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Matching pennies
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== Real-life data == The conclusions of laboratory experiments have been criticized on several grounds.<ref name=clg02>{{cite journal |first1=P. |last1=Chiappori |first2=S. |last2=Levitt |author-link2=Steven Levitt |first3=T. |last3=Groseclose |year=2002 |url=http://pricetheory.uchicago.edu/levitt/Papers/ChiapporiGrosecloseLevitt2002.pdf |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=1138β1151 |doi= 10.1257/00028280260344678|jstor=3083302 |citeseerx=10.1.1.178.1646 }}</ref><ref name=p03>{{cite journal |first=I. |last=Palacios-Huerta |year=2003 |title=Professionals Play Minimax |journal=[[Review of Economic Studies]] |volume=70 |issue=2 |pages=395β415 |doi=10.1111/1467-937X.00249 |citeseerx=10.1.1.127.9097 }}</ref> * Games in lab experiments are artificial and simplistic and do not mimic real-life behavior. * The payoffs in lab experiments are small, so subjects do not have much incentive to play optimally. In real life, the market may punish such irrationality and cause players to behave more rationally. * Subjects have other considerations besides maximizing monetary payoffs, such as to avoid looking foolish or to please the experimenter. * Lab experiments are short and subjects do not have sufficient time to learn the optimal strategy. To overcome these difficulties, several authors have done statistical analyses of professional sports games. These are zero-sum games with very high payoffs, and the players have devoted their lives to become experts. Often such games are strategically similar to matching pennies: * In [[soccer]] [[Penalty kick (association football)|penalty kick]]s, the kicker has two options{{snd}}kick left or kick right{{snd}}and the goalie has two options{{snd}}jump left or jump right.<ref>There is also the option of kicking/standing in the middle, but it is less often used.</ref> The kicker's probability of scoring a goal is higher when the choices do not match, and lower when the choices match. In general, the payoffs are asymmetric because each kicker has a stronger leg (usually the right leg) and his chances are better when kicking to the opposite direction (left). In a close examination of the actions of kickers and goalies, it was found<ref name=clg02/><ref name=p03/> that their actions do not deviate significantly from the prediction of a Nash equilibrium. * In [[tennis]] serve-and-return plays, the situation is similar. It was found<ref name="ww01">{{cite journal|jstor=2677937|title=Minimax Play at Wimbledon|journal=The American Economic Review|volume=91|issue=5|pages=1521β1538|last1=Walker|first1=Mark|last2=Wooders|first2=John|year=2001|doi=10.1257/aer.91.5.1521|citeseerx=10.1.1.614.5372}}</ref> that the win rates are consistent with the minimax hypothesis, but the players' choices are not random: even professional tennis players are not good at randomizing, and switch their actions too often.
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