Game Theory: an Overview 2.1 Introduction

Transcription

1 2 Game Theory: an Overvew 2.1 Introducton Game theory s a branch of mathematcs that s concerned wth the actons of ndvduals who are conscous that ther actons affect each other. As such, game theory (hereafter GT) deals wth nteractve optmsaton problems. Whle many economsts n the past few centures have worked on what can be consdered game-theoretcal (hereafter G-T) models, John von Neumann and Oskar Morgenstern are formally credted as the fathers of modern game theory. Ther classc book Theory of Games and Economc Behavor [1] summarses the basc concepts exstng at that tme. GT has snce enjoyed an exploson of developments, ncludng the concept of equlbrum [2], games wth mperfect nformaton [3], cooperatve games [4, 5], and auctons [6], to name just a few. Ctng Shubk [7], In the 50s... game theory was looked upon as a curosum not to be taken serously by any behavoural scentst. By the late 1980s, game theory n the new ndustral organsaton has taken over: game theory has proved ts success n many dscplnes. GT s dvded nto two branches, called the non-cooperatve and cooperatve branches. The two branches of GT dffer n how they formalse nterdependence among the players. In the non-cooperatve theory, a game s a detaled model of all the moves avalable to the players. By contrast, the cooperatve theory abstracts away from ths level of detal, and descrbes only the outcomes that result when the players come together n dfferent combnatons. Though standard, the terms noncooperatve and cooperatve game theory are perhaps unfortunate. They mght suggest that there s no place for cooperaton n the former and no place for conflct, competton etc. n the latter. In fact, nether s the case. One part of the non-cooperatve theory (the theory of repeated games) studes the possblty of cooperaton n ongong relatonshps. And the cooperatve theory embodes not just cooperaton among players, but also competton n a partcularly strong, unfettered form. The non-cooperatve theory mght be better termed procedural game theory, the cooperatve theory combnatoral game theory. Ths would ndcate the real dstncton between the two branches of the subject, namely that the frst specfes

2 14 Producton Plannng n Producton Networks varous actons that are avalable to the players whle the second descrbes the outcomes that result when the players come together n dfferent combnatons. The goal of ths chapter s to gve a bref overvew about GT and, specfcally, about G-T concepts and tools. Obvously, due to the need of short explanatons, all proofs wll be omtted, and we wll only focus on the ntuton behnd the reported results. 2.2 Game Setup To break the ground for next secton on non-cooperatve games, basc GT notaton wll be ntroduced: the reader can refer to Fredman [8] and Fudenberg and Trole [9] f a more deep knowledge s requred. A game n the normal form conssts of: players (ndexed by = 1,2,..., n ), a set of strateges (denoted by x, = 1,2,..., n) avalable to each player and payoffs (π ( x1, x2,..., x n), = 1, 2,..., n ) receved by each player. Each strategy s defned on a set X, x X, so we call the Cartesan product X1 X2... X n the n strategy space (typcally the strategy space s R ). Each player may have a onedmensonal strategy or a mult-dmensonal strategy. However, n smultaneousmove games each player s set of feasble strateges s ndependent from the strateges chosen by the other players,.e., the strategy choce of one player does not lmt the feasble strateges of another player. A player s strategy can be thought of as the complete nstructon for whch actons to take n a game. For example, a player can gve hs or her strategy to a person that has absolutely no knowledge of the player s payoff or preferences and that person should be able to use the nstructons contaned n the strategy to choose the actons the player desres. Because each player s strategy s a complete gude to the actons that are to be taken, n the normal form the players choose ther strateges smultaneously. Actons, whch are adopted after strateges, are thus chosen and those actons correspond to the gven strateges. The normal form can also be descrbed as a statc game, n contrast to the extensve form whch s a dynamc game. If the strategy has no randomly determned choces, t s called a pure strategy; otherwse t s called a mxed strategy. There are stuatons n economcs and marketng n whch mxed strateges have been appled: e.g., search models [10] and promoton models [11]. In a non-cooperatve game the players are unable to make bndng commtments regardng whch strategy they wll choose before they actually choose ther strateges. In a cooperatve game players are able to make these bndng commtments. Hence, n a cooperatve game players can make sdepayments and form coaltons. The overvew here reported starts wth noncooperatve statc games.

