy is.  The mixed strategy, which has a symmetric result, is more plausibly played than either of the pure strategies.  There are three reasons for this.

  1. If we allow communication before play begins then Miller will think about what happens when Bud plays tough as a pure strategy in an asymmetric equilibrium and try to persuade Bud that it is better off playing a mixed strategy.  Similar reasoning would be used by Bud.  In the end both  arrive at the conclusion that a mixed strategy is the better bet.
  2. Earlier it was argued that Nash equilibrium can be the result of reasoned introspection.  In the Miller v. Bud game each player will expect his twin to play as he plans to play.
  3. A symmetric equilibrium is more likely to be a focal point of a game.
Two Case Studies
Monopoly Bankruptcy