game th slides

7/31/2019 Game Th Slides

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Mergers andAcquisitionsStrategic Games

A (short) Introduction to Game Theory

(based on M.J. Osborne An Introduction to GameTheory Oxford University Press 2004)


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Pedagogic approach

More seriously I assume nothing known

Any question is welcome (if possible, in English)

The goal is to enter into the field, not to cover as much aspossible stuff (I dont care about going to the end of the announced

program but I care a lot on an in-depth understanding)

The mathematical level of the lecture should not be too challenging(basic equations resolution, some mathematical optimization)

We will play many games during the lecture

You MUST work each week to prepare the lecture (read the slidesin advance, prepare questions, review concept definitions, )

I rest on you to correct my numerous mistakes

This is a theoretical lecture !

Game Theory - A (Short) Introduction 29/12/2011


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Outline

1 Introduction

1.1 What is game theory?

1.2 The theory of rational choice

1.3 Coming attractions: interacting decision-makers

2 Nash Equilibrium Theory (perfect information)

2.1 Strategic games

2.2 Example: the Prisoners Dilemma

2.3 Example: Bach or Stravinsky?

2.4 Example: Matching Pennies

2.5 Example: the Stag Hunt 2.6 Nash equilibrium

2.7 Examples of Nash equilibrium

2.8 Best response functions


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Outline

2.9 Dominated actions

2.10 Equilibrium in a single population: symmetric games andsymmatric equilibria

3 Nash Equilibrium: Illustrations

3.5 Auctions

4 Mixed Strategy Equilibrium (probabilistic behavior)

4.1 Introduction

4.2 Strategic games in which players may randomize

4.3 Mixed strategy Nash equilibrium

4.4 Dominated actions 4.5 Pure equilibria when randomization is allowed

4.7 Equilibrium in a single population


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Outline

4.9 The formation of players beliefs

4.10 Extension: finding all mixed strategy Nash equilibria

4.11 Extension: games in which each player has a continuum ofactions

4.12 Appendix: Representing preferences by expected payoffs

9 Bayesian Games (imperfect information)

9.1 Motivational examples

9.2 General definitions

9.3 Two examples concerning information

9.6 Illustration: auctions



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Outline

5 Extensive Games (Perfect Information): Theory

5.1 Extensive games with perfect information

5.2 Strategies and outcomes

5.3 Nash equilibrium

5.4 Subgame perfect equilibrium

5.5 Finding subgame perfect equilibria of finite horizon games:backward induction

10 Extensive Games (Imperfect Information)

10.1 Extensive games with imperfect information

10.2 Strategies 10.3 Nash equilibrium

10.4 Beliefs and sequential equilibrium

10.5 Signaling games

10.8 Illustration: strategic information transmission



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1 Introduction


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Game theory aims to help understand situations in whichdecision-makersinteract.

The main fields of applications are:

Economic analysis

Social analysis

Politic

Biology

Typical applications:

Competing firms

Bidders in auctions

Main tool: model development. This is an arbitrage between:

Realistic assumptions

Simplicity


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An outline of the history of game theory

First major development in the 1920s

Emile Borel

John von Neumann

Decisive publication: Theory of Games and Economic Behavior,

von Neumann and Morgenstern (1944)

Early 1950s: John Nash

Nash equilibrium

Game-theoric study of bargaining

1994 Nobel Prize in Economic Sciences Harsanyi (1920-2000) Bayesian games (Harsanyi doctrine)

Nash (1928-) Nash equilibrium

Selten (1930-) Bounded rationality, extensive games


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Modeling process

Step 1: selecting aspects of a given situation (that appear to berelevant) and incorporating them into a model. This step is mostlyan art

Step 2: model analysis (using logic and mathematic) Step 3: studying models implications to determine whether our

ideas make sense. This may point towards a revision of themodels assumptions in order to better capture stylized facts.


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Rational choice: The decision-maker chooses the best action according to her preferences, among

all the actions available to her

No qualitative restriction is place on preferences

Rationality means consistency of her decisions when faced with different sets ofavailable actions.

The theory is based on two components: Actions andPreferences

1.2.1 Actions Set A consisting of all actions that, under some circumstances, are available to the

decision-maker

In any given situation, the decision-makerknows the subset ofavailable choices,and takes it as given (the subset is not influenced by the decision-makerpreferences)


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1.2.2 Preferences and payoff functions

We assume that the decision-maker, when presented with any pair ofactions, knows which of the pair she prefers

We assume further that these preferences are consistent (ifa > band

b > c, then a> c).

Preferences representation: preferences can be represented by apayoff function:

the payoff function associates a number with each action in such a waythat actions with higher numbers are preferred.

More precisely:u(a) > u(b) if and only if the decision-maker prefers ato b

(Economists often speak about utility function)


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Exercise 5.3

Person 1 cares about both her income and person 2 income.

Precisely, the value she attaches to each unit of her own income is

the same as the value she attaches to any two units of person 2sincome. For example, she is indifferent between a situation inwhich her income is 1 and person 2s is 0, and one in which her

income is 0 and person 2s is 2. How do her preferences order the

outcomes (1,4), (2,1) and (3,0), where the first component in eachcase is her income and the second component is person 2s

income? Give a payoff function consistent with these preferences.


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Note that, as decision-makers preferences convey only ordinalinformation, the payoff function also conveys only ordinal preference.

Eg.: ifu(a)=0, u(b)=1 and u(c)=100, it doesnt mean that the decision-maker likes ca lot more than b! A payoff function contains no such

information.

Note that, as a consequence, a decision-makers preferences can berepresented by many different payoff functions.

Ifurepresents a decision-makers preferences and vis another payofffunction for which

v(a) > v(b) if and only ifu(a) > u(b)then valso represents the decision-makers preferences.

More succinctly: ifurepresents a decision-markers preferences, thenany increasing function ofualso represents these preferences.


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Exercice 6.1

A decision-makers preferences over the set A={a,b,c} arerepresented by the payoff function ufor which u(a)=0, u(b)=1 and

u(c)=4.Are they also represented by the function vfor which v(a)=-1,v(b)=0, and v(c)=2? How about the function wfor whichw(a)=w(b)=0and w(c)=8?


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1.2.3 The theory of rational choice

The theory of rational choice is the action chosen by a decision-maker is at least as good, according her preferences, as everyother available action.

Note that not every collection of choices for different sets ofavailable actions is consistent with the theory.

Eg. : we observe that a decision chooses awhenever she faces the set {a,b}, but

sometimes chooses bwhen facing the {a,b,c}. This is inconsistent:- always choosing awhen facing {a,b} means that the decision-maker prefers a tob

-when facing {a,b,c}, she must choose aorc.

(Independence of irrelevant alternatives)


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1.3 Coming attractions

Up to now, the decision-maker cares only about her own choice.

In the real world, a decision-maker often does not control all thevariables that affect her.

Game theory studies situations in which some of the variables thataffect the decision-marker are the actions ofother decision-markers.


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2 Nash Equilibrium:Theory


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2.1 Strategic games

Terminology:

we refer to decision-makers as players

each player has a set of possible actions

the action profileis the list of all players actions

each player has preferences about the action profiles

Definition 13.1 (Strategic game with ordinal preferences)

A strategic game with ordinal preferences consists of

a set of players

for each player, a set of actions for each player, preferences over the set of action profiles


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2.1 Strategic games

Note that:

This allows to model a very wide range of situations:

players = firms, actions = prices, preferences = profits

players = animals, actions = fighting for a prey, preferences =

winning or loosing

It is frequently convenient to specify the payers preferences by

giving payoff functions that represent them. Keep however inmind that a strategic game with ordinal preferences is defined bythe players preferences, not by the payoffs that represent thesepreferences

Time is absent from the model : each player chooses her actiononce and for all and the players choose their actionssimultaneously (no player is informed of the action chosen by anyother player)


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2.2 Example: the Prisoners

Dilemma

Example 14.1

Two suspects in a major crime are held in separate cells. There isenough evidence to convict each of them of a minor offense, but

not enough evidence to convict either of them of the major crimeunless one of them acts as an informer against the other (finks). Ifthey both stay quiet, each will be convicted of the minor offenseand spend one year in prison. If one and only one of the finks, shewill be freed and used as a witness against the other, who willspend four years in prison. If the both fink, each will spend threeyears in prison.

