Do Mixed Strategy Nash Equilibria Exist in Sports?
Mixed Strategy Nash Equilibria make sense in stylized games with clear-cut payoff matrices, but do players in real games actually follow the predictions of these stylized models? Sports provide many real world scenarios where we can check the theoretical predictions of these models. Consider penalty kicks in soccer. Kickers want to score while goalies wish to defend the goal, and both players face essentially three strategy choices: kick/defend right, kick/defend center, and kick/defend left. This game is similar to the matching pennies game because: 1) goalies want to defend the same area that the player will kick, while players want to kick where goalies are not defending; 2) choices are simultaneous, since the ball is normally moving too fast for goalies to decide where to jump after it is kicked. Several academic papers have examined footage of penalty kicks and have found that MSNE analysis does a good job of predicting the distribution of strategies—both goalies and kickers randomize in such a way that equalizes the expected payoff between the three strategies.
Tennis service strategies may also be predicted by MSNE. A server has the option of serving the the returner’s forehand or backhand, and the returner must chose whether to anticipate a serve to his or her right/left. While the payoffs are not as clear cut as they are with penalty kicks- correctly anticipating where the serve is going does not guarantee winning the point- if players play according to a MSNE, then servers should randomize in such a way that equalizes the expected payout between strategies. One recent undergraduate paper has empirically tested and, for the most part, confirmed this prediction.
MSNE may also exist in ice hockey shootouts. When an NHL game is tied after five minutes of overtime, the game is decided by players from each team taking successive shootout attempts against the other team’s goalie. Skaters have two general strategies to choose from: dekeing or shooting. While shooting basically involves skating the puck down the ice and shooting at an open area of the net, dekeing consists of stick-handling the puck in a way that moves the goalie out of position. The opposing goalie also faces two general strategy options: skating out to challenge the player or staying back and protecting the net. Shooters have an easier time dekeing on goalies who are out challenging, and goalies back in their nets are less easy to move out of position. So in theory there should be some MSNE where players deke with probability p and shoot with probability 1-p, while goalies challenge with probability q and stay back in the net with probability 1-q.
But without even looking into the data, two characteristics of hockey shootouts suggest that MSNE analysis might not be the best method for predicting strategies. First, unlike the penalty kick example, dekeing and shooting are not easily bucketed into distinct strategies like kicking right vs. kicking left. Every shooter’s strategy involves some stick-handling and some sort of shot, and many players begin by dekeing before taking a shot. Second, the choices in this game are not simultaneous. While a goalie may initially orient himself on staying in the net, he may notice the player positioning for a shot and decide to come out and challenge. Thus the 2×2 payoff matrix may not be the best way to ultimately model this game.