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Introduction to Microeconomics Game theory Chapter 9

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Basic elements – The players. – The strategies. – The payoffs. Payoff matrix – A table that describes the payoffs in a game for each possible combination of strategies. LO1: Basic Elements of A Game Elements of a Game © 2012 McGraw-Hill Ryerson Limited Ch9 -2

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Strategy Dominant strategy – One that yields the highest payoff no matter what the other players in the game choose Dominated strategy – Any other strategy available to a player that has a dominant strategy

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The Prisoner’s Dilemma – A classic example of potential conflict between the narrow self-interest of individuals and the broader interest of larger communities. The Prisoner’s Dilemma – Each player has a dominant strategy. – The dilemma is: Payoffs are smaller than they would be if each player had played a dominated strategy. Lo4: The Effect of Dominant Strategy Prisoner’s Dilemma © 2012 McGraw-Hill Ryerson Limited Ch9 -4

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Will the prisoners confess? – Two prisoners, Horace and Jasper, are being held in separate cells for a serious crime that they did in fact commit. – The prosecutor, has only enough hard evidence to convict them of a minor offence. The Payoff Matrix for the original Prisoner’s Dilemma © 2012 McGraw-Hill Ryerson Limited Ch9 -5 Lo4: The Effect of Dominant Strategy

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Example 9.3: Will the prisoners confess? – The dominant strategy for each prisoner is to confess. Table 9.3: The Payoff Matrix for the original Prisoner’s Dilemma © 2012 McGraw-Hill Ryerson Limited Ch9 -6 √ √ Dominate strategy Lo4: The Effect of Dominant Strategy

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Will the prisoners confess? – The dominant strategy for each prisoner is to confess. The Payoff Matrix for the original Prisoner’s Dilemma © 2012 McGraw-Hill Ryerson Limited Ch9 -7 √√ Dominate strategy Lo4: The Effect of Dominant Strategy

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Will the prisoners confess? – When each follows his dominant strategy and confesses, both will do worse than if each had shown restraint. Table 9.3: The Payoff Matrix for the original Prisoner’s Dilemma © 2012 McGraw-Hill Ryerson Limited Ch9 -8 Nash Equilibrium Nash Equilibrium Better Outcome Lo4: The Effect of Dominant Strategy

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Terminology When a player tries to choose the "best" strategy among a multitude of options, that player may compare two strategies A and B to see which one is better. The result of the comparison is one of: B dominates A: choosing B always gives as good as or a better outcome than choosing A. There are 2 possibilities: – B strictly dominates A: choosing B always gives a better outcome than choosing A, no matter what the other player(s) do. – B weakly dominates A: There is at least one set of opponents' action for which B is superior, and all other sets of opponents' actions give B at least the same payoff as A. B and A are intransitive: B neither dominates, nor is dominated by, A. Choosing A is better in some cases, while choosing B is better in other cases, depending on exactly how the opponent chooses to play. For example, B is "throw rock" while A is "throw scissors" in Rock, Paper, Scissors.Rock, Paper, Scissors B is dominated by A: choosing B never gives a better outcome than choosing A, no matter what the other player(s) do. There are 2 possibilities: – B is weakly dominated by A: There is at least one set of opponents' actions for which B gives a worse outcome than A, while all other sets of opponents' actions give A at least the same payoff as B. (Strategy A weakly dominates B). – B is strictly dominated by A: choosing B always gives a worse outcome than choosing A, no matter what the other player(s) do. (Strategy A strictly dominates B). This notion can be generalized beyond the comparison of two strategies. – Strategy B is strictly dominant if strategy B strictly dominates every other possible strategy. – Strategy B is weakly dominant if strategy B dominates all other strategies, but some are only weakly dominated. – Strategy B is strictly dominated if some other strategy exists that strictly dominates B. – Strategy B is weakly dominated if some other strategy exists that weakly dominates B. Source: Wikipedia

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ThePrisonersDilemma.cdf

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Imagine that Pepsi and Coca Cola are the only makers of cola drinks. Both are earning economic profits of $6000/day. Assume the following: – If Pepsi increases its advertising expenditures by $1000/day and Coca Cola spends no more on advertising, Pepsi’s profit will increase to $8000/day and Coca Cola’s will decrease to $2000. – If both spend $1000 on advertising, each will earn an economic profit of $5500/day. – If Pepsi stands pat while Coca Cola increases its spending by $1000, Pepsi’s economic profit will fall to $2000/day, and Coca Cola’s will increase to $8000. – The payoffs are symmetric. Will Pepsi spend more money on advertising? LO1: Basic Elements of A Game Ch9 -12 © 2012 McGraw-Hill Ryerson Limited

