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(1)

130 ^Chapter^4.Mixed Strategy Equilibrium

St. Continue

Stop _I ^'--i, 1 ^-- _£ ^- 1 - E, _u

.---•... --_._---,- ,,'--._.0-

I

Continue

__

_{.�=_�_,}

____

_Q�.Q.

_ __' Figure 130.1 he gme n Execse 130.2.

equilibrium, as does any smmetric game in which each player has initely many acions, by he following result (he proof of which requires relaively advanced mahemaical tools).

• PROPOSITION

130.1

(Existence of smmeric mixed strategy Nash equilibrium in smmeric nite games)

_Evy strategic game with vNM prferences in which each

player has the same inite set of actions has a symmetric mixed strategy Nash equilibrium.

!EXERCISE

130.2

(Approaing cars) Members of a single populaion of car drivers are randoly matched in pairs when hey simultaneously approach intersecions rom diferent directions. n each interaction, each driver can eiher stop or con inue. he drivers' preferences are represented by he expected value of he payof funcions given in Figure

130.1;

he parameter ^E,wih Q

< E <

I, relects he fact hat each driver dislikes being he only one to stop. Find he symmetric Nash ebrium (equilibria?) of the game (ind boh the equilibrium strategies and he equilibrium payoffs). .

Now suppose hat drivers are (re)educated to feel guilty about choosing

Con

tinue,

wih he consequence hat heir payofs when choosing

Continue

fall by

0

> O. hat is, he enry

(2, 1- E ₎

in Figure

130.1

is replaced by

(2 - 0, 1 - E ₎ ,

he enry

(1- E,2 ₎

is replaced by

(1- E, 2 - 0),

and he enry

(0, 0)

is replaced by

(-0, -0).

Show hat all drivers are

better of

in he symmetric equilibrium of

_s

game han hey are in he smmeric equilibrium of he original game. Why is he society better of if everyone feels guilty about being aggressive? (he equilibrium of

_s

game, lke hat of he equilibrium of he game of epert diagnosis n Secion

4.6,

may aracively be nterpreted as represening a steady state in which some mem bers of he populaion always choose one acion and oher members always hoose he oher acion.)

j

^EXERCISE

^130.3

(Bargaing) Pairs of players rom a single populaion bargan over he ivision of a pie of sze

^10.

he members of a pair smultaneously make demands; he possible demands are he nonegaive

ven

integers up to

10. f

he demands sum to

10,

then each player receives her demand; if he demands sum to less

_n 10,

hen each player receives her demand plus half of he pie hat re mains after boh demands have been saisied; if the demands sum to more

_n

10,

hen neiher player receives any payof. Find all he symmeric mixed srategy Nash equilibria n which each player assigns positive probability to at most two demands. (Many siuaions in which each player assigns positive probability to two actions, say

a'

and ^a^"^,can be ruled out as equilibria because when one player uses such a strategy, some acion yields he oher player a payof higher han does one or boh of he acions ^a^'and

a"

.)

4.8 illustation: reporting a crime

131 4.8 Illustration: reporting a crime

A crime is observed by a group of

n

people. Each person would like the I' _b infomed but prefers hat someone else make the phone call Specii llPO Ice to e h at ea person attaches he value ch

_v

to he police being informed d b ^. ca y, suppose

if h ^an ^{ears he}

cost

c

s e makes he phone call, where

_v

>

c

>

0

hen he situai _' _d b h y e followmg srategic game wih vNM preferences. ^. ^. on IS mo eled

Players

_he

_n

_{people .}

Actions

Each player's set of actions is

{Call, Don't call}.

Preferences

.^Ea

^�

player's pr

_�

ferences are represented by he expected value of a payof funcion hat assIgns

0

to he proile in wich no one calls _

to any proile in which she cals, and

_v

to any proile in which at le

_�

s

_�

on

_�

person calls, but she does not.

�

_.^g

�

e is a variant of he one in Exercise

33.1,

with

k

⁼

1.

It has

n

pure Nash eq�bna, m each of which exactly one person calls.

