University of Hong Kong

Transcription

1 University of Hong Kong ECON6036 Game Theory and Applications Problem Set I 1 Nash equilibrium, pure and mixed equilibrium 1. This exercise asks you to work through the characterization of all the Nash equilibria of general two-player strategic games in which each player has two actions (i.e. 2x2 matrix games). This process is time consuming but straightforward and is recommended to the student who is unfamiliar with the mechanics of determining Nash equilibria. Let the game be as illustrated in figure 1. The pure-strategy Nash equilibria are easily found by testing each cell of the matrix; e.g. (U, L) is a Nash equilibrium if and only if a b and b d. To determine the mixedstrategy equilibria requires more work. Let x be the probability player 1playsU and let y be the probability player 2 plays L. Weprovidean outline, which the student should complete: (a) Compute each player s best response function as a function of his opponent s randomizing probability. (b) For which parameters is player i indifferent between his two strategies regardless of the play of his opponent? (c) For which parameters does player i have a strictly dominant action? (In a 2x2 game, an action for player i is his strictly dominant action if that action gives him a strictly greater payoff than the other action, regardless of the action of the other player.) (d) Show that if neither player has a strictly dominant action and the game has a unique equilibrium, then the equilibrium must be in mixed strategies. (e) Consider the particular example illustrated in figure 2. (f) Derive the best-response functions graphically by plotting player i s payoff to his two pure strategies as a function of his opponent s mixed strategy. 1

2 (g) Plot the two reaction functions in the (x, y) space. What are the Nash equilibria? Figure 1 L R U a,b c,d D e,f g,h Figure 2 L R U 1,-1 3,0 D 4,2 0,-1 2. O&R (A war of attrition) (focus on pure strategy Nash equilibrium only) 3. (Hotelling s model) In O&R exercise 19.1, but replace "Nash equilibrium" by "pure strategy Nash equilibrium". 4. Consider a variant of Hotelling s model (a la exercise 19.1 in O&R) that captures features of a U.S. presidential election. Voters are divided between two states. State 1 has more electoral college votes than does state 2. The winner is the candidate who obtains the most electoral college votes. Denote by m i the median favorite position among the citizens of state i, fori =1, 2; assume that m 2 <m 1.Eachoftwocandidates chooses a single position. Each citizen votes (nonstrategically) for the candidate whose position is closest to her favorite position. The candidate who wins a majority of the votes in a state obtains all the electoral college votes of that state; if for some state the candidates obtain the same number of votes, they each obtain half of the electoral college votes of that state. Find the Nash equilibrium (equilibria?) of the strategic game that models this situation. (Only focus on pure strategy equilibrium, please.) 5. (Guess the average) O&R exercise General A is defending territory accessible by two mountain passes against an attack by General B. General A has three divisions at her disposal, and general B has two divisions. Each general allocates her 2

3 divisions between the two passes. General A wins the battle at a pass if and only if she assigns at least as many divisions to the pass as does General B; she successfully defends her territory if and only if she wins the battle at both passes. Formulate this situation as a strategic game and find all its mixed strategy equilibria. In an equilibrium, do the generals concentrate all their forces at one pass, or spread them out? 7. A new political party, A, is challenging an established party, B. The race involves three localities of different sizes. Party A can wage a strong campaign in only one locality; B must commit resources to defend its position in one of the localities at the same time Party A is deciding which locality to target. If A targets district i and B devotes its resources to some other district, then A gains a i votes at the expense of B; leta 1 >a 2 >a 3 > 0. IfB devote resources to the district that A targets, then A gains no votes. Each party s preferences are represented by the expected number of votes it gains. (Perhaps seats in a legislature are allocated proportionally to vote shares.) Formulate this situation as a strategic game and find its mixed strategy equilibria. 8. (Bertrand competition) There are two firms producing an identical product with same constant marginal cost c<1without any fixed cost. Suppose the quantity demanded for firm i s product is 1 P i when his price P i < P j,andis(1 P i )/2 when Pi = Pj,andis zero if P i > P j. Suppose the two firms compete by choosing prices simultaneously. What is the pure strategy Nash equilibrium? 9. A crime is observed by a group of n people. Each person would like the police to be informed but prefers that someone else makes the phone call. Specifically, suppose that each person attaches the value v to the police being informed and bears the cost c if she makes the phone call, where v>c>0. Each person s preferences are risk neutral expected payoff maximizer. Find all the pure strategy Nash equilibria. Find a symmetric mixed strategy Nash equilibrium. In this later equilibrium, what is the probability that at least one person calls the police? Does this probability go up or go down with n? 10. (Mixed strategy equilibrium in Bertrand competition) Two risk-neutral firms produce identical products at zero marginal cost. Market demand is unity for any finite price, and the firm that charges the lowest price 3

