EconS 44 Strategy and Game Theory Homework #5 Answer Key Exercise #1 Collusion among N doctors Consider an infinitely repeated game, in which there are nn 3 doctors, who have created a partnership. In each period, each doctor decides how hard to work. Let ee tt ii be the effort chosen by doctor i in period t, and ee tt ii = 1,,, 10. Doctor i s discount factor is δδ ii. Total profit for the partnership: (ee tt 1 + ee tt + ee tt 3 + + ee tt nn ) A doctor i s payoff: 1 nn (ee 1 tt + ee tt + ee 3 tt + + ee nn tt ) ee ii tt a. Assume that the history of the game is common knowledge. Derive a subgame perfect NE in which each player chooses effort ee > 1. To begin, note that doctor s payoff can be rearranged to: nn (ee 1 + ee + + ee ii 1 + ee ii+1 + + ee nn ) nn nn ee ii Since a doctor s payoff is strictly decreasing in her own effort, she wants to minimize it. ee ii = 1 is then a strictly dominant strategy for doctor i and therefore there is a unique stage game Nash equilibrium in which each doctor chooses the minimal effort level of 1. Next, note that each doctor s payoff from choosing a common effort level of e is: 1 nn (ee + ee + + ee) ee = 1 nnnn ee = ee nn To determine a doctor s best deviation, we must take a partial derivative with respect to ee ii of their payoff function when all other (n-1) players select e, yielding uu ii = nn ee ii nn This suggests a corner solution where doctor i wants to minimize effort by playing the lowest e possible, i.e., e i=1. We can now describe a grim-trigger strategy. When conditions are met and the strategy is played symmetrically, that will guarantee cooperation at an effort level e>1. 1
Consider the symmetric grim-trigger strategy: o o In period 1: choose ee 1 ii = ee In period t : choose ee tt ii = ee when ee ττ jj = ee for all j, for all ττ tt 1; and choose 1 otherwise. This is a subgame perfect Nash equilibrium if and only if ee 1 δδ ii nn 1 nn ee nn nn 1 + δδ ii 1 δδ ii ffffff aaaaaa ii. That is to say, the equilibrium will only hold so long as the payoff from remaining in the equilibrium is greater than or equal to the one period payoff from deviating plus the payoff from the punishment equilibrium played every period thereafter. Solving for δδ ii yields: δδ ii nn (nn 1) The following figure depicts this cutoff of δδ ii, shading the region of discount factors above δδ ii which would support collusion. It is now possible to see how the equilibrium responds to changes in n. Differentiating the about cutoff of δδ ii with respect to n, we obtain
δδ ii = 1 (nn 1) This partial is positive, indicating that as the group size n increases, δδ ii has to increase to maintain the cooperative equilibrium. So it is more difficult to support cooperation as the group size increases. b. Assume that the history of the game is not common knowledge, i.e., in each period, only the total effort is observed. Find a subgame perfect NE in which each player chooses effort ee > 1. Consider the strategy profile in part (a), except that it now conditions on total effort. Let ee tt denote total effort for period t. In period 1: choose ee 1 ii = ee In period t : choose ee tt ii = ee when ee ττ = nnee for all j, for all ττ tt 1; and choose 1 otherwise. This is a subgame perfect Nash equilibrium under the exact same conditions as in part (a). Exercise # - Collusion when firms compete in quantities Consider two firms competing as Cournot oligopolists in a market with demand: pp(qq 1, qq ) = aa bbqq 1 bbqq Both firms have total costs, TTTT(qq ii ) = ccqq ii where c > 0 is the marginal cost of production. a. Considering that firms only interact once (playing an unrepeated Cournot game), find the equilibrium output for every firm, the market price, and the equilibrium profits for every firm. Firm 1 chooses q 1 to solve maxq1 ππ ii = pp(aa bbqq 1 bbqq ) ccqq 1 Taking first order conditions with respect to q1, we find ππ 1 qq 1 = aa bbqq 1 bbqq cc = 0 and solving for q1 we obtain firm 1 s best response function 3
