The document provides an overview of game theory, covering its history, key concepts, and applications, particularly in strategic decision-making. It explores various types of games, such as zero-sum and non-zero-sum games, and introduces essential strategies like Nash equilibrium. Additionally, it discusses the limitations of game theory, including assumptions about rationality and the quantification of payoffs.

1. Introduction to
Game Theory

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.
 AII 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 opponent’s
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 player’s
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

15. Matrix Notation
(Column) Player II
Strategy A Strategy B
(Row) Player I
Strategy A (P1, P2) (P1, P2)
Strategy B (P1, P2) (P1, P2)
Notes: Player I's strategy A may be different from Player II's.
P2 can be omitted if zero-sum game

16. Prisoner’s Dilemma &
Other famous games
A sample of other games:
Marriage
Disarmament (my generals are
more irrational than yours)

17. Prisoner’s Dilemma
Notes: Higher payoffs (longer sentences) are bad.
Non-cooperative equilibrium  Joint maximum
Institutionalized “solutions” (a la criminal organizations, KSM)
NCE
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
Prisoner’s Dilemma
with Cooperation

20. Prisoner’s 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

23. If you have a dominating strategy,
use it
Use
strategy 1

24. Eliminate any dominated
strategy
Eliminate
strategy 2 as
it’s dominated
by strategy 1

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

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/mccain/game/game.html

29. Is this a Nash Equilibrium?

30. Cost to press
button = 2 units
When button is pressed,
food given = 10 units
Boxed Pigs Example

31. Decisions, decisions...

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

34. Mixed Strategy Solution
Value in
Safe
Probability
of being
Guarded
Expected
Loss
Safe 1 10,000
$ 1 / 11 9,091
$
Safe 2 100,000
$ 10 / 11 9,091
$
Both 110,000
$

35. The Payoff Matrix
for Holmes & Moriarity
P
l
a
y
e
r
#
1
Player #2
Strategy #1 Strategy #2
Strategy #1
Strategy #2
Payoff (1,1) Payoff (1,2)
Payoff (2,1) Payoff (2,2)

36. Where is game theory
currently used?
–Ecology
–Networks
–Economics

37. 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

38. 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 thinking agent

39. 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/i
ndex.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.