While all sports naturally have complex strategy to determine the victor, soccer specifically has a situation in which the game theory behind it is comparatively very easy to analyze, penalty kicks. These penalty shots are a one on one game, shooter against goal keeper; the shooter trying to guess which side the goalkeeper will not defend, and the goalkeeper trying to guess the side the shooter will shoot. Another nice simplicity of this game is that it occurs simultaneously; the ball only takes .2-.3 seconds from the kick to reach the net, so if the goalie were to make a decision after the kick it would be too late.

As a result this game can be represented by a simple 3 X 3 matrix of scoring percentage for the left, center, and right sides. A kicker’s shot depends on his dominant foot and the side opposite of that foot is his strongest shot. This side has an increased percent of scoring since it is harder to save and less likely to miss the net. Similarly goalies may have a dominant side for which they are better at saving goals (independent on a kicker’s dominant side) however this is almost insignificant compared to the shooters dominant side. Another key point to the game is that all professional goalies know of the shooters dominant foot and previous penalty shot experience (and thus is able to create the shot percent matrix).

So while each shooter has a strongly dominant side that is more likely to score than the other, he still has no dominant strategy of only shooting to that side since the goalie would then just always jump that way also. We know in theory there should exists some mixed strategy for both the goalie and the shooter such that the probability of scoring is the same at each of the three sides. The question however is do kickers and goalies follow this strategy in order to maximize their outcome. To test this data from the French and Italian first-leagues was analyzed by A. Chappori, S. Levitt, and T. Groseclose in Testing Mixed-Strategy Equilibria When Players Are Heterogeneous: The Case of Penalty Kicks in Soccer, and were condensed in the tables below.

TABLE 3—OBSERVED MATRIX OF SHOTS TAKEN

|                                                  Kicker

|                              Left    Middle    Right    Total

|              Left           117      48          95        260

|Goalie  Middle           4         3            4          11

|              Right          85       28          75         188

|              Total          206     79         174        459

TABLE 4—OBSERVED MATRIX OF OUTCOMES:

PERCENTAGE OF SHOTS IN WHICH A GOAL IS SCORED

|                                                 Kicker

|                                Left    Middle    Right    Total

|               Left            63.2   81.2       89.5      76.2

|Goalie    Middle          100     0           100       72.7

|               Right          94.1   89.3        44.0     74.4

|               Total          76.7    81.0        70.1    74.9

“Notes: The sample includes all French first-league penalty kicks from 1997–1999 and all Italian first-league kicks (1997–2000). For shots involving left-footed kickers, the directions have been reversed so that shooting left corresponds to the “natural” side for all kickers.”

According to the data kickers shooting to the left have a 76.7 % chance of scoring, middle 81%, and right 70.1% with a average of 74.9%; goalies defending left have a 23.8% of saving, middle 27.3% and right 25.6% with a average of 25.1%. Based on these numbers the percentages are quite similar, close enough to be within statistical margin of error. We therefore can conclude that both shooters and goalies in penalty shots use mixed strategies to maximize their chance at a goal and a save respectively. So next time you watch a penalty shot understand there is a lot more than physical strength speed and skill, these players also have to be experts in mixed strategies and the numbers behind it.