Games with more than round Reeated risoner s dilemma Suose this game is to be layed 0 times. What should you do? Player High Price Low Price Player High Price 00, 00-0, 00 Low Price 00, -0 0,0 What if there are multile equilibria in the stage game Examle Player High Middle Low Player High 00,00,0 0, -0 Middle 0, 70,70,0 Low -0,0 0, 0,0 What are the Nash equilibria (ure strategies) if this game is layed twice? Why is there only equilibrium in this game when it is layed twice? What if it is layed N times? Player High Middle Low Player High 00,00 0,0 0, -0 Middle 0,0 70,70,0 Low -0,0 0, 0,0 strategy.doc
Games without a definite end-date Consider a risoner s dilemma game. There are two firms. In each eriod, t=,... each sets either a High or Low rice. Profits in the eriod are as follows. firm High Low firm High, 0,9 Low 9,0, Table : Prisoners Dilemma As we have seen, if there are only a finite number of eriods, the unique strategic equilibrium is to choose Low in the last eriod, hence Low in the second last eriod, hence... Low in the first eriod. But what if there is no end-date? Instead, let us assume that the game goes on forever. Suose that both firms in fact choose to cooerate by setting the High rice. The stream of ayoffs is then as deicted in Figure. ayoff 3 4 6 eriod Figure : Stream of ayoffs from continuing to choose High If firm starts out in this way, can firm do better? This deends not only on firm s immediate ayoff, but also on what firm will do if firm should choose to lay non-cooeratively (to cheat ) in the first eriod. One ossibility is that firm will resond by never again trusting firm and hence choosing the Low rice. If this is the case, firm s ayoff stream is as deicted in Figure. 9 ayoff 3 4 6 eriod Figure : Stream of ayoffs from choosing Low
To comute the resent value of this strategy, we let V be the resent value of all ayoffs following eriod, that is, to the right of the first dotted vertical line. Then the resent value of all the ayoffs is 9+V. Now move to the second dotted line just after eriod. Looking ahead, the infinite stream is exactly the same as it was after eriod. Thus it also has a value of V when discounted to just after eriod. Adding in the eriod ayoff, the value of the stream discounted to eriod is +V. Then if we discount this stream back to eriod, the resent value is (+V)/(+r), where r is the interest rate er eriod. But we began by assuming that the resent value from just after eriod is V. Thus V must satisfy: V V, or after rearranging, V = + + r =. r Adding in the first eriod ayoff of 9, the resent value of not cooerating is 9 + r. We now comare this with the ayoff if both firms cooerate. Arguing exactly as above, the resent value of the stream discounted to just after eriod is /r. Then adding in the first eriod ayoff, the resent value of cooerating is + r. Then cooerating by choosing the High rice has a higher ayoff if 3 4 3 ( + ) ( 9 + ) = 4 = ( r ) > 0. r r r r 4 Thus in this examle, it is better to cooerate as long as the eriod to eriod interest rate is less than 0.7. But this is not quite the end of the story. Suose firm indicates that it will lay in the way described above. That is, it will trust initially but if it is ever crossed, will never trust again. Is this threat credible? To answer this question we must ask whether the threat is an equilibrium strategy of the game continuing after eriod. Suose firm chooses Low in the first eriod. If firm believes that firm will carry out the threat, firm s best resonse is to lay Low. But with firm laying Low, firm s best resonse is also to lay Low. Thus Low is an equilibrium strategy of the continuing game. That is, the threat is credible. RULE: The threatened resonse to a layer that cheats is a credible threat if it is a strategic equilibrium strategy of the continuing game.
