Axiomatizations of Pareto Equilibria in Multicriteria Games

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1 ames and Economc Behavor 28, Artcle ID game , avalable onlne at on Axomatzatons of Pareto Equlbra n Multcrtera ames Mark Voorneveld,* Dres Vermeulen, and Peter Borm Department of Econometrcs, Tlburg Unersty, P.O. Box 90153, 5000 LE Tlburg, The Netherlands Receved October 3, 1997 We focus on axomatzatons of the Pareto equlbrum concept n noncooperatve multcrtera games based on consstency. Some axomatzatons of the Nash equlbrum concept have mmedate generalzatons. For strategc games t was shown that there exst no consstent refnements of the Nash equlbrum concept that satsfy ndvdual ratonalty and nonemptness on a reasonably large class of games. We show that such refnements of the Pareto equlbrum concept on multcrtera games do exst. Journal of Economc Lterature Classfcaton Number: C Academc Press 1. INTRODUCTION In a recent manfesto, Bouyssou et al observe that wthn multcrtera decson makng. a systematc axomatc analyss of decson procedures and algorthms s yet to be carred out. In ths paper, several axomatzatons of the Pareto equlbrum concept for multcrtera games are provded. In multcrtera games, a player can have more than one crteron functon. Shapley ntroduced Pareto. equlbrum ponts for the type of games that are a straghtforward generalzaton of the Nash equlbrum concept for uncrteron games. Axomatc propertes of the Nash equlbrum concept based on the noton of consstency have been studed n several artcles, ncludng Peleg and Tjs 1996., Peleg, Potters, and Tjs 1996., and Norde et al Informally, consstency requres that f a strategy combnaton x s a soluton of a game wth player set N, and players outsde a coalton S of players commt to playng accordng to x N S,.e., the strategy combnaton restrcted to the players n N S, then x s a soluton of the reduced S *Correspondng author. E-mal: M.Voorneveld@kub.nl $30.00 Copyrght 1999 by Academc Press All rghts of reproducton n any form reserved. 146

2 AXIOMATIZATIONS OF PARETO EQUILIBRIA 147 game. Several of these axomatzatons carry over to multcrtera games. The strong result of Norde et al , characterzng the Nash equlbrum concept on the set of mxed extensons of fnte strategc form games by nonemptness, the selecton of utlty maxmzng strateges n one person games, and consstency, does not have such an analogon n multcrtera games: We show that nonemptness, consstency and an mmedate extenson of utlty maxmzaton are not suffcent to axomatze the Pareto equlbrum concept. An addtonal property s provded to establsh an axomatzaton. 2. PRELIMINARIES A fnte multcrtera game s a tuple ² N, X., u. : N N, where N s a fnte set of players, X s the fnte set of pure strateges of player N, and for each player N, the functon u : Ł j N Xj r. maps each strategy combnaton to a pont n r. -dmensonal Eucldean space. The nterpretaton of ths last functon s that player N consders not just one, but r. dfferent crtera. For notatonal convenence, the set of fnte multcrtera games s denoted fnte. The payoff functons are extended to mxed strateges n the obvous way. The set of mxed extensons of fnte multcrtera games s denoted by. Ths set contans the set strategc of mxed extensons of fnte games n strategc form, snce these are smply multcrtera games n whch each player has only one crteron. For notatonal convenence, let X. denote the set of probablty m m measures on a fnte set X, and for m, Ý 14 m 1 s the unt smplex n m. Let ² N, X., u. : N N be a multcrtera game, let. N x X be a strategy profle n, and let S 2, N 4 N be a proper subcoalton of the player set N. The reduced game S, x of wth respect to S and x s the multcrtera game n whch the player set s S; each player S has the same set X of pure strateges as n ; the payoff functons u. are defned by u y. u y, x. S S S N S for all y X.. S S Notce that ths s the game that arses f the players n N S commt to playng accordng to x N S, the strategy combnaton restrcted to the players n N S. Defntons for reduced games on fnte and strategc are completely analogous.

