The ex ante incentive compatible core of the assignement game

T he Ex A nt e I ncent ive Compat ible Core of

t he A ssignment Game

¤

Françoise For ges

T H EM A , U niversit é de Cer gy-Pont oise,

and I nst it ut U niver sit aire de Fr ance

¤_{I wish t o t hank B . D ut t a, J.-F . M ert ens, E. M inelli and most part icular ly, R. Vohra,}

Résumé

N ous consi dér ons un marché bipar t it e dans lequel les agent s disposent d’infor -mat ions pr ivées sur un ét at de la nat ur e qui dét er mine leur s ut ilit és d’appa-r iement . L es t d’appa-ransfed’appa-r t s monét aid’appa-r es sont ped’appa-r mis et les fonct ions d’ut ilit é sont quasi-linéair es. L e modèle ét end donc les j eux d’allocat ion int r odui t s par Shapley et Shubik. N ous démont r ons que le coeur ex ant e inci t at if du j eu d’appar iement est non-vide. D es exemples simples illust r ent deux di¤ér ences avec l’infor mat ion compl èt e: t out d’abor d, les mécanismes d’appar iement aléat oi res dé…nissent une fonct ion car act ér ist ique ( à ut ilit é t r ansfér able) plus élevée que les mécanismes dét er minist es; de pl us, les solut ions du coeur ex ant e incit at if ne coï ncident pas nécessair ement avec les r ésult at s st ables cor -r espondant s, et ce même si les valeu-r s sont p-r ivées et indépendant es. N ot -r e appr oche s’ét end au coeur br ut ( int er im) incit at if, qui est lui aussi non-vide.

A bst r act

We consider t wosided mat ching mar ket s in which agent s have pr ivat e infor -mat ion on a st at e of nat ur e which det er mines t he agent s’ ut ilit ies of -mat ching. M onet ar y t r ansfer s ar e all owed and ut ilit y funct ions ar e quasi-linear . T he model t hus ext ends t he assignment game int r oduced by Shapley and Shubik. We pr ove t hat t he ex ant e incent ive compat ible cor e of t he mat ching game is nonempt y. Simpl e examples illust r at e t wo di¤er ences wit h complet e infor -mat ion: …rst , r andom -mat ching mechanisms de…ne a higher T U char act er ist ic funct ion t han det er minist ic ones; fur t her mor e, ex ant e incent ive compat ible cor e solut ions need not coincide wit h ex ant e incent ive compat ible st able out comes, even if values ar e independent and pr ivat e. Our appr oach ext ends t o t he ( int er im) i ncent ive compat ible coar se cor e, which is also non-empt y. JEL classi…cat ion number s: C78, C71, D 82

K eywor ds: assignment game, core, incent ive compat ible mechanism, mat ch-ing, pr ivat e infor mat ion.

1

I n t r o d u ct i on

T wo-sided mat chi ng mar ket s have been ext ensively st udied under t he as-sumpt ion of complete i nfor mati on (see, e.g., Rot h and Sot omayor [23]) . Rot h [22] also invest igat ed ex post st abl e mechanisms in mar r i age pr oblems wi t h incomplet e infor mat ion about ot her s’ prefer ences ( i.e., pr ivat e values) .

We consider t wo-sided mat ching mar ket s wit h ar bit r ar y incomplet e in-for mat ion ( i.e., possibl e common values) . T he agent s ar e divided int o t wo disj oint set s ( e.g., pot ent ial buyer s and seller s, …rms and wor ker s, et c.) . Ev-er y agent has pr ivat e infor mat ion on a st at e of nat ur e which ent Ev-er s agent s’ ( ex post ) ut ilit y of being mat ched wit h a par t ner fr om t he ot her side of t he mar ket . T he ( ex post ) wor t h of a coalit ion is det er mined by pair wise com-binat ions of agent s ( fr om di¤er ent sides of t he mar ket ) and ar bit r ar y money t r ansfer s ( e.g., in t he auct ions fr amewor k, we allow for ent r y fees, bidder r ings, et c.) . U t ilit y funct ions ar e assumed t o be linear in money. Our model is t hus an ext ension of t he ( complet e infor mat ion) assi gnment game int r o-duced by Shapley and Shubik [26]. U nder incomplet e infor mat ion, a number of par t icular cases ( bilat er al t r ading, auct ions, et c.) have been analyzed in great det ails ( see, among ot hers, [7], [8], [18], [21],[29]) .

We ext end Shapley and Shubik [26]’s r esult s by pr oving t hat t he ex ant e i ncenti ve compati ble core of t he assignment game is non-empt y. T his solu-t ion concepsolu-t has been mossolu-t ly applied in di ¤er ensolu-t i al infor masolu-t ion exchange economi es ( see [13]) . I t is appr opr iat e if coalit ions can for m befor e agent s know t heir pr ivat e infor mat ion. T he member s of a coalit ion or ganize mat ch-ings and monet ar y t ransfer s by means of ( r andom) B ayesian incent ive com-pat ible mechanisms. T his gener at es a well -de…ned T U char act er ist ic funct ion, namely t he maximal sum of ( ex ant e) expect ed payo¤s t hat every coal it ion can guar ant ee by r elying on an incent ive compat ible mechanism. T he ex ant e incent ive compat ible cor e is de…ned as t he ( st andar d) cor e of t his char act er is-t ic funcis-t ion. T o esis-t ablish iis-t s non-empis-t iness, we apply is-t he B ondar eva-Shapley t heor em.