Model this situation as a strategic game.


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Dilemma

Solution

Players: the two suspects

Actions: Each players set of actions is {Quiet, Fink}

Preferences: Suspect 1s ordering of the action profiles (from

best to worse): (Fink,Quiet) free

(Quiet,Quiet) one year in prison

(Fink,Fink) three years in prison

(Quiet,Fink) four years in prison

(and vice-versa for player 2)

We can adopt a payoff function for each player:

u1(Fink,Quiet)>u1(Quiet,Quiet)>u1(Fink,Fink)>u1(Quiet,Fink)

Eg.:FQFFQQQF ,,,, 10111213


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Dilemma

Graphically, the situation is the following :

(numbers are payoffs of payers)

Suspect 1

Suspect 2

Quiet

Fink

Quiet Fink

(2,2)

(3,0)

(0,3)

(1,1)

The prisoners dilemma models a situation in which there are gains from cooperation

(each player prefers that both players choose Quietthan they both choose Fink) buteach player has an incentive to free ride whatever the other play does.


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Dilemma

2.2.1 Working on a joint project

You are working with a friend on a joint project. Each of youcan either work hard or goof off. If your friend works hard, thenyou prefer to goof off (the outcome of the project would be

better if you worked hard too, but the increment in its value toyou is not worth the extra effort). You prefer the outcome ofyour both working hard to the outcome of your both goofing off(in which case nothing gets accomplished), and the worstoutcome for you is that you work hard and your friend goofs off(you hate to be exploited).



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Dilemma

2.2.2 Duopoly

In a simple model of a duopoly, two firms produce the samegood, for which each firm charges either a low price or a highprice. Each firm wants to achieve the highest possible profit. Ifboth firms choose High, then each earns a profit of $1000. Ifone firm chooses Highand the other chooses Low, then the firmchoosing Highobtains no customers and makes a loss of $200,whereas the firm choosing Lowearns a profit of $1200 (its unit

profit is low, but its volume is high). If both firms choose Low,the each earns a profit of $600. Each firm cares only about itsprofit.



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Dilemma

Exercise 17.1

Determine whether each of the following games differs from thePrisoners Dilemma only in the names of the players actions

X

Y

X

Y

X Y X Y

3,3

5,1

1,5

0,0

2,1

3,-2

0,5

1,-1

An application to M&As: the Grossman & Hart free riding argument.


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2.3 Example: Back or Stravinsky?(Battle of the Sexes or BoS)

Situation:

Players agree that it is better to cooperate

Players disagree about the best outcome

Example 18.2

Two people wish to go out together. Two concerts are available:one of music by Bach, and one of music by Stravisky. One personprefers Bach and the other prefers Stravinsky. If they go to differentconcerts, each of them is equally unhappy listening to the music ofeither composer.


An application to merging banks: two banks are merging. Bothagree that they will be better off using the same information

system technology but they disagree on which one to choose.


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Google versus Microsoft/Yahoo



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2.3 Example: Back or Stravinsky?(Battle of the Sexes or BoS)

Solution

Player 1

Player 2

Bach

Stravinsky

Bach Stravinsky

(2,1)

(0,0)

(0,0)

(1,2)


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Situation:

A purely conflictual situation

Example 19.1

Two people choose, simultaneously, whether to show the head orthe tail of a coin. If they show the same side, person 2 pays person1 a dollar. I they show different sides, person 1 pays person 2 adollar. Each person cares only about the amount of money shereceives (and is a profit maximizer!).


An application to choices of appearances fornew products by an establishedproduced and a new entrant in a market of fixed size: the established producedprefers the newcomers product to look different from its own (to avoid confusion)

while the newcomer prefers that the products look alike.


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IPhone iOS versus Android



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Solution

Player 1

Player 2

Head

Tail

Head Tail

(1,-1)

(-1,1)

(-1,1)

(1,-1)


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2.5 Example: the stag Hunt

Situation:

Cooperation is better for both but not credible.

Example 20.2

Each of a group of hunters has two options: she may remainattentive to the pursuit of a stag, or she may catch a hare. If allhunters pursue the stag, they catch it and share it equally. If anyhunter devotes her energy to catching a hare, the stag escapes,and the hare belongs to the defecting hunter alone. Each hunterprefers a share of the stag to a hare.



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2.5 Example: the stag Hunt

Solution

Player 1

Player 2

Stag

Hare

Stag Hare

(2,2)

(1,0)

(0,1)

(1,1)


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Question:

What actions will be chosen by players in a strategic game?

(assuming that each player chooses the best available action)

Answer:

To make a choice, each player must form a beliefabout other playersaction.

Assumption:

We assume in strategic games that players beliefs are derived from

their past experience playing the game: they know how their opponent will behave.

note however that they do not know which specific opponent they are faced to andso, they can not condition their behavior on being faced to a specific opponent.Beliefs are about typical opponents, not any specific set of opponents.


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In this setup, a Nash equilibrium is action profile a*with theproperty that no playerican do better by choosing an actiondifferent from a*i, given that every other playerjadheres to a*j.

Note:

A Nash equilibrium corresponds to a steady state: if, whenever thegame is played, the action profile is the same Nash equilibrium a*,then no player has a reason to choose any action different from hercomponent ofa*.

Players beliefsabout each others actions are (assumed to be)

correct. This implies, in particular, that two players beliefs about athird players action are the same (expectations are coordinated Harsanyi Doctrine).

Two key ingredients: rational choices and correct beliefs


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Notations and formal definition:

Let aibe the action of playeri

Let abe an action profile: a=(a1, a2, an)

Let ai be any action of playeri(different from ai )

Let (ai,a-i) be the action profile in which every playerjexcept ichooses her action ajas specified by a, whereas playerichoosesai(the subscriptistands for except i).

(ai,a-i) is the action profile in which all the players other than iadhere to awhile ideviates to ai.

Note that ifai=ai, then (ai,a-i) = (ai,a-i) =a


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Note:

This definition implies neither that a strategic game necessarily has aNash equilibrium, nor that it has at most one.

This definition is designed to model a steady state among experienced

players. An alternative approach (called rationalizability) is: to assume that players know each others preferences

to consider what each player can deduce about the other players action

from their rationality and their knowledge of each others rationality

Nash equilibrium has been studied experimentally.

The keys to conceive suited experiment are:

to ensure that players are experienced playing the game

to ensure that players do not face repeatedly the same opponents (as eachgame must played in isolation)

The key to correctly interpret results is to remember that Nashequilibrium is about equilibrium: the outcome must have converged (andthe theory says nothing about the necessary for convergence toappear).


