Editing Strategy (game theory) (section)

=== Interpretations of mixed strategies ===
During the 1980s, the concept of mixed strategies came under heavy fire for being "intuitively problematic", since they are weak Nash equilibria, and a player is indifferent about whether to follow their equilibrium strategy probability or deviate to some other probability.<ref name="Aumann1985">{{cite book |author-link=Robert Aumann |last=Aumann |first=R. |chapter-url=http://www.ma.huji.ac.il/raumann/pdf/what%20is%20game%20theory.pdf |chapter=What is Game Theory Trying to accomplish? |title=Frontiers of Economics |editor1-first=K. |editor1-last=Arrow |editor2-first=S. |editor2-last=Honkapohja |pages=909–924 |publisher=Basil Blackwell |location=Oxford |year=1985}}</ref> 
<ref name="Rubinstein1991">{{cite journal |author-link=Ariel Rubinstein |last=Rubinstein |first=A. |title=Comments on the interpretation of Game Theory |journal=[[Econometrica]] |year=1991 |volume=59 |issue=4 |pages=909–924 |jstor=2938166|doi=10.2307/2938166 }}</ref> Game theorist [[Ariel Rubinstein]] describes alternative ways of understanding the concept. The first, due to Harsanyi (1973),<ref>{{cite journal |last1=Harsanyi |first1=John |author1-link=John Harsanyi |title=Games with randomly disturbed payoffs: a new rationale for mixed-strategy equilibrium points |journal=Int. J. Game Theory |volume=2 |pages=1–23 |year=1973 |doi=10.1007/BF01737554|s2cid=154484458 }}</ref> is called ''[[purification theorem|purification]]'', and supposes that the mixed strategies interpretation merely reflects our lack of knowledge of the players' information and decision-making process. Apparently random choices are then seen as consequences of non-specified, payoff-irrelevant exogenous factors.<ref name="Rubinstein1991" />
A second interpretation imagines the game players standing for a large population of agents. Each of the agents chooses a pure strategy, and the payoff depends on the fraction of agents choosing each strategy. The mixed strategy hence represents the distribution of pure strategies chosen by each population. However, this does not provide any justification for the case when players are individual agents.

Later, Aumann and Brandenburger (1995),<ref>{{cite journal |last1=Aumann |first1=Robert |author1-link=Robert Aumann |last2=Brandenburger |first2=Adam |author2-link=Adam Brandenburger |title=Epistemic Conditions for Nash Equilibrium |journal=Econometrica |volume=63|pages=1161–1180 |year=1995 |doi=10.2307/2171725 |jstor=2171725 |issue=5 |citeseerx=10.1.1.122.5816 }}</ref> re-interpreted Nash equilibrium as an equilibrium in ''beliefs'', rather than actions. For instance, in [[rock paper scissors]] an equilibrium in beliefs would have each player ''believing'' the other was equally likely to play each strategy. This interpretation weakens the descriptive power of Nash equilibrium, however, since it is possible in such an equilibrium for each player to ''actually'' play a pure strategy of Rock in each play of the game, even though over time the probabilities are those of the mixed strategy.