Game Theory

“Life must be understood backward, but … it must be lived forward.”

- Soren Kierkegaard

James Bednar

Drawing a decision table: What ACTIONS are you try to choose between?

• Put your actions on the rows.• United States: Search North or Search South• Japan: Go North or Go South

The OUTCOMES of your actions depend upon what?

• They depend upon what your rival does.• Put your rival’s actions on the columns.

What are the potential OUTCOMES?

• Number of days of bombing. • U.S. prefers higher numbers. Japan prefers lower numbers.

What is YOUR best action? What is your RIVAL’s best action?

Battle of the Bismarck Sea

Summary: A U.S. Army general knew that the Japanese were about to send a supply convoy to Papua-New Guinea. He knew that they would take either a rainy northern route or a sunny southern route but didn’t know which route they would take. He had to decide which route to search first.

Japan

Take North

Take South

U.S.

Search North

2 2

Search South

1 3

Japan’s Goal: Minimize days of being bombed; thus, lower numbers are better.

U.S.’s Goal: Maximize days of bombing; thus, higher numbers are better

Battle of the Bismarck Sea

Summary: A U.S. Army general knew that the Japanese were about to send a supply convoy to Papua-New Guinea. He knew that they would take either a rainy northern route or a sunny southern route but didn’t know which route they would take. He had to decide which route to search first.

Japan

Take North

Take South

U.S.

Search North

2 2

Search South

1 3

Japan’s Goal: Minimize days of being bombed; thus, lower numbers are better.

U.S.’s Goal: Maximize days of bombing; thus, higher numbers are better

Northern or southern route?

Japan

Take North

Take South

U.S.

Search North

2 2

Search South

1 3

Japan has a dominant act, so they “should” take it. Go North.

U.S. expects Japan to take its dominant act, so they should search north.

In effect:

Japan

Take North

Take South

U.S.

Search North

2 2

Search South

1 3

In effect:

Japan

Take North

Take South

U.S.

Search North

2 2

Search South

1 3 In effect, the U. S. now has a dominant act as well.

Game Theory:

“When you think carefully before you act--when you are aware of your objectives or preferences and of any limitations or constraints on your actions, and choose your actions in a calculated way to do the best thing according to your own criteria--you are said to be behaving rationally. Game theory adds another dimension to rational behavior, namely interaction with other equally rational decision makers. In other words, game theory is the science of rational behavior in interactive situations.”

Dixit and Skeath, Games of Strategy

Definition of a game

Must consider the strategic environment• Who are the decision makers (PLAYERS)? • What are the feasible ACTIONS? • What are the outcomes (PAYOFFS)?

Rules of the game• What is the nature of the conflict?

• Zero-sum or• No zero-sum

• What is the nature of interaction? • One time or• Iterated

• What information is available?• Simultaneous or• Sequential

Simultaneous or Sequential?

Simultaneous Players must move without knowledge of what their rivals have

chosen to do. Try to figure out what opponent is going to do right now;

opponent is trying to do the same, and recognizes this, and recognizes that you recognize this.....

Battle of the Bismarck Sea

Sequential There is a strict order of play, and players know what players

who have gone before them have done. Each player must consider: if I make this move, how will my

opponent respond? Senate Race Game, Tic-Tac-Toe

Player 1

You have $100 that you must share between yourself and player 2. You must make an offer as to how to split the $100. Player 2 must either reject or accept the offer. If player 2 rejects the offer, neither of you will get anything. If player 2 accepts the offer, you divide the money accordingly.

What is your offer?

Player 2

Do you accept or reject the offer?

Sequential Rationality

COMMANDMENT

Look forward and reason back.

Anticipate what your rivals will do tomorrow

in response to your actions today

You have $100,000 that you must divide between yourself and another person (player 2) in $1,000 increments. You must make an offer as to how to divide the money. If player 2 accepts the proposal, you divide the money accordingly. If player 2 rejects the proposal, he or she gets $80,000.

What do you offer?

Sequential Rationality

COMMANDMENT

Look forward and reason back.

Anticipate what your rivals will do tomorrow

in response to your actions today