This is the cynical, total campaign spending buys the election model from class. The election is...

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This is the cynical, "total campaign spending buys the election" model from class. The election is between party A and party B. The players are two campaign donors who favor party A (call them Donors A₁ and A₂ and one donor who favors party B (Donor B). They choose dollar amounts to donate to their respective parties: a₁, a₂, b≥ 0. As shorthand, write a = a₁ + a₂ for the total amount given by the two party A donors. The probability of each party winning the election is a function of money donated: Pr(A wins) =and Pr(B wins) = a+b The country has natural resources V = 100. To be even more cynical, this "prize" gets split equally among the winning party's donors. The cost of a donation is just its dollar amount, so the players' payoffs are: ₁ (a₁, a₂, b) = Pr(A wins) - α₁, ₂(a₁, a₂, b) = . Pr(A wins) - a₂ ug (a₁, a₂, b) = V Pr(B wins) - b a. Solve for a Nash equilibrium in which Donors A₁ and A₂ give equal amounts. In doing so, you should demonstrate the claim from class that the party with fewer donors wins more often: Pr(B wins) = 3. Use economic logic to explain this result. b. Extend the analysis to a situation where there are n donors for party A. Solve for a Nash equilibrium where these n donors give equally: a₁ = a₂ = = a₁. The main purpose of this is to determine the probability of party A winning, show how that probability depends on n, and give some intuition. This is the cynical, "total campaign spending buys the election" model from class. The election is between party A and party B. The players are two campaign donors who favor party A (call them Donors A₁ and A₂ and one donor who favors party B (Donor B). They choose dollar amounts to donate to their respective parties: a₁, a₂, b≥ 0. As shorthand, write a = a₁ + a₂ for the total amount given by the two party A donors. The probability of each party winning the election is a function of money donated: Pr(A wins) =and Pr(B wins) = a+b The country has natural resources V = 100. To be even more cynical, this "prize" gets split equally among the winning party's donors. The cost of a donation is just its dollar amount, so the players' payoffs are: ₁ (a₁, a₂, b) = Pr(A wins) - α₁, ₂(a₁, a₂, b) = . Pr(A wins) - a₂ ug (a₁, a₂, b) = V Pr(B wins) - b a. Solve for a Nash equilibrium in which Donors A₁ and A₂ give equal amounts. In doing so, you should demonstrate the claim from class that the party with fewer donors wins more often: Pr(B wins) = 3. Use economic logic to explain this result. b. Extend the analysis to a situation where there are n donors for party A. Solve for a Nash equilibrium where these n donors give equally: a₁ = a₂ = = a₁. The main purpose of this is to determine the probability of party A winning, show how that probability depends on n, and give some intuition.

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