Microeconomics 2 Game Theory Lecture notes. 1. Simultaneous-move games: Solution in (iterated) strict dominance and rationalizable strategies

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1 Sapenza Unversty of Rome. Ph.D. Program n Economcs a.y Mcroeconomcs 2 Game Theory Lecture notes. Smultaneous-move games: Soluton n (terated) strct domnance and ratonalzable strateges. Smultaneous-move games n normal form: defnton and notaton.. Games n pure strateges..2 Games n mxed strateges.2 Alternatve approaches to the determnaton of the equlbrum.3 Strct domnance.3. Strct and weak domnance: defntons.3.2 The prsoners dlemma: soluton n strct domnance.4 Iterated strct domnance.4. Iterated elmnaton of strctly domnated strateges.4.2 Iterated weak domnance.5 Ratonalzable strateges.5. Approach and defnton.5.2 Iterated elmnaton of non ratonalzable strateges.5.3 Relaton between the set of strctly undomnated and ratonalzable strateges The soluton, or equlbrum of a game s a predcton that we make as to the choces that the agents, nvolved n the strategc stuaton descrbed by the game, wll make. Two keyu assumptons enter n the formulaton of ths predcton. We frst s that players behave ratonally; we generally attach to ths assumpton the specfc meanng that players choose an acton that maxmzes ther payoff. But n a context of strategc nteracton the best acton of any player would depend on the actons of the other players. We further assume, therefore, that each player formulates a conjecture of the other players choces and evaluates, on ths bass, the expected payoff of hs alternatve strateges. Players ratonalty and belefs of other players choces have a decsve role n the soluton concepts of smultaneous-move games that we are gong to examne n these Lecture Notes. In ncreasng order of the deductve and ntrospectve capactes asked of the players, these are bascally the followng: - strct domnance; - terated strct domnance; - ratonalzable strateges survvng the terated elmnaton of non ratonalzable strateges; - Nash equlbrum. D. Tosato Game Theory Lecture Notes a.y

2 Ths Lecture Note s dedcated to the presentaton and the crtcal mplcatons of the solutons based on the applcaton of the notons of domnance and ratonalzablty (sectons.3-.5). Nash equlbrum s analyzed n Lecture Note n. 2; extensons and refnements of Nash equlbrum n statc games are presented n separate Lecture Notes. Sectons. and.2 contan some prelmnary notons: n the former the defnton of smultaneous-move games, both n pure and mxed strateges, s presented and notaton establshed; n the latter alternatve conceptual approaches to the determnaton of the equlbrum of the game are brefly llustrated.. Smultaneous-move games n normal form: defnton and notaton A smultaneous-move or statc game descrbes a stuaton of strategc nteracton n whch the agents nvolved henceforth, to be named the players are supposed to take decsons not knowng the actons smultaneously chosen by the other players. The nterpretaton to be gven of the term smultaneous should not be reserved to stuatons n whch the players lterally take decsons at the same tme, but more generally to stuatons of gnorance of the choces made by the other players, possbly at dfferent moments n tme. The lack of knowledge of the other players moves characterzes the game as one of mperfect nformaton, as opposed to stuatons of perfect nformaton, whch obtan f each player knows, when called to play, the decsons taken by all the other players n the precedng rounds of the game... Games n pure strateges The assumpton that players move smultaneously leads to the descrpton of the game n terms of the normal or strategc form, whch conssts n the ndcaton of: () the number of players; (2) the strateges avalable to each player; (3) the payoff receved by each player for each possble combnaton of strateges chosen by the players. In standard notaton: - I s the number of players; players are specfed by the subscrpt,..., I ; a standard compact notaton dstngushes player from all other players, formally ndcated as ; - S fnte number s the space of strateges avalable to player. The strategy space may conssts of a, representng K dstnct pure K of elements, k,...,,..., K s s s s S strateges, or of an nfnte number of elements representng a contnuum of possble pure strateges,.e. of possble values of player s decson varable. In ths latter case the strategy space s represented by the nterval S s, s. In statc games strateges concde wth actons; the two terms wll therefore be used nterchangeably. In dynamc games a strategy s The strategc nteracton problem may call for the presence of several varables capable of assumng a contnuum of values. D. Tosato Game Theory Lecture Notes a.y

3 a rule of acton ndcatng the specfc acton of the player at each round of the game n whch he must take a decson; a strategy s therefore an ordered sequence of actons; - S S S S I, wth,...,...,, I S S j j, s the strategy space of the game and s s s s s s S s a profle of strateges, where the compact notaton for all players dfferent from player - has been used; u s, s : S s player s payoff resultng from all possble profles of actons of all players, formally a mappng from the strategy space of the game nto the real lne. Strategc nteracton among players shows up n ths defnton of the payoff functon. The payoff functon s assumed to be, n general, an ordnal utlty functon; unless otherwse ndcated, ths s the nterpretaton to be gven to the players payoffs n the matrx games used to llustrate dfferent game stuatons. When the players are frms, nvolved for nstance n an olgopoly game, payoff functons are better consdered to represent money profts. A game s fnte f the strategy space S s fnte; ths requres that both the number of players and the strategy space of each player be fnte. In several game stuatons of specfc nterest to economc theory, the strategy space of the players s naturally modeled wth contnuous varables, such as prces and quanttes, for nstance, n the classc olgopoly models of Cournot and Bertrand. The presentaton n ths Note bascally refers to fnte games; separate consderaton wll be gven, when approprate, to contnuous games. Defnton.. The normal form representaton of a game G conssts n the specfcaton of the number of players, the pure strategy space and the payoff functon of each player: I (.),,, G I S u s s We say that the game s of complete nformaton f all players know the structure of the game, namely all ts consttuent elements: the number of players, the strategy space of every player and everybody s payoff functons. Ths last element s partcularly relevant: ncomplete nformaton typcally occurs when the payoff functon of one or several players s not known by all the other players. A Bayesan approach to the descrpton of the game s n that case necessary; the noton of equlbrum must be correspondngly extended...2 Games n mxed strateges Not all games n normal form have soluton n pure strateges. Take the followng game n bmatrx form, known as Matchng Pennes (see Fg..). Each player s strategy space contans two actons S Heads, Tals, for short S H, T,,2. The payoffs reflect a game n whch each player has a penny and must choose how to dsplay t. If the two pennes match wth heads or tals facng up, player, the row player, wns player 2 s penny; f the D. Tosato Game Theory Lecture Notes a.y

