# The Game Theory of Soccer Penalty kicks

It is reasonable to claim that when a soccer game has penalty kicks, these are one of the most important moments of the game and in that moment everyone tries to guess which side will the striker kick. Indeed penalty kicks is a type of game that game theory can solve.

To solve this game, we first need to think about the type of game, namely if it is a simultaneous game or a sequential game meaning that it is one in which the players effectively make their decisions at the same time or one in which the players take alternate turns to make their choices, respectively. In penalty kicks the players move simultaneously since the goalie cannot wait until the ball comes off the foot of the kicker to decide what to do. So, both players have to choose a side to play before the “game” starts. We will look at a simple example, where the players can only choose left or right (so, for simplicity, we ignore the possibility of choosing the middle). In principle, the goalie wants to match sides, while the striker wants to mismatch sides.

 Kicker Kick left Kick right Goalie Dive left 1;-1 -1;1 Dive right -1;1 1;-1

In this game there is no pure strategy Nash equilibrium, i.e. it is not possible to determine the move a player will make for any situation he could face. Instead there is a mixed strategy Nash equilibrium, i.e. the solution is for both players to pick each side with equal probability. This means that the striker kicks left half the time and right half the time and the goalie dives left half the time and dives right also half the time. These strategies are optimal because neither player can be exploited under these circumstances. So we conclude that it is important to randomize in order not to be exploited.

In real life, however, we tend to think that a player should always kick to his strong side. But this choice would lead the goalie to expect the ball to come from the player’s strongest side and he would be able to stop the shot. So this is not a good strategy.

Once again, game theory suggests that the best strategy is to choose randomly, which side to use, using a “mixed strategy”. So, in a penalty kick there is no single strategy that will always work (as we have seen there in no pure strategy Nash Equilibrium). Indeed, a study (by Palacios-Huerta) showed that players tend to act using a “mixed strategy” or, using “common language”, intuitively. Although some players tend to go to their strongest side more often, on average, the players tend to use both sides as often. Furthermore, using the strong side resulted almost in the same percentage of goals as when using the weak side.

Finally, game theory also predicts that shots will be “serially independent”, meaning that a given decision won’t affect those that follow. For example, playing left first doesn’t mean that the player will play right the next time.

In conclusion, game theory teaches us something about penalty kicks: players should employ a “mixed strategy” in penalty kicks and hence not employ the same strategy over and over again. In other words, the best strategy for both kickers and goalies is to have no strategy, just play randomly. Indeed, players will reach an equilibrium in which going to the weak side is as valid as going to the strong side, and where one shot is not related to the next one.

So the next time you look at a penalty kick you should just pray for the goalie not to catch the ball!

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