4 getstheentiremarket. Intheeventofatie,eachfirm has an equal chance of servicing the entire market. Thus, the expected profits of firm i if it charges any price p i [0, ) when the rival charges p j [0, ) is given by π i (p 1,p 2 )= p i if p i <p j < 1 2 p i if p i = p j < 0 otherwise. These payoff functions arise in the Hotelling model when the firms are located atthesamepointinproducespace. (Bytheusualreasoningp 1 = p 2 =0 is the unique pure strategy Nash equilibrium.) Show that for every k (0, ) there also exists a symmetric mixed-strategy Nash equilibrium that is atomless on [k, ) in which each firm earns expected profits of k and prices according to the distribution function F (p) = ½ 0 if p k 1 k p if p>k over the support [k, ). 2 Bayesian game 1. Two people are involved in a dispute. Person 1 does not know whether person 2 is strong or weak; she assigns probability α to person 2 s being strong. Person 2 is fully informed. Each person can either fight or yield. Each person get the payoff of 0 if she yields (regardless of the other person s action) and a payoff of 1 if she fights and her opponent yields; if both people fight, then their payoffs are (-1,1) if person 2 is strong and (1,-1) if person 2 is weak. Formulate this situation as a Bayesian game and find its Nash equilibria if α<1/2 and if α>1/2. 2. Whether candidate 1 or candidate 2 is elected depends on the votes of two citizens. The economy may be in one of two states, A and B. The citizens agree that candidate 1 is best if the state is A and candidate 2 is best if the state is B. Each citizen gets a payoff of 1 ifthebestcandidateforthestatewins(obtainsmorevotesthanthe other candidate), a payoff of 0 if the other candidate wins, and payoff of 4

5 1/2 if the candidates tie; both citizens are expected value maximizers. Citizen 1 is informed of the state, whereas citizen 2 believes it is A with probability 0.9 and B with probability 0.1. Each citizen may either vote for candidate 1, vote for candidate 2, or not vote. (a) Formulate this situation as a Bayesian game. (Construct the table of payoffs for each state.) (b) Show that the game has exactly two pure Nash equilibria, in one of which citizen 2 does not vote and in the other of which she votes for 1. (c) Show that an action of one of the players in the second equilibrium is weakly dominated. (d) Why is swing voter s curse an appropriate name for the determinant of citizen 2 s decision in the first equilibrium? 3. (double auction) A single seller and a single buyer may trade 0 or 1 unit of a good. The seller (player s) has cost v s, and the buyer (player b) has valuation v b,wherev s and v b both are private information and belong to the interval [0, 1]. The seller and the buyer simultaneously and independently choose bids b s,b b [0, 1]. If b s b b, the two parties tradeatpricet =(b s + b b )/2. If b s >b b, the parties do nto trade the good and do not transfer money. The seller s utility is thus u s = (b s + b v )/2 c if b s b v, and 0 if b s > b v ; the buyer s utility is u b = v (b s + b b )/2 if b s b b, and 0 if b s >b b. Suppose each player s strategy is a linear function of his type and is strictly increasing and continuously differentiable. Suppose also that v s and v b are uniformly distributed in [0,1]. Solve for the equilibrium. 4. (Stag hunt) Consider N villagers i =1,..., N, each of whom is privately informed of the cost he must incur if he goes hunting with the pack. This cost, denoted c i, is a priori uniformly distributed on [0, 1+ε],where ε is some positive number, and the c i are independently distributed across villagers. If all hunt together, they will catch a stag, which yields a value 1 to each of them. On the other hand, if only one villager decides to stay at home, the other will not be able to catch the stag. Show that in the unique equilibrium no one will hunt. 5

6 3 Purification and Correlated Equilibirum (difficult) 1. O&R: Exercise 42.1 (purification) 2. O&R: Exercise 48.1 (correlated equilibrium) 4 Rationalizability 1. Find the set of rationalizable actions of each player in the game in figure below. L C R T 2,1 1,4 0,3 B 1,8 0,2 1,3 2. Use iternated elimination of strictly (or weakly) dominated actions to find a Nash equilibrium. Verify the outcome is indeed a Nash equilibrium. N C J N 73, 25 57, 42 66, 32 C 80, 26 35, 12 32, 54 J 28, 27 63, 31 54, CanaNashequilibriumtothegamebefoundbyiteratedelimination of strictly dominated strategies? If so, describe exactly in what order you delete actions. A B C D E A 63, -1 28, -1-2, 0-2, 45-3, 19 B 32, 1 2, 2 2, 5 33, 0 2, 3 C 54, 1 95, -1 0,2 4, -1 0, 4 D 1, -33-3, 43-1, 39 1, -12-1, 17 E -22, 0 1, -13-1, 88-2, -57-3, A fighter command has four strategies, and its opponent bomber command has three counterstrategies. The diagram below shows the probability that the fighter destroys the bomber. The fighter s objective is to maximize the probability of destroying the bomber, while the bomber s 6

7 objective is to minimize it. Use the iterated elimination of strictly dominated actions to simplify the game. Does your final outcome depend on the way you eliminate the actions? Bomber command Full fire partial fire no fire guns Fighter rockets command toss-bombs ramming Consider a variant of Hotelling s model of electoral competition (a la 19.1 in O&R) in which there are two candidates and the set of possible positions is {0, 1, 2,..., l}, wherel is even. Assum that there is only one position m with the property that exactly half of the voters favorite positions are at most m, and half of the voters favorite positions are at least m. Show that this position is the unique rationalizable action of each player. 6. For exercise 35.1 (guess the average) in O&R, show that only annoucing 1 is rationalizable for each player. 7