Compute pure strategy and mixed strategy equilibria of game, Game Theory

Ronaldo (Brazil) kicks a penalty against Casillas (Spain) in the 2006 World Cup nal. Sup- pose that Ronaldo can kick the ball to Casillas' upper left (UL), lower left (LL), upper right (UR) and lower right (LR) side. Casillas simultaneously decides whether to jump to his left (L), stay in the center (C), or jump to his right(R). Players do not care about the cost of jumping or kicking the ball to one side or the other. Ronaldo only cares about the probability of scoring and Casillas only about the probability of avoiding a goal.

Ronaldo sometimes misses the goal even if Casillas jumps to the other side. - The probability of scoring when Ronaldo kicks to Casillas' upper left is: 0.7 if Casillas jumps to the right, 0.5 if Casillas stays in the center and 0.5 if Casillas jumps to the left. - The probability of scoring when Ronaldo kicks to Casillas' lower left is: 0.9 if Casillas jumps to the right, 0.8 if Casillas stays in the center and 0.4 if Casillas jumps to the left. - The probability of scoring when Ronaldo kicks to Casillas' upper right is: 0.4 if Casillas jumps to the right, 0.8 if Casillas stays in the center and 1 if Casillas jumps to the left. - The probability of scoring when Ronaldo kicks to Casillas' lower right is: 0.3 if Casillas jumps to the right, 0.5 if Casillas stays in the center and 0.7 if Casillas jumps to the left.

(a) What are the strategies available for Ronaldo and Casillas?

(b) Write down the Normal form of this game (the bimatrix of strategies and payoffs).

(c) Compute all the pure strategy and mixed strategy equilibria of the game. Find the expected payoff of each player in each equilibrium.

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