Introduction to Game Theory
Advanced Microeconomics II
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What is Game Theory?
Game theory is a field of Mathematics,
analyzing strategically inter‐dependent
situations in which the outcome of your
actions depends also on the actions of others
actions depends also on the actions of others.
–Is strategic thinking really important?
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Is strategic thinking relevant?
• In price theory, the market outcome is derived by the intersection of the demand curve and the supply curve. (demand‐supply analysis)
• There is no strategic inter‐dependence in its framework.
Q: What’s the underlying assumption? A: Each economic agent is a “price‐taker.”
The birth of Game Theory
Q: Are most of problems in Economics indeed
mere applications of “constrained
von Neumann and Morgenstern (1944)
“We need essentially new mathematical theory
to solve variety of problems in social
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Strategic Thinking: Example
Example: Nintendo vs. Sony
• Nintendo’s action depends on how Nintendo predicts the Sony’s action.
• Nintendo’s action depends on how Nintendo predicts
h S di h Ni d ’ i
how Sony predicts the Nintendo’s action.
• Nintendo’s action depends on how Nintendo predicts how Sony predicts how Nintendo predicts the Sony’s action.
and so on… (this is called “infinite regress”)
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Revolution by Game Theory
• Game theory can solve the problem of strategic inter‐dependence by pinning down how to predict other players’ action.
Th f h
Therefore, game theory
• Provides us tools for analyzing important economic phenomena beyond market economy (with perfect competition).
• Enables us to compare different resource allocation mechanisms.
New Areas Pioneered by Game Theory
• How does economy function if market is immature or not existing?
– Economic History, Development Economics
• How do governments behave? Political Economics
– Political Economics
• What’s going on inside private companies? – Organizational Economics
• How to compare different types of market economy? – Comparative Institutional Analysis
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Discovery by vNM
• Any social problem can be formalized as a “game,” consisting of three elements:
Players: i=1,2,…,N i’s Strategy: gy si
Si i’s Payoff:
Q: What’s the solution of games?
They failed to establish a general solution concept…
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John Nash: “A Beautiful Mind” (movie)
• The solution of games must satisfy the following criterion.
Nash equilibrium(mathematical definition):
) , ( ) ( ,
) ,..., (
i i i i
s s s
s s s
Interpretations of NE.
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.
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Q: Does NE heavily depend on rationality of players? A: Not necessarily so.
Example: NE in Evolutionary Biology
Strategy = “phenotype” : a character of each animal determined by its gene
Payoff = “fitness” : a number of the offspring
• NE is a stable situation reached not by rationality but by evolutionary dynamics.
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Q: Does NE always exist?
A: Yes (in almost every cases).
Theorem (Nash, 1950)
“There exists at least one Nash equilibrium in
any finite games in which the numbers of
players and strategies are both finite.”(we will consider this issue in lecture 5)
Impact of Nash
“Soon after Nash’s work, game‐theoretic models began to be used in economic theory and political science, and psychologists began studying how human subjects behave in experimental games. In the 1970s game theory was first used as a tool in g y evolutionary biology. Subsequently, game‐ theoretic methods have come to dominate microeconomic theory and are used also in many other fields of economics and a wide range of other social and behavioral sciences.”
(from An Introduction to Game Theory by Osborne)
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Games in Two Forms
In principle, static (/ dynamic) games can also
be analyzed in an extensive‐form (/a normal‐
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The normal‐form (strategic‐form) representation
of a game specifies:
1 h l i h
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).
• In a normal‐form representation, each player simultaneouslychooses 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
• 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:
–If one prisoner confesses and the other one does not, p , the former will be given a reward of 1 and the latter will receive a fine equal to 2.
–If both confess, each will receive a fine equal to 1. –If neither confesses, both will be set free.
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Player 2 Player 1
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:
• Nash equilibrium is defined over strategy profiles, NOT over individual strategies.
) , ( ) ( ,
i i i i
s s s
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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 thePlaying 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).
• A strategy s is called a dominant strategy if playing s is optimal for anycombination of other players’ strategies.
• 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 polic
Countries Lower trade barriers
trade policy barriers
Couple Obedient Demanding
Provision of public goods
Citizen Contribute Free-ride
Deforestation Woodmen Restrain cutting Cut down maximum
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Example: Coordination Game
Player 2 Player 1
Solving Coordination Game
• There are two equilibria, (W,W) and (M,M). –Games, in general, can have more than oneNash
• 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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Battle of the sexes–“Corruption” Game
Stag Hunt Game
“Mi i ” G
Hawk‐Dove (Chicken) Game–“Land Tenure” Game
(Chapter 2 in Games in Economic Development by Wydick)
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Two Frameworks of Games
• Non‐cooperative Game Theory
– examine individual decision makingin strategic settings. – assume a person decides her action on her own. – does NOT rule out cooperative behaviors.
• Cooperative Game Theory
i d i i ki i i i
– examine group decision makingin strategic settings. – assume players can agree on their joint action, or can make
– simplifies strategic analysis by NOT modeling the negotiation process explicitly.
The two tools are complements to one another, but this lecture focuses (almost) entirely on Non‐cooperative game.
Games in Different Settings
Incomplete Information Static Nash Equilibrium
(L t 3 5)
Bayesian NE (L t 9 10) (Lecture 3-5) (Lecture 9,10) Dynamic Subgame Perfect
Equilibrium (Lecture 6-8)
Perfect Bayesian Equilibrium (Lecture, 11,12)
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