A s in most paper s on auct ions and bar gaining under incomplet e infor -mat ion, we assume t hat -mat ching may r esult fr om a lotter y. T his pr ocedur e is nat ur al, especially if one r elies on t he r evel at ion pr inciple ( as will be t he case her e) , but is not needed i n Shapley and Shubik [26]’s or iginal model (see also [23]) . Random assignment pr ocedures can never t hel ess be useful under complet e infor mat ion, for inst ance t o guar ant ee fair ness when money is not available ( see, e.g., [1], [5], [6], [16]) . We shall show on an example t hat ,

when incent ive const r aint s mat t er , t he T U char act er ist ic funct ion associat ed wit h r andom mechanisms can t ake higher values t han t he one associat ed wi t h det er minist ic mechanisms.

U nder complet e informat i on, it is also well -known ( see again [26]) t hat t he cor e coincides wit h t he out comes which ar e stable, i.e., cannot be blocked by any single agent nor pai r of agent s ( fr om di¤er ent sides of t he mar ket ) . H owever , as we show on simple examples, no such pr oper t y hol ds under incomplet e infor mat ion, and t his even under pr ivat e, independent val ues. M or e precisely, one can const r uct i ncent ive compat i ble mechanisms which cannot be blocked by any single agent nor any pair of agent s, but ar e blocked by t hr ee agent s coalit ions.

T he non-empt iness of t he ex ant e incent i ve compat ible cor e means t hat t her e exist s an incent ive compat ible mechanism

¹

for t he gr and coal it ion ( which select s mat chings and t r ansfer s) such t hat no coalit ion can pr opose an incent ive compat ible mechanism impr oving t he expect ed payo¤ of all of it s member s. I n par t icular ,

¹

is i ncent ive ex ant e e¢ cient (i n t he sense of H olmst r öm and M yer son [15]) and ex ant e individually r at ional.

¹

is t hus also incent ive int er im e¢ cient . B ut

¹

is not necessari ly int er im individually r at ional.

Given t he impor t ance of t he lat t er pr oper t y in t he mechanism desi gn lit er -at ur e, a n-at ur al quest ion concer ns t he exist ence of incent ive comp-at ible cor e solut ions t hat would be int er im individually rat ional, i.e., such t hat all t ypes of agent s would like t o par t icipat e in t he mechanism at t he int er im st age. A sat isfact or y solut ion concept in t his r espect i s t he i ncent i ve compat i ble coar se core, namely Vohr a [27]’s incent ive compat ible ver sion of W ilson [28]’s coar se cor e. I n t his appr oach, agent s for m coalit ions aft er having lear nt t heir t ypes but can only exchange infor mat i on inside coali t i ons, once t hese have for med. T hey t hus use exact ly t he same incent ive compat ible mechanisms as above. H owever , blocking t akes place at t he int er im st age so t hat , basically, ever y possible t ype of an agent should get a bet t er payo¤. I n par t icular, incent i ve compat ible coar se cor e solut ions ar e int er im individually r at ional.

I n a similar way as in W il son [28], we const r uct an auxiliar y ( N T U ) char act er ist ic funct ion such t hat t he incent ive compat ible coar se cor e is t he st andar d cor e of t his charact er ist ic funct ion. Given t he N T U st r uct ur e of t he game, we rely on on Scarf [24]’s t heor em t o pr ove t hat t he i ncenti ve compat -i ble coar se core -is non-empty. T h-i s r esult guar ant ees t hat , -in a lar ge class of t r ading models wit h incomplet e i nfor mat ion, t here exist incent ive compat i-ble mechanisms t hat ar e not only int er im e¢ cient and int er im individually

r at ional ( as est ablished in a number of paper s) , but cannot even be impr oved by any coalit ion. I n t hat sense, t hese mechanisms ar e “ collusion-pr oof ” .

T he posit ive r esult s obt ained her e cont r ast wit h t he ones t hat pr evail in di¤er ent ial infor mat ion exchange economies. A s shown by For ges, M er t ens and Vohr a [11], in t his model, bot h t he ex ant e i ncent ive compat ible cor e and t he incent ive compat ible coar se cor e can be empt y, even if random mecha-nisms and monet ar y t r ansfer s ar e allowed. T he li near st r uct ur e of t he assign-ment game is cer t ainly helpful but t his explanat ion of t he r esult s should be used wit h car e. T he incent ive compat ible cor es ar e not empt y in exchange economi es wit h linear ut il it y funct ions ( see [27]) but r andom mechanisms can be dispensed wit h i n t his model. On t he cont r ar y, her e, t hey enable t he agent s t o achieve higher expect ed payo¤s. H ence, t he r esult on linear ex-change economies cannot be applied dir ect ly. I t is also t r ue t hat random mechanisms gener at e li near it y in t he sense t hat expect ed payo¤s ar e linear in t he mechanism pr obabilit ies. H owever , t his does not su¢ ce for t he non-empt iness of t he incent ive compat ible cor es, as t he count er -example in [11] shows.

