## Friday, April 29, 2011

### Mixed Strategies Highlights

Choosing the Right Strategy?

In our section on mixed strategies, we studied games that had no set of deterministic best responses. For instance, in soccer penalty kicks, the best response to a keeper who defends left is to kick right, but then the keeper's best response is to defend to the right, which leads to kicking to the left as being optimal and we end up going in a circle. The solution to the "circle" problem is to study mixed strategy equilibria. These strategies can be understood literally as using randomization schemes to determine which strategy to play.

An aside: The can also be understood as representing population measures of play. For instance, suppose that the keeper's strategy is to defend left when he wakes up on the left side of the bed and to defend right otherwise. If half of the keeper's wake up on the left and half wake up on the right, then it is as if each is randomizing his strategy even though all are playing pure strategies.

The goal of randomization is not pure payoff maximization. Indeed, in equilibrium, all strategies yield the same (expected) payoff. Rather it is to keep the rival off balance and thereby avoid being exploited. In strategy terms, one can view mixed strategies as providing a rationale for "strategic ambiguity"---remaining intentionally vague about one's strategy. In terms of inward/outward thinking, mixed strategies represent the culmination of outward thinking since the mixture is done purely for the sake of the rival.

Mixed strategies can sometimes lead to surprising insights as we saw in the Volunteer's Dilemma. In this situation, a group of n people can decide to help or not. Helping provides a public benefit B but requires a private cost c. If a person knew that she was pivotal in providing effort (i.e. no effort would be undertaken without her), then since B > c, she would volunteer. In that sense, there is no apparent free rider problem. But what if there is uncertainty about whether you are pivotal? Here, we solved the game using mixed strategies. Not surprisingly, volunteerism declines as n increases. But more strikingly, the chance that anyone helps also declines with n. That is, the phenomenon that social psychologists refer to as "diffusion of responsibility" can be understood as a rational response to the volunteer's dilemma game using a mixed strategy equilibrium.