微观经济理论英文版课件：ch15 GAME THEORY MODELS

1、Chapter 15GAME THEORY MODELSOF PRICINGCopyright 2005 by South-Western, a division of Thomson Learning. All rights reserved.1Game TheoryGame theory involves the study of strategic situationsGame theory models attempt to portray complex strategic situations in a highly simplified and stylized settinga

2、bstract from personal and institutional details in order to arrive at a representation of the situation that is mathematically tractable2Game TheoryAll games have three elementsplayersstrategiespayoffsGames may be cooperative or noncooperative3PlayersEach decision-maker in a game is called a playerc

3、an be an individual, a firm, an entire nationEach player has the ability to choose among a set of possible actionsThe specific identity of the players is irrelevantno “good guys” or “bad guys”4StrategiesEach course of action open to a player is called a strategyStrategies can be very simple or very

4、complexeach is assumed to be well-definedIn noncooperative games, players are uncertain about the strategies used by other players5PayoffsThe final returns to the players at the end of the game are called payoffsPayoffs are usually measured in terms of utilitymonetary payoffs are also usedIt is assu

5、med that players can rank the payoffs associated with a game6NotationWe will denote a game G between two players (A and B) byGSA,SB,UA(a,b),UB(a,b) whereSA = strategies available for player A (a SA)SB = strategies available for player B (b SB)UA = utility obtained by player A when particular strateg

6、ies are chosenUB = utility obtained by player B when particular strategies are chosen7Nash Equilibrium in GamesAt market equilibrium, no participant has an incentive to change his behaviorIn games, a pair of strategies (a*,b*) is defined to be a Nash equilibrium if a* is player As best strategy when

7、 player B plays b*, and b* is player Bs best strategy when player A plays a*8Nash Equilibrium in GamesA pair of strategies (a*,b*) is defined to be a Nash equilibrium ifUA(a*,b*) UA(a,b*) for all aSAUB(a*,b*) Ub(a*,b) for all bSB9Nash Equilibrium in GamesIf one of the players reveals the equilibrium

8、 strategy he will use, the other player cannot benefitthis is not the case with nonequilibrium strategiesNot every game has a Nash equilibrium pair of strategiesSome games may have multiple equilibria10A Dormitory GameSuppose that there are two students who must decide how loudly to play their stere

9、os in a dormeach may choose to play it loudly (L) or softly (S)11A Dormitory GameALSA chooses loud (L) or soft (S)BBLSLSB makes a similarchoice7,55,46,46,3Payoffs are interms of As utilitylevel and Bsutility levelNeither playerknows the othersstrategy12A Dormitory GameSometimes it is more convenient

10、 to describe games in tabular (“normal”) form13A Dormitory GameA loud-play strategy is a dominant strategy for player Bthe L strategy provides greater utility to B than does the S strategy no matter what strategy A choosesPlayer A will recognize that B has such a dominant strategyA will choose the s

11、trategy that does the best against Bs choice of L14A Dormitory GameThis means that A will also choose to play music loudlyThe A:L,B:L strategy choice obeys the criterion for a Nash equilibriumbecause L is a dominant strategy for B, it is the best choice no matter what A doesif A knows that B will fo

12、llow his best strategy, then L is the best choice for A15Existence of Nash EquilibriaA Nash equilibrium is not always present in two-person gamesThis means that one must explore the details of each game situation to determine whether such an equilibrium (or multiple equilibria) exists16No Nash Equil

13、ibriaAny strategy is unstable because it offers the other players an incentive to adopt another strategy17Two Nash EquilibriaBoth of the joint vacations represent Nash equilibria18Existence of Nash EquilibriaThere are certain types of two-person games in which a Nash equilibrium must existgames in w

14、hich the participants have a large number of strategiesgames in which the strategies chosen by A and B are alternate levels of a single continuous variablegames where players use mixed strategies19Existence of Nash EquilibriaIn a game where players are permitted to use mixed strategies, each player

15、may play the pure strategies with certain, pre-selected probabilitiesplayer A may flip a coin to determine whether to play music loudly or softlythe possibility of playing the pure strategies with any probabilities a player may choose, converts the game into one with an infinite number of mixed stra

16、tegies20The Prisoners DilemmaThe most famous two-person game with an undesirable Nash equilibrium outcome21The Prisoners DilemmaAn ironclad agreement by both prisoners not to confess will give them the lowest amount of joint jail timethis solution is not stableThe “confess” strategy dominates for bo

17、th A and Bthese strategies constitute a Nash equilibrium22The Tragedy of the CommonThis example is used to signify the environmental problems of overuse that occur when scarce resources are treated as “common property”Assume that two herders are deciding how many of their yaks they should let graze

18、on the village commonproblem: the common is small and can rapidly become overgrazed23The Tragedy of the CommonSuppose that the per yak value of grazing on the common isV(YA,YB)=200 (YA + YB)2 where YA and YB = number of yaks of each herderNote that both Vi 0 and Vii 3 + 1/(1-) 46The Tragedy of the C