4 pennes do not match, player 2 wns player s penny. 2 Snce each player wns what the other player looses, ths s a compettve or zero-sum game. It s mmedately clear that the game has no soluton n pure strategy; each player would try to outguess the other, but wth no element on the bass of whch to make one s choce of strategy. 2 Heads Tals Fgure. Payoff matrx of the Matchng Pennes game The uncertanty each player faces n the choce of strategy has led to the dea that players may randomze between ther pure strateges S and thus adopt a mxed strategy, whch represents a probablty dstrbuton over the set of pure strateges. Although the mxed extenson of the game has a crucal role n the proof of exstence of a equlbrum soluton, the tradtonal vew of mxtures as conscous randomzaton carred out by each player has, nonetheless, come under heavy crtcsm. To suppose, n fact, that ratonal economc agents, frms n partcular, should delegate the determnaton of ther behavor to the outcome of a pure chance mechansm s dsturbng and utterly unrealstc. Harsany (973) and Aumann (987) have, however, convncngly argued for a dfferent vew of mxng, namely for an approach whch reflects the uncertanty that each player has on the strategy choce of the other player - more generally, of the other players. 3 Accordng to ths vew, players do not randomze; each player chooses a defnte strategy. But other players need not know whch one, and the mxture represents ther uncertanty, ther conjecture about ths choce (Aumann and Brandenburger, 995, p. 62, emphass added). Ths uncertanty can be analytcally expressed n terms of a probablstc conjecture, or belef, concernng the opponent s behavor. Let us then ndcate wth 2 the probablty that player assgns to player 2 playng strategy s Heads and wth 2 player 2 playng strategy s Tals the probablty the he assgns to. Let us smlarly ndcate wth conjecture of player 2 as to the strategy choces 2 s Heads and s Tals The probablty dstrbutons, Heads +, - -, + Tals -, + +, - and, 2 2 and the of player. defne a new set of possble strateges that each player attrbutes, by way of conjecture, to the opponent. We wll refer to these strateges as mxed strateges to dstngush them from the pure strateges Heads and Tals, of whch they are a probablty combnaton. For short, as n most textbooks, we wll 2 In the Matchng Pennes game payoffs are naturally taken to be a quantty of money, rather than the utlty of wnnng or losng a penny. 3 Actually Harsany s so called purfcaton theorem has the stll further mplcaton that players play pure strateges and mxed strateges represent the probabltes wth whch the pure strateges are effectvely played. D. Tosato Game Theory Lecture Notes a.y

5 wrte that player j. s the mxed strategy of player, omttng to say that ths s a conjecture made by Defnton.2. The mxed strategy k K,...,,..., of player s a probablty dstrbuton over the set of pure strateges S. The set of mxed strateges of player K k k 0, k (.2) S S s therefore a pure strateges K S the pure strategy K The mxed strategy dmensonal smplex of probablty dstrbutons over the set of k ; s k s k ; S dmensonal vector space. s the probablty assgned by the mxed strategy s a non empty, closed, bounded and convex subset of a s then a vector wth as many component as there are pure strateges. When the strategy space contans only two pure strateges, as n the case of the Matchng Pennes game, t s suffcent to ndcate the frst component of the vector set. to ; we wll drectly The set of mxed strateges S of the game Matchng Pennes s represented n Fg..2 by and the segment AB. The ponts A and B correspond to the mxed strateges,0 B 0,, that s respectvely to the case of playng wth certanty strategy Heads and to the case of playng Tals for certan. Pure strateges thus correspond to degenerate mxed strateges,.e. to mxed strateges that assgn probablty one to just one pure strategy and probablty zero to all the others. Wth the same notaton as used above wth regard to pure strategy games, let S Sj e j be the sets of mxed strateges of the players S S S A Fg..2 The set of mxed strateges of the Matchng Pennes D. Tosato Game Theory Lecture Notes a.y

