Penalty Kick Shootout

In this clip, we see the high stress moments of a the USA vs. Brazil soccer game when the two teams participate in a penalty kick shoot out in order to determine the winner. If a match ends in a tie, and the score remains tied in over time, games will often go into penalty kicks to break the tie. In the shootout, each team chooses 5 players to take a penalty kick, designating a certain order for them to kick in. A coin toss decides which team shoots first. Once the first shooter from team 1 goes, the 1^st shooter from team 2 goes, and they continue in this alternating pattern until either (1) all five players form both teams have kicked, or (2) one team makes a certain number of shots while the other team has missed a certain amount so that the later team cannot possibly win. In the end, whichever team has the most successful PK shots wins the game. If the case occurs where all 5 players have kicked, and the teams have an equal number of “makes” and “misses”, the shootout continues (either with the same kickers or new kickers) until one team makes their shot and the other team misses (becomes a sudden death shootout).

A penalty kick shootout can be viewed as representing a version of game theory. In class we discussed game theory with a specific focus on the prisoner’s dilemma in which a dominant strategy equilibrium exists where both players play their dominant strategy even though this equilibrium may not be Pareto optimal. The penalty kick shootout is slightly different; it is a two-strategy game, which can illustrate a mixed strategy, and results in a mixed strategy Nash equilibrium. A Nash equilibrium   exists when each player is making the best decision he/she can, taking into account the other player’s decisions, and therefore has no motivation to change their strategy. In the shootout, we can still use a payoff matrix to represent the payoffs to each player for each choice:

Player A/Player B	Right	Left
Right	(1) +1, -1	(2) -1, +1
Left	(3) -1, +1	(4) +1, -1

There are two players, A and B. Player A represents the goalie from team A, while player B represents the shooter from team B. Boxes 1 and 4 represent the situation in which the shooter shoots to the same side that the goalie dives, in which case we assume the goalie saves the shot. We can say the result is team A gets a point (prevented the other team from getting a point that round) while team B loses a point (missed the chance to get the point that round). Boxes 2 and 3 represent the situation in which the shooter shoots one way and the goalie goes the other way, in which case we are assuming the shot goes in. Here we can say team B gets a point (gets the point that round), while team A loses a point (does not prevent team B from getting the point that round). These numbers can be changed so that a goal gives a team a point, a goal missed gives 0 points, and a goal saved gives 0 points. Either way, the point is that there are only 2 results, and each player prefers exactly one of them. Each player has an equal chance of choosing left or right, so if each player is basing their choice off the other player’s choice, they are essentially indifferent (it is a matter of luck).  If neither player has an incentive to switch their strategy, because they know the other player is equally as likely to pick one strategy over the other, there is no dominant strategy equilibrium, but rather the game results in an equilibrium of mixed strategies.