Abstract
We prove that an equilibrium of a nondegenerate bimatrix game has index + 1 if and only if it can be made the unique equilibrium of an extended game with additional strategies of one player. The main tool is the “dual construction”. A simplicial polytope, dual to the common best-response polytope of one player, has its facets subdivided into best-response regions, so that equilibria are completely labeled points on the surface of that polytope. That surface has dimension m − 1 for an m×n game, which is much lower than the dimension m + n of the polytopes that are classically used.
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References
Bárány, I., Vempala, S., Vetta, A.: Nash equilibria in random games. In: Proc. 46th FOCS, pp. 123–131 (2005)
Chen, X., Deng, X.: Settling the complexity of 2-player Nash-equilibrium. In: Proc. 47th FOCS, pp. 261–272 (2006)
Conitzer, V., Sandholm, T.: Complexity results about Nash equilibria. In: Proc. 18th International Joint Conference on Artificial Intelligence (IJCAI), pp. 765–771 (2003)
Demichelis, S., Germano, F.: On the indices of zeros of Nash fields. Journal of Economic Theory 94, 192–217 (2000)
Gilboa, I., Zemel, E.: Nash and correlated equilibria: some complexity considerations. Games and Economic Behavior 1, 80–93 (1989)
Govindan, S., Wilson, R.: Uniqueness of the index for Nash equilibria of two-player games. Economic Theory 10, 541–549 (1997)
Hofbauer, J.: Some thoughts on sustainable/learnable equilibria. In: 15th Italian Meeting on Game Theory and Applications, Urbino, Italy, July 9–12 (2003) (accessed on October 29, 2007), http://www.econ.uniurb.it/imgta/PlenaryLecture/Hofbauer.pdf
Johnson, C.R., Olesky, D.D., van den Driessche, P.: Matrix classes that generate all matrices with positive determinant. SIAM J. Matrix Anal. Appl. 25, 285–294 (2003)
Lemke, C.E., Howson Jr., J.T.: Equilibrium points of bimatrix games. Journal of the Society for Industrial and Applied Mathematics 12, 413–423 (1964)
Myerson, R.B.: Sustainable equilibria in culturally familiar games. In: Albers, W., et al. (eds.) Understanding Strategic Interaction: Essays in Honor of Reinhard Selten, pp. 111–121. Springer, Heidelberg (1997)
Papadimitriou, C.H.: On the complexity of the parity argument and other inefficient proofs of existence. Journal of Computer and System Sciences 48, 498–532 (1994)
Savani, R., von Stengel, B.: Exponentially many steps for finding a Nash equilibrium in a bimatrix game. In: Proc. 45th FOCS, pp. 258–267 (2004)
Shapley, L.S.: A note on the Lemke–Howson algorithm. Mathematical Programming Study 1: Pivoting and Extensions, pp. 175–189 (1974)
von Schemde, A.: Index and Stability in Bimatrix Games. Lecture Notes in Economics and Mathematical Systems, vol. 1853. Springer, Berlin (2005)
von Stengel, B.: Computing equilibria for two-person games. In: Aumann, R.J., Hart, S. (eds.) Handbook of Game Theory, vol. 3, pp. 1723–1759. North-Holland, Amsterdam (2002)
Ziegler, G.M.: Lectures on Polytopes. Springer, New York (1995)
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von Schemde, A., von Stengel, B. (2008). Strategic Characterization of the Index of an Equilibrium. In: Monien, B., Schroeder, UP. (eds) Algorithmic Game Theory. SAGT 2008. Lecture Notes in Computer Science, vol 4997. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-79309-0_22
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DOI: https://doi.org/10.1007/978-3-540-79309-0_22
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