Micro - Game Theory

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    GAME THEORY

    1

    Lectures 21 and 22

    A. Madestam

    What is game theory?

    We focus on games where:

    There are at least two rational players

    Each player has more than one choice

    The outcome depends on the strategies chosen by all

    2

    players; there is strategic interaction

    Example 1: Six people go to a restaurant

    Each person pays his/her own meal a simple decision

    problem

    Before the meal, every person agrees to split the bill

    evenly among them a game

    Example 2: Prisoners Dilemma

    John and Peter have beenarrested for possession of guns.However, the police suspects thatJohn and Peter also havecommitted a major robbery butthe police lacks the evidence toprove this

    Ifno one confesses the robbery,they will both be jailed for 1years

    Ifonly one confesses, he will bereleased and his partner ends upin jail for 20 years

    If they both confess, they bothget 5 year

    3

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    Example 3: What to Do?

    Philip Morris

    No Ad Ad

    ReynoldsNo Ad 50 , 50 20 , 60

    4

    If you are advising Reynolds, what strategy do you

    recommend?

    4

    , ,

    Game theory has many applications

    Economics

    Politics

    Business, etc, etc

    5

    Examples

    Prisoners dilemma

    The entry or predation game

    Battle of the sexes

    Example 1: the prisoners dilemma revisited

    Two robbers, X and Y, have been caught by the police and put

    in separate interrogation rooms; the district attorney has

    enough evidence to convict both for a lesser charge, but she

    wants to convict them for a more serious crime, for which she

    needs additional evidence

    6

    The district attorney and her assistant go simultaneously (at

    the same time) to each robber offering them a plea bargain a

    reduced prison term in return for testifying against the other

    robber (which will increase the latters prison term)

    Each robber must choose between confess or keep quiet,

    without knowing what the other is doing

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    If only one of the robbers confess, he will be released, while

    the non-confessing prisoner will go to jail for 20 yearsIf both confess, they will serve a 5 years sentence

    If both keep quiet, they will both be convicted for a lesser

    crime that carries a sentence of1 year

    7

    Each player whishes to minimize the time he spends in jail,

    hence we can represent the payoff as the negative of this

    8

    Diagram to be drawn by YOU!

    Example 2: the entry, or predation game

    Firm X is considering entering a market that currently has a

    single incumbent, firm Y. If X enters, the incumbent, Y, can

    respond in one of two ways: it can either

    i) accommodate the entrant, giving up some of its sales, i.e. it

    can produce a low output level

    9

    ,

    the market price, i.e. it can produce a high output level

    IfXstays out of the market, it gets no profits, while Ygets 2 if

    it produces a low output level and 3 otherwise

    If Xenters the market and Yfights back and produces a high

    output level, they both get -1; if Y instead accommodates the

    entrant and produce a low output level, they both get 1

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    The predation game

    Y

    (i) (ii) (iii) (iv)E

    XNE

    1 , 1

    0 , 2

    1 , 1

    0 , 3

    -1 , -1

    0 , 2

    -1 , -1

    0 , 3

    10

    i) L if E, L if NE

    ii) L if E, H if NE

    iii) H if E, L if NE

    iv) H if E, H if NE

    Example 3: the battle of the sexes

    Pete and Maria are trying to coordinate on attending the ballet

    or a (boxing) fight in the evening

    They work at separate workplaces and cannot communicate -

    yes, this was before the days of cell phones and for some

    reason the fixed phone line is down

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    ere w ey mee

    Both Pete and Maria know the following:

    Both would like to spend the evening together, if they

    dont they both receive 0

    But Pete prefers the ballet and receives 2 in this case,

    while Maria receives 1 if she joins Pete

    Maria prefers the fight and receives 2 in this case,

    while Pete receives 1 if he joins Maria

    12

    Diagram to be drawn by YOU (again)!

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    Noncooperative game theory studies decision making in

    situations where strategic behavior is important (e.g. shouldthe robbers confess or not?)

    Situations in which strategic considerations are an essential

    part of decision making are called games

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    A decision maker is behaving strategically when she takes

    into account what she thinks the other agents are going to do

    The theory is labeled noncooperative because each decision

    maker acts solely in her own self-interest; this does not mean

    that cooperation is not a possible outcome of strategic behavior

    A central feature of multi-agent interaction is the potential for

    the presence ofstrategic interdependence

    In multi-agent situations with strategic interaction, each agent

    recognizes that the payoffs she receives (in utility or profits)

    depends not only on her own actions but also on the actions of

    other agents

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    In her decision-making process, each agent should take into

    account:

    1) actions the other agents have already taken

    2) actions she expects them to be taking at the same time

    3) future actions they may (or may not) take as a result of her

    current actions

    Basic elements of a game

    To describe a situation of strategic interdependence, we need

    four basic elements:

    i) Players: the decision makers in a game (who is involved?)

    ii) Actions: the possible moves available to the players (what

    can the do?

