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Game Theory. M i c r o e c o n o m y. João Castro Miguel Faria Sofia Taborda Cristina Carias. Master in Engineering Policy and Management of Technology 24 th February. Estratégia de Soares é minimizar Alegre

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Game theory

Game Theory

M

i

c

r

o

e

c

o

n

o

m

y

João Castro

Miguel Faria

Sofia Taborda

Cristina Carias

Master in Engineering Policy and Management of Technology 24th February


Game theory

Estratégia de Soares é minimizar Alegre

“Mário Soares pretende ignorar tanto quanto possível a candidatura de Manuel Alegre...”

“...garantiu que "não muda nada" na sua estratégia por causa de Alegre e repetiu que o seu adversário é "o candidato da direita, que não sei ainda se é, mas que espero que seja o Prof. Cavaco Silva".

Diário de Noticias 26/09/2005

Sondagem inicial:

Cavaco Silva 53,0%

Mário Soares 16,9%

Manuel Alegre 16, 2%

Soares agita meios políticos

E deixa Alegre fora da corrida a Belém. Sócrates afirmou preferir ex-Presidente da República. Cavaco não se deixa inibir. «PS ficou dividido», diz PSD. Alegre não comenta «reflexão» de Soares, que quer «escutar o sentimento» dos portugueses antes de avançar.

Portugal Diário, 24/07/2005

Resultados:

Cavaco Silva 50,6%

Manuel Alegre 20,7%

Mário Soares 14,3%

José Sócrates proíbe represálias sobre Manuel Alegre

“...o apoio a Alegre, se viesse a ocorrer uma segunda volta, foi mesmo aprovado por unanimidade na reunião do secretariado do PS que se realizou no domingo à tarde no Largo do Rato. Nesse encontro, José Sócrates analisou os vários cenários possíveis e deixou claro que, se houvesse segunda volta e o candidato de esquerda a passar fosse Manuel Alegre, o PS daria o seu apoio incondicional para a eleição do vice-presidente da Assembleia da República.

Público 24/01/2006


Introduction

Introduction

“If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is.”

(John von Neumann)

“We, humans, cannot survive without interacting with other humans, and ironically, it sometimes seems that we have survived despite those interactions.”

(Levent Koçkesen)


Elements of a game strategic environment

Elements of a Game:Strategic Environment

Players

decision makers

Strategies

feasible options

Payoffs

objectives


Elements of a game the rules

Elements of a Game:The Rules

Timingof moves

Simultaneous or sequential?

Nature of conflictand interaction

Are players’ interests in conflict or in cooperation?

Will players interact once or repeatedly?

Informationalconditions

Is there full information or advantages?

Enforceability ofagreements orcontracts

Can agreements to cooperate work?


Elements of a game assumptions

Elements of a Game:Assumptions

“I can calculate the motions of heavenly bodies, but not the madness of people”

Isaac Newton

(upon losing £20,000 in the South Sea Bubble in 1720)

Rationality

Players aim to maximize their payoffs

Players are perfect calculators

CommonKnowledge

“I Know That You Know That I Know…” (popular saying)


Interests

Interests

  • Zero sum: a game in which one player's winnings equal the other player's losses

  • Variable-sum (non-zero sum): a game in which one player's winnings may not imply the other player's losses


Type of games

Type of Games

Static Games ofComplete Information

yes

Is it a one-move game?

Dynamic Games ofComplete Information

no

yes

Are all the payoffs known?

Static Games ofIncomplete Information

no

yes

Is it a one-move game?

no

Dynamic Games ofIncomplete Information


Static games of complete information

Characteristics

Static Games ofComplete Information

  • all the payoffs are know

  • players simultaneously choose a strategies

  • the combinations of strategies may be represented in a normal-form representation

How to predict the solution of a game?

Player B

Strategy B1

Strategy B2

Payoff 1

Payoff 2

Player A

Strategy A1

Payoff 3

Payoff 4

Strategy A2


Static games of complete information1

InvisibleHand

Characteristics

Static Games ofComplete Information

“Every individual necessarily labours to render the annual revenue of the society as great as he can. He generally neither intends to promote the public interest, nor knows how much he is promoting it (...) By pursuing his own interest he frequently promotes that of the society more effectually than when he really intends to promote it.”

