Today, I’m going to examine, through the lens of Game Theory, the most important game in the world, soccer (according to Yale professor Ben Polak)!

One of the most game changing moments in soccer is the penalty kick in which a player is allowed to take a shot at the goal from the penalty line while it is only defended by the goalkeeper. Especially in low scoring games, the penalty kick is often the decisive goal (check out the seven most defining penalty kicks in history). Such crucial moments can be regarded as a strategic game between two players: the kicker and the goalkeeper.

The game can be simplified as such: the kicker has three choices of directions: left, middle, and right. To start our analysis, let’s further simplify the problem and assume that the goalkeeper only has the choice of diving left or right. If the kicker and the goalkeeper choose the same direction, the kicker has 40% chance of scoring. If they choose different directions, the kicker has 90% chance of scoring. Yet, if the kicker chooses middle, s/he always has 60% chance of scoring. The goalkeeper’s payoff is considered to be the negative of chance of scoring. Thus, we have this simple payoff matrix:

Kicker (left)/Keeper (Top) | left | right |

left | 4,-4 | 9,-9 |

middle | 6,-6 | 6,-6 |

right | 9,-9 | 4,-4 |

We can see that there is no strictly or weakly dominated strategy to delete for either party. What should we do?

In this case, we need to introduce a new concept in Game Theory: **Best Response**. A best response is a strategy for a player that produce the best outcome for that player given a particular combination of strategies of other players. For example, if the keeper chooses left, the best response for the kicker is right.

Yet, this concept is based on the idea that we know the choice of the other player. In the penalty kick, the keeper’s choice is not given. What should we do then?

We have to base our decisions on our expectations of the keeper’s action. We will be calculating our **expected payoff** instead. The formula for expected payoff is rational and simple. It is the sum of expected possibilities of individual strategies times their respective payoffs. For example, if we think that the goalkeeper has a 30% chance of diving left then our expected payoff for kicking left is:

0.3×4+0.7×9= 7.5

To make this clearer, let’s draw a graph of our expected payoff with respect to our belief of keeper’s chance of diving right. Thus, we have:

From this graph, we can see that the green curve representing the expected payoff of kicking to the middle is never the highest. Thus, kicking to the middle is never the best response, no matter of your belief of the keeper’s action. Generally, game theory would advise people NOT to choose a strategy that’s NEVER the best response if you want to maximize your expected payoff. In other words, according to game theory, one should NEVER choose middle in a penalty kick.

Yet, following the skeptic tone set up in the last post, is this model a perfect representation of the reality? The answer is NO.

In this particular analysis, we have to ignore a lot of important factors in penalty kick. For example, for our convenience, we assumed that the goalie will not stay in the middle which is definitely a choice in real soccer. Also, we failed to take into account the speed of the ball. If a player can kick the ball very hard towards the middle and increase his probability of scoring to 80%, how will that affect our analysis? Try to draw the expected payoff graph for this player and let me know in the comments.

Work Cited: Yalecourses. (2008, November 20). *Game theory* [Video file]. Retrieved from https://www.youtube.com/watch?v=YYUPc-cfPyc

Image: [Penalty kick]. (n.d.). Retrieved from http://www.laughinggif.com/gifs/fxu3molktn %5BExpected payoff of penalty kick]. (2011, June 6). Retrieved from http://philosophicaldisquisitions.blogspot.com/2011/06/game-theory-part-6-penalty-kick-game.html

perlinefengIt is very impressive that you analyzed this penalty in such detailed and scientific way. I’ve never thought soccer or sports, in general, to be this rational especially when it comes to a goal kick. Whenever I watch a soccer game or other sports, there is always so much excitement and mixtures of different emotions. So it is really interesting to analyze a PK in this numeral and scientific way.

trayhammondHaving played soccer for most of my life, I always love a detailed analysis of the game! I think it would also be interesting to take into account that at the professional level, teams will have statistics on the other team’s penalty taker/keeper. How does that knowledge affect the matrix? I also wonder if there is a correlation between dominant footing and preferred direction of a kick. Thanks for the interesting blog!

Summer CaiPost authorHi Tray! Thanks for your comment! I think team statistics, dominant footing, preferred direction will help make the data in the model more accurate. The best response functions will shift around, results of the analysis will change, but the principle about choosing the best response will not.