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Solutions to two game theory problems. The first problem involves a baseball manager making decisions about which player to use and which pitcher to face based on success rates. The second problem deals with two rock hounds splitting and selling their meteor chunks. Game trees, payoff matrices, and equilibrium points.
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The nameís ; but my friends call me :
Pitcher X Pitcher Y Pitcher Z Player A 20% 29% 27% Player B 35% 23% 24%
(a) Construct a game tree for the managersíchoices.
(b) Determine the strategies each manager should follow, assuming best play by both man- agers. If Manager 1 opts to go with Player B at the beginning, Manager 2 will bring in Pitcher Y, and the batter will be successful only 23% of the time. If Manager 1 stays with Player A, then if Manager 2 stays with Pitcher X, Manager 1 will bring in Player B be successful 35% of the time. If Manager 2 brings in Pitcher Y, then Manager 1 will stay with Player A and be successful 29% of the time. If Manager 2 brings in Pitcher Z, Manager 1 will stay with Player A and be successful 27% of the time. So, Manager 1 will not switch to Player B immediately because the outcomes for staying with Player A at the start are better. Given that Manager 1 will stay with Player A, Manager 2 will switch to Pitcher Z because this gives the lowest percentage of success to the other team. Finally, Manager 1 will stick with Player A to face Pitcher Z.
(a) Construct the payo§ matrix that gives the proÖts for each rock hound for all the possi- bilities.
Rock Hound 2 4 5 6 7 4 (100; 100) (88; 110) (76; 114) (64; 112) Rock Hound 1 5 (110; 88) (95; 95) (80; 96) (65; 91) 6 (114; 76) (96; 80) (78; 78) (60; 70) 7 (112; 64) (91; 65) (70; 60) (49; 49)
(b) Find all the equilibrium points of this matrix. The equilibrium points are (80; 96) and (96; 80) : (c) Find the reduced payo§ matrix.
Rock Hound 2 5 6 Rock Hound 1 5 (95; 95) (80; 96) 6 (96; 80) (78; 78)
(d) Describe the best strategies for each rock hound. The best strategy for each rock hound is to select to make either 5 or 6 pieces.
Aaron Alex Strategy Percentage of Strategy Percentage of Success Success 60 paces 20% 50 paces 25% 40 paces 40% 30 paces 35% 20 paces 70% 10 paces 75%
We can view the duel as a zero-sum game, with the payo§s computed as follows. Decide who will Öre Örst and compute that personís payo§. For example, if Aaron choose to Öre at 60 paces, then Aaron shoots Örst, no matter what. He wins 20% of the time and loses 80% of the time, meaning his payo§ will be 20% 80% = 60%:
(a) Find the payo§ matrix, with Aaron as the row player.
Alex 50 30 10 60 60% 60% 60% Aaron 40 50% 20% 20% 20 50% 30% 40%