For our examle, we have seen that, for r < 0.7 it is a strategic equilibrium for each firm to start out setting a High rice and to switch forever, if the other firm even once chooses Low. Now let us look at the risoner s dilemma game more generally. Each firm can get a ayoff of g, the good ayoff, or b the bad ayoff. Moreover, if one tries to be good while the other deviates for a short-run gain, the latter gets d, while the former gets s. firm High Low firm High g,g s,d Low d,s b,b Table 3: General x Prisoners Dilemma Arguing exactly as above, the ayoff from trying to steal the market is b d + r while the ayoff from cooerating is g g +. r Thus cooeration is the referred strategy if g b g b d g g b ( g + ) ( d + ) = ( d g) = ( r) > 0 r r r r d g For a risoner s dilemma game we require d > g > b > s thus the first term in the final arentheses is ositive. It follows that as long as the interest rate is sufficiently low, cooeration is ossible in any infinitely reeated risoner s dilemma game. My Scottish ancestral clan seems to have understood this rincile well. The clan motto is Never forget a friend, never forgive an enemy! 3
We next consider a slightly more comlicated situation where there are several alternatives to ricing non-cooeratively. For examle, suose, as in Table 4 that the two firms are considering three ossible ricing strategies. As before, it is readily confirmed that both ricing Low is an equilibrium. Moreover, just as before, there is a cooerative equilibrium in which any cheating is unished forever. firm High Middle Low High 7,3,6,7 firm Middle 0, 6, 3,3 Low,- 7,- 4,0 Table 4: Prisoners Dilemma with three alternatives But these are not the only two equilibria. Firm might argue that, because it is more rofitable, it should get a better deal from cooeration. It announces a strategy of setting a Middle rice and holding there as long as firm kees its rice High. If firm ever chooses a rice other than High, firm will switch to Low forever. It is left to the reader to confirm that this is also a strategic equilibrium. RULE: In games without a definite end-date, there are many strategic equilibria. The choice of an equilibrium thus hinges on the ability of the arties to agree on which air of equilibrium strategies they will lay. Whenever there are multile equilibria, the ower of the theory is greatly weakened unless there is some other reason why one equilibrium is more lausible than another. There is another equilibrium in which firm chooses Middle initially and firm chooses High. Does this seem as lausible? Why, or why not? 4
Cournot Duooly Choosing outut levels firm Low Middle High Low 7,7 60,80 4,8 firm Middle 80,60 64,64 6,63 High,- 63,6 64,64 For what interest rates does it ay to cooerate? Consider the following examle = 30 q q, C( q ) = 6q i i
Cooeration with an uncertain end-date As an alternative to the above model, suose that the game will end at some oint but neither layer knows exactly when this will be. Perhas at some oint a large firm will enter the market and eliminate the rofit of both layers. We model this by assuming that, if the game gets to eriod t there is a robability that the game will end and a robability - that the game will continue to eriod t+. Suose that once again firm threatens to never cooerate again if firm deviates even once. Let c be the ayoff if both cooerate and let n be the Nash equilibrium ayoff in the stage game. Finally let d > c be the biggest one eriod ayoff if someone defects. Reducing what might be a much larger game to its essential elements, we have the following ayoff matrix. firm Left Right firm To c,c s,d Bottom d,s n,n The ayoff tree for layer if he deviates is as deicted below. d n n n - - - - 0 0 0 0 3 4 eriod Figure 3: Payoff tree from choosing Low 6
Let V be the resent exected value of the stream of benefits after eriod. Arguing as before, if we move forward to eriod, the discounted exected value of the stream from then on must be n+ V. Discounting back to eriod, this stream has n+ V eriod resent value of. This outcome occurs with robability -. With + r robability the game ends and the rofit is zero. Then n+ V V = ( )( ) + r Rearranging,. V( + r) V( ) = n( ) hence V = ( ) r+ Suose for simlicity that the interest rate is zero. The exected resent value of the entire tree is thus d + ( ) n. Arguing as above, if both layers cooerate, the exected resent value is c+ ( ) c The net gain to cooerating is therefore ( )( c n) ( d c) and this is ositive as c n long as <. Thus as long as the robability of the current round being the last d n is not too high, cooeration yields a higher ayoff than one short-run gain followed by a low ayoff thereafter. Once again this is not the end of the story since we must check that the lanned resonse to cheating is a strategic equilibrium of the continuing game. It is left to the reader to check that the argument made in the revious section holds here as well. 7