3 Game Theory: an Overvew Non-cooperatve Statc Games In non-cooperatve statc games the players choose strateges smultaneously and are thereafter commtted to ther chosen strateges. The soluton concept for these games was formally ntroduced by John Nash [2] although some nstances of usng smlar concepts date back to a couple of centures. The concept s best descrbed through best response functons. Defnton 1. Gven the n-player game, player s best response (functon) to the strateges x of the other players s the strategy x * that maxmzes player s payoff π ( x, x ): x * ( x ) = arg max π ( x, x ). x If π s quas-concave n x the best response s unquely defned by the frstorder condtons. Clearly, gven the decsons of other players, the best response s the one that the best player can hope for. Naturally, an outcome n whch all players choose ther best responses s a canddate for the non-cooperatve soluton. Such an outcome s called a Nash equlbrum (hereafter NE) of the game. * * * * Defnton 2. An outcome ( x1, x2,..., xn ) s a Nash equlbrum of the game f x s * a best response to x for all = 1, 2,..., n. One way to thnk about an NE s as a fxed pont of the best response mappng n n R R. Indeed, accordng to the defnton, the NE must satsfy the system of equatons π x =0, for all. Recall that a fxed pont x of mappng f(x), n n R R s any x such that f( x) = x. Defne f( x1, x2,..., xn) = π x + x. By the defnton of a fxed pont, * * * * * * * * f( x1, x2,..., xn) = π( x1,..., xn) x + x π( x1,..., xn) x = 0, all. * Hence, x solves the frst-order condtons f and only f t s a fxed pont of mappng f(x) defned above. The concept of the NE s ntutvely appealng. Indeed, t s a self-fulfllng prophecy. To explan, suppose a player s able to guess the strateges of the other players. A guess would be consstent wth payoff maxmsaton (and therefore reasonable) only f t presumes that strateges are chosen to maxmse every player s payoff gven the chosen strateges. In other words, wth any set of strateges that s not an NE there exsts at least one player that s choosng a non payoff maxmzng strategy. Moreover, the NE has a selfenforcng property: no player wants to unlaterally devate from t snce such behavour would lead to lower payoffs. Hence the NE seems to be the necessary condton for the predcton of any ratonal behavour by players. Although attractve, numerous crtcsms of the NE concept exst. Two partcularly vexng problems are the non-exstence of equlbrum and the multplcty of equlbra. Wthout the exstence of an equlbrum, lttle can be sad regardng the lkely outcome of the game. If there are multple equlbra, then t s not clear whch one wll be the outcome. Indeed, t s possble the outcome s not even an equlbrum because the players may choose strateges from dfferent equlbra. In some stuatons t s possble to ratonalse away some equlbra va a refnement of the NE concept: e.g., tremblng hand perfect equlbrum [12], sequental equlbrum [13] and proper equlbra [14]. In fact, t may even be

4 16 Producton Plannng n Producton Networks possble to use these refnements to the pont that only a unque equlbrum remans. An nterestng feature of the NE concept s that the system optmal soluton (a soluton that maxmses the total payoff to all players) need not be an NE. In fact, an NE may not even be on the Pareto fronter: the set of strateges such that each player can be made better off only f some other player s made worse off. A set of strateges s Pareto optmal f they are on the Pareto fronter; otherwse a set of strateges s Pareto nferor. Hence, an NE can be Pareto nferor. The Prsoner s Dlemma game s the classc example of ths: only one par of strateges s Pareto optmal (both cooperate ), and the unque Nash equlbrum (both defect ) s Pareto nferor. 2.4 Exstence of Equlbrum An NE s a soluton to a system of n equatons (frst-order condtons), so an equlbrum may not exst. Non-exstence of an equlbrum s potentally a conceptual problem snce n ths case t s not clear what the outcome of the game wll be. However, n many games an NE does exst and there are some reasonably smple ways to show that at least one NE exsts. As already mentoned, an NE s a fxed pont of the best response mappng. Hence fxed-pont theorems can be used to establsh the exstence of an equlbrum. There are three key fxed pont theorems, named after ther creators: Brouwer, Kakutan and Tarsk. (see [15] for detals and references.) However, drect applcaton of fxed-pont theorems s somewhat nconvenent and hence generally not done (see [16] for exstence proofs that are based on Brouwer s fxed-pont theorem). Alternatve methods, derved from these fxed-pont theorems, have been developed. The smplest (and the most wdely used) technque for demonstratng the exstence of an NE s through verfyng concavty of the players payoffs, whch mples contnuous best response functons. Theorem 1. [17]. Suppose that for each player the strategy space s compact and convex and the payoff functon s contnuous and quas-concave wth respect to each player s own strategy. Then there exsts at least one pure strategy NE n the game. If the game s symmetrc (.e., f the players strateges and payoffs are dentcal), one would magne that a symmetrc soluton should exst. Ths s ndeed the case, as the next theorem ascertans. Theorem 2. Suppose that a game s symmetrc and for each player the strategy space s compact and convex and the payoff functon s contnuous and quasconcave wth respect to each player s own strategy. Then there exsts at least one symmetrc pure strategy NE n the game.