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2.7 Examples of Nashequilibrium

2.7.1 Prisoners Dilemma

Suspect 1

Quiet

Fink

Quiet Fink

(2,2)

(3,0)

(0,3)

(1,1)

Suspect 2


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Detailed explanation

(Fink, Fink) is a Nash equilibrium because:

given that player 2 chooses Fink, player 1 is better off choosingFinkthan Quiet

given that player 1 chooses Fink, player 2 is better off choosingFinkthan Quiet

No other action profile is a Nash equilibrium. Eg, (Quiet, Quiet) isnot a Nash equilibrium because:

if player 2 chooses Quiet, player 1 is better off choosing Fink

(moreover), if player 1 chooses Quiet, player 2 is also better off

choosing Fink

The incentive to free ride eliminates the possibility thatthe mutually desirable outcome (Quiet, Quiet) occurs.


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Note that:

in the present case, the Nash equilibrium action is the best actionfor each player:

if the other player chooses her equilibrium action (Fink)

but also if the other player chooses her other action (Quiet)In this sense, this equilibrium is highly robust. But, this is not arequirement of the Nash equilibrium. Only the first conditionmust be met.


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Exercise 27.1

Each of two players has two possible actions, Quietand Fink;each action pair results in the players receiving amounts of

money equal to the numbers corresponding to that action pair in

the following figure:

Quiet

Fink

Quiet Fink

(2,2)

(3,0)

(0,3)

(1,1)Player 1

Player 2


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Players are not selfish: the preferences of each playeriarerepresented by the payoff function mi(a)+mj(a), where mi(a) isthe amount of money received by playeri,jis the other player,and is a given non-negative number. Player 1s payoff to the

action pair(Quiet,Quiet) is, for example, 2 + 2.

1. Formulate the strategic game that models this situation in the case=1. Is this game the Prisoners dilemma?

2. Find the range of values offor which the resulting game is thePrisoners dilemma. For values offor which the game is not the

Prisoners dilemma, find the Nash equilibria.


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2.7.2 BoS

Player 1

Player 2

Bach

Stravinsky

Bach Stravinsky

(2,1)

(0,0)

(0,0)

(1,2)

Nash equilibria are (B,B) and (S,S). Why?

Note that this means that BoS has two steady states!


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2.7.3 Matching Pennies

Player 1

Player 2

Head

Tail

Head Tail

(1,-1)

(-1,1)

(-1,1)

(1,-1)

There is no Nash equilibrium. Why?


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2.7.4 The Stag Hunt

Player 1

Player 2

Stag

Hare

Stag Hare

(2,2)

(1,0)

(0,1)

(1,1)

Nash equilibria are (S,S) and (H,H). Why?

Note that, despites (S,S) is better for both players than (H,H), thishas no bearing on the equilibrium status of(H,H).


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Exercise 30.1 (extension to n players)

Consider the variants of the n-hunter Stag Hunt in which only mhunters, with 2mn, need to pursue the stag in order to catch it(continue to assume that there is a single stag). Assume that a

captured stag is shared only by the hunters who catch it. Undereach of following assumptions on the hunters preferences, find

the Nash equilibria of the strategic game that models thesituation.

a. As before, each hunter prefers the fraction 1/mof the stag toa hare;

b. Each hunter prefers a fraction 1/kof the stag to a hare, butprefers a hare to any smaller fraction of the stag, where kis aninteger with mkn.


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Note

In games with many Nash equilbria, the theory isolates more thanone steady state but says nothing about which one is more likely toappear.

In some games, however, some of these equilibria seem morelikely to attract the players attentions than others. These equilibria

are called focal.

Example: (B,B)seems here more likely than (S,S)

Player 1

Player 2

Bach

Stravinsky

Bach Stravinsky

(2,2)

(0,0)

(0,0)

(1,1)


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2.7.8 Strict and nonstrict equilibria

The definition 23.1 requires only that the outcome of a deviation (bya player) be no better for the deviant than the equilibrium outcome.

A equilibrium is strictif each players equilibrium action is better

than all her other actions, given the other players actions:

ui(a*) > ui(ai, a*-i) for every action ai a*iof playeri

(Note the strict inequality, contrasting with definition 23.1)


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2.8 Best Response Functions

2.8.1 Definition

In more complicated games, analyzing one by one each actionprofile quickly becomes intractable.

Let us denote the set of playeribest actions when the list of the

other players actions is a-iby Bi(a-i) or, more precisely:

iiiiiiiiiiii AaaauaauAaaB in'allfor),'(),(:in)(

Any action in Bi(a-i) is at least as good for playeriasevery other action of playeriwhen the other players

actions are given by a-i.


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2.8.2 Using best response functions to define Nash equilibrium

Proposition 36.1: The action profile a*is a Nash equilibrium of astragetic game with ordinal preferences if and only if every players

actions is a best response to the other players actions:

If each playerihas a single best response to each list a-i

(Bi(a-i) = {bi(a*-i)}), then this is equivalent to:

The Nash Equilibrium is then characterized by a set ofnequations inthe nunknowns a*i:

iaBa iii playereveryfor)(inis**

iaba iii playereveryfor)(**

),...(

...

),...(

*

1

*

1

*

**

21

*

1

nnn

n

aaba

aaba


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2.8.3 Using the best response functions to find Nash equilibria

Procedure:

1. find the best response function of each player

2. find the action profiles that satisfy proposition 36.1

Exercise 37.1.b

Find the Nash Equiliria of the game in Figure 38.1

Represents graphically the solution

2,2 1,3 0,1

3,1 0,0 0,0

1,0 0,0 0,0

TM

B

L C R


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Solution

2,2 1*,3* 0*,13*,1* 0,0 0*,0

1,0* 0,0* 0*,0*

TM

B

L C R

Player 1

Player 1

T M B

L

C

R

Player 2

Player 2


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Example 39.1

Two individuals are involved in a synergistic relationship. If bothindividuals devote more effort to the relationship, they are bothbetter off. For any given effort of individualj, the return to individual

is effort first increases, then decreases. Specifically, an effort levelis a nonnegative number, and individual is preferences (fori=1,2)are represented by the payoff function ai(c+aj-ai), where ai is i effortlevel, ajis the other individuals effort level, c>0 is a constant.

Questions:

Model the situation as a strategic game

Find players best response functions Find the Nash equilibrium

Represent graphically the situation


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Strategic game:

Players: the two individuals

Actions: each players set of actions is the set of effort levels (non

negative numbers)

Preferences: playeris preferences are represented by payofffunction ai(c+aj-ai), for i=1,2

Note that each player has infinitely many actions, so the game can notbe represented by a matrix of payoff, as previously.


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Best response function:

Intuitive construction

Given aj, individual ipayoff is a quadratic function ofai, that iszero when ai=0 and when ai=c+aj. As quadratic function are

symmetric, this implies that the playeribest response to aj is:

)(2

1)( jji acab

0

Payoff

aic+aj


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Nash equilibrium:

To find the Nash equilibrium, following proposition 36.1, we have tosolve the following system of equations:

By substitution, we get:

)(2

1

)(2

1

12

21

aca

aca

ca

ac

acca

1

1

11

:So

4

1

4

3

))(

2

1(

2

1 The unique Nash equilibrium

is (c,c)


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Graphical representation

Player 1

Player 2

0 a1

a2

c

c

c

c

b1(a2)

b2(a1)


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Note that:

The best response of a player to actions of other players needs notto be unique. If a player has many best responses to some of theother players actions, then her best response function is thick (a

surface) at some points;

Nash equilibrium needs not to exist: the best response functionmay not cross;

If best response functions are not linear, the Nash equilibria neednot to be unique;

Best response function can be discontinuous, generating anotherset of difficulties


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Exercice 42.1

Find the Nash Equilibria of the two-player strategic game in whicheach players set of actions is the set of nonnegative numbers and

the players payoff functions are u1(a1,a2)=a1(a2-a1) andu

2(a

1,a

2)=a

2(1-a

1-a

2)


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2.9.1 Strict dominations

In any game, a players action strictly dominates another action if

it is superior, no matter what the other player do.

