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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. Strategy. Dominant strategy - PowerPoint PPT Presentation
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Introduction to Microeconomics
Game theoryChapter 9
• 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
Strategy
Dominant strategy – One that yields the highest payoff no matter what
the other players in the game chooseDominated strategy
– Any other strategy available to a player that has a dominant strategy
• 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
• 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 -5Lo4: The Effect of Dominant
Strategy
• 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
• 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
• 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
Better Outcome
Lo4: The Effect of Dominant Strategy
• ..\..\..\..\..\..\Users\gmason.PRAINC\Documents\Pavtube\youtube_converter\Nash Equilibrium - YouTube.mp4
TerminologyWhen 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
• 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
The Payoff Matrix for an Advertising Game
© 2012 McGraw-Hill Ryerson Limited Ch9 -13LO1: Basic Elements of A Game
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
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
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
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 StrategyLO3: Find an Equilibrium for a Game
• 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
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 increaseits advertising, so raising the advertising budget is a dominant strategy for Coca Cola.
Payoff is higher
Payoff is higher
Dominate strategy
LO3: Find an Equilibrium for a Game
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
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
LO3: Find an Equilibrium for a Game
• 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
– 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
DMR
© 2012 McGraw-Hill Ryerson Limited Ch9 -22LO5: Games with Equilibrium Like Prisoner’s
Dilemma
– 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
DMR
0.90
1100
© 2012 McGraw-Hill Ryerson Limited Ch9 -23LO5: Games with Equilibrium Like Prisoner’s
Dilemma
– 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
LO5: Games with Equilibrium Like Prisoner’s Dilemma
• ..\..\..\..\..\..\Users\gmason.PRAINC\Documents\Pavtube\youtube_converter\Game Theory 101 What Is a Nash Equilibrium (Stoplight Game) - YouTube_0.mp4