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# Ch 13 Game Theory ECO 420K

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This 8 page Class Notes was uploaded by Natalie Strawn on Friday July 8, 2016. The Class Notes belongs to ECO 420K at 1 MDSS-SGSLM-Langley AFB Advanced Education in General Dentistry 12 Months taught by John Thompson in Summer 2016. Since its upload, it has received 19 views. For similar materials see MICROECONOMIC THEORY in Economcs at 1 MDSS-SGSLM-Langley AFB Advanced Education in General Dentistry 12 Months.

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Date Created: 07/08/16

July 6, 2016 Ch 13 – Game Theory Game theory is the use of simplified strategic situations to explain real-world strategic scenarios. -used in business, negotiations, military strategy, political science, psychology, economics -pricing, output, location, advertising, investment, new product introduction Each game has players whose strategic decisions result in payoffs. A strategy is a rule or plan of action for playing the game. An optimal strategy maximizes a player’s expected payoff. Generally, games are either cooperative (cartel) or non-cooperative (prisoners’ dilemma). In cooperative games, binding contracts are possible; in non-cooperative games, they are not. The Cartel Revisited The Cournot output problem: -Daily demand for spring water is Q = 12 – P -Jack and Jill have access to the spring; MC1 = MC2 = 0 -They form a cartel to maximize joint profits. What is the cartel’s total output, price and profits? What is the Cournot output, price and profits? Look for a “dominant strategy” … A dominant strategy is optimal no matter what an opponent does. A mixed strategy involves changing your choice depending on what the other player does. J&J each have a dominant strategy of Q = 4 Both players have mixed strategies. Are there any Nash Equilibriums? One way to solve mixed strategy games is to assign probability and to maximize expected payoffs. Suppose GM has signaled that they are going to introduce a sweet cereal, and Post believes that is 70% likely. Post’s expected payoffs: Profit(crispy) = 0.3(-5) + 0.7(10) = 5.5 Profit (sweet) = 0.3(10) + 0.7(-5) = -0.5 How confident must Post be that GM will produce a sweet cereal in order to justify producing a crispy cereal? Profit (crispy) = x(-5) + (1-x)(10) -5x + 10 - 10x Profit (sweet) = x(10) + (1-x)(-5) 10x – 5 + 5x X = 50%; 1-x = 50% Kroger vs Safeway Likely to be a single period game? In a repeated game, are we more likely to avoid the Prisoners’ Dilemma “trap”? Suppose we’re in period 1, and Kroger is considering offering coupons in period 2. If they do, Safeway offers coupons in period 3… A perpetuity is a constant stream of identical cash flows with no end. Present value of perpetuity: payment/discount rate PV0 = C/(1+r)^1 + C/(1+r)^2 + C/(1+r)^3 + … = C/r *Aka PV0 = C/r C= cash flow, r = discount rate Calculate the expected payoff, in period 1, from “no coupons” and “coupons”, assuming r = 10% Profit (no coupons) = 3/0.1 = 30 Profit (coupons) = [6 + (1/0.1)]/1.1 = 14.55 At r = 10%, the future “cost” of coupons outweighs the present “benefit”. Recalculate, assuming r = 90% Profit (no coupons) = 3/0.9 = 3.33 Profit (coupons) = [6 + (1/0.9)]/1.9 = 3.74 1.1and 1.9 are the period plus interest rate* Calculate the “breakeven” discount rate. 3/r = [6 + 1/r]/(1+r) 3 + 3r = 6r +1 >>> 2 = 3r >>> r = 66.7% At a discount rate below 66.7%, triggering a price war will be unattractive. Triggering a price war becomes more attractive with a higher discount rate. Likewise, cheating on a collusive agreement is more attractive with a higher discount rate. Either way, avoiding the Prisoners’ Dilemma “trap” is more likely in a repeated game. What if we start in the Prisoners’ Dilemma “trap”? Would it be worth it to unilaterally raise price? (repeated game) Sequential Games In a sequential game, each player moves in turn, responding to each other. Model with a decision tree. Entry problem -Solve sequential games by backward induction -How could the monopolist deter entry? Entry problem w/ threat -Is the monopoly’s threat credible? Investment Problem -If Boeing could move first, what would they do? - A simultaneous game can be modeled as a sequential game. -Set up the sequential form and solve by backward induction. Bargaining – cars, employment contracts, profit sharing, etc. Simultaneous game or sequential, single period or repeated. Bargaining is often modeled as a game of “chicken”. Chicken (simultaneous) We could solve using probability. One or the other player might consider a maximin strategy. A maximin maximizes the minimum payoff – it is a strategy of caution. -If a player follows a maximin, they will swerve. What if James could commit first? Set up and solve by backward