Definition 45.1 (Strict domination): in a strategic game with ordinal

preferences, playeris action aistrictly dominatesher action aiif:

Action aiis said to be strictly dominated.

Example: in the Prisoners Dilemma,

the action Finkstrictly dominates

the action Quiet

Quiet

Fink

Quiet Fink

(2,2)

(3,0)

(0,3)

(1,1)


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Note that, as a strictly dominated action is not a best response toany actions of the other players, a strictly dominated action is notused in any Nash equilibrium.

When looking for Nash equilibria of a game, we can thereforeeliminate from consideration all strictly dominated actions.

2.9.2 Weak domination

In any game, a players action weakly dominates another action if

the first action is at least as good as the second action, no matterwhat the other players do, and is better than the second action forsome actions of the other players.


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Definition 46.1 (Weak domination) : In a strategic game with ordinalpreferences, playeris action aiweakly dominates her action ai if:

Note that is a strict Nash equilibrium, no players equilibrium action

is strictly dominated but in a nonstrict Nash equilibrium, an actioncan be weakly dominated.

actionsplayers'otheroflisteveryfor),(),('''

iiiiiii aaauaau

actionsplayers'otheroflistsomefor),(),('''

iiiiiii aaauaau


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Exercise 47.1 (Strict equilibria and dominated actions)

For the game in Figure 48.1, determine, for each player,whether any action is strictly dominated or weakly dominated.Find the Nash equilibria of the game. Detemine whether any

equilibrium is strict.

0,0 1,0 1,1

1,1 1,1 3,0

1,1 2,1 2,2

T

M

B

L C R


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2.9.4 Illustration: collective decision-making

The members of a group of people are affected by a policy,modeled as a number. Each person ihas a favorite policy, denotedx*i. She prefers the policy yto the policy zif and only ifyis closer tox*

ithan is z. The number ofnpeople is odd. The following

mechanism is used to choose the policy:

each person names a policy

the policy chosen is the median of those named

Eg.: if there are five people, and they name the policies -2, 0,0.6,5and 10, the policy 0.6 is chosen.

Questions: Model this situation as a strategic game

Find the equilibrium strategy of the players

Does anyone have an incentive to name her favorite policy?


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Strategic game:

Players: n people

Actions: each persons set of actions is the set of policies

(numbers)

Preferences: each person iprefers the action profile ato the actionprofile aif and only if the median policy named in ais closer to x*ithan is the median policy named in a.

Equilibrium strategy of the players:

Claim: for each playeri, the action of naming her favorite policy x*i

weakly dominates all her other actions. Why ?


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Proof:

Take xi> x*I(reporting a higher policy than the preferred one) a. for all actions of the other players, playeriis at least as well off

naming x*ias she is naming xi

for any list of actions of the players other than player i, denote the valueof the (n-1)th highest action by a-and the value of (n+1)th highestaction a+(so that half of the remaining players actions are at most a-and half of them are at least a+).

ifx*i a+ : the median policy is the same whether playerinames x*i orxi (asxi> x*i ).

ifxi a- : the same hold true (as x*i < xi )

ifx*i < a+ and xi> a-, then

when the playerinames x*i, the median policy is at most the greater ofx*i and a-

when the play inames xi, the median policy is at least the lesser ofxiand a+.

Thus, playeriis worse off naming xi than naming x*i .

a- at (n-1)th

a+ at (n+1)th

n

n

0


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b. for some actions of the other players, playeriis better of naming x*ithan she is naming xi

Suppose that half of the remaining players name policies less than x*iand half of them name policies greater than xi. Then the outcome is x*i ifplayerinames x*i and xi if she names xi . Thus player iis better off

naming xi than she is naming x*i .

A symmetric argument applies when xi < x*i.

Telling the truth weakly dominates all other action.


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2.10 Equilibrium in a singlepopulation: symmetric games

We focus here in cases where we want to model the interactionbetween members of a single homogenous population of players.Players interact anonymously and symmetrically.

Definition 51.1(Symmetric two-player game with ordinal preferences)

A two-player strategic game with ordinal preferences is symmetric if theplayers sets of actions are the same and the players preferences are

represented by payoff functions u1 and u2 for which u1(a1,a2)=u2(a2,a1)for every action pair(a1,a2)

Definition 52.1(Symmetric Nash equilibrium)

An action profile a*in a strategic game with ordinal preferences in whicheach player has the same set of actions is a symmetric Nashequilibrium if it is a Nash equilibrium and a*i is the same for everyplayeri.


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2.10 Equilibrium in a singlepopulation: symmetric games

Exercise 52.2

Find all the Nash equilibria of the game in Figure 53.1. Which ofthe equilibria, if any, correspond to a steady state if the gamemodels pairwise interactions between the members of a single

population?

1,1 2,1 4,1

1,2 5,5 3,6

1,4 6,3 0,0

A

BC

A B C


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3 Nash Equilibrium:Illustrations


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3.5 Auctions

3.5.1 Introduction Auctions are used to allocate significant economic resources, from works of art to

short-term government bonds to radio spectrum

Auctions are of many form:

Sequential or sealed bid (simultaneous)

First or Second price Ascending (English) or Descending (Dutch)

Single or Multi-Units

With or without reservation price

With or without entry costs

Auctions:

exist since long ago (annual auction of marriageable womans in Babyloniansvillages

and remain up-to-date (EBay on Internet)

Main questions

What are the designs likely to be the most effective at allocating resources?

What are the designs more likely to raise the most revenue?


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3.5 Auctions

Main assumption: we discuss here auctions in which everybuyer knows her own valuation and every other buyers

valuation of the item being sold

Buyers are perfectly informed.

This assumption will be dropped in Chapter 9.


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3.5 Auctions

3.5.2 Second-price sealed-bid auctions

In a common form of auction, people sequentially submit increasingbids for an object. When no one wish to submit a higher bid than thecurrent bid, the person making the current bid obtains the object at theprice shed bid.

Given that every person is certain of her valuation (perfect valuation) ofthe object before the bidding begins, during the bidding, no one canlearn anything relevant to her actions.

Thus we can model the auction by assuming that each person decides,before bidding begins, the most she is willing to bid (her maximal bid).

During the bidding, eventually, only the person with the maximal bid and

the one with the second highest maximal bid will be left competingagainst each other.

To win, the person with the highest maximal bid needs therefore to bidslightly more than the second highest maximal bid.


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3.5 Auctions

We can therefore model such an ascending auction as a strategicgame in which each player chooses an amount of money (themaximal amount she is willing to bid) and the player who choosesthe highest amount obtains the object and pays a price equal to thesecond highest amount.

This game model also a situation in which the peoplesimultaneously put bids in sealed envelopes, and the person whosubmits the highest bid wins and pays a price equal to the secondhighest bid.

In a perfect information context, ascending auctions (or Englishauctions) and second-price sealed bid auction are modeled by thesame strategic game.


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3.5 Auctions

Notations

vi: the value playeriattaches to the object

p: price paid for the object

vi-p: winning player payoff

n: number of players number the players such that v1>v2> > vn>0

bi: sealed bid submitted by each player

Rules

Each player submit a sealed bid bi

Ifbiis the highest bid, playeriwins the auction, get the object and paysthe second highest bid (sayj). In such a case, playeripayoff is vi-bj

In case of tie, it is the player with the smallest number (the highestvaluation) who wins. She pays her own bid (as there is a tie)


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3.5 Auctions

Strategic game representation:

Players: the nbidders, where n 2

Actions: the set of actions of each player is the set of possible bids(nonnegative numbers)

Preferences: denote by bithe bid of playeriand by b+the highestbid submitted by a player other than i. If eitherbi>b+orbi=b+andthe number of every other player who bids b+is greater than i, then

playeris payoff is vi-b+. Otherwise playeris payoff is 0.


