Competitive Equilibrium with Indivisibilities

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Ma, Jinpeng

Working Paper Competitive Equilibrium with Indivisibilities

Working Paper, No. 1998-09

Provided in Cooperation with: Department of Economics, Rutgers University

Suggested Citation: Ma, Jinpeng (1998) : Competitive Equilibrium with Indivisibilities, Working Paper, No. 1998-09, Rutgers University, Department of Economics, New Brunswick, NJ

This Version is available at: http://hdl.handle.net/10419/94341

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Comp etitive Equilibrium with Indivisibilities

Jinp eng Ma

Rutgers University

Revised January 25, 1998

Abstract

This pap er studies an exchange economy with a nite numb er of agents in which each

agent is initially endowed with a nite numb er of p ersonalized indivisible commo dities. We

observe that the equivalence theorem of the core and the comp etitive equilibrium may not hold

for this economy when the coalitional form game is generated in the standard manner. We

provide an alternative de nition of the coalitional form game to resolve this problem so that

the balancedness of the new de ned game provides a useful necessary and sucient condition

for the existence of comp etitive equilibrium for the original economy.

We also observe that the nice strategy pro of prop erty of the minimum comp etitive price

mechanism in the assignment problem and the Vickrey auction mo del do es not carry over to the

ab ove economy.We show that examples of exchange economies exist for which no comp etitive

price mechanism is individually coalitionally strategy pro of.

JEL classi cation numb ers: D41, D44, D50

I thank Sushil Bikhchandani , Vince Crawford and editor Karl Shell for helpful comments and suggestions. I

am grateful to an asso ciate editor for many fruitful comments that greatly improve the exp osition of the pap er. Of

course, any remaining errors are mine.

Department of Economics, Rutgers University, Camden, NJ 08102, jinp [email protected] 1

1 Intro duction

We consider an exchange economy with a nite numb er of agents. Each agent is initially endowed

with a nite numb er of indivisible homogeneous or heterogeneous commo dities and may consume

as many items as he wishes. An agent's utility function is a set function over the set of all

commo dity bundles. This exchange economy is fairly broad. Some related examples of it include

the exchange economy in Bikhchandani and Mamer [1] and Gul and Stacchetti [7], the assignment

problem Ko opmans and Beckman [12], Shapley and Shubik [20] and the auction mo del in Vickrey

[21]. It is also analogous to the job-matching market in Kelso and Crawford [11] .

It is known that the assignment problem has a comp etitive equilibrium. Indeed Shapley and

Shubik [20] de ned a market game for the assignment problem and showed that the market game is

totally balanced and thus has a nonempty core. It follows from the core equivalence theorem that

a comp etitive equilibrium exists in the assignment problem. This existence result dep ends on the

assumption that each buyer consumes at most one unit. If buyers can consume as many as they wish,

then a comp etitive equilibrium may not exist; see Bikhchandani and Mamer [1], Kelso and Crawford

[11] and Section 3 for examples. This motivates a natural question: Under what circumstance do es

a comp etitive equilibrium exist under the general situation? Kelso and Crawford [11] studied a

job-matching market and discovered that if rm's utility function satis es their gross substitutes

condition, their matching mo del has a nonempty core and thus has a comp etitive equilibrium since

a core matching is also comp etitive Kelso and Crawford [11, pp.1487 & 1502]. Bikhchandani

and Mamer [1] used a di erent approach and obtained a necessary and sucient condition for the

existence of comp etitive equilibrium. In their approach, they designed two linear programmings

and showed that a comp etitive equilibrium exists if and only if the two linear programmings have

solutions in common. Lately, Gul and Stacchetti [7] found two new sucient conditions, the no

complementarities and the single improvement prop erty, on buyer's utility function for the existence

of comp etitive equilibrium. They showed that their two new conditions are equivalent to the gross

substitutes condition in Kelso and Crawford [11] and then invoked the nonempty core theorem in

Kelso and Crawford [11] to obtain their existence theorem.

This pap er uses the core approach in Shapley and Shubik [20] to study the same existence issue.