T he quasi-linear ut i lit y funct ions ar e useful in [11], as i n many paper s on mechanism design ( see, e.g. [2], [3], [4], [7], [8] and [17]) t o const r uct incent ive compat ible, ex post e¢ ci ent mechanisms, which t ur n out t o achieve expect ed payo¤s in t he ex ant e incent ive compat i ble core. Such r esult s also apply her e: under appr opr iat e assumpt i ons, t he ex ant e incent ive compat ible cor e cont ains ex post e¢ cient allocat ions. H owever , no speci…c assumpt ions ( e.g., on t he agent s’ bel iefs) are needed her e t o est ablish t he non-empt iness of t he ex ant e incent ive compat ible cor e. Even mor e, t his r esult ( as well as t he non-empt iness of t he incent ive compat ible coar se cor e) st ill holds if t her e ar e no monet ar y t r ansfer s at all. T he model cover s t hen incomplet e infor mat ion ver sions of t he mar r iage pr oblem and t he indivisible good economy of Shapley and Scar f [25]. Obviously, i n t his case, t he under lying char act er ist ic funct ion is always N T U , even in t he ex ant e fr amewor k. T he main r eason t o consider monet ar y t r ansfer s is t o connect our analysis wit h t he lit er at ur e on t r ading wit h incomplet e infor mat ion (e.g., auct ions) , in which B ayesian incent i ve compat ible mechanisms have become a st andar d t ool.

T he next sect ion descr ibes t he model. Sect ion 3 de…nes t he ex ant e incen-t ive compaincen-t ible cor e and esincen-t abli shes iincen-t s non-empincen-t iness. Secincen-t ion 4 i s devoincen-t ed t o t he incent ive compat ible coar se cor e. Examples illust r at e our solut ion con-cept s in sect ion 5. Sect ion 6 concludes wit h fur t her r emar ks on possible ex post proper t ies.

2

I n cen t i v e com p at i b l e m at ch i n g m ech an i sm s

T he economy consist s of t wo, …nit e, disj oint set s of agent s

I

and

J

:

I

_{\J = ;}

; let

K = I

_{[ J}

. Ever y agent

k

_{2 K}

has pri vat e infor mat ion, r epr esent ed by his type

t

k

2 T

k, wher e

T

k is a …nit e set1; let

T =

Q

_k2K

T

k and let

q

be a

pr obabilit y dist r ibut ion over

T

, t he common pr ior of t he agent s, such t hat , wit hout loss of gener alit y,

q(t

k

) > 0

for ever y

t

k. Ever y agent der ives ( von

N eumann - M or genst er n) ut ili t y fr om being mat ched wit h an agent of t he ot her side of t he mar ket and fr om monet ar y t r ansfer s. For ever y

i

_{2 I}

,

j

_{2 J}

and

t

_{2 T}

, let

u

i

(t; j)

( r esp.,

u

j

(t; i)

) denot e agent

i

( r esp.,

j

) ’s ut i lit y of being

mat ched wit h agent

j

( r esp.,

i

) when t he t ypes ar e

t

. L et also

u

k

(t; 0)

denot e

agent

k

’s ut ilit y of being unmat ched (

k

_{2 K}

,

t

_{2 T}

) . M onet ar y t r ansfer s are j ust added linearl y t o t hese ut il it y number s. A gent s t hus have addit ively separabl e ut i lit y funct ions for mat ching and t r ansfer s.

Except per haps for t he l at t er assumpt ions, t he framewor k is quit e gen-er al: any infor mat ional ext gen-er nalit y is allowed ( t he whole vect or of t ypes

t

ent er s all ut ilit y numbers, so t hat common values ar e possibl e) . Fur t her -mor e, any mat ching ext er nalit y is also al lowed, since ever y agent car es about his par t ner .

A s a t ypical example, ext ending Shapley and Shubik [26]’s housi ng mar -ket , t he

I

-agent s ar e seller s, each wi t h a single it em for sale, and t he

J

-agent s are pot ent ial buyer s;

u

i

(t; j)

r epr esent s seller

i

’s ut ilit y of selling his it em t o

buyer

j

at t he st at e of infor mat ion

t

, while

u

j

(t; i)

r epr esent s buyer

j

’s ut ilit y

of buying an it em fr om seller

j

at

t

. St andar d assumpt ions ar e t hat t he sell-er s ar e indi¤sell-er ent t o sell t o a buysell-er or anot hsell-er (

u

j

(t; i) = u

j

(t; i

0

)

for ever y

i; i

0

_{2 I}

_{) , or t hat t he pr ivat e infor mat ion of t r ader s coi ncides wit h t hei r r eser}

-vat ion pr ices ( pr i -vat e values:

u

i

(t; j) = u

i

(t

i

; j)

r epr esent s buyer

i

’s r eser

va-t ion pr ice for seller

j

’s i t em when his t ype is

t

i,

u

j

(t; i) = u

j

(t; i

0

) = u

j

(t

j

)

r epr esent s seller

j

’s r eser vat ion pri ce for his it em) . T his example cover s most auct ion models. M onet ar y t r ansfer s ar e nat ur al in t his example; t hey basi-cally cor r espond t o sale pr ices, but may also involve compensat ions for not get t ing an it em, ent r y fees, et c.

