Game Theory
By:
Ali Farahani Rad
Benjamin Ghassemi
What is Game Theory?

Game theory is a branch of applied
mathematics that is often used in the
context of economics. It studies strategic
interactions between agents.

In economics, an agent is an actor in a
model that (generally) solves an
optimization problem. In this sense, it is
equivalent to the term player, which is
also used in economics, but is more
common in game theory.
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What is Game Theory?

In strategic games, agents choose
strategies that will maximize their return,
given the strategies the other agents
choose.

The essential feature is that it provides a
formal modeling approach to social
situations in which decision makers
interact with other agents. Game theory
extends the simpler optimization approach
developed in neoclassical economics.
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What is Game Theory?

Neoclassical economics refers to a general
approach in economics focusing on the
determination of prices, outputs, and income
distributions in markets through supply and
demand. These are mediated through a
hypothesized maximization of income-constrained
utility by individuals and of cost-constrained
profits of firms employing available information
and factors of production.
Antonietta Campus (1987), "marginal economics," The New
Palgrave: A Dictionary of Economics, v. 3, p. 323.
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Applications of Game Theory
Mathematics
 Computer Science
 Biology
 Economics
 Political Science
 International
Relations

Psychology
 Law
 Military Strategy
 Management
 Sports
 Game Playing
 Philosophy

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Representation of games
The games studied by game theory are welldefined mathematical objects. A game
consists of a set of players, a set of moves
(or strategies) available to those players,
and a specification of payoffs for each
combination of strategies.
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Extensive form
Games here are often presented as trees.
Here each vertex (or node) represents a
point of choice for a player. The player is
specified by a number listed by the vertex.
The lines out of the vertex represent a
possible action for that player. The payoffs
are specified at the bottom of the tree.
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Extensive form
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Normal form

The normal (or strategic form) game is usually
represented by a matrix which shows the players,
strategies, and payoffs. More generally it can be
represented by any function that associates a
payoff for each player with every possible
combination of actions.
Player 2
chooses Left
Player 2
chooses Right
Player 1
chooses Up
4, 3
–1, –1
Player 1
chooses Down
0, 0
3, 4
Normal form or payoff matrix of a 2-player, 2strategy game
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Normal form

When a game is presented in normal
form, it is presumed that each player acts
simultaneously or, at least, without
knowing the actions of the other. If
players have some information about the
choices of other players, the game is
usually presented in extensive form.
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Types of games

Cooperative or non-cooperative

Symmetric and asymmetric

Zero sum and non-zero sum

Simultaneous and sequential

Perfect information and imperfect information

Infinitely long games

Discrete and continuous games

Meta games
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Cooperative or non-cooperative



A game is cooperative if the players are able to form binding
commitments. For instance the legal system requires them
to adhere to their promises. In non-cooperative games this
is not possible.
Often it is assumed that communication among players is
allowed in cooperative games, but not in non-cooperative
ones. This classification on two binary criteria has been
rejected (Harsanyi 1974).
Of the two types of games, non-cooperative games are able
to model situations to the finest details, producing accurate
results. Cooperative games focus on the game at large.
Considerable efforts have been made to link the two
approaches. The so-called Nash-program has already
established many of the cooperative solutions as noncooperative equilibrium.
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Symmetric and asymmetric

A symmetric game is a game where the
payoffs for playing a particular strategy
depend only on the other strategies
employed, not on who is playing them. If
the identities of the players can be
changed without changing the payoff to
the strategies, then a game is symmetric.
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Zero sum and non-zero sum

Zero sum games are a special case of
constant sum games, in which choices by
players can neither increase nor decrease
the available resources. In zero-sum
games the total benefit to all players in
the game, for every combination of
strategies, always adds to zero (more
informally, a player benefits only at the
expense of others).
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Simultaneous and sequential

Simultaneous games are games where both players move
simultaneously, or if they do not move simultaneously, the
later players are unaware of the earlier players' actions
(making them effectively simultaneous). Sequential games
(or dynamic games) are games where later players have
some knowledge about earlier actions. This need not be
perfect information about every action of earlier players; it
might be very little knowledge. For instance, a player may
know that an earlier player did not perform one particular
action, while he does not know which of the other available
actions the first player actually performed.

