Game Theory Approach to Competitive Economic Dynamics

Transcription

1 Scuola d Dottorato n Economa Dottorato d Rcerca n Matematca per le Applcazon Economco-Fnanzare XXV cclo Game Theory Approach to Compettve Economc Dynamcs Thess submtted n partal fulfllment of the requrements for the degree of Doctor of Phlosophy n Mathematcs for Economc-Fnancal Applcatons by Ilara Poggo Program Coordnator Prof. Dr. Mara B. Charolla Thess Advsors Prof. Dr. Lna Mallozz Dr. Arsen Palestn

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3 To my father

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5 Contents Introducton x I Non Cooperatve Approach xv 1 Basc concepts on games and equlbra Fnte Games Non-Fnte Games Cournot-NE Approxmate Nash Equlbra Potental Games Bayesan Games Supermodular Games Multcrtera games Weak and Strong Pareto Equlbra Scalarzaton Relaton between Supermodular Multcrtera Games and Potental Multcrtera Games 35 3 Bayesan Pareto Equlbra n Multcrtera Games Introducton Bayesan Multcrtera Games (BMG)

6 v CONTENTS 3.3 Bayesan Potental Multcrtera Games (BPMG) Approxmate Bayesan Pareto Equlbra Exstence results: the scalarzaton approach An economc applcaton: the Cournot duopoly The contracton approach The potental approach A Bayesan Potental Game to Illustrate Heterogenety n Cost/Benet Characterstcs Introducton The setup of the model Man characterstcs of the game The Bayesan potental game Monotoncty of strateges Examples wth derent producton functons II Cooperatve Approach 81 5 TU-Games: an overvew Prelmnary Dentons Imputaton and Core Indces of Power A new perspectve on cooperatve games Extended cooperatve games Power ndces for extended cooperatve games Polluton-Control Game Derent Approaches

7 CONTENTS v Man Features Welfare allocaton among players A numercal smulaton Bblography 137

8 v CONTENTS

9 Introducton Game theory s a mathematcal theory that studes models of conct and cooperaton between ntellgent and ratonal decson-makers. Speccally t deals wth all real-lfe stuatons n whch ratonal people nteract each-other, that s when an ndvdual's sngle strategy depends on what other ndvduals choose to do. In ths sense t should not be suprsng that economcs s the eld n whch game theorsts develop ther man deas: the narrowness of economc world resources and the concts between countres to get them both create all the necessary ngredents for a game stuaton. In lterature game theory's brth concdes wth the book Theory of Games and Economc Behavor publshed n 1944 by the mathematcan John Von Neumann and the economst Oskar Morgenstern (see [123]). In game theory there s a classcal dstncton between non-cooperatve games and cooperatve games. In a non-cooperatve game, player's agreements ether do not occur or are not bndng, even f pre-play communcaton between players s possble. In contrast, n cooperatve game theory, player's commtments are bndng and enforceable. In non-cooperatve game theory the focus s manly on ndvdual behavor whle n cooperatve game the emphass s on the group of coaltons of players and on how to dvde the gans among coaltons. Ths thess deals both wth non-cooperatve and cooperatve games n order to apply the mathematcal theory to compettve dynamcs arsng from economcs, partcularly quantty competton n olgopoles and polluton reducton models n IEA (Internatonal Envronmental Agreements). In Chapter 3 a new model of game s dened: the Bayesan multcrtera game. In our opnon ths class of game s a very useful tool to model economc stuatons as Cournot duopoly game n

10 x Introducton whch rms produce two derent goods and a rm may have derent producton costs accordng to a gven probablty dstrbuton. The new dea s to thnk that a rm can produce two (or more) knd of products. For example rms produce two types of mneral water: wthout bubbles and wth bubbles or we can consder the damond market whch s tpcally dvded n two lnes of producton: one coverng the luxury market and the other for an ndustral use. Ths leads to optmze the derent prots at the same tme. On the other sde t s naturally magne that each rm prot can be aected by uncertanty: for example, the cost could be derent dependng on the used technology. We extend the denton of Bayesan game when the players have many objectves to optmze as dened by Shapley n [102] and nvestgate the exstence of strong and weak Bayesan Pareto equlbra ([30] and [31]). In the specal case of potental games ([74]) t s extended the result obtaned n [94] to Bayesan multcrtera game. In general t s used a scalarzaton approach to obtan an exstence theorem for weak and strong Bayesan Pareto equlbra (wbpe and sbpe for short, respectvely). The exstence of approxmate equlbra for Bayesan games (see ([73]) s also dscussed n the multcrtera case. There s a eld of game theory lterature whch deals wth envronmental ssues, n partcular a bg number of contrbutons have been publshed on polluton reducton models n recent years, see for example [1], [2], [20], [29], [42], [46], [59], [70], [85], [116] and [127]. The typcal ssues analyzed n ths lterature are the ncentve schemes of countres whch sgn a treaty and the stablzaton of Internatonal Envronmental Agreements. There are two man lnes of thought. The rst lne of research, exempled by [9], [10], [11], [22], [23], [35] and [45], sees the problem of desgnng (or sgnng) an IEA from the perspectve of coaltons stablty, a concept that has ts root n the cartel problem n ndustral organzaton lterature. The stablty of an IEA s ensured by two tests: the entry test that ntends to see whether t s n nterest of an already formed group of sgnatores to enlarge the IEA wth new members; the ext test that ntends to check whether t s n the nterest of a player to reman n the coalton. The general message carred out n ths lterature s that only a small number of countres wll end up sgnng an IEA,.e. only a small stable coalton

11 Introducton x can emerge. Ths approach s also known as the small coalton approach. The second lne of thought adopts cooperatve game theory as the analytcal framework. The allocaton problem s solved followng a two-step methodology. Frst, one computes the Pareto-optmal emsson levels and second, one uses a soluton concept based on cooperatve game theory (Core, Shapley value, etc) to allocate each player hs share of the total optmzed cooperatve payo. The remanng ssue s to nd the rght allocaton functon that guarantees the stablty of the formed soluton n the core sense. Contrary to the rst approach, here the stablty of the coalton s passve n the sense that the number of partecpatng countres s exogenous. In other words, ths approach supposes the exstence of a large number of countres that are predsposed to sgn the agreement, from whch the namng grand coalton approach orgnates. (See for example [26], [27], [28], [41], [44], [51], [52], [65] and [96]). Chapter 4 s devoted to llustrate a polluton-reducton model. In ths chapter an applcaton of Bayesan game s shown n the eld of envronmental economcs. Speccally we apply the model of Bayesan olgopoly games to an envronmental game where countres choose ther optmal emssons strategy maxmzng ther own prots, havng to take nto account that ther aggregate emssons amount to an envronmental cost suered by all of them. Here the type structure, whch s about margnal gans and producton functon, s nte and partally ordered. Under some hypothess the Bayesan game has a potental functon and, n ths way, t s smple to compute optmal pure strateges n classcal examples: n ths chapter we deal wth three derent models, whose respectve payos were endowed wth lnear, lnear-quadratc and lnear-logarthmc cost functons. The startng ponts are [5] and [18], whch on ther turn are related to [52] and [53]. In the above models, the nvolved countres am at maxmzng ther utlty functons by manoeuvrng ther emssons strateges, whch aect both ther revenues and the damage provoked by the pollutng actons. The countres are derentated based on these two crucal characterstcs: margnal gans and margnal damage, the former expressng compettveness and ntensty of producton, the latter nvolvng the negatve mpact of the economc actvtes on the envronment. Such double formulaton of uncertanty s somewhat smlar to the uncertanty n nverse demand functons and cost

