Considering an ultimatum game (T- 1 bargaining game) where before player 1 makes

Question

Considering an ultimatum game (T- 1 bargaining game) where before player 1 makes his offer to player 2, player 2 can invest in the size of the pie. If player 2 chooses a low level of investment (L) then the size of the pie is small, equal to vi while ent (H) then the size of the pie is large, equal to vH. The cost to player 2 of choosing L is cL, while the cost of choosing H is cH. Assume that if player 2 chooses a high level of investm (a) What is the unique subgame-perfect equilibrium of this game? Is it Pareto Optimal? (b) Can you find a Nash equilibrium of the game that results in an outcome that is better for both players as compared to the unique subgame-perfect equilibrium?

Explanation / Answer

a) we need to understand that a sub game perfect equilibrium is where both players win something. In this case, the second player is the one who choses to accept or reject the offer of the firs player. We will have two tipes of equilibriums, these will depend if the second player choses a high or a low level of investment. The high level of optimal investmet will be equal to   if there is no other set that brings a higher return. In a simiar way, the optimal low level investmet strategy for both players will be , if again there is no other level of endowment that maximize the return of both players.

b) There is another Nash equilibrium for this game, this equilibrium will be where the second player rejects every offer from player and both players do not receive any return; if this is the case, none of them could get more by changing their strategy in an unilaterally way. This will be an equilibrium, but not a sub-game perfect equilibrium, therefore, a Nash equilibrium is the best possible outcome for the players in this type of game.

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