GAME THEORY

History 1928 –

Game theory did not really exist as a unique field until John von Neumann &

Morgenstern published a paper.

1950 -

John Nash developed a criterion for mutual consistency of players' strategies, known as

Nash equilibrium, applicable to a wider variety of games than the criterion proposed

by von Neumann and Morgenstern.

1965 –

Reinhard Selten introduced his solution concept of subgame perfect equilibria, which

further refined the Nash equilibrium.

1970 –

Game theory was extensively applied in biology, largely as a result of the work of John

Maynard Smith and his evolutionarily stable strategy.

2007 –

Leonid Hurwicz, together with Eric Maskin and Roger Myerson, was awarded the

Nobel Prize in Economics "for having laid the foundations of mechanism

design theory

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What's IT?? Game theory is a method of

studying strategic decision

making. More formally, it is "the

study of mathematical models of

conflict and cooperation

between intelligent rational

decision-makers.

Strategic- It is a setting where

the outcomes that affect you

depends on actions not just on

your own actions but on actions

of others. 3

Why game theory ??? Because the press tells us to …

―Managers have much to learn from game theory —provided they use it to clarify their thinking, not as a substitute for business experience.‖

The Economist, 15 June 2007

―Game Theory, long an intellectual pastime, came into its own as a business tool.‖

Forbes, 3 July 2009

―Game theory is hot.‖

The Wall Street Journal, 13 February 2011

Because recruiters tell us to …

―Game theory forces you to see a business situation over many periods from two perspectives: yours and your competitor’s.‖

Judy Lewent – CFO, Merck

―Game theory can explain why oligopolies tend to be unprofitable, the cycle of over capacity and overbuilding, and the tendency to execute real options earlier than optimal.‖

Tom Copeland – Director of Corporate Finance, McKinsey 4

Terminology

Players

Strategies

◦ Choices available to each of the players

Might be conditioned on history

Payoffs

◦ Some numerical representation of the

objectives of each player

Could take account fairness/reputation, etc.

Does not mean players are narrowly selfish

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Standard Assumptions

Rationality

◦ Players are perfect calculators and

implementers of their desired strategy

Common knowledge of rules

◦ All players know the game being played

Equilibrium

◦ Players play strategies that are mutual best

responses

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Strategies for Studying Games of

Strategy Two general approaches

◦ Case-based

Pro: Relevance, connection of theory to application

Con: Generality

◦ Theory

Pro: General principle is clear

Con: Applying it may not be

Types of firms :

1)Perfect competition

2)Monopolistic or Imperfect competition.

ex- Motor car industry , mobile markets etcetera.

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An easy strategy

Asking your friend out for dinner?

A complex strategy

Geopolitical talks?

Where It Exists?

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Representation of Games The games studied in game theory are well-defined

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.

Categories :

Extensive form

Normal form

Characteristic function form

Partition function form

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Extensive form

Games here are played on trees (as pictured to the left).

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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Normal/Strategic 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.

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Types of Games

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. Often it is assumed that communication among players is allowed in cooperative games. Cooperative games focus on the game at large.

Non-cooperative :

In game theory, a non-cooperative game is one in which players make decisions independently. Non-Cooperative games are able to model situations to the finest details, producing accurate results.

Hybrid games :

contain cooperative and non-cooperative elements.

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Symmetric :

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. Prisoner's dilemma, and the stag hunt are all symmetric

games.

Asymmetric :

Most commonly studied asymmetric games are games where there

are not identical strategy sets for both players.

Zero-sum games :

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 equal expense of others).

Non Zero-sum games :

These type of games have outcomes whose net results are greater or less

than zero

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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) :

They 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.

Perfect information and Imperfect 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.

Many card games are games of imperfect information, for instance poker.

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Combinatorial games :

Games in which the difficulty of finding an optimal strategy stems from the multiplicity of possible moves are called combinatorial games. Examples include chess and go. Games that involve imperfect or incomplete information may also have a strong combinatorial character. A related field of study, drawing from computational complexity theory, is game complexity, which is concerned with estimating the computational difficulty of finding optimal strategies.

Discrete and continuous games :

Continuous games includes changing strategy set.