5 Game Theory: an Overvew Multple Equlbra Many games are just not blessed wth a unque equlbrum. The next best stuaton s to have a few equlbra. (The worst stuaton s ether to have an nfnte number of equlbra or no equlbrum at all.) The obvous problem wth multple equlbra s that the players may not know whch equlbrum wll preval. Hence, t s entrely possble that a non-equlbrum outcome results because one player plays one equlbrum strategy whle a second player chooses a strategy assocated wth another equlbrum. However, f a game s repeated, then t s possble that the players eventually fnd themselves n one partcular equlbrum. Furthermore, that equlbrum may not be the most desrable one. If one does not want to acknowledge the possblty of multple outcomes due to multple equlbra, one could argue that one equlbrum s more reasonable than the others. For example, there may exst only one symmetrc equlbrum and one may be wllng to argue that a symmetrc equlbrum s more focal than an asymmetrc equlbrum. In addton, t s generally not too dffcult to demonstrate the unqueness of a symmetrc equlbrum. If the players have one-dmensonal strateges, then the system of n frst-order condtons reduces to a sngle equaton and one need only show that there s a unque soluton to that equaton to prove the symmetrc equlbrum s unque. If the players have m-dmensonal strateges, m > 1, then fndng a symmetrc equlbrum reduces to determnng whether a system of m equatons has a unque soluton (easer than the orgnal system, but stll challengng). 2.6 Dynamc Games The smplest possble dynamc game was ntroduced by Stackelberg [18]. In a Stackelberg duopoly model, player 1 chooses a strategy frst (the Stackelberg leader) and then player 2 observes ths decson and makes hs own strategy choce (the Stackelberg follower). To fnd an equlbrum of a Stackelberg game (often called the Stackelberg equlbrum) we need to solve a dynamc two-perod * problem va backwards nducton: frst fnd the soluton x2 ( x 1) for the second player as a response to any decson made by the frst player: * π 2( x2, x1) x2( x1): = 0. x2 Next, fnd the soluton for the frst player antcpatng the response by the second player: * * * dπ π π 1( x1, x2( x1)) 1( x1, x2) 1( x1, x2) x 2 = + = 0. dx1 x1 x2 x1 Intutvely, the frst player chooses the best possble pont on the second player s best response functon. Clearly, the frst player can choose an NE, so the leader s always at least as well off as he would be n NE. Hence, f a player were allowed to choose between makng moves smultaneously or beng a leader n a game wth complete nformaton he would always prefer to be the leader.