aa cc qq 1 (qq ) = bb 1 qq and similarly for firm, since firms are symmetric, qq (qq 1 ) = aa cc bb 1 qq 1 Plugging qq (qq 1 ) into qq 1 (qq ), we find firm 1 s equilibrium output: qq 1 = aa bb aa bbqq 1 cc cc bb = aa bb bb aa bbqq 1 cc cc aa cc aa + cc aa cc = = 4bb bb 3bb 3bb Similarly, firm s equilibrium output is cc aa bb aa cc qq = 3bb = aa bb bb Therefore, the market price is cc cc 3aa aa + cc 3cc aa cc aa = = 6bb bb 6bb 3bb aa cc cc pp = aa bb bb aa 3bb 3bb 3aa aa + cc aa + cc = 3 and the equilibrium profits of each firm are = aa + cc 3 ππ cccccccccccccc 1 = ππ cccccccccccccc aa + cc aa cc cc (aa cc) = cc aa = 3 3bb 3bb 9bb b. Now assume that they could form a cartel. Which is the output that every firm should produce in order to maximize the profits of the cartel? Find the market price, and profits of every firm. Are their profits higher when they form a cartel than when they compete as Cournot oligopolists? Since the cartel seeks to maximize their joint profits, they choose output levels q 1 and q that solve which simplifies to max ππ 1 + ππ = (aa bbqq 1 bbqq )qq 1 ccqq 1 + (aa bbqq 1 bbqq )qq ccqq max (aa bbqq 1 bbqq )(qq 1 + qq ) cc(qq 1 + qq ) 4
Notice that this maximization problem can be further reduced to the choice of aggregate output Q=qq 1 + qq that solves max (aa bbbb)qq cccc Interestingly, this maximization problem coincides with that of a regular monopolist. In other words, the overall production of the cartel of two symmetric firms coincides with that of a standard monopoly. Indeed, taking first order conditions with respect to Q, we obtain aa bbbb cc 0 And solving for Q, we find QQ = aa cc, which is an interior solution given that a>c by definition. bb Therefore, ach firm s output level in the cartel is qq 1 = qq = QQ aa cc = bb = aa cc 4bb And the market price is Therefore, each firm s profits in the cartel are aa cc aa + cc pp = aa bbbb = aa bb = bb aa + cc cc cc (aa cc) ππ 1 = ppqq 1 TTTT(qq 1 ) = aa cc aa = 4bb 4bb ππ 1 cccccccccccc = ππ cccccccccccc = (aa cc) Comparing the profits that every firm makes in the cartel, (aa cc), against those under Cournot competition, (aa cc), we can conclude that 9bb ππ 1 cccccccccccc = ππ cccccccccccc > ππ 1 cccccccccccccc = ππ cccccccccccccc c. Can the cartel agreement be supported as the (cooperative) outcome of the infinitely repeated game? 1. First, we find the discounted sum of the infinite stream of profits when firms cooperate in the cartel agreement (they do not deviate). 5
Payoff of cartel when they cooperate is (aa cc) As a consequence, the discounted sum of the infinite stream of profits from cooperating in the cartel is (aa cc) (aa cc) + δδ + = 1 (aa cc) 1 δδ. Second, we need to find the optimal deviation that, conditional on firm choosing the cartel output, maximizes firm 1 s profits. That is, which is the output that maximizes firm 1 s profits, and which are its corresponding profits from deviating? Since firm sticks to cooperation (i.e., produces the cartel output qq = aa cc ), if firm 1 seeks to 4bb maximize its current profits (optimal deviation), we only need to plug qq = aa cc 4bb response function, as follows qq dddddd aa cc 1 qq 1 4bb = aa cc bb 1 aa cc 4bb 3(aa cc) = into firm 1 s best which provides us with firm 1 s optimal deviation, given that firm is still respecting the cartel agreement. In this context, firm 