(f

hat person swhes to not calling, her payoff falls from

_{v -} _c

_to

_0;

if any other person switches to callng, her payofls ro

_� _v

to

_v

. ^. ^c. ^� ^f

he members of he group difer in some respect, hen the

_�

e as

_�

erlc

.eqillbna may be compelling as steady states. For example, the sooal norm m which the oldest person in the group makes the phone call is stable.

f

_he

. members of he group eiher do not difer signiicanly or are not aware of any dffere

_�

ces among he

_'

elves-if hey are drawn rom a single homoge neo� p

_?

pul

_�

ion�hen here 5 no way.for them to coordinate, and a symmetric eqilibrlm, m which every player uses he same srategy, is more compelng.

he game has no, symmeric pure Nash equilibrium.

(f

everyone calls, hen any person is better of SWitching to not calling.

f

no one calls, then any person is beter of Switcing to calling.)

However, it has a symmeric mixed srategy equilibrium in which each person calls wih posiive probability less than one.

n

any such equilibrim, each per son's epected payof to callng is equal to her epected payof to not calling. Each person's payof to calling is

_v - c,

and her payof to not calling is

0

if no one else calls and

_v

if at least one oher person calls, so he equilibrium condiion is

or or

v ^{- c}

⁼

⁰

. Pr{ no one else calls} +

_v

. Pr{ at least one oher person calls},

v - ^c

⁼

v· ^(1-

Pr{no one else calls}),

c/v

⁼Pr{no one else calls}.

(131.1)

Denote by

p

he probability wih which each person calls. he probabiliy hat no one else calls is he probability hat every one of he oher

n _- 1

people does not call, namely

(1- _p)n-l.

Thus he equilibrium condition is

_c/v

⁼

(1- p)n-l,

or

p

⁼

1 - (c/v)l/(n-l).

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132 ^Chapter^4.Mixed Strategy Equilibrium

is number

p

is between

0

and I, so we conclude hat he game has a ique smmeric �ed srategy equilibrim, in which each person cals wih probability

l_'(c/v)l/(n-l).

hat is, here is a steady state in which whenever a person is in a group of

n

people facing he situation modeled by he game, she calls wih

probabiliy

1- (c/v)l/(n-l). ^.

How does his equilibrim change as he size of he group increases? We see hat as

ⁿ

increases, he probability

p

hat any given person calls decreases. (As

n

increases,

l/(n -1)

decreases, so hat

(c/v)lI(n-l)

inreases.) What about he probability hat

at least

one person cals? Fix any player

i.

hen he event "no one calls" is he same as he event

"i

does not call and no one

other than i

calls". hus

Pr{no one calls} ⁼Pr{i does not call}· Pr{no one else calls}.

(132.1)

Now, he probability hat any given person calls decreases as

n

increases, or equiv alently he probabiy hat she does not call increases as

n

ncreases. Furher, rom he ebrium condition

(131.1),

Pr{no one else calls} is equal to

c/v, independent

of n.

We conclude hat he probability hat no one calls

increases

as

n

increases. hat is, the larger he group, he

less

likely he police ar:: iformed of he crime!

The condiion deining a ixed srategy equilibrium is responsible for

_s

re sult. For any given person to be indfferent between calling and not callng,

_s

condiion requires hat he probability hat no one else calls be independent of he size of he group. hus each person's probability of not calling is larger in a larger group, and hence, by he laws of probability relected in

(132.1),

he probability hat no one calls is larger in a larger group.

he result hat he larger he group, he less likely any given person calls is not surprising. he result hat he larger he group, he less lkely at least one person calls is a more subtle implicaion of he noion of equilibrium. In a larger group no individual is any less concened hat the police should be caled, but in a steady state he behavior of he group drives down he chance hat he police are noied of he rime.

�

^EXERCISE

^132.2

(Reporing a crme when he winesses are heterogeneous) Con sider ^avariant of he model stuied in

_s

secion n which

_nl

winesses ncur he cost

_cl

to report he crme, and

n2

winesses incur he cost

C2,

where

0 < Cl < V,

o

< C2 < v,

and

nl

+

n2

⁼

n.

Show hat

f _cl

and

C2

are suficiently close, hen he game has a mixed strategy Nash eqibri� in which every winess's sr�tegy assigns posiive probabiliies to boh reporing and not reporting.

}

EXERCISE

132.3

(Contributing to a public good) Consider an extension of he anal ysis in is secion to he game in Exercise

33.1

for

k :: 2.

