Editing Strategy (game theory) (section)

== Mixed strategy ==
=== Illustration ===

In a soccer penalty kick, the kicker must choose whether to kick to the right or left side of the goal, and simultaneously the goalie must decide which way to block it. Also, the kicker has a direction they are best at shooting, which is left if they are right-footed. The matrix for the soccer game illustrates this situation, a simplified form of the game studied by Chiappori, Levitt, and Groseclose (2002).<ref>{{Cite journal | doi = 10.1257/00028280260344678| title = Testing Mixed-Strategy Equilibria when Players Are Heterogeneous: The Case of Penalty Kicks in Soccer| journal = American Economic Review| volume = 92| issue = 4| pages = 1138| year = 2002| last1 = Chiappori | first1 = P. -A. | last2 = Levitt | first2 = S. | last3 = Groseclose | first3 = T. | url = http://pricetheory.uchicago.edu/levitt/Papers/ChiapporiGrosecloseLevitt2002.pdf| citeseerx = 10.1.1.178.1646}}</ref> It assumes that if the goalie guesses correctly, the kick is blocked, which is set to the base payoff of 0 for both players. If the goalie guesses wrong, the kick is more likely to go in if it is to the left (payoffs of +2 for the kicker and -2 for the goalie) than if it is to the right (the lower payoff of +1 to kicker and -1 to goalie).

{| class="wikitable" style="background:white;color:maroon;text-align:center; float:right;" 
| style="background:white; border:1px solid white;" colspan="2" rowspan="2"| || style="background:White; color:black; border:1px solid white; font-weight:bold;" align="center" colspan="2"| Goalie
|-
| style="color:cadetblue;"|Lean Left||style="color:cadetblue;"|Lean Right
|-
| style="background:white; color:black; border:1px solid white; font-weight:bold;" valign="center" rowspan="2"|Kicker || style="color:cadetblue;" align="left"|Kick Left || &nbsp;0,&nbsp;&nbsp;0 || +2, -2
|-
| style="color:cadetblue;" align="left"|Kick Right || +1, -1 || &nbsp;0,&nbsp;&nbsp;0
|-
| style="background:white; border:1px solid white;" colspan="4"|
|-
| style="background:white; border:1px solid white;" colspan="4"|
|-
| style="background:white; border:1px solid white;" align="left" colspan="4"| &nbsp;Payoff for the Soccer Game (Kicker, Goalie)
|-
|}
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This game has no pure-strategy equilibrium, because one player or the other would deviate from any profile of strategies—for example, (Left, Left) is not an equilibrium because the Kicker would deviate to Right and increase his payoff from 0 to 1.

The kicker's mixed-strategy equilibrium is found from the fact that they will deviate from randomizing unless their payoffs from Left Kick and Right Kick are exactly equal. If the goalie leans left with probability g, the kicker's expected payoff from Kick Left is g(0) + (1-g)(2), and from Kick Right is g(1) + (1-g)(0). Equating these yields g= 2/3. Similarly, the goalie is willing to randomize only if the kicker chooses mixed strategy probability k such that Lean Left's payoff of k(0) + (1-k)(-1) equals Lean Right's payoff of k(-2) + (1-k)(0), so k = 1/3. Thus, the mixed-strategy equilibrium is (Prob(Kick Left) = 1/3, Prob(Lean Left) = 2/3).

In equilibrium, the kicker kicks to their best side only 1/3 of the time. That is because the goalie is guarding that side more. Also, in equilibrium, the kicker is indifferent which way they kick, but for it to be an equilibrium they must choose exactly 1/3 probability.

Chiappori, Levitt, and Groseclose try to measure how important it is for the kicker to kick to their favored side, add center kicks, etc., and look at how professional players actually behave. They find that they do randomize, and that kickers kick to their favored side 45% of the time and goalies lean to that side 57% of the time. Their article is well-known as an example of how people in real life use mixed strategies.

=== Significance ===
In his famous paper, [[John Forbes Nash]] proved that there is an [[Nash equilibrium|equilibrium]] for every finite game. One can divide Nash equilibria into two types. ''Pure strategy Nash equilibria'' are Nash equilibria where all players are playing pure strategies. ''Mixed strategy Nash equilibria'' are equilibria where at least one player is playing a mixed strategy. While Nash proved that every finite game has a Nash equilibrium, not all have pure strategy Nash equilibria. For an example of a game that does not have a Nash equilibrium in pure strategies, see [[Matching pennies]]. However, many games do have pure strategy Nash equilibria (e.g. the [[Coordination game]], the [[Prisoner's dilemma]], the [[Stag hunt]]). Further, games can have both pure strategy and mixed strategy equilibria. An easy example is the pure coordination game, where in addition to the pure strategies (A,A) and (B,B) a mixed equilibrium exists in which both players play either strategy with probability 1/2.

=== 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.