ECO290E Game Theory
Lecture 2: Static Games and Nash Equilibrium
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.
made a revolution in Economics.
Two Frameworks of Game Theory
Non-cooperative Game Theory
examine individual decision making in strategic settings.
assume a person decides her action on her own.
does NOT rule out cooperative behaviors.
Cooperative Game Theory
examine group decision making in strategic settings.
assume players can agree on their joint action, or can make binding contracts.
simplifies strategic analysis by NOT modeling the negotiation process explicitly.
⇒ The two tools are complements to one another, but this lecture focuses mainly on Non-cooperative games.
Timing and Information
Complete
Information
Incomplete
Information
Static Nash Equilibrium
(Lecture 2-4)
Bayesian NE
(Lecture 11-12)
Dynamic Subgame Perfect
Equilibrium
(Lecture 5, 7-10)
Perfect Bayesian
Equilibrium
(not covered)
Look into Games
We consider static situations in which each player
simultaneously chooses a strategy, and the combination of
strategies determines a payoff for each player.
The players need not literally act simultaneously.
Each chooses her own action without knowing others’ choices.
Formally, representation of a game specifies:
1.
Players in the game.
2.
Strategies available to each player.
3.
Payoff received by each player (for every possible
combination of strategies chosen by the players).
Review: Prisoners’ Dilemma
Here we set payoffs as (negative of) years.
Other numbers such that larger number shows better outcomes can express the same situations.
Player 2 Player 1
Silent Confess
Silent -1 -1
0 -5
Confess -5 0
-3 -3
Prisoners’ Dilemma: Analysis
(Silent, Silent) looks mutually beneficial outcomes, though
Playing Confess is optimal regardless of other player’s choice!
Acting optimally (Confess, Confess) rends up realizing!!
This is why the game is called Prisoners’ “Dilemma”. Player 2
Player 1
Silent Confess
Silent -1 -1
0 -5
Confess -5 0
-3 -3
Prisoners’ Dilemma: Remarks
Playing “Confess” is optimal no matter how the
opponent takes “Confess” or “Silent.”
“Confess” is an optimal (dominant) strategy.
Combination of dominant strategies is Nash equilibrium.
There are many games where no dominant strategy exists.
Individually best decision ≠ Socially efficient outcome
Optimality for individuals does not necessary imply optimality (Pareto efficiency) for a group or society.
Terminology
Dominant strategy:
A strategy X is called a dominant strategy if playing X
is optimal for any combination of other players’
strategies.
Pareto efficiency:
An outcome of games is Pareto efficient (or Pareto
optimal) if it is not possible to make one person
better off (through moving to another outcome)
without making someone else worse off.
Prisoners’ Non-Dilemma
If plea bargain is not allowed: (Silent, Silent) realizes
Playing Silent becomes optimal regardless of other player’s choice!
Police cannot achieve (Confess, Confess) any more… Player 2
Player 1
Silent Confess
Silent -1 -1
-3 -3
Confess -3 -3
-3 -3
Application: Price Competition
(Keep, Keep) are joint-profit maximizing outcomes, though
Cprice-cut is optimal regardless of other player’s choice!
Acting optimally (Price-cut, Price-cut) realize!!
Competing firms may be difficult to escape from price wars.
Player 2 Player 1
Keep Price-cut
Keep 2 2
3 -1
Price-cut -1 3
0 0
General Formulation of PD
The larger the payoff, the better the corresponding result.
Desirability of outcomes for each player:
g > c > d > l, that is, (D, C) > (C, C) > (D, D) > (C, D) Player 2
Player 1
Cooperation Defection
Cooperation c c
g l
Defection l g
d d
Applications of PD
Examples Players “Cooperation” “Defection”
Arms races Countries Disarm Arm
International trade policy
Countries Lower trade barriers
No change
Marital cooperation
Couple Obedient Demanding
Provision of public goods
Citizen Contribute Free-ride Deforestation Woodmen Restrain cutting Cut down
maximum
Example: Coordination Game
Two students need a new computer each for joint-work.
Having different OS generates no value.
They (are assumed to) prefer (Mac, Mac) to (Win, Win) Player 2
Player 1
Windows Mac
Windows 1 1
0 0
Mac 0
0
2 2
Coordination Game: Optimal Play (1)
If the opponent chooses Win,
Selecting Win is optimal since 1 > 0. Player 2
Player 1
Windows Mac
Windows 1 1
0 0
Mac 0
0
2 2
Coordination Game: Optimal Play (2)
If the opponent chooses Mac,
Selecting Mac is optimal since 2 > 0. Player 2
Player 1
Windows Mac
Windows 1 1
0 0
Mac 0
0
2 2
Coordination Game: Analysis
There is NO dominant (optimal) strategy.
Choosing Mac (Win) is optimal if the others is Mac (Win).
Best strategy varies depending on other’s choice.
Game cannot be solved merely from individual rationality.
Coordination game is NOT like Prisoner’s dilemma.
We need to look at Nash equilibrium!
(Mac , Mac) looks a unique reasonable outcome…
Let’s review the definition of Nash equilibrium!
The Solution: Nash Equilibrium
John Nash discovered a path breaking solution
concept, called Nash equilibrium!
No one can benefit if she unilaterally changes her
action from the original Nash equilibrium.
⇒
NE describes a stable situation.
Everyone correctly predicts other players’ actions
and takes best-response against them.
⇒
NE serves as a rational prediction.
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.”
Finding NE: Underline Method
Draw underlines for optimal strategy for each player.
If both numbers are underlined => Nash equilibrium
Otherwise => outcome is NOT Nash equilibrium Player 2
Player 1
Windows Mac
Windows 1 1
0 0
Mac 0
0
2 2
Example: Battle of the Sexes
A wife (player 1) and a husband (player 2) are supposed
to choose between going to a musical and a soccer game.
Presumably, the wife prefers the musical and the husband
the soccer game.
Player 2 Player 1
Musical Soccer
Musical 1 3
0 0
Soccer 0 0
3 1
Battle of the Sexes: Underline Method
There are two Nash equilibria:
(Musical, Musical) and (Soccer, Soccer)
Unlike Coordination Game, two equilibria cannot be
(Pareto) comparable: conflict of interests
Player 2 Player 1
Musical Soccer
Musical 1 3
0 0
Soccer 0 3
Example: Chicken Game
Two teenagers take their cars to opposite ends of main street and start to drive toward each other.
The one who swerves to prevent a collision is the chicken and the one who keeps going straight (tough) is the winner.
Player 2 Player 1
Chicken Tough
Chicken 2 2
3 0
Tough 0 3
-1 -1
Chicken Game: Optimal Play (1)
If the opponent chooses Chicken,
Playing Tough is optimal since 3 > 2.
This is the best outcome!
Player 2 Player 1
Chicken Tough
Chicken 2 2
3 0
Tough 0 3
-1 -1
Chicken Game: Optimal Play (2)
If the opponent chooses Tough,
Playing Chicken is optimal since 0 > -1.
Low payoff, but still better than the worst outcome.
Player 2 Player 1
Chicken Tough
Chicken 2 2
3 0
Tough 0 3
-1 -1
Chicken Game: Underline Method
There are two Nash equilibria:
(Chicken, Tough) and (Tough, Chicken).
They are asymmetric and look unfair, but still equilibrium!
Player 2 Player 1
Chicken Tough
Chicken 2 2
3 0
Tough 0 3
-1 -1
Further Exercises
Find a social problem which can be described as a
Coordination game,
Battle of sexes, and
Chicken game.