3 148 VOORNEVELD ET AL. A soluton concept on s a functon whch assgns to each element a subset. X. N of strategy combnatons. Analo- gously one defnes a soluton concept on strategc or fnte. For strategc form games, we recall the followng axoms. A soluton concept on satsfes: strategc Nonemptness ( NEM ),f. for all strategc; Utlty Maxmzaton ( UM ), f for each one player game ² 4, X, u : we have that. x X. u x. u y. strategc y X.4 ; Consstency ( CONS ), f for each game strategc, each proper N subcoalton S 2, N 4, and each element x., we have S, that x x.. S. Norde et al prove: PROPOSITION 2.1. A soluton concept on strategc satsfes NEM, UM, and CONS f and only f NE, the Nash equlbrum concept. Equlbrum ponts for multcrtera games are ntroduced by Shapley Let ² N, X., u. : N N be a multcrtera game. A Pareto equlbrum s a strategy combnaton x X. N such that for each N, there does not exst an x X. such that: u x, x. u x, x., where for two vectors a, b m, we wrte a b f aj bj for all j 1,..., m. The soluton concept on assgnng to each the set of Pareto equlbra s denoted by PE. The Pareto equlbrum concept PE on fnte s, of course, defned n a smlar way by restrctng attenton to pure, rather than mxed, strateges. Consder a multcrtera game n whch player has r. crtera. For each N, let r. be a vector of weghts for the crtera,. N. The -weghted game s the strategc form game wth player set N, mxed strategy spaces X.. N, and payoff. functons defned for all N and x X. by x. N N ².: r., u x Ý u x. k1 k k. If each player assgns equal weght to all 1. r. hs crtera,.e., r. 1,..., 1 for all N, the weghted game s denoted by. Shapley e proves that all Pareto equlbra can be found by a sutable weghng of the crtera of the players: LEMMA 2.2. For each : x PE. f and only f there exsts for each N a ector of weghts such that x NE.. r. As a corollary, Pareto equlbra always exst n mxed extensons of fnte multcrtera games, snce for any vector of weghts the game has Nash equlbra n mxed strateges.

4 AXIOMATIZATIONS OF PARETO EQUILIBRIA FINITE MULTICRITERIA AMES Peleg and Tjs and Peleg, Potters, and Tjs provde several axomatzatons of the Nash equlbrum concept for fnte strategc form games. In ths secton two of these axomatzatons are extended to fnte multcrtera games. We use the followng axoms. A soluton concept on satsfes: Restrcted Nonemptness ( r-nem ), f for every fnte wth PE. we have. ; One Person Effcency ( OPE ), f for each one player game ² 4, X, u : we have that. x X y X : u y. fnte u x.4 ; Consstency ( CONS ), f for each fnte, each proper subcoal- N ton S 2, N 4, and each element x., we have that S, x x. S ; Converse Consstency ( COCONS ), f for each fnte wth at least two players, we have that.., where fnte ½ 5 N S, x. x Ł X S 2, N 4: x S.. N Accordng to restrcted nonemptness, the soluton concept provdes a nonempty set of strateges whenever Pareto equlbra exst. One person effcency clams that n games wth only one player, the soluton concept pcks out all strateges whch yeld a maxmal payoff wth respect to the order. Consstency means that a soluton x of a game s also a soluton of each reduced game n whch the players that leave the game play accordng to the strateges n x. Converse consstency prescrbes that a strategy combnaton whch gves rse to a soluton n every reduced game s also a soluton of the orgnal game. Our frst result ndcates that the axomatzaton of the Nash equlbrum concept on fnte strategc games of Peleg, Potters, and Tjs 1996, Theorem 3. n terms of restrcted nonemptness, one person ratonalty and consstency can be generalzed to multcrtera games. THEOREM 3.1. A soluton concept on fnte satsfes r-nem, OPE, and CONS f and only f PE. Proof. It s clear that PE satsfes the axoms. Let be a soluton ² concept on fnte satsfyng r-nem, OPE, and CONS. Let N, X., u. :. We frst show that. PE. N N fnte. Let x..if N 1, then x PE. by OPE. If N 1, then CONS