T he r est ri ct ions imposed on t he mar ket will be r e‡ect ed by t he feasible mat ching mechanisms. T he out comes of such a mechanism consist of disj oint pair s

fi; jg ; i 2 I; j 2 J

of mat ched agent s, a set of unmat ched agent s and

1

T his assumpt ion (also made in, e.g., [7], [8], [11], [17], [27], et c.) i s mainly for simplic-it y. M ost of our r esult s go t hr ough if t he set s of t ypes ar e r eal int er vals and t he beliefs have a densit y, as in [18], [19], [21], et c.

monet ar y t r ansfer s

m

k

; k

2 K

such t hat

P

_k2K

m

k

· 0

. I n par t icular , an

agent can be mat ched t o at most one agent of t he ot her side of t he mar ket and ar bit r ar y t r ansfers ar e allowed, incl uding bet ween agent s who ar e not mat ched t oget her .

We assume t hat agent s exchange infor mat ion and mat ch aft er having for med coalit ions2_{. A coalit ion}

_S

_{is j ust a non-empt y subset of}

_K

_{. T he}

member s of

S

can r ely on any mechanism ( in par t icular , any non-cooper at i ve bar gaini ng game) t o or ganize mat ching and t r ansfer s. B y t he revelat i on pr i n-ci ple ( see, e.g., [18] for an applicat ion t o auct i ons and [22] for an appli cat ion t o mat ching) , all feasible collect ive decisions of

S

can be repr esent ed as t he out comes of a random, di rect, i ncent i ve compat i ble, mat chi ng mechani sm, which we de…ne pr ecisely below.

A s in most paper s on col lect ive decisions under incomplet e infor mat ion ( see [7], [8], [11], [12], [15], [18], [22], et c.) , we allow t hat mat ching r esult s fr om lot t er ies, which is usual in many auct ions mechanisms. L ot t er ies also appear as a consequence of t he r evelat ion pri nciple if agent s ar e allowed t o use mixed bar gaining st r at egies. U nder complet e infor mat ion, it is well-known t hat r andom mat ching mechanisms do not achieve mor e expect ed payo¤s t han t he det er minist ic ones ( see [26], [23]) . T his is no longer t he case under asymmet r ic infor mat ion, as we show in sect ion 5.

L et

S

be a coalit ion; let us set

T

S

=

Q

k2S

T

k. For ever y

t = (t

k

)

k2K

2 T

,

let

t

S

= (t

k

)

k2S. A ( r andom) mat ching mechanism3 for

S

consist s of

map-pings

¹

_S

= (x

S

; m

S

) : T

! [0; 1]

I£J

£ R

K

: t

! [(x

ijS

(t))

(i;j)2I£J

; (m

kS

(t))

k2K

]

such t hat

² ¹

S is measur abl e w.r .t .

T

S, namely

¹

S

(t) = ¹

S

(t

0

)

for ever y

t; t

0

2 T :

t

S

= t

0S, and for ever y

t

2 T; x

ij S

(t) = 0

if

i =

2 S

or

j =

2 S

and

m

kS

(t) = 0

if

k =

2 S

²

P

j2J

x

ij S

(t) =

P

j2J\S

x

ij

S

(t)

· 1

for ever y

i

2 I

and

P

i2I

x

ij S

(t) =

P

i2I\S

x

ij S

(t)

· 1

for ever y

j

2 J

²

P

k2K

m

kS

(t) =

P

k2K\S

m

kS

(t)

· 0

2_{T ypically, coalit ions for m at t he ex ant e st age, i.e., before t he agent s know t heir t ypes.}

I n sect ion 4, we will assume t hat coali t ions for m at t he int erim st age, but t hat agent s do not exchange infor mat ion unt il t hey ar e in a coali t ion.

3

W e wi ll use t he t er m “ mat ching mechanism” t o r efer t o a “ feasible ( mat ching) mech-anism” sat isfying t he feasibilit y const r aint s.

T he int er pr et at ion of t hese feasibili t y condi t i ons is t hat ever y member of

S

is invit ed t o r epor t his pr ivat e informat i on t o t he mechanism

¹

_S, which, as a funct ion of t he repor t ed t ypes

t

, mat ches

i

and

j

wit h pr obabi lit y

x

ij_S

(t)

, and dist r ibut es t he ( expect ed) t r ansfer s

m

k

S

(t)

.4 T he feasi bilit y condit ions fur t her

asser t t hat a coal it ion can only use t he infor mat ion of it s member s, t hat some agent s may be left unmat ched, and t hat t r ansfer s must balance. Obser ve t hat a mat ching mechanism

¹

_S, associat ed wit h an ar bit r ar y coalit ion

S

, can be viewed as a mat ching mechanism for t he gr and coalit ion

K

, which leaves unmat ched all agent s in

K

_nS

.

