Introduction to Microeconomics

Introduction to Microeconomics Game theory Chapter 9

Elements of a Game • 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

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

Prisoner’s Dilemma • 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

The Payoff Matrix for the original Prisoner’s Dilemma • 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. Lo4: The Effect of Dominant Strategy

Table 9.3: The Payoff Matrix for the original Prisoner’s Dilemma • Example 9.3: Will the prisoners confess? • The dominant strategy for each prisoner is to confess. Dominate strategy √ √ Lo4: The Effect of Dominant Strategy

The Payoff Matrix for the original Prisoner’s Dilemma • Will the prisoners confess? • The dominant strategy for each prisoner is to confess. Dominate strategy √ √ Lo4: The Effect of Dominant Strategy

Table 9.3: The Payoff Matrix for the original Prisoner’s Dilemma • Will the prisoners confess? • When each follows his dominant strategy and confesses, both will do worse than if each had shown restraint. 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. • 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

ThePrisonersDilemma.cdf

Will Pepsi spend more money on advertising? • 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. LO1: Basic Elements of A Game

The Payoff Matrix for an Advertising Game LO1: Basic Elements of A Game

Table 9.1: The Payoff Matrix for an Advertising Game 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

Table 9.1: The Payoff Matrix for an Advertising Game 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

Table 9.1: The Payoff Matrix for an Advertising Game No matter which strategy Pepsi chooses, Coca Cola will earn a higher economic profit by increasing its spending on advertising. 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 Nash equilibrium LO2: Identifying Dominant Strategy LO3: Find an Equilibrium for a Game

Strategies • 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

Example 9.2: The Payoff Matrix for an Advertising Game When One Player Lacks a Dominant Strategy 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. Dominate strategy Payoff is higher Payoff is higher LO3: Find an Equilibrium for a Game

Example 9.2: The Payoff Matrix for an Advertising Game When One Player Lacks a Dominant Strategy Pepsi does not have a dominate strategy in this game. Payoff is higher Payoff is higher LO3: Find an Equilibrium for a Game

Example 9.2: The Payoff Matrix for an Advertising Game When One Player Lacks a Dominant Strategy 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 LO3: Find an Equilibrium for a Game

Cartels • 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

FIGURE 9.1: The Market Demand for Mineral Water • 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. D MR LO5: Games with Equilibrium Like Prisoner’s Dilemma

FIGURE 9.2: The Temptation to Violate a Cartel Agreement • 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. 0.90 D MR 1100 LO5: Games with Equilibrium Like Prisoner’s Dilemma

TABLE 9.4: The Payoff Matrix for a Cartel Agreement • 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. Nash Equilibrium LO5: Games with Equilibrium Like Prisoner’s Dilemma

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Introduction to Microeconomics