Game theory, only typed answers

Exercise 5.15: Challenging Question. A team of n professional swimmers (n > 2) - from now on called players - are partying on the bank of the Sacramento river on a cold day in January. Suddenly a passerby shouts "Help! My dog fell into the water!". Each of the swimmers has to decide whether or not to jump into the icy cold water to rescue the dog. One rescuer is sufficient: the dog will be saved if at least one player jumps into the water; if nobody does, then the dog will die. Each player prefers somebody else to jump in, but each player prefers to jump in himself if nobody else does. Let us formulate this as a game. The strategy set of each player i = 1,...," is S, = (J,-/}, where J stands for 'jump in' and - for 'not jump in'. The possible basic outcomes can be expressed as subsets of the set / = (1,..., "} of players: outcome Nc / is interpreted as 'the players in the set / jump into the water'; if N=O the dog dies, while if NO the dog is saved. Player i has the following ordinal ranking of the outcomes: (1) N - N', for every N + 0, N'=0 with ig N andie N', (2) N > N' for every N + 0, N' = 0 with i 0. (a) Find all the pure-strategy Nash equilibria. (b) Suppose that each player i has the following von Neumann-Morgenstern payoff function (which is consistent with the above ordinal ranking): if N + 3 and i = N #,(M)= v-c ifNO andie N with 0