how does game theory, specifically the concepts of zero-sum and non-zero-sum games, applies to this supply chain management?
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You play the following bargaining game. Player $A$ moves first and makes Player $B$ an offer for the division of $\$ 100$. (For example, Player $A$ could suggest that she take $\$ 60$ and Player $B$ take $\$ 40 .$ ) Player $B$ can accept or reject the offer. If he rejects it, the amount of money available drops to $\$ 90,$ and he then makes an offer for the division of this amount. If Player $A$ rejects this offer, the amount of money drops to $\$ 80$ and Player $A$ makes an offer for its division. If Player $B$ rejects this offer, the amount of money drops to 0 Both players are rational, fully informed, and want to maximize their payoffs. Which player will do best in this game?
Oluwadamilola A.
6.27 Two players are involved in a competitive game. One player, called the row player, has two strategies available; the other player, called the column player, has three strategies available. If the row player selects strategy i and the column player selects strategy j, the payoff to the row player is cij and the payoff to the column player is -cij. Thus, the column player loses what the row player wins and vice versa; this is a two-person zero-sum game. The following matrix gives the payoffs to the row player: Let x1, x2, and x3 be probabilities with which the column player will select the various strategies over many plays of the game. Thus, x1 + x2 + x3 = 1, x1, x2, x3 >= 0. If the column player applies these probabilities to the selection of her strategy for any play of the game, consider the row player's options. If the row player selects row 1, then her expected payoff is 2x1 - x2. If the row player selects row 2, her payoff is -3x1 + 2x2 + x3. Wishing to minimize the maximum expected payoff to the row player, the column player should solve the following linear program: Minimize z subject to x1 + x2 + x3 = 1 2x1 - x2 <= z -3x1 + 2x2 + x3 <= z x1, x2, x3 >= 0 z unrestricted. Transposing the variable z to the left-hand-side, we get the column player's problem: Maximize z subject to x1 + x2 + x3 = 1 z - 2x1 + x2 >= 0 z + 3x1 - 2x2 - x3 >= 0 x1, x2, x3 >= 0 z unrestricted. a. Give the dual of this linear program. b. Interpret the dual problem in Part (a). (Hint: Consider the row player's problem.) c. Solve the dual problem of Part (a). (Hint: This problem may be solved graphically.) d. Use the optimal dual solution of Part (c) to compute the column player's probabilities. e. Interpret the complementary slackness conditions for this two-person zero-sum game.
Akash M.
a) Discuss the property, prove/disprove if Nash solution satisfies property. b) Determine Nash solution for this problem and discuss.
Ayushi S.
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