Penalty kicks

The 2022 FIFA World Cup final between Argentina and France will likely go down in history as one of the greatest matches in history.  Argentina scored twice in the first half, but France scored twice in the second half.  Both teams scored again in extra time, leaving the match tied at 3 goals apiece.  Argentina won the ensuing penalty shoot-out to become the 2022 FIFA World Cup champions for the third time in history (and the first time since 1986).  It was a nice capstone for one of the all-time greats, Argentina's Lionel Messi, who won the FIFA World Cup for the first time in his legendary career.

For those of us who do not regularly follow football (known as soccer here in the United States), deciding a game by penalty kicks seems a lot like deciding an American football game by a field goal in sudden death overtime - it's far too anti-climactic.  As it turns out, there is a lot of strategy - and hence, game theory - in determining the success of a penalty kick, both from the kicker's (i.e. success means that a goal was scored) and goalkeeper's (i.e. success means that a goal was prevented) perspective.  It's a classic zero sum game - the total winnings are always fixed, so that one player’s gains are equal to the other player’s losses.

According to the FIFA rules, the defending goalkeeper must remain on the goal line, facing the kicker, between the goalposts until the ball has been kicked (the goalkeeper can't move until after the ball has been kicked).  The opposing player (i.e. the one trying to score) places the ball on the penalty mark located in the penalty area.  The penalty mark is located 12 yards away from the goal line, so in the typical kick, the ball takes about 0.3 seconds to reach the goal line (the ball travels at around 125 mph).  There is just not enough time for the goalkeeper to react to the placement of the ball - in other words, the goalkeeper chooses which direction to defend before the opposing player kicks the ball!  The goalkeeper therefore has two choices - move to the right or move to the left.  Similarly, the kicker has two choices - kick to the right or kick to the left.  

It's relatively straightforward to analyze the different strategies from a game theoretic perspective.  For those of you who are interested in the mathematics behind this, there are at least two published articles that provide a good explanation (see "Professionals play minimax" by Ignacio Palacios-Huerta and "Testing mixed-strategy equilibria when players are heterogeneous: The case of penalty kicks in soccer" by Chappori, Levitt, and Groseclose).  I have set up the 2x2 matrix below with the possible strategies and associated pay-offs (note that when the kicker scores a goal, the pay-off is +1 to the kicker and -1 to the goalkeeper, as this is a zero sum game):

  

Note that the goalkeeper's best strategy is to move towards the same side as the kicker kicks (resulting in a blocked goal), while the kicker's best strategy is to kick towards the opposite side as the goalkeeper moves (resulting in a score).  For those of you who have been following along the last several posts, there is no pure or dominant strategy Nash equilibrium (recall that in a Nash equilibrium, no player can improve based on a unilateral change in strategy).  Essentially, both players should use what is called a mixed strategy (as opposed to a pure strategy) - basically, rather than always going right or always going left, they should choose which direction to kick or defend based upon a probability distribution.  

I talked about John von Neumann's minimax theorem in my last post ("Cutting cake and matching pennies").  According to the minimax theorem, the best strategy for the kicker is to kick the ball in the opposite direction that the goalkeeper moves, while the goalkeeper's best strategy is to move in the same direction as the kicker kicks the ball.  If either player always moves to one side of the goal, then the opposing player's best minimax strategy becomes quite easy to determine.  It's easy to see why that would be a very poor strategy.

We can use basic rules of probability and expected value to determine each player's best strategy.  I won't provide the full details here, but if you are interested you can go to the two studies linked above (or to the website here).  What is amazing is that the best strategy, according to the minimax theorem corresponds to within 1% of the actual strategy that football players apparently use in real-life penalty kick situations!  Specifically, game theory predicted that the best strategy for kickers was to kick to the left 39% of the time.  Based upon five years worth of penalty kicks in professional football league play in Europe, players kicked to the left 40% of the time!  Similarly, game theory predicted that the best strategy for goalkeepers was to move to the left 42% of the time, and again in league play goalkeepers actually moved to the left 42% of the time!

It's doubtful that football players are running through these calcuations in their head at the time of the penalty kick.  Regardless, these are remarkable results and provide real-world evidence for the utility of game theory in making decisions!