    15

    iii) Strategies: the players plans of action at any stage of the

    game (what are they planning to do?)

    iv) Payoffs: the possible rewards enjoyed by the players (what

    will they gain?)

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    Strategy

    A strategy is a complete plan, or decision rule, that specifies

    how the player will act in

    EVERY POSSIBLE DISTINGUISHABLE

    CIRCUMSTANCE

    in which she might be called upon to move

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    Intuition: when a player specifies her strategy, it is as if she

    had to write down an instruction book prior to play so that a

    representative could act on her behalf merely by consulting

    that book

    Being a complete contingent plan, a strategy often specifies

    actions for a player at circumstances that may not be reached

    during the actual play of the game

    Some definitions

    Games in which all players move simultaneously (at the same

    time) are known as simultaneous games

    (e.g. prisonersdilemma; battle of the sexes)

    Games in which players moves may precede one another are

    known as sequential games

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    (e.g. entry/predation game)

    Games in which all players move knowing the earlier orsimultaneous moves of the other players are known as games

    of perfect information

    (e.g. entry/predation game)

    Definitions contd

    Games in which some players must move without knowing the

    earlier or simultaneous moves of the other players are known

    as games of imperfect information

    (e.g. prisonersdilemma; battle of the sexes)

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    Text to be added by YOU !

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    Text to be added by YOU (and again)!

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    Text to be added by YOU (and again)!

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    Representation of games

    Games can be represented in two forms, the normal form and

    the extensive form

    The normal form presents the game directly in terms of

    strategies and their associated payoffs

    When describin a ame in its normal form there is no need to

    22

    ,

    keep track of the specific moves associated with each strategy

    The prisoners dilemma in normal form

    Y

    C NC

    C -5 , -5 0 , -20X

    NC -20 , 0 -1 , -1

    23

    The payoff matrix summarizes the payoffs associated with

    each combination of strategies

    Note that the normal form is practically useful only when there

    are two players and the set of possible strategies is limited

    The extensive formThe extensive form captures who moves when, what actions

    each player can take, what players know when they move,

    what the outcome is as a function of the actions taken by the

    players, and the players payoff from each possible outcome

    The extensive form relies on the conceptual apparatus known

    24

    as a game ree

    The circumstances in which agents are, or might be, called

    upon to move are represented by decision nodes (little gray

    squares in previous picture)

    Each of the choices available at a particular decision node is

    represented by a branch from the decision node itself

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    The prisoners dilemma in extensive form

    )5,5(

    )20,0(

    )0,20(

    X

    YC

    NC

    C NC

    C

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    : Decision node : Information set

    The dashed oval around the two decision nodes for Y, known

    as information set, is used to represent Ys inability to

    distinguish between these two points at the time it makes her

    decision; from Ys point of view, the entire information set is a

    single decision node

    )1,1( NC

    26

    Diagram to be drawn by YOU (again)!

    The dominant strategy equilibrium

    A dominant strategy* is the best strategy regardless of what

    any other player does

    There is no reason for players to use anything other than their

    dominant strategy, IF they have one (often dominant strategies

    simply do not exist)

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    Hence, when each player has a dominant strategy, the only

    reasonable equilibrium outcome is for each player to use its

    dominant strategy

    A dominant strategy equilibrium is an outcome in a game in

    which each player follows a dominant strategy

    -------------------------------------------------

    * The correct definition is actually strictly dominant

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    Dominant strategies in the prisoners dilemma

    Y

    C NC

    C -5 , -5 0 , -20X

    NC -20 , 0 -1 , -1

    Has Xa dominant strategy?

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    Y

    C NC

    C -5 , -5 0 , -20X

    NC -20 , 0 -1 , -1

    Has Ya dominant strategy?

    NB: playing Cis dominant for both players! Hence, {C,C} is a

    dominant strategy equilibrium for the prisoners dilemma

    The Nash equilibrium

    The Nash equilibrium is the most widely used solution

    concept in applications of game theory to economics

    (http://www.princeton.edu/mudd/news/faq/topics/Non-

    Cooperative_Games_Nash.pdf)

    Consider a game with two players, Xand Y; a pair of strategies

    form a Nash equilibrium if:

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    actually played by Y

    ANDii) the strategy played by Y is optimal given the strategy

    actually played by X

    In general, in a Nash equilibrium, each players strategy choice

    is her best response to the strategies actually played by her

    rivals

    In simultaneous games of imperfect information, playerscannot directly observe the rivals moves (e.g. prisoners

    dilemma; battle of the sexes)