6 dfferent from player strateges s therefore a vector, and the space of mxed strateges of the game. 4 A profle of mxed. S To complete the descrpton of a mxed strategy game we must defne players payoffs. Returnng to Matchng Pennes, let us determne the expected payoffs of player, frst separately for hs two pure strateges Heads and Tals and then for the mxed strategy over Heads and Tals that represents player 2 s conjecture. Each pure strategy s now a lottery over the consequences, (.3) 2 2. The expected payoffs are,,,, 2 2 u Tals, 2 u Tals, Heads 2 u Tals, Tals 2 = wth probabltes u Heads u Heads Heads u Heads Tals These payoffs correspond to von Neumann and Morgenstern s expected utltes. The expected payoff to player s mxed strategy, s therefore the weghted average of the expected payoffs of the two pure strateges Heads and Tals: (.4),,, u u Heads u Tals The expected payoff of player 2 can, obvously, be determned n a wholly smlar way. The payoff functon of player s then a mappng from the mxed strategy space of the game to the real lne. 2 Heads Tals Heads 2 2 Tals 2 2 Fg.3 Probablty dstrbuton nduced by the mxed strateges and 2 The defnton (.4) of the expected payoff of player and smlarly of player 2 shows that the players mxed strateges nduce a probablty dstrbuton over the cells of the payoff matrx of the game, as ndcated n Fg..3. If, as apparently obvous, each player attrbutes 4 The defnton of the mxed strategy space of players j s based on the assumpton that each player s randomzaton be ndependent of that of all other players or better, that each player beleves that the mxed strategy choces of the other players are ndependent,. e. uncorrelated. Ths possblty and the consequences thereof are, for an mportant aspect, analyzed at the end of ths Lecture Note. D. Tosato Game Theory Lecture Notes a.y

7 probablty be equal to 2 4 playng Heads, the probablty on each of the cells of the payoff matrx would. We are now n a poston to gve a formal defnton of a mxed strategy game, whch parallels Defnton.. Defnton.3 The normal form representaton of a mxed strategy game conssts n the specfcaton of the number of players, of each player s mxed strategy space and expected payoff functon I I S u (.5),,, To sum up the notaton used n the mxed strategy extenson of the game, we have: - I,2,..., I s the fnte number of payers; - k S,...,,..., K s s s s the fnte set of pure strateges or actons of player ; K k 0, s a k - S S dstrbuton over the set of pure strateges by the mxed strategy and convex subset of a - S j Sj K S to the pure strategy K k k ; k s dmensonal vector space; S S S and s dmensonal smplex of probablty s therefore the probablty assgned ; S s a non empty, closed, bounded ; as the Cartesan product of non empty, S S S closed, bounded and convex sets, and convex subset of a -, S K s a non empty, closed, bounded dmensonal vector space; s a profle of mxed strateges of all players; - u, s the payoff to player from the strategy profle,..2 Alternatve approaches to the determnaton of the equlbrum The equlbrum of a game s a profle of strateges such that no player has the ncentve to unlaterally devate from hs choce. How s ths choce determned n a stuaton of strategc nteracton, of whch all players are fully aware? It s commonly assumed, n a context of ndvdual decson-makng, that agents behave ratonally, namely that the acton chosen by each agent s at least as good, accordng to the agent s preferences, 5 as any other avalable acton. 6 We wll assume ths prncple to hold also 5 Ths noton s sometmes referred to as nstrumental ratonalty. D. Tosato Game Theory Lecture Notes a.y

8 n game theoretc stuatons. 7 But n a strategc context the best acton of any player generally depends on the actons that the other players wll choose. Each player wll then be compelled to antcpate these actons and base hs decson on a conjecture, or belef, of the other players choce of acton. In hs PhD dssertaton John Nash (950) suggests two approaches to the determnaton of an equlbrum profle of strateges and thus to the formaton of conjectures. The frst s by deducton and ntrospecton, that s through a process of deductve reasonng based on the assumpton of ratonalty and common knowledge of ratonalty and of the game structure. Ths s the approach followed n ths Lecture Note and throughout the course. It wll be amply llustrated n the study of Nash Equlbrum. The second approach s based on the mass-acton nterpretaton of the equlbrum ponts of the game. Nash presents ths approach envsagng a large populaton of potental players for each poston of the game; wth reference to a two-person game, we thus assume a large populaton of partcpants n the poston of each of the two players. A partcpant s selected at random from each populaton and plays the game wth no possblty of communcaton wth the other player. The repeated play by randomly selected players, so that the probablty that the same players meet agan can be gnored, determnes an average playng of the game. Hence, the probablty that a partcular [set] of strateges wll be employed n a playng of the game should be the product of the probabltes ndcatng the chance of each of n pure strateges to be employed n a random playng (Nash, PhD dssertaton, 950, p. 22). One natural specfcaton of each player s conjecture about the play of hs opponent s that t converges to the probablty dstrbuton correspondng to the average playng of the game. In such dealzed settng of play, the equlbrum of the game s a steady state of the process of repeated playng n a random matchng of players (Osborne, 2004, p. 22). The dea underlyng Nash s mass-acton nterpretaton of equlbrum ponts of a game s that players learn each other s strateges from ther experence playng the game. An alternatve approach to Nash s mass-acton model s to assume that players start from some ntal, possble unexplaned pror conjecture, whch they subsequently revse n response to nformaton receved. Ths model envsages a personal learnng process whch, f convergent, leads to a steady state soluton of the game. Cournot (838, Ch. VII) reles on such a process for the attanment of the duopoly equlbrum n hs quantty strategy game; he assumes, n partcular, that each player adjusts hs strategy assumng that the other frm wll mantan the level of output chosen n the prevous perod. 6 Bounded ratonalty departs from ths prncple. As remarked by Smon (955), economc agents do not have the knowledge and the means to make optmzng choces and settle, therefore, for satsfycng decsons, on the bass of routnes that have proved n the past to delver satsfactory results. 7 We wll shortly see how the prncple of ratonal choce s adapted n the theory of evolutonary games amng to portray the process of natural selecton among anmals and plants. D. Tosato Game Theory Lecture Notes a.y