5 150 VOORNEVELD ET AL. 4, mples that for each N : x x.,so x y X z X : u z, x. u y, x.4 by OPE. Hence, x s a Pareto equlbrum: x PE.. Snce fnte was chosen arbtrarly, conclude that PE. ² To prove the converse ncluson,.e., that PE, agan let N, X., u. : and let x PE. N N fnte ˆ. Construct a fnte multcrtera game H as follows: let m N ; the player set s N m 4; players N have the same strategy set X as n ; player m has strategy set 0, 14; payoff functons to players N are defned, for all x, x. m 0, 14 X, by: N u x. f xm 1 e r. x m, x. f x m 0, x ˆx e r. f x 0, x xˆ m where e r. r. s the vector wth each component equal to one. the payoff functon to player m s defned, for all x, x. m m 0, 14 X,by N 0 f xm 0 x, x m m. 1 f x m 1, x xˆ 1 f x m 1, x xˆ Smple verfcaton ndcates that 1, ˆx. s the unque Pareto equlbrum of H. Snce H. PE H. by the prevous part of the proof, we conclude.. N, 1, xˆ. by r-nem that 1, ˆx H. Then by CONS, ˆx H.., N, 1, xˆ. snce by defnton H. Hence ˆx., fnshng our proof. Our second result shows that the axomatzaton of the Nash equlbrum concept on fnte strategc games of Peleg and Tjs 1996, Theorem n terms of one person ratonalty, consstency and converse consstency can also be generalzed to multcrtera games. THEOREM 3.2. A soluton concept on fnte satsfes OPE, CONS, and COCONS f and only f PE. Proof. PE satsfes the axoms. Let be a soluton concept on fnte that also satsfes them. As n the proof of Theorem 3.1, we have that. PE. for each by OPE and CONS. To prove that fnte

6 AXIOMATIZATIONS OF PARETO EQUILIBRIA 151 PE.. for each fnte, we use nducton on the number of players. In one player games, the clam follows from OPE. Now assume the clam holds for all fnte multcrtera games wth at most n players and let be an n 1. -player game. By CONS of PE: PE. PE. fnte. By nducton: PE... By COCONS of :... Combnng these three nclusons: PE... These results seem to llustrate that the axomatzatons that exst n the lterature for the Nash equlbrum concept generalze to the Pareto equlbrum concept for multcrtera games. Ths analogy, however, breaks down when we consder mxed extensons of fnte multcrtera games, as s done n the next secton. 4. MIXED EXTENSIONS OF FINITE MULTICRITERIA AMES Norde et al characterze the Nash equlbrum concept on mxed extensons of fnte strategc form games by nonemptness, utlty maxmzaton, and consstency cf. Proposton In ths secton t s shown that analogons of these propertes are not suffcent to characterze the Pareto equlbrum concept n mxed extensons of fnte multcrtera games. Frst, we lst some of the axoms used n ths secton. A soluton concept on satsfes: Nonemptness ( NEM ),f. for each ; Weak One Person Effcency ( WOPE ), f for each one player game ² 4, X, u : we have that. x X. y X. : u y. u x.4 ; Consstency ( CONS ), f for each, each proper subcoalton N S 2, N 4, and each element x., we have that x S S, x.; Converse Consstency ( COCONS ), f for each wth at least two players, we have that.., where ½ 5 N S, x. x Ł X. S 2, N 4: xs.. N. It s easy to see that PE on satsfes NEM See Lemma 2.2, WOPE, and CONS. LEMMA 4.1. If a soluton concept on satsfes WOPE and CONS, then PE. Hence, PE s the unque maxmal-under-ncluson soluton concept that satsfes WOPE and CONS.