6 18 Producton Plannng n Producton Networks 2.7 Smultaneous Moves: Repeated and Stochastc Games A dfferent type of dynamc game arses when both players take actons n multple perods. Two major types of ths game exst: wthout and wth tme dependence. In the mult-perod game wthout tme dependence the exact same game s played over and over agan (hence the term repeated games). The strategy for each player s now a sequence of actons taken n all perods. Consder one repeated game verson of the competng newsvendor game n whch the newsvendor chooses a stockng quantty at the start of each perod, demand s realsed and then leftover nventory s salvaged. In ths case, there are no lnks between successve perods other than the players memory about actons taken n all the prevous perods. A fascnatng feature of repeated games s that the set of equlbra s much larger than the set of equlbra n a statc game and may nclude equlbra that are not possble n the statc game. At frst, one may assume that the equlbrum of the repeated game would be to play the same statc NE strategy n each perod. Ths s, ndeed, an equlbrum but only one of many. Snce n repeated games the players are able to condton ther behavour on the observed actons n the prevous perods, they may employ so-called trgger strateges: the player wll choose one strategy untl the opponent changes hs play, at whch pont the frst player wll change the strategy. Ths threat of revertng to a dfferent strategy may even nduce players to acheve the best possble outcome (.e., the centralsed soluton) whch s called an mplct colluson. Many such threats are, however, non-credble n the sense that once a part of the game has been played, such a strategy s not an equlbrum anymore for the remnder of the game. To separate out credble threats from non-credble, Selten [19] ntroduced the subgame, a porton of the game (that s a game n tself) startng from some tme perod and a related noton of subgameperfect equlbrum (ths noton also apples n other types of games, not necessarly repeated), and equlbrum for every possble subgame (see Hall and Porteus [20] and van Meghem and Dada [21] for solutons nvolvng subgameperfect equlbra n dynamc games). 2.8 Cooperatve Games The dea behnd cooperatve game theory has been expressed n ths way: Cooperatve theory starts wth a formalzaton of games that abstracts away altogether from procedures and concentrates, nstead, on the possbltes for agreement. There are several reasons that explan why cooperatve games came to be treated separately. One s that when one does buld negotaton and enforcement procedures explctly nto the model, then the results of a non-cooperatve analyss depend very strongly on the precse form of the procedures, on the order of makng offers and counter-offers and so on. Ths may be approprate n votng stuatons n whch precse rules of parlamentary order preval, where a good strategst can ndeed carry the day. But problems of negotaton are usually more amorphous; t s dffcult to pn down just what the procedures are. More fundamentally, there s a feelng that procedures are not really all that relevant; that t s the possbltes for coalton formng, promsng and threatenng that are decsve, rather than whose

7 Game Theory: an Overvew 19 turn t s to speak. Detal dstracts attenton from essentals. Some thngs are seen better from a dstance; the Roman camps around Metzada are ndscernble when one s n them, but easly vsble from the top of the mountan [22]. The subject of cooperatve games frst appeared n the semnal work of von Neumann and Morgenstern [1]. However, for a long tme cooperatve game theory dd not enjoy as much attenton n economcs lterature as non-cooperatve GT. Cooperatve GT nvolves a major shft n paradgms as compared to noncooperatve GT: the former focuses on the outcome of the game n terms of the value created through cooperaton of (a subset of) players but does not specfy the actons that each player wll take, whle the latter s more concerned wth the specfc actons of the players. Hence, cooperatve GT allows us to model outcomes of complex busness processes that otherwse mght be too dffcult to descrbe (e.g., negotatons) and answers more general questons (e.g., how well s the frm postoned aganst competton). In what follows, we wll cover transferable utlty cooperatve games (ncludng two soluton concepts: the core of the game and the Shapley value). 2.9 N-Person Cooperatve Games Recall that the non-cooperatve game conssts of a set of players wth ther strateges and payoff functons. In contrast, n ths case, although players are autonomous decson makers, they may have an nterest n makng bndng agreements n order to have a bgger payoff at the end of the game. Ths agreement or partnershp s the basc ngredent of the mathematcal model of a cooperatve game, and t s called a coalton. Mathematcally, a coalton s a subset of the set of players N and we can denote t by S. To form a coalton S, t s requred that agreements take place nvolvng all players n the future coalton S. Whenever all players approve jonng n a new entty called coalton, we can say that the new coalton s formed. Jonng a coalton S also mples that there s no possble agreement between any member of S and any member not n S (set N\S). In short, the essental feature of a coalton s ts foundatonal agreement that bnds and reconsttutes the ndvduals as a coordnated entty. The grand coalton of all n players wll be referred as coalton N (there s a total of 2 n 1 possble coaltons); The empty coalton s a coalton made up of no members (the null set ). A coalton structure s a means of descrbng how the players dvde themselves nto mutually exclusve coaltons. Any exhaustve partton of the players can be descrbed by a set S { S1, S2,..., S m} of the m coaltons that are formed. The set S s a partton of N that satsfes three condtons: S j, j = 1,..., m S Sj =, for all j, and Sj = N. These condtons state that each player belongs to one and only one of the m non-empty coaltons wthn the coalton structure, and also specfes that none of