9 Convergence fundamentally depends on the model of learnng. Opposte results can easly emerge wth reasonable changes of assumptons. Cournot s duopoly model offers examples of convergent a well as of dvergent results. A fnal menton s n order to evolutonary game theory. Evolutonary game theory orgnated as an applcaton of the mathematcal theory of games to bologcal contexts. In lne wth Nash s mass-acton nterpretaton, t studes the behavor of a large populaton of agents who repeatedly engage n strategc nteracton. In evolutonary game theory strateges are genercally nherted trats that control an ndvdual s acton somethng akn to a computer program. Payoffs are specfed n unts of ftness, whch represent the capacty of a spece to reproduce tself and thus survve n the Darwnan process of natural selecton. The mathematcal bologst John Maynard Smth (972) ntroduced the noton of evolutonary stable strategy (ESS), as a tool of explanng the exstence of an equlbrum outcome of anmal conflct, n the sense that, f every member of the populaton follows an ESS, no mutant can successfully nvade and radcally change the equlbrum of the spece. An explct dynamc foundaton, n terms of dfferental equatons, of ths equlbrum concept was later provded by the model of the replcator dynamcs. Replcator dynamcs reflects the rule that the ftter players wll generate more replcas of themselves than the less ft, whch wll be thus culled out of the populaton. The approach and the tools elaborated by evolutonary game theory have been of nterest to the development of game theory n general, n partcular n the applcatons to the problem of equlbrum selecton n the presence of multple equlbra..3 Strct domnance We start the search of equlbrum solutons of a game on the bass of a negatve crteron: no ratonal player would ever choose an acton that s payoff domnated by another acton avalable to hm. Ths crteron requres each player to consder only hs own payoffs; no knowledge s requred of the structure of the game; no conjecture need be formulated as to choces of the other players..3. Strct and weak domnance: defntons Defnton.4 The pure strategy s S s a strctly domnant strategy for player n the I G I S u s s f for all s swe have game,,, (.6) u s, s u s, s for s S A double condton dentfes a strctly domnant strategy for player : ) t offers the hghest payoff among all strateges avalable to the player and ) ths hghest payoff s for all possble D. Tosato Game Theory Lecture Notes a.y

10 strateges of the other players. A weaker defnton of domnance results f the strct nequalty sgn n (.6) s changed nto a greater than or equal sgn, as n the followng defnton Defnton.5 The pure strategy I game,,, s S G I S u s s f for all s swe have (.7) s a weakly domnant strategy for player n the u s, s u s, s for s S Weak domnance requres therefore that strategy avalable strateges s and at least one payoff strctly greater. s have a payoff no less than that of all other The converse of the noton of domnant strategy s the noton of a domnated strategy. Defnton.6 The pure strategy s Ss strctly domnated for player n the game I,,, G I S u s s f there exsts a strategy (.8) s S u s, s u s, s for s S Defnton (.6) obvously mples that strategy s such that strctly domnates strategy s. We can at ths pont restate the defnton of a strctly domnant strategy as a strategy that strctly domnates all other strateges s S. The extenson of Defnton.6 to the case of weakly domnated strategy parallels that of a weakly domnant strategy. We wll examne n the next secton an example of applcaton of the noton of domnance to determne the soluton of a most renown game, the Prsoners Dlemma. These defntons of strctly (weakly) domnant strategy and strctly (weakly) domnated strategy apply also to the mxed extenson of a game. Defnton.7 The mxed strategy S I S u s strctly domnated for player n the I game,,, f there exsts a strategy S (.9) u, u, for S such that We then have: Defnton.8 The mxed strategy S I S u s strctly domnant for player n the I game,,, f t strctly domnates all other strateges n. An mplcaton of Defnton (.7) s that a pure strategy may not be strctly domnated by any other pure strategy, but by a mxed strategy of other pure strateges. S D. Tosato Game Theory Lecture Notes a.y

11 Consder, for example, the followng game (Mas-Colell 995, p.24) n whch player has three pure strateges Top, Mddle, Bottom, for short and player 2 has two pure strateges Left, Rght, for short LR, normal form representaton of the game n Fg..4. T, M, B. The payoffs of the two players are ndcated n the Left Rght Top 0, 0, 4 Mddle 4, 2 4, 3 Bottom 0, 5 0, 2 Top Bottom 5, 3 5, Fg..4 Game wth pure strategy M domnated by the mxed strategy,0, 2 2 Note that strategy M s not domnated by ether pure strategy T or B: t has a greater payoff than strategy T (B) f player 2 plays strategy R (L). M s, however, strctly domnated by the mxed strategy,0, 2 2 wth a payoff of 5 whatever the strategy played by player 2. Ths s not the only mxed strategy of player whch domnates the pure strategy M. It can be easly checked that all mxed strateges wth Top 4,6 strctly domnate M..3.2 The prsoners dlemma: soluton n strct domnance The Prsoners Dlemma game s depcted n Fg..5. The story behnd the game s the followng. Two suspects of havng commtted a crme are taken nto jal by the polce and held n separate cells, so that communcaton among the prsoners s made mpossble. The Dstrct Attorney has only ndrect evdence, whch wll lead the jury to a sentence of only 3 month jal unless the prsoners admt of havng commtted the crme they are accused of. To obtan a confesson, the Attorney meets each prsoner separately and nforms hm of the consequences of hs possble actons. If he s the only one to confess, he wll be rewarded wth a lght sentence of, say, month jal; but f he negates partcpatng to the crme whle the other prsoner confesses, he wll be heavly punshed wth a 2 month sentence to jal. If they both confess, the jury wll have mercy on them and sentence both to 6 month jal. D. Tosato Game Theory Lecture Notes a.y

