problem 1: Fighting for survival
Two animals are ghting 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 out the Nash Equilibria for all possible parameter congurations and given the following restrictions: v > 0, c_{1} > c_{2} > 0, v≠ 2c_{1} and v ≠ 2c_{2}.
problem 2: Split the dollar
Players 1 and 2 are bargaining over how to split one dollar. Both players simultaneously name shares they would like to keep s_{1} and s_{2}. Furthermore, players' choices have to be in increments of 25 cents, that is, s_{1} ≡ {0, 0.25, 0.50, 0.75, 1.00} and s_{2} ≡ {0, 0.25, 0.50, 0.75, 1.00}. If s_{1} + s_{2} > 1, then player 1 gets s_{1} and player 2 gets s_{2}. If s_{1} + s_{2} > 1, then both players get 0.
A) prepare down the normal form of the game (the bimatrix of strategies and payoffs).
B) Find the Nash Equilibria of this game.
problem 3: Tragedy of Commons
Two individuals use a common resource (a river or a forest, for ex) to produce output. The more the resource is used, the less output any given individual can produce. Denote by x_{i} the amount of the resource used by individual i (where i = 1, 2). Assume specically that individual i's output is x_{i}(1 - (x_{1} + x_{2})) if x_{1} + x_{2} ≤ 1 and zero otherwise. Each individual i chooses x_{i} ≡ [0, 1] to maximize her output.
a) Formulate this situation as a strategic game.
b) Find the best response correspondences of the players.
c) Find its Nash equilibria.
d) Does the Nash equilibrium value of x_{1}, x_{2} maximize the total output? (Is there any other output prole that results in a higher total output than the Nash equilibrium?)
e) Suppose now there are n individuals and hence the payo function of individual i (where i = 1, 2, ....., n) is given by x_{i}(1 - x_{1} + x_{2} + .... +x_{n})) if x_{1} + x_{2} + .... + x_{n} ≤ 1 and zero otherwise. Find the Nash equilibria of this game.
problem 4: Fight!
Two people are involved in a dispute. Person 1 does not know whether person 2 is strong or weak; she assigns probability to person 2 being strong. Person 2 is fully informed. Each person can either ght or yield. Each person obtains a payo of 0 if she yields (regardless of the other persons action) and a payo of 1 if she ghts and her opponent yields. If both people ght then their payos are (-1, 1) if person 2 is strong and (1,-1) if person 2 is weak. Formulate the situation as a Bayesian game and nd its Bayesian equilibria if α < 1/2 and if α > 1/2 .
problem 5: Let's Work Together
Two people are engaged in a joint project. If each person i puts in the effort x_{i}, a non-negative number equal to at most 1, which costs her c(x_{i}), the outcome of the project is worth f(x_{1}, x_{2}). The worth of the project is split equally between the two people, regardless of their effort levels.
a) Formulate this situation as a strategic game.
b) Find its Nash equilibria when
i) f(x_{1}, x_{2}) = 3x_{1}x_{2}, c(x_{i}) = x^{2}_{i}, for i = 1, 2.
ii) f(x1, x2) = 4x_{1}x_{2}, c(x_{i}) = x_{i}, for i = 1, 2.
c) In each case, is there a pair of effort levels that yields both players higher payoffs than the Nash equilibrium effort levels?