What is the normal-form representation of this game -nbsp

Question

Negative Ad Campaigns Revisited:

In which each one of two political parties can choose to buy time on commercial radio shows to broadcast negative ad campaigns against its rival. These choices are made simultaneously. Government regulations forbid a party from buying more than 2 hours of negative campaign time, so that each party cannot choose an amount of negative campaigning above 2 hours. Given a pair of choices (a₁, a₂), the payoff of party i is given by the following function: v_i(a₁, a₂) = a_i - 2a_j + a_ia_j - (a_i)².

a. Find the unique pure-strategy Nash equilibrium of the one-shot game.

b. If the parties could sign a binding agreement on how much to campaign, what levels would they choose?

c. Now assume that this game is repeated infinitely often, and the foregoing demonstrates the choices and payoffs per period. For whichdiscount factors δ ∈ (0, 1) can the levels you found in (b) be supported as a subgame-perfect equilibrium of the infinitely repeated game?

d. Despite the parties' ability to coordinate, as you have demonstrated in your answer to (c), the government is concerned about the parties' ability to run up to 2 hours a day of negative campaigning, and it is considering limiting negative campaigning to 1/2 hour a day, so that now a_i ∈ [0, 1/2 ]. Is this a good policy to limit negative campaigns further? Justify your answer with the relevant calculations. What is the intuition for your conclusion?

Exercise

Negative Ad Campaigns: Each one of two political parties can choose to buy time on commercial radio shows to broadcast negative ad campaigns against its rival. These choices are made simultaneously. Government regulations forbid a party from buying more than 2 hours of negative campaign time, so that each party cannot choose an amount of negative campaigning above 2 hours. Given a pair of choices (a₁, a₂), the payoff of party i is given by the following function: v_i(a₁, a₂) = a_i - 2a_j + a_ia_j - (a_i)².

a. What is the normal-form representation of this game?

b. What is the best-response function for each party?

c. What is the pure-strategy Nash equilibrium? Is it unique?

d. If the parties could sign a binding agreement on how much to campaign, what levels would they choose?