1 s profit is while that of firm is 3(aa cc) aa cc cc) ππ 1 = aa bb bb cc 3(aa 4bb aa cc cc) = 3 3(aa = 8 9(aa cc) 64bb 3(aa cc) aa cc cc ππ = aa bb bb cc aa 4bb 4bb aa cc cc 3(aa cc) = 3 aa = 8 4bb 3bb Hence, firm 1 has incentives to unilaterally deviate since its current profits are larger by deviating (while firm respects the cartel agreement) than by cooperating. That is, ππ 1 dddddddddddddd = 9(aa cc) 64bb > ππ 1 cccccccccccc = (aa cc) 3. Finally, we can now compare the profits that firms obtain from cooperating in the cartel agreement (part i) with respect to the profits they obtain from choosing an optimal deviation (part 6
ii) plus the profits they would obtain from being punished thereafter (discounted profits in the Cournot oligopoly). In particular, for cooperation to be sustained we need Firm 1: Solving for discount factor δ, we obtain 1 (aa cc) 9(aa cc) > + δδ (aa cc) 1 δδ 64bb 1 δδ 9bb 1 > 9 + δδ 9, which implies δδ > 8(1 δδ) 64 9(1 δδ) 17 Hence, firms need to assign a sufficiently high value to future payoffs, δδ 9, 1, for the cartel agreement to be sustained. Finally, note that firm has incentives to carry out the punishment. Indeed, if it does not revers to the NE of the stage game (producing the Cournot equilibrium output), firm obtains profits of 3(aa cc) 3bb, since firm 1 keeps producing its optimal deviation of qq 1 dddddd = 3(aa cc) 17 while firm produces the cartel output qq cccccccccccc = aa cc. If, instead, firm practices the punishment, producing the Cournot 4bb output aa cc (aa cc) 3(aa cc), its profits are, which exceed for all parameter values. Hence, upon 3bb 9bb 3bb observing that firm 1 deviates, firm prefers to revert to the production of its Cournot output level than being the only firm that produces the cartel output. Exercise #3 Collusion among N firms Consider n firms producing homogenous goods and choosing quantities in each period for an infinite number of periods. Demand in the industry is given by, Q being the sum of individual outputs. All firms in the industry are identical: they have the same constant marginal costs, and the same discount factor. Consider the following trigger strategy: Each firm sets the output that maximizes joint profits at the beginning of the game, and continues to do so unless one or more firms deviate. After a deviation, each firm sets the quantity, which is the Nash equilibrium of the one-shot Cournot game. 7
(a) Find the condition on the discount factor that allows for collusion to be sustained in this industry. First find the quantities that maximize joint profits π = (1 Q) Q cq. It is easily 1 c checked that this output level isq =, yielding profits of ππ = 1 1 cc 1 cc 1 cc cc = (1 cc) 4 for the cartel. m 11 c Therefore, at the symmetric equilibrium individual quantities are q = and n m 1 (1 c) individual profits under the collusive strategy are π =. n 4 As for the deviation profits, the optimal deviation by a firm is given by d m m q ( q ) = argmax q 1 ( n 1) q q q cq. m 11 c where note that all other n-1 firms are still producing their cartel output q =. n It can be checked that the value of q that maximizes the above expression is (1 ) q d ( q m ) = ( n+ 1) c, and that the profits that a firm obtains by deviating from the 4n collusive output are, hence, 11 1 1 1 d 1 ( 1) c ( 1) c n n ( n 1) c π = + cn ( 1) c n 4n + +, 4n 4n which simplifies to d (1 c) ( n+ 1) π = 16n Therefore, collusion can be sustained in equilibrium if 1 m d d cn π π + π, 1 d 1 d (1 + n) which after solving for the discount factor, δ, yieldsδ. For compactness, we 1 + 6n + n (1 + n) cn denote this ratio as δ. 1+ 6n+ n Hence, under punishment strategies that involve a reversion to Cournot equilibrium forever after a deviation takes place, tacit collusion arises if and only if firms are sufficiently patient. cn The following figure depicts cutoff δ, as a function of the number of firms, n, shading the region of δ that exceeds such a cutoff. 8