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3.5 Auctions

Nash equilibrium

The game has many Nash equilibria:

One equilibrium is (b1,b2, bn)=(v1,v2, vn):each player bid is equal toher valuation of the object:

because v1>v2> > vn, the outcome is that player 1 obtains the object andpays b2. Her payoff is v1-b2. Every other players payoff is zero.

if player 1 changes he bid to some other price at least equal to b2, then theoutcome does not change. If she changes her bid to a price less than b2, thenshe loses and obtains a zero payoff

if some other player lowers her bid or raises her bid to some price at mostequal to b1, the she remains a loser. If she raises her bid above b1, then shewins but, in paying the price b1, she makes a loss (because her valuation isless then b1).


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3.5 Auctions

Another equilibrium is (b1,b2, bn)=(v1,0, 0): the player 1 obtains theobject and pays 0. Sad is issue for the auctioneer

Another equilibirum is (b1,b2, bn)=(v2,v1, 0 0): the player 2 bids v1and obtains the object at price v2and every players payoff is zero:

if player 1 raises her bid to v1 or more, she wins the object but her payoffremains zero (she pays the price v1, bid by player 2)

if player 2 changes her bid to some other prices greater than v2, the outcomedoes not change. If she changes her bid to v2or less, she loses, and herpayoff remains zero.

if any other player raises her bid to a most v1, the outcome does not change.If she raises her bid above v1, then she wins but get a negative payoff.

Note that, in this equilibrium, player 2 bids more than her valuation. Thismight seem strange. This is due to the fact that, in a Nash equilibrium, a

player does not consider the risk that another player will take an actiondifferent from her equilibrium action. Each player simply chooses anaction that is optimal, given the other players actions.

This however suggests that this equilibrium is less plausible as anoutcome of the auction than the equilibrium in which each bidder bidsher valuation.


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3.5 Auctions

This is due to the fact that:

in a second-price sealed-bid auction (with perfect information),a players bid equal to her valuation weakly dominates all herother bids.

That is:

for any bid bi vi, playeribid viis at least as good as bi, no

matter what the other players bid, and is better than bi for

some actions of the other players.


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3.5 Auctions

The precise argument is given by Figure 85.1

0 b+

vi-b+

vi 0 b+

vi-b+

vibi

viis better than biin this region

0 b+

vi-b+

vi

bi

viis better than biin this region

The Figure compares player ipayoffs to the bid vi(left panel) with her payoff to a bid bi< vi(middlepanel) and with her payoff to a bid bi > vi,as a function of the highest of the other players bids (b+).

We see that:-for all value ofb+, playeripayoff to a bid vi is a least as large as her payoffs to any other bid;-for some values of the b+, her payoffs to viexceed her payoff to any other bid.


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3.5 Auctions

Exercise 84.1

Find a Nash equilibrium of a second-price sealed bid auction inwhich player n obtains the object.

Exercise 86.1 (Auctioning the right to choose)

An action affects each of two people. The right to choose theaction is sold in a second-price auction. That is, the two peoplesimultaneously submit bids, and the one who submits the higherbid chooses her favorite action and pays (to a third party) theamount bid by the otherperson, who pays nothing. Assume that ifthe bids are the same, person 1 is the winner.

Fori=1,2, the payoff of person iwhen the action is aand person i

pays mis ui(a)-m. In the game that models this situation, find for each player a bid

that weakly dominates all the players other bids (and thus find a

Nash equilibrium in which each players equilibrium action weakly

dominates all her other actions).


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3.5 Auctions

3.5.3 First-price sealed-bid auctions

Difference with as second-price auction: the winner pays the priceshe bids

Strategic game representation: Players: the nbidders, where n2

Actions: the set of actions of each player is the set of possiblebids (nonnegative numbers)

Preferences: denote by bithe bid of playeriand by b+thehighest bid submitted by a player other than i. If either (a) bi>

b+or (b) bi= b+and the number of every other player whobids b+is greater than i, then playeripayoff is vi-bi. Otherwise,playeripayoff is 0.


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3.5 Auctions

Note that this game models:

a sealed-bid auction where the highest bid wins

but also

a dynamic auction in which the auctioneer begins by

announcing a high price, which she gradually lowers untilsomeone indicates her willingness to buy the object (aDutch auction)

(this equivalence is even, in some sense, stronger thanthe one between an ascending auction and second-pricesealed-bid auction does not depend on private values).

Nash equilibrium

One Nash equilibrium is (b1,b2, bn)=(v2,v2, vn), in whichplayer 1 bid is player 2 valuation and every other players bid is

her own valuation. The outcome is that player 1 obtains theobject at price v2.


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3.5 Auctions

Exercise 86.2

Show that (b1,b2, bn)=(v2,v2, vn) is a Nash equilibriumof a first-price sealed-bid auction.

A first-price sealed-bid auction has many other equilibria, but in allequilibria the winner is the player who values the object most highly(player 1), by the following argument:

in any action profile (b1, bn) in which some playeri1 wins,we have bi > b1.

Ifbi> v2, then ipayoff is negative, so that she can do

better by reducing her bid to 0 ifbi v2, then player 1 can increase her payoff from 0 to

v1-biby bidding bi, in which case she wins.


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3.5 Auctions

Exercise 87.1(First-price sealed-bid auction)

Show that in a Nash equilibrium of a first-price sealed-bidauction the two highest bids are the same, one of thesebids is submitted by player 1, and the highest bid is at

least v2and at most v1. Show also that any action profilesatisfying these conditions is a Nash equilibrium.


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3.5 Auctions

As in the second-price auction sealed-bid auction, the potentialriskiness to playeriof a bid bi> vi is reflected in the fact that it isweakly dominated by the bid vi, as shown by the following argument:

if the other players bids are such that player iloses when she bidsbi, then the outcome is the same whether she bids biorvi

it the other players bids are such that player iwins when she bidsb

i, then her payoff is negative when she bids b

iand zero when she

bids vi(regardless of whether this bid wins)

However, unlike a second-price auction, in a first-price auction, a bid bi< viof playeriis not weakly dominated by the bid vi(it is in fact notweakly dominated by any bid):

it is not weakly dominated by a bid bibibecause if the otherplayers highest bid is less than bi, then both biand bi win and biyield a lower price.


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3.5 Auctions

Note also that, though the bid viweakly dominates higher bids,this bid is itself weakly dominated by a lower bid! Theargument is the following:

if playeribids vi, her payoff is 0 regardless of the otherplayers bids

whereas, if she bids less than vi, her payoff is either 0 (ifshe loses) or positive (if she wins)

In a first-price sealed-bid auction (with perfect information), a playersbid of at least her valuation is weakly dominated, and a bid of less thanher valuation is not weakly dominated.


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3.5 Auctions

Note finally that this property of the equilibria depends on theassumption that a bid may be any number. In the variant of the game inwhich bids and valuations are restricted to be multiples of some discretemonetary unit ,

an action profile (v2-, v2-, b3, bn) for any bj vj-forj = 3, nis a Nash equilibrium in which no players bid is weakly dominated.

further, every equilibrium in which no players bid is weakly

dominated takes this form.

Ifis small, this is very close to (v2, v2, b3, bn) : this equilibrium istherefore (on a somewhat ad-hoc basis) considered as the distinguishedequilibria of a first-price sealed-bid auction.

One conclusion of this analysis is that, while both second-price and first-price

auctions have many Nash equilibria, their distinguished equilibria yield thesame outcome: in every distinguished equilibrium of each game, the object isold to player 1 at the price v2. This is notion ofrevenue-equivalence is acornerstone of the auction theory and will be analyzed in depth later.


