What Should Chivas USA do to Win the Playoff Game?
The soccer team Chivas USA has a tough decision to make in their upcoming playoff game against the San Jose Earthquakes. Last time they played each other, San Jose was able to score three points off of free kicks, and those were the only points they scored that game. However, normally San Jose relies on their strong offense to score. The Chivas know that either one of these strategies by San Jose will defeat them, but the Chivas are too weak of a team to defend both cases. To determine what the best defensive strategy would be for the Chivas, we can model this game as a game of prisoner’s dilemma. There are two choices for Chivas: defend the offense, or prevent free kicks. Meanwhile, San Jose knows that two ways to beat the other team are with their strong offense or by making Chivas give away free kicks.
So our table would look like this:
|
San Jose Uses Strong Offense |
San Jose Goes For Free Kicks |
|
|
Chivas Defend Offense |
+1, -1 |
+3, -3 |
|
Chivas Prevent Free Kicks |
+4, -4 |
+1, -1 |
Where the first value is the payoff for San Jose and the second value is the payoff for Chivas.
Since Chivas is a weak team, lets assume that if they defend correctly then San Jose only scores once. There is no Nash Equilibrium for this problem, because if Chivas defends and San Jose uses offense, San Jose can just instead go for free kicks to improve. Then Chivas would prevent free kicks, but then San Jose would use strong offense, and then Chivas would prevent strong offense. So no matter what, either one can improve so there is no Nash Equilibrium. We can find a mixed Nash equilibrium for this. Let p be the probability of Chivas defending the offense, and 1-p is the probability of them defending free kicks. Therefore the payoff to San Jose to use offense is 1(p) + (1-p)(4) and the payoff to San Jose to go for free kicks is 3(p) + (1-p) (1). Since these should be equal, p=3/5.
Therefore Chivas should defend offense 3/5 of the time and prevent free kicks 2/5 of the time in order to have the best chance of winning the game.