Much of game theory is concerned with finite, discrete games, that

have a finite number of players, moves, events, outcomes, etc.

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Infinitely long games :

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. The focus of attention is usually not so much on what is the

best way to play such a game, but simply on whether one or the other

player has a winning strategy.

Meta-games :

These are games the play of which is the development of the rules for

another game, the target or subject game. Metagames seek to maximize

the utility value of the rule set developed.

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Lesson 1

Dominance Principle :

Do Not Play A Strictly Dominated Strategy.

Player 2

α β

α

Player 1

β

We say that my strategy α strictly dominates my strategy β if my

payoff from α is strictly greater than that from β regardless of what

others do.

Here playing strategy β is a strictly dominated strategy.

0,0

3,-1

-1,3

1,1

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GAME 1

Put in the box below either the letter α (alpha) or β (beta) .Think of this as a

great bid. I will randomly pair your form with another form and neither your

pair or you would know with whom you were paired.

Here is how grades maybe assigned :-

you pair

α, β – A,C

α, α – B minus , B minus

β, α – C , A

β, β –B plus , B plus

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α

β

α B

minus

A

β C B

plus

α

β

α B

minus

C

β A B plus

ME

(X)

MY

PAIR

(X`)

MY PAIR

(X`)

ME

(X)

α, β – A, C

α, α – B minus , B minus

β, α – C , A

β, β –B plus , B plus

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α

β

α B minus ,

B minus

A,

C

β C ,

A B plus ,

B plus

MY PAIR

(X`)

Me

(X)

Outcome matrix

1) We cared about our own grade

2) We cared about other grades

} 2 Different payoffs

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α

β

α 0,0 3, -1

β -1, 3 1,1

MY PAIR

(X`)

ME

(X)

We try to put numbers into perspective and these are called

# utilities

just represent that we optimise our own gain and others loss.

(A,C) – 3

(B minus , B minus )- 0

If the pair chooses

alpha and I choose

alpha then I get

zero and if player

chooses alpha and I

choose beta I get -1.

0 > -1

If the player

chooses beta and I

choose alpha I get 3

and if player

chooses beta and I

choose beta I get 1

3>1

Hence in both

cases she receives a

higher payoff if she

chooses alpha.

α

β

α 0,0 -1, -3

β -3,-1 1,1

MY PAIR

(X`)

ME

(X)

Replacing (A,C) = -1,(C,A) = -3

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Penalty kick game

l r

L 4,-4 9,-9

M 6,-6 6,-6

R 9,-9 4,-4

Goal keeper

Shooter

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FINALS OF UEFA 2008

MAN Vs CHELSEA

It seemed that Chelsea’s strategy of going to Van der Sar’s left had been hatched by someone on the Utd bench. As Anelka prepared to take Chelsea’s 7th penalty Van der Sar pointed to the left corner. Now Anelka had a terrible dilemma. This was game theory in its rawest form. So Anelka knew that Van der Sar knew that Anelka knew that Van der Sar tended to dive right against right footers. Instead Anelka kicked right but it was at mid-height which Ignacio warned against.

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Lesson 2

Put yourself in others shoes and try to

figure out what they will do. Destroyer

Easy Hard

Easy

Savior

Hard

Here savior should look upon the destroyers payoff before

making a move.

1,1

1,1

0,2

2,0

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Prisoners Dilemma

Two Prisoners are on trial for a crime and

each one faces a choice of confessing to the

crime or remaining silent.

The payoffs are given below: Prisoner 2

Confess Silent

Confess

Prisoner 1

Silent

4,4

1,5

5,1

2,2

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Solution to Prisoners Dilemma

Prisoner 2

Confess Silent

Confess

Prisoner 1

Silent

The only stable solution in this game is that

both prisoners confess; in each of the other 3

cases, at least one of the players can switch

from ―silent‖ to ―confess‖ and improve his

payoff.

4,4

1,5

5,1

2,2

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Odds to Prisoners Dilemma What would the Prisoner 1 and Prisoner 2 decide if

they could negotiate?

They could both become better off if they reached the

cooperative solution….

which is why police interrogate suspects in separate

rooms.

Equilibrium need not be efficient. Non-cooperative

equilibrium in the Prisoner’s dilemma results in a

solution that is not the best possible outcome for the

parties.

