Short Introduction of the Game theory.

1. Vikram Singh Slathia
2011MAI025
Dept. of Computer Science

2. CONTENTS
 Introduction
 History of Game Theory.
 Elements of games.
 Basic Concepts of Game Theory
 Kinds of Strategies.
 Nash Equilibrium.
 Types of Games.
 Applications of Game Theory.
 Conclusion .
 References .

3. INTRODUCTION
 Game theory is the mathematical analysis of a
conflict of interest to find optimal choices that
will lead to a desired outcome under given
conditions. To put it simply, it's a study of ways
to win in a situation given the conditions of the
situation. While seemingly trivial in name, it is
actually becoming a field of major interest in
fields like economics, sociology, and political
and military sciences, where game theory can
be used to predict more important trends.

4. CONT ….
 In the broadest terms, game theory analyses how
groups of people interact in social and economic
situations. An accurate description of game theory
is the term used by psychologists ? the theory of
social situations. There are two main branches of
game theory: co-operative and non-co-operative
game theory. Most of the research in game theory
is in the field of non-co-operative games, which
analyses how intelligent (or rational) people interact
with others in order to achieve their own goals

5. HISTORY OF GAME THEORY.
 The ideas underlying game theory have appeared
throughout history, apparent in the bible, the
Talmud, the works of Descartes and Sun Tzu, and
the writings of Chales Darwin. The basis of modern
game theory, however, can be considered an
outgrowth of a three seminal works:
 Augustin Cournot’s Researches into the Mathematical
Principles of the Theory of Wealth in 1838, gives an
intuitive explanation of what would eventually be
formalized as the Nash equilibrium, as well as provides
an evolutionary, or dynamic notion of best-responding to
the actions of others

6. CONT…..
 Francis Ysidro Edgeworth’s Mathematical
Psychics demonstrated the notion of competitive equilibria
in a two-person (as well as two-type) economy
 Emile Borel, in Algebre et calcul des probabilites, Comptes
Rendus Academie des Sciences, Vol. 184, 1927, provided
the first insight into mixed strategies - randomization may
support a stable outcome.
 A modern analysis began with John von
Neumann and Oskar Morgenstern's book, Theory of
Games and Economic Behavior and was given its
modern methodological framework by John Nash
building on von Neumann and Morgenstern's results.

7. ELEMENTS OF GAMES
The essential elements of a game are:
a. Players: The individuals who make decisions.
b. Rules of the game: Who moves when? What can
they do?
c. Outcomes: What do the various combinations of
actions produce?
d. Payoffs: What are the players’ preferences over
the outcomes?
e. Information: What do players know when they
make decisions?
f. Chance: Probability distribution over chance
events, if any.

8. BASIC CONCEPTS OF GAME THEORY
1. Game
2. Move
3. Information
4. Strategy
5. Extensive and Normal Form
6. Equilibria

9. 1. GAME
 A conflict in interest among n individuals or groups
(players). There exists a set of rules that define the
terms of exchange of information and pieces, the
conditions under which the game begins, and the
possible legal exchanges in particular conditions.
The entirety of the game is defined by all the moves
to that point, leading to an outcome.

10. 2. MOVE
 The way in which the game progresses between
states through exchange of information and pieces.
Moves are defined by the rules of the game and
can be made in either alternating fashion, occur
simultaneously for all players, or continuously for a
single player until he reaches a certain state or
declines to move further. Moves may be choice or
by chance. For example, choosing a card from a
deck or rolling a die is a chance move with known
probabilities. On the other hand, asking for cards in
blackjack is a choice move.

11. 3. INFORMATION
A state of perfect information is when all
moves are known to all players in a game.
Games without chance elements like chess
are games of perfect information, while
games with chance involved like blackjack
are games of imperfect information.

12. 4. STRATEGY
A strategy is the set of best
choices for a player for an entire game. It is an
overlying plan that cannot be upset by occurrences
in the game itself.
 Strategy combination
A strategy profile is a set of strategies for each
player which fully specifies all actions in a game. A
strategy profile must include one and only one
strategy for every player.