Adam Smith in The Wealth of Nations

Does the invisible hand exist?


Static games of complete information2

Characteristics

DominantStrategy

InvisibleHand

Static Games ofComplete Information

Dominant Strategy:

A strategy that outperforms all other choices no matter what opposing players do

Dominant Strategy Equilibrium


Static games of complete information3

InvisibleHand

Prisoner’sDilemma

DominantStrategy

Characteristics

Static Games ofComplete Information

Prisoner's Dilemma

Suspect B

Confess

Deny

Suspect A

Confess

-36

-60

-36

-1

-3

Deny

-1

-60

-3

Not a Pareto efficiency!


Static games of complete information4

Characteristics

NashEquilibrium

DominantStrategies

Prisoner’sDilemma

InvisibleHand

Static Games ofComplete Information

  • a player’s best decision is dependent on the other players’ decisions

Nash Equilibrium:

Each player chooses its best strategy according to the other players’ best strategy


Static games of complete information5

NashEquilibrium

Characteristics

DominantStrategies

Prisoner’sDilemma

InvisibleHand

Static Games ofComplete Information

Company B

High

Low

Then how to overcome several Nash equilibriums of a game?

10,10

6,4

High

Company A

2,2

5,5

Low

Player B

Left

Right

2,2

0,2

several Nash Equilibriums may coexist in the same game…

Left

Player A

2,0

1,1

Right


Static games of complete information6

DominatedStrategies

Characteristics

DominantStrategies

Prisoner’sDilemma

NashEquilibrium

InvisibleHand

Static Games ofComplete Information

Player B

Left

Right

Middle

1,0

1,2

0,1

Top

Player A

0,3

0,1

2,0

Bottom

  • Verify the existence of dominated strategies of one player

  • Re-design the normal-form representation

  • Verify the existence of dominated strategies of the other player

  • Re-design the normal-form representation

And is it possible that a game doesn’t have a single Nash Equilibrium?


Static games of complete information7

Characteristics

DominatedStrategies

DominantStrategies

Prisoner’sDilemma

NashEquilibrium

MixedStrategies

InvisibleHand

Static Games ofComplete Information

Mixed Strategy:

A strategy in which the players judge their decision based on a degree of probability

Company B

High

Low

pA

3,6

6,2

pA= 5/7

High

Company A

5,1

1,4

Low

pB= 3/7

pB


Dynamic games of complete information

Characteristics

Dynamic Games ofComplete Information

Imperfect: although the players know the payoffs, playing simultaneously disables them to have the perfect information

The information can be

Perfect: occurs when the players know exactly what has happened every time a decision needs to be made


Dynamic games of complete information1

Characteristics

Dynamic Games ofComplete Information

One Shot

  • Players don’t know much about one another

  • Players interact only once

Repeated

Indefinitely versus Finitely?

  • Finite

    • No incentive to cooperate

    • There's a future loss to worry about in the last period

  • Infinite

    • Cooperation may arise!

    • Reputation concerns matter

    • The game doesn’t need to be played forever, what matters is that the players don’t realize when the game is going to end


Dynamic games of complete information2

Characteristics

SimultaneousDecision

Dynamic Games ofComplete Information

Simultaneous Decision

(imperfect information)

How to think?

Must anticipate what your opponent will do right now, recognizing that your opponent is doing the same

Rationality of the Players

  • Put yourself in your opponent’s shoes

  • Iterative reasoning


Dynamic games of complete information3

Cooperation

Characteristics

SimultaneousDecision

Dynamic Games ofComplete Information

Keep in mind

If you plan to pursue an aggressive strategy ask yourself whether you are in a one-shot or in a repeated game. If it’s a repeated game:

THINK AGAIN


Dynamic games of complete information4

Cooperation

Characteristics

SimultaneousDecision

Dynamic Games ofComplete Information

“If it’s true that we are here to help others, then what exactly are the others here for?”

George Carlin

Is cooperation impossible if the relationship between players is for a fixedandknown length of time?

Answer: We never know when “the game” (interaction between players) will end!

