PROBLEM SOLVING
ECON 306 Homework: Sequential Move Games and Repeated Games 2021 PROBLEM SOLVING. 1. Alternating Bargaining Game In the infinitely repeated 2-player Rubinstein alternating bargaining game, we stated that the unique subgame perfect equilibrium exhibits no delay. We will double check that result here in more detail. Recall the equil. strategies are Player 1 demands s1,t = 1 (1+ δ) , Player 2 responds by d2,t = A iff s1,t ≤ 1 (1+ δ) ; Player 2 offers s2,t+1 = δ (1+ δ) , Player 1 responds by d1,t+1 = A iff s2,t+1 ≥ δ (1+ δ) , for any odd periods t = 1, 3, 5, . . ., so t + 1 = 2, 4, 6, . . . are the even periods. Note: “iff” means “if and only if”. (a) In odd periods, given d2,t = A if s1,t ≤ 1(1+δ) , can player 1 get a deal by demand- ing s1,t = 1(1+δ)? (b) Expecting there will be a deal reached in t = 2, is it profitable for player 1 to demand more than 1(1+δ)? (c) Consider period 1(b) where player 1 has made some demand s1,t. Should player 2 accept any demand s1,t > 1(1+δ)? Why or why not? 2. 3-Period Alternating Bargaining Game This question walks through the 3-period Rubinstein alternating bargaining game we covered in class. Below is the setup. • players 1 and 2 (P1 and P2) bargain over one dollar by making alternating offers • players are impatient—they discount payoffs received in later periods by the factor δ per period • timing 1a P1 makes an demand s1 ∈ [0, 1] for P1’s share; subscript 1 denotes P1’s demand 1b P2 makes a decision d2 ∈ {A, R} – if d2 = A, payoffs realize π1 = s1, π2 = 1− s1; – if d2 = R, move on to stage 2... 2a P2 makes an offer s2 ∈ [0, 1] for 1’s share; subscript 2 denotes P2’s offer for P1’s share 2b P1 makes a decision d1 ∈ {A, R} – if d1 = A, payoffs realize π1 = s2, π2 = 1− s2; – if d1 = R, move on to stage 3... 3 payoffs determined exogenously π1 = s, π2 = 1− s ECON 306 Page 2 of 5 2021 • Solve the game using backward induction. (a) In period 2b, what is P1’s strategy on decision rule, given P2’s offer s2? (b) In period 2a, what offer is P2’s optimal strategy, expecting P1’s responses in period 2b? (Hint: need to show that inducing P1 to accept is worthwhile for P2) (c) In period 1b, what is P2’s strategy on decision rule, given P1’s demand s1? (d) In period 1a, what offer is P1’s optimal strategy, expecting P2’s responses in period 1b? (Hint: need to show that inducing P2 to accept is worthwhile for P1) (e) What is the outcome of the game in the subgame perfect equilibrium? 3. Supporting Cooperation in PD through Infinitely Repeated Game Consider the following prisoner’s dilemma (PD) game P2 C D P1 C (4,4) (0,5)D (5,0) (1,1) P1 and P2 play an infinitely repeated PD game. Specifically, • P1 and P2 play the above stage game simultaneously, once the payoffs are realized, the players observe the outcome and move on to the next period, when the game is played again. • Both players have a discount factor of δ per period. Consider the strategy of “tit-for-tat”. Specifically, P1 and P2 promise to choose strategies conditioning on the previous outcomes in the following way • play C in period 1; • if (C,C) has always been played in the past, then play “C” this period; • if the opponent ever played “D”, then play “D” forever. (a) If • “D” has never been played in previous periods; • P1 believes P2 plays the “tit-for-tat” strategy as above; • P1 will play “tit-for-tat” in the future. What is the net present value (NPV) of P1’s payoff from playing “C” in the current period? (Note: P2’s NPV under the same belief is identical) (b) If • “D” has never been played in previous periods; • P1 believes P2 plays the “tit-for-tat” strategy as above; • P1 will play “tit-for-tat” in the future. What is the net present value (NPV) of P1’s