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apProf. J. Ajith Kumar, TAPMI, Manipal GAME THEORY NOTES

GAME THEORY & APPLICATIONS

J. Ajith Kumar, TAPMI, Manipal

Pure strategy, Nash Equilibrium and Dominance

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apProf. J. Ajith Kumar, TAPMI, Manipal GAME THEORY NOTES 4

INTRODUCTION

E.g. cricket, where batsman has options of multiplestrokes, and bowler has option of multiple types of ball.

Simplistically speaking, the number of runs scored by

the batsman can be thought of as a function of the

pairing between the type of ball bowled and type ofstroke played.

Analysis of a game can also be thought of as solving a

decision tree with more than one agent, with each

agent trying to maximize her returns.

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apProf. J. Ajith Kumar, TAPMI, Manipal GAME THEORY NOTES

Game theoretic analyses can help us understand realitybetter and

explain current reality why things are the way they

are.

prescribe actions what should be done to shapereality in some desirable manner.

HOW GAME THEORY IS USEFUL

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apProf. J. Ajith Kumar, TAPMI, Manipal GAME THEORY NOTES

* Simultaneous game

a model that assumes playersdont get to see each othersactions before they

perform their own actions.

SIMULTANEOUS vs. SEQUENTIAL GAMES

* Sequential game a model that accommodates the

time dimension, and assumes players get to see whatthe other players did before performing their own

actions.

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apProf. J. Ajith Kumar, TAPMI, Manipal GAME THEORY NOTES

A game is typically characterized bya) the context,

b) the players,

c) actions (strategies) available to the players,

d) the payoff functions of the players , and

e) whether the game is simultaneous or sequential.

CHARACTERISTICS OF A GAME

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apProf. J. Ajith Kumar, TAPMI, Manipal GAME THEORY NOTES 8

Context PlayersActions

availablePayoffs

How the game is

played*

Pricing in the

telecom

industry

Telecomfirms

Different

levels of price(e.g. high,

medium, low)

Revenues

Profits

Market share

Simultaneous

(over multiple

rounds)

Writing an

Examination

Invigilators,

Students

Invigilators:

(strict,

lenient),

Students:

(cheat, dont

cheat)

Invigilators:

how well the

exam isconducted.

Students:

performance

in the exam

Simultaneous /

sequential

EXERCISE (10 minutes)

List real life situations that you can model as games. See

examples below.

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apProf. J. Ajith Kumar, TAPMI, Manipal GAME THEORY NOTES 9

Theory of Rational Choice(Choice Theory, Rational Actor Theory)

Decision makers are rational actors. While making a

decision, they choose the alternative(s) that maximizes

their utility under given circumstances. They also have

adequate knowledge and ability to estimate their utilities

and to use them to discriminate between alternatives.

Axioms of Rationality

For rationality to be possible, we assume these to be true:

1. Reflexivity (trivial)2. Completeness

3. Transitivity

4. Continuity (sufficient but not necessary)

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apProf. J. Ajith Kumar, TAPMI, Manipal GAME THEORY NOTES

App Development Project Game

10

Each student is given a slip in which s/he indicates

her/his choice after reading the problem and thinking

over it. No one should talk to each other.

ReadApp Development Project

in Exercise Set A

How many marked Work hardand how many Hardly

Work? Why did you mark what you did?

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apProf. J. Ajith Kumar, TAPMI, Manipal GAME THEORY NOTES

App Development Project Game

Player 1s utilities are:

u1(a1 = WH | a2 = WH) = 30

u1(a1 = HW | a2 = WH) = 70

u1(a1 = WH | a2 = HW) = 10

u1(a1 = HW | a2 = HW) = 20

Player 2s utilities will be similar.

