Assignment 1.∗

by ECON6901†

Problem 1. Suppose there are 100 citizens and three candidates for a mayor: A, BandC,and no citizen is indifferent between any two of them. Each citizen can cast a vote for one ofthe candidate, the candidate who gets the majority wins. In case there is a tie for the firstplace there is a second round. Each citizen prefers his favorite candidate winning to a tieand prefers a tie to some other candidate winning. Show that a citizen’s only weakly domi-nated action is a vote for her least preferred candidate. Find a Nash Equilibrium in whichsome citizen does not vote for her favorite candidate, but the action she takes is not weaklydominated.

Problem 2. (Finding Nash Equilibria using best response functions) Find the NashEqulibria of the two-player strategic game in which each player’s set of actions is the set ofnonnegative numbers and the players’ payoff functions are u1(a1, a2) = a1(a2 − a1) andu2(a1, a2)= a2(1− a1− a2).

Problem 3. (Games with mixed strategy equlibria) Find all the mixed strategy Nash Equi-libria of the strategic game in Figure 1.

L R L R

T 6,0 0,6 T 0,1 0,2

B 3,2 6,0 B 2,2 0,1

Figure 1.

Problem 4. (A coordination game, solve for the case of c < 1) Two people can perform atask if, and only if, they both exert effort. They are both better off if they both exert effortand perform the task than if neither exerts effort (and nothing is accomplished); the worstoutcome for each person is that she exerts effort and the other person does not (in whichcase again nothing is accomplished). Specifically, the players’ preferences are represented bythe expected value of the payoff functions in Figure 2, where c is a positive number less than1 that can be interpreted as the cost of exerting effort. Find all the mixed strategy Nashequilibria of this game. How do the equilibria change as c increases? Explain the reasons forthe changes.

No effort Effort

No effort 0, 0 0, − c

Effort − c, 0 1− c, 1− c

Figure 2.

Problem 5. Consider an all pay auction with two bidders who value the object at 2 and 1respectively. Find the mixed strategy equilibrium in this setting.

∗. This document has been written using the GNU TEXMACS text editor (see www.texmacs.org).

†. Document transcribed by Jorge Rojas.

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Problem 6. (Silverman’s game) Each of two players chooses a positive integer. If player i’sinteger is greater than player j’s integer and less than three times this integer, then player j

pays 1 dollar to player i. If player i’s integer is at least three times player j’s integer, thenplayer i pays 1 dollar to player j. If the integers are equal, no payment is made. Eachplayer’s preferences are represented by her expected monetary payoff. Show that the gamehas no Nash equilibrium in pure strategies and that the pair of mixed strategies in whicheach player chooses 1, 2 and 5 each with probability

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3is a mixed strategy Nash equilibrium.

(In fact, this pair of mixed strategies is the unique mixed strategy Nash equilibrium.) (Youcannot appeal to proposition 116.2 in Osborne because the number of actions of each playeris not finite. However, you can use the argument for the “ if ” part of this result.)

Problem 7. (Rock, Paper, Scissors) Each of two players simultaneously announces eitherRock , or Paper , or Scissors . Paper beats (wraps) Rock , Rock beats (blunts) Scissors , andScissors beats (cuts) Paper . The player who names the winning object receives 1 dollarfrom her opponent; if both players make the same choice, then no payment is made. Eachplayer’s preferences are represented by the expected amount of money she receives. (Anexample of the variant of Hotelling’s model of electoral competition considered in Exercise75.3 has the same payoff structure. Suppose there are three possible positions, A, B, and C,and three citizens, one of whom prefers A to B to C, one of whom prefers B to C to A, andone of whom prefers C to A to B. Two candidates choose different positions, each citizenvotes for the candidate whose position she prefers; if both candidates choose the same posi-tion, they tie for first place.)

a) Formulate this situation as a strategid game and find all its mixed strategy equilibria(give both the equilibrium strategies and the equilibrium payoffs).

b) Find all the mixed strategy equilibria of the modified game in which player 1 is pro-hibited from announcing Scissors .

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