12 The prsoners Dlemma lends tself to a presentaton n the terms of a game n normal form. The number I of players s equal to 2; the strategy space of each players contans two actons Confess, Don ' t Confess, for short ; 8 the payoffs reflect to story behnd the game. C, NC 2 Confess Don t Confess Confess -6, -6 -, -2 Don t Confess -2, - -3, -3 Fg..5 Payoff matrx of the Prsoners Dlemma The frst step n search of a soluton s to analyze the payoff matrx of the game to see f one of the two strateges s strctly domnant say for player. From defnton., we must, therefore, consder only the payoffs of ths player. At the rsk of beng overfastdous, these payoffs are reproduced n Fg..6. It s mmedately evdent that the strategy Confess strctly domnates the alternatve strategy Don t Confess. Ths means that the ratonal choce of player, based only on the knowledge of hs own payoffs (and not of the entre game) and wth no need to make any conjecture as to the decson of the other prsoner, s to play Confess, whch offers hm a hgher payoff than Don t Confess whchever the choce of the other player. Snce player s 2 ratonal choce s, for the same reasons, Confess, the equlbrum strategy profle of the game s Confess, Confess. 2 Confess Don t Confess Confess -6 - Don t Confess -2-3 Fg..6 Payoff of player n the Prsoners Dlemma Needless to say, the soluton n strct domnance has a paradoxcal flavor: the strategy profle Don ' t Confess, Don ' t Confess s preferable for both prsoners. The conflct between the game theoretc soluton, rgorously based on the applcaton of the exclusve prncple of ratonal behavor, and the effcent soluton, whch Pareto domnates the game theoretc soluton, s ndcatve of a stuaton that may arse n very dfferent and very real contexts, typcally n problems of ndustral organzaton. Ths crcumstance paves the way to heavy crtcsm of game theory solutons concepts and to proposals amed at fndng ways to reconcle equlbrum wth effcent solutons. We wll dedcate attenton to some of these problems further on n the course. 8 Fnk for Confess and Quet or Mum for Don t Confess are also used to ndcate the two strateges of the Prsoners Dlemma. D. Tosato Game Theory Lecture Notes a.y

13 .4 Iterated strct domnance The smple crteron that ratonal players would never use a strctly domnated strategy has led us to the determnaton of a unque soluton of the Prsoners Dlemma. Our purpose s now to understand f substantally the same dea that a ratonal player wll never play a strctly domnated strategy opens the way to the determnaton of a soluton n games n whch the drect elmnaton of strctly domnated strateges s, by tself, not suffcent to do t. The technque conssts n consderng the possblty that the terated elmnaton of strctly domnate strateges may reduce the payoff matrx to a sngle survvng cell as n the Prsoners Dlemma. As we wll see, ths s ndeed possble n some nstances, but the process of terated elmnaton requres a substantal change n the deductve capactes of the players: t s not enough for each player to be ndvdually ratonal; we have to ask players to know that all players are ratonal a property known as common knowledge of ratonalty (CKR). We wll separately examne the problems that may arse when the process of terated elmnaton s extended to weakly domnated strateges. To avod repeatng a long sentence, we wll dstngush between the soluton approach based on the exstence of a strctly domnant strategy and the approach based on the terated elmnaton of strctly domnated strateges n terms of drect domnance and terated domnance..4. Iterated elmnaton of strctly domnated strateges Consder the game depcted n Fg..6, n whch both players have three strateges, Top Mddle Bottom, for short S T M B, and S Left Center Rght S,, S2 L, C, R.,, 2,,, for short 2 Left Center Rght Top 0, 3, 0, 2 Mddle 2, 2, 0 2, Bottom, 0 0,, 0 Fg..6 Whle no pure strategy of player 2 s domnated by any other of hs pure strateges, strategy Bottom of player s strctly domnated by strategy Mddle. The elmnaton of strategy Bottom of player renders, n turn, strategy Center of player 2 strctly domnated by both hs other two strateges. The further elmnaton of ths strategy nduces player to dscard strategy Top, whch has now become strctly domnated by strategy Mddle. The fnal step of the process s now for player 2 to elmnate strategy Rght that gves hm a payoff of, as opposed to the payoff of 2 of strategy Left: By successve elmnaton of strctly domnated D. Tosato Game Theory Lecture Notes a.y