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apProf. J. Ajith Kumar, TAPMI, Manipal GAME THEORY NOTES 12

App Development Project Game

PARTNER

WH HW

YOU

WH

30, 30 10, 70

HW 70, 10 20, 20

Strategic form or normal form representation

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apProf. J. Ajith Kumar, TAPMI, Manipal GAME THEORY NOTES

App Development Project Game

What is the best action for you? Why?

What is the best action for your partner? Why?

We see that regardless of what your partner does, you

get higher payoffs if you chose HW. Thus HW is the

best action for you.

By the same reasoning as above, HW is the best action

for your partner too.

If both players are rational, then what will the

outcome be?

(HW, HW): also called the Nash Equilibrium of this

game.

13

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apProf. J. Ajith Kumar, TAPMI, Manipal GAME THEORY NOTES 14

NASH EQUILIBRIUM

A Nash Equilibrium is an action profile in which no

player can improve his payoff by changing his action,given that the other playersactions remain the same

as in that action profile.

An action profile is a Nash equilibrium if and only if

each players action is his best response to the actionsof all other players in that action profile.

In other words, iffui(ai, a-i) ui (ai, a-i) for all i, ai, is

satisfied, then (ai

, a-i

) is a Nash Equilibrium.

Examine if (HW, HW) fulfils the above condition.

Examine if any of the other three action profiles in this game

fulfil the condition and can be called a Nash Equilibrium.

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apProf. J. Ajith Kumar, TAPMI, Manipal GAME THEORY NOTES 15

NASH EQUILIBRIUM

Important Caveat:

An NE is an action profile, not a payoff profile.

To say that (HW, HW) is a NE correct

But to say that (20, 20) is an NE is incorrect.

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apProf. J. Ajith Kumar, TAPMI, Manipal GAME THEORY NOTES 16

STRICT vs. NON-STRICT NASH EQUILIBRIUM

If in a NE, ui(ai, a-i) > ui (ai, a-i) for all i, ai,

then (ai, a-i) is said to be a Strict Nash Equilibrium.

If not, then it is a nonstrict NE.

Player 2

H T

Player1

H 1, 2 3, 0

T 2, 1 3, 2

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apProf. J. Ajith Kumar, TAPMI, Manipal GAME THEORY NOTES 17

NASH EQUILIBRIUM: A SOLUTION CONCEPT

1. Indicates possible actions that players might take in agiven situation.

2. Indicates possible stable steady statethat a system

might reach after a period of time, over several trials

of the game.

3. An NE can be thought of as a stable social norm. If

everyone else adheres to it, no individual would wish

to deviate from it.

4. Give us a language to discuss what can be done to

change the system behavior (alter playersbehaviors,

alter the equilibrium). Mechanism Design.

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apProf. J. Ajith Kumar, TAPMI, Manipal GAME THEORY NOTES 18

If the players had chosen WH instead of HW, then

both would have got higher payoffs. Yet, why is (HW,

HW) the rational outcome of this game?

App Development Project Game

Lack of adequate trust, understanding? If I choose

WH, how do I know for sure that my partner will alsochoose WH? He might choose HW. If he does that

then my payoff will be just 10 points. On the other

hand, I can be assured of getting at least 20 points if I

choose HW and might even get 70 if my partner

chooses WH.

This reasoning implicitly assumes that each player

thinks purely selfishly.

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apProf. J. Ajith Kumar, TAPMI, Manipal GAME THEORY NOTES 19

Lesson: When everyone acts rationally, it can lead tosuboptimal outcomes for everyone. The combined

effect of individual rationality can sometimes be

collective irrationality.

App Development Project Game

Can control and coordination mechanisms help?

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apProf. J. Ajith Kumar, TAPMI, Manipal GAME THEORY NOTES 20

They did not understand the question correctly.

They did not calculate all the payoffs correctly.

...other reasons?

Our model predicts that HWwould be the rational

choice. Why then did everyone not choose HW?

Lesson: Bounded Rationality(Herbert Simon)the

NE may not always be reached in all real-life

situations.

