Transcript ECO290E: Game Theory

ECO290E: Game Theory
Lecture 2
Static Games and Nash
Equilibrium
2008/02/06
Lecture 2
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Review of Lecture 1
Game Theory
• studies strategically inter-dependent
situations.
• provides us tools for analyzing most of
problems in social science.
• employs Nash equilibrium as a solution
concept.
• creates a revolution in Economics.
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What is Game?
Complete
Information
Incomplete
Information
Static
Nash
Equilibrium
(Lec. 2-5)
Bayesian Nash
Equilibrium
(Lec. 10-11)
Dynamic
Subgame
Perfect
Equilibrium
(Lec. 6, 8-9)
Perfect Bayesian
Equilibrium
(Lec. 12-14)
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Games in Two Forms
• Static games
 The normal/strategic-form
representation
• Dynamic games
 The extensive-form representation
• In principle, static (/ dynamic) games
can also be analyzed in an extensiveform
(/a
normal-form)
representation.
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Normal-form Games
The normal-form (strategic-form)
representation of a game specifies:
1. The players in the game.
2. The strategies available to each player.
3. The payoff received by each player
(for each combination of strategies
that could be chosen by the players).
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Static Games
• In a normal-form representation, each player
simultaneously chooses a strategy, and the
combination of strategies chosen by the
players determines a payoff for each player.
• The players do not necessarily act
simultaneously: it suffices that each chooses
her own action without knowing others’
choices.
 We will also study dynamic games in an
extensive-form representation later.
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Example: Prisoners’
Dilemma
• Two suspects are charged with a joint clime,
and are held separately by the police.
• Each prisoner is told the following:
1) If one prisoner confesses and the other one
does not, the former will be given a reward of
1 and the latter will receive a fine equal to 2.
2) If both confess, each will receive a fine equal
to 1.
3) If neither confesses, both will be set free.
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Payoff Bi-Matrix
Player 2
Silent
Confess
Player 1
Silent
0
0
Confess
-2
-2
1
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-1
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How to Use Bi-Matrices
• Any two players game (with finite number of
strategies) can be expressed as a bi-matrix.
• The payoffs to the two players when a
particular pair of strategies is chosen are
given in the appropriate cell.
• The payoff to the row player (player 1) is
given first, followed by the payoff to the
column player (player 2).
 How can we solve this game?
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Definition of Nash
Equilibrium
Nash equilibrium (mathematical definition)
• A strategy profile s* is called a Nash
equilibrium if and only if the following
condition is satisfied:
i, si
 i ( s )   i ( s , si )
*
*
i
• Nash equilibrium is defined over strategy
profiles, NOT over individual strategies.
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Solving PD Game
• For each player, u(C,C)>u(S,C) holds.
 (Confess, Confess) is a NE.
• There is no other equilibrium.
• Playing “Confess” is optimal no matter how
the opponent takes “Confess” or “Silent.”
 “Confess” is a dominant strategy.
• The NE is not (Pareto) efficient.
 Optimality for individuals does not
necessary imply optimality for a group
(society).
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Terminology
Dominant strategy:
• A strategy s is called a dominant strategy if
playing s is optimal for any combination of
other players’ strategies.
Pareto efficiency:
• An outcome of games is Pareto efficient if it
is not possible to make one person better off
(through moving to another outcome)
without making someone else worse off.
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Applications of PD
Examples
Players
Silent
Confess
Arms
races
Countries
Disarm
Arm
International
trade policy
Countries
Lower trade
barriers
No change
Marital
cooperation
Couple
Obedient
Provision of
public goods
Citizen
Demandin
g
Contribute Free-ride
Deforestatio Woodmen
n
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Restrain
cutting
Lecture 2
Cut down
maximum
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Example: Coordination
Game
Player 2
Player 1
Windows
Windows
Mac
1
1
0
Mac
0
0
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2
2
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Solving Coordination Game
• There are two equilibria, (W,W) and (M,M).
 Games, in general, can have more than one
Nash equilibrium.
• Everybody prefers one equilibrium (M,M) to
the other (W,W).
 Several equilibria can be Pareto-ranked.
• However, bad equilibrium can be chosen.
 This is called “coordination failure.”
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Other Examples
• Battle of the sexes
 Corruption Game
• Stag Hunt Game
 Migration Game
• Hawk-Dove (Chicken) Game
 Land Tenure Game
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