14 strateges of the two players only the strategy profle Mddle, Left remans. Ths profle of strateges s the soluton of the game. We have carred the process of terated elmnaton of strctly domnated strateges n an utterly mechancal way. However, what s the reasonng that justfes the players to successvely dscard some strateges wthout drectly knowng the prevous decsons of the other player? Here come the mportant logcal underpnnngs of terated domnance, whch make the dstncton wth regard to the drect domnance approach of the Prsoners Dlemma. Frst, the drect domnance approach demands the players to know only ther own payoffs: ndrect domnance assumes, on the contrary, that players know the entre structure of the game, n partcular the complete payoff matrx. Ths assumpton mples that the structure of the game s common knowledge: everybody knows that everybody knows that everybody knows and so on. Second, t s necessary to explan why player 2 decdes to elmnate hs pure strategy Center, whch s not strctly domnated by ether pure strategy Left or Rght. To ths end, we must assume not only, as we have already done, that the structure of the game s common knowledge, but also that player 2 knows that player s ratonal and that, as a consequence, he wll not play hs strctly domnated strategy Mddle. The further step n the process of terated elmnaton requres that player knows that player 2 knows that he s ratonal. The fnal step requres, n turn, that player 2 knows that player knows that he (player 2) knows that player s ratonal. The term for ths assumpton s common knowledge of ratonalty: everybody knows that everybody knows that everybody knows ( and so on) that players are ratonal. If the strategy space of every player s fnte the process of terated elmnaton of strctly domnated strateges necessarly ends n a fnte, though perhaps very long, number of steps. If the strategy space s nfnte, terated domnance leads to a regress to nfnty, as wll become apparent n the subsequent applcaton to Cournot s duopoly game. The process of terated elmnaton of strctly domnated strateges can be formalzed followng the descrpton gven above, wth player begnnng the process and players takng turns n the successve steps. 9 Let S 0 S 0 and S S be player s ntal pure and mxed strategy spaces. Defne the frst found of elmnaton of strctly domnated strateges n terms of the survvng undomnated strateges 0 (.0) S s S there does not S s.t. u, s u s, s, s S Movng to the followng round, defne (.) there does not 2 2 s.t. 2, 2, 2, 2, S s S S u s u s s s S Proceedng n ths way we can defne the subset of pure strateges survvng terated elmnaton respectvely of player and player 2 as n n S and S 2. At each step of the process 9 A more elegant formal descrpton, ndependent of the player who starts the process, has the players to move smultaneous at each step. D. Tosato Game Theory Lecture Notes a.y

15 the survvng subset of unelmnated pure strateges s strctly contaned n the precedng one; we then have n n3 0 n n S S S and S S S. Indcatng wth obvously non empty, the fnal stages of the process, we can conclude that the game admts a soluton n terated domnance f and only f both Nothng, of course, guarantees that the sets S S and and S 2 S 2 S contan only one element. and S 2 contan only one element, as the matrx game depcted n Fg..7 of the followng Secton.5. shows, snce none of the pure strateges of the players s strctly domnated. In ths case, we have S S,,2.,.4.2 Iterated weak domnance The process of terated domnance apples to the elmnaton of strctly domnated and not to weakly domnated strateges. The basc motvaton for ths excluson s that, when there are more than one pure strategy wth s weakly domnated by another pure strategy, the outcome of the process of terated elmnaton may be path dependent, that s t may depend on the order of elmnaton to the weakly domnated strateges. Consder the game of Fg..7 (Mas-Colell, p. 238). 2 Left Rght Top 5, 4, 0 Mddle 6, 0 3, Bottom 6, 4 4, 4 Fgura.7 Whle nether strategy of player 2 s domnated, the strateges Top and Mddle of player are weakly domnated by strategy Bottom. We can start the process of terated elmnaton by ether one. Suppose we start by elmnatng strategy Top suppose, more correctly, that player 2, knowng that player s ratonal, assumes that he wll not play Top. The sequence of terated elmnaton proceeds at ths pont by player assumng that the ratonal player 2 wll n turn elmnate the now weakly domnated strategy Rght. The remanng profle of strateges s then Bottom, Left, whch would accordngly be the soluton of the game. It can easly be checked that f the process of terated elmnaton of weakly domnated strateges starts wth the elmnaton of strategy Mddle, the resultng equlbrum profle of strateges would be Bottom, Rght..5 Ratonalzable strateges D. Tosato Game Theory Lecture Notes a.y

16 The noton of ratonalzable strateges was ndependently proposed by Bernhem (984) and Pearce (984) n a context of dffused crtcsm of Nash equlbrum and ts varous refnements. The nature of the crtcsm mpled by the ratonalzable strateges approach wll be consdered further on n the course. Our concern here s wth the defnton and the relatonshp wth domnance..5. Approach and defnton As above underlned, the ratonale for the elmnaton drect and terated - of strctly domnated strateges s the crteron that no ratonal player would choose a strategy that would delver a lower payoff than another, pure or mxed, strategy avalable to hm, regardless of the strateges chosen by the other players. To ths negatve approach of elmnatng poor strateges, ratonalzable strateges oppose a postve approach to the soluton of a game, namely of searchng for good strateges. These are the strateges that a player would be justfed playng, by an approprate chan of reasonng, when the structure of the game and the players ratonalty are common knowledge. By mplcaton, snce a player would never choose a strategy whch s not good, non ratonalzable strateges can be elmnated, n a way smlar to terated domnance, when lookng for a soluton of the game. I S u I Defnton.9 In the game,,, the mxed strategy S s ratonalzable f t s a best response to some conjectures (belefs) that player as to the strateges S exsts S of the other players. Formally: such that s ratonalzable f there (.2) u, u, for S By mplcaton, the mxed strategy S conjecture of player as to the strateges S s a best response. s never a best response f there s no of the other players for whch There are aspects concernng the role of conjectures that deserve beng stressed: ) player s supposed to formulate a conjecture as to the possble choces of the other players and determne hs best response on that bass; these conjectures correspond to ) the defnton we have adopted for mxed strateges - an expresson of player s uncertanty as to the strategy choces of hs rvals; s ratonalzable f there s some conjecture that makes t a best response. As regards the practcal approach to follow to test for razonalzable strateges n typcal two-player b-matrx games, t s convenent to proceed as follows. Start from the pure strateges of D. Tosato Game Theory Lecture Notes a.y