12 x Introducton functons analyzed by [39] and [40] n ther papers on the exstence of Bayesan Cournot equlbra n duopoles. Derently from ther approach, here the focus s on monotoncty wth respect to the partal order of the type spaces rather than on exstence and unqueness of equlbra. In the second part, the envronmental aspect s faced wth a cooperatve pont of vew. Chapter 6 proposes a new perspectve on cooperatve games, by assumng that the nvolved players are supposed to face a common damage. The agents can choose to make an agreement and form a coalton or to defect and face such damage ndvdually. When such dsadvantage s modeled by a dynamc state varable evolvng over tme, cooperatng and non-cooperatng agents solve derent optmzaton problems, but they all must take nto account such state varable, as f t represented an externalty n all ther respectve value functons. Even f we just consder the cooperatve and statc aspects of such a game, the externalty has a key role n the worth of coaltons. The approach reles on a class of cooperatve games ncludng an external eect, such that the characterstc value functon s splt n two parts: one of them s standard, the other one s aected by externalty. It s worth descrbng ths new dea of externalty, whch bascally ders from the prevous characterzatons n lterature. Transferable utlty games wth postve externaltes were dened by [99], whch related such externalty to an ncrease n pay-o for the players n a specc coalton when the remanng coaltons commtted to mergng. That s, n presence of a partton of the set of agents and of multple coaltons, a group of players may enjoy a postve spllover orgnatng from a merger of external coaltons rather than from a strategc choce. On the other hand, the role of externalty s played, and ts amount s measured, by a derent state varable, not drectly dependng on the possbly undertaken agreements. Loosely speakng, externaltes arse n the same way as they do n standard dynamc olgopoly models (see [64]). When we relate ths dea to the welfare of a country dealng wth an emsson reducton strategy, we stress that the clean share of welfare s always postve, whereas the share ncludng the polluton eect s negatve, then the total welfare must be globally evaluated.

13 Introducton x The tools whch allowed us to study economc applcatons are dscussed n the rest of the thess. In partcular the rst part s devoted to non-cooperatve games. Chapter 1 shows classcal tools of non-cooperatve game theory. More precsely we underlne the dstncton between nte games and non-nte games dscussng Nash equlbra and approxmate Nash equlbra. A secton s dedcated to potental games: n such games, ntroduced n [83], the ncentve of all players to change ther strategy can be expressed usng a sngle global functon called the potental functon. Secton 1.5 deals wth Bayesan games. Harsany n [58] ntroduces games wth ncomplete nformaton. He proves the exstence of Bayesan equlbra for the case when the pure strategy spaces are nte. Many aspects of Bayesan games have been studed n lterature. Some of them regard the exstence of equlbra n these games. Mlgrom and Weber n [82] noted that the usual xed pont argument of Nash n [86] wth the standard assumptons s not applcable n provng the exstence of Bayesan equlbrum and hence ntroduced sucent condtons for the exstence. Balder n [6] and [7] generalzed ther result and Radner and Rosenthal n [98] presented sucent condtons for the exstence of pure strategy Bayesan equlbrum. Km and Yannels n [67] provde equlbrum exstence results for Bayesan games wth nntely many agents. Reny n [100] generalzes Athey's and McAdams results n [4] and [76] respectvely, on the exstence of monotone pure strategy equlbra n Bayesan games. Mallozz, Pusllo and Tjs n [73] consder stuatons where one of the players may have an nnte set of pure strateges, one crteron and a nte number of types and get an exstence theorem of approxmate equlbra. As for mxed strateges they are usually regarded as unappealng because they are not only hard to nterpret, but also, consdered as too complex for real players to use. Motvated by ths vew, game-theorsts have provded several purcaton theorems that descrbe when mxed strateges can be replaced by equvalent pure strateges. Several purcaton results have been obtaned for games wth a large number of players, see for example Cartwrght and Wooders n [24] and Carmona n [21]. As concerns the economcs lterature, Bayesan games play a key role: ndeed several types of uncertanty are consdered, and ther mplcatons on the provson of publc goods are dscussed. Gradsten n [54] assumes that consumers are uncertan about the contrbuton of other ndvduals. Under ths uncertanty,

14 xv Introducton the tme dynamcs of the prvate provson of publc good s derved. Gradsten et al. n [55] reexamne Warr's neutralty of the provson of a publc good wth respect to ncome dstrbuton (see [126]) n the context of uncertanty. In the model, uncertanty s about the consumers' ncome: each consumer knows her own endowment, but her nformaton regardng the endowments of other consumers s ncomplete. Keenan et al. n [66] examne the mpact of ncreased uncertanty on the provson of the publc good under a non-nash response and symmetrc equlbrum. Here agan, uncertanty s about the response of other contrbutors to a contrbuton to the publc good. In [61] the authors consder a publc good economy wth derental nformaton regardng consumers ncome and preferences. The prvate nformaton of each consumer s gven by her nformaton partton: that s, a consumer cannot dstngush between derent states of nature that belong to the same element n her nformaton partton. In [62] the authors apply the concept of nformaton advantage n [38] to a model of a publc good economy ntroduced n [61]: they consder a publc good economy where the consumers' state-dependent utltes have a multplcatve structure. Also as regards Cournot olgopoly n [38] authors study the value of nformaton: n an olgopoly where the market demand and the lnear cost are uncertan, a rm wth superor nformaton obtans hgher expected prots than a rm whose nformaton s nferor. Eny et al. n [38] also present an example of a Cournot duopoly wth quadratc costs where superor nformaton s dsadvantageous. Also n [32] and n [68] the authors show that n equlbrum a less nformed rm earns hgher expected prots than a more nformed rm. Fnally, the last secton of Chapter 1 s devoted to supermodular games ntroduced by Topks n [109] and very useful to descrbe, for example, olgopoly stuatons. Chapters 2 deals wth multcrtera games. In recent years, many authors have studed the game problem wth vector payos, for example, see [3] and [14]. Although many concepts have been suggested to solve multcrtera games, the noton of Pareto equlbrum, ntroduced by Shapley n [102], s the most studed concept n game theory. In [125], Voorneveld et al. ntroduced the new concept of deal Nash equlbrum for nte multcrtera games whch has the best propertes and Radjef and Fahem n [97] provde an exstence theorem for ths new soluton concept. Patrone, Pusllo, Tjs n [94] lnk the concept of multcrtera game wth that one of potental game. For