8 20 Producton Plannng n Producton Networks the players n any coalton m s connected to other players not n the coalton; fnally, the mutually exclusve unon of all coaltons m forms the grand coalton Characterstc Functon and Imputaton von Neumann and Morgenstern [23] ntroduced the term characterstc functon for the frst tme. More formally, we can defne that: Defnton 3. For each subset S of N, the characterstc functon ν of a game gves the bggest amount ν ( S) that the members of S can be sure of recevng f they act together and form a coalton, wthout any help from other players not n S. A restrcton on ths defnton s that the value of the game to the empty coalton s zero, that s, ν ( ).A further requrement that s generally made s called superaddtvty. Superaddtvty can be expressed as follows: ν ( S T) ν( S) + ν( T) for all ST, Nsuch that S T =. Ths means that the total payoff for the grand coalton s collectvely ratonal, because the total payoff to the players s always as much as what they would get ndvdually. Ths suggests the followng defnton. Defnton 4. A game n characterstc functon form conssts of a set of players, together wth a functon ν defned for all subsets of N, such that ν ( S T) ν( S) + ν( T) whenever S and T are dsjont coaltons of players. Games n whch at least one possble coalton can ncrease the total payoff of ts members are called essental, and those n whch there s no coalton that mproves the total payoff are called nessental. Mathematcally, an essental game s one n whch at least one of the superaddtve nequaltes ν ( S T) ν( S) + ν( T ) s strct. The specfc actons that players have to take to create ths value are not specfed: the characterstc functon only defnes the total value that can be created by utlsng all players resources. Hence, players are free to form any coaltons that are benefcal to them and no player s endowed wth power of any sort. We wll further restrct our attenton to the transferable utlty games n whch the outcome of the game s descrbed by real numbers π, = 1,..., Nshowng how the total created value (or utlty or pe) N π( N ) = π = 1 was dvded among players. Of course, one could offer a very smple rule prescrbng dvson of the value; for example, a fxed fracton of the total pe can be allocated to each player. However, such rules are often too smplstc to be a good soluton concept. A much more frequently used soluton concept of the cooperatve game theory s the core of the game. Ths concept can be compared to the NE for non-cooperatve games:

9 Game Theory: an Overvew 21 Defnton 5. The utlty vector π 1,..., π N s n the core (and wll be called mputaton) of the cooperatve game f t satsfes π( N) = ν ( N), group ratonalty, x ν ( ), ndvdual ratonalty. and { } The core of the game, ntroduced by Glles n 1953 [24], can be nterpreted through the added-value prncple. Defne (N\S) as a set of players excludng those n coalton S (coalton can nclude just one player). Then the contrbuton of a coalton S can be calculated as ν ( N) ν ( N\ S ). Clearly, no coalton should be able to capture more than ts contrbuton to the coalton (otherwse the remanng N\S players would be better off wthout the coalton S). Defnton 5 clearly satsfes the added-value prncple. Typcally, when analysng a game, one has to calculate an added value from each player: f the value s zero, the player s not n the core of the game. If the core s non-empty, the added values of all players n the core comprse the total value that the players create. As s true for NE, the core of the game may not exst (.e., t may be empty) and the core s often not unque. When the core s non-empty, the cooperatve demands of every coalton can be granted, but when the core s empty, at least one coalton wll be dssatsfed. Shubk [7] noted that a game wth a non-empty core s socologcally neutral,.e. every cooperatve demand by every coalton can be granted, and there s no need to resolve conflcts. On the other hand, n a coreless game, the coaltons are too strong for any mechansm to satsfy every coaltonal demand. However, a core set wth too many elements s not desrable, and t has lttle predctve power [25]. Imputatons n the core, where they exst, have a certan stablty, because no player or subset of players has any ncentve to leave the grand coalton. But snce many games have empty core, the core fals to provde a general soluton for n- person games n characterstc form. von Neumann and Morgenstern [24] proposed a dfferent soluton concept more generally applcable than the core. That proposal s called the von Neumann Morgenstern soluton or the stable set. The stable set s based on the concept of domnance, whch s explaned as follows. One mputaton s sad to domnate another f there s a subset of players who prefer the frst to the second and can enforce t by formng a coalton Shapley Value The concept of the core, though ntutvely appealng, also possesses some unsatsfyng propertes. As we mentoned, the core mght be empty or qute large or ndetermnstc. As t s desrable to have a unque NE n non-cooperatve games, t s desrable to have a soluton concept for cooperatve games that results n a unque outcome and hence has a reasonable predctve power. Shapley [26] offered an axomatc approach to the soluton concept that s based on three rather ntutve axoms. Frst, the value of the player should not change due to permutatons of players,.e., only the role of the player matters and not names or ndces assgned to players. Second, f a player s added value to the coalton s zero then ths player should not get any proft from the coalton, or n other words

