FIG. 3.

(A) Umbrella game for one person, with 60% chance of rain and 40% chance of sunshine. The best strategy is to always carry an umbrella; all stochastic strategies result in more mistakes. (B) Umbrella game for two people with identical strategies. As there is a collective reward for carrying an umbrella, if the chance of rain is significant, the ideal situation is that in which one person brings an umbrella, leaving the other person free from carrying one. This is possible only with the use of a stochastic strategy, because both individuals are using the same strategy. If the chance of rain is below a certain threshold (p < 1/3), however, then the deterministic strategy of never carrying an umbrella is best.