Lesson: To be perfectly selfish, a person needs tohave complete knowledge and intelligence...!

Otherwise, it may not be possible. Can we say the

same of being perfectly unselfish?

App Development Project Game

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apProf. J. Ajith Kumar, TAPMI, Manipal GAME THEORY NOTES

IMPORTANT LESSONS

21

Lesson: The payoffs that each player gets are afunction of her actions in combination with the actions

of all other players in the game.

Lesson: The payoffs need not always be in terms of atangible measure such as money, market share etc...

Utilityis a more general conceptualization of payoff.

Covers intangibles like satisfaction, happiness,

frustration etc... Utility: assessed by the modeller andis thus, subjective in nature.

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apProf. J. Ajith Kumar, TAPMI, Manipal GAME THEORY NOTES 22

Action: decision choice available to a player. More

generally, also referred to as pure strategy. Actions are

mutually exclusive and collectively exhaustive. In this

model, a player can play only one action at a time and

not a combination of actions. Often denoted as ai.

Action set: defined for each player in the game, as a

listing of all possible actions that the player can choose.

Denoted by Ai. In the above game, A1 = {WH, HW} and

A2 = {WH, HW}.In this example, equal numberand identicalactions

available to all players (i.e. two) but not necessary, in

general.

Some Terminology

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apProf. J. Ajith Kumar, TAPMI, Manipal GAME THEORY NOTES

Some Terminology

23

Action profile: a distinct combination of actions of all

the players, in which for each player, one action islisted, and collectively, the actions are listed in a

standard order. In a two player game, the row players

action is listed first.

An action (strategy) profile is also referred to as anoutcome.

A 2 x 2 game (two players, two actions each) has four

possible action profiles. Likewise a m x n xp game will

have mnp possible action (pure strategy) profiles.

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apProf. J. Ajith Kumar, TAPMI, Manipal GAME THEORY NOTES 24

Utility (payoff) profile: Corresponding to each strategy

profile is an utility profile: the distinct combination ofutilities received by all players if the corresponding

strategy profile is played.

Some Terminology

For each player, one utility value is listed, and

collectively, the utilities are listed in the same order inwhich actions are listed in the action profile.

In a two player game, the row players utility is listed

first.

A 2 x 2 game has four possible utility profiles. E.g. (30,

30), (70, 10), (10, 70), (20, 20). Likewise a m x n xp

game will have mnp possible utility profiles.

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apProf. J. Ajith Kumar, TAPMI, Manipal GAME THEORY NOTES 25

The NE represents a possible outcome

But is it the best, or optimal outcome?

What would be an optimal outcome,

when there are multiple agents?

How do we determine optimality,

when there are trade-offs?

WHAT IS THE BEST OUTCOME?

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apProf. J. Ajith Kumar, TAPMI, Manipal GAME THEORY NOTES 26

PARETO OPTIMALITY: ANOTHER SOLUTION

CONCEPT

Pareto Domination:

Action profile a Pareto dominates action profile a if for

all i N, ui (a) ui (a), and there exists somej N for

which uj(a) > u

j(a).

Pareto Optimal:

Action profile a is Pareto optimal (or, Pareto efficient) if

there does not exist another action profile a S, which

Pareto dominates a.

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apProf. J. Ajith Kumar, TAPMI, Manipal GAME THEORY NOTES 27

PARETO OPTIMALITY: ANOTHER SOLUTION

CONCEPT

Player 2

H T

Player

1 H1, 2 3, 0

T 2, 1 3, 2

Shyam

Run Stand

Ram R

un

1, 1 2, 2

Stan

d3, 3 4, 4

Ashwin

R Q

Murali M 3, 2 2, 3

L 2, 1 1, 2

India

Fight Dont

Pakistan

Fig

ht

5, 4 5, 4

Don

t5, 4 5, 4

Find the NE(s) as well as the Pareto optimal solution(s)

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