Duersch, Peter and Oechssler, Joerg and Schipper, Burkhard C (2010): Pure Saddle Points and Symmetric Relative Payoff Games.

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Abstract
It is well known that the rockpaperscissors game has no pure saddle point. We show that this holds more generally: A symmetric twoplayer zerosum game has a pure saddle point if and only if it is not a generalized rockpaperscissors game. Moreover, we show that every finite symmetric quasiconcave twoplayer zerosum game has a pure saddle point. Further sufficient conditions for existence are provided. We apply our theory to a rich collection of examples by noting that the class of symmetric twoplayer zerosum games coincides with the class of relative payoff games associated with symmetric twoplayer games. This allows us to derive results on the existence of a finite population evolutionary stable strategies.
Item Type:  MPRA Paper 

Original Title:  Pure Saddle Points and Symmetric Relative Payoff Games 
Language:  English 
Keywords:  symmetric twoplayer games; zerosum games; RockPaperScissors; singlepeakedness; quasiconcavity; finite population evolutionary stable strategy; increasing differences; decreasing differences; potentials; additive separability 
Subjects:  C  Mathematical and Quantitative Methods > C7  Game Theory and Bargaining Theory > C72  Noncooperative Games C  Mathematical and Quantitative Methods > C7  Game Theory and Bargaining Theory > C73  Stochastic and Dynamic Games ; Evolutionary Games ; Repeated Games 
Item ID:  20864 
Depositing User:  Burkhard C Schipper 
Date Deposited:  22. Feb 2010 11:28 
Last Modified:  01. Sep 2015 11:33 
References:  AlosFerrer, C. and A.B. Ania (2005). The evolutionary stability of perfectly competitive behavior, Economic Theory 26, 497516. Ania, A. (2008). Evolutionary stability and Nash equilibrium in finite populations, with an application to price competition, Journal of Economic Behavior and Organization 65, 472488. Branzei, R., Mallozzi, L., and S. Tijs (2003). Supermodular games and potential games, Journal of Mathematical Economics 39, 3949. Bryant, J. (1983). A simple rational expectations Keynestype coordination model, Quartely Journal of Economics 98, 525528. Debreu, G. (1952). A social equilibrium existence theorem, Proceedings of the National Academy of Sciences 38, 886893. Duersch, P., Oechssler, J., and B.C. Schipper (2010). Unbeatable imitation, mimeo., University of Heidelberg and the University of California, Davis. Hehenkamp, B., Leininger, W., and A. Possajennikov (2004). Evolutionary equilibrium in Tullock contests: Spite and overdissipation, European Journal of Political Economy 20, 10451057. Hehenkamp, B., Possajennikov, A., and T. Guse (2010). On the equivalence of Nash and evolutionary equilibrium in finite populations, Journal of Economic Behavior and Organization 73, 254258. Leininger, W. (2006). Fending off one means fending off all: evolutionary stability in quasisubmodular games, Economic Theory 29, 713719. Matros, A., Temzilidis, T., and J. Duffy (2009). Competitive behavior in market games: Evidence and theory, mimeo. Milgrom, P. and J. Roberts (1990). Rationalizability, learning, and equilibrium in games with strategic complementarities, Econometrica 58, 12551277. Monderer, D. and L.S. Shapley (1996). Potential games, Games and Economic Behavior 14, 124143. Nash, J. (1953). Twoperson cooperative games, Econometrica 21, 128140. Nash, J. (1951). Noncooperative games, Annals of Mathematics 54, 286295. von Neumann, J. (1928). Zur Theorie der Gesellschaftsspiele, Mathematische Annalen 100, 295320. Nydegger, R.V. and G. Owen (1974). Twoperson bargaining: An experimental test of the Nash axioms, International Journal of Game Theory 3, 239249. Osborne, M. (2004). An introduction to game theory, Oxford University Press. Possajennikov, A. (2003). Evolutionary foundation of aggregativetaking behavior, Economic Theory 21, 921928. Radzik, T. (1991). Saddle point theorems, International Journal of Game Theory 20, 2332. Roth, A.E. and M.W.K. Malouf (1979). Gametheoretic models and the role of information in bargaining, Psychological Review 86, 574594. Schaffer, M.E. (1989). Are profitmaximizers the best survivors?, Journal of Economic Behavior and Organization 12, 2945. Schaffer, M.E. (1988). Evolutionary stable strategies for a finite population and a variable contest size, Journal of Theoretical Biology 132, 469478. Shapley, L.S. (1964). Some topics in twoperson games, in: Dresher, M., Shapley, L.S. and A.W. Tucker (eds.), Advances in Game Theory, Annals of Mathematical Studies 52, 128. Tanaka, Y. (2000). A finite population ESS and a long run equilibrium in an nplayers coordination game, Mathematical Social Sciences 39, 195206. Topkis, D. M. (1998). Supermodularity and complementarity, Princeton, New Jersey: Princeton University Press. Tullock, G. (1980). Effcient rent seeking, in: Buchanan, Tollison, Tullock (eds.), Towards a theory of the rent seeking society, Texas A & M University Press, 315. Van Huyck, J., Battalio, R., and R. Beil (1990). Tacit coordination games, strategic uncertainty and coordination failure, American Economic Review 80, 234248. VegaRedondo, F. (1997). The evolution of Walrasian behavior, Econometrica 65, 375384. Walker, J.M., Gardner, R., and E. Ostrom (1990). Rent dissipation in a limitedaccess CommonPool resource: Experimental evidence, Journal of Environmental Economics and Management 19, 203211. 
URI:  https://mpra.ub.unimuenchen.de/id/eprint/20864 