15 Introducton xv some applcatons see for example [31]. Chapter 5, n the second part of thess, provdes the tools of cooperatve game theory. In partcular n ths chapter TU-games are nvestgated.

16 xv Introducton

17 Part I Non Cooperatve Approach

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19 Chapter 1 Basc concepts on games and equlbra The theory of non-cooperatve games studes the behavor of agents n any stuatons where each agent's optmal choce may depend on her forecast of the choces of her opponents. In noncooperatve games the emphass s manly on the ndvdual behavor. 1.1 Fnte Games Let us denote wth N = {1,..., n} the players' nte set of cardnalty n. Denton 1.1. A non-cooperatve game wth a nte number n of players s a tuple G = (N, X 1,..., X n, u 1,..., u n ) where N, X s a non-empty set and represents the pure strategy space of player ; u : X := N X R s the utlty functon of player. If we also assume that N, X s a nte set, we say that G s a nte game, and we denote wth Γ fnte the class of nte games. Notaton 1.1. Take N, we denote wth (x, x ) the element belongng to X such that: x X ; x j N\{} X j =: X.

20 2 Basc concepts on games and equlbra Denton 1.2. Let G = (N, X 1,..., X n, u 1,..., u n ) be a non-cooperatve game. Player s strategy x X domnates strategy x X f u ( x, x ) u (x, x ) x X, x X, (1.1) wth a strct nequalty for at least one x X. A player s strategy s domnated f there exsts at least another strategy whch domnates t. Player s strategy x X strctly domnates strategy x X f u ( x, x ) > u (x, x ) x X, x X, (1.2) The most mportant soluton concept for non-cooperatve games are Nash equlbra. Denton 1.3. Let G = (N, X 1,..., X n, u 1,..., u n ) be a non-cooperatve game. A n-tuple ( x 1,..., x n ) X s a Nash Equlbrum (NE for short) [Strong Nash Equlbrum (sne for short)] for G f N we have: u ( x, x ) [>] u (x, x ) x X. (1.3) We denote wth NE(G) [sne(g)] the set of Nash equlbra [strong Nash equlbra] for G. We can observe that the condton (1.3) s equvalent to say that N we have: u ( x, x ) sup x X u (x, x ). (1.4) Nash equlbra are characterzed as xed ponts of partcular correspondences called best response correspondences. In general gven two sets X, Y a correspondence from X to Y s a map assocatng to each element of X a subset of Y. Denton 1.4. Let X, Y be topologcal spaces and F : X Y a correspondence. We say that: F s upper hemcontnuous (u.h.c. for short) n x X f for any open neghbourhood V of F (x), there exsts a neghbourhood U of x such that F (x ) V x U;

21 1.1 Fnte Games 3 F s lower hemcontnuous (l.s.c. for short) n x X f y F (x) and for any open neghbourhood V of y n Y, there exsts a neghbourhood U of x n X such that F (x ) V x U; F s upper hemcontnuous n X f t s upper hemcontnuous x X; F s lower hemcontnuous n X f t s lower hemcontnuous x X; F s contnuous n x X f t s upper hemcontnuous and lower hemcontnuous n x X; F s contnuous n X f t s upper hemcontnuous and lower hemcontnuous n X. Recall that, gven a correspondence F : X Y, we say that k X s a xed pont for F f k F (k). For further explanatons see [69]. Denton 1.5. Let G = (N, X 1,..., X n, u 1,..., u n ) be a non-cooperatve game. We dene for all N the correspondence R : X X where R ( x ) = arg max x X u (x, x ) = { x X : u ( x, x ) u (x, x ) x X }, that s the player s best reply when the other players play x. Let us call the correspondence such that R : X X R (x) = (R 1 (x 1 ),..., R n (x n )), x X. Then R s sad best reply for Nash equlbra of G. The followng theorem lnks xed ponts and Nash equlbra. Theorem 1.1. Let G = (N, X 1,..., X n, u 1,..., u n ) be a non - cooperatve game, and x X a strategy prole. Then

22 4 Basc concepts on games and equlbra x NE(G) x R ( x). Proof. It follows from Dentons 1.3 and 1.5. Now let us consder the followng example. Example 1.1. Let us consder the game G = (N, X 1, X 2, u 1, u 2 ) Γ fnte wth two players and payos' matrx gven by Table1.1 n whch X 1 = X 2 = {T, B} are the nte strategy spaces of player I and II respectvely. The utlty functons u 1, u 2 : X 1 X 2 R of player I and II respectvely are dened n the followng way: u 1 (T, T ) = 2 u 1 (T, B) = 0 u 1 (B, T ) = 4 u 1 (B, B) = 1; u 2 (T, T ) = 2 u 2 (T, B) = 4 u 2 (B, T ) = 0 u 2 (B, B) = 1. Table 1.1: Prsoner's dlemma I \ II T B T B We have that NE(G) = {(B, B)}. In the Example 1.1 we have seen that there exsts a unque NE. However the exstence and unqueness property are not ensured for ths knd of soluton as the followng examples show. In partcular such propertes are not ensured for the class Γ fnte. Example 1.2. Let us consder, as n the prevous example, the game G = (N, X 1, X 2, u 1, u 2 ) Γ fnte wth two players and payos' matrx gven by Table1.2 Table 1.2: Matchng Pennes I \ II P D P D Here there are not Nash equlbra.

