We provide an example of an outcome game form with two players for which there is in an open set of
utilities for both players such that, in each of the associated games, the set of Nash equilibria induce a
continuum of outcome distributions. The case for three or more players has been settled by Govindan and
McLennan [3]

Kukushkin, Nicolai S.

Litan, Cristian M.

Marhuenda, Francisco

English

e-Archivo (Univ. Carlos III de Madrid e-Archivo)

   Working Paper 07-35 Departamento de Economía  Economic Series 20 Universidad Carlos III de Madrid April 26, 2007 Calle Madrid, 126  28903 Getafe (Spain)  Fax (34-91) 6249875   ON THE GENERIC FINITENESS OF OUTCOME DISTRIBUTIONS FOR BIMATRIX GAME FORMS  NIKOLAI S. KUKUSHKIN, CRISTIAN M. LITAN, AND FRANCISCO MARHUENDA1  Abstract   We provide an example of an outcome game form with two players for which there is in an open set of utilities for both players such that, in each of the associated games, the set of Nash equilibria induce a continuum of outcome distributions. The case for three or more players has been settled by Govindan and McLennan [3].   Keywords and phrases: Generic finiteness, Nash equilibrium.                                                           1 This problem was communicated to us by Francesco De Sinopoli. We thank Andreu Mas-Colell for providing us with the manuscript [7]. The authors aknowledge financial support from the project No. SEJ2004-00968/ECON from the Spanish Ministry of Education and Science. C. Litan acknowledges financial support from “Plan de Formación de Personal Investigador” of the Comunidad de Madrid. 1. IntroductionIf one perturbs independently the values of the utility functions ina finite normal form game, then generically there is a finite number ofequilibria (see Harsanyi [4]).However, when the normal form is derived from an extensive form,Harsanyi’s result has no immediate implications, because many strate-gies lead to the same final node. Even if the payoffs of the final nodescan be perturbed independently, the finiteness of the number of equi-libria is not necessarily a generic property. There can be constructedmany examples of extensive forms where it is not. On such grounds,Kreps and Wilson [5] criticized Harsanyi’s result as not being usefulfor extensive forms.As noted by Govindan and McLennan [3], “Kreps and Wilson [5]proved a theorem that responds to this critique.” Govindan and McLen-nan [2] made a direct proof of the same result. This result states thatif the payoffs of the final nodes can be perturbed independently, thefiniteness of the distributions on the terminal nodes (paths) inducedby sequential equilibria is a generic property. And this remains true ifone considers paths induced by Nash equilibria.However, as noted by Mas-Colell [7], “the Kreps and Wilson [5] crit-icism to the normal form result can be reiterated.” Govindan andMcLennan [3] pointed out in the same direction: “Many games arisingin economic models, and in other contexts, have sets of terminal nodesthat are naturally regarded as a priori equivalent for reasons arisingout of the nature of the phenomenon being modeled.”Therefore, the case in which the payoffs of the final nodes are tied to-gether by convex linear constraints (see Govindan and McLennan [3], [1]and Mas-Colell [7]) is of high conceptual interest. As further noted byGovindan and McLennan [3], in this case “...the ‘relevant’ space of ter-minal node utilities is a linear subspace of the space of all terminalnode utilities, and generic finiteness in the subspace does not followfrom Kreps and Wilson’s theorem.”This note continues the above line of research in that it addresses thefollowing question: is it the case that, in the relevant space of terminalnode utilities (i.e. when the payoffs of the final nodes are tied togetherby convex linear constraints), generically there are only finitely manyoutcome distributions (i.e. distributions on the relevant set of terminalnodes) induced by the Nash equilibria?When the number of players is strictly greater than two, Govindanand McLennan [3] provide an example of a game form that shows that,in open set of utilities, the Nash equilibria of the associated game inducea continuum of outcome distributions. In that work, they also provethat in some special cases (e.g. with two outcomes and any number ofplayers) the answer to the above question is positive.However, for games with two players, the above issue has been onlypartially addressed and the general question for this case has remainedunanswered until now. Mas–Colell [7] shows that for two person games2the equilibrium payoffs are generically