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3.5 Auctions

3.5.4 Variants

Uncertain valuation: we have assumed that each bidder is certainof both her own valuation and every other bidders valuation, which

is highly unrealistic. We will study the case of imperfect informationin Chap. 9 (in the framework of Bayesian games)

Interdependent/Common valuations: in some auction, the maindifference between bidders is not that they value the objectdifferently but that they have different information about its value(eg, oil tract auctions). As this also involve informationalconsiderations, we will again study this in Chap. 9.

All-pay auctions: in some auctions, every bidder pay, not only the

winner (eg, competition of loby groups for government attention).


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3.5 Auctions

Mutiunit auctions: in some auctions, many units of an object areavailable (eg, US Treasury bills auctions) and each bidder mayvalue positively more than one unit. Each bidder chooses thereforea bid profile (b1,b2,bk) if there are kunits to sell. Different auctionmechanisms exist and are characterized by the rule governing the

price paid by the winner: Discriminatory auction: the price paid for each unit is the

winning bid for that unit

Uniform-price auction: the price paid for each unit is thesame, equal to the highest rejected bid among all the bids forall unit

Vickrey auction (of the name of Nobel prize): a bidder wins kobjects pays the sum of the khighest rejected bids submittedby the other bidders.


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4. Mixed StrategyEquilibrium


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4.1. Introduction

4.1.1. Stochastic steady state

Nash Equilibrium in a strategic game: action profile in which everyplayers action is optimal given every other players action (see def.

23.1)

This corresponds to a steady state of the game:

every players behavior is the same whenever she plays

the game

no player wishes to change her behavior, knowing (fromexperience) the other players behavior

In such a framework, the outcome ofevery play of the game isthe same Nash equilibrium

More general notion of steady state exists


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4.1. Introduction

players choices are allowed to vary:

different members of a given population may choosedifferent actions, each player choosing the same actionwhenever she plays the game

each individual may, on each occasion she plays the

game, choose her action probabilistically according tothe same, unchanging, distribution

these situations are equivalent:

in the first case, a fraction pof the populationrepresenting playerichooses the action a

in the second case, each member of the population

representing playerichooses the action awithprobability p

These notion of (stochastic) steady state ofmodeled as mixed strategy Nash equilibrium


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4.1. Introduction

4.1.2 Example: Matching Pennies

Player 1

Player 2

Head

Tail

Head Tail

(1,-1)

(-1,1)

(-1,1)

(1,-1)

Outcomes

The game has no Nash equilibrium: no pair ofaction is compatible with a steady state.


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4.1. Introduction

The game has howeverstochastic steady state in which eachplayer chooses each of her actions with probability 1/2 :

Suppose that player 2 chooses each of her actions with probability

If player 1 chooses Headwith probability pand Tailwith probability (1-

p), then:

each outcome (Head,Head) and (Head,Tail) occurs withprobability p x

each outcome (Tail,Head) and (Tail,Tail) occurs withprobability (1-p) x

Thus, the probability that the outcome is either (Head,Head) or

(Tail,Tail) (in which case player 1 wins 1$) p+ (1-p) = . The other two outcomes (Head,Tail) and (Tail,Head) (which correspond

to a loss of 1$) have also probability


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4.1. Introduction

the probability distribution over outcome is independent ofp!

every value ofpis optimal (in particular )!

the same analysis hold for player 2. We conclude that thegame has a stochastic steady state in which each playerchooses each action with probability .

Moreover (under a reasonable assumption on the players

preferences), the game has no other steady state :

Assumption: each player wants the probability of her gaining1$ to be as large as possible (maximization of expected profit)

Denote qthe probability with which player 2 chooses Head(she chooses Tail with probability (1-q) )

If player 1 chooses Head with probability p, she gains 1$ withprobability pq+ (1-p)(1-q) (outcomes Head,Head or Tail,Tail)and she looses 1$ with probability (1-p)q+ p(1-q).


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4.1. Introduction

Note that:

Player 1 wins 1$ : pq+ (1-p)(1-q) = 1-q + p(2q-1)

Player 1 loses 1$:(1-p)q+ p(1-q).= q + p (1-2q)

Ifq< , the first probability (winning 1$) is decreasing in pandthe second probability (loosing 1$) is increasing in p. Player 1chooses therefore p= 0.

Thus, if player 2 chooses Head with probability less than ,the best response of player 1 is to choose Tail with certainty.

A similar argument shows that if player 2 chooses Head withprobability superior to , the best response of player 1 is tochoose Head with certainty.

We already have shown that is one player is choosing a givenaction with certainty (Nash Equilibrium), there is no steadystate.


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4.1. Introduction

4.1.3 Generalizing the analysis: expected payoffs

The matching pennies case is particularly simple because it hasonly two outcomes for each player, allowing to deduce playerspreferences regarding lotteries (probability distributions) overoutcomes from their preferences regarding deterministic

outcomes: if a player prefers ato band ifp> q, he most likely prefers alottery in which aoccurs with probability p(and bwith probability(1-p)) to a lottery in which aoccurs with probability q(and bwithprobability (1-q))

To deal with more general cases (eg, more than two outcomes),we need to add to the model a description of her preferencesregarding lotteries (probability distribution) over outcomes


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4.1. Introduction

The standard approach is to restrict attention to preferencesregarding lotteries (probability distribution) over outcomes that maybe represented by the expected value of a payoff function overdeterministic outcomes:

for every playeri, there is a payoff function ui, with the

property that playeriprefers one probability distribution overoutcomes to another if and only if, according to ui, theexpected value of the first probability distribution exceeds theexpected value of the second probability distribution.

eg. :

three outcomes: a, b, c

two prob. dist.: P(pa,pb,pc) and Q(qa,qb,qc) for each player i, prob. dist. Pis preferred to prob. dist. Qif and

only ifpaui(a) + pbui(b) + pcui(c) > qaui(a) + qbui(b) + qcui(c)

Preferences that can be represented by the expected value of apayoff function over deterministic outcomes are called vNM (vonNeumann Morgenstern) preferences.A payoff function whose expected value represents suchpreferences is called a Bernouilli payoff function.


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4.1. Introduction

Note that if the outcomes are amount of money and if the preferences arerepresented by the expected value of the amount of money, the player is riskneutral.

Two classic utility functions: CARA & CRRA

In the reality: the fact that people buy insurance (the expected payoff is inferior to the

insurance fee) show that economic agents are risk averse. the fact that people buy lottery tickets shows that, in some circumstance, than

can be risk preferring (small investment, extremely high payoff. in both cases, the preferences can be represented by the expected value of a

payoff function: concave in case of risk aversion convex in case of risk preference

Note finally that given preferences, many different payoff functions can be used torepresented them. It is the ordering that matter.

4 2 Strategic games in which


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4.2 Strategic games in whichplayers may randomize

Definition 106.1 (Strategic game with vNM preferences)

A strategic game with vNM preferences consists of

a set of players

for each player, a set of actions

for each player, preferences regarding prob. dist. over actionprofiles that may be represented by the expected value of a(Bernoulli) payoff function over action profiles.

Representation: a two-player strategic game with vNM preferences inwhich each player has finitely many actions may be represented in a

table like in Chapter 2. However, the interpretation of the number isdifferent:

in Chapter 2, numbers are values of payoff functions that represent theplayers preferences over deterministic outcome

here, numbers are values of (Bernoulli) payoffs whose expected valuesrepresent the players preferences over prob. dist..

