Mathematics of Operations Research November 1989 [antikvár]

MATHEMATtCS OF OPERATIONS RESEARCH Vol. 14. No. 4, November 1989 Primed ia U.S.A. STABLE EQUILIBRIA—A REFORMULATION Part I. Definition and Basic Properties*^ jean-françois mertens Vniversite Catholique de Louvai» A reformulation of stable equilibria is given, yielding a number of additional properties like backwards induction while still giving "typically", i.e., except in "made up" examples, the same solutions. 1. Introduction. 1.1. Tiie paper is part of a programme to verify whether it is possible to obtain a game theoretically founded, ordinal concept of equilibrium ("strategic stability"), that, in actual economic applications, selects the type of equilibrium we have been selecting on ad hoc—i.e., game-specific—grounds up to now. This programme was started by E. Kohlberg and this author, in "On the Strategic Stability of Equilibria". We refer to §§1 and 2 of that paper for the motivation of this programme and to the introduction of Mertens (1987) for the ordinality issue. A major desideratum was not achieved in that paper: stable equilibria did not yield backwards induction—as an example by F. Gul demonstrated (Kohlberg and Mertens 1986, p. 1028): some stable set failed to include a sequential equilibrium. Thus we said there "We give an incomplete definition of what seems to us the "right" concept In order not to further complicate our terminology, we still refer in this paper to equilibria satisfying the incomplete definition as "stable" We hope that in the future some appropriately modified definition of stability will, in addition, imply connectedness and backwards induction." I presented this modified definition at the 1985 meeting of the Econometric Society in Boston—essentially §2 and §3 of this paper (but restricted to the case (M, M') = (Z, 0)—cf. infra). This definition was also given in Mertens (1987). Since then, many papers have appeared on the subject, both apphed [e.g., K. Chatteijee and L. Samuelson 1987, J. Glazer and A. Weiss 1987, U. Schweizer 1986b] and theoretical [e.g. Ph. Reny 1987, J. Hillas 1988, E. Van Damme 1987a—and concerning specific classes of games: J. Banks and J. Sobel 1987, In-Koo Cho and D. Kreps 1987, In-Koo Cho and J. Sobel 1987, M. Okuno-Fujiwara and A. Postlewaite 1987, M. Osborne 1987, E. Van Damme 1987b], making the availability of a written version of the definition and its properties all the more pressing. This is an unfortunately huge paper. We wiU first describe briefly its organisation, and the nature of the definitions and of the properties achieved. Then we will return to the questions of why we think of this as essentially the right definition, and what do we hope to learn from this. The question "where do we go from here" is addressed in the concluding remarks (§7). Part I contains 1, 2 and 3—i.e., the definition and the basic properties—; §4 to §7 will appear as Part II; besides improved forms of the definition and some additional •Received September 28, 1988; revised April 15, 1989. AMS 1980 subject dassijicalion. Primary; 90D40. Secondary; 90D10. lAOR 1973 subject classification. Main; Games. Cross References; Economics. OR/MS Index 1978 subject classijication. Primary; 231 Games. Key words. Games, n-players, equilibria, stable sets, perturbations. ^Supported in part by NSF Grant SES 8619652. 575 0364-765X/89/1404/0575$01.25 Copyrighl © 1989, The Institute of Management Sciences/Operations Research Society of America