finite. Govindan and McLen-nan [1] prove that for every bimatrix game form, the equilibrium dis-tributions on outcomes are generically finite in the space of zero sumand common interest utility games.In this note, we show that the answer to the previous question con-tinues to be negative for two player games. We provide an example ofa game form and an open set of utilities, such that in each of the as-sociated games there is a continuum of outcome distributions inducedby the Nash equilibria.2. The conjectureLet Ω be a finite outcome space and let S1, S2 denote the finitestrategy spaces of each of the two players. An outcome game form isa mapping θ : S1 × S2 → Ω. The utilities of the players are defined bythe functions ui : Ω → R, i = 1, 2.For each i = 1, 2, let Δi = {μ ∈ RSi++ :∑x∈Si μ(x) = 1}. A pair ofstrategy vectors pi ∈ Δi of the players induces a probability distributionin Ω.An outcome game form θ and the utilities of the players ui, i =1, 2 determine a bimatrix game. We denote by A(ui) the associatedmatrices of utilities of the players in this game. That is, the entry ajlof A(ui) is ui(θ(sj, sl)).A completely mixed Nash equilibrium in this game consists of twostrategy vectors p1 ∈ Δ1 and p2 ∈ Δ2 such thatA(u2) p1 = β e for some β ∈ R.p2 A(u1) = α e for some α ∈ R.where e denotes the vector (in the appropriate Euclidean space) withall of its entries equal to 1.Conjecture 1. For every bimatrix game form θ, there is a generic setof utilities in RΩ × RΩ for which the set of distributions induced onoutcomes by the completely mixed Nash equilibria is finite.For the case of two person games, this conjecture corresponds toConjecture T in Govindan and McLennan [3]. They disprove the con-jecture for games with more than two players.3. The exampleWe now give an example that shows that Conjecture 1 does not hold.Let there be four outcomes, denoted by Ω = {a, b, c, d}. And considerthe outcome matrixA(a, b, c, d) =(c a b bd a a bc d b c)Let us use the notation ai = ui(a), bi = ui(b), ci = ui(c), di = ui(d),and ui = (ai, bi, ci, di) ∈ R4 for the utilities of agent i = 1, 2. We denote3G = {(u1, u2) ∈ R8 | d1, b1 < a1, c1 & d2 < b2 < a2, c2}, an open subsetin the space of utilities. For each (u1, u2) ∈ G and t ∈ R, we definep1(u2) =1a2 − b2 + c2 − d2 (b2 − d2, c2 − b2, a2 − b2) ∈ R3andp2(u1; t) =( a1 − b1a1 − b1 + c1 − d1 −(a1 − b1)ta1 − d1 ,(c1 − d1)ta1 − d1 ,c1 − d1a1 − b1 + c1 − d1 −(c1 − d1)ta1 − d1 , t)∈ R4.One checks immediately that the pair 〈p1(u2), p2(u1; t)〉 is a completelymixed Nash equilibrium provided t is positive and small enough. Theprobability of outcome a induced by the equilibrium is computed easilyaspa =(b2 − c2)(c1 − d1)(a1 − b1 + c1 − d1)(−a2 + b2 − c2 + d2)+b2(d1 − c1) + b1(c2 − d2) + c1d2 − c2d1(a1 − d1)(−a2 + b2 − c2 + d2) tTherefore, there is a continuum of equilibrium probability distributions(i.e. for t positive and small enough) on the set of outcomes as longas (u1, u2) ∈ G and b2(d1 − c1) + b1(c2 − d2) + c1d2 − c2d1 = 0; suchvalues of u1 and u2 form an open subset in the space of utilities. Re-mark that when u1 = ±u2, the coefficient of t in the above expressionfor pa (as well as for the other outcomes) disappears and there is aunique probability distribution on outcomes, as proved in Govindanand McLennan [1] and Litan [6].References1. S. Govindan and A. McLennan, Generic Finiteness of Outcome Distributionsfor Two Person Game Forms with Zero Sum and Common Interest Utilities,Mimeo, University of Western Ontario (1998).2. , Direct Proofs of Generic Finiteness of Nash Equilibrium Outcomes,Econometrica 69 (2001), 765–769.3. , On the Generic Finiteness of Equilibrium Outcome Distributions inGame Forms, Econometrica 69 (2001), 455–471.4. J.C. Harsanyi, Oddness of the Number of Equilibrium Points: a New Proof,International Journal of Game Theory 2 (1973), 235–250.5. D.M. Kreps and R. Wilson, Sequential Equilibria, Econometrica 50 (1982), 863–894.6. C. M. Litan, An Elementary Proof of the Generic Finiteness of Equilibrium Out-come Distributions for Zero Sum Games, Mimeo, University Carlos III (2007).7. A. Mas-Colell, Generic Finiteness of Equilibrium Payoffs for Bimatrix Games,Mimeo, Harvard University (1994).4Russian Academy of Sciences, Dorodnicyn Computing Center, MoscowE-mail address: ququ@bluebottle.comUniversity Carlos III de Madrid, Department of Economics, Madrid.E-mail address: clitan@eco.uc3m.esUniversity Carlos III de Madrid, Department of Economics, Madrid.E-mail address: marhuend@eco.uc3m.es5

On the generic finiteness of outcome distributions for bimatrix game forms

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