23 1.1 Fnte Games 5 Example 1.3. Let us consder, as n the prevous example, the game G = (N, X 1, X 2, u 1, u 2 ) Γ fnte wth two players and payos' matrx gven by Table1.3 Table 1.3: Battle of the sexes I \ II T B T B We have that NE(G) = {(T, T ), (B, B)}. To get a result that ensures the exstence of at least a NE we have to consder the mxed extensons of nte games. Denton 1.6. Take G = (N, X 1,..., X n, u 1,..., u n ) Γ fnte. We dene mxed extenson of G the game G = (N, (X 1 ),..., (X n ), ũ 1,..., ũ n ), where for all N we have: (X ) = { p R X : p j 0 j = 1,..., X, } X p j = 1 that s the probablty space on X ; ũ : (X) := N (X ) R dened n the followng way: ũ (p) = where p (X). X 1 k 1 =1 X X n k n=1 p 1k1 p j p nkn u ( x 1k1,..., x j,..., x nkn ), We denote wth Γ mxed the class of mxed extenson of nte game. Nash equlbra and best reples for mxed extensons of a nte game are dened n a smlar way. Let us consder the followng results. Denton 1.7. Let X, Y be subsets of R n and F : X Y a correspondence. We say that F has a closed graph f the set {(x, y) X Y : y F (x)} s a closet subset of X Y.

24 6 Basc concepts on games and equlbra Theorem 1.2. Let K be a compact, convex and non-empty subset of R n and F : K K a corrspondence wth closed graph and where F (K) s a non-empty and convex set. Then there s a x K such that x F (x). Proof. See [69]. Theorem 1.3. Let S, T be metrc spaces and f : S T R a contnuous functon. Then the correspondence M : S T such that has closed graph. M(s) = arg max f (s, t) t T Proof. See [69]. Let us consder the followng denton: Denton 1.8. Let X R n be a convex set. A functon f : X R s sad quas concave f t R the set {x X : f(x) t} s convex. Theorem 1.4. Take G = (N, X 1,..., X n, u 1,..., u n ) Γ fnte then the mxed extenson of G, G has a NE. Proof. For every N the set R ( p ) s non-empty snce ũ s contnuous and (X ) s compact, and t s convex snce ũ s quas-concave on (X ) ; R s upper hemcontnuous (that s equvalent to have closed graph snce R s compact-valued), snce each ũ s contnuous. Thus by Theorem 1.2 R has a xed pont. Let us calculate Nash equlbra n mxed strateges of Example 1.2. Example 1.4. Let us consder the mxed extenson G = (N, (X 1 ), (X 2 ), ũ 1, ũ 2 ) n Example 1.2. We can dentfy the mxed-strategy space of player I and II, (X 1 ), (X 2 ) respectvely as the

25 1.1 Fnte Games 7 nterval [0, 1]. Let us call p = (p, 1 p) the mxed-strategy of player I and q = (q, 1 q) the mxedstrategy of player II. The utlty functons n mxed-strategy ũ 1, ũ 2 : (X 1 ) (X 2 ) R for player I and II respectvely are dened n the followng way: ũ 1 ( p, q ) = pq p (1 q) q (1 p) + (1 p) (1 q), Moreover ũ 2 ( p, q ) = pq + p (1 q) + q (1 p) (1 p) (1 q). R I ( q ) = argmaxp [0,1] pq p (1 q) q (1 p) + (1 p) (1 q) = argmax p [0,1] p (4q 2) + 1 2q. Then Smlarly ( ) Then NE G = {( 1 2, 2)} 1. ( ) R I q = ( ) R II p = {1} f q > 1 2 {0} f q < 1 2 [0, 1] f q = 1 2 {0} f p > 1 2 {1} f p < 1 2 [0, 1] f p = 1 2 We can note that Theorem 1.4 s only an exstence-theorem and does not ensure the unqueness as the followng example shows. Example 1.5. Let us consder the mxed extenson G = (N, (X 1 ), (X 2 ), ũ 1, ũ 2 ) n Example 1.3.

26 8 Basc concepts on games and equlbra We can dentfy the mxed-strategy space of player I and II, (X 1 ), (X 2 ) respectvely as the nterval [0, 1]. Let us call p = (p, 1 p) the mxed-strategy of player I and q = (q, 1 q) the mxedstrategy of player II. The utlty functons n mxed-strategy ũ 1, ũ 2 : (X 1 ) (X 2 ) R for player I and II respectvely are dened n the followng way: ũ 1 ( p, q ) = 3pq + (1 p) (1 q), ũ 2 ( p, q ) = pq + 3 (1 p) (1 q). Moreover R I ( q ) = argmaxp [0,1] 3pq + (1 p) (1 q) = argmax p [0,1] p (4q 1) + 1 q. Then Smlarly {1} f q > 1 4 ( ) R I q = {0} f q < 1 4 [0, 1] f q = 1 4 {1} f p > 3 4 ( ) R II p = {0} fp < 3 4 [0, 1] f p = 3 4 ( ) In Fgure1.1 the contnuous lne descrbes the graph of R I q, whle the dotted lne descrbes the ( ) ( ) graph of R II p. The crcles represent the Nash equlbra. Then NE G = { (0, 0), ( 3 4, 1 ) } 4, (1, 1) We can note that the equlbra (0, 0), (1, 1) correspond to pure-equlbra (T, T ), (B, B) respectvely. The last remark of Example 1.5 s true for all game G Γ fnte but the vceversa does not hold as we can see from Example 1.5.

27 1.2 Non-Fnte Games Non-Fnte Games Fgure 1.1: Nash Equlbra A non-cooperatve game wth non-nte strategy-spaces s called non-nte game. In partcular the mxed-extenton of a nte game s a non-nte game. In ths sense we have a corollary of Theorem 1.4 for non-nte game n general. Corollary 1.1. A non-cooperatve game G = (N, X 1,..., X n, u 1,..., u n ) has a NE f for each player : the strategy set X s a non-empty, compact and convex subset of an Eucldean space; the payo functon u s contnuous and quas-concave n x. Proof. It follows from Theorem Cournot-NE A soluton very smlar to NE was rst used by Cournot as early as 1838 n the framework of duopoly model. Ths model s consdered rghtly as one of the major classc examples of appled game theory n economcs. In ths model, the rms are supposed to choose smultaneously ther

28 10 Basc concepts on games and equlbra volume of output. See [108]. Two rms produce and sell a homogeneous good. Let us call q 1 and q 2 the quanttes produced by rm 1 and rm 2, respectvely. To smplfy matters, assume that there are not xed costs and that margnal costs are constant and equal to c, so that the total cost s: C = cq. Frms face an nverse demand functon gven by: P = max {a Q, 0}, where Q = q 1 + q 2, P s the prce of the good and a s a postve constant and, n generally, t s assumed to be the reservaton prce of the homogeneous good. In order to avod a corner soluton assume that a > c. Frms are supposed to choose smultaneously the quanttes q 1 and q 2. In ths model those varable are thus the players' strateges. The strategy sets of the player are dentcal and gven by: X 1 = X 2 = [0, a c]. The players' payo functons are here the prot functons of the rms: u 1 (q 1, q 2 ) = P (q 1, q 2 ) q 1 cq 1 u 2 (q 1, q 2 ) = P (q 1, q 2 ) q 2 cq 2. Or,more generally, after a clear change of notatons: If [a (q + q j ) c] q f 0 q a c q j u (q, q j ) = 0 f 2 c q j q a c. ( ) q, q j s a NE of ths game, then N : (1.5) u ( q, q j ) u ( q, q j ),

