problem 1: Fighting for survival
Two animals are fighting over a prey. The prey is worth v to each animal. The cost of fighting is c_{1} for the first animal (player 1) and c_{2} for the second animal (player 2). If they both act aggressively (hawkish) and get into a fight, they split the prey in two equal parts but suffer the cost of fighting. If both act peacefully (dovish) then they also split the prey in two equal parts but without incurring any cost. If one acts dovish and the other hawkish, there is no fight and the hawkish gets the prey.
A) prepare down the normal form of the game (the bimatrix of strategies and payoffs).
B) Find the Nash Equilibria for all possible parameter configurations and given the following restrictions:
v> 0, c_{1} > c_{2} > 0, v = 2c_{1} and v = 2c_{2}.
problem 2: Choosing courses
Pat and Ben have to decide which class to enroll in. There are two options: Wine Tasting and Theoretical Physics. Each individual chooses one class (it can be the same or different). The utility for each person is 4 for attending wine tasting and 2 for attending physics. Furthermore, Pat enjoys Ben’s company and gets an extra utility of x > 0 if they are both enrolled in the same class. By contrast, Ben dislikes Pat’s company and gets an extra disutility of −3 if they are both enrolled in the same class. Suppose that they make their choice simultaneously.
A) prepare down the bimatrix of players actions and payoffs.
B) For each x > 0, find all the Pure strategy and Mixed strategy Nash equilibrium (equilibria?) of the game.
C) Interpret.
problem 3: Bayesian Cournot
Consider the Cournot duopoly model in which two firms, 1 and 2, simultaneously choose the quantities they will sell in the market, q_{1} and q_{2}. The price each receives for each unity given these quantities is P(q_{1}, q_{2})= a − b(q_{1} + q_{2}). Suppose that each firm has probability µ of having unit costs of c_{L} and (1 − µ) of having unit costs of c_{H}, where c_{H} > c_{L}. Solve for the Bayesian Nash equilibrium.