payoff from playing “D” in the current period? (Note: P2’s NPV under the same belief is identical) ECON 306 Page 3 of 5 2021 (c) Derive the condition under which playing C in period 1 is a best response if players expect each other to play “tit-for-tat”. (d) If (C,C) is played in period 1, derive the condition under which playing C in period 2 is a best response if players expect each other to play “tit-for-tat”. (e) If (C,C) is played in all previous periods up to some period t ≥ 2, derive the condition under which playing C in the next period t + 1 is a best response if players expect each other to play “tit-for-tat”. (f) Use your previous results to show: both players playing “tit-for-tat” is indeed a Nash equilibrium, and the outcome of the game is (C,C) being played in every period if δ ≥ 14 . (g) Use your previous results to show: both players playing “tit-for-tat” is not a Nash equilibrium if δ < 14="" .="" 4.="" review:="" one-shot="" cournot="" competition="" versus="" one-shot="" collusion="" consider="" a="" cournot="" competition="" model.="" •="" firm="" 1="" and="" 2="" (f1="" and="" f2)="" compete="" by="" simultaneously="" choosing="" output="" levels,="" q1,="" q2,="" to="" maximize="" profit.="" •="" the="" two="" products="" are="" perfect="" substitutes="" for="" the="" consumers="" who="" have="" inverse="" demand="" p="a" −="" b(q1="" +="" q2).="" •="" each="" firm="" pays="" a="" constant="" marginal="" cost="" of="" c="" to="" produce="" every="" unit="" of="" good.="" (a)="" solve="" for="" the="" one-shot="" ne="" of="" this="" game.="" (hint:="" the="" equilibrium="" is="" (q∗1="a−c" 3b="" ,="" q="" ∗="" 2="a−c" 3b="" )="" and="" their="" profits="" are="" (π="" ∗="" 1="(a−c)2" 9b="" ,="" π="" ∗="" 2="(a−c)2" 9b="" ))="" (b)="" if="" the="" two="" firms="" were="" integrated="" into="" one="" firm,="" i.e.="" a="" monopoly.="" solve="" for="" the="" profit-maximizing="" monopolistic="" output="" qm="" and="" profit="" level="" πm.="" (hint:="" the="" monopolistic="" output="" is="" qm="and" its="" profit="" is="" πm=".)" (c)="" if="" the="" two="" firms="" can="" collude="" with="" each="" other="" by="" agreeing="" act="" as="" if="" they="" were="" one="" firm,="" i.e.="" a="" monopoly,="" and="" share="" the="" market="" equally.="" what="" is="" each="" firm’s="" output="" level="" and="" profit="" level?="" is="" it="" more="" profitable="" to="" collude?(hint:="" use="" your="" result="" in="" the="" previous="" part,="" each="" firm="" shares="" half="" of="" the="" monopoly="" market="" and="" monopoly="" profit.)="" (d)="" this="" part="" investigates="" if="" the="" two="" firms="" are="" able="" to="" maintain="" collusion="" without="" a="" binding="" contract.="" if="" firm="" 1="" believes="" that="" firm="" 2="" will="" honor="" the="" collusion="" agree-="" ment="" and="" produce="" half="" the="" monopoly="" output="" level="" qm2="a−c" 4b="" .="" •="" what="" is="" firm="" 1’s="" best="" response="" qd1="" how="" does="" the="" best="" response="" compare="" against="" the="" collusive="" output="" level="" qm1="" •="" if="" firm="" 2="" indeed="" produces="" the="" collusive="" output="" level="" qm2="" ,="" while="" firm="" 1="" ex-="" pected="" this="" and="" plays="" its="" best="" response="" qd1="" ,="" what="" are="" each="" firm’s="" profit="" lev-="" els?="" •="" how="" do="" their="" profits="" compare="" to="" collusive="" profits?="" •="" does="" firm="" 1="" have="" incentive="" to="" deviate="" from="" collusion?="" econ="" 306="" page="" 4="" of="" 5="" 2021="" (hint:="" in="" this="" part,="" we="" assume="" firm="" 2="" is="" naive="" and="" produces="" qm2="a−c" 4b="" regardless="" of="" firm="" 1’s="" play.)