Game Theory

Master in Engineering Policy and Management of Technology 24th February

Microeconomy

João CastroMiguel Faria

Sofia TabordaCristina Carias

Game Theory

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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

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

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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)

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Elements of a Game:Strategic Environment

Players decision makers

Payoffs objectives

Strategies feasible options

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Elements of a Game:The Rules

Timingof moves Simultaneous or sequential?

Informationalconditions Is there full information or advantages?

Nature of conflictand interaction

Are players’ interests in conflict or in cooperation?

Will players interact once or repeatedly?

Enforceability ofagreements or

contractsCan agreements to cooperate work?

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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)

RationalityPlayers aim to maximize their payoffs

Players are perfect calculators

CommonKnowledge

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

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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

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Type of Games

Static Games ofComplete Information

Dynamic Games ofComplete Information

Static Games ofIncomplete Information

Dynamic Games ofIncomplete Information

Is it a one-move game?


Are all the payoffs known?

yes

no

yes

no

yes

no

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• all the payoffs are know

• players simultaneously choose a strategies

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

Characteristics

Strategy B1 Strategy B2

Player B

Strategy A1

Strategy A2Pla

yer

A Payoff 1 Payoff 2

Payoff 3 Payoff 4

How to predict the solution of a game?

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“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?

InvisibleHand

Characteristics

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Characteristics

DominantStrategy

Dominant Strategy:

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

Dominant Strategy Equilibrium

InvisibleHand

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Confess Deny

Suspect B

Confess

Deny

Su

sp

ect

A

Prisoner's Dilemma

-36 -1

-60 -3

-36 -60

-3-1

Characteristics

DominantStrategy

Prisoner’sDilemma

Not a Pareto efficiency!

InvisibleHand

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

Characteristics

Nash Equilibrium:

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

NashEquilibrium

DominantStrategies

Prisoner’sDilemma

InvisibleHand

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2,2

2,0 1,1

0,2


NashEquilibrium

Characteristics

DominantStrategies

Prisoner’sDilemma

InvisibleHand10,10

2,2 5,5

6,4

High Low

High

Co

mp

any

A

Company B

Low

several Nash Equilibriums may coexist in the same game…

Left Right

Left

Right

Player B

Pla

yer

A

Then how to overcome several Nash equilibriums of a game?

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DominatedStrategies

Characteristics

DominantStrategies

Prisoner’sDilemma

NashEquilibrium

InvisibleHand1,0

0,3 0,1

0,11,2

2,0

Player B

Left RightMiddle

Top

BottomPla

yer

A

1. Verify the existence of dominated strategies of one player

2. Re-design the normal-form representation

3. Verify the existence of dominated strategies of the other player

4. Re-design the normal-form representation

5. …

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

Equilibrium?

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075 B

pdpd

A

A

CharacteristicsMixed Strategy:

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

DominatedStrategies

DominantStrategies

Prisoner’sDilemma

NashEquilibrium

MixedStrategies

InvisibleHand

3,6

5,1 1,4

6,2

High Low

High

Co

mp

any

A

Company B

Low

pB

pA pA = 5/7

pB = 3/7

)1(151163 BBABBAA pppppp

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The information can be

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

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

Characteristics

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• Players don’t know much about one another• Players interact only once


Indefinitely versus Finitely?

CharacteristicsOne Shot

Repeated

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

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Simultaneous Decision(imperfect information)

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

doing the same

Rationality of the Players

Characteristics

How to think?

• Put yourself in your opponent’s shoes

• Iterative reasoning

SimultaneousDecision

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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

Cooperation

Characteristics


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“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 fixed and known length of time?

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

Cooperation

Struggle between high profits today and

a lasting relationship into the future

Characteristics


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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

Strategies

Cooperation

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

Characteristics


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SequentialGames

Sequencial Decision(Perfect Information)

Strategies

Cooperation“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

Characteristics


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Corollary:

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

• represented in extensive form, using a game tree

SequentialGames

Strategies

Cooperation

Kuhn´s Theorem:

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

strategy A1 Payoff A1

strategy A2str

ategy B1

strategy B2

Payoff B1

Payoff B2

Characteristics


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• 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

1. Start with the last move in the game2. Determine what that player will do3. Trim the tree

• Eliminate the dominated strategies4. This results in a simpler game5. Repeat the procedure

SequentialGames

Strategies

Cooperation

Strategy

Rollback or Backward Induction

How to solve the game?

Characteristics


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SequentialGames

Strategies

Cooperation

Strategy

E out

in M fight

acc

0 , 100

-50 , -50

50 , 50

Entrant makes the first move

(must consider how monopolist will respond)

If Entrant enters

Monopolist accommodates

Characteristics


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SequentialGames

Strategies

Cooperation

Strategy

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

Characteristics


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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?