We use the economy to generate a coalitional form game and wanttoshow that the balancedness

of the game provides a useful necessary and sucient condition for the existence. Typically the

numb er of players in a coalitional form game generated by an economy is the same as the number

But it should b e aware that a worker's utility function in Kelso and Crawford [11] is more complicated and it

dep ends on not only the received salary but also the name of a rm with whom he is matched. This is quite di erent

from the exchange economy of this pap er in which agents are not concerned with the names of the owners of the

commo dities.

Gale [5], Kaneko [8,9], Kaneko and Yamamoto [10] and Quinzii [15] obtained several existence theorems for quite

general situations with indivisi bl e go o ds. 2

of agents in the economy. But we observe that when the coalitional form game is generated in this

standard manner, its core do es not coincide with the set of comp etitivepayo s and the balancedness

do es not provide sucient information on the existence of comp etitive equilibrium for the original

economy; as shown in Example 1. To resolve the problem, we provide an alternativeway to generate

a coalitional form game. The approach is as follows. Supp ose that an economy has one agent i with

no indivisible commo dities and one agent j with two indivisible commo dities, 1 and 2. Given such

an economy,we consider a bilateral exchange economy with four \agents", i; j; 1 and 2, such that

agents i and j have the same utility functions as they do in the original economy but b oth haveno

indivisible commo dities even though agent j owns two commo dities in the original economy, and

commo dity \agents" 1 and 2 have zero utility functions but own commo dities 1 and 2 resp ectively.

Then we use this bilateral exchange economy to generate a coalitional form game for the original

exchange economy. It follows from the argument made in Kelso and Crawford [11, pp.1487 & 1502]

that the core equivalence theorem holds for this bilateral exchange economy, i.e., the core of the

coalitional form game coincides with the set of comp etitive equilibrium payo s. Our observation is

that a comp etitive equilibrium exists in the bilateral exchange economy if and only if it exists in

the original exchange economy. Therefore a comp etitive equilibrium in the original economy exists

if and only if the generated game on fi; j; 1; 2g is balanced, since the balancedness of the game is

the necessary and sucient condition for it to have a nonempty core Bondareva [2], Shapley [18].

The ab ove idea is in fact not completely new and it has b een implicitly used in Kelso and

Crawford [11] b efore. In their one-sided \matching" market, they start with the coalitional form

game and construct a job-matching market by adding dummy buyers, each with an identical utility

function. Here we consider an exchange economy with heterogeneous utility functions and construct

a bilateral exchange economy to generate a game by adding dummy commo dity sellers. This

approach is quite useful b ecause it delivers information on existence or nonexistence of a comp etitive

equilibrium at the same time. Moreover, we can say exactly what are the comp etitive prices in

the original exchange economy. They are nothing but the core payo s of the coalitional form game

\received" by those commo dity \sellers".

Our second concern is the incentive asp ects of comp etitive price mechanisms. A mechanism

is individually coalitionally strategy pro of if it is always the b est for each agent each agentin

each coalition to reveal his true information. It is well-known that the minimum comp etitive

price mechanism in the auction mo del in Vickrey [21] exists and it is individually strategy pro of.

Demange [3] and Leonard [13] studied the assignment problem and indep endently showed that

the minimum comp etitive price mechanism is individually strategy pro of. Demange and Gale

[6] studied a generalized assignment problem and showed that the minimum comp etitive price

mechanism exists and it is individually strategy pro of as well. The minimum comp etitive price

mechanism is coalitionally strategy pro of in these mo dels if side payments are not allowed. These 3

three mo dels and their outcomes are bilateral in nature. One maywonder if this bilateral feature

is the source of the nice incentive prop erty of the minimum comp etitive price mechanism. We will

show that this is not the case. Indeed we use a bilateral exchange economy and show that the

minimum comp etitive price mechanism exists but it is not immune to the misrepresentation of the

true information by individuals or coalitions. In fact we show that examples of exchange economies

exist for which no comp etitive price mechanism is individually or coalitionally strategy pro of.

