arXiv:2005.12832v1 [cs.GT] 26 May 2020 sec lyrpasete taeyrnol ihprobabil with randomly equili Nash either one plays player only each has as game with play pennies actions player matching their each the among is, instance, choose players that some pure, where be mixed, can or advantage equilibrium an Nash gain consisten a could is player Such single then no equilibrium it, a Nash plays as The everybody knowledge, player. common each is of players tr payoffs all they of that o rationality sense the this use in bifur and the rationally about act possesse facts avoids players topological that game that assumed simple form needs proof, only strategic and alternative theorem every an that (for showed equilibrium [1] Nash John Introduction 1 ∗ † [email protected] [email protected] [email protected], eidcSrtge I eeaiain n Extensions and Generalizations II: Strategies Periodic neitmcgm hoyfaeok n ics eea featur several discuss periodic and the framework, incorporate theory we Finally, game strateg epistemic are. periodic an equilibria and p way, Nash the the non-cooperative as fact, purely erative a In betwe in theory. differences obtained game we the are cooperative discuss information; and we incomplete dem formalism Moreover with we strategies games. games addition, Bayesian in In on investiga exist cus we strategies. may and clas strategies periodic strategy, this such periodic periodic in of also one properties that least show matical at simult We exists multi-player n always games. to does there form this payoff generalize strategic own strat we information the periodic Here, contrast, fect a in In action. [3], opponent’s his. paper to the previous sticks a opponent in her developed as long as action, own tamxdNs qiiru,tepyffo lyrde o eedo h on depend not does player a of payoff the equilibrium, Nash mixed a At 1 a lnkIsiuefrMteaisi h Sciences the in Mathematics for Institute Planck Max nesrse2,013Lizg Germany Leipzig, 04103 22, Inselstrasse 2 at eIsiue e eio USA Mexico, New Institute, Fe Santa .K Oikonomou K. V. a 7 2020 27, May Abstract 1 1 .Jost J. , ∗ 1 , 2 † omxmz hi payoffs, their maximize to y t 1 ity rmauiaea deviation. unilateral a from ain,se[].Hr,i is it Here, [2]). see cations, etepsil cin and actions possible the re ru,adti smixed, is this and brium, eti rbblte.For probabilities. certain ntesneta when that sense the in t rue’ xdpoint fixed Brouwer’s f oedfiiestrategy, definite some s so hsapproach. this of es / .Sc ie Nash mixed a Such 2. roi strategies eriodic tlatoeNash one least at s e r scoop- as are ies nteperiodic the en g,aconcept a egy, etemathe- the te ntaethat onstrate tdpn on depend ot taeisin strategies nosper- aneous fgames, of s hl fo- shall er equilibrium has a curious property. To see this, for simplicity, we consider a game with two players i = A, B who have two possible actions 1, 2 each. When A and B play actions α and β with respective probabilities pα and qβ (with p1 + p2 =1= q1 + q2), then the (expected) utility of A is (in obvious notation)

U = U (α, β)p q . (1) A X A α β α,β

When now A wants to maximize her payoff, she adjusts her probabilities p1 and applies calculus to get as a first order necessary condition at a mixed value 0

0= U (1, β)q − U (2, β)q . (2) X A β X A β β β

This then is a condition about the probabilities qβ of her opponent which is independent of her own probabilities pα. That is, when the opponent plays according to those values, it is irrelevant for A what she plays. She will always get the same payoff. Thus, at a mixed , when every player has a mixed strategy, no single player can change her by changing her strategy, as long as all others stick to their probabilities. Of course, this is well known. The phenomenon is simply a consequence of the fact that the utility U depends linearly on t