17 ) player and suppose (conjecture) that also the other player uses a pure strategy (n other terms, that he plays a degenerate mxed strategy). If a pure strategy of player s not a best response to any pure strategy of another player, t wll not be best response to any non degenerate mxed strategy of the other player; The assumpton of common knowledge of the structure of the game and of the ratonalty of the players has the mportant mplcaton that each player s conjecture about the other players strateges should not be arbtrary. He should expect the other players to use only strateges that are best response to some of ther conjectures. The argument of common knowledge of ratonalty leads now to a potental regress to nfnty about the recprocal conjectures of the players and supples the bass for the process of terated elmnaton of strateges that are never a best response (Fudenberg and Trole, 998, p. 49). I I S u Defnton.0 In the game,,, the strateges survve the terated elmnaton of never best response strateges represent the set S R of ratonalzable strateges. A convenent analytcal defnton of a best response strategy s by means of the correspondence BR : S S S to S. 0 S that, that s the mappng from some conjecture Defnton. A best response strategy BR correspondence s an element of the best response (.3),, and some BR S u u S We wll analyze the propertes of best response correspondences n Lecture Nore n. 2 when we wll examne the problem of exstence of a Nash equlbrum. It s tme to work out an example of the determnaton of best response strateges and of the set of ratonalzable strateges. Consder Bernhem s game depcted n Fg.7 Both players have 4 pure strateges: S a, a, a, a and S b, b, b, b In order to determne whch strateges of player are best response to some strateges of player 2 we proceed as ndcate n pont ) above. Suppose that player conjectures that player 2 s gong to play for certan strategy b,.e. that he adopts the degenerate mxed strategy 2,0,0,0 ; comparng the payoffs of the dfferent strateges of player n the column correspondng to player 2 playng b (respectvely 0,5,7,0 ) we mmedately see that 0 The reason for the defnton of the mappng as a correspondence,.e. a multvalued functon, wll become apparent n the constructon of the best response functon n mxed strateges. D. Tosato Game Theory Lecture Notes a.y

18 strategy a 3 s player s best response. We sgnal ths n the payoff matrx by a bar over the hghest payoff. As we contnue n ths way, modfyng the conjecture of player as to the pure strategy played for certan by player 2, we realze that all hs pure strateges are best response to some pure strategy of hs rval. We do the same for player 2, ndcate hs best response pure strateges by a bar under the hghest payoff and notce that strategy b 4 s never a best response. In fact, as ndcated n the last column of the payoff matrx of Fg..7, strategy b 4 s also strctly domnated by an equal probablty mx of strateges b and b 3. We proceed, therefore, to the elmnaton of ths strategy. When ths s done, we realze that strategy a 4 s no longer a best response, whch was prevously justfed by the presence of Ths s a short and mprecse wordng for the longer, but correct statement: player correctly conjectures that strategy b 4 s never best response; player 2, whose ratonalty s by assumpton common knowledge, wll therefore never use t; at ths pont player 2 correctly conjectures, on the same bass, that player wll never choose strategy a 4 b 4, whch ceases to be a best response. The set of ratonalzable strateges of the two players are, therefore,,, and S b, b, b S a a a Cartesan product SR S S2. ; the set of ratonalzable strateges of the game s the. b b 2 b 3 b 4 b b a 0, 7 a 2 a 3 a 4 5,2 7,0 0, 0 2, 5 3, 3 7,0 5,2 2, 5 0, 7 0, -2 0, 0 0,3 3,5;3,5 0, 2,5;2 0, 3,5;3,5 0, 0; 0 Fg..7 Bernhem s game As stated at the begnnng of ths Secton, ratonalzable strateges are strateges that a player would be justfed playng, by an approprate chan of reasonng, when the structure of the game and the players ratonalty are common knowledge. The chan of reasonng that leads to the concluson that a strategy s ratonalzable deserves to be spelled out n detal: mportant dfferences emerge among ratonalzable strateges from ths pont of vew. Consder strategy a of player. Ths strategy s ratonalzable by the conjecture that player 2 wll play strategy b 3, whch s n turn ratonalzable for player 2 by the conjecture that player wll play strategy a 3. Proceedng further n our chan of reasonng, a 3 s razonalzable by player s conjecture that player 2 wll play b, whch s ratonalzable by player 2 s D. Tosato Game Theory Lecture Notes a.y

19 conjecture that player plays a. At ths pont, we are back where we started and the loop leadng to ratonalzaton contnues to nfnty. In extreme synthess: a s ratonalzable by b 3 ; b 3 s ratonalzable by a 3 ; a 3 s ratonalzable by b ; b s ratonalzable by a ; a s ratonalzable by b 3 ; and so on n an nfnte regress wth players conjectures always dsproved. Consder, on the contrary, strategy a 2, whch s justfable by player s conjecture that player 2 wll play strategy wll play strategy b 2 a 2, whch s turn razonalzable for player 2 by the conjecture that player. In ths case, the loop s mmedately closed; both players conjectures are verfed. Ths property of conjectures to be verfed consttutes the equlbrum assumpton that leads to the defnton of Nash equlbrum, whch s a subset of ratonalzable strateges. In Defnton.9 we have assocated the set of ratonalzable strateges to the set of strateges survvng the elmnaton of never best response strateges. From Defnton.0 we have further drawn the concluson that the set of ratonalzable strateges concdes wth the set of best response strateges. In games wth a fnte set of pure strateges, as n Berghem s game, there s no analytcal way to dentfy razonalzable strateges: we have to check whch strateges are actually best response to some best responses of the other players. In the mxed extenson of a game, best response correspondences can be formally determned as wll be shown n Lecture Note n. 2, where examples of dervaton wll be gven and, n connecton wth the proof of exstence of Nash equlbrum, propertes of mxed best response strateges are defned..6 Relaton between the set of strctly undomnated and ratonalzable strateges It can be readly verfed that n the game depcted n Fg..7 the set of razonalzable strateges concdes wth the set of terated undomnated strateges. Whle ths s a result of general valdty n 2-person games, t need not hold n games wth more than 2 players. Let us start by provng the followng result Proposton. A strctly domnated strategy can never be a ratonalzable strategy. The proof of the converse proposton, namely that a ratonalzable strategy cannot be strctly domnated, can be found n Fudenberg and Trole (99, pp ). D. Tosato Game Theory Lecture Notes a.y

