But there would be two possible situations. First, they make agreement so that they can believe each other. In this case, if Player 1 believed Player 2’s announcement, the Player 1’s payoff would be like this.
Situation Expected payoffs
When Player 1 chose Rock Ep R =0· (0.4) +10· (0.3) -10· (0.3) 0
When Player 1 chose Scissors Ep S =-10(0.6) +0· (0.3) +10· (0.3) -1
When Player 1 chose Paper Ep P =10· (0.4)-10(0.3) +0· (0.3) 1
Therefore, Player 1 is going to put Paper as it draws the highest payoff of 1. But it is quite unrealistic assumption. In practical situation, Player 2’s announcement cannot be believed by rival because they are in zero-sum game so both players want to better off by deceiving the opponent. As a result, Player 1 has to
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Assume that slowing down the pitched fastball changes the payoff to the hitter in the “anticipate fastball/throw fastball” cell from 4 to 3. Explain carefully how you determine the answer here and show your work. Also explain why slowing the fastball can or cannot improve the pitcher’s expected payoff in the game.
PITCHER Throw curve(q) Throw fastball(1-q)
BATTER Anticipate curve(p) 2, -2 -1, 1 Anticipate fastball(1-p) -1, 1 3,- 3 Calculate p value: -3p+1=4p -3 (So, p*=4/7)
Pitcher’s expected payoff is
Ep C (P) + Ep F (P) = -2p +1(1-p) +1p-3(1-p) = -3p+1+4p -3 =p-2= -1.43
In the same way, if we calculate Batter’s expected payoff, it is 10/7
So, it shows that slowing the fastball cannot improve the pitcher’s expected payoff. [Ch.8]
1. Find Nash equilibrium in mixed strategies for the following game by using the method of best-response analysis. Draw a best-response diagram and show the equilibrium mixture on the diagram. Also indicate each player’s expected payoff in equilibrium.
1) Give probability to each option
COLUMN Left(q) Right(1-q)
ROW Up(p) 4, 0 -1, 2 Down(1-p) 1, 1 2, -1 2) Derive player’s expected pay-off
Row’s Expected Payoffs Column’s expected payoffs
EpU(R) = 4(q) - 1(1-q) = 5q-1 EpL(C) = 0(p) + 1(1-p) =
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