Therefore, our result reveals that it is the assumption of unit demand in the assignment problem

and the Vickrey auction mo del that contributes to the strategy pro of prop erty of the minimum

comp etitive price mechanism. The intuition is as follows: An agent in the current mo del has the

opp ortunity to misrepresent his utility function to undercut the commo dity bundles in his demand

corresp ondence. This typ e of misrepresentation decreases comp etition in the demand side and thus

lowers the comp etitive prices for his consumption bundle. Since coalitional misrepresentation do es

not need any side payments in the current mo del, our result may also provide a new insight for

the formation of bidder rings, a very fact in the auction practice which is often interpreted as a

by-pro duct of side payments or of rep eated auctions.

This pap er is structured as follows. Section 2 de nes the exchange economy. Section 3 proves

the main result on the existence of comp etitive equilibrium. Section 4 studies the incentive prop-

erties of comp etitive price mechanisms.

2 The Exchange Economy

Let N = f1; 2; ;ng denote the set of agents. Each agent i 2 N is initially endowed with a nite

numb er of indivisible commo dities = f! ; ;! ; ;! g, where k 2f0; 1; 2; g is a nite

i i1 ij ik i

integer. Thus some agents may not have initial physical endowments k = 0 and some agents may

j =k

i=n

have units of them k 1. Let = [ [ ! denote the set of all commo dities in the economy

i ij

j =1 i=1

and 2 denote the set of all commo dity bundles. For an agent i 2 N , his utility function u is a

set function u :2 ! R satisfying u ; = 0. Henceforth we assume that agents' utility functions

i i

are weakly monotone. Following Bikhchandani and Mamer [1] and Gul and Stacchetti [7], we

assume that each agent i is initially endowed with wealth W u that enables him to buy any

i i

commo dity bundle A . We denote this economyby E .

Vickrey noted its imp ortance for the incentive prop erty of his second-price auction mechanism to hold. However,

he did not provide evidence how buyers may misrepresent the mechanism if it is not satis ed. The auction literature

thereafter often considers the situation that there is one item for sale.

A utility function u is weakly monotone if for all A; B 2 such that A B , uA uB . Free disp osal is a

sucient condition for this assumption.

The NBA lab or market may provide a go o d example for this economy if players only care ab out money. In the

NBA lab or market a player maybeowned by a team or by himself, and a team typically owns many players. In this

lab or market, a player is a seller but a team may b e a seller or a buyer or b oth. The other example of this economy

is the secondary sale market for the sp ectrum licenses in which rms may trade their licenses after they obtain the 4

A feasible allo cation Y in the economy E is a partition Y 1; ;Y n of , where agent i

is allo cated the commo dity bundle Y i. Weintro duce some notation from Gul and Stacchetti [7]

b elow. Let T = f1; 2; ;tg b e a set and x 2 R beavector, de ne

= x ; where A T:

a2A

j j j j

! R is de ned by , agent i's trading pro t function v :2 R Given a price vector P 2 R

+ +

v A; P = u A+ < ;P >

i i i

j j

and his demand corresp ondence D : R ! 2 is de ned by

D P =fA : v A; P v B; P ; 8B g:

i i i

A pair Y; P of a feasible allo cation Y and a price vector P is a comp etitiveWalrasian

equilibrium if Y i 2 D P for all agents i 2 N .At a comp etitive equilibrium, each agent obtains

the greatest trading pro t.

De ne v P = u A+ < ;P > for A 2 D P and v P = v P ; ;v P :

i i i i 1 n

Let

j j jN j

W = fP; v P 2 R R : P is comp etitiveg

denote the set of all comp etitive equilibrium payo s.

3 The Existence of Comp etitive Equilib ri um

An economy can b e used to generate a coalitional form game. Typically the set of agents in the

coalitional form game is the same as in the economy. In many economic situations the equivalence

theorem of the core and the comp etitive equilibria holds, i.e., the core of the coalitional form game

coincides with the set of comp etitive equilibrium payo s. Therefore a sucient condition for the

nonempty core provides a useful condition for the existence of comp etitive equilibrium, and vice

versa. But we observe that the equivalence theorem fails in the bilateral exchange economyin

Bikhchandani and Mamer [1] and Gul and Stacchetti [7] and in the exchange economy E in Section

2 when the coalitional form game is generated in the standard manner.