Alternative: Implied contract

if there were a long relationship between the parties—(partners in crime) are more likely to back each other

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CPU Competition

Intel and AMD compete fiercely to develop innovations

in CPUs for PCs

As a result of this, CPU speeds have increased

dramatically, but there are few differences between the

products of the two companies

Even though both companies expended huge amounts

of money to gain a competitive advantage, their relative

competitive position ends up unchanged.

Both companies are worse off than if they had each

slackened the pace of innovation

This is an example of a prisoner’s dilemma

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Sale Price Guarantees

Globus and many other stores like

Pantaloons offer sale price guarantees

◦ If an item comes on sale in the time period

after you bought it, they will match the

difference in prices

◦ Thus, consumers wishing to buy now are

―protected‖ against regrets from future price

reductions

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Walking Down the Demand Curve

Globus would like to sell goods at high prices

to those with high willingness to pay for them

and then lower prices to capture those with

lower willingness to pay.

They might do this by running sales after new

items have been out for a while

But high value consumers will anticipate this

and wait for the sale to occur.

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The Power of Commitment

By offering to rebate back the difference

in prices, Globus makes the sale strategy

less profitable for its ―future‖ self.

This commits it to less discounting in the

future

In fact It enables it to charge higher prices

today…and tomorrow.

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Coordination Game

Coordination game involves multiple outcomes that can be stable. A simple

Coordination game involves two players choosing between two options,

wanting to choose the same.

Battle of the Sexes :

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Solution to Battle Of the Sexes

Solution: Both should play the same game.

Everyone benefits from being in

cooperative group, but each can do better

by exploiting cooperative efforts of

others.

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Anti Coordination Game

Routing Congestion Game :

Solution: The players must coordinate to send

traffic on different connection points.

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Rules of the Game

The strategic environment

◦ Players – I , j

◦ Strategies- S(i) – strategy of player I

◦ S(i) – set of possible strategies of player I

◦ S – strategy profile

◦ Payoffs- µ (i)

◦ µi (s) – profile utilities

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The rules

◦ Timing of moves

◦ Nature of conflict and interaction

◦ Informational conditions

◦ Enforceability of agreements or contracts

The assumptions

◦ Rationality

◦ Common knowledge

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Prisoner's Dilemma (communication is not legal)

communication wont help

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Investor game

(where both can communicate legally.)

A communication can help

L

R

U 1,1 0,0

D 0,0 1,1

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So far we have used the dominant strategy solution and iterative

elimination of dominated strategy (IEDS) solution concepts to solve

strategic form games. A third approach is to use the Nash equilibrium

concept.

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Nash Equilibrium- Necessary requirements

for a Nash equilibrium are that each player

play a best response against a conjecture,

and the conjecture must be correct.

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Going to movies

Person 1 Person 2

Person 3

Movie1 2,1 0,0 0,-1

Movie 2 0,0 1,2 0,-1

Movie 3 -1,0 -1,0 -2,-2

She

He

Either both people go for movie 1 or Both go for Movie 2 You call this each persons ―equilibrium‖ or ―Nash equilibrium ―

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How do we identify the Nash equilibria in a game?

Look for the dominant strategy.

Eliminate the dominated strategies.

Play a minimax strategy. In a zero sum game you choose that strategy in

which your opponent can do you the least harm from among all of the

'bad' outcomes.

Cell-by-cell inspection or trial and error

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1- no regrets as we choosing the best

move

2- Self fulfilling believes

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Coordination game –

1) Your farewell party

2) LED technology

3) Tv shows

4) Software platforms coordination

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selfish routing in large networks like the Internet

theoretical basis to the field of multi-agent systems.

Use of NLP for optimizing results.

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Diverse applications

1)Economics and business

2) Political science

-game-theoretic models in which the players are often voters, states, special

interest groups, and politicians.

3) Biology

-Evolutionary game theory (EGT) is the application of game theory to

evolving populations of lifeforms in biology.

4) Computer science and logic

-Algorithmic game theory and within it algorithmic mechanism design

combine computational algorithm design and analysis of complex systems

with economic theory.

5) Philosophy-common beliefs or knowledge

6) Legal problems and behavioral economics.

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