The difference between simultaneous and sequential games
is captured in the different representations discussed
above. Normal form is used to represent simultaneous
games, and extensive form is used to represent sequential
ones.
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Perfect information and imperfect
information

An important subset of sequential games consists
of games of perfect information. A game is one of
perfect information if all players know the moves
previously made by all other players. Thus, only
sequential games can be games of perfect
information, since in simultaneous games not
every player knows the actions of the others.

Perfect information is often confused with
complete information, which is a similar concept.
Complete information requires that every player
know the strategies and payoffs of the other
players but not necessarily the actions.
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Infinitely long games

Games, as studied by economists and real-world
game players, are generally finished in a finite
number of moves. Pure mathematicians are not
so constrained, and set theorists in particular
study games that last for infinitely many moves,
with the winner (or other payoff) not known until
after all those moves are completed.
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Discrete and continuous games

Most of the objects treated in most branches of
game theory are discrete, with a finite number of
players, moves, events, outcomes, etc. However,
the concepts can be extended into the realm of
real numbers. This branch has sometimes been
called differential games, because they map to a
real line, usually time, although the behaviors
may be mathematically discontinuous. Much of
this is discussed under such subjects as
optimization theory and extends into many fields
of engineering and physics.
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Meta games

These are games the play of which is the
development of the rules for another
game, the target or subject game. Meta
games seek to maximize the utility value
of the rule set developed. The theory of
meta games is related to mechanism
design theory.
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Key Elements of a Game

Players: Who is interacting?

Strategies: What are their options?

Payoffs: What are their incentives?

Information: What do they know?

Rationality: How do they think?
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Cigarette Advertising on TV

All US tobacco companies advertised heavily on
television

Surgeon General issues official warning
•Cigarette smoking may be hazardous

Cigarette companies’ reaction
•Fear of potential liability lawsuits

Companies strike agreement
•Carry the warning label and cease TV advertising in exchange for
immunity from federal lawsuits.
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Strategic Interactions

Players:
Reynolds and Philip Morris

Strategies:

Payoffs:

Each firm earns $50 million from its customers

Advertising costs a firm $20 million

Advertising captures $30 million from
competitor

How to represent this game?
{ Advertise , Do Not Advertise}
Companies’ Profits
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Payoff Table
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Best responses

Best response for Reynolds:
•If Philip Morris advertises:
•If Philip Morris does not advertise:
advertise
advertise

Advertise is dominant strategy!

This is another Prisoners’ Dilemma
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What Happened?

After the 1970 agreement, cigarette
advertising decreased by $63 million

Profits rose by $91 million

Why/how were the firms able to escape
from the Prisoner’s Dilemma?
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Changing the Game through
Government-Enforced Collusion

The agreement with the government
forced the firms not to advertise.

The preferred outcome (No Ad, No Ad)
then was all that remained feasible
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Rationality?

Most economic analysis assumes
“rationality” of decision-makers, that you
make decisions by
1.forming a belief about the world
2.choosing an action that maximizes your
welfare given that belief
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And Common Knowledge of
Rationality??

Most game-theoretic analysis makes the
further assumption that players’ rationality
is common knowledge;
•Each player is rational
•Each player knows that each player is rational
•Each player knows that each player knows that each
player is rational
•Each player knows that each player knows that each
player knows thateach player is rational
•Each player knows that each player knows that each
player knows that each player knows that each player is
rational
•Etc. etc. etc.
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And Correct Beliefs?!?

Nash equilibrium assumes that each player has
correct beliefs about what strategies others will
follow

Implicitly this is saying that, in novel strategic
situations, each player knows what the other
believes

Requires all players to thoroughly understand
each other
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Nash Equilibrium

Nash Equilibrium:
•A set of strategies, one for each player, such
that each player’s strategy is a best response to
others’ strategies

Best Response:
•The strategy that maximizes my payoff given
others’ strategies.

Everybody is playing a best response
•No incentive to unilaterally change my strategy
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Dominant Strategies
Recall: Cigarette Ad Game

Reynolds’ best strategy is Ad regardless of
what Philip Morris does 􀃆Ad is “dominant
strategy”
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Dominant Strategies and Rationality

If you are rational, you should play your
dominant strategy. Period.