Example 1 Consider the following economy with three agents, i, j and k . Agent i is endowed

with three ob jects, 1, 2 and 3, and has zero utilityover any bundle of the three ob jects. Agents j

and k have utility functions as follows see Kelso and Crawford [11]:

licenses from the auction; see McMillan [14] for an example. 5

u f1g= 4; u f2g= 4; u f3g= 4+

j j j 1

u f1; 2g= 7+; u f1; 3g= 7; u f2; 3g= 7

j j j

u f1; 2; 3g= 9

u f1g= 4+ ; u f2g= 4; u f3g= 4

k 2 k k

u f1; 2g= 7; u f1; 3g= 7; u f2; 3g= 7+

k k k

u f1; 2; 3g= 9

where 2 [0; 1] and ; 2 [0; 3]. Clearly, these utility functions are weakly monotone.

1 2

Wenow de ne the coalitional form game w on fi; j; k g in the standard manner. Then wehave

that

w fig= w fj g= w fk g= w fj; k g= 0;wfi; j g= w fi; k g= 9

and

11 + 0 ; 0

1 2

w fi; j; kg= 11 + ;

1 1 2 1

11 + ; .

2 2 1 2

The core C w is nonempty since w fi; j; k g 2; 1; 1 is in C w for any , and . But

1 2

1 1

we knew from Kelso and Crawford [11] that the economy with parameters = = and =

1 2

4 2

do es not have a comp etitive equilibrium. Therefore, this example shows that the core equivalence

theorem fails for the coalitional form game w . The balancedness condition of the game w is not

sucient for the existence of comp etitive equilibrium. Q.E.D.

To resolve the problem, we construct from the original exchange economy E a bilateral one E

such that N and are the two disjoint sets of agents in E such that each agent i in N has the same

utility functions as he do es in E but with no initial endowments and each agent ! 2 owns one

single indivisible ob ject, namely the ob ject itself, but she has zero utility functions. We use the

economy E to generate a coalitional form game V and make use of the core of this game to study

the existence of comp etitive equilibrium in the economy E .

De nition A feasible allo cation in E is a map X : [ N ! 2 such that i for all ! 2 ,

X ! 2 N [f! g; ii for all ! 2 such that X ! 62 N , X ! = ! ; iii for all i 2 N , X i ;

iv for all ! 2 and i 2 N , X ! = i if and only if ! 2 X i.

Let T ;N denote the set of all feasible allo cations in E . Given a subset S N [ , let

T S \ ;S \ N denote the set of all feasible allo cations for the coalition S . Also let D P denote

the demand corresp ondence in the economy E .Thus a pair X; P of a feasible allo cation X and a 6

price vector P is a comp etitive equilibrium if i P 0; ii P = 0 for all ! such that X ! = ! ;

and iii X i 2 D P for all i 2 N .

Let

j j jN j

~ ~

W = fP; v P 2 R R : P is comp etitivein Eg

+ +

denote the set of comp etitive equilibrium payo s in the economy E , where v P =u A

i i

for A 2 D P .

Wenow use the economy E to generate a coalitional form game V .For S [ N , de ne

V S = max u X i;

i S

X 2T S \ ;S \N

i2S \N

satisfying V ;= 0. V S is the maximum amount of utilities generated by the coalition S . This

game V is a natural extension of the coalitional form game in Shapley and Shubik [20].

j j jN j

The core C V of the game V consists of all payo vectors x; y 2 R R such that

< ;x > + = V [ N ;

+ V S for all S [ N:

Our observation is quite simple. First, it follows from Kelso and Crawford [11, pp.1487 & 1502]

that the core equivalence theorem holds in the economy E when the coalitional form game V is

de ned on N [ asabove. Second, we show that a comp etitive equilibrium exists in E if and

only if it exists in E . Finally, it follows from Bondareva [2] and Shapley [18] that the core C V is

nonempty if and only if the game V is balanced.