19、ommon RevisitedThe overgrazing of yaks on the village common may not persist in an infinitely repeated gameAssume that each herder has only two strategies availablebringing 4 or 5 yaks to the commonThe Nash equilibrium (A:5,B:5) is inferior to the cooperative outcome (A:4,B:4)47The Tragedy of the Co

20、mmon RevisitedWith an infinite number of repetitions, both players would find it attractive to adopt cooperative trigger strategies if544/(1-) 595 + 500(1-) 551/595 = 0.9348Pricing in Static GamesSuppose there are only two firms (A and B) producing the same good at a constant marginal cost (c)the st

21、rategies for each firm consist of choosing prices (PA and PB) subject only to the condition that the firms price must exceed cPayoffs in the game will be determined by demand conditions49Pricing in Static GamesBecause output is homogeneous and marginal costs are constant, the firm with the lower pri

22、ce will gain the entire marketIf PA = PB, we will assume that the firms will share the market equally50Pricing in Static GamesIn this model, the only Nash equilibrium is PA = PB = cif firm A chooses a price greater than c, the profit-maximizing response for firm B is to choose a price slightly lower

23、 than PA and corner the entire marketbut Bs price (if it exceeds c) cannot be a Nash equilibrium because it provides firm A with incentive for further price cutting51Pricing in Static GamesTherefore, only by choosing PA = PB = c will the two firms have achieved a Nash equilibriumwe end up with a com

24、petitive solution even though there are only two firmsThis pricing strategy is sometimes referred to as a Bertrand equilibrium52Pricing in Static GamesThe Bertrand result depends crucially on the assumptions underlying the modelif firms do not have equal costs or if the goods produced by the two fir

25、ms are not perfect substitutes, the competitive result no longer holds53Pricing in Static GamesOther duopoly models that depart from the Bertrand result treat price competition as only the final stage of a two-stage game in which the first stage involves various types of entry or investment consider

26、ations for the firms54Pricing in Static GamesConsider the case of two owners of natural springs who are deciding how much water to supplyAssume that each firm must choose a certain capacity output levelmarginal costs are constant up to that level and infinite thereafter55Pricing in Static GamesA two

27、-stage game where firms choose capacity first (and then price) is formally identical to the Cournot analysisthe quantities chosen in the Cournot equilibrium represent a Nash equilibriumeach firm correctly perceives what the others output will beonce the capacity decisions are made, the only price th

28、at can prevail is that for which quantity demanded is equal to total capacity56Pricing in Static GamesSuppose that capacities are given by qA and qB and thatP = D -1(qA + qB) where D -1 is the inverse demand functionA situation in which PA = PB total capacity so one firm could increase its profits b

29、y raising its price and still sell its capacity57Pricing in Static GamesLikewise, a situation in which PA = PB P is not a Nash equilibriumtotal quantity demanded c in period T (the final period) offers B the option of choosing PA PB cAs threat to charge PA in period T is noncrediblea similar argumen

30、t applies to any period prior to T62Repeated Games and Tacit CollusionIf the pricing game is repeated over infinitely many periods, twin “trigger” strategies become feasibleeach firm sets its price equal to the monopoly price (PM) providing the other firm did the same in the prior periodif the other

31、 firm “cheated” in the prior period, the firm will opt for competitive pricing in all future periods63Repeated Games and Tacit CollusionSuppose that, after the pricing game has been proceeding for several periods, firm B is considering cheatingby choosing PB PA = PM it can obtain almost all of the s

32、ingle period monopoly profits (M)64Repeated Games and Tacit CollusionIf firm B continues to collude tacitly with A, it will earn its share of the profit stream(M + M + 2M + nM +)/2= (M /2)1/(1-) where is the discount factor applied to future profits65Repeated Games and Tacit CollusionCheating will b

33、e unprofitable ifM 1/2Providing that firms are not too impatient, the trigger strategies represent a subgame perfect Nash equilibrium of tacit collusion66Tacit CollusionSuppose only two firms produce steel bars for jailhouse windowsBars are produced at a constant AC and MC of $10 and the demand for

34、bars isQ = 5,000 - 100PUnder Bertrand competition, each firm will charge a price of $10 and a total of 4,000 bars will be sold67Tacit CollusionThe monopoly price in this market is $30each firm has an incentive to colludetotal monopoly profits will be $40,000 each period (each firm will receive $20,0

35、00)any one firm will consider a next-period price cut only if $40,000 $20,000 (1/1-) will have to be fairly high for this to occur68Tacit CollusionThe viability of a trigger price strategy may depend on the number of firmssuppose there are 8 producerstotal monopoly profits will be $40,000 each perio

36、d (each firm will receive $5,000)any one firm will consider a next-period price cut if $40,000 $5,000 (1/1-)this is likely at reasonable levels of 69Generalizations and LimitationsThe viability of tacit collusion in game theory models is very sensitive to the assumptions madeWe assumed that:firm B c