29 1.2 Non-Fnte Games 11 for all q X. Then for each player, q must be a soluton of: max q u = [a (q + q j ) c] q. It s easy to check that, by Corollary 1.1, n ths game there always exsts at least a NE. Wth the assumpton that q < a c, the rst-order condtons of ths optmzaton problem are necessary and sucent: u q = 0, = 1, 2, whch gves: (a q j c ) q = 2, = 1, 2. Solvng ths par of equatons leads nally to the outcome of the game: q 1 = q 2 = a c 3. The Cournot duopoly model can be extended to the case n whch there are many rms (n > 2): n ths case we speak of Cournot olgopoly. In general the early lterature on Cournot olgopoly has been concerned wth three man ssues: whether the model s quas-compettve,. e., ndustry output rses and prce falls wth addtonal rms (see for example [13], [49] and [78]); whether the model converges to perfect competton wth an nnte number of rms (see [25], [49], [60], [78], [103] and [119]). The thrd ssue concernng the queston whether the equlbrum soluton tself s dynamcally stable (see [56], [89] and [105]). Ths model has many varants n whch cost structures, nverse demand and value of nformaton change. For example n [12] authors consder a duopoly model wth quadratc cost functons. They show exstence and unqueness of ane equlbrum strateges and that, n equlbrum, expected prots of rm ncrease wth the precson of ts nformaton and decrease wth the precson of the rval's nformaton. Novshek and Sonnenschen n [88] consder a duopoly model wth constant costs and examne the ncentves for the rms to acqure and release prvate nformaton. Clarke n [33] consders an n-rm olgopoly model and shows that there s never a mutual ncentve for all rms n the ndustry to share nformaton unless they may cooperate on strategy once nformaton

30 12 Basc concepts on games and equlbra has been shared. Vves n [120] observed that more nformaton can be undesrable n the settng of Cournot olgopoly. More recently papers about the value of nformaton n ths framework are [39] and [68]. 1.3 Approxmate Nash Equlbra In ths secton we deal wth a derent concept soluton. We consder the followng denton ntroduced by Tjs n [106]. Denton 1.9. Let G = (N, X 1,..., X n, u 1,..., u n ) be a non-cooperatve game and ɛ > 0. Then ( x 1,..., x n ) X s an approxmate Nash equlbrum (ɛ-ne for short) for G f for each N we have: u ( x, x ) u (x, x ) ɛ x X. (1.6) We denote wth ɛ NE(G) the set of approxmate Nash equlbra of G. Obvously for ɛ = 0 the set of approxmate Nash equlbra s equal to the set of Nash equlbra. The condton (1.6) means that for each N we have: u ( x, x ) sup x X u (x, x ) ɛ. (1.7) Moreover f ɛ 1 < ɛ 2 then ɛ 1 NE(G) ɛ 2 NE(G) for each game G. Example 1.6. Let us consder G = (N, X 1, X 2, u 1, u 2 ) wth X 1 = X 2 = R and u 1 (x 1, x 2 ) = u 2 (x 1, x 2 ) = x 2 2 x2 1. By condton 1.7 we have that, by xng ɛ > 0, the par ( x 1, x 2 ) R 2 s a ɛ-ne f x 2 2 x sup u 1 (x 1, x 2 ) ɛ = sup ( x 2 x 2 ) 1 ɛ x 1 R x 1 R = x 2 2 ɛ,

31 1.3 Approxmate Nash Equlbra 13 and x 2 1 x sup u 2 ( x 1, x 2 ) ɛ = sup ( x 1 x 2 ) 2 ɛ x 2 R x 2 R That s f ( x 1, x 2 ) [ ɛ, ɛ] [ ɛ, ɛ]. = x 2 1 ɛ. ɛ NE (G) s the square wth center (0, 0) and sde 2 ɛ. In partcular NE(G) = {(0, 0)}. Example 1.7. Let G = (N, X 1, X 2, u 1, u 2 ) be a non-cooperatve game wth X 1 = X 2 = R and u 1 (x 1, x 2 ) = u 2 (x 1, x 2 ) = x 1 x 2. We have that: and sup u 1 (x 1, x 2 ) ɛ = sup x 2 x 1 ɛ x 1 R x 1 R < + x 2 = 0, sup u 2 ( x 1, x 2 ) ɛ = sup x 1 x 2 ɛ x 2 R x 2 R < + x 1 = 0. So the unque ɛ NE s the par ( x 1, x 2 ) = {(0, 0)}. In ths case ɛ NE(G) = NE(G). Next example shows that for some values of ɛ the exstence of approxmate Nash equlbra s not ensured. Example 1.8. Let us consder G = (N, X 1, X 2, u 1, u 2 ) a non-cooperatve game wth X 1 = X 2 = { 1, 1} and u 1 (x 1, x 2 ) = u 2 (x 1, x 2 ) = x 1 x 2, (see 1.2). By condton (1.7) we have that, by xng ɛ > 0, the par ( x 1, x 2 ) { 1, 1} { 1, 1} s a ɛ-ne f: and x 1 x 2 max x 1 { 1,1} u 1 (x 1, x 2 ) ɛ = max x 1 { 1,1} x 2x 1 ɛ = 1 ɛ, x 1 x 2 max x 2 { 1,1} u 2 ( x 1, x 2 ) ɛ = max x 2 { 1,1} x 1x 2 ɛ = 1 ɛ.

