Introduction to Game Theory

1
Introduction to Game Theory
Yale BraunsteinSpring 2007
2
General approach

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

3
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

4
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

5
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.

6
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

7
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.

8
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

9
Types of Games

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

10
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

11
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.

12
Games of Perfect Information

• The information concerning an opponents move is
  well known in advance.
• All sequential move games are of this type.

13
Imperfect Information

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

14
Key Area of Interest

• chance
• strategy

Non-zero Sum
Imperfect Information
15
Matrix Notation
Notes Player I's strategy A may be different
from Player II's. P2 can be omitted if zero-sum
game
16
Prisoners Dilemma Other famous games
A sample of other games Marriage Disarmament
(my generals are more irrational than yours)
17
Prisoners Dilemma
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.
18
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

19
Games of Co-operation

• Players may improve payoff through
• communicating
• forming binding coalitions agreements
• do not apply to zero-sum games
• Prisoners Dilemma
• with Cooperation

20
Prisoners Dilemma with Iteration

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

21
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

22
Plan ahead and look back
Opponent
Strategy 2
Strategy 1
150
1000
Strategy 1
You
25
Strategy 2
- 10
23
If you have a dominating strategy, use it
Opponent
Strategy 2
Strategy 1
150
1000
Strategy 1
You
25
Strategy 2
- 10
24
Eliminate any dominated strategy
Opponent
Strategy 2
Strategy 1
150
1000
Strategy 1
You
Strategy 2
25
- 10
Strategy 3
-15
160
25
Look for any equilibrium

• Dominating Equilibrium
• Minimax Equilibrium
• Nash Equilibrium

26
Maximin Minimax Equilibrium

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

27
Maximin Minimax Equilibrium Strategies
Opponent
Strategy 2
Strategy 1
Min
150
150
1000
Strategy 1
You
25
Strategy 2
- 10
- 10
Strategy 3
-15
160
-15
160
Max
1000
28
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/mcca
  in/game/game.html

29
Is this a Nash Equilibrium?
Opponent
Strategy 2
Strategy 1
Min
150
150
1000
Strategy 1
Strategy 2
You
25
- 10
- 10
Strategy 3
-15
160
-15
160
Max
1000
30
Boxed Pigs Example
Cost to press button 2 units
When button is pressed, food given 10 units
31
Decisions, decisions...
Little Pig
Press
Wait
5 , 1
Press
4 , 4
Big Pig
Wait
9 , -1
0 , 0
32
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?

33
Mixed Strategy
Safe 2
Safe 1
10,000
0
Safe 1
100,000
0
Safe 2
34
Mixed Strategy Solution
35
The Payoff Matrix for Holmes Moriarity
36
Evolutionary Game Theory

• Natural selection replaces rational behavior
• Survival of the fittest
• Why use evolution to determine a strategy?

37
Hawk / Dove Game
38
Evolutionary Stable Strategy

• Introduced by Maynard Smith and Price (1973)
• Strategy becomes stable throughout the population
• Mutations becoming ineffective

39
Hawk
Dove
Dove
Hawk
40
(No Transcript)
41
Hawk
Dove
Dove
Hawk
42
(No Transcript)
43
Where is game theory currently used?

• Ecology
• Networks
• Economics

44
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

45
Summary

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

46
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.