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### Game Theory

This results in a simpler game Repeat the procedure

### Game Theory

M

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João Castro

Miguel Faria

Sofia Taborda

Cristina Carias

Master in Engineering Policy and Management of Technology 24th February

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

“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

Players

decision makers

Strategies

feasible options

Payoffs

objectives

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

“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

- 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

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 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

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?

DominantStrategy

InvisibleHand

Static Games ofComplete InformationDominant Strategy:

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

Dominant Strategy Equilibrium

InvisibleHand

Prisoner’sDilemma

DominantStrategy

Characteristics

Static Games ofComplete InformationPrisoner's Dilemma

Suspect B

Confess

Deny

Suspect A

Confess

-36

-60

-36

-1

-3

Deny

-1

-60

-3

Not a Pareto efficiency!

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

NashEquilibrium

Characteristics

DominantStrategies

Prisoner’sDilemma

InvisibleHand

Static Games ofComplete InformationCompany 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

DominatedStrategies

Characteristics

DominantStrategies

Prisoner’sDilemma

NashEquilibrium

InvisibleHand

Static Games ofComplete InformationPlayer 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?

DominatedStrategies

DominantStrategies

Prisoner’sDilemma

NashEquilibrium

MixedStrategies

InvisibleHand

Static Games ofComplete InformationMixed 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 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 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

SimultaneousDecision

Dynamic Games ofComplete InformationSimultaneous 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

Characteristics

SimultaneousDecision

Dynamic Games ofComplete InformationKeep 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

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

Cooperation

Characteristics

SimultaneousDecision

Dynamic Games ofComplete InformationTit-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

SequentialGames

Strategies

Cooperation

Characteristics

SimultaneousDecision

Dynamic Games ofComplete InformationSequencial 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

SequentialGames

Strategies

Cooperation

Characteristics

SimultaneousDecision

Payoff A1

strategy A1

Payoff B1

strategy B1

strategy A2

strategy B2

Payoff B2

Dynamic Games ofComplete InformationKuhn´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

SequentialGames

Strategies

Cooperation

Strategy

Characteristics

SimultaneousDecision

Dynamic Games ofComplete InformationRollback 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

SimultaneousDecision

SequentialGames

Strategies

Cooperation

Strategy

Characteristics

out

0 , 100

E

in

-50 , -50

fight

M

acc

50 , 50

Dynamic Games ofComplete InformationEntrant makes the first move(must consider how monopolist will respond)

If Entrant enters

Monopolist accommodates

SequentialGames

Strategies

Cooperation

Strategy

Characteristics

SimultaneousDecision

Dynamic Games ofComplete InformationIs 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

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?

SequentialGames

Strategies

Cooperation

Strategy

Bargaining

Characteristics

SimultaneousDecision

Dynamic Games ofComplete InformationTake-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?”

SequentialGames

Strategies

Cooperation

Strategy

Bargaining

Characteristics

SimultaneousDecision

Dynamic Games ofComplete InformationWho 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

SimultaneousDecision

Dynamic Games ofComplete InformationCOMMANDMENT:

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!)

SequentialGames

Strategies

Cooperation

Strategy

Bargaining

Characteristics

SimultaneousDecision

Dynamic Games ofComplete InformationLessons

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 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

Bayes’ Law

Static Games ofIncomplete Information1 .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

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 InformationBayes’ Law:

Bayes’ Law

Revisingjudgments

Static Games ofIncomplete InformationStrategy

Type

Action

Beliefs

Payoffs

Separating

Strategy

Pooling

Strategy

Strategy

Spaces

Bayes’ Law

Revisingjudgments

MixedStrategies

Static Games ofIncomplete InformationMixed 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

- 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?

Bayes’ Law

Revisingjudgments

MixedStrategies

RevelationPrinciple

Static Games ofIncomplete InformationThe 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

Bayes’ Law

Revisingjudgments

MixedStrategies

RevelationPrinciple

Static Games ofIncomplete InformationTheorem

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

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.

Perfect Bayesianequilibrium

Characteristics

Dynamic Games of Incomplete InformationRequirements

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.

Perfect Bayesianequilibrium

Characteristics

SignalingGames

Dynamic Games of Incomplete Information1. 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.

Perfect Bayesianequilibrium

Characteristics

SignalingGames

SELL

BUY

BUY

PASS

BUY

Dynamic Games of Incomplete Information3. 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.

Perfect Bayesianequilibrium

Characteristics

SignalingGames

Dynamic Games of Incomplete InformationJob 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

Perfect Bayesianequilibrium

Characteristics

SignalingGames

Perfect BayesianEquil. in SG

Dynamic Games of Incomplete InformationRequirements

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.

Perfect Bayesianequilibrium

Characteristics

SignalingGames

Perfect BayesianEquil. in SG

CorporateInvestment

Dynamic Games of Incomplete InformationCorporate 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?

Perfect Bayesianequilibrium

SignalingGames

CorporateInvestment

Perfect BayesianEquil. in SG

Dynamic Games of Incomplete InformationHow 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.

Perfect Bayesianequilibrium

Characteristics

SignalingGames

Perfect BayesianEquil. in SG

CorporateInvestment

JobSignaling

Dynamic Games of Incomplete InformationJob 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

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

- 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

- 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

M

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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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