32 14 Basc concepts on games and equlbra then the par ( x 1, x 2 ) { 1, 1} { 1, 1} s a ɛ-ne ɛ 2. So f ɛ < 2, there are not approxmate Nash equlbra and, n partcular N E(G) =. From these examples we have seen that, as for Nash equlbra, nether the exstence (see Example 1.8) nor the unqueness (see Example 1.6) of approxmate Nash equlbra s guaranteed. Also n ths case there are exstence theorems. We quckly show a result due to Tjs (see [106]), but many other papers have been wrtten about ths topc (see for example [16], [104]). Theorem 1.5. Take G = (N, X 1,..., X n, u 1,..., u n ) Γ fnte such that for each player, u s an upper bounded functon on X = N X, then for each ɛ > 0, the mxed extenson of G, G has ɛ NE. Proof. See [106]. 1.4 Potental Games Potental games were ntroduced by Monderer and Shapley n [83] and studed for example n [124]. A game s sad a potental game f the ncentve of all players to change ther strategy can be expressed usng a sngle global functon called the potental functon. Ths potental functon provdes the necessary nformaton for the computaton of the pure Nash equlbra. Thus a potental functon s an economcal way to summarze the nformaton concernng pure Nash equlbra nto a sngle functon. Moreover, every nte game wth a potental functon has an equlbrum n pure strateges: snce the strategy space s nte, the potental functon acheves ts maxmum at a certan strategy prole. Denton Let G = (N, X 1,..., X n, u 1,..., u n ) be a non-cooperatve game. We say that G s a potental game f there s a functon (called potental functon) Π : X := N X R such that for each N, x, y X, x X we have: u (x, x ) u (y, x ) = Π (x, x ) Π (y, x ).

33 1.4 Potental Games 15 Table 1.4: A potental functon T B T 2 4 B 4 5 Example 1.9. Let us consder the Prsoner's dlemma game (see Example 1.1). It s a potental game, where a potental functon Π s gven n Table1.4. From Denton 1.10 t follows that a potental functon s not unque: f Π s a potental functon for a game G also Π + c, wth c R, s a potental functon for G. Then all potental functons of the Prsoner's dlemma game are Π k, wth k R, gven by Table1.5. Table 1.5: All potental functons T B T 2+k 4+k B 4+k 5+k Interestng classes of potental games are the coordnaton games and the dummy games. Denton Let G = (N, X 1,..., X n, u 1,..., u n ) be a non-cooperatve game. We say that G s a coordnaton game f u = u j, j = 1,..., n. That s the utlty functons are equal for each player. Denton Let G = (N, X 1,..., X n, u 1,..., u n ) be a non-cooperatve game. We say that G s a dummy game f u (x, x ) = u (y, x ) = 1,..., n, x, y X, x X. That s player s strategy choce does not aect her payo. A potental functon for a coordnaton game G = (N, X 1,..., X n, u 1,..., u n ) s Π = u 1 whle a potental functon for a dummy game s the null functon. Not all nte games admt a potental functon as we can conclude from:

34 16 Basc concepts on games and equlbra Theorem 1.6. Take G = (N, X 1,..., X n, u 1,..., u n ) Γ fnte a potental game, Π a potental functon for G, and G 1 = (N, X 1,..., X n, Π,..., Π) a coordnaton game. Then ) NE(G) = NE(G 1 ); ) G has a NE. Proof. See [107]. By Theorem 1.6 the game n the Example 1.2 s not a potental game. For non-nte games we have the followng theorem: Theorem 1.7. Let G = (N, X 1,..., X n, u 1,..., u n ) be a potental game and Π an upper bounded potental functon for G. Then ɛ > 0, G has ɛ NE. Proof. See [83]. The next results, dealng wth derentable games (.e. such that ther utlty functons are derentable) are well-known. Lemma 1.1. Let G = (N, X 1,..., X n, u 1,..., u n ) be a game n whch for each player, X are ntervals of real numbers. Suppose the utlty functons u are contnuously derentable N, and let Π : X R. Then Π s a potental functon for G Π s contnuously derentable, and u x = Π x, N. Theorem 1.8. Let G = (N, X 1,..., X n, u 1,..., u n ) be a game n whch for each player, X are ntervals of real numbers. Suppose the utlty functons u are twce contnuously derentable N. Then G s a potental game 2 u x x j = 2 u j x x j,, j N. (1.8) Moreover, f the utlty functons satsfy (1.8) and z s an arbtrary (but xed) strategy prole n X, then a potental functon for G s gven by Π (x) = N 1 0 u x (y (t)) (y ) (t) dt, (1.9)

35 1.5 Bayesan Games 17 where y : [0, 1] X s a pecewse contnuously derentable path n X that connects z to x. Example Let G c = (N, X 1,..., X n, u 1,..., u n ) be a Cournot olgopoly game n whch there s a lnear nverse demand functon P = max {a Q, 0}, where Q = n =1 q, and cost functons c 1,..., c n wth contnuous dervatves. We take X = [0, + ) and u (q 1,..., q n ) = P (q 1,..., q n ) q c (q ), N. It s smple to prove that G c s a potental game wth potental functon n n Π (q 1,..., q n ) = a q q 2 n q q j c (x ). 1.5 Bayesan Games =1 =1 1 <j n =1 We frequently wsh to model stuatons n whch some of the partes are not certan of the characterstcs of some of the other partes. The model of a Bayesan game s desgned for ths purpose: ndeed the case of perfect knowledge of payos s a smplfyng assumpton that may be a good approxmaton n some cases. A Bayesan game, or game wth ncomplete nformaton, s a game n whch, at the rst pont n tme when the players can begn to plan ther moves n the game, some players already have prvate nformaton about the game that other players do not know. The ntal prvate nformaton that a player has at the rst pont n tme s called the type of the player. The type of a player embodes any prvate nformaton (more precsely, any nformaton that s not common knowledge to all players) that s relevant to the player's decson makng. Ths may nclude, n addton to the player's utlty functon, her belefs about other players' utlty functons, her belefs about what other players beleve her belefs are, and so on. To dene a Bayesan game, see for example [50], we must specfy a set of players N and, for each player N, we must specfy a set of possble actons A, a set of possble types T, a probablty functon p and a utlty functon u. We let A = N A, T = N T. That s, A s the set of all possble proles of actons that the players may use n the game, and T s the set of all possble proles of types that the players may have n the game. For each player, we let T denote the set