Struggle between high profits today and a lasting relationship into the future


Dynamic games of complete information5

Strategies

Cooperation

Characteristics

SimultaneousDecision

Dynamic Games ofComplete Information

Tit-for-Tat Strategy

  • Players cooperate unless one of them fails to cooperate in some round of the game.

  • The others do in the next round what the uncooperative player did to them in the last round

Trigger Strategy

  • Begin by cooperating

  • Cooperate as long as the rivals do

  • After a flaw, strategy reverts to a period of punishment of specified length in which everyone plays non-cooperatively

Grim Trigger Strategy

  • Cooperate until a rival deviates

  • Once a deviation occurs, play non-cooperatively for the rest of the game


Dynamic games of complete information6

SequentialGames

Strategies

Cooperation

Characteristics

SimultaneousDecision

Dynamic Games ofComplete Information

Sequencial Decision

(Perfect Information)

“Loretta’s driving because I’m drinking and I’m drinking because she’s driving”

in “The Lockhorns Cartoon”

Games in which players make at least some of their decision at different times


Dynamic games of complete information7

SequentialGames

Strategies

Cooperation

Characteristics

SimultaneousDecision

Payoff A1

strategy A1

Payoff B1

strategy B1

strategy A2

strategy B2

Payoff B2

Dynamic Games ofComplete Information

Kuhn´s Theorem:

Every game of perfect information with a finite number of nodes, has a solution to backward induction

  • represented in extensive form, using a game tree

Corollary:

If the payoffs at all terminal nodes are unequal (no ties) then the backward induction solution is unique


Dynamic games of complete information8

SequentialGames

Strategies

Cooperation

Strategy

Characteristics

SimultaneousDecision

Dynamic Games ofComplete Information

Rollback or Backward Induction

  • Must look ahead in order to know what action to choose now

  • The analysis of the problem is made from the last play to the first

  • Look forward and reason back

How to solve the game?

  • Start with the last move in the game

  • Determine what that player will do

  • Trim the tree

    • Eliminate the dominated strategies

  • This results in a simpler game

  • Repeat the procedure


  • Dynamic games of complete information9

    SimultaneousDecision

    SequentialGames

    Strategies

    Cooperation

    Strategy

    Characteristics

    out

    0 , 100

    E

    in

    -50 , -50

    fight

    M

    acc

    50 , 50

    Dynamic Games ofComplete Information

    Entrant makes the first move(must consider how monopolist will respond)

    If Entrant enters

    Monopolist accommodates


    Dynamic games of complete information10

    SequentialGames

    Strategies

    Cooperation

    Strategy

    Characteristics

    SimultaneousDecision

    Dynamic Games ofComplete Information

    Is there a First Mover advantage?

    Depends on the game!

    Normally there's a first move advantage:

    First player can influence the game by anticipation

    But there are exceptions!

    Example:

    Cake-cutting: one person cuts, the other gets to decide how the two pieces are allocated


    Dynamic games of complete information11

    SequentialGames

    Strategies

    Cooperation

    Strategy

    Bargaining

    Characteristics

    SimultaneousDecision

    Dynamic Games ofComplete Information

    “Necessity Never Made a Good Bargain”

    Benjamin Franklin

    • The “Bargaining Problem” arises in economic situations where there are gains from trade:

    • the size of the market is small

    • there's no obvious price standards

    • players move sequentially, making alternating offers

    • under perfect information, there is a simple rollback equilibrium

    • Example: when a buyer values an item more than a seller.

    • I value a car that I own at 1000€. If you value the same car at 1500€, there is a 500€ gain from trade (M).

    The question is how to divide the gains, for example, what price should be charged?


    Dynamic games of complete information12

    SequentialGames

    Strategies

    Cooperation

    Strategy

    Bargaining

    Characteristics

    SimultaneousDecision

    Dynamic Games ofComplete Information

    Take-it-or-leave-it Offers

    • Consider the following bargaining game for the used car:

    • I name a take-it-or-leave-it price

    • If you accept, we trade

    • If you reject, we walk away

    • Advantages

    • Simple to solve

    • Unique outcome

    • Disadvantages

    • Ignore “real” bargaining (too trivial)

    • Assume perfect information; we do not necessarily know each other’s values for the car

    • Not credible: “If you reject my offer, will I really just walk away?”