13. DIFFERENCE BETWEEN
Move Strategy
 A Move is a single step  A strategy is a complete
a player can take during set of actions, which a
the game. player takes into account
while playing the game
throughout

14. CONT …..
Example

15. KINDS OF STRATEGIES
I. Pure strategy
II. Mixed Strategy
III. Totally mixed strategy.

16. I. PURE STRATEGY
A pure strategy provides a complete
definition of how a player will play a game. In
particular, it determines the move a player
will make for any situation he or she could
face.
 A player‘s strategy set is the set of pure
strategies available to that player.
 select a single action and play it
 Each row or column of a payoff matrix represents
both an action and a pure strategy

17. II. MIXED STRATEGY
A strategy consisting of possible moves
and a probability distribution (collection of
weights) which corresponds to how
frequently each move is to be played. A
player would only use a mixed strategy
when she is indifferent between several
pure strategies, and when keeping the
opponent guessing is desirable - that is,
when the opponent can benefit from
knowing the next move.

18. III. TOTALLY MIXED STRATEGY.
A mixed strategy in which the player assigns
strictly positive probability to every pure strategy
 In a non-cooperative game, a totally mixed strategy
of a player is a mixed strategy giving positive
probability weight to every pure strategy available
to the player.

19. 5. PAYOFF
 The payoff or outcome is the state of the
game at it's conclusion. In games such as
chess, payoff is defined as win or a loss. In
other situations the payoff may be material
(i.e. money) or a ranking as in a game with
many players.

20. 5. EXTENSIVE AND NORMAL FORM
 Extensive Form
The extensive form of a game is a complete
description of:
1. The set of players
2. Who moves when and what their
choices are
3. What players know when they move
4. The players’ payoffs as a function of the
choices that are made.
In simple words we also say it is a graphical
representation (tree form) of a sequential
game.

21.  The normal form
The normal form is a matrix representation of
a simultaneous game. For two players, one is the
"row" player, and the other, the "column" player. Each
rows or column represents a strategy and each box
represents the payoffs to each player for every
combination of strategies. Generally, such games are
solved using the concept of a Nash equilibrium. .

22. 6. EQUILIBRIUM
Equilibrium is fundamentally very complex and
subtle. The goal to is to derive the outcome when
the agents described in a model complete their
process of maximizing behaviour. Determining
when that process is complete, in the short run and
in the long run, is an elusive goal as successive
generations of economists rethink the strategies
that agents might pursue.

24. NASH EQUILIBRIUM
 A Nash equilibrium, named after John Nash, is a
set of strategies, one for each player, such that
no player has incentive to unilaterally change her
action. Players are in equilibrium if a change in
strategies by any one of them would lead that
player to earn less than if she remained with her
current strategy. For games in which players
randomize (mixed strategies), the expected or
average payoff must be at least as large as that
obtainable by any other strategy

25. CONT ……..
 A strategy profile s = (s1, …, sn) is a Nash
equilibrium if for every i,
 si is a best response to S−i , i.e., no agent can do better
by unilaterally changing his/her strategy
 Theorem (Nash, 1951): Every game with a finite
number of agents andaction profiles has at least
one Nash equilibrium

26. EXAMPLE
BATTLE OF THE SEXES
 Two agents need to coordinate their actions, but they
have different preferences
 Original scenario:
• husband prefers football
• wife prefers opera
 Another scenario:
• Two nations must act together to deal with an
international crisis
• They prefer different solutions
 This game has two pure-strategy Nash equilibria and
one mixed-strategy Nash equilibrium
 How to find the mixed-strategy Nash equilibrium?

28. TYPES OF GAMES
A. One-Person Games
B. Zero-Sum Games
C Non zero sum game
D. Two-Person Games
E. Repeated Games

29. A. ONE-PERSON GAMES
 A one-person games has no real conflict of interest.
Only the interest of the player in achieving a
particular state of the game exists. Single-person
games are not interesting from a game-theory
perspective because there is no adversary making
conscious choices that the player must deal with.
However, they can be interesting from a
probabilistic point of view in terms of their internal
complexity.