    Hence, each player forms ideas/conjectures (really guesses)

    about what the rivals will do, and reacts consequently by

    choosing her best response to the conjectured rivals strategies

    30

    In a Nash equilibrium, these conjectures turn out to be

    correct: each players strategy reveal itself to be the best

    response to the rivals actual moves

    In sum: players do not have incentives to unilaterally deviate

    from the equilibrium once the rivals move become observable

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    Y

    The Nash equilibrium in the prisoners dilemma

    Y

    C NC

    C -5 , -5 0 , -20XNC -20 , 0 -1 , -1

    31

    C NC

    C -5 , -5 0 , -20X

    NC -20 , 0 -1 , -1

    Hence, {C,C} is not only a dominant strategy equilibrium, but

    also a Nash equilibrium for the prisoners dilemma

    NB: all dominant strategy equilibria are Nash equilibria (by

    definition), while the converse is false

    Y

    C NC

    C -5 , -5 0 , -20X

    NC -20 , 0 -1 , -1

    Y

    32

    In this case, the conjectures are not mutually correct! Hence,

    {C,C} is the unique Nash equilibrium for this game

    C NC

    C -5 , -5 0 , -20

    XNC -20 , 0 -1 , -1

    IfPete goes to the ballet, Marias best response is to go to the

    Nash equilibria in the battle of the sexes

    MB F

    B 2 , 1 0 , 0P

    F 0 , 0 1 , 2

    33

    a e as we ; ar a goes o e a e , e e s es

    response is to go to the ballet as well (2 > 0)

    IfPete goes to the fight,Marias best response is to go to the

    fight as well (2 > 0); ifMaria goes to the fight, Petes best

    response is to go to the fight as well (1 > 0)

    Hence, { B , B } and { F , F } are both equally plausible Nash

    equilibria for this game

    NB: there may be several Nash equilibria in a given game

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    The battle of the sexes is a coordination game

    Two equilibria exist

    Pete and Marie prefer different equilibria

    How to achieve the most desirable outcome for you?

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    - sequential moves: the bossier one in the relationship

    chooses the equilibrium by leaving the office a bit earlier.

    Who do you think leaves early?

    - strategic moves: can you commit? Suppose Pete goes to

    the ballet early but knows that Marie will go to the fight

    regardless of what he does, will he stay to watch the Swan

    Lake?

    Nash equilibria in the Predation Game

    Y

    (i) (ii) (iii) (iv)

    E 1 , 1 1 , 1 -1 , -1 -1 , -1X

    NE 0 , 2 0 , 3 0 , 2 0 , 3

    35

    IfYplays (i), Xplays E; ifXplays E, Yplays (i) or (ii)

    IfYplays (ii), Xplays E; ifXplays E, Yplays (i) or (ii)

    IfYplays (iii), Xplays NE; ifXplays NE, Yplays (ii) or (iv)

    IfYplays (iv), Xplays NE; ifXplays NE, Yplay (ii) or (iv)

    NB: { E , (i) }, { E , (ii) }, and { NE , (iv) } are Nash equilibria

    Consider the three Nash equilibria:

    1 E L ifE L ifNE outcome: 1 1

    Y

    (L,L) (L,H) (H,L) (H,H)

    E 1 , 1 1 , 1 -1 , -1 -1 , -1X

    NE 0 , 2 0 , 3 0 , 2 0 , 3

    36

    ,

    2) { E ; L ifE , HifNE } ; outcome: (1,1)

    3) { NE ; HifE , HifNE } ; outcome: (0,3)

    NB: the first two equilibria generate the SAME outcome,

    because Ys strategies differ only at the decision node that is

    NOT reached during the actual play of the game

    Does this really make sense?!

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    To get the intuition, consider the predation game in extensive

    form:

    )1,1(

    )1,1(

    30

    X

    Y H

    EL

    H

    37

    ,

    )2,0(YNE

    L

    Suppose that Xhas already chosen to enter (E): what would Ys

    optimal response be?

    Y would clearly choose to accommodate the entrant and

    produce a low output level (L), since 1 > -1!

    )1,1(

    )1,1(

    )3,0(

    )2,0(

    X

    Y

    Y

    H

    NE

    E

    L

    L

    H

    38

    what would Ys optimal response be?

    This time, Y would clearly choose to produce a high outputlevel (H), since 3 > 2!

    Once X has made its move, Y finds it optimal to play L if X

    played E, and HifXplayed NE

    Hence, the strategy L ifE , H ifNE is the credible strategy

    for Y, and Xhas to take this into account!

    From Xs point of view, the sequential game reduces to the

    following:

    E)1,1(

    Y

    )(L

    39

    NE )3,0(Y

    )(H

    If X takes into account that L if E , H if NE is the only

    credible strategy for Y, X will evidently choose to enter (E),

    because 1 > 0

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    The procedure used, which involves solving first for optimal

    behavior at the end of the game and then determining what

    the optimal behavior is earlier in the game given the

    anticipation of this later behavior, is known as backwardinduction

    Using this procedure, we realized that:

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    ; ,

    is the only Nash equilibrium in the predation game that seems

    credible what do I mean with credibility?

    The Subgame Perfect Nash equilibrium

    In a sequential game, sometimes not all Nash equilibria are

    equally plausible: some of them may be based on non-credible

    threats

    Consider the three Nash equilibria in our predation game:

    when Xplays E, actions at the decision node that is unreached

    by play of the equilibrium strategies do not affect Ys payoff

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    (i.e. the first two Nash equilibria generate the same outcome)

    Th...

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