(In is case a player may contribute even houh he good is not provided; he player's payoff in

_s

case is

-c,)

Denote by

_Qn-l,m(P)

he probability hat exactly

m

of a group of

n -1

players conribute when each player conributes wih probability

p.

What condiion must be satisied by

Qn-l,k-l(P)

in a symeric mixed srategy equilib im (in wich each player conributes wih he same probability)? (When does

4.8 illustration: reporting a crime

133 a player'S conribuion make a diference to he outcome?) For he case ^{v :} ¹

n

⁼

4, k

⁼

2,

and

c

⁼

_�

,nd he eqibria eplicitly. (You need to use he fact tha

_;

Q3,1(P)

⁼

3p(1-p)2,

and do a bit of algebra.)

REPORTING A CRIME: SOCIAL PSYCHOLOGY AND GAME THEORY

y-eight people winessed he brutal murder of Caherine

("iy")

_Genovese over a period of half an hour in New York City in March

194.

During his period no one sigicanly responded to her sreams for help; no one even called h

_�

police. Jonaliss, psychiarists, soCiologists, and ohers subsequently sruggled to nderstand he winesses' inacion. Some ascribed t to apahy engendered by life in a large city: "Indfference to one's neighbor and

s

roubles is a conditioned relex of fe in New York as it is in other big ciies" (Rosenhal

194, 81-82).

he event paricularly interested social psychologists. It led them to try to un derstand he circmstances under which a bystander would help someone in trou ble. Experiments quickly suggested that, conrary to the popular theory, people even hose living in large ciies-are not in general apathetic to ohers' plights. An epermental subject who is he lone winess of a person n disress is very likely to

y

to help. But as he size of he group of winesses inreases, here is a decline not only in he probability hat any iven one of hem ofers assistance, but also n he probability hat at least one of hem ofers assistance. Soial psychologists hypohesize hat hree factors explain hese experimental indings. First," difu sion of responsibiity": he larger he group, the lower he psychological cost of not helpng. Second, "audience ibition": he larger he group, he greater he ebarrassment sufered by a helper in case he event

_s

out to be one in wih help is inappropriate (because, for example, t is not in fact an emergency). ird,

"social inluence": a person infers he appropriateness of helping rom ohers' be havior, so hat in a large group everyone else's lack of intervenion leads any given person to

^k

ntervenion is less likely to be appropriate.

In terms of he model in Secion 4.8, hese hree factors raise he expected cost and/or reduce he expected beneit of a person's intervening. hey all seem plausible. However, hey are not needed to explain he phenomenon: our game heoreic analysis shows hat even

f

he cost and beneit are

indpendent

of group size, a decrease in he probability hat at least one person ntervenes is an implica ion of equlibrim. is game-heoreic analysis has an advantage over he socio psyhological one: it derives he conclusion rom he same principles hat underlie all he oher models studied so far (oigopoly, aucions, voing, and elections, for example), raher han posing special features of he specic environment in wich a group of bystanders may come to he aid of a person in disress.

he criical element missing rom he socio-psychological analysiS is he notion of an

equilibrium.

Wheher any given person intervenes depends on he probabiliy she assigns to some oher person's ntervening. n an equilibrium each person

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134 ^Chapter^4.Mixed Stategy Equilibrium

must be ndfferent beween intervng nd not ntervng, and as we have seen

is condition leads inexorably to he conclusion at an nrease in group size

reduces he probability hat at least one person intervenes.

4.9 The formation of players' beliefs

n

a Nash eqilibrium, each player chooses a srategy hat maxizes h _� r expected

payof, knowing the oher players' strategies. So far we _� ave not co _: ldered how

players may 'acquire such information. nformally, he Id _� a under _� ymg he pr _� -,

vious analysis is hat he players have leaned each oher s strategIes ro _� herr

experience playng he game.

n

he idealized situation to wi _� he _� al _! S _� S cor

responds, for eah player in he game here is a large populaion of m _� l _� du _� s

who may tke he role of hat player; n any play of he game, one pariClpant IS

dran randoly rom each population.

n

is situaion, a new ndividual who

joins a populaion hat is in a steady state (i.e. is using a Nash equilibrium srategy

prole) cn learn he oher players' strategies by obsev _� g heir acions over _�� y

plays of he gme. As long as he nover in players

⁵

. ^s �n ^enough, � ^1S�g

players' encounters wih neophytes (who may use noneq�bnum str _� tegIes) will

be sficiently rare hat heir beliefs about he steady state _ll not be disturbed, so

hat a new player'S problem is simply to lean he oher players' acions. _.

s

analysis leaves open he quesion of what might happen

f

new players

simltaneously join more hn one population in suicient numbers hat hey have

a siicant chance of facng opponents who are themselves new. n paricular,

can we expect a steady state to be reached when no one has experience playng he

game?