10 22 Producton Plannng n Producton Networks only players generatng added value should share the benefts. Fnally, the thrd axom requres addtvty of payoffs: for any two characterstc functons ν 1 and ν 2 t must be that π( ν1 + ν2, N) = πν ( 1, N) + πν ( 2, N ). The surprsng result obtaned by Shapley s that there s a unque equlbrum payoff (called the Shapley value) that satsfes all three axoms. Theorem 3. There s only one payoff functon π that satsfes the three axoms. It s defned by the followng expressons for N and all ν : S!( N S 1)! π( ν) = ( ν ( S {} ) ν ( S) ). S N\ N! The Shapley value assgns to each player hs margnal contrbuton ν ( S {} ) ν ( S ) when S s a random coalton of agents precedng and the orderng s drawn randomly. To explan further, (see Myerson [14]), suppose players are pcked randomly to enter nto a coalton. There are N! dfferent orderngs for all players, and for each set S that does not contan player there are S!( N S 1)! ways to order players so that all of the players n S are pcked ahead of player. If the orderngs are equally lkely, there s a probablty of S!( N S 1)! N! that when player s pcked he wll fnd S players n the coalton already. The margnal contrbuton of addng player to coalton S s ν ( S {} ) ν( S ). Hence, the Shapley value s nothng more than a margnal (expected) contrbuton of addng player to the coalton. Due to ts unqueness, the concept of the Shapley value has found numerous applcatons n economcs and poltcal scences The Barganng Game Model To better understand the negotatonal mechansm and theory, whch wll be shown more specfcally n the next chapter, we here consder the former approach to ths ssue showng how to face the problem addressed by the barganng n cooperatve game theory. In order to do ths, consder a group of two or more agents facng wth a set of feasble outcomes, any one of whch wll be the result f t s accepted by unanmous agreement of all partcpants. In the event that no unanmous agreement s reached, a gven dsagreement outcome s the result. If the feasble outcomes are such that each partcpant can do better than the dsagreement outcome, then there s an ncentve to reach an agreement; however, so long as at least two of the partcpants dffer over whch outcome s most preferable, there s a need for barganng and negotaton over whch outcome should be agreed upon. Note that snce unanmty s requred, each partcpant has the ablty to veto any outcome dfferent from the dsagreement outcome. To model ths atomc negotaton process, we use the cooperatve barganng process ntated by Nash [27]. It s pertnent to menton that expermental barganng theory ndcates stronger emprcal evdence of ths barganng theory than any others. Nash engaged n an axomatc dervaton of the barganng soluton. The soluton refers

11 Game Theory: an Overvew 23 to the resultng payoff allocaton that each of the partcpants unanmously agrees upon. The axomatc approach requres that the resultng soluton should possess a lst of propertes. The axoms do not reflect the ratonale of the agents or the process n whch an agreement s reached but only attempts to put restrctons on the resultng soluton. Further, the axoms do not nfluence the propertes of the feasble set. Before lstng the axoms, we wll now descrbe the constructon of the feasble set of outcomes. Formally, Nash defned a two-person barganng problem (whch can be extended easly to more than two players) as consstng of a par 2 Fd, where F s a closed convex subset of d d, d s a vector n R, and ( ) = R. F s convex, closed, non-empty, and bounded. Here, F, the feasble set, represents the set of all feasble utlty allocatons and d represents the dsagreement payoff allocaton or the dsagreement pont. The dsagreement pont may capture the utlty of the opportunty proft. Nash looked for a barganng soluton,.e., an outcome n the feasble set that satsfed a set of axoms. The axoms ensure that the soluton s symmetrc (dentcal players receve dentcal utlty allocatons), feasble (the sum of the allocatons does not exceed the total pe), Pareto optmal (t s mpossble for both players to mprove ther utltes over the barganng solutons), the soluton be preserved under lnear transformatons and be ndependent of rrelevant alternatves. Due to constrants on space, the reader can refer to Roth [28] for a very good descrpton of the soluton approach and a more detaled explanaton of the axoms. The remarkable result due to Nash s that there s a barganng soluton that satsfes the above axoms and t s unque. Theorem 4 [27]. There s a unque soluton that satsfes all the axoms. Ths soluton, for every two-person barganng game Fd, s obtaned by solvng: arg max ( x d )( x d ). x= ( x, x ) F, x d The axomatc approach, though smple, can be used as a buldng block for much more complex barganng problems. Even though the axomatc approach s prescrptve, descrptve non-cooperatve models of negotaton such as the Nash demand game [29] and the alternatng offer game [30], reach smlar conclusons as Nash barganng. Ths somehow justfes the Nash barganng approach to model negotatons. In our dscusson, we have only provded a descrpton of the barganng problem and ts soluton between two players. However, ths result can easly be generalsed to any number of players smultaneously negotatng for allocatons n a feasble set References [1] von Neumann J, Morgenstern O (1944) Theory of games and economc behavour. Prnceton Unversty Press [2] Nash JF (1950) Equlbrum ponts n n-person games. Proc Nat Acad Sc USA 36, [3] Kuhn HW (1953) Extensve games and the problem of nformaton. In Contrbutons to the theory of games, vol II, Kuhn HW, Tucker AW, edtors. Prnceton Unversty Press

