Question 1: Fighting for survival
Two animals are ghting over a prey. The prey is worth v to each animal. The cost of fighting is c1 for the first animal (player 1) and c2 for the second animal (player 2). If they both act aggressively (hawkish) and get into a fi ght, they split the prey in two equal parts but suff er 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) Write down the normal form of the game (the bimatrix of strategies and payo ffs).
B) Find out the Nash Equilibria for all possible parameter con gurations and given the following restrictions: v > 0, c1 > c2 > 0, v≠ 2c1 and v ≠ 2c2.
Question 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 s1 and s2. Furthermore, players' choices have to be in increments of 25 cents, that is, s1 ≡ {0, 0.25, 0.50, 0.75, 1.00} and s2 ≡ {0, 0.25, 0.50, 0.75, 1.00}. If s1 + s2 > 1, then player 1 gets s1 and player 2 gets s2. If s1 + s2 > 1, then both players get 0.
A) Write down the normal form of the game (the bimatrix of strategies and payoffs).
B) Find the Nash Equilibria of this game.
Question 3: Tragedy of Commons
Two individuals use a common resource (a river or a forest, for example) to produce output. The more the resource is used, the less output any given individual can produce. Denote by xi the amount of the resource used by individual i (where i = 1, 2). Assume speci cally that individual i's output is xi(1 - (x1 + x2)) if x1 + x2 ≤ 1 and zero otherwise. Each individual i chooses xi ≡ [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 x1, x2 maximize the total output? (Is there any other output pro le 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 xi(1 - x1 + x2 + .... +xn)) if x1 + x2 + .... + xn ≤ 1 and zero otherwise. Find the Nash equilibria of this game.
Question 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 payo s 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 .
Question 5: Let's Work Together
Two people are engaged in a joint project. If each person i puts in the e ffort xi, a non-negative number equal to at most 1, which costs her c(xi), the outcome of the project is worth f(x1, x2). The worth of the project is split equally between the two people, regardless of their eff ort levels.
a) Formulate this situation as a strategic game.
b) Find its Nash equilibria when
i) f(x1, x2) = 3x1x2, c(xi) = x2i, for i = 1, 2.
ii) f(x1, x2) = 4x1x2, c(xi) = xi, for i = 1, 2.
c) In each case, is there a pair of eff ort levels that yields both players higher payoff s than the Nash equilibrium eff ort levels?