Actually, this post is about game theory in Penalty kicks.
Although peculiar, it is straightforward how the strategic decision making in the interaction of goal-keeper and striker is a matter of study for game theorists. Both players move simultaneous, choosing a side wether to shoot or dive. From the striker perspective, when the keeper dives to his left, he prefers to shoot right, and when the keeper dives to his right, he prefers to shoot left. On the other hand, when the striker shoots left, the keeper wants to dive to the striker’s left, and when the striker shoots right, the keeper wants to dive to the striker’s right.
Kevin Leyton-Brown, a professor in UBC’s Dept. of Computer Science, developed the following game to model penalty kicking. There is a superhuman goal-keeper, who always saves the penalty kick when he manages to match the side that both players choose. There is a right-foot striker who, as every right-foot players, has more accuracy and control in the shoot when aiming to his left side. When the goal-keeper mismatches the chosen sides, this guy scores every time he shoots left, and scores X% of the times he shots right. Below is depicted the payoff matrix.
It is clear that there is no pure strategy. If they could perfectly foresee what the other would do, as soon as the goal-keeper knew what the striker would do, he would want to choose an action which would make the striker want to change what he would do in the first place.
Confusing? The Nash equilibrium will be a mixed strategy. The equilibrium will be a randomization of the choice. If the Goal-Keeper faces the same average payoff when choosing left or right, he may randomize its choice, using a mixed strategy. If that randomization leaves the striker indifferent between choosing left or right, then he will get no payoff increase from deviating from that strategy, and the Nash Equilibrium is found. In softer words, the Nash equilibrium will be got when the goal keeper’s expected utility of diving left equals the expected utility of diving right, and the striker’s expected utility of shooting left equal the expected utility of shooting right. As this is a strategic decision game, each player’s expected utility will depend on the other player’s mixed strategy. The equilibrium mixed strategies, computed through the algorithm just presented, are depicted in the payoff matrix, next to each player’s decision. The striker will shoot right 1/(1+X)*100 percent of the times, and left X/(1+X)*100. The goal-keeper will dive left X/(1+X)*100 percent of the times, and right 1/(1+X).
An important result of this game is met when analyzing how the striker’s mixed strategy changes as he improves his ability on shooting to his weak side. Taking the first derivative of the optimal proportion of shooting to the right [1/1+x] in respect to x, the ability to shoot to the right side, we get as result -1/(1+X^2), a negative function. This means that the higher the ability to shoot right, the less he will do so. What may seem counter-intuitive is explained by the strategic behavior. If the keeper knows that X is low, he will dive more often to the left, and the striker will compensate for his weakness shooting right more often. If X increases, the keeper will respond by diving more frequently to the right, which in turn, will make the striker shoot less to the right. He will no longer need to compensate for his weakness.
A second result worth standing out is that, when players equalize their expected utility to get the Nash mixed strategy, they end up equalizing their expected payoff. So, this analysis points out that players will have the same rate of success when shooting to their weaker or to their stronger side.
A 2002 study by Ignacio Palacios-Huerta published in the Review of Economic Studies tested this results by analyzing thousands of penalty kicks taken in professional league. His findings, depicted in the plots below, are enlightening. Surprisingly, players tend to meet naturally and spontaneously their Nash mixed strategy. They instinctively rock in strategic decision making.
For something completely different, and if it interests you, you can also go through this extremely useful guide on how to take a penalty kick.
Gonçalo Pessa, 750