I n or der t o de…ne incent ive compat ibilit y of

¹

_S, consider

i

_{2 I \ S}

; let

t

i ( r esp.,

t

0i) denot e agent

i

’s t r ue ( r esp., repor t ed) t ype; agent

i

’s expect ed

ut ilit y is

U

i

(¹

S

jt

i

; t

0i

) =

X

t_¡i

q(t

_¡i

_jt

i

)

2

4

X

j2J[f0g

x

ij_S

(t

0_i

; t

_¡i

)u

i

((t

i

; t

¡i

); j) + m

iS

(t

0i

; t

¡i

)

3

5

wher e

t

_¡i

= t

Knfig and for every

t

2 T

,

x

i0S

(t) = 1

¡

P

j2J

x

ij

S

(t)

is t he

pr obabilit y t hat agent

i

is left unmat ched by

¹

S at

t

. L et us denot e as

U

i

(¹

S

jt

i

)

t he ( int er im) expect ed ut ilit y of agent

i

when he t rut hfully r epor t s

his t ype t o

¹

_S, namely

U

i

(¹

S

jt

i

) = U

i

(¹

S

jt

i

; t

i

)

¹

_S is i ncenti ve compati ble if

U

i

(¹

S

jt

i

)

¸ U

i

(¹

S

jt

i

; t

0i

)

for ever y

i

2 I \ S; t

i

; t

0i

2 T

i ( 1)

and similar ly for t he agent s

j

_{2 J \ S}

on t he ot her side of t he mar ket .

4_{Ever y x}

S(t)2 [0; 1]I£J sat isfying t he above condit ions is a subst ochast ic mat r ix and

t hus a convex combi nat ion of det er minist ic mat chings in_{f0; 1g}I£J ( t his is a var iant of t he well-known B irkho¤-von N eumann t heor em, see, e.g., [26] or [6] for a r ecent applicat ion) . T he descript i on of r andom mechanisms by mat ching pr obabilit ies (as opposed t o pr oba-bilit y dist r ibut i ons over det ermi nist ic mat chings) su¢ ces here, t hanks t o t he propert ies of t he ut ili t y funct ions (such a simpli …cat ion is not always possible in exchange economies, see [11], [12] and t he discussi on following pr oposit ion 1). I n t he same way, t ransfers ar e det erminist ic wit hout loss of gener alit y.

We denot e as

U

k

(¹

S

)

t he ( ex ant e) expect ed ut ilit y of agent

k

2 S

fr om

t he mat ching mechani sm

¹

_S, namel y

U

k

(¹

S

) =

X

tk

q(t

k

)U

k

(¹

S

jt

k

)

A ll t he not ions int r oduced in t his sect i on ar e illust r at ed in sect ion 5.

3

T h e ex an t e i n cen t i v e com p at i b l e cor e

We de…ne a cooper at ive game, namel y a charact er ist ic funct ion5

V

¤

A, fr om t he

pr evious set up. A coalit ion

S

can achieve a payo¤ vect or

v = (v

k

)

k2S

2 R

S,

i.e.,

v

2 V

¤

A

(S)

, i¤ t her e exist s an incent ive compat ible mechanism

¹

S such

t hat

v

k

· U

k

(¹

S

)

for ever y

k

2 S

. T he char act er ist ic funct ion

V

A¤ is consi st ent

wit h t he fol lowing scenar io:

²

Ever y coalit ion

S

chooses a mat ching mechanism

¹

_S, coali t ions possibly for m

²

A vect or of t ypes

t

2 T

is select ed accor ding t o

q

; ever y agent

k

2 K

is informed of his own t ype

t

k

2 T

k

²

I f

S

has for med,

¹

_S is implement ed, member s of

S

r epor t t heir t ypes t o

¹

_S, mat chings and t r anfer s ar e r ealized accor ding t o

¹

_S.

One immediat ely checks t hat if

v = (v

k

)

k2S

2 V

A¤

(S)

and

w = (w

k

)

k2S

2

R

S _{is such t hat}

P

k2S

w

k

=

P

k2S

v

k, t hen

w

2 V

A¤

(S)

. I ndeed,

w

can be

achieved by modifying t he t r ansfer s used for

v

independent ly of t ypes, i.e., in an incent ive compat i ble way. T he cooper at i ve game is t hus T U and one can descr ibe t he char act eri st ic funct ion as

v

_A¤

(S) = max

¹S IC

[

X

k2S

U

k

(¹

S

)]

( 2)

wher e t he maximum6 is over all incent ive compat ible ( I C) mat ching mech-anisms

¹

S. T hi s de…nes t he ex ante assi gnment game. Obser ve t hat t he

ex-pect ed sum of t r ansfer s, namely

P

_t

q(t)

P

_k2S

m

k

S

(t)

· 0

, is par t of t he above

5

A s in [15], we put a “ * ” for t he incent ive compat ible concept s.

6_{(2) de…nes a “ gener al linear problem” (see, e.g., Gale [14]). I t i s easily checked t hat}

ob j ect ive funct ion, and t hat , i mplicit ly,

v

¤

A

(

fkg) =

P

t

q(t)u

k

(t; 0)

for any

single agent

k

.

