Rationalizable actions in a public good game:-

(More difficult, but also more interesting.) Show the following results for the variant of the game in Exercise in which contributions are restricted to be nonnegative.

a. Any contribution of more than wi/2 is strictly dominated for player i.

b. If n = 3 and w1 = w2 = w3 = w then every contribution of at most w/2 is rationalizable. [Show that every such contribution is a best response to a belief that assigns probability one to each of the other players' contributing some amount at most equal to w/2.]

c. If n = 3 and w1 = w2 3 then the unique rationalizable contribution of players 1 and 2 is 0 and the unique rationalizable contribution of player 3 is w3. [Eliminate strictly dominated actions iteratively. After eliminating a contribution of more than wi/2 for each player i (by part a), you can eliminate small contributions by player 3; subsequently you can eliminate any positive contribution by players 1 and 2.]