Bargaining Consider some oortunity which will yield a otential artnershi a total rofit of $V er year for Y years. (For simlicity we will ignore discounting.) The two otential artners, Alex and Bev, have to first reach agreement on the share of the rofit each will receive. Alex might start out by demanding (say) 70% of the gain and saying that if Bev does not immediately agree, she will ever-after raise her demand to 80%. If Bev believes this she has a best rely of Accet any offer of at least 70%. And if Bev has this strategy, Alex s offer is a best resonse to Bev. Of course Bev might start out with her own outrageous offer - -demanding (say a 90% share and indicating that she will refuse to budge. Again we have a strategic equilibrium. It aears, therefore that game theory roduces an embarrassment of riches - -almost any outcome can be a strategic equilibrium. However, if each round of bargaining takes time, and hence there is a cost of delay, a very different conclusion emerges. Let there be T eriods with one round of bargaining er eriod. Let v be the rofit er eriod (so that vt = VY). Bargaining takes the form of alternating offers and counteroffers. As in any alternating move game, we look forward to the end of the game and figure out what layers would do if they get to that oint. We then work backwards. Rounds Offer by total rofit offer ayoffs if share to go rejected Alex v (v,0) (0,0) (00,0) Bev v (v,v) (v,0) (0,0) 3 Alex 3v (v,v) (v,v) (67,33) 4 Bev 4v (v,v) (v,v) (0,0)... Alex v (6v,v) (v,v) (,4) 8
It is readily checked that with an even number of rounds to go, the offer is always a 0:0 slit. Moreover, as the number of rounds grows, the share offered in the odd numbered rounds aroaches the equal slit as well. Thus if the time between rounds is fairly short and thus the otential number of rounds is large, the advantage to moving last is very small and the two arties agree to an equal share. Outside alternative Suose Alex can earn v A er round in some alternative activity if in each eriod where agreement is not reached, while Bev can earn v B., where both are less than v /. How would this affect the discussion above? Hint: In each eriod there is a otential surlus of x= v va vb. In the last eriod Alex can claim all of this surlus since if Bev rejects she earns v B. Costly delay Another cost of bargaining is the cost of time. That is, if you agree to a 0:0 slit of $ million a eriod from now this is only worth δ = million today. + r Suose that if agreement is reached at time t the value of the agreement discounted to that eriod will be v. Suose Alex moves at t= first and Bev at t=. Let v A be the resent value of Alex at t=3 given the equilibrium bargaining strategies and let v B be the resent value to Bev. 9
Bev δ( v δv A ) v δ v A v B Alex v δ( v δv A ) δ va v A t= t= t=3 Since the future at t=3 is identical to the future at t= the equilibrium ayoff at t= must be the same. Thus v = v δ( v δv ) A Rearranging, A δ δ v va = v= v= δ ( δ)( + δ) + δ Since the values add u to, v B v δ v = = + δ + δ. Thus if the time between rounds is short so the discount factor is close to, the shares are aroximately equal. 0
War of Attrition Payoff to the victor V Cost er eriod (litigation) c Seek a symmetric equilibrium in which exit occurs each round with robability. If both exit each has an equal chance of winning. Suose that firm adots this strategy. If firm exits immediately its exected ayoff is Pr{oonent exits immediately} V = V. If firm lans to exit after eriod it incurs round litigation costs. Its exected ayoff is c+ Pr{oonent exits immediately} V + Pr{oonent exits in round }( V) = c+ V + ( ) V. For a mixed strategy equilibrium firm must be indifferent. Rearranging, V = c+ V + ( ) V c + = Hence V c ( ) = + =. V Therefore c =. V The larger the ratio of c to V the larger the robability of quitting and hence the less wasteful battling.
c/v 0. 0.3 0. 0.6 0. 0. 0.0 0.0 0.0 0.0