induction. Strategic bargaining depends on: a) who moves first b) who can commit to a position Spatial Games Beach Location Game Two vendors selling snacks at identical prices. Customers are distributed uniformly along the beach, and prefer the closest vendor. Where does each vendor locate? July 6, 2016 Ch 13 – Game Theory Game theory is the use of simplified strategic situations to explain real-world strategic scenarios. -used in business, negotiations, military strategy, political science, psychology, economics -pricing, output, location, advertising, investment, new product introduction Each game has players whose strategic decisions result in payoffs. A strategy is a rule or plan of action for playing the game. An optimal strategy maximizes a player’s expected payoff. Generally, games are either cooperative (cartel) or non-cooperative (prisoners’ dilemma). In cooperative games, binding contracts are possible; in non-cooperative games, they are not. The Cartel Revisited The Cournot output problem: -Daily demand for spring water is Q = 12 – P -Jack and Jill have access to the spring; MC1 = MC2 = 0 -They form a cartel to maximize joint profits. What is the cartel’s total output, price and profits? What is the Cournot output, price and profits? Look for a “dominant strategy” … A dominant strategy is optimal no matter what an opponent does. A mixed strategy involves changing your choice depending on what the other player does. J&J each have a dominant strategy of Q = 4 Both players have mixed strategies. Are there any Nash Equilibriums? One way to solve mixed strategy games is to assign probability and to maximize expected payoffs. Suppose GM has signaled that they are going to introduce a sweet cereal, and Post believes that is 70% likely. Post’s expected payoffs: Profit(crispy) = 0.3(-5) + 0.7(10) = 5.5 Profit (sweet) = 0.3(10) + 0.7(-5) = -0.5 How confident must Post be that GM will produce a sweet cereal in order to justify producing a crispy cereal? Profit (crispy) = x(-5) + (1-x)(10) -5x + 10 - 10x Profit (sweet) = x(10) + (1-x)(-5) 10x – 5 + 5x X = 50%; 1-x = 50% Kroger vs Safeway Likely to be a single period game? In a repeated game, are we more likely to avoid the Prisoners’ Dilemma “trap”? Suppose we’re in period 1, and Kroger is considering offering coupons in period 2. If they do, Safeway offers coupons in period 3… A perpetuity is a constant stream of identical cash flows with no end. Present value of perpetuity: payment/discount rate PV0 = C/(1+r)^1 + C/(1+r)^2 + C/(1+r)^3 + … = C/r *Aka PV0 = C/r C= cash flow, r = discount rate Calculate the expected payoff, in period 1, from “no coupons” and “coupons”, assuming r = 10% Profit (no coupons) = 3/0.1 = 30 Profit (coupons) = [6 + (1/0.1)]/1.1 = 14.55 At r = 10%, the future “cost” of coupons outweighs the present “benefit”. Recalculate, assuming r = 90% Profit (no coupons) = 3/0.9 = 3.33 Profit (coupons) = [6 + (1/0.9)]/1.9 = 3.74 1.1and 1.9 are the period plus interest rate* Calculate the “breakeven” discount rate. 3/r = [6 + 1/r]/(1+r) 3 + 3r = 6r +1 >>> 2 = 3r >>> r = 66.7% At a discount rate below 66.7%, triggering a price war will be unattractive. Triggering a price war becomes more attractive with a higher discount rate. Likewise, cheating on a collusive agreement is more attractive with a higher discount rate. Either way, avoiding the Prisoners’ Dilemma “trap” is more likely in a repeated game. What if we start in the Prisoners’ Dilemma “trap”? Would it be worth it to unilaterally raise price? (repeated game) Sequential Games In a sequential game, each player moves in turn, responding to each other. Model with a decision tree. Entry problem -Solve sequential games by backward induction -How could the monopolist deter entry? Entry problem w/ threat -Is the monopoly’s threat credible? Investment Problem -If Boeing could move first, what would they do? - A simultaneous game can be modeled as a sequential game. -Set up the sequential form and solve by backward induction. Bargaining – cars, employment contracts, profit sharing, etc. Simultaneous game or sequential, single period or repeated. Bargaining is often modeled as a game of “chicken”. Chicken (simultaneous) We could solve using probability. One or the other player might consider a maximin strategy. A maximin maximizes the minimum payoff – it is a strategy of caution. -If a player follows a maximin, they will swerve. What if James could commit first? Set up and solve by backward induction. Strategic bargaining depends on: a) who moves first b) who can commit to a position Spatial Games Beach Location Game Two vendors selling snacks at identical prices. Customers are distributed uniformly along the beach, and prefer the closest vendor. Where does each vendor locate?

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