    Dynamic games of complete information13

    SequentialGames

    Strategies

    Cooperation

    Strategy

    Bargaining

    Characteristics

    SimultaneousDecision

    Dynamic Games ofComplete Information

    Who has the advantage in playing first?

    Depends…

    Value of the money in the future(discount factor)

    • Patience

    • If players are patient:

    • - Second mover is better off!

    • - Power to counteroffer is stronger than power to offer

    • If players are impatient

      • - First mover is better off!

      • - Power to offer is stronger than power to counteroffer


    Dynamic games of complete information14

    SequentialGames

    Strategies

    Cooperation

    Strategy

    Bargaining

    Characteristics

    SimultaneousDecision

    Dynamic Games ofComplete Information

    COMMANDMENT:

    In any bargaining setting, strike a deal as early as possible!

    Why doesn’t it happen naturally?

    • “Time has no meaning”

    • Lack of information about values!

    • (bargainers do not know one another’s discount factors)

    • Reputation-building in repeated settings!

    • (looks like “giving in”)

    Nevertheless, bargaining games could continue indefinitely… In reality they do not.

    Why not?

    • Both sides have agreed to a deadline in advance

    • The gains from trade, M, diminish in value over time (at a certain date M=0)

    • The players are impatient (time is money!)


    Dynamic games of complete information15

    SequentialGames

    Strategies

    Cooperation

    Strategy

    Bargaining

    Characteristics

    SimultaneousDecision

    Dynamic Games ofComplete Information

    Lessons

    Buyer:

    Good guy

    - see the seller’s points of view (“put yourself in the other’s shoes”)

    Seller:

    - create the “invisible buyer” (put pressure on the buyer)

    • Both:

    • achieve a “win-win” trade

    • signal that you are patient, even if you are not

    • For example, do not respond with counteroffers right away. Act unconcerned that time is passing-have a “poker face.”

    • remember that the more patient a player gets the higher fraction of the amount M that is on the table takes


    Static games of incomplete information

    Assumptions

    Static Games ofIncomplete Information

    Properties

    • at least one player is uncertain about another player’s payoff function

    • the importance of these analysis is related with beliefs, uncertainty and risk management

    Practical applications

    • R&D and development of products

    • banking and financial markets

    • defense - rooting terrorists


    Static games of incomplete information1

    Assumptions

    Bayes’ Law

    Static Games ofIncomplete Information

    1 .Which side of the court should I choose?

    2 . The other player tries to confuse you… he moves softly to the other side

    Bayes’ Law is used whenever update of new information is necessary

    R

    L


    Static games of incomplete information2

    Assumptions

    Bayes’ Law

    Revisingjudgments

    probability that the player 2 has a poor reception on the left

    probability that player 2 choose a position on the right

    probability that the player has a poor reception on the left giving he is on the right

    probability that player 2 moves to the right given that he has a poor reception on the left

    Static Games ofIncomplete Information

    Bayes’ Law:


    Static games of incomplete information3

    Assumptions

    Bayes’ Law

    Revisingjudgments

    Static Games ofIncomplete Information

    Strategy

    Type

    Action

    Beliefs

    Payoffs

    Separating

    Strategy

    Pooling

    Strategy

    Strategy

    Spaces


    Static games of incomplete information4

    Assumptions

    Bayes’ Law

    Revisingjudgments

    MixedStrategies

    Static Games ofIncomplete Information

    Mixed strategies revisited

    j’s mixed

    strategy

    i’s uncertainty about j’s

    choice of a pure strategy

    depends on the realization

    of a small amount of private

    information

    j’s choice

    randomization

    Nash

    equilibrium

    uncertainty

    Incomplete information


    Static games of incomplete information5

    Assumptions

    • Myerson (1979) – important tool for designing games when players have private information

    Assumptions

    Bayes’ Law

    Revisingjudgments

    MixedStrategies

    RevelationPrinciple

    Examples

    • used in auction and bilateral-trading problems

    • bidder paid money to the seller and received the good

    • bidder must to pay an ENTRY FEE

    • the seller might set a RESERVATION PRICE

    Possibilities

    Static Games ofIncomplete Information

    • How to simplify the problem?


