Editing Strategy (game theory) (section)

=== 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.