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
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
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
Ch9 -4
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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
Ch9 -5
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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
Ch9 -6
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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
√
√
Ch9 -7
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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
Ch9 -8
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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 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 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 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
Ch9 -12
© 2012 McGraw-Hill
Ryerson Limited
The Payoff Matrix for an Advertising Game
LO1: Basic Elements of A
Game
Ch9 -13
© 2012 McGraw-Hill
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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
Ch9 -14
© 2012 McGraw-Hill
Ryerson Limited
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
Ch9 -15
© 2012 McGraw-Hill
Ryerson Limited
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
-16
LO3: Find anCh9
Equilibrium
for a Game
© 2012 McGraw-Hill
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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
Ch9 -17
© 2012 McGraw-Hill
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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
Ch9 -18
© 2012 McGraw-Hill
Ryerson Limited
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
Ch9 -19 for a Game
© 2012 McGraw-Hill
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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
Ch9 -20
© 2012 McGraw-Hill
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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
Ch9 -21
Dilemma
© 2012 McGraw-Hill
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FIGURE 9.1: The Market Demand for Mineral Water
MR
D
– 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.
LO5: Games Ch9
with-22
Equilibrium Like Prisoner’s Dilemma
© 2012 McGraw-Hill
Ryerson Limited
FIGURE 9.2: The Temptation to Violate a Cartel Agreement
0.90
D
MR
1100
– 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.
© 2012 McGraw-Hill
LO5: Games Ch9
with-23
Equilibrium Like Prisoner’s Dilemma
Ryerson Limited
TABLE 9.4: The Payoff Matrix for a Cartel Agreement
Nash Equilibrium
– 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.
LO5: Games Ch9
with-24
Equilibrium Like Prisoner’s Dilemma
© 2012 McGraw-Hill
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101 What Is a Nash Equilibrium (Stoplight
Game) - YouTube_0.mp4