Theorem 1 Assume that u is weakly monotone for all i 2 N . Then a comp etitive equilibrium

exists in E if and only if V is balanced.

Pro of. In fact Step 1 b elow follows from Kelso and Crawford [11, pp.1487 & 1502]. Here we

provide a pro of for the sake of completeness.

Step1. C V = W .

Let x; y 2 C V . It follows from the de nition of the core C V that there exists an optimal

allo cation Y in the economy E such that

X X

[y + ]= V i [ Y i = V [ N :

i2N i2N

It follows from the core that for all A ,

y + u A:

i i 7

In particular,

y + u Y i = V i [ Y i

i i

for all i 2 N . Therefore,

y + = V i [ Y i

for all i 2 N . Clearly x 0. It follows that Y; x is a comp etitive equilibrium in E .

Let P; v P 2 W .We show that P; v P is in the core C V . Clearly,

< ;P > + = V [ N

since any comp etitive allo cation Y must b e optimal. Now supp ose that there exists a coalition

S [ N such that

+ < V S :

It follows that 9i 2 S \ N and 9A S \ such that

+v P < V i [ A = u A:

i i

This implies that v P , contradicting P is comp etitive.

i i

Step 2. W 6= ; i W 6= ;.

First note that every comp etitive price vector in the economies E and E is nonnegative. This

follows from the assumption that agents haveweakly monotone utility functions. Let S be

any commo dity bundle. Then for all agents i 2 N ,

u S + < ;P > u C + < ;P >

i i i i

if and only if

u S u C

i i

for all C .

Let Y; P b e a comp etitive equilibrium in the economy E . Since Y is a partition, Y is an

~ ~

allo cation in E . Because Y i 2 D P for all i 2 N , it follows from * that Y i 2 D P for all

i i

i 2 N .Thus Y; P is a comp etitive equilibrium in the economy E .

Let X; P b e a comp etitive equilibrium in the economy E . Then by de nition, wehave

i P = 0 for all ! such that X ! = ! and ii X i 2 D P for all i 2 N . It follows from *

! j j j i

and ii that X i 2 D P for all i 2 N . Let W = f! 2 : X ! = ! g and Y i= X i [ W .

i i j i j j i

Clearly Y is a partition of . Since X i 2 D P and u is weakly monotone for all i 2 N ,it

i i

follows that Y i 2 D P for all i 2 N note that P = 0 for every w 2 W . Hence Y; P isa

i w i

comp etitive equilibrium in the economy E . 8

Finally, it follows from Bondareva [3] and Shapley [18] that the core C V is nonempty if and

only if the game V is balanced. This together with Steps 1 and 2 completes the pro of. Q.E.D.

Theorem 1 shows that the balancedness of the game V do es provide a useful necessary and

sucient condition for the existence of comp etitive equilibrium in the exchange economy E .From

the construction of the economy E ,we know precisely what is a comp etitive price vector. A

comp etitive price vector in the economy E is nothing but a core payo vector received by the

commo dities \sellers" in the game V .

Next we apply Theorem 1 to study Example 1 to nd out which set of parameters admits a

comp etitive equilibrium and which do es not. This example may b e helpful to showwhy Theorem

1 is useful. The game V for Example 1 is de ned as follows:

V fj; C g= u C for all C

V fk; C g= u C for all C

4+ C = f1g

4 C = f2g

4+ C = f3g

V fC ; j; kg=

8+ C = f1; 2g

8+ + C = f1; 3g

1 2

8+ C = f2; 3g

V [fj; k g= w fi; j; k g

and V C = 0 for all C orC N , where = f1; 2; 3g and w fi; j; k g is de ned as in Example

1. De ne F = f0; 0 [f ; 2 [0; 3] [0; 3] :j j 2gg:

1 2 1 2

u -

3 2

Figure 1 F in Claim ii.

