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## Introduction to Game Theory

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Introduction to Game Theory

Yale BraunsteinSpring 2007

General approach

- Brief History of Game Theory
- Payoff Matrix
- Types of Games
- Basic Strategies
- Evolutionary Concepts
- Limitations and Problems

Brief History of Game Theory

- 1913 - E. Zermelo provides the first theorem of game theory; asserts that chess is strictly determined
- 1928 - John von Neumann proves the minimax theorem
- 1944 - John von Neumann & Oskar Morgenstern write "Theory of Games and Economic Behavior”
- 1950-1953 - John Nash describes Nash equilibrium

Rationality

Assumptions:

- humans are rational beings
- humans always seek the best alternative in a set of possible choices

Why assume rationality?

- narrow down the range of possibilities
- predictability

Utility Theory

Utility Theory based on:

- rationality
- maximization of utility
- may not be a linear function of income or wealth

It is a quantification of a person's preferences with respect to certain objects.

What is Game Theory?

Game theory is a study of how to mathematically determine the best strategy for given conditions in order to optimize the outcome

Game Theory

- Finding acceptable, if not optimal, strategies in conflict situations.
- Abstraction of real complex situation
- Game theory is highly mathematical
- Game theory assumes all human interactions can be understood and navigated by presumptions.

Why is game theory important?

- All intelligent beings make decisions all the time.
- AI needs to perform these tasks as a result.
- Helps us to analyze situations more rationally and formulate an acceptable alternative with respect to circumstance.
- Useful in modeling strategic decision-making
- Games against opponents
- Games against "nature„
- Provides structured insight into the value of information

Types of Games

- Sequential vs. Simultaneous moves
- Single Play vs. Iterated
- Zero vs. non-zero sum
- Perfect vs. Imperfect information
- Cooperative vs. conflict

Zero-Sum Games

- The sum of the payoffs remains constant during the course of the game.
- Two sides in conflict
- Being well informed always helps a player

Non-zero Sum Game

- The sum of payoffs is not constant during the course of game play.
- Players may co-operate or compete
- Being well informed may harm a player.

Games of Perfect Information

- The information concerning an opponent’s move is well known in advance.
- All sequential move games are of this type.

Imperfect Information

- Partial or no information concerning the opponent is given in advance to the player’s decision.
- Imperfect information may be diminished over time if the same game with the same opponent is to be repeated.

Matrix Notation

Notes: Player I's strategy A may be different from Player II's.

P2 can be omitted if zero-sum game

Prisoner’s Dilemma & Other famous games

A sample of other games:

Marriage

Disarmament (my generals are more irrational than yours)

NCE

Prisoner 2

Blame

Don't

Blame

10 , 10

0 , 20

Prisoner 1

Don't

20 , 0

1 , 1

Notes: Higher payoffs (longer sentences) are bad.

Non-cooperative equilibrium Joint maximumInstitutionalized “solutions” (a la criminal organizations, KSM)

Jt. max.

Games of Conflict

- Two sides competing against each other
- Usually caused by complete lack of information about the opponent or the game
- Characteristic of zero-sum games

Games of Co-operation

Players may improve payoff through

- communicating
- forming binding coalitions & agreements
- do not apply to zero-sum games

Prisoner’s Dilemma

with Cooperation

Prisoner’s Dilemma with Iteration

- Infinite number of iterations
- Fear of retaliation
- Fixed number of iteration
- Domino effect

Basic Strategies

1. Plan ahead and look back

2. Use a dominating strategy if possible

3. Eliminate any dominated strategies

4. Look for any equilibrium

5. Mix up the strategies

If you have a dominating strategy, use it

Opponent

Strategy 1

Strategy 2

Strategy 1

150

1000

You

Strategy 2

25

- 10

Eliminate strategy 2 as it’s dominated by strategy 1

Eliminate any dominated strategyOpponent

Strategy 1

Strategy 2

Strategy 1

150

1000

You

Strategy 2

25

- 10

Strategy 3

160

-15

Look for any equilibrium

- Dominating Equilibrium
- Minimax Equilibrium
- Nash Equilibrium

Maximin & Minimax Equilibrium

- Minimax - to minimize the maximum loss (defensive)
- Maximin - to maximize the minimum gain (offensive)
- Minimax = Maximin

Maximin & Minimax Equilibrium Strategies

Opponent

Strategy 1

Strategy 2

Min

150

Strategy 1

150

1000

You

Strategy 2

25

- 10

- 10

Strategy 3

160

-15

-15

Max

160

1000

Definition: Nash Equilibrium

“If there is a set of strategies with the property that no player can benefit by changing her strategy while the other players keep their strategies unchanged, then that set of strategies and the corresponding payoffs constitute the Nash Equilibrium. “

Source: http://www.lebow.drexel.edu/economics/mccain/game/game.html

Is this a Nash Equilibrium?

Opponent

Strategy 1

Strategy 2

Min

150

Strategy 1

150

1000

You

Strategy 2

25

- 10

- 10

Strategy 3

160

-15

-15

Max

160

1000

Time for "real-life" decision making

- Holmes & Moriarity in "The Final Problem"
- What would you do…
- If you were Holmes?
- If you were Moriarity?
- Possibly interesting digressions?
- Why was Moriarity so evil?
- What really happened?
- What do we mean by reality?
- What changed the reality?

Moriarty

Strategy #1

Strategy #2

Canterbury

Dover

Player #1

0

50

Strategy #1

Canterbury

Payoff (1,1)

Payoff (1,2)

Holmes

0

100

Dover

Strategy #2

Payoff (2,1)

Payoff (2,2)

The Payoff Matrix for Holmes & MoriarityWhere is game theory currently used?

- Ecology
- Networks
- Economics

Limitations & Problems

- Assumes players always maximize their outcomes
- Some outcomes are difficult to provide a utility for
- Not all of the payoffs can be quantified
- Not applicable to all problems

Summary

- What is game theory?
- Abstraction modeling multi-person interactions
- How is game theory applied?
- Payoff matrix contains each person’s utilities for various strategies
- Who uses game theory?
- Economists, Ecologists, Network people,...
- How is this related to AI?
- Provides a method to simulate a thinkingagent

Sources

- Much more available on the web.
- These slides (with changes and additions) adapted from: http://pages.cpsc.ucalgary.ca/~jacob/Courses/Winter2000/CPSC533/Pages/index.html
- Three interesting classics:
- John von Neumann & Oskar Morgenstern, Theory of Games & Economic Behavior (Princeton, 1944).
- John McDonald, Strategy in Poker, Business & War (Norton, 1950)
- Oskar Morgenstern, "The Theory of Games," Scientific American, May 1949; translated as "Theorie des Spiels," Die Amerikanische Rundschau, August 1949.

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