The penalty shootout is an excellent example of a game with no Nash equilibrium in pure strategy - where the equilibrium is a mixed strategy equilibrium. That is, it is best for the players to randomise their strategy choice...

Notice that there are no outcomes (cells) where both players are playing a best response, so there is no Nash equilibrium in pure strategy. To work out the mixed strategy equilibrium, we would need to know a bit more about the probability of success if both goalkeeper and shooter chose the same (here we assume there was a 0% chance of a goal), and the probability of success if they chose differently (here we assumed there was a 100% chance of a goal).Mark Johnston (HOD economics at King's College in Auckland) pointed me to several blog posts he has written on penalty shootouts. This post in particular gives us the probability of success or failure depending on which way the kicker shoots and which way the goalkeeper dives (the data comes from Ignacio Palacios-Huerta's book Beautiful Game Theory):

In the table above, NS represents the kicker's 'natural side' (to the right for a right-footed kicker), and OS represents the kicker's opposite side. Solving for the mixed strategy equilibrium requires some algebra. Let

*p*be the probability that the kicker kicks to their natural side (and

*1-p*will be the probability that the kicker kicks to the opposite side). And let

*q*be the probability that the goalkeepers dives to the kicker's natural side (and

*1-q*will be the probability that the goalkeeper dives to the kicker's opposite side).

Now, for each player we set the expected values of their two strategy choices to be equal, but those expected values depend on the probability that the

*other*player chooses each strategy (this is analagous to our definition of Nash equilibrium, where both players are doing the best they can, given the choice of the other player).

So, for the kicker:

EV[natural side] = 0.7

*q*+ 0.95(

*1-q*) = 0.95 - 0.25

*q*

EV[opposite side] = 0.92

*q*+ 0.58(

*1-q*) = 0.58 + 0.34

*q*

0.95 - 0.25

*q =*0.58 + 0.34

*q*

0.59

*q*= 0.37

*q*= 0.627

And, for the goalkeeper (remembering that the payoffs to the goalkeeper are the complement of the payoffs to the kicker):

EV[natural side] = 0.3

*p*+ 0.08(

*1-p*) = 0.08 + 0.22

*p*

EV[opposite side] = 0.05

*p*+ 0.42(

*1-p*) = 0.42 - 0.37

*p*

0.08 + 0.22

*p*

*=*0.42 - 0.37

*p*

0.59

*p*= 0.34

*p*= 0.576

So, the mixed strategy equilibrium is for the kicker to kick to their natural side 58% of the time (

*p*= 0.576) and the goalkeeper to dive to the kicker's natural side 63% of the time (*q*= 0.627). Which makes sense - the kicker has a higher probability of scoring on their natural side, and knowing this the goalkeeper should go to that side more often.For more on solving for mixed strategy equilibrium, try this video by William Spaniel:

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