(Swimming with sharks) You and a friend are spending two days at the beach and would like to go…

(Swimming with sharks) You and a friend are spending two days at the beach and would like to go for a swim. Each of you believes that with probability π the water is infested with sharks. If sharks are present, anyone who goes swimming today will surely be attacked. You each have preferences represented by the expected value of a payoff function that assigns −c to being attacked by a shark, 0 to sitting on the beach, and 1 to a day’s worth of undisturbed swimming. If one of you is attacked by sharks on the first day then you both deduce that a swimmer will surely be attacked the next day, and hence do not go swimming the next day. If no one is attacked on the first day then you both retain the belief that the probability of the water’s being infested is π, and hence swim on the second day only if −πc + 1 − π ≥ 0. Model this situation as a strategic game in which you and your friend each decides whether to go swimming on your first day at the beach. If, for example, you go swimming on the first day, you (and your friend, if she goes swimming) are attacked with probability π, in which case you stay out of the water on the second day; you (and your friend, if she goes swimming) swim undisturbed with probability 1 − π, in which case you swim on the second day. Thus your expected payoff if you swim on the first day is π(−c + 0)+(1 − π)(1 + 1) = −πc + 2(1 − π), independent of your friend’s action. Find the mixed strategy Nash equilibria of the game (depending on c and π). Does the existence of a friend make it more or less likely that you decide to go swimming on the first day? (Penguins diving into water where seals may lurk are sometimes said to face the same dilemma, though Court (1996) argues that they do not.)