Claim i The game V ab ove is balanced if and only if

1 1 1

V f1; 3;j;kg+ V f1; 2;jg+ V f2; 3;kg V [ N ;

2 2 2 9

ii A comp etitive equilibrium exists in Example 1 if and only if ; 2F.

1 2

Claim ii is an application of Claim i and Theorem 1. The sets of parameters in Example

1 that admit a comp etitive equilibrium consist of the two triangles and the origin in Figure 1

describ ed by the set F .Any other parameters do not have a comp etitive equilibrium.

1 1

Consider the economy E in Example 1 with = = and = . Kelso and Crawford [11]

1 2

4 2

showed that the constructed economy E has an empty core they showed this by the core condi-

tions. Therefore the economy E do es not have a comp etitive equilibrium. Their result together

with Step 2 shows that the economy E in Example 1 do es not have a comp etitive equilibrium.

1 1

Alternatively, one can use the condition in Claim i to show that when = = and = , the

1 2

4 2

game V is not balanced. Therefore a comp etitive equilibrium do es not exist with these parameters.

Pro of of Claim i. Since the game V is sup eradditive, Theorem 3 in Shapley [18] shows

that the core C V 6= ; i the game V is balanced for every prop er minimal balanced collection.

To nd all prop er minimal balanced collections, we consider a prop er balanced collection C =

fC ; C ; C g of [ N such that each elementof C , C and C contains fj; k g, fj g

fj;k g fj g fk g fj;k g fj g fk g

and fk g, resp ectively. Let x; y and z b e the balanced weightof fj; k g of the collection C , the

fj;k g

balanced weightoffj g of the collection C and the balanced weightoffk g of the collection C ,

fj g fk g

resp ectively. It follows that y = z since x + y =1=x + z . Clearly x> 0;y > 0 and z>0 since

C is prop er. Thus each C , C and C must contain at least one element. Now it follows

fj;k g fj g fk g

from Shapley [18] that ff1; 2g; f1; 3g; f2; 3gg is the unique minimal balanced collection of f1; 2; 3g:

Therefore, a minimal balanced collection C that is also prop er must b e one of such one-to-one maps

: ffj; k g; fj g; fkgg ! ff1; 2g; f1; 3g; f2; 3gg:

Thus, wemust have x = y = z = . It follows that V is balanced if and only if

1 1 1

V f1; 3;j;kg+ V f1; 2;jg+ V f2; 3;kg V [ N ;

2 2 2

since for any coalition C such that C 6= ,V fC ; j; kg V f1; 3;j;kg, V fC; j g

V f1; 2;jg and V fC; kg V f2; 3;kg. Q.E.D.

In Shapley [18] it is shown that the core is nonempty for any sup eradditive three players

f1; 2; 3g coalitional form gamew ~ if and only if

1 1 1

w~ f1; 3g+ w~ f1; 2g+ w~ f2; 3g w~ f1; 2; 3g:

2 2 2

Our condition ab ove for the economy in Example 1 is analogous to the result in Shapley [18]. In

any general exchange economy with N = fj; k g and = f1; 2; 3g, it is sucient and necessary to

check six such inequalities see the map ab ove for the existence of comp etitive equilibrium.

A balanced collection is prop er if no two elements in it are disjoint Shapley [18]. 10

In the close of this section we remark that if agents havemultiple units of the same go o d or

di erent agents have the same go o d, then Theorem 1 still holds. This is due to two observations.

The rst one is that multiple copies of the same go o d will have the same price at any comp etitive

equilibrium. Otherwise there exist arbitrage opp ortunities at equilibrium. The second one is that

each copy, as a commo dity seller, of the same go o d will have the same payo in all core outcomes of

the game V . This is b ecause all copies of the same go o d are symmetric in the game V and any core

outcome will assign the same payo to these symmetric sellers according to the core conditions.

Therefore the conditions that identical go o ds would havetohave the same price will b e satis ed

at all comp etitive equilibria and in all core outcomes. Also see Bikhchandani and Mamer [1] for a

detailed discussion.