4 2 Strategic games in which


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4.2 Strategic games in whichplayers may randomize

The change is subtle but important (figure 107.1)

The 2 games represent the same game with ordinal preferences (theprisoners dilemma).

However, the 2 games represent different strategic games with vNM

preferences: left game: players 1 payoff to (Q,Q) is the same as her expected payoff

to the prob. dist. that yield (F,Q) with probability and (F,F) withprobability

right game: her payoff to (Q,Q) is higher than her expected payoff to thisprob. dist.

Q

F

Q

F

Q F Q F

2,2

3,0

0,3

1,1

3,3

4,0

0,4

1,1

4 3 Mixed strategy Nash


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4.3 Mixed strategy Nashequilibrium

4.3.1 Mixed strategies

We allow now each player to choose a probability distribution overher set of actions (rather than restricting her to choose a singledeterministic action)

Definition 107.1 (Mixed strategy)

A mixed strategy of a player in a strategic game is a probabilitydistribution over the players actions.

Notations:

: profile of mixed strategies (matrix) i(ai): probability assigned by playeris mixed strategy i to her

action ai



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eg: in Matching pennies, the strategy of player 1 that assignsprobability to each action is the strategy 1(Head)= and1(Tail) = .

Shortcut: mixed strategies are often written as a list ofprobabilities (one for each action), in the order the actions are

given in the table (see table 107.1).eg.: ( , ) assigns, in table 107.1, probability to Q andprobability to F.

Note that a mixed strategy may assign probability 1 to a singleaction. In that case, such as strategy is referred as a pure

strategy.



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4.3.2 Equilibrium

The mixed strategy Nash equilibrium extend the concept ofNash equilibrium to the probabilistic setup.

Definition 108.1 (Mixed strategy Nash equilibrium of strategicgame with vNM preferences)

The mixed strategy profiles *in a strategic game with vNMpreferences is a mixed strategy Nash equilibrium if, for eachplayeriand every mixed strategy iof playeri, the expectedpayoff to playeriof*is at least as large as the expected payoffpayoff to playeriof (i,-i*), according to a payoff functionwhose expected value represents playeris preferences overprob. dist.

.profilestrategymixedthetopayoffexpecteds'playeris)(where

playerofstrategymixedeveryfor),*,(*)(

iU

iUU

i

iiiii


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111/335



Two players two actions games

Player 1 has action Tand B

Player 2 has action L and R

ui (i=1,2) denotes a Bernoulli payoff function for playeri(payoffover action pair whose expected value represents playerispreferences regarding prob. dist. over action pairs)

Player 1 mixed strategy 1 assigns probability 1(T) to heraction T(denoted p) and probability 1(B) to her action B(denoted 1-p), with 1(T) + 1(B) = 1.

Similarly, denotes qthe probability that player 2s mixedstrategy assigns to L et 1-qto R.

We take the players choices to be independent (when players

choose the mixed strategies 1 and 2, the probability of anyaction pair (a1,a2) is the product of the correspondingprobabilities assigned by mixed strategies).

4.3 Mixed strategy Nash


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From this probability distribution, we can compute player 1s

expected payoff to the mixed strategy pair (1, 2):

which can be written as:

So, the probabilities of the four outcomes are:

T(p)

B(1-p)

L(q) R(1-q)

pq

(1-p)q

p(1-q)

(1-p)(1-q)

(Figure 109.1)

),()1)(1(),()1(),()1(),( 1111 RBuqpLBuqpRTuqpLTupq



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which can be written more compactly as:

),()1(),()1(),()1(),( 1111 RBuqLBuqpRTuqLTuqp

Player 1 expected payoff

when she uses a purestrategy that assignsprobability 1 to Tand player 2uses a mixed strategy 2

Player 1 expected payoff

when she uses a purestrategy that assignsprobability 1 to Band player 2uses a mixed strategy 2

2121 ,)1(, BEpTpE

Player 1 expected payoff to the mixed strategy pair (1,2)as a weighted average of her expected payoffs to Tand Bwhen player 2 uses the mixed strategy 2, with weights

equal to the probabilities assigned to Tand Bby 1.



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In particular, player 1s expected payoff is a linear function ofp

0 p 1

2121 ,)1(, BEpTpE 21 ,BE

2121 ,, BETE

21 ,TE

(Figure 110.1)



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A significant implication of this linearity form of the player 1s

expected payoff is that there is only three possibilities for herbest response to a given mixed strategy of player 2:

player 1s unique best response is the pure strategy T(ifE1(T,2) >E1(B,2) ): see figure 110.1

player 1s unique best response is the pure strategy B(ifE1(T,2) , her expected payoff to Head exceeds herexpected payoff to Tail.

ifq= , then both Head and Tail (and all her mixed strategies)lead to the same payoff.

we conclude that player 1s best responses to player 2s

strategy are her mixed strategy that assigns probability 0 toHead ifq< , her mixed strategy that assigns probability 1 toHead ifq > and all her mixed strategies ifq= .



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3 ed st ategy asequilibrium

The best response function of player 2 is similar (see figure 112.1)

The set of mixed strategy Nash equilibria corresponds (as before)to the set of intersections of the best response functions in figure112.1.

Matching Pennies has no Nash Equilibrium if players are notallowed to randomize !



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gyequilibrium

Exercise 114.2

Find all the mixed strategy Nash equilibria of the strategicgames in Figures 114.2

T

B

T

B

L R L R

6,0

3,2

0,6

6,0

0,1

2,2

0,2

0,1

(Figure 114.2)



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gyequilibrium

Exercise 114.3Two people can perform a task if, and only if, they both exert effort. They are bothbetter off if they both exert effort and perform the task than if neither exerts effort(and nothing is accomplished); the worst outcome for each person is that sheexerts effort and the other person does not (in which case again nothing isaccomplished). Specifically, the players preferences are represented by the

expected value of the payoff functions in Figure 115.1, which cis a positivenumber less than 1 than can be interpreted as the cost of exerting effort. Find allthe mixed strategy Nash equilibria of this game. How do the equilibria change as c

increase? Explain the reasons for the changes.

No Effort

Effort

No Effort Effort

0,0

-c,0

0,-c

1-c,1-c

(Figure 115.1)



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gyequilibrium

4.3.4 A useful characterization of mixed strategy Nashequilibrium

The method used up to now to find Mixed strategy Nash equilibriainvolves constructing players best response functions. In

complicated games, this method may be intractable. There is a

characterization of mixed strategy Nash equilibria that is aninvaluable tool in the study of generale game.

The key is the following observation: a players expected payoffto a mixed strategy profile is a weighted average of herexpected payoffs to all pure strategy profiles of the type (ai,-i),where the weights attached to each pure strategy (ai,-i) is the

probability i(ai)assigned to that strategy aiby the playersmixed strategy i(see section 4.3.3).



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gyequilibrium

Symbolically:

where: Ai is playeris set of actions (pure strategies)

Ei(ai,-i) is her expected payoff when she uses the pure strategythat assign probability 1 to aiand every other playerjuses hermixed strategy j.

ii Aa

iiiiii aEaU ),()()(



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gyequilibrium

This leads to the following analysis:

Let *be a mixed strategy Nash equilibrium

Denote by E*iplayeris expected payoff in the equilibrium

Because *is an equilibrium, payeris expected payoff, given *-i,to all her strategies (including all her pure strategies), is at most E*i

But E*i is a weighted average of playeris expected payoffs to thepure strategies to which *iassigns a positive probability

Thus, playeris expected payoffs to these pure strategies are allequal to E*i(if any smaller, then the weighted average would besmaller!).