="" 5.="" tacit="" collusion="" of="" cournot="" duopoly="" through="" repeated="" play="" if="" we="" focus="" on="" the="" collusive="" arrangement="" (“collude”)="" versus="" competitive="" outcomes="" (“dumping”="" for="" lack="" of="" a="" better="" word)="" in="" the="" previous="" question,="" we="" have="" a="" payoff="" matrix="" f2="" (c)ollude="" (d)umping="" f1="" (c)ollude="" (="" 1="" 8="" (a−c)2="" b="" ,="" 1="" 8="" (a−c)2="" b="" )="" (="" 3="" 32="" (a−c)2="" b="" ,="" 9="" 64="" (a−c)2="" b="" )="" (d)umping="" (="" 964="" (a−c)2="" b="" ,="" 3="" 32="" (a−c)2="" b="" )="" (="" 1="" 9="" (a−c)2="" b="" ,="" 1="" 9="" (a−c)2="" b="" )="" f1="" and="" f2="" repeatedly="" play="" this="" 2-by-2="" game="" infinitely.="" specifically,="" •="" p1="" and="" p2="" play="" the="" above="" stage="" game="" simultaneously,="" once="" the="" payoffs="" are="" realized,="" the="" players="" observe="" the="" outcome="" and="" move="" on="" to="" the="" next="" period,="" when="" the="" game="" is="" played="" again.="" •="" both="" players="" have="" a="" discount="" factor="" of="" δ="" per="" period.="" consider="" the="" strategy="" of="" “tit-for-tat”.="" specifically,="" f1="" and="" f2="" promise="" to="" choose="" strategies="" conditioning="" on="" the="" previous="" outcomes="" in="" the="" following="" way="" •="" play="" c="" in="" period="" 1;="" •="" if="" (c,c)="" has="" always="" been="" played="" in="" the="" past,="" then="" play="" “c”="" this="" period;="" •="" if="" the="" opponent="" ever="" played="" “d”,="" then="" play="" “d”="" forever.="" (a)="" what="" is="" the="" net="" present="" value="" (npv)="" of="" f1’s="" payoff="" from="" playing="" “c”="" in="" the="" current="" period?="" (note:="" f2’s="" npv="" under="" the="" same="" belief="" is="" identical)="" (b)="" if="" •="" “d”="" has="" never="" been="" played="" in="" previous="" periods;="" •="" p1="" believes="" p2="" plays="" the="" “tit-for-tat”="" strategy="" as="" above;="" •="" p1="" will="" play="" “tit-for-tat”="" in="" the="" future.="" what="" is="" the="" net="" present="" value="" (npv)="" of="" p1’s="" payoff="" from="" playing="" “d”="" in="" the="" current="" period?="" (note:="" p2’s="" npv="" under="" the="" same="" belief="" is="" identical)="" (c)="" derive="" the="" condition="" under="" which="" playing="" c="" in="" period="" 1="" is="" a="" best="" response="" if="" players="" expect="" each="" other="" to="" play="" “tit-for-tat”.="" (d)="" if="" (c,c)="" is="" played="" in="" period="" 1,="" derive="" the="" condition="" under="" which="" playing="" c="" in="" period="" 2="" is="" a="" best="" response="" if="" players="" expect="" each="" other="" to="" play="" “tit-for-tat”.="" (e)="" if="" (c,c)="" is="" played="" in="" all="" previous="" periods="" up="" to="" some="" period="" t="" ≥="" 2,="" derive="" the="" condition="" under="" which="" playing="" c="" in="" the="" next="" period="" t="" +="" 1="" is="" a="" best="" response="" if="" players="" expect="" each="" other="" to="" play="" “tit-for-tat”.="" (f)="" use="" your="" previous="" results="" to="" show:="" both="" firms="" playing="" “tit-for-tat”="" is="" indeed="" a="" nash="" equilibrium,="" and="" the="" outcome="" of="" the="" game="" is="" (c,c)="" being="" played="" in="" every="" period="" if="" δ="" ≥="" 917="" .="" econ="" 306="" page="" 5="" of="" 5="" 2021="" (g)="" use="" your="" previous="" results="" to="" show:="" both="" players="" playing="" “tit-for-tat”="" is="" not="" a="" nash="" equilibrium="" if="" δ="">< 14 . (h) interpret your results in the context of collusion. 14="" .="" (h)="" interpret="" your="" results="" in="" the="" context="" of="">