20 Proof. By drect comparson of defntons we have: from defnton.9, the pure strategy s not razonalzable f there exsts S such that u s, u, s for all s S. But ths condton s dentcal to the condton, gven n Defnton.7, of a strctly domnated strategy. We can draw a frst concluson, namely that the set of razonalzable strateges S R s weakly contaned n the set of undomnated strateges S UD : S R S UD. In games wth 3 or more players, on the contrary, the relaton between ratonalzable and strctly undomnated strateges may depend on the exstence of a possble correlaton among the choces of the opponents of a gven player. It s a standard assumpton n game theory that the mxed strateges, wth 2, are statstcally ndependent, n the sense that each player chooses hs mxed strategy ndependently of the smultaneous choces of the other players. The followng example, taken from Brandenburger (992), llustrates the pont- Consder the 3-player game n whch the strategy spaces of the players are: S U D S L R and S A B C 2,,, 3,,. We can represent the game n normal form by means of 3 payoff matrces, n whch player s the raw player, player 2 the column player and player 3 chooses the matrx. In each cell of the matrces (see Fg..8), the payoffs of the three players are ndcated n the proper order: frst that of player, then that of player 2 and fnally that of player 3. 2 Left Rght 2 Left Rght 2 Left Rght Up,,,0, Up 2,2,.7 0,0,0 Up,,0,0,0 Down 0,,0 0,0,0 Down 0,0,0 2,2,.7 Down 0,, 0,0, Player 3: plays A Player 3: plays B Player 3: plays C Fg..8 Payoff matrces of the 3-players game We frst check for strctly domnated strateges. To ths end we consder the payoffs of player from all the possble combnatons of strateges of player 2 and 3. The result s presented n Fg LA, LB, LC, RA, RB, RC, Up 2 0 Down Fg..9 Payoffs of player to the possble combnatons of strateges of players 2 and 3 D. Tosato Game Theory Lecture Notes a.y

21 The payoff matrx depcted n Fg..9 shows that nether strategy of player s strctly domnated. The same s true for player 2, snce the payoffs to hs strateges Left and Rght to all the possble combnatons of strateges of players and 3 are dentcal to the payoff matrx of Fg A A 2 2 UL, UR, DL, DR, 0 0 B C 0 0 C Fg..0 Payoffs of player 3 to the possble combnatons of strateges of players and 2 The payoff matrx of Fg..0 s constructed n the same way as that of Fg..9; t shows the payoffs of player 3 to all possble combnatons of strateges of players and 2. Inspecton reveals that no pure strategy s no strctly domnated ether by another pure strategy or by non degenerate mxed strategy. Note, n partcular, that strategy B s not domnated, for nstance, by an equal chance mxture of strateges A and C. However, strategy B s not a best response to any possble conjecture of ndependent choces of players and 2, as the bar over the payoffs show. Actually, only the pure strateges A and C can be best response to the possble combnatons of strateges of players and 2. Take, for nstance, the ndependent mxed strateges of players and 2 whch nduces a 2 2 probablty dstrbuton of over the columns of the matrx.0. the resultng payoffs to 4 strateges A, B and C of player 3 are respectvely.5,.3.5,.5. We conclude that the set of razonalzable strateges s strctly contaned n the set of undomnated strateges. Assume, on the contrary, that the mxed strateges of players and 2 are correlated, ether on account of preplay communcatons between the players or by the observaton of a common sgnal and are, e.g., U, L D, R and U R D L to strateges A, B and C 2,, 0. The expected payoffs of player 3 are now respectvely.5,.7,.5. So that we can conclude that, f correlated randomzaton s admtted, the set of razonalzable strateges does concde wth the set of undomnated strateges. D. Tosato Game Theory Lecture Notes a.y

22 References Aumann, R. (987), Correlated Equlbrum as an Expresson of Bayesan Ratonalty, Econometrca, vol 55, pp. -8 Aumann, R. and A. Brandenburger (995), Epstemc Condtons for Nash Equlbrum, Econometrca, vol. 63, n. 5, pp Brandenburger, A. (992), Knowledge and Equlbrum n Games. Journal of Economc Perspectves, vol. 6, no. 4, pp Fudenberg, D. and J. Trole (99), Game Theory, the MIT Press, Cambrdge, Mass, USA, pp. Harsany, J. (973), Games wth Randomly Dsturbed Payoffs: A New Ratonale for Mxed Strategy Equlbrum Ponts, Internatonal Journal of Game Theory, vol. 2, pp.-23, Maynard Smth, J. (972), On Evoluton, Ednburgh Unversty Press, Ednburgh D. Tosato Game Theory Lecture Notes a.y