12 24 Producton Plannng n Producton Networks [4] Aumann RJ (1959) Acceptable ponts n general cooperatve n-person games. In Contrbutons to the theory of games, vol IV, Kuhn HW, Tucker AW, edtors. Prnceton Unversty Press [5] Shubk M (1962) Incentves, decentralzed control, the assgnment of jont costs and nternal prcng. Manag Sc 8: [6] Vckrey W (1961) Counter speculaton, auctons, and compettve sealed tenders. J Fnance, Vol.16, 8 37 [7] Shubk M (2002) Game theory and operatons research: some musngs 50 years later. Oper Res 50: [8] Fredman JW (1986) Game theory wth applcatons to economcs. Oxford Unversty Press [9] Fudenberg D, Trole J (1991) Game theory. MIT Press. [10] Varan H (1980) A model of sales. Am Econ Rev 70: [11] Lal R (1990) Prce promotons: lmtng compettve encroachment. Market Sc 9: [12] Selten R (1975) Reexamnaton of the perfectness concept for equlbrum ponts n extensve games. Int J Game Theory 4:25 55 [13] Kreps D, Wlson R (1982) Sequental equlbra. Econometra 50: [14] Myerson RB (1997) Game theory. Harvard Unversty Press. [15] Border KC (1999) Fxed pont theorems wth applcatons to economcs and game theory. Cambrdge Unversty Press. [16] Lederer P, L L (1997) Prcng, producton, schedulng, and delvery-tme competton. Oper Res 45: [17] Debreu D (1952) A socal equlbrum exstence theorem. Proc Nat Acad Sc USA 38: [18] Stackelberg H (1934) Markform and glechgewcht. Venna: Julus Sprnger. [19] Selten R (1965) Speltheoretsche behaundlung enes olgopolmodells mt nachfragetraghet. Z gesamte staatswss 12: [20] Hall J, Porteus E (2000) Customer servce competton n capactated systems. Manuf Serv Oper Manag 2: [21] van Meghem J, Dada M (1999) Prce versus producton postponement: capacty and competton. Manag Sc 45: [22] Aumann RJ (1989) Game theory. In Eatwell J, Mlgate M, Newman P (eds), The new Palgrave, New York, Norton, 8 9 [23] von Neumann J, Morgenstern O (1947) Theory of games and economc behavour. 2nd ed, Prnceton, NJ, Prnceton Unversty Press [24] Glles DB (1953) Some theorems on n-person games. PhD dssertaton, Department of Mathematcs, Prnceton Unversty, Prnceton, NJ [25] Kahan JP, Rapoport A (1984) Theores of coalton formaton, Lawrence Erlbaum, Hllsdale, NJ [26] Shapley LL (1953) A value for n-person games. In Kuhn HW, Tucker W (eds), Contrbutons to the theory of games II. Ann Math Studes n.28. Prnceton NJ, Prnceton Unversty Press [27] Nash JF (1951) Noncooperatve games. Ann Math 54: [28] Roth A (1979) Axomatc models n barganng, Sprnger-Verlag [29] Roth A (1995) Handbook of expermental economcs, Prnceton Unversty Press [30] Rubnsten A (1982) Perfect equlbrum n a barganng model. Econometrca 50:97 110

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