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Game Theory
“Loretta’s Driving Because I’m Drinking and I’m Drinking
Because She’s Driving”- The Lockhorns Cartoon
Mike ShorLecture 3
Game Theory - Mike Shor 2
Review Understand the game you are in
Note if the rules are flexible
Anticipate your opponents’ reactions
Understand the assumptions• Recognize that not everyone else understands them
Game Theory - Mike Shor 3
Game Theory - Mike Shor 4
Equilibrium Nash Equilibrium:
• A set of strategies, one for each player, such that each player’s strategy is best for her given that all other players are playing their equilibrium strategies
Best Response:• The best strategy I can play given the strategy
choices of all other players
Everybody is playing a best response• No incentive to unilaterally change my strategy
Game Theory - Mike Shor 5
Identifying the Equilibrium Pure strategy equilibrium
• Consider mixed later
Dominance• Dominance solvable• Only one dominant strategy
Successive elimination of dominated strategies
Cell-by-cell inspection
Game Theory - Mike Shor 6
Cigarette Advertising on TV All US tobacco companies advertised
heavily on TV
Surgeon General issues official warning• Cigarette smoking may be hazardous
Cigarette companies’ reaction• Fear of potential liability lawsuits
Companies strike agreement• Carry the warning label and cease TV
advertising in exchange for immunity from federal lawsuits.
1964
1970
Game Theory - Mike Shor 7
Strategic Interactions Players: Reynolds and Philip Morris Strategies: { Advertise , Do Not Advertise } Payoffs: Companies’ Profits
Each firm earns $50 million from its customers Advertising costs a firm $20 million Advertising captures $30 million from competitor
How to represent this game?
Game Theory - Mike Shor 8
Normal (Strategic) Form
PLAYERS
STRATEGIESPAYOFFS
Philip Morris
No Ad Ad
Reynolds No Ad 50 , 50 20 , 60
Ad 60 , 20 30 , 30
Game Theory - Mike Shor 9
Normal Form
Best reply for Reynolds:• If Philip Morris advertises: advertise• If Philip Morris does not advertise: advertise
Regardless of what you think Philip Morris will do
Advertise!
Philip Morris
No Ad Ad
ReynoldsNo Ad 50 , 50 20 , 60
Ad 60 , 20 30 , 30
Game Theory - Mike Shor 10
Dominant StrategyA strategy that outperforms all other choices
no matter what opposing players do Firm 1’s strategies: { A, B, C } Firm 2’s strategies: { X, Y, Z } C is strictly dominant for Firm 1 if:
(C,X)>(A,X) (C,X)>(B,X) (C,Y)>(A,Y) (C,Y)>(B,Y) (C,Z)>(A,Z) (C,Z)>(B,Z)
C is weakly dominant for Firm 1 if: Some inequalities are weak (), at least one is strong(>)
Game Theory - Mike Shor 11
Dominance Solvable
If each player has a dominant strategy, the game is dominance solvable
What is the equilibrium of the cigarette advertising game?
COMMANDMENT
If you have a dominant strategy, use it.
Expect your opponent to use her dominant strategy if she has one.
Game Theory - Mike Shor 12
Cigarette Advertising After the 1970 agreement, cigarette
advertising decreased by $63 million Profits rose by $91 million Prisoner’s Dilemma An equilibrium is NOT necessarily efficient
What if the game is not dominance solvable?
Game Theory - Mike Shor 13
A Strategic SituationTwo firms competing over sales
Time and The Economist must decide upon the cover story to run some week.
The big stories of the week are:• A presidential scandal (labeled S), and• A proposal to deploy US forces to Grenada (G)
Neither knows which story the other magazine will choose to run
Game Theory - Mike Shor 14
One Dominant Strategy
Who has a dominant strategy? Assume it will be played! Other player can plan accordingly.
The Economist
G S
TimeS 100 , 100 0 , 90
G 95 , 100 95 , 90
Game Theory - Mike Shor 15
Dominated Strategies
For The Economist: G dominant = S dominated
Dominated Strategy:• There exists another strategy which always does
better regardless of opponents’ actions
The Economist
G S
TimeS 100 , 100 0 , 90
G 95 , 100 95 , 90
Game Theory - Mike Shor 16
Successive Elimination of Dominated Strategies If a strategy is dominated,
eliminate it The size and complexity of the game
is reduced Eliminate any dominant strategies
from the reduced game Continue doing so successively
Game Theory - Mike Shor 17
Example: Tourists & Natives• Two bars (bar 1, bar 2) compete• Can charge price of $2, $4, or $5• 6000 tourists pick a bar randomly• 4000 natives select the lowest price bar
$2 $4 $5
Bar 1
$2 10 , 10 14 , 12 14 , 15
$4 12 , 14 20 , 20 28 , 15
$5 15 , 14 15 , 28 25 , 25
Bar 2
Game Theory - Mike Shor 18
Successive Elimination of Dominated Strategies Does any player have a dominant
strategy? Does any player have a
dominated strategy?• Eliminate the dominated strategies• Reduce the normal-form game• Iterate the above procedure
What is the equilibrium?
Game Theory - Mike Shor 19
Successive Elimination of Dominated Strategies
$4 $5
Bar 1$4 20 , 20 28 , 15
$5 15 , 28 25 , 25
25 , 25
28 , 15
14 , 15
$5$4
15 , 2815 , 14$520 , 2012 , 14$4Bar 114 , 1210 , 10$2
$2
,
,
,
, ,
, , Bar 1, ,
Bar 2
Bar 2
Game Theory - Mike Shor 20
No Dominated Strategies Often there are no dominated strategies
• Or: reducing the game is not sufficient
There may be multiple equilibria Method:
Cell-by-cell inspection Ask:
Is each player playing the best response to the other player?
Game Theory - Mike Shor 21
Types of Games Games of Assurance Games of Coordination Games of Chicken
Game Theory - Mike Shor 22
Games of Assurance Two firms each earning $45,000 Both can invest the $45,000 into R&D R&D successful only if both invest If R&D successful, each earns $95,000
Invest Don’t
Firm 1Invest 50 , 50 0 , 45
Don’t 45 , 0 45 , 45
Firm 2
Game Theory - Mike Shor 23
Consider { Invest , Don’t }
Both players have an incentive to change their strategy: NOT an equilibrium
Cell-by-cell Inspection
Invest Don’t
Firm 1Invest 50 , 50 0 , 45
Don’t 45 , 0 45 , 45
Firm 2
Game Theory - Mike Shor 24
Assurance Outcomes Two equilibria exist Both firms prefer (I ,I) to (D,D)
• Payoffs of 50 to each firm instead of 45 However, investing is risky
• Must have assurances How to achieve assurance?
• Strategic moves: commit to choosing I• Sequential moves: leader chooses
the equilibrium
Game Theory - Mike Shor 25
Games of Coordination Joint ventures and the choice of supplier Two firms engaged in joint venture Must use the same supplier, but
each firm has a preferred supplier
Firm 2A B
Firm 1A 100 , 50 0 , 0
B 0 , 0 50 , 100
Game Theory - Mike Shor 26
Coordination Outcomes Two equilibria exist Firms prefer different equilibria How to achieve the most
desirable outcome for you?• Strategic moves: commit to choosing A• Sequential moves: leader chooses
the equilibrium
Game Theory - Mike Shor 27
Summary You must put yourself in
your rival’s shoes
Recognize dominant and dominated strategies
Anticipate that your opponent will recognize them as well