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gyequilibrium

We conclude that:

expected payoff to each action to which *iassigns positiveprobability is E*i

the expected payoff to every other action is at most E*i

Proposition 116.2

A mixed strategy profile *in a strategic game with vNMpreferences in which each player has finitely many actions is amixed strategy Nash equilibrium if and only if, for each playeri,

the expected payoff, given *-i, to every action to which *iassignsa positive probability is the same

the expected payoff, given *-i, to every action to which *iassignsa zero probability is at most the expected payoff to any action towhich *iassigns a positive probability

Each players expected payoff in an equilibrium is her expected

payoff to any of her actions that she uses with positiveprobability



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This proposition allows to check whether a mixed strategyprofile is an equilibrium.

Example 117.1

L(0) C(1/3) R(2/3)

T(3/4)

M(0)

B(1/4)

.,2 3,3 1,1

.,. 0,. 2,.

.,4 5,1 0,7

(Figure 117.1)



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gyequilibrium

For the game in Figure 117,1 (in which the dots indicateirrelevant payoffs), the indicated pair of strategies ((3/4,0,1/4)for player 1 and (0,1/3,2/3) for player 2) is a mixed strategyNash equilibrium.

To verify this claim, it suffices, by proposition 116.2, to studyeach players expected payoffs to her three pure strategies. For

player 1, these payoffs are:

350

325

31:

342

320

31:

351

323

31:

B

M

T



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gyequilibrium

Player 1s mixed strategy assigns positive probability to Tand Band probability zero to M. So, the two conditions of proposition116.2 are satisfied for player 1.

The same verification is easily done for player 2. Note howeverthat, for player 2, the action L (which she uses with probability0), has the same expected payoff to her other two actions. Thisequality is consistent with proposition 116.2 (no greater than).



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Exercise 117.2 (Choosing numbers)

Players 1 and 2 each choose a positive integer up to K. If theplayers choose the same number, then player 2 pays $1 toplayer 1; otherwise no payment is made. Each players

preferences are represented by her expected monetary payoff.

Show that the game has a mixed strategy Nash equilibrium inwhich each player chooses each positive integer up to Kwithprobability 1/K

Show that the game has no other mixed strategy Nash equilibria(Deduce from the fact that player 1 assigns positive probability tosome action kthat player 2 must do so; then look at the implied

restriction on player 1s equilibrium strategy)



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yequilibrium

Note finally that

an implication of Proposition 116.2 is that a nondegenerate mixedstrategy equilibrium (a mixed strategy equilibrium that is not also apure strategy equilibrium) is never a strictNash equilibrium: everyplayer whose mixed strategy assigns a positive probability to more

than one action is indifferent between her equilibrium mixedstrategy and every action to which this mixed strategy assignspositive probability.

The theory of mixed Nash equilibrium does not state that playersconsciously choose their strategies at random given the equilibriumprobabilities. Rather, the conditions for equilibrium are designed to

ensure that it is consistent with a steady state. The question of howa steady state may come about remains to be studied at this stage.



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equilibrium

4.3.5 Existence of equilibrium in finite games

Proposition 119.1 (Existence of mixed strategy Nashequilibrium in finite games)

Every strategic game with vNM preferences in which each playerhas finitely many actions has a mixed strategy Nash equilibrium.

This proposition does not help to find the equilibrium but it is auseful fact.

Note also that: the finiteness of the number of actions is a sufficient condition for

the existence of an equilibrium, not a necessary one.

that a players strategy in mixed strategy Nash equilibrium may

assign probability 1 to a single action.

4 4 Dominated actions


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Definition 120.1 (Strict Domination)

In a strategic game with vNM preferences, playeris mixedstrategy istricly dominates her action ai if:

where ui is a Bernoulli payoff function and Ui(i,a-i) is playerisexpected payoff underuiwhen she uses the mixed strategy iand the actions chosen by the other players are given by a-i.

actionsplayers'otherfor thelisteveryfor),'(),( iiiiiii aaauaU



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An action not strictly dominated by any pure strategy may bestrictly dominated by a mixed strategy (see Figure 120.1)

T

M

L R

1

4

1

0

(Figure 120.1)

B 0 3

The action Tof player 1 is not strictly (or weakly) dominatedby MorB, but it is strictly dominated by the mixed strategythat assigns probability to Mand probability to B.



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Exercise 120.2 (Strictly dominated mixed strategy)

In Figure 120.1, the mixed strategy that assigns probability toMand to Bis not the only mixed strategy that strictlydominates T. Find all the mixed strategy that do so.

Exercise 120.3 (Strict domination for mixed strategies)

Determine whether each of the following statements is true offalse:

A mixed strategy that assigns positive probability to a strictlydominated action is strictly dominated.

A mixed strategy that assigns positive probability only to actionsthat are not strictly dominated is not strictly dominated.



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A strictly dominated action is not a best response to anycollection of mixed strategies of the other players Suppose that playeris action ai is strictly dominated by her mixed

strategy i Playeris expected payoffUi(i,-i) when she uses the mixed strategy

iand the other players use the mixed strategies -iis a weightedaverage of her payoffs Ui(i,a-i) as a-ivaries over all the collections ofaction for the other players, with the weight on each a-iequal to theprobability with which it occurs when the other players mixed

strategies are -i.

Playeris expected payoff when she uses the action aiand the otherplayers use the mixed strategies -i is a similar weighted average; the

weights are the same but the terms take the form ui(ai,a-i), rather thanUi(i,a-i).

The fact that ai is strictly dominated by imeans that Ui(i,a-i) >ui(ai,a-i) for every collection a-iof the players actions.

Hence playeris expected payoff when she uses the mixed strategy iexceeds her expected payoff when she uses the action ai, given -i.



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Consequently, a strictly dominated action is not used withpositive probability in any mixed strategy Nash equilibrium.

Definition 121.1 (Weak domination)

In a strategic game with vNM preferences, playeris mixedstrategy iweakly dominates her action ai if:

and

where ui is a Bernoulli payoff function and Ui(i,a-i) is playerisexpected payoff underuiwhen she uses the mixed strategy iand the actions chosen by the other players are given by a-i.

actionsplayes'othertheoflisteveryfor),'(),( iiiiiii aaaUaU

actionsplayes'othertheoflistsomefor),'(),( iiiiiii aaaUaU


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4.5 Pure equilibria when


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randomization is allowed

Equilibria when the players are not allowed to randomizeremain equilibria when they are allowed to randomize

Proposition 122.2 (Pure strategy equilibria survive when

randomization is allowed)Let a* be a Nash equilibrium ofGand for each playeri, let *ibe the mixed strategy of playerithat assigns probability one tothe action a*i. Then * is a mixed strategy Nash equilibrium ofG.



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Any pure equilibria that exist when the players are allowed torandomize are equilibria when they are not allowed torandomize.

Proposition 123.1 (Pure strategy equilibria survive whenrandomization is prohibited)

Let * be a mixed strategy Nash equilibrium ofGin which themixed strategy of each playeriassigns probability one to thesingle action a*i. The a* is a Nash equilibrium ofG.


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Proposition 122.2

Let a*be a Nash equilibrium ofG, and for each playerilet *ibe the mixed that assigns probability 1 to a*i. Since a*is a Nashequilibrium ofG, we know that in Gno playerihas an actionthat yields her a payoff higher than does a*iwhen all otherplayers adhere to *-i. Thus *satisfies the two conditions inProposition 116.2. So, it is a mixed strategy equilibrium ofG.

Proposition 123.1

Let *be a mixed strategy Nash equilibrium ofGin which everyplayers mixed strategy is pure. For each playeri, denote a*i the

action to which iassigns probability one. Then, no mixedstrategy of playeriyields her a payoff higher than does *i. Thusa*is Nash equilibrium ofG.

4.7 Equilibrium in a single


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