“Necessity Never Made a Good Bargain”

Benjamin Franklin

SequentialGames

Strategies

Cooperation

Strategy

Bargaining

Characteristics


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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

Take-it-or-leave-it Offers

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?”

SequentialGames

Strategies

Cooperation

Strategy

Bargaining

Characteristics


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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

SequentialGames

Strategies

Cooperation

Strategy

Bargaining

Characteristics


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• “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”)

• 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!)

COMMANDMENT:

In any bargaining setting, strike a deal as early as possible!Why doesn’t it happen naturally?

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

Why not?

SequentialGames

Strategies

Cooperation

Strategy

Bargaining

Characteristics


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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 notFor 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

SequentialGames

Strategies

Cooperation

Strategy

Bargaining

Characteristics


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Assumptions

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

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Bayes’ Law

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

L R

Assumptions

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Bayes’ Law: )|()()(

)|( 222

222 LR

R

LRL PRPp

PpPRp

PPRp

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

)( 2LPRp

probability that player 2 choose a position on the right

)( 2RPp

)|( 22 RL PPRpprobability 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

)|( 22 LR PRPp

Bayes’ Law

Assumptions

Revisingjudgments

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Strategy

Action Type Beliefs Payoffs

SeparatingStrategy

PoolingStrategy

StrategySpaces

Bayes’ Law

Assumptions

Revisingjudgments

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j’s mixedstrategy

i’s uncertainty about j’schoice of a pure strategy

j’s choicedepends on the realizationof a small amount of privateinformation

Nashequilibrium

uncertainty

randomization

Incomplete information

Mixed strategies revisited

Bayes’ Law

Assumptions

Revisingjudgments

MixedStrategies

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

Examples• used in auction and bilateral-trading problems

Possibilities

• bidder paid money to the seller and received the good

• bidder must to pay an ENTRY FEE

• the seller might set a RESERVATION PRICE

How to simplify the problem?

Bayes’ Law

Assumptions

Revisingjudgments

MixedStrategies

RevelationPrinciple

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TW

O W

AY

S

1st . 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

2nd

directmechanism

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

The seller can restrict attention to:

incentive-compatible

Bayes’ Law

Assumptions

Revisingjudgments

MixedStrategies

RevelationPrinciple

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•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

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

Bayes’ Law

Assumptions

Revisingjudgments

MixedStrategies

RevelationPrinciple

Theorem

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Dynamic Games of Incomplete Information

Dynamic games: Sequential GamesOne player plays after the other

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

Revision

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.

Characteristics

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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.

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

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

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

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.

Perfect Bayesianequilibrium

Characteristics

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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.


Characteristics

SignalingGames

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

SELL

BUY

PASS

BUY

4. Payoffs are given to the Sender and Receiver.

BUY


Characteristics

SignalingGames

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Job Signaling

Sender: worker Type: worker’s productive ability Message: worker’s education choiceReceiver: 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 stakeReceiver: potential investor Action: decision about whether to invest


Characteristics

SignalingGames

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Requirements1. After observing any message the Receiver must have a belief about which types could have sent m.

p q …∑p(ti|mj)=1

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.

Perfect BayesianEquil. in SG


Characteristics

SignalingGames

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Corporate InvestmentSituation: 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?



Characteristics

SignalingGames

CorporateInvestment

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How to turn this problem into a signaling game?

Probability(=L)=p

João offers an equity stake s to a potential investor

Investor accepts

Investor rejects

IP= %i of the profit EP= %e of the profit

IP= the investment saved in a bankEP=not giving up the company

Required investment I

The investor will accept if and only if:

His share of the expected profit≥investment saved in a bank

The investor will accept if and only if:

Stake offered≤Relative return of the project

Pooling equilibrium Separating equilibriumThe high-profit type must subsidize the low profit type.

Examples

Different types offer different stakes.



Characteristics

SignalingGames

CorporateInvestment

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Job SignalingSituation: An employer wants to sort among future employees.Sender: Employees Type: Bright or Dull Msg: Beach or CollegeReceiver: Employer Action: Hire or Reject

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

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.



Characteristics

SignalingGames

CorporateInvestment

JobSignaling

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Summary




Dynamic Games ofIncomplete Information



Are all the payoffs known?

yes

no

yes

no

yes

no

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Recommended Readings

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

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

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

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

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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

Master in Engineering Policy and Management of Technology 24th February

Microeconomy

João CastroMiguel Faria

Sofia TabordaCristina Carias

Game Theory

Discussion

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Game Theory

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