    Static games of incomplete information6

    Assumptions

    Bayes’ Law

    Revisingjudgments

    MixedStrategies

    RevelationPrinciple

    Static Games ofIncomplete Information

    The seller can restrict attention to:

    • . The bidders simultaneously make (possibly dishonest) claims about their type (their valuations)

    • . For each possible combinations of claims, the sum of possibilities must be less than or equal to one

    1st

    • TWO WAYS

    direct

    mechanism

    2nd

    • . Restrict attention to those direct mechanisms in which it is a Bayesian Nash equilibrium for each bidder to tell the truth

    incentive-

    compatible


    Static games of incomplete information7

    Assumptions

    Bayes’ Law

    Revisingjudgments

    MixedStrategies

    RevelationPrinciple

    Static Games ofIncomplete Information

    Theorem

    Any Bayesian Nash equilibrium of any Bayesian game can be represented by an incentive-compatible direct mechanism

    • If all other players tell the truth, then they are in effect playing the strategies

    • Truth-telling is an equilibrium, it is a Bayesian Nash equilibrium of the static Bayesian game


    Dynamic games of incomplete information

    Characteristics

    Dynamic Games of Incomplete Information

    Revision

    Dynamic games:Sequential Games

    One player plays after the other

    Incomplete Information:

    At least one player doesn’t know the other players’ payoff.

    They hold Beliefs about others’ behavior – which are updated using Bayes’ Law …

    They may try to mislead, trick or communicate…

    To solve this games a new equilibrium has to be found.


    Dynamic games of incomplete information1

    Perfect Bayesianequilibrium

    Characteristics

    Dynamic Games of Incomplete Information

    Requirements

    At each information set the player with the move must have a belief about each node in the information set has been reached by the play of the game.

    Given their beliefs, the player’s strategies must be sequentially rational.

    Beliefs are determined by Bayes’ rule and the players’ equilibrium strategies.

    Belief – Probability distribution over the nodes in the information set.

    Sequentially rational – the action taken by the player with the move must be optimal given the player’s belief.

    Information set for a player – it’s a collection of decision nodes satisfying: the player has the move at every node and that he has same set of feasible actions at each node.


    Dynamic games of incomplete information2

    Perfect Bayesianequilibrium

    Characteristics

    SignalingGames

    Dynamic Games of Incomplete Information

    1. Nature draws a type t for the Sender from a set of feasible types.

    2. The Sender observes t and then chooses a message m from a set of feasible messages.


    Dynamic games of incomplete information3

    Perfect Bayesianequilibrium

    Characteristics

    SignalingGames

    SELL

    BUY

    BUY

    PASS

    BUY

    Dynamic Games of Incomplete Information

    3. The Receiver observes m (but not t) and then chooses an action a from a set of feasible actions.

    4. Payoffs are given to the Sender and Receiver.


    Dynamic games of incomplete information4

    Perfect Bayesianequilibrium

    Characteristics

    SignalingGames

    Dynamic Games of Incomplete Information

    Job Signaling

    Sender: worker

    Type: worker’s productive ability

    Message: worker’s education choice

    Receiver: market of prospective employers

    Action: wage paid by the market

    Corporate Investment

    Sender: firm needing capital to finance new project

    Type: the profitability of the firm’s existing assets

    Message: firm’s offer of an equity stake

    Receiver: potential investor

    Action: decision about whether to invest


    Dynamic games of incomplete information5

    Perfect Bayesianequilibrium

    Characteristics

    SignalingGames

    Perfect BayesianEquil. in SG

    Dynamic Games of Incomplete Information

    Requirements

    1. After observing any message the Receiver must have a belief about which types could have sent m.

    ∑p(ti|mj)=1

    p

    q

    2. For each m, the Receiver’s action must maximize the Receiver’s expected utility. The Sender’s action must maximize the Sender’s Utility.

