Question

Goalie Left Right 30, 70 80,20 70,30 40,60 Kicker Left Right (a) Derive the expected payoffs to the two players at the equilibrium/equilibria. (e) Explain why the players may want to play mixed strategies in this game. (f) The assurance game also has a mixed strategy equilibrium. Does the reasoning from part (e) apply in this case? Explain your answer.

Answer 1

(d)

As we can see that both kicker and Goalie has two options

So we have 2*2 = 4 payoff combinations

As we can see that

If kicker kicks to left

The goalie should jump to left as it will give a higher payoff 70 to him instead of 20 if he jumps right

Similarly,

if Kicker kicks right

Then Goalie must jump to the right to get a better payoff.

On the other hand, if Goalie jumps to left then, the kicker must kick right for higher payoff and vice -versa.

Thus, we can conclude that for both the players there is no strictly dominant or strictly dominated strategy. Hence, there is no pure-strategy Nash equilibrium.

(e)

Now, as we can see that there is no pure-strategy Nash equilibrium. Therefore, both the players will think of a mixed strategy and hence we will get payoff related to a mixed strategy, Nash equilibrium.

For mixed strategy, let's suppose that p time the goalie jumps to right and 1- p times it jumps to left.

To find Goalie's mix, we will use kicker's payoff

left jump = p*30 + (1-p)*80 = 80 - 50p

Right jump= p*70 + (1-p)*40 = 30p +40

Since they are using a mixed strategy, it is the best response. So, each of the pure strategies in the mix must themselves be the best responses. Both left jump and right jump should have the same payoff.

Thus,

80 - 50p = 30p + 40

80p = 40

p = 0,5

So, half of the time Goalie should jump left and half of the time he should jump right

Similarly,

For kicker's mix, we will use Goalie's payoff

if q times kicker kicks to the right and 1-q times it kicks to the left

then,

Left kick = q*70 + (1-q)*30 = 40q + 30

Right kick = q*20 + (1-q)*60 = 60 - 40q

Considering the implications of a mixed strategy

40q + 30 = 60 - 40q

80q =30

q = 3/8

(1-q) = 5/8

So, he should kick right 3/8 of the times and 5/8 of the times he should kick left.

Expected payoff = (15+30, 80 - 25)= (45,55)