Question

In a simplified penalty-taking game presented below, the striker can only kick to the left or the right, and the goalkeeper can only move to the left or the right (from the striker’s perspective). If the striker and goalkeeper choose the same direction, the goalkeeper saves the penalty; otherwise, the striker scores. Derive the mixed-strategy equilibrium for this game.

.................................Goalkeeper

..............................Left ....................Right

Striker ....Left ......(0, 1) ....................(2, 0)

.............Right ......(1, 0) ....................(0, 1)

Answer 1

Let Striker choose Left with probability 'p' and Right with probability '1-p'. So, if Goalkeeper best responds with a mixed strategy then striker must make him indifferent between his strategies so that his expected payoff from his two strategies is equal. That is E_G(Left) = E_G(Right),
So, 1p + 0(1-p) = 0p + 1(1-p)
So, p = 1-p
So, p+ p = 2p = 1
So, p = 1/2

Let Goalkeeper choose Left with probability 'q' and Right with probability '1-q'. So, if Striker best responds with a mixed strategy then goalkeeper must make him indifferent between his strategies so that his expected payoff from his two strategies is equal. That is E_S(Left) = E_S(Right),
So, 0q + 2(1-q) = 1q + 0(1-q)
So, 2 - 2q = 1q
So, 1q+ 2q = 3q = 2
So, q = 2/3

Thus, mixed strategy equilibrium for this game = (p, q) = (1/2, 2/3)
That is, Striker choose left with probability 1/2 and right with probability 1-(1/2) = 1/2
And, goalkeeper choose left with probability 23 and right with probability 1-(2/3) = 1/3