30. B. ZERO-SUM GAMES
 A zero-sum game is one in which no wealth is created or
destroyed. So, in a two-player zero-sum game,
whatever one player wins, the other loses. Therefore,
the player share no common interests. There are two
general types of zero-sum games: those with perfect
information and those without.
 If the total gains of the participants are added up, and
the total losses are subtracted, they will sum to zero
 Example
a. Rock, Paper, Scissors
b. Poker game

31. EXAMPLE OF POKER GAME
 Let’s there are three players, Rajesh, Suresh and
Varun each starting with Rs100, a total of Rs300.
 They meet at Rajesh’s house and play for a couple
of hours. At the end of the evening Rajesh has
Rs200, Suresh has Rs60 and Varun has Rs40. The
total amount of money between them is still Rs300.
 Rajesh is up Rs100, Suresh is down Rs40 and
Varun is down Rs60.
 The total of these three numbers is zero (100-40-
60), so it is a zero-sum game.

32. EXAMPLE
 In non-zero-sum games, one player's gain needn't
be bad news for the other(s). Indeed, in highly non-
zero-sum games the players' interests overlap
entirely. In 1970, when the three Apollo
13 astronauts were trying to figure out how to get
their stranded spaceship back to earth, they were
playing an utterly non-zero-sum game, because the
outcome would be either equally good for all of
them or equally bad.

33. C. NON ZERO SUM GAME
 In game theory, situation where
one decision maker's gain (or loss) does not
necessarily result in the other decision
makers' loss (or gain). In other words,
where the winnings and losses of all players
do not add up to zero and everyone can
gain: a win-win game.
 Example
 Prisoner's dilemma

34. C. TWO-PERSON GAMES
 Two-person games are the largest category of
familiar games. A more complicated game derived
from 2-person games is the n-person game. These
games are extensively analyzed by game theorists.
However, in extending these theories to n-person
games a difficulty arises in predicting the interaction
possible among players since opportunities arise
for cooperation and collusion.

35. D. REPEATED GAMES
 In repeated games, some  Examples
game G is played multiple
times by the same set of 1. Iterated Prisoner’s
agents Dilemma
 G is called the stage game 2. Repeated
 Each occurrence of G is called Ultimatum Game
an iteration or a round 3. Repeated
 Usually each agent knows Matching Pennies
what all the agents did in the
previous iterations, but not 4. Repeated Stag
what they’re doing in the Hunt
current iteration 6. Roshambo
 Usually each agent’s payoff
function is additive

36. E. SEQUENTIAL GAMES
 A sequential game is a game where one player
chooses his action before the others choose theirs.
Importantly, the later players must have some
information of the first's choice, otherwise the
difference in time would have no strategic
effect. Extensive form representations are usually
used for sequential games, since they explicitly
illustrate the sequential aspects of a game.
 Combinatorial games are usually sequential games.
 Sequential games are often solved by backward
induction.

37. F. SIMULTANEOUS GAMES
A simultaneous game is a game where
each player chooses his action without
knowledge of the actions chosen by other
players. Normal form representations are
usually used for simultaneous games.
 Example
 Prisoner dilemma .

38. APPLICATIONS OF GAME THEORY
 Philosophy
 Resource Allocation and Networking
 Biology
 Artificial
Intelligence
 Economics
 Politics

39. CONCLUSION
 Byusing simple methods of game theory,
we can solve for what would be a confusing
array of outcomes in a real-world situation.
Using game theory as a tool for financial
analysis can be very helpful in sorting out
potentially messy real-world situations,
from mergers to product releases.

40. REFERENCES
 Books ;
 Game theory: analysis of conflict ,Roger B. Myerson,
Harvard University Press
 Game Theory: A Very Short Introduction, K. G.
Binmore- 2008, Oxford University Press.
 Links :
 http://library.thinkquest.org/26408/math/prisoner.shtml
 http://www.gametheory.net