As we move into the knock out stages of the World Cup the spectre of penalty kicks raises it ugly head. Penalty kicks are both awarded as punishments during the game and are used to break ties at the end of the game. They are especially interesting for researchers as they represent an excellent chance to test whether predictions from game theory apply in the real world. Penalty kicks are mini-games within a game. These mini-games have a kicker and a goalie facing a zero sum game; i.e. one of the players will win while the other will lose. Clearly being better at taking penalty kicks or saving them can help but the key insight is, at any given ability level, to win the mini-game it helps for the kicker to be unpredictable.
In general terms the kicker needs to decide whether to kick left, right or down the centre. If they always make the same choice the goalie will know what is going to happen. This significantly increases the goalie’s chance of saving the penalty. That means that although the kicker, assuming they are right footed, will probably be better at kicking to the left the kicker won’t always kick left. As a kicker always playing to your strengths makes the goalie’s prediction task too easy. Thus the kicker mixes up what they do. Sometimes the kicker makes shots that are a little harder to do well to confuse the goalie’s prediction of what they will do. This is the essence of a mixed strategy. (The classic mixed strategy game is Rock, Paper, Scissors where unpredictablity is the key to success).
I use the idea of mixed strategies in my research so it is great to see attempts to “test mixed-strategy behavior using data generated outside of a controlled experiment.” (Chiappori, S, Levitt and T .Groseclose. 2002, page 1150). Their work provides important evidence that mixed strategies matter in a real-world environment.
If you see a player from your team taking a penalty they are probably using a sophisticated strategy based upon game theory. Or they might just be panicking given billions of people are watching. Either way they are likely to quite unpredictable.
Read: P.A. Chiappori, S, Levitt and T .Groseclose (2002) Testing Mixed-Strategy Equilibria When Players Are Heterogenous: The Case of Penalty Kicks in Soccer, The American Economic Review, Volume 92, Number 4, September, pages 1138-1151.