7 152 VOORNEVELD ET AL. Proof. Let be a soluton concept on, satsfyng WOPE and CONS. Let and x..if N 1, then x PE. by WOPE. If 4, N 1, then for each player N : x x. by CONS, so x y X. z X.: u z, x. u y, x.4 by WOPE. Hence, x s a Pareto equlbrum: x PE.. Obvously, PE s the largest soluton concept on satsfyng NEM, WOPE, and CONS, but not the only one, as our next result shows. THEOREM 4.2. There exsts a soluton concept on whch satsfes NEM, WOPE, and CONS, such that PE. Proof. Defne as follows. Let ² N, X., u. :. N N If N 1, take x X y X : u y u x PE.. Ths guarantees that satsfes WOPE. If N 1, take ½. x Ł X. N : y X. such that N u y, x. u x, x. 5, where for a, b m we wrte a b f aj bj for all j 1,..., m and a b. Shapley calls ths the set of strong equlbrum ponts and provdes an exstence theorem smlar to Lemma 2.2, thereby establshng NEM. It s easy to see that s also consstent. To show that PE, consder the game n Fg. 1, where both players have two pure strateges and two crtera. Obvously B, L. PE., but B, L.., snce u T, L. 1 u B, L.. 1 In order to arrve at an axomatzaton of PE, we requre an addtonal axom. A soluton concept on satsfes: WEIHT f for every game and each vector. N Ł of weghts:... N r. FIURE 1

8 AXIOMATIZATIONS OF PARETO EQUILIBRIA 153 The soluton concept satsfes WEIHT f for every weght vector, the solutons of the assocated weghted strategc form game are solutons of the underlyng multcrtera game. Our man result, usng the strong theorems of Norde et al and Shapley 1959., shows that the Pareto equlbrum concept s the unque soluton concept on satsfyng NEM, WOPE, CONS, and WEIHT. THEOREM 4.3. A soluton concept on satsfes NEM, WOPE, CONS, and WEIHT f and only f PE. Proof. Straghtforward verfcaton and applcaton of Lemma 2.2 ndcates that PE ndeed satsfes the four axoms. Now let be a soluton concept on satsfyng NEM, WOPE, CONS, and WEIHT. By Lemma 4.1, PE. Now let, and x PE.. Remans to show that x.. By Lemma 2.2, there exsts a vector. Ł N N r. of weghts such that x NE.. Notce that restrcted to strategc, the set of mxed extensons of strategc form games, satsfes NEM, UM, and CONS, and hence by Proposton 2.1, H. NE H. for all H strategc. Consequently,. NE. x. So by WEIHT: x.. Fnally, wthout proof, we menton that the analogon of Theorem 3.2 also holds when we consder mxed extensons: THEOREM 4.4. A soluton concept on satsfes OPE, CONS, and COCONS f and only f PE. It s an easy exercse to show that the axoms used n our theorems are logcally ndependent. Remark 4.5. In the proof of Theorem 4.2 we mentoned the strong Pareto equlbrum concept. By slght modfcatons n the axoms n partcular, to weak. one person strong effcency and a weght axom concernng strctly postve, rather than nonnegatve, weghts., all axomatzatons n Secs. 3 and 4 have analogons for the strong Pareto equlbrum concept. Also, a result analogous to Theorem 4.2 holds. To see ths, defne a soluton concept on as follows. Let ² N, X., u. : N N If N 1, take x X y X : u y u x. Ths guarantees that satsfes weak. one person strong effcency. If N 1, take. NE. e, the set of Nash equlbra of the scalarzed game n whch the players assgn equal weght to ther crtera. By the exstence of Nash equlbra n mxed extensons, satsfes NEM. It s easy to see that s also consstent. To show that s not equal to the strong Pareto equlbrum concept, refer agan to the game n Fg. 1.

9 154 VOORNEVELD ET AL. T, L. s a strong Pareto equlbrum of, but the weghted payoff to player 2 ncreases from 2 to 2 f he devates to R, ndcatng that T, L.. NE.. e REFERENCES Bouyssou, D., Perny, P., Prlot, M., Tsoukas, A., and Vncke, P A Manfesto for the New MCDA Era, J Mult-Crt. Decson Anal. 2, Norde, H., Potters, J., Rejnerse, H., and Vermeulen, D Equlbrum Selecton and Consstency, ames Econ. Behaor 12, Peleg, B., Potters, J., and Tjs, S Mnmalty of Consstent Solutons for Strategc ames, n Partcular for Potental ames, Econ. Theory 7, Peleg, B. and Tjs, S The Consstency Prncple for ames n Strategc Form, Int. J. ame Theory 25, Shapley, L. S Equlbrum Ponts n ames wth Vector Payoffs, Naal Res. Log. Quarterly 6, 5761.