4.9.1

Eliminating dominated actions

n

some games he players may reasonably be expected to choose heir Nash equi

librium acions rom an ntrospecive analysis of he game. At an exreme, each

player'S best acion may be ndependent of he oher players' actions, as in he

Prisoner's Dilemma (Example 14.1).

n

such a game no player needs to worry about

e oher players' actions.

n

a less exreme case, some player's best acion may

depend on he oher players' acions, but he acios he oher players _ll hoose

may be clear because each of hese players has an acion hat srictly donates

all ohers. For example, n he game in Figure 135.1, player 2's action _R sricly

donates _L, so hat no mater what player 2 s player 1 _ll do, she should

choose _R. Consequently, player

I,

who can deduce by _s argument hat player 2

will choose _R, may reason hat she should choose B. hat is, even nexperienced

players may be led to he nique Nash equilibrium ₍ B, _R) in his game.

his line of argument may be extended. For example, in he game n Fig

ure 135.2, player l's acion

T

is stricly dominated, so player 1 may reason hat

4.9 The formation of players' beliefs

L R

T _/ ^1,2 ! ^0,3 _I

B _[§JILIJ

Figure 135.1 A game n which player 2 has a sricly dominant acion and player 1 does _not.

L

_R

T ! ^0,2 _I 0,0 ^I

M _[,T[C2--j

B _L!!;�J

135

Figure 135.2 A game

,n which player 1 may reason hat she should choose B because player 2 wll reason hat player 1 will not choose T, so that player 2

_ll

choose R

player 2 wll deduce hat player 1 _ll not choose

T.

Consequently player 1 may

deduce hat player 2 will choose _R, making B a better acion for her than M.

he set of acion proiles hat reman at he end of such a reasong process

contns all Nash eqbia; for many games (nlike hese examples) he set con

tains many oher acion proiles as well.

n

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St. Continue

Stop I '--i, 1 -- £ - 1 - E, u

I

Continue

.�=_�_,

Q�.Q.

130.1

Evy strategic game with vNM prferences in which each

player has the same inite set of actions has a symmetric mixed strategy Nash equilibrium.

130.2

130.1;

< E <

Con­

tinue,

Continue

0

(2, 1- E )

130.1

(2 - 0, 1 - E ) ,

(1- E,2 )

(1- E, 2 - 0),

(0, 0)

(-0, -0).

better of

s

s

4.6,

j

130.3

10.

ven

10. f

10,

n 10,

n

10,

a'

a"

n

v

c

v

c

0

Players

n

Actions

{Call, Don't call}.

Preferences

�

�

0

v

�

�

�

�

�

33.1,

k

1.

n

(f

v - c

0;

� v

v

. . c. � f

�

�

f

�

'

?

�

(f

f

n

v - c,

Stop _I ^'--i, 1 ^-- _£ ^- 1 - E, _u

_{.�=_�_,}

_Q�.Q.

_Evy strategic game with vNM prferences in which each

Con

(2, 1- E ₎

(2 - 0, 1 - E ₎ ,

(1- E,2 ₎

_s

_s

^130.3

^10.

_n 10,

_n

_v

_v

_n

^�

_�

_v

_�

_�

_�

_{v -} _c

_0;

_� _v

_v

. ^. ^c. ^� ^f

_�

_�

_�

_'

_?

_�

_v - c,

_v

v ^{- c}

⁰

_v

v - ^c

v· ^(1-

n _- 1

(1- _p)n-l.

_c/v

1- (c/v)l/(n-l). ^.

ⁿ

_s

_s

^132.2

_s

_nl

_cl

f _cl

_s

_Qn-l,m(P)