2008/01/30 Lecture 1 1

Game Theory

2008/01/30 Lecture 1 2

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.

Are mathematical models helpful? Is strategic thinking really important?

2008/01/30 Lecture 1 3

Are mathematical models helpful?

Physical phenomena• Follow natural laws.• We cannot ask objects

for the reason behind the event.

Mathematical models are necessary.

Economic phenomena• Each person acts

anyway she wants.• She can explain why

she took a specific action.

No mathematical model is needed (?)

2008/01/30 Lecture 1 4

Two alternative approaches

• Institutional knowledge: Look into the “facts” in detail.

Superficial knowledge alone cannot explain economic movement.

• Economic Theory: Look for the “laws” behind facts.

Q: What’s the fundamental economic law?

A: Each person acts for her own interest.

2008/01/30 Lecture 1 5

Advantages of math models

Mathematical model can be served as a• Guideline for fact-finding.• Device of checking consistency

between different explanations.• Common language shared by

researchers.

Two approaches are both important: not substitute, but complement.

2008/01/30 Lecture 1 6

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.”

2008/01/30 Lecture 1 7

The birth of Game Theory

Q: Are most of problems in Economics indeed mere applications of “constrained optimization”?

A: NO!

von Neumann and Morgenstern (1944)“We need essentially new mathematical

theory to solve variety of problems in social sciences.”

2008/01/30 Lecture 1 8

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 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…

2008/01/30 Lecture 1 9

Revolution by Game Theory

• Game theory can solve the problem of strategic inter-dependence by pinning down how to predict other players’ action.

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.

2008/01/30 Lecture 1 10

New Areas Pioneered by GT

• How does economy function if market is immature or not existing?

Economic History, Development Economics

• How do governments behave? Political Economics

• What’s going on inside private companies? Organizational Economics

• How to compare different types of market economy? Comparative Institutional Analysis

2008/01/30 Lecture 1 11

Discovery by vN=M

• Any social problem can be formalized as a “game,” consisting of three elements:

Players: i=1,2,…,N

i’s Strategy:

i’s Payoff:

Q: What’s the solution of games? They failed to establish a general solution

concept…

ii Ss ),...,( 1 Ni ss

2008/01/30 Lecture 1 12

Nash Equilibrium

John Nash: “A Beautiful Mind” (movie) • The solution of games must satisfy the

following criterion.

Nash equilibrium (mathematical definition)

),()(

,

),...,(

**

**1

*

iiii

i

N

sss

si

sss

2008/01/30 Lecture 1 13

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.

2008/01/30 Lecture 1 14

Rationality Question

Q: Does NE heavily depend on rationality of players?

A: Not necessarily so.

Example: NE in Evolutionary BiologyStrategy = “phenotype” : a character of each

animal determined by its genePayoff = “fitness” : a number of the offspring• NE is a stable situation reached not by

rationality but by evolutionary dynamics.

2008/01/30 Lecture 1 15

Existence Question

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.”