36 18 Basc concepts on games and equlbra of all possble combnatons of types for the players other than. The probablty functon p must be a functon from T nto (T ), the set of probablty dstrbutons over T. That s, for any possble type t T, the probablty functon must specfy a probablty dstrbuton p ( t ) over the set T, representng what player would beleve about the other players' types f her own type were t. Thus, for any t T, p (t t ) denotes the subjectve probablty that would assgn to the event that t s the actual prole of types for the others players, f her own type were t. For any player N, the utlty functon u n the Bayesan game must be a functon from A T to R. These structures together dene a Bayesan game G, so we may wrte G = (N, A 1,..., A n, T 1,..., T n, p 1,..., p n, u 1,..., u n ). G s nte the sets N, A and T are nte N. When we study such a Bayesan game G, we assume that each player knows the entre structure of the game and her own actual type n T and ths fact s common knowledge among all the players n N. A strategy for a player n the Bayesan game G s dened to be a functon from her set of types T nto her set of acton A. We say that belefs (p ) N n a Bayesan game are consstent there s some common pror dstrbuton over the set of type prole t such that each players' belefs gven her type are just the condtonal probablty dstrbuton that can be computed from the pror dstrbuton by Bayes's formula. That s, belefs are consstent there exsts some probablty dstrbuton p (T ) such that: p (t t ) = p (t, t ) p (t ) N. (1.10) Because we consder n the followng consstent belefs under condton 1.10 we denote G = (N, A 1,..., A n, T 1,..., T n, p, u 1,..., u n ) nstead of G = (N, A 1,..., A n, T 1,..., T n, p 1,..., p n, u 1,..., u n ). A play of such a game proceeds as follows: before the types are announced each player chooses a strategy x A T 1. If the type prole s t = (t 1,..., t n ) then player s payo s 1 In general, gven two sets X and Y, the notaton X Y ndcates the set of functons from Y to X, that s X Y = {f f : Y X}

37 1.5 Bayesan Games 19 u (x 1 (t 1 ), x 2 (t 2 ),..., x n (t n ), t 1,..., t n ). The a pror expected payo for player when the players use strateges x 1,..., x n respectvely s a functon U : A T 1 1 ATn n R such that U (x 1,..., x n ) = t T p(t)u (x 1 (t 1 ), x 2 (t 2 ),..., x n (t n ), t 1,..., t n ), beng p(t) the probablty dstrbuton of player when her type prole s t. Denton Let G = (N, A 1,..., A n, T 1,..., T n, p, u 1,..., u n ) be a Bayesan game. We say that a strategy prole x = ( x 1, x 2,..., x n ) A T 1 1 AT 2 2 ATn n (for short BNE) f N, x A T we have s a Bayesan Nash equlbrum U ( x) U (x, x ). Example Let G = (N, A 1, A 2, T 1, T 2, p, u 1, u 2 ) be a nte Bayesan game where: N = {1, 2} ; A 1 = {a 1, b 1 }, A 2 = {a 2, b 2 } ; T 1 = { t 1 1}, T2 = { t 1 2, t2 2}. The functons u 1, u 2 : (A 1 A 2 ) (T 1 T 2 ) R are represented by the bmatrces Table 1.6 and Table 1.7. Table 1.6: u 1 1 \ 2.1 a 2 b 2 a b The rst one represents the case n whch player 2's type s t 1 2, whle the second one represents the case n whch player 2's type s t 2 2. We can note that the player 1's payos are the same n both matrces.

38 20 Basc concepts on games and equlbra Table 1.7: u 2 1 \ 2.2 a 2 b 2 a b We suppose that Nature extracts wth probablty P [0, 1] the type t 1 2 (obvously wth probablty 1 P the type t 2 2 ). So we have that p ( t 1 1, t1 2) = P and p ( t 1 1, t 2 2) = 1 P. Then the values of U 1 and U 2 are gven n the bmatrx Table 1.8, Now we want to compute the Table 1.8: A pror expected payo functons U 1 \U 2 a 2 b 2 a 1 1 P P b 1 0 2P Bayesan Nash equlbra for ths game dependng on P. If P 2 3 BNE = {(a 1, a 2 ), (b 1, b 2 )} ; If P < 2 3 BNE = {(b 1, b 2 )}. Now we ntroduce the noton of mxed extenson of a Bayesan game. Denton Let G = (N, A 1,..., A n, T 1,..., T n, p, u 1,..., u n ) be a Bayesan game. Then the mxed extenson of G s the Bayesan game G = (N, Ã1,..., Ãn, T 1,..., T n, p, ũ 1,..., ũ n ), where à s the famly of probablty measures (on the σ algebra of all subsets of A ) wth nte support. Such probablty measures are the form µ = s k=1 p ke ak where a 1,..., a s A, p k 0, for all k {1,..., s} and s k=1 p k = 1, where e ak (B) = { 1 f B A, a k B 0 f B A, a k / B, Furthermore ũ (µ,..., µ n, t) = u (a 1,..., a n, t) dµ 1 (a 1 )... dµ n (a n ) for all N and (µ 1,..., µ n ) à = N Ã.

39 1.6 Supermodular Games Supermodular Games The class of supermodular games was ntroduced by [109] and further studed by [81], [110], [121] and [122]. Supermodular games are games n whch each player's margnal utlty of ncreasng her strategy rses wth ncreases n her rvals' strateges. In such games the best response correspondences are ncreasng, so that the players' strateges are strategc complements. When there are two players, a change n varables allows ths framework to also accomodate the case of decreasng best responses. Supermodular games are partcularly well behaved: they have pure-strategy Nash equlbra. The upper bound of player s Nash-equlbrum strateges exsts and t s a best response to the upper bounds of her rvals' sets of Nash-equlbrum strateges, and smlarly for the lower bounds. The smplcty of supermodular games makes convexty and derentablty assumptons unnecessary, although they are satsed n many applcatons, for example n the Cournot duopoly. Let us recall some dentons about supermodular games. Denton A partally ordered set (POSET) s a set X on whch there s a bnary relaton that s reexve, antsymmetrc and transtve. Denton Let us consder a partally ordered set X and a subset Y of X. If y X and y x for each x Y, then y s a lower bound for Y ; If z X and x z for each x Y, then z s an upper bound for Y. When the set of lower bounds of Y has a greatest element, then ths greatest lower bound of Y s the nmum of Y n X. When the set of upper bounds of Y has a least element, then ths least upper bound of Y s the supremum of Y n X. Denton If two elements x 1 and x 2 of a partally ordered set X have a supremum n X, t s called the meet of x 1 and x 2 and t s denoted by x 1 x 2 ; If two elements x 1 and x 2 of a partally ordered set X have a nmum n X, t s called the jon of x 1 and x 2 and t s denoted by x 1 x 2.

