Battle of the sexes: Mixed strategies in games of coordination and cooperation

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Battle of the sexes: Mixing strategies in games of coordination and cooperation

Battles, mixing, and coordination

Previously, we showed that mixing is a viable strategy and solution to games of conflict. Mixing, however, can work in games of coordination too. In this post, we look at the Battle of the Sexes and a potential role for mixed strategies when coordination and cooperation matters.

Battle of the Sexes

In our version of the Battle of the Sexes, there are two friends: Luce and Raiffa. Both of them have to decide independently, without prior communication or coordination, whether to go to a basketball game or the cinema. Luce and Raiffa are happiest when they’re together. While Luce enjoys basketball more than the movies, Raiffa is the opposite. The table below summarizes the players, strategies and payoffs for each player.

Battle of the Sexes (basketball and cinema)

What happens in this game?

Nash equilibria in pure strategies

This game has two Nash equilibria in pure strategies. The first equilibrium sees both players choosing Basketball, while the second sees both choosing Cinema. A strategic profile in which one player chooses Basketball and the other Cinema is unstable. Their best response is to do what they believe the other will do. Of course, without more information, we cannot say which of the two Nash equilibria will result.

Elements of conflict

The Battle of the Sexes is similar to the Stag Hunt game in that there are two Nash equilibria in pure strategies. But unlike the Stag Hunt game, there is no unique socially optimum equilibrium in pure strategies. Going from Basketball to Cinema, or vice versa, results in someone losing and someone winning. That is, this is a coordination game with elements of conflict. Luce and Raiffa’s interests are not perfectly aligned.

Nash equilibrium in mixed strategies

There is also a Nash equilibrium in mixed strategies. To see this, lets specify their randomisation strategies and expected payoffs.

Randomisation strategies:

Probability that Luce (L) chooses Basketball (B) or Cinema (C):

1 = BL + CL

Probability that Raiffa (R) chooses Basketball (B) or Cinema (C):

1 = BR + CR

We can now specify Luce and Raiffa’s payoffs from mixing strategies.

Luce’s payoffs:

Luce’s expected payoff (LB) when she chooses Basketball:

LB = (3 x BR) + (0 x CR)

Luce’s expected payoff (LC) when she chooses Cinema:

LC = (0 x BR) + (2 x CR) 

Noting that 1 = BR + CR , Luce is indifferent between Basketball and Cinema when:

LB = LC

(3 x BR) + (0 x CR) = (0 x BR) + (2 x CR)

BR = 2/5 and CR = 3/5

In other words, Luce is indifferent between her choices when Raiffa randomises between Basketball with 2/5 probability, and Cinema with 3/5 probability.

Raiffa’s payoffs:

Raiffa’s expected payoff (RB) when she chooses Basketball:

RB = (2 x BL) + (0 x CL)

Raiffa’s expected payoff (RC) when she chooses Cinema:

RC = (0 x BL) + (3 x CL)

Noting that 1 = BL + CL, Raiffa is indifferent between Basketball and Cinema when:

RB = RC

(2 x BL) + (0 x CL) = (0 x BL) + (3 x CL)

BL = 3/5 and CL = 2/5

In other words, Raiffa is indifferent between her choices when Luce randomises between Basketball with 3/5 probability, and Cinema with 2/5 probability.

Nash equilibrium in mixed strategies

Hence, we have a Nash equilibrium in mixed strategies when: Raiffa randomizes between Basketball with 2/5 probability, and Cinema with 3/5 probability; and Luce randomizes between Basketball with 3/5 probability, and Cinema with 2/5 probability. It is a Nash equilibrium because neither has an incentive to change their strategy when both are using their mixed strategies. If any other pair of probabilities were played, the other player would have the best response in pure strategy (try to check this for yourself). And the best response to a pure strategy returns us to the game in pure strategies that we had originally.

Uncoordinated equilibrium

Notice though, that the expected payoff from their Nash equilibrium in mixed strategies is lower than the Nash equilibria in pure strategies. For example, the expected payoff for Luce is:

EP = (BR CL x 0) + (CR CL x 2) + (BR BL x 3) + (CR BL x 0) = 1.2

Likewise, the expected payoff for Raiffa is:

EP = (BR CL x 0) + (CR CL x 3) + (BR BL x 2) + (CR BL x 0) = 1.2

While the Nash equilibrium in mixed strategy is theoretically stable, the results appear undesirable. It’s also fragile in the sense that players must mix with specific probabilities. What’s more, because Luce and Raiffa are mixing in uncoordinated fashions, it’s possible for them to end up at separate venues (hence the lower expected payoffs). This is most problematic when coordination is valuable – much like allies on the battlefield, pondering about their joint military response to a common enemy.

Coordinated mixing

One alternative to uncoordinated mixing, is coordinated mixing. That is, to introduce a common randomisation device and decision rule. Maybe they’ll flip a coin and agree to go to basketball if heads, and to the cinema if tails. Here, the expected payoff to each player is a more favourable 2.5.

Perhaps that’s a fair and efficient resolution, particularly if they found the uncoordinated Nash equilibrium in mixed strategies undesirable and complicated; and if they could not deliberate beforehand as to which Nash equilibrium in pure strategies they should pursue.

Randomization devices is only one of many potential solutions to coordination. In practice, social interactions between Luce and Raiffa are likely ongoing. To preserve their friendship, they may establish other rules, norms and conventions. Maybe they’ll settle on alternations between basketball and cinema?

Do people randomise?

Do we expect players to randomize and solve for Nash equilibrium in mixed strategies in such a way? Yes and no. Yes, in the sense that, like a game of penalty kicks in football, mixing between strategies keeps your opponent on their toes. Predictability in a game of pure conflict implies failure.

But I’d fashion a hesitant no for several reasons. Firstly, people do not like uncertainty. In politics, business and warfare, leaders don’t want to be perceived as acting randomly. Secondly, war generals and chief executives are unlikely to solve simultaneous equations to make their opponent indifferent between their strategies. It’s hard enough as to know what the options are, let alone the payoffs and probabilities.

It’s possible though, through repeated interactions, that mixing leads to a solution that is consistent with a Nash equilibrium in pure or mixed strategies. But even in studies of competitive of rock-paper-scissors, there are large variances in behaviors between players. (Of course, there are many games theoretic extensions to these problems. But they start to get complicated and tedious. The point of this post though is to highlight the viability of mixing strategies in an introductory setting.)

Further reading

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