A s a benchmar k, we consider t he …rst best char act er ist ic funct ion

v

A in

which agent s ar e not submit t ed t o incent ive compat ibilit y const r aint s. I n t his case, at an opt imum, t he sum of monet ar y t r ansfer s i s exact l y

0

:

v

A

(S) = max

¹S

[

X

k2S

U

k

(¹

S

)]

= max

xS

X

t

q(t)

2

4

X

i2I

X

j2J[f0g

x

ij_S

(t)u

i

(t; j) +

X

j2J

X

i2I[f0g

x

ij_S

(t)u

j

(t; i)

3

5

Obviously,

v

_A¤

(S)

· v

A

(S)

Shapl ey and Shubik [26] ( see also [23]) have shown t hat under complet e infor mat ion, one can r est ri ct oneself on det er minist ic mechanisms wit hout loss of gener alit y. H ence, in t he above expr ession of

v

A

(S)

, t he maximum is

in fact over all det er minist ic mechanisms

x

S, wit h values in

f0; 1g

. A s we

alr eady ment ioned, t his pr oper t y is no longer t r ue under incomplet e infor -mat ion. I n example 3 ( sect i on 5) , t he best expect ed payo¤ a coal it ion

S

can get by r elying on det er minist ic incent ive compat i ble mat ching mechanisms is st r ict l y less t han

v

_A¤

(S)

.

A not her int er est ing pr oper t y of t he assignment game wit h complet e in-for mat ion is t hat all gai ns ar e generat ed by pair s of agent s fr om t he di¤er ent sides of t he mar ket ( see [23]) . M or e precisely, t he wor t h of any coal it ion

S

can t hen be comput ed dir ect ly from t he wort hs of al l singlet ons and pair s cont ained in

S

and, as a consequence, t ransfer s ar e onl y needed bet ween pair s of mat ched agent s. T his st r uct ur e does not sur vi ve under incomplet e infor mat ion, even in t he simplest sellersbuyer s models wit h i ndependent pr i -vat e values ( see example 1 sect ion 5) . T he int uit ive r eason for t his is t hat in our model, t he player s exchange infor mat ion aft er coalit ions have for med. I f t hey par t it ion int o pair s at t he coal it ion for mat ion st age, t hey may loose t he int er act ion possibilit ies which ar e o¤er ed by lar ger coalit ions’ mechanisms.

Recall t hat t he core of a T U game

v

over

K

i s t he set of all vect or payo¤s

w = (w

k

)

k2K which can be achieved by t he gr and coalit ion, i.e.,

P

k2K

w

k

·

v(K)

, and cannot be i mpr oved upon by any coal it ion, i.e.,

P

k2S

w

k

¸ v(S)

cor e” of t he assignment game. U nder complet e infor mat ion, Shapley and Shubik [26] have pr oved t hat t he cor e of t he assignment game is not empt y. A st r aighfor war d applicat ion of t he B ondar eva-Shapley t heor em will enable us t o show t hat Shapl ey and Shubik’s r esult ext ends, namely

C(v

¤

A

)

6= ;

.

H owever , as we point ed out above, many nice pr oper t ies of t he char act er ist ic funct ion disappear . I n par t icular, t he ex ant e incent ive compat ible cor e does not coinci de wit h t he ( payo¤s of ) stable mat chings ( which ar e individually r at ional and cannot be blocked by any pai r

(i; j)

of agent s in

I

£ J

; see examples 1 and 2 i n sect ion 5) .

P r op osi t i on 1 T he ( T U) assi gnment game de…ned by

v

_A¤ i s balanced, so that the ex ante i ncenti ve compati ble core

C(v

¤

A

)

i s non-empty.

T he r esult is a dir ect consequence of t he fol lowing lemma:

L em m a 1 Let

_S

be a balanced fami ly of coali ti ons wi t h associ at ed wei ght s

¸

S,

S

2 S

, and let

¹

S

= (x

S

; m

S

)

be a mechani sm for

S

,

S

2 S

. T hen

¹ =

P

_S2S

¸

S

¹

S i s a matchi ng mechani sm for

K

and sat i s…es

U

k

(¹

jt

k

; t

0k

) =

X

S2S:S3k

¸

S

U

k

(¹

S

jt

k

; t

0k

)

8k; t

k

; t

0k ( 3)

I n par ti cular , i f ever y

¹

S i s i ncenti ve compati ble, so i s

¹

.