40 22 Basc concepts on games and equlbra Denton A partally ordered set that contans the jon and the meet of each par of ts elements s a lattce. If a subset Y of a lattce X contans the jon and the meet (wth respect to X) of each par of elements of Y, then Y s a sublattce of X. Remark 1.1. The real lne R wth the natural orderng s a lattce wth x y = max {x, y} and x y = mn {x, y} x, y R. Also R n, (n > 1) wth the usual partal order s a lattce wth x y = (x 1 y 1,..., x n y n ) and x y = (x 1 y 1,..., x n y n ), x = (x 1,..., x n ), y = (y 1,..., y n ) R n. Any subset of R s a sublattce of R, and a subset X of R n s a sublattce of R n f x, y X we have that x y, x y X. Denton A supermodular game G = (N, X 1,..., X n, u 1,..., u n ) s a tuple where N = {1,..., n} s a nte set of players; N, X R m() (for some m () N) and X s the strategy space of player, X = N X s the cartesan product of the strategy spaces; u : X R s the payo functon of player ; N, X s a sublattce of R m() ; N, u have ncreasng derences on X,.e. x = (x 1,..., x n ), y = (y 1,..., y n ) X such that x y, we have u (x 1,..., x 1, x, x +1,..., x n ) u (y 1,..., y 1, x, y +1,..., y n ) u (x 1,..., x 1, y, x +1,..., x n ) u (y 1,..., y 1, y, y +1,..., y n ) ; N, u s supermodular n the th coordnate,.e. x = (x 1,..., x n ), y = (y 1,..., y n ), z = (z 1,..., z n ) X we have u (z 1,..., z 1, x, z +1,..., z n ) + u (z 1,..., z 1, y, z +1,..., z n )

41 1.6 Supermodular Games 23 u (z 1,..., z 1, x y, z +1,..., z n ) + u (z 1,..., z 1, x y, z +1,..., z n ). Increasng derences pont out that an ncrease n the strateges of player s rvals rases the desrablty of playng a hgh strategy for player. We can observe that supermodularty s automatcally satsed f for each N, X s sngledmensonal. We wll need supermodularty n the case of mult-dmensonal strategy spaces to prove that each player's best responses are ncreasng wth her rvals' strateges. For example the Cournot duopoly dened n Subsecton of Chapter 1 s a supermodular game. From [110] we have the followng propostons. Proposton 1.1. Let f : R n R be a derentable functon on R n, then f has ncreasng derences on R n f x s ncreasng n x j for each, j = 1,..., n wth j and x = (x 1,..., x n ). Proof. See [110]. Proposton 1.2. Let f : R n R be a twce derentable functon on R n, then f has ncreasng derences on R n Proof. See [110]. 2 f x x j 0 for each, j = 1,..., n wth j and x = (x 1,..., x n ). The followng exstence theorem s due to Topks n [109]. Theorem 1.9. Let G = (N, X 1,..., X n, u 1,..., u n ) be a supermodular game. If, for each N, X s compact and u s upper hemcontnuous n x for each x X, then the set of pure-strategy Nash equlbra s nonempty and possesses greatest and least equlbrum ponts. Proof. See [109].

42 24 Basc concepts on games and equlbra

43 Chapter 2 Multcrtera games Multcrtera (or multobjectve) optmzaton problems typcally have conctng objectves, and a gan n one objectve s, sometmes, a loss for another. Therefore the denton of optmalty s not obvous as n the one-crteron case. However n many settngs, mathematcal models nvolvng more than one objectve seem much more adherent to the real problems. Formally, a multcrtera optmzaton problem can be formulated as Optmze f 1 (x),..., f r (x) (2.1) subject to x D, where D denotes the feasble set of alternatves and r N the number of crteron functons f k : D R, k = 1,..., r. See for example [30], [31], [101] and [124]. In ths chapter we study the stuaton n whch there s not only a conct between crtera, but there are, also, many optmzaton problems to solve smultaneously: that s we deal wth multcrtera games. In recent years, many authors have studed the game problem wth vector payos, for example, see [3] and [14]. Although many concepts have been suggested to solve multcrtera games, the noton of Pareto equlbrum, ntroduced by Shapley n [102], s the most studed concept n game theory. In [125], Voorneveld et al. ntroduced the new concept of deal Nash equlbrum for nte multcrtera games whch has the best propertes and Radjef and Fahem n [97] provde an exstence

44 26 Multcrtera games theorem for ths new soluton concept. Patrone, Pusllo, Tjs n [94] lnk the concept of multcrtera game wth that one of potental game. For some applcatons see for example [31]. 2.1 Weak and Strong Pareto Equlbra Denton 2.1. A non-cooperatve multcrtera game s a tuple G m = (N, X 1,..., X n, u 1,..., u n ) where for each N N s a nte set and represents the set of players ; X s a non-empty set and represents the pure-strategy space of player ; u : X := N X R m s the utlty functon of player, where m s the number of objectves. Let us denote wth Γ m fnte the class of nte multcrtera games. Remark 2.1. We recall the partal order denton n R m. For all a, b R m, we say that: a = b f a = b = 1,..., m; a b f a b = 1,..., m; a b f a b = 1,..., m, and a b; a > b f a > b = 1,..., m. Denton 2.2. Let G m = (N, X 1,..., X n, u 1,..., u n ) be a multcrtera game. Then the strategy prole x X s a weak Pareto equlbrum for G m f N x X such that u (x, x ) > u ( x, x ); a strong Pareto equlbrum for G m f N x X such that u (x, x ) u ( x, x ).

45 2.1 Weak and Strong Pareto Equlbra 27 Let us denote wth wp E(G m ) and sp E(G m ) weak and strong Pareto equlbra of G m. From Denton 2.2 we can note that, n one-crteron case, weak [strong] Pareto equlbra correspond to NE [sne] respectvely, for the game. It s clear that a strong Pareto equlbrum s also a weak Pareto equlbrum but the vceversa does not hold as the followng example shows. Example 2.1. Let us consder the nte bcrtera game G 2 = (N, X 1, X 2, u 1, u 2 ) wth payo matrx gven by Table2.1 where X 1 = {T, B}, X 2 = {L, R} are the strategy spaces of player I and II respectvely. The utlty functons u 1, u 2 : X 1 X 2 R 2 of player I and II respectvely are dened n the followng way: u 1 (T, L) = (3, 4) u 1 (T, R) = (4, 3) u 1 (B, L) = (3, 5) u 1 (B, R) = (1, 2), u 2 (T, L) = (3, 2) u 2 (T, R) = (2, 3) u 2 (B, L) = (1, 1) u 2 (B, R) = (2, 2). We have that wp E(G) = {(T, L), (T, R)} whle sp E(G) = {(T, R)}.Then sp E(G) wp E(G). Table 2.1: Weak Pareto Equlbra I \ II L R T (3, 4) (3, 2) (4, 3) (2, 3) B (3, 5) (1, 1) (1, 2) (2, 2) Pareto equlbra can be characterzed as xed ponts of best reply correspondences. Denton 2.3. Let G m = (N, X 1,..., X n, u 1,..., u n ) be a multcrtera game. We dene for each N wb : X X where wb (x ) = {x X y X : u (y, x ) > u (x, x )}. Call X := N X, and dene wb : X X