37、an easily detect that firm A has cheatedfirm B responds to cheating by adopting a harsh response that not only punishes A, but also condemns B to zero profits forever70Generalizations and LimitationsIn more general models of tacit collusion, these assumptions can be relaxeddifficulty in monitoring o

38、ther firms behaviorother forms of punishmentdifferentiated products71Entry, Exit, and StrategyIn previous models, we have assumed that entry and exit are driven by the relationship between the prevailing market price and a firms average costThe entry and exit issue can become considerably more compl

39、ex72Entry, Exit, and StrategyA firm wishing to enter or exit a market must make some conjecture about how its actions will affect the future market pricethis requires the firm to consider what its rivals will dothis may involve a number of strategic ploys especially when a firms information about it

40、s rivals is imperfect73Sunk Costs and CommitmentMany game theoretic models of entry stress the importance of a firms commitment to a specific marketlarge capital investments that cannot be shifted to another market will lead to a large level of commitment on the part of the firm74Sunk Costs and Comm

41、itmentSunk costs are one-time investments that must be made to enter a marketthese allow the firm to produce in the market but have no residual value if the firm leaves the marketcould include expenditures on unique types of equipment or job-specific training of workers75First-Mover Advantage in Cou

42、rnots Natural SpringsUnder the Stackelberg version of this model, each firm has two possible strategiesbe a leader (qi = 60)be a follower (qi = 30)76First-Mover Advantage in Cournots Natural SpringsThe payoffs for these two strategies are:77First-Mover Advantage in Cournots Natural SpringsThe leader

43、-leader strategy for each firm proves to be disastrousit is not a Nash equilibriumif firm A knows that firm B will adopt a leader strategy, its best move is to be a followerA follower-follower choice is profitable for both firmsthis choice is unstable because it gives each firm an incentive to cheat

44、78First-Mover Advantage in Cournots Natural SpringsWith simultaneous moves, either of the leader-follower pairs represents a Nash equilibriumBut if one firm has the opportunity to move first, it can dictate which of the two equilibria is chosenthis is the first-mover advantage79Entry DeterrenceIn so

45、me cases, first-mover advantages may be large enough to deter all entry by rivalshowever, it may not always be in the firms best interest to create that large a capacity80Entry DeterrenceWith economies of scale, the possibility for profitable entry deterrence is increasedif the first mover can adopt

46、 a large-enough scale of operation, it may be able to limit the scale of a potential entrantthe potential entrant will experience such high average costs that there would be no advantage to entering the market81Entry Deterrence in Cournots Natural SpringAssume that each spring owner must pay a fixed

47、 cost of operations ($784)The Nash equilibrium leader-follower strategies remain profitable for both firmsif firm A moves first and adopts the leaders role, Bs profits are relatively small ($116)A could push B out of the market by being a bit more aggressive82Entry Deterrence in Cournots Natural Spr

48、ingSince Bs reaction function is unaffected by the fixed costs, firm A knows thatqB = (120 - qA)/2 and market price is given byP = 120 - qA - qBFirm A knows that Bs profits areB = PqB - 78483Entry Deterrence in Cournots Natural SpringWhen B is a follower, its profits depend only on qATherefore,Firm

49、A can ensure nonpositive profits for firm B by choosing qA 64 Firm A will earn profits of $2,80084Limit PricingAre there situations where a monopoly might purposely choose a low (“limit”) price policy to deter entry into its market?In most simple situations, the limit pricing strategy does not yield

50、 maximum profits and is not sustainable over timechoosing PL 0 90Limit Pricing and Incomplete InformationRegardless of its true costs, firm A is better off if B does not enterOne way to ensure this is for A to convince B that bB and0 if bA bB)Since A believes that VB is distributed normally,prob(bA

51、bB) = prob(bA kBVB) = prob(bA /kB VB) = bA /kBTherefore,A = (VA - bA) (bA /kB)107An Oil Tract AuctionNote that A is maximized whenbA = VA /2Similarly,bB = VB /2The firm with the highest valuation will win the bid and pay a price that is only 50 percent of the valuation108An Oil Tract AuctionThe pres

52、ence of additional bidders improves the situation for the sellerIf firm A continues to believe that each of its rivals valutions are uniformly distributed over the 0,1 interval,prob(bA bi) = prob(bA kiVi) for i = 1,n109An Oil Tract AuctionThis means thatA = (VA - bA)(bAn -1/kn -1) and the first-orde

53、r condition for a maximum isbA = (n-1)/nVAAs the number of bidders rises, there are increasing incentives for a truthful revelation of each firms valuation110Dynamic Games with Incomplete InformationIn multiperiod and repeated games, it is necessary for players to update beliefs by incorporating new information provided by each round of playEach player is aware that his opponent will be doing such updatingmust take