P r o of of t h e l em m a: Recal l t hat a mechanism

¹

_S for an ar bit r ar y coal it ion

S

can be viewed as a mechani sm for t he gr and coalit ion. T he mechanism

¹

de…ned i n t he st at ement is

T

-measur abl e as a linear combinat ion of

T

-measur able mechanisms. B y de…nit i on of balancedness,

P

_S2S:S3k

¸

S

= 1

for

ever y

k

2 K

;

¹ = (x; m)

is not , as such, a convex combi nat i on of t he

¹

S’s

but sat is…es

x

ij

(t) =

X

S2S:S¶fi;jg

¸

S

x

ijS

(t)

8i 2 I; j 2 J; t 2 T

( 4)

m

k

(t) =

X

S2S:S3k

¸

S

m

kS

(t)

8k 2 K; t 2 T

( 5)

so t hat

¹

is feasi ble. I ndeed, let

j

_{2 J}

and

t

_{2 T}

:

X

i2I

x

ij

(t) =

X

i2I

X

S2S:S¶fi;jg

¸

S

x

ijS

(t)

=

X

S2S:S3j

¸

S

X

i2I\S

x

ij_S

(t)

· 1

and similar ly on t he ot her side of t he mar ket . T he t r ansfer s sat isfy

X

k2K

m

k

_{(t) =}

X

k2K

X

S2S:k2S

¸

S

m

kS

(t) =

X

S2S

¸

S

X

k2S

m

k S

(t)

· 0

T he equalit ies (3) can be checked in a similar way:

X

j2J

x

ij

(t

0_i

; t

_¡i

)u

i

((t

i

; t

¡i

); j)

=

X

j2J

[

X

S2S:S¶fi;jg

¸

S

x

ijS

(t

0i

; t

¡i

)]u

i

((t

i

; t

¡i

); j)

=

X

S2S:S3i

¸

S

X

j2J\S

x

ij_S

(t

0_i

; t

_¡i

)u

i

((t

i

; t

¡i

); j)

Simil ar ly,

x

i0

(t

0_i

; t

_¡i

)u

i

((t

i

; t

¡i

); 0)

= [1

_¡

X

j2J

X

S2S:S¶fi;jg

¸

S

x

ijS

(t

0i

; t

¡i

)]u

i

((t

i

; t

¡i

); 0)

=

X

S2S:S3i

¸

S

[1

¡

X

j2J\S

x

ij_S

(t

0_i

; t

_¡i

)]u

i

((t

i

; t

¡i

); 0)

and t he t r ansfer s sat isfy ( 5) .

Q.E.D . P r o of of t h e p r op osi t i on : L et

S

be a bal anced fami ly of coalit ions wi t h associat ed weight s

¸

S,

S

2 S

. We must show t hat

v

¤A

(K)

¸

P

S2S

¸

S

v

A¤

(S)

.

L et , for ever y

S

_{2 S}

,

¹

_S be a mechanism achieving t he maximum in ( 2) , namely

v

¤

A

(S) =

P

k2S

U

k

(¹

S

)

and de…ne

¹

as in t he lemma.

¹

is incent i ve

compat ible. B y t he linear it y of ut ilit y funct ions,

v

_A¤

(K)

¸

X

k2K

U

k

(¹) =

X

k2K

X

S2S:S3k

¸

S

U

k

(¹

S

)

=

X

S2S

¸

S

X

k2S

U

k

(¹

S

) =

X

S2S

¸

S

v

¤A

(S)

Q.E.D . L et us compar e t he pr evious pr oposit ion wit h t he r esult s obt ained for t he ex ant e incent ive compat ible core of an exchange economy in Vohr a [27],

For ges and M inelli [12] and Forges, M er t ens and Vohr a [11]. T he seller s-buyer s example ( wit hout mat ching ext er nalit ies for t he sell er s) is basi cally a par t icular case of t he economies consider ed in [11], wher e ar bit r ar y monet ar y t r ansfer s ar e allowed and ut ili t y funct i ons ar e quasi -linear. A s shown in [11], t hese condi t i ons do not guar ant ee t he non-empt iness of t he ex ant e incent i ve compat ible cor e. H er e, t he ut ili t y funct ions ar e in fact linear , and t his even if one r est ri ct s oneself t o det er mini st ic mechanisms ( in which all pr obabilit ies of mat ching ar e

0

or

1

) .

T he ex ant e incent ive compat ible cor e of an exchange economy wit h linear ut ilit y funct ions is known t o be non-empt y; r andom mechanisms ar e clear ly useless in t hat fr amewor k ( see, e.g., [27], [13]) . L et us show t hat t his r esult cannot be applied dir ect ly here. I t is t empt i ng t o vi ew t he mat ching pr oba-bilit ies of sect ion 2 as quant it ies of i ndivi sible goods, e.g., by appealing t o a t ime-shar ing int er pr et at ion. H owever , t he st andar d r esour ce const r aint s do not account for t he feasibi lit y const r aint s st at ing t hat each buyer is mat ched wit h ( t ot al) pr obabilit y less t han one at ever y st at e of nat ur e. T his should r esult fr om an assumpt i on on t he buyer s’ init ial ut ilit y funct ions, namely t hat , at every st at e of nat ur e, a buyer ’s ut ilit y for sever al houses is j ust his ut ilit y for his favor it e house. Such ut i lit y funct ions ar e obviously not li near . To r est r ict on t ime-shar es ( as opposed t o pr obabilit y dist r ibut ions over de-t er minisde-t ic al locade-t ions) and ude-t il ide-t y funcde-t ions de-t hade-t ar e linear in de-t hese, as in sect ion 2 ( recall foot not e 4) , a const r uct ion much mor e t edious t han above is needed. T his appr oach gener at es cor e mechanisms which ar e feasi ble in expect at ion. T hese ar e always feasi ble her e ( see again foot not e 4) , unlike in gener al exchange economies wi t h asymmet r ic infor mat ion.