4 Misrepresentation

Let U denote the set of all pro les of weakly monotone utility functions such that a comp etitive

equilibrium exists in the economy E u for every u 2U . Given an exchange economy E u, a

comp etitive price vector P is the minimum comp etitive price vector if it satis es P P for every

comp etitive price vector P in the economy. A mechanism '; P is a comp etitive price mechanism if

'u;Pu is a comp etitive equilibrium for each pro le u 2U and it is the minimum comp etitive

price mechanism if P u is the minimum comp etitive price vector for each pro le u 2U . Similarly

one may de ne the maximum comp etitive price vector and the maximum comp etitive price mech-

anism. A mechanism '; P induces a strategic form game and it is individually coalitionally

strategy pro of on the domain U if it is always the b est for every agent every memb er in each

coalition to reveal his/her true utility function u for every pro le u 2U .

In the Vickrey auction mo del and the assignment problem, the minimum comp etitive price

mechanism is individually strategy pro of and it is also coalitionally strategy pro of if side payments

are prohibited. We show that the nice incentive prop erties of the minimum comp etitive price mech-

anism in these two mo dels do not carry over to the current exchange economy. Indeed examples of

exchange economies exist for which no comp etitive price mechanism is individually coalitionally

strategy pro of.

Prop osition 1 There exists an exchange economy such that no comp etitive price mechanism

is individually coalitionally strategy pro of.

Pro of. Let =0, = 0 and = in Example 1. Also let u =u ;u ;u b e the underlying

1 2 i j k

true pro le of utility functions de ned in Example 1 with these parameters. Note that this economy

E u is bilateral in nature as in the assignment problem. 11

There are two optimal allo cations Y and Y in the economy E u de ned by

0 0

Y j = f1; 2g;Y j = f1g and Y k = f3g;Y k = f2; 3g:

Let P be any comp etitive price vector in the economy E u. It follows from Y j ;Y j 2 D P

. It also follows that u f1; 2g P P = u f1g P .Thus P = u f1; 2g u f1g= 3

j 1 2 j 1 2 j j

from Y j 2 D P that u f1g P u f1; 3g P P .Thus P 3. By symmetry,wehave

j j 1 j 1 3 3

that P 3 and P = P .We claim that P =3; 3 ; 3 is the minimum comp etitive price vector.

1 1 3

To see this, it is sucienttoshow that P =3; 3 ; 3 is comp etitive. But this is an easy task to

check.

At the minimum comp etitive price vector P , agents j and k each obtain trading pro ts 1.

Since they are buyers not sellers, the minimum comp etitive price mechanism is the most favorable

one to them. Thus the trading pro ts for agents j and k are at most 1 for any comp etitive price

mechanism when they rep ort truthfully.

Given agents i and k rep orting their truth, we show that agent j can obtain more than 1 for

any comp etitive price mechanism by misrep orting her utility functions as follows:

u~ f1g=u ~ f2g=u ~ f3g= 1

j j j

u~ f1; 2g=u ~ f1; 3g=u ~ f2; 3g= 3; u~ f1; 2; 3g= 4:5:

j j j j

Note thatu ~ is monotone. Letu ~ =u ; u~ ;u . There are two optimal allo cations X and X in the

j i j k

economy E ~u de ned by

0 0

X j = ;;X j = f1g and X k = f1; 2; 3g;X k = f2; 3g:

Note that agent j reduces the comp etition in the demand side by such misrepresentation.

Let Q be any comp etitive price vector in the economy E ~u. It follows from X k ;X k 2 D Q

, and that Q =1

1 1

Q Q Q + and Q Q Q + :

1 2 1 1 3 1

2 2

Thus agent j obtains true trading pro ts 2 in the economy E ~u for any comp etitive price

mechanism. But she obtains at most 1 when she reveals her true utility functions for any comp etitive

price mechanism in the economy E u. This shows that no comp etitive price mechanism in the

economy E u is individually strategy pro of.

Wenow show that no comp etitive price mechanism in E u is also coalitionally strategy pro of.

