A game theoretic mixed strategy involves deliberately randomizing one’s own choice in a strategic situation. In football, this could mean flipping a coin to decide to call a running play or a passing play – the mixed strategy allows randomization to determine which play is actually called.
How can this be useful if you aren’t actually picking a strategy? Jeff at Cheap Talk discusses mixed strategies in the context of icing a kicker at the end of a game. Should an opposing team call a time out prior to a game-tying or game-winning field goal attempt to make the kicker more nervous? First, as Jeff says,
According to this article in the Wall Street Journal, icing the kicker has almost no effect and if anything only backfires. Among all field goal attempts taken since the 2000 season when there were less than 2 minutes remaining, kickers made 77.3% of them when there was no timeout called and 79.7% when the kicker was “iced.”
But then Jeff highlights the potential gain of randomization in strategies:
So much for icing? No! Icing the kicker is a successful strategy because it keeps the kicker guessing as to when he will actually have to prepare himself to perform. The optimal use of the strategy is to randomize the decision whether to call a timeout in order to maximize uncertainty. We’ve all seen kickers, golfers, players of any type of finesse sport mentally and physically prepare themselves for a one-off performance. The mental focus required is a scarce resource. Randomizing the decision to ice the kicker forces the kicker to choose how to ration this resource between two potential moments when he will have to step up.
If you ice with probability zero he knows to focus all his attention when he first takes the field. If you ice with probability 1 he knows to save it all for the timeout. The optimal icing probability leaves him indifferent between allocating the marginal capacity of attention between the two moments and minimizes his overall probability of a successful field goal. (The overall probability is the probability of icing times the success probability conditional on icing plus the probability of not icing times the success probability conditional on icing.)
Essentially, the added uncertainty can cause the kicker to expend more energy than otherwise – since the kicker will not know whether or not a timeout will be called. So, in this case, it is optimal to mix it up!