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Gtap Sessions 01-03

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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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