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 ß > 0.5?