No need to think about whether others are
rational, etc.

Rationality + dominant strategies implies Nash
equilibrium
•no need for common knowledge or correct beliefs
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Dominant Strategies and Rationality

Nash equilibrium is not the right concept
for some strategic situations


Real players make mistakes or, for other reasons, may
fail to be “rational”
Yet dominant strategies give a clear
prescription of what to do, regardless.
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Example: SUV Price Wars
“General Motors Corp. and Ford Motor Co.
slapped larger incentives on popular sportutility vehicles, escalating a discounting
war in the light-truck category… Ford
added a $500 rebate on SUVs, boosting
cash discounts to $2,500. The Dearborn,
Mich., auto maker followed GM, which
earlier in the week began offering $2,500
rebates on many of its SUVs.”
--Wall Street Journal, January 31, 2003
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SUV Price Wars: The Game
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SUV Price Wars: Outcome

Each firm has a unilateral incentive to
discount but neither achieves a pricing
advantage.
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Prisoners’ Dilemma
SUV Price War is a “prisoners’ dilemma”
game:
1.
Both firms prefer to Discount
regardless of what the other does.
(Discount is a dominant strategy.)
2.
But both firms are worse off when they
both Discount than if they both Don’t.
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Prisoners’ Dilemma Game

Key features:
•Both players have a dominant strategy to
Confess
•BUT both players better off if they both don’t
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Prisoners’ Dilemma Game
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Reaction Curves in
Prisoners’ Dilemma
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Evolution in Prisoners’ Dilemma (One
Population)

Row and Col players are drawn from the
same population

Those who Confess get higher payoff, so
Confess dominates the population
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Loyal Servant Game

Key features:
•One player (Master) has dominant strategy
•Other player (Servant) wants to do the same
thing as Master
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Loyal Servant Game
43
Reaction Curves in
Loyal Servant Game
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Evolution in Loyal Servant Game (Two
Populations)
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Bluffing in Poker: Set-Up
Player A will be drawing on an inside
straight flush
 Player A will have the best hand if:
•flush (another club: 9 cards total) or
•straight (any 2 or 7: additional 6 cards)

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Winning Cards
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Bluffing Game: Rules

Each player has put $100 into the pot

After receiving the fifth card, player A will
either Raise $100or Not

If Raise, Player B then either Calls (adds
$100 more) or Folds (automatically losing
$100 already in pot)

Player A wins the pot if either A gets
winning card or B folds
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Bluffing Game: Rules
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Analysis of Bluffing Game
You get Good Card 15/48, about 1/3
 What do you do with Bad Card?


If you never raise, player B will always Fold
when you have a Good Card.
•get +100 when Good, -100 when Bad
•average payoff about –33

If you always raise, player B will always Call
you on it (even worse!)
•get + 200 when Good, -200 when Bad
•average payoff about -67
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How Often to Raise in Equilibrium?

Need to Raise enough for Player B to be
indifferent between Fold and Call

B gets –100 if Folds

B gets either –200 or +200 if Calls
•By Call, B “risks 100 to gain 300” relative to Fold
•So we need Prob(Bluff| Raise) = 25%

15 Good Cards so we Bluff on 5 Bad Cards
•So, Raise with 5/33 Bad Cards
•When 1/3 chance of Good Card, Bluff with prob. 1/6
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How Often to Fold in Equilibrium?

Need to Fold enough for Player A to be
indifferent between Raise and Not with
Bad Card

A gets –100 if Not Raise

A gets either –200 or +100 if Raise
•By raising, A “risks 100 to gain 200”

So we Fold 33%
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Payoffs in Equilibrium

Player B Folds 33% of time


…& Player A indifferent to Raise or Not
given a Bad Card


Good Card: 33%(+100)+67%(+200), so get
167 when Good Card
–100 when Bad Card
Overall payoff is about –11for A

much better than always/never bluffing
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Best responses in bluffing
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Next Step …
Strategies and game theory!
from
to
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References
1. David McAdams,"Game Theory for
Managers", MIT openCourseWare.
 2. Adam M. Brandenburger and Barry J.
Nalebuff, "The Right Game: Use Game
Theory to Shape Strategy", Harvard
Buseness Review, July - August 1995.
 3. The Internet

www.wikipedia.org

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