# solution

Consider the following game (sometimes called the Nash Bargaining game). Two players with linear utility for money have to share \$10. Each player makes a bid b1 and b2, which can be any number between 0 and 10. If b1 + b2 = 10, then each player receives their bid, otherwise they each receive zero. These bids are made simultaneously.

1. Assuming standard preferences, show that a pair of strategies {b1,b2} is a Nash Equilibrium if b1 + b2 = 10. Explain if there exists further Nash equilibrium of this game. (Remember, a Nash Equilibrium is a pair of strategies {b1,b2} such that b1 is the best that player 1 can do, given b2, and b2 is the best that player 2 can do given b1)

2. Assume that player 1 has standard preferences, and player 2 has inequality averse preferences with a > 0. Show that there is a threshold for B such that, if b2 < B, then {b1, b2} such that b1 + b2 = 10 is not a Nash Equilibrium. Calculate B as a function of a.

3. For the case b2 > B, explain if the strategy {b1, b2} with b1 +b2 = 10 can be a Nash Equilibrium of the game. Reconsider your explanation for the case with ÃŸ > 0.5?

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