    3. The Receiver’s Belief, at any given point, follows from Bayes’ Rule.


    Dynamic games of incomplete information6

    Perfect Bayesianequilibrium

    Characteristics

    SignalingGames

    Perfect BayesianEquil. in SG

    CorporateInvestment

    Dynamic Games of Incomplete Information

    Corporate Investment

    Situation: João Silva is an entrepreneur and wants to undertake a new project in his enterprise. He has information about the profitability of the existing company, but not about the new project.

    He needs outside financing.

    Question: What will the equity stake be?


    Dynamic games of incomplete information7

    Characteristics

    Perfect Bayesianequilibrium

    SignalingGames

    CorporateInvestment

    Perfect BayesianEquil. in SG

    Dynamic Games of Incomplete Information

    How to turn this problem into a signaling game?

    Investor accepts

    IP= %i of the profit

    EP= %e of the profit

    João offers an equity stake s to a potential investor

    Investor rejects

    Examples

    IP= the investment saved in a bank

    EP=not giving up the company

    Required investment I

    Probability(=L)=p

    The investor will accept if and only if:

    The investor will accept if and only if:

    Stake offered≤

    Relative return of the project

    His share of the expected profit≥

    investment saved in a bank

    Separating equilibrium

    Pooling equilibrium

    The high-profit type must subsidize the low profit type.

    Different types offer different stakes.


    Dynamic games of incomplete information8

    Perfect Bayesianequilibrium

    Characteristics

    SignalingGames

    Perfect BayesianEquil. in SG

    CorporateInvestment

    JobSignaling

    Dynamic Games of Incomplete Information

    Job Signaling

    Situation: An employer wants to sort among future employees.

    Sender: Employees

    Type: Bright or Dull Msg: Beach or College

    Receiver: Employer

    Action: Hire or Reject

    No sender wishes to deviate from the strategy, given the Receiver’s hiring policy;

    Hiring is better for the Receiver given the Sender’s contingent strategy.

    Question: What is the perfect Bayesian Equilibrium of this game?


    Summary

    Summary

    Static Games ofComplete Information

    yes

    Is it a one-move game?

    Dynamic Games ofComplete Information

    no

    yes

    Are all the payoffs known?

    Static Games ofIncomplete Information

    no

    yes

    Is it a one-move game?

    no

    Dynamic Games ofIncomplete Information


    Recommended readings

    Recommended Readings

    • MATA, José (2000) - Economia da Empresa, Fundação Calouste Gulbenkian (http://josemata.org/ee)

    • GIBBONS, Robert (1992) - A primer in game theory, Harvester Wheatsheaf, New York

    • VARIANT, H. (2003) - Intermediate Microeconomics: A modern approach, 6th ed., W.W. Norton & Co.

    • HARSANYI, John C. (1994) - Games with incomplete information, Nobel Lecture, December 9, 1994.


    References

    References

    • MATA, José (2000) - Economia da Empresa, Fundação Calouste Gulbenkian

    • GIBBONS, Robert (1992) - A primer in game theory, Harvester Wheatsheaf, New York

    • HARSANYI, John C. (1994) - Games with incomplete information, Nobel Lecture, December 9, 1994.

    • VARIANT, H. (2003) - Intermediate Microeconomics: A modern approach, 6th ed., W.W. Norton & Co.

    • www.columbia.edu/~lk290/teaching/uggame/lecture/intro.pdf

    • Economics Department, Princeton University Princeton Economic Theory Papers - http://ideas.repec.org/s/wop/prinet.html

    • The Bank of Sweden Prize in Economic Sciences in Memory of Alfred Nobel http://nobelprize.org/economics

    • Game theory - www.gametheory.net

    • The Center for Game Theory in Economics - www.gtcenter.org

    • The Game Theory Society - www.gametheorysociety.org

    • The International Society Of Dynamic Games - www.isdg.tkk.fi

    • http://william-king.www.drexel.edu/top/eco/game/patent.html

    • http://www.unc.edu/depts/econ/byrns_web/HET/Pioneers/smith.htm


    Game theory1

    Game Theory

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    Discussion

    João Castro

    Miguel Faria

    Sofia Taborda

    Cristina Carias

    Master in Engineering Policy and Management of Technology 24th February


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