(b) Indicate how the number of firms in the industry affects the possibility of reaching the tacit collusive outcome. By carrying out a simple exercise of comparative statics using the critical threshold for the discount factor, one concludes that (This could be anticipated from our previous figure, where the critical discount factor increases in n.) Intuition: Other things being equal, as the number of firms in the agreement increases, the more difficult it is to reach and sustain tacit collusion. Since firms are assumed to be symmetric, an increase in the number of firms is equivalent to a lower degree of market concentration. Therefore, lower levels of market concentration are associated ceteris paribus with less likely collusion.. 9
WATSON CHAPTER 4 EXERCISE 1 Ex. 1 Chapter 4 Watson Bet, - -1,1 F Player 1 B Prob=.5 Ace Fold 1, -1 Nature Player Prob=.5 King Bet -, -1,1 f Player 1 b Fold 1, -1 Player has only two available strategies SS = {BBBBBB, FFFFFFFF} But player 1 has four available strategies SS 1 = {BBBB, BBBB, FFFF, FFFF} This implies that the Bayesian normal form representation of the game is 10
Player 1 Player Bet Fold Bb Bf Fb Ff In order to find the expected payoffs for strategy profile (Bb, Bet) top left-hand side cell of the matrix, we proceed as follows: EEUU 1 = 1 + 1 ( ) = 0 Similarly for strategy profile (Bf, Bet), EEUU = 1 ( ) + 1 () = 0 (0,0) EEUU 1 = 1 ( 1) + 1 ( ) = 3 For strategy profile (Ff, Bet), EEUU = 1 1 + 1 = 3 EEUU 1 = 1 ( 1) + 1 ( 1) = 1 For strategy profile (Bb, Fold), EEUU = 1 1 + 1 1 = 1 11
EEUU 1 = 1 1 + 1 1 = 1 EEUU = 1 ( 1) + 1 ( 1) = 1 For strategy profile (Bf, Fold), EEUU 1 = 1 1 + 1 ( 1) = 0 EEUU = 1 ( 1) + 1 1 = 0 Similarly for (Fb, Fold), EEUU 1 = 1 ( 1) + 1 1 = 0 EEUU = 1 1 + 1 ( 1) = 0 Finally, for (Ff, Fold), EEUU 1 = 1 ( 1) + 1 ( 1) = 1 EEUU = 1 1 + 1 1 = 1 We can now insert these expected payoffs into the Bayesian normal form, Player Player 1 Bet Fold Bb 0, 0 1, -1 Bf 1, 1 0, 0 Fb 3, 3 0, 0 Ff -1, 1-1, 1 1
WATSON CHAPTER 6 EXERCISE 6 Following the methods used above to convert this game into normal form, we see that players 1& have the following strategies spaces: Player has only two available strategies SS = {UU, DD} But player 1 has four available strategies SS 1 = {LLLL, LLRR, RRLL, RRRR } The Bayesian normal form game would be constructed as such: Player Player 1 U D LL LR RL RR Making similar calculations as before, we may find the expected values of the different strategy profiles for each person and fill in the normal form table as such: Player Player 1 U D LL, 0, 0 LR 1, 0 3, 1 RL 1, 3, 0 RR 0, 4, 1 13
It is evident by finding each player s Best Responses that the BNE is at {LL, U} WATSON CHAPTER 6 EXERCISE 7 Watson Chapter 6 Ex. 7 Nature c= c=0 Prob=/3 Prob=1/3 Player Player Player 1 A 0, 1 1, 0 Player 1 A 0, 1 1, 0 B 1, 0, 1 B 1, 0 0, 1 A) Player (uninformed) for only two strategies SS = {XX, YY} Player 1 (informed) has four strategies, depending on his type, SS 1 = {AAAA, AABB, BBAA, BBBB } Where the first component of every strategy pair denotes what player 1 does when his type is c= while the second component reflects his choice when his type is c=0. 14
The Bayesian normal form representation is: Player Player 1 X Y AA 0, 1 1, 0 AB 1/3, /3 /3, 1/3 BA /3, 1/3 5/3, /3 BB 1, 0 4/3, 1 B) Underlining players best responses as usual, we find a unique BNE: {BA, Y} 15