Game Theory - Exercises

1. Eliminating Dominated Strategies

Test your knowledge on the following questions.

a) Solve the normal form game in of Fig. 1. by eliminating dominated strategies. Verify that theresulting solution is a Nash equilibrium of the game.

N C JN 73,25 57,42 66,32C 80,26 35,12 32,54J 28,27 63,31 54,29

Figure 1 Eliminating dominated strategies in a 3 x 3 normal form game.

b) Can a Nash equilibrium to the game in Fig. 2 be found by the iterated elimination of dominatedstrategies? If so, describe exactly in what order you delete strategies.

a b c d eA 63, -1 28, -1 -2, 0 -2, 45 -3, 19B 32, 1 2, 2 2, 5 33, 0 2, 3C 54, 1 95, -1 0, 2 4, -1 0, 4D 1, -33 -3, 43 -1, 39 1, -12 -1, 17E -22, 0 1, -13 -1, 88 -2, -57 -3, 72

Figure 2 Eliminating dominated strategies in a 5 x 5 normal form game.

2. Second- Price Auction

A single object is to be sold at auction. There are n > 1 bidders, each submitting a single bid,in secret, to the seller. The value of the object to bidder i is vi. The winner of the object is thehighest bidder, but i pays only the next highest bid.

a) Show that ”truth-telling” (i.e., each player i bids vi) is a dominant strategy for each player. Tosimplify the argument, you can assume, where convenient, that there are no ties.

b) Does your analysis depend on whether or not others tell the truth?

c) What are some reasons that real second-price auctions might not conform to the assumptionsof this model?

3. An Increasing-Bid Auction

A Ming vase is to be sold at auction. There are n bidders, and the value of the vase to bidderi is vi > 0, i = 1, ...., n. The auctioneer begins the bidding at zero, and raises the price at the rateof $ 1 per second. All bidders who are willing to buy the vase at the stated price put their handsup simultaneously.This last and highest bidder is the winner of the auction and must buy the vasefor the stated price. Show that this auction has the same optimal strategies and the same outcomeas the Second-Price Auction.

HINT: First show that the only undominated pure strategies for bidder i take the form ofchoosing a price pi and keeping a hand up until the auctioneer’s price goes higher than this. Thenfind the optimal pi.

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4. Eliminating Dominated Strategies adAbsurdum

Consider an n-player game in which each player announces simultaneously an integer between1 and 1000. The winner is the player whose announcement is closed to 1/2 of the average of allthe announcements. In the case of a tie, the price is given by a random draw among the winners.

a) Show that if players iterate the elimination of dominated strategies, the only remaining strategyis to announce 1.

b) To most people, this does not sound like what would actually happen if n people play this game,probably because people do, at most, a few iterations of dominated strategies. How many levelsof iteration do you think people actually engage in?

c) Extra credit: Stage the game with some friends, with the payoff such that the winner receives$10.00. Can you estimate how many stage of elimination dominated strategies people will gothrough?

5. A Pure Coordination Game

Three people independently choose an integer between two and nine. If the three choices arethe same, each person receives the amount chosen. Otherwise each person loses the amount theperson chose.

a) What are the pure strategy Nash equilibria of this game?

b) How do you think people will actually play this game?

c) What doses the game look like if you allow communication among the players before they maketheir choices? How would you model such communication, and how do you think communicationwould change the behavior of the players?

6.Variations on Duopoly

In a certain market there are two firms, which we label a and b. If the firms produce outputqa and qb, then the price they will receive for their goods is given by p = α − β(qa + qb) or zeroif p would otherwise be negative. Each firm has marginal cost c > 0 and no fixed costs. Supposeα > 3c (youll see why we make this assumption as you go through the problem.)

a) Suppose the firm choose the quantity qa and qb independently in each period, without regardto their behavior in previous periods, and each maximizes profits Πa = (p − c)qa and Πb =(p − c)qb.Find the unique pure strategy Nash equilibrium to this game. This is called theCournotDuopoly model.

b) Suppose the two firms collude by agreeing that each will produce an amount q∗ = qa = qb,and they have some way to enforce the agreement. What should they choose for q∗? Whatare the profits of the two firms? Compare this with the Cournot duopoly profits. This Nashequilibrium is called the Monopolymodel or the Cartelmodel.

c) Suppose firm a reneges on its promise on the previous part but b doses not. What should a

choose for qa? What are a’s profits, and what are b’s profits?

d) Suppose firm b finds out that firm a is going to do in the previous part, and chooses qb tomaximize profits, given what firm a is going to do. What is qb now? What do you thinkhappens if they go back and forth this way forever?

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e) Suppose the two firms choose price as opposed to quantity, where costumers all go to the lowest-price firm, and the firm split the market if they choose the same price. What is the unique Nashequilibrium of this game? This is called the Bertrandduodolymodel. Bertand’s result is oftencalled a ”paradox”. In what sense do you think this is an accurate description of this solution?How do you explain the dramatic difference in outcomes between this and the Cournot model?

f) Suppose firm a gets to choose its output first and only afterwards doses firm b get to choose itsoutput. Find the equilibrium choices of qa and qb in this case, and compare the profits of thetwo firms with the Cournot duopoly case. This is called the Stackelbergduopolymodel. HINT:Firm a should find its profit for every qa, given that firm b will choose qb to maximize its profitsgiven qa. Then, among all these profits, firm a choose qa to maximize profits.

7. The Rotten Kid Theorem

This problem is core of Gary Becker’s (1981) famous theory of the family. You might check theoriginal, through, since I’m not sure I got the genders right.1

A certain family consists of a mother and a son, with increasing, concave utility funktions u(y)for the mother and v(z) for the sun. The sun can affect both his income and of the mother bychoosing a level of familial work commitment a, so y = y(a) and z = z(a). The mother, however,feels a degree of altruism α > 0 towards the son, so given y and z, she transfers an amount t tothe son to maximize the objective function.

u(y − t) + αv(z + t) (1)

The son, however is perfectly selfish (”rotten”), and chooses the level of a to maximize his ownutility v(z(a) + t). However he knows that his mother’s transfer depends on y and z, and henceon a.

a) Use backward induction to show, what the son chooses a to maximize total family incomey(a) + z(a).

b) Show that t is an increasing function of α. Also if we write y = y(a) + y, and t is an increasingfunction of the mother’s exogenous wealth y.

c) Show that for sufficiently small α > 0, t < 0; i.e., the transfer is from the son to the mother.

HINT: For each a, the mother observes y(a) and z(a), and chooses t = t(a) to maximize (1).This gives the first-order condition −u′+αv′ = 0.Differentiate the first-order condition with respectto a to show that the solution to the first-order condition for the maximization of z(a) + t(a) isthe same as for the maximization of y(a) + z(a) = 0.

1 The Rotten Kid Theorem has been empirically tested and receives some support, throughthe evidence falls short (often way short) of confirming that cross-generational families maximizejoint income. See Cox 1987, 1990; Cox and Rank 1992; and Altonji et al. 1992, 1997.

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