; 2; 2 is comp etitive, it must b e the maximum comp etitive price vector in the economy Since 1

E ~u, at which agent k obtains trading pro ts 3 . Since agent k is a buyer not a seller, he can

obtain at least 3 for any comp etitive price mechanism in the economy E ~u. But agent k obtains

at most 1 in the economy E u when agent j rep orts her true utility functions. Thus agent k gets 12

b etter o when agent j misrep orts her utility functions by~u for any comp etitive price mechanism

in the economy E ~u. Thus each memb er in the coalition fj; k g can do b etter for any comp etitive

price mechanism when agent j misrep orts her utility functions by~u , without side payments. This

completes the pro of. Q.E.D. 13

References

[1] Sushil Bikhchandani and John W. Mamer, Comp etitive Equilibrium in an Exchange Economy

with Indivisibilities, Anderson Graduate Scho ol of Management, UCLA, 1994. Mimeo. To

app ear in Journal of Economic Theory.

[2] O.N. Bondareva, Some Applications of Linear Programming Metho ds to the Theory of Co op-

erative Games, Problemy Kybernetiki 10 1963, 119-139.

[3] Gabrielle Demange, Strategypro ofness in the Assignment Market Game, Lab oratoire

d'Econometrie de l'Ecole Polytechnique, Paris, 1982. Mimeo.

[4] Gabrielle Demange and David Gale, The Strategy Structure of Two-sided Matching Markets,

Econometrica 53 1985, 873-88.

[5] David Gale, Equilibrium in a Discrete Exchange Economy with Money, Int. J. of Game Theory

13 1984, 61-4.

[6] David Gale and Lloyd Shapley, College Admissions and the Stability of Marriage, Amer. Math.

Monthly 69 1962, 9-15.

[7] Faruk Gul and Ennio Stacchetti, Walrasian Equilibrium without Comp ementarities, Princeton

University, 1996. Mimeo.

[8] Mamoru Kaneko, The Central Assignment Game and the assignment Markets, J. of Math.

Econ. 10 1982, 205-32.

[9] Mamoru Kaneko, Housing Markets with Indivisibilities, J. of Urban Econ. 13 1983, 22-50.

[10] Mamoru Kaneko and Yoshitsugu Yamamoto, The Existence and Computation of Comp etitive

Equilibria in Markets with an Indivisible Commo dity, J. of Econ. Theory 38 1986, 118-36.

[11] Alexander S. Kelso, Jr. and VincentP. Crawford, Job Matching, Coalition Formation, and

Gross Substitutes, Econometrica 50 1982, 1483-1504.

[12] T. Ko opmans and M. Beckman, Assignment Problems and the Lo cation of Economic Activi-

ties, Econometrica 25 1957, 53-76.

[13] Herman B. Leonard, Elicitation of Honest Preferences for the Assignment of Individuals to

Positions, J. of Polit. Econ. 91 1983, 461-79.

[14] John McMillan, Selling Sp ectrum Rights, The J. of Econ. Perspectives 8 1994, 145-162. 14

[15] Martine Quinzii, Core and Comp etitive Equilibria with Indivisibilities, Int. J. of Game Theory

13 1984, 41-60.

[16] Alvin E. Roth, Stability and Polarization of Interests in Job Matching, Econometrica 52

1984, 47-57.

[17] Alvin E. Roth and Marilda Sotomayor, Two-sided Matching: a Study in Game-theoretic

Mo deling and Analysis, Cambridge: Cambridge University Press, 1990.

[18] Lloyd Shapley, On Balanced Sets and Cores, Navel Res. Logistics Quarterly 14 1967, 453-460.

[19] Lloyd Shapley and Herb ert Scarf, On Core and Indivisibility, J. of Math. Econ. 1 1974, 23-8.

[20] Lloyd Shapley and Martin Shubik, The Assignment Game I: the core, Int. J. of Game Theory

1 1972, 111-30.

[21] W. Vickrey, Countersp eculation, Auctions, and Comp etitive Sealed Tenders, J. of Finance 16

1961, 8-37. 15