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The Payoff Matrix for an Advertising Game © 2012 McGraw-Hill Ryerson Limited Ch9 -13 LO1: Basic Elements of A Game

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Table 9.1: The Payoff Matrix for an Advertising Game © 2012 McGraw-Hill Ryerson Limited Ch9 -14 Suppose Coca Cola assumes that Pepsi will raise its spending on advertising, in that case, Coca Cola’s best option would be to follow suit. Payoff is higher LO1: Basic Elements of A Game

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Table 9.1: The Payoff Matrix for an Advertising Game © 2012 McGraw-Hill Ryerson Limited Ch9 -15 Suppose Coca Cola assumes that Pepsi will do nothing, in that case, Coca Cola’s best option would be to raise its spending on advertisements. Payoff is higher LO1: Basic Elements of A Game

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Table 9.1: The Payoff Matrix for an Advertising Game © 2012 McGraw-Hill Ryerson Limited Ch9 -16 No matter which strategy Pepsi chooses, Coca Cola will earn a higher economic profit by increasing its spending on advertising. Nash equilibrium Nash equilibrium Since this game is perfectly symmetric, a similar conclusion holds for Pepsi: No matter which strategy Coca Cola chooses, Pepsi will do better by increasing its spending on advertisements. Dominate strategy LO2: Identifying Dominant Strategy LO3: Find an Equilibrium for a Game

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Dominant strategy: – A strategy that yields a higher payoff no matter what the other players in a game choose. Dominated strategy: – Any other strategy available to a player who has a dominant strategy. Nash Equilibrium: – Any combination of strategies in which each player’s strategy is his best choice, given the other players’ strategies. LO2: Identifying Dominant Strategy Strategies © 2012 McGraw-Hill Ryerson Limited Ch9 -17

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Example 9.2: The Payoff Matrix for an Advertising Game When One Player Lacks a Dominant Strategy © 2012 McGraw-Hill Ryerson Limited Ch9 -18 No matter what Pepsi does, Coca Cola will do better to increase its advertising, so raising the advertising budget is a dominant strategy for Coca Cola. Payoff is higher Dominate strategy LO3: Find an Equilibrium for a Game

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Example 9.2: The Payoff Matrix for an Advertising Game When One Player Lacks a Dominant Strategy © 2012 McGraw-Hill Ryerson Limited Ch9 -19 Pepsi does not have a dominate strategy in this game. Payoff is higher Payoff is higher LO3: Find an Equilibrium for a Game

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Example 9.2: The Payoff Matrix for an Advertising Game When One Player Lacks a Dominant Strategy © 2012 McGraw-Hill Ryerson Limited Ch9 -20 Nash equilibrium: If Pepsi believes that Coca Cola will spend more on advertisements, Pepsi’s best strategy is to keep its own spending constant. Dominate strategy Nash Equilibrium Nash Equilibrium LO3: Find an Equilibrium for a Game

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Cartel: – A coalition of firms that agree to restrict output for the purpose of earning an economic (excess) profit. – Normally cartels involve several firms. This makes retaliation against a dissenter difficult. – Agreements are not legally enforceable and hence may be unstable. – Constant temptation for each participant to cheat on the agreement. Example: OPEC oil cartel production quotas. – Economic naturalist 9.1: Why might cartel agreements be unstable? LO5: Games with Equilibrium Like Prisoner’s Dilemma Cartels © 2012 McGraw-Hill Ryerson Limited Ch9 -21

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– Faced with the demand curve shown, a monopolist with zero marginal cost would produce 1000 bottles/day (the quantity at which marginal revenue equals zero) and sell them at a price of $1.00/bottle. FIGURE 9.1: The Market Demand for Mineral Water D MR © 2012 McGraw-Hill Ryerson Limited Ch9 -22 LO5: Games with Equilibrium Like Prisoner’s Dilemma

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– By cutting its price from $1/bottle to $0.90/bottle, Aquapure can sell the entire market quantity demanded at that price, 1100 bottles/day, rather than half the monopoly quantity of 1000 bottles/day. FIGURE 9.2: The Temptation to Violate a Cartel Agreement D MR © 2012 McGraw-Hill Ryerson Limited Ch9 -23 LO5: Games with Equilibrium Like Prisoner’s Dilemma

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– Each firm’s dominant strategy is to sell at the lower price, yet in following that strategy, each earns a lower profit than if each had sold at the higher price. TABLE 9.4: The Payoff Matrix for a Cartel Agreement © 2012 McGraw-Hill Ryerson Limited Ch9 -24 Nash Equilibrium Nash Equilibrium LO5: Games with Equilibrium Like Prisoner’s Dilemma

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