Micro - Game Theory

8/3/2019 Micro - Game Theory

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GAME THEORY

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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

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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

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Example 3: What to Do?

Philip Morris

No Ad Ad

ReynoldsNo Ad 50 , 50 20 , 60

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If you are advising Reynolds, what strategy do you

recommend?

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, ,

Game theory has many applications

Economics

Politics

Business, etc, etc

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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

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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

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

hence we can represent the payoff as the negative of this

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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

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,

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

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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

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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?

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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

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,

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

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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

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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

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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

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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

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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

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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

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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

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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

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,

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(

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X

Y H

EL

H

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,

)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

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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

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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)

Therefore, Y can plan to do anything at this decision node:

given Xs strategy of choosing E, Ys payoff remains

maximized

To rule out non-credible outcomes, we introduce the principleof sequential rationality: a players strategy should specify

optimal actions at every point in the game tree

In other words, given that a player finds herself at some point

in the tree, her strategy should prescribe play that is optimal

from that point on given her opponents strategies

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What is the link between sequential rationality and credibility:

under sequential rationality, each time an agent is

called upon to make a move (i.e. at each of the

agents decision nodes), it is in the agents self-

interest to carry out the action called for by its

strategy (this means in terms of our example)

strategies that respect the principle of sequential

rationality are credible


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A Nash equilibrium that satisfies the principle of sequential

rationality, i.e. specifies only credible strategies, is known as a

(subgame) perfect Nash equilibrium

Hence { E ; L if E , H if NE } is the only subgame perfect

Nash equilibrium in the predation game

The other two Nash equilibria

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{ E ; L ifE , L ifNE } , { NE ; HifE , HifNE }

are clearly based on non-credible threats: Y would NEVER

play L ifNE or HifE!

Summing up

)1,1( Y HIn this

subgame, Y

IfYplays L ifE,HifNE,

XplaysE

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X

NE )3,0(

)2,0(Y L

H

In this

subgame, Y

playsH

,

{E;L ifE, HifNE} is the (subgame) perfect Nash equilibrium

Note that the perfect Nash equilibrium generates an outcome,

(1,1), that is not the best possible outcome for Y

The Nash equilibrium { NE ; H ifE , H ifNE } generates a

much better outcome from Ys point of view: (0,3)

This Nash equilibrium, however, is based on a non-credible

threat, since Ywould never play HifXdecides to enter

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a comm s o p ay no ma er w a oes

X

NE

E)1,1(

Y

)(H

)3,0(

Y

)(H

IfXtakes the threat

seriously, it playsNE,

and Ygets its max.

payoff


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Extensive form or normal form?

To find the (subgame) perfect Nash equilibrium of sequential

games we should focus on their extensive form representation

and solve them using backward induction

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To find all Nash equilibria of simultaneous or sequential games

we should focus on their normal form representation

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Micro - Game Theory