A quick recap: the striker has the option to kick to his left or right, and the goalie to dive likewise. The payoff matrix is:
The striker’s probability of shooting to the left is ps, and he aims to provide no incentive for the goalie to dive either to the left or right.
dive to left = dive to right
-0.3*ps -0.8*(1-ps) = -0.9*ps -0.2*(1-ps)
(-0.3 + 0.8 + 0.9 – 0.2)*ps = 0.8 – 0.2
ps = 0.6/1.2 = 0.5
The goalie’s probability of diving to the left is pg, and he aims to provide no incentive for the striker to shoot either to the left or right.
strike to left = strike to right
0.3*pg + 0.9*(1-pg) = 0.8*pg + 0.2*(1-pg)
(0.3 – 0.9 – 0.8 + 0.2)*pg = 0.2 – 0.9
pg = 0.7/1.2 = 0.58
The number of goals =
ps*pg*corresponding chance of a goal+
ps*(1-pg)*corresponding chance of a goal+
(1-ps)*pg*corresponding chance of a goal+
(1-ps)*(1-pg)*corresponding chance of a goal
The number of goals = 0.5*0.58*0.3 + 0.5*0.42*0.9 + 0.5*0.58*0.8 + 0.5*0.42*0.2 = 0.55
About a 55% chance of scoring a goal.