L et us fur t her illust r at e t he di ¤er ences bet ween t he assignment game and an exchange economy by consider ing a simple example. L et

I =

_{f1; 2g}

and

J =

_{f3; 4g}

; consider t he balanced family

_{S = ff1; 2; 4g; f1; 3g; f2; 3g; f4gg}

wit h all weight s equal t o 1₂ and t he following det er mi nist ic mat chings for some st at e of nat ur e

t

:

2

and

4

in

_{f1; 2; 4g}

,

1

and

3

in

_{f1; 3g}

,

2

and

3

in

f2; 3g

( and null t ransfer s) . T he mechanism const r uct ed in t he lemma is, at

t

,

x(t) =

µ

₁ 2

0

1 2 1 2

¶

=

1

2

µ

1 0

0 1

¶

+

1

2

µ

0 0

1 0

¶

which is feasible as expect ed. L et us now view t he example as an exchange economy, in which agent

1

and agent

2

, t he seller s, bot h init iall y have

1

unit of good. T he fol lowing det er minist i c mechanisms ar e now feasible: agent

2

get s

2

unit s in

_{f1; 2; 4g}

; agent

3

get s

1

unit in

_{f1; 3g}

and

1

unit as wel l in

f2; 3g

. Pr oceeding as in t he lemma yields t he mechani sm which al locat es

0

or

2

unit s, each wit h pr obabili t y 1₂, t o agent

2

and

1

unit wit h pr obabilit y

1

t o agent

3

. T his mechanism cannot be feasible since it di st r ibut es

3

unit s wit h posit ive pr obabilit y. H owever , t he average mechanism, which gives

1

t o agent

2

and

1

t o agent

3

i s feasible7_{. I f t he ut ilit y funct ions ar e l inear , t he}

aver age mechanism is obviously equivalent t o t he r andom one, but t hen, by cont r ast wit h t he assignment game, r andom mechanisms ar e not useful at all.

4

T h e i n cen t i v e com p at i b l e coar se cor e

U p t o now, we have assumed t hat coal it ions for m at t he ex ant e st age, be-for e agent s know t heir t ypes. Such a scenar io is obviously not always feasible ( see, e.g.,[15]) : if t ypes r epr esent int r insi c char act er ist ics of t he agent s, t he mat ching pr ocedur e cannot st ar t befor e t he i nter i m st age, in which ever y agent only knows his own t ype. I f we mai nt ain t he assumpt ion t hat agent s do not exchange infor mat ion unt il t hey ar e in a coalit ion, t he incent ive com-pat ible ver sion of W ilson [29]’s coar se core pr oposed by Vohr a [27] is t hen an appr opr iat e not ion of t he cor e. I n par t icular , solut ions in t he incent i ve compat ible coar se cor e ar e i nt er i m i ndi vi dual ly rati onal in t he st andar d sense ( see [18], [19], [21] et c.) and i ncenti ve i nter i m e¢ ci ent ( see [15]) . A gent s wil l exchange infor mat ion only aft er having for med coalit ions, by means of t he same mechanisms as in sect ion 2, but will possi bly block pr oposals at t he int er im st age. A s a consequence, t hey will base t heir ob j ect ions on event s t hat ar e common knowledge inside t he coalit ion at t he int er i m st age.

I n or der t o make t he for mal de…nit ion of common knowledge event s ( and t hus of t he incent i ve compat ible coar se core) as simple as possible, we shal l make, t hr oughout t his sect ion, t he assumpt ion t hat all vect or of t ypes occur wit h posit ive pr obabilit y:

q(t) > 0

for ever y

t

2 T

, e.g., t hat t ypes ar e independent .8 I n t his case, t he set of all t ypes

T

is t he only event t hat can be common knowledge i n a coalit ion

S

of t wo agent s or mor e at t he int er im

7_{T he present example is simplist ic since t her e is only one good. [11] fully de…nes a}

T U exchange economy in which t he ex ant e incent i ve compat ible core is empt y. H ence, t he corr esponding T U game cannot be balanced (which is obviously much st r onger t han ver ifying t hat t he most naive const ruct ion t o est ablish balancedness does not work, as we did above) .

st age ( since t he infor mat ion part it ion of agent

k

of t ype

t

0

kis t he set of al l t ype

vect or s

t

such t hat

t

k

= t

0k) . Obviously, if

S =

fkg

, “ common knowledge” is

synonymous wi t h “ knowledge” , and at t he i nt er im st age, agent

k

knows his own t ype

t

k.

L et

¹

be a mat ching mechanism for t he gr and coalit ion and let

º

S be a

mat ching mechanism for coalit ion

S

,

_{jSj ¸ 2}

, as de…ned in sect ion 2.

º

S is a

coar se obj ecti on t o

¹

i¤

U

k

(º

S

jt

k

) > U

k

(¹

jt

k

)

8k 2 S; t

k

2 T

k ( 6)

I n a similar way, agent

k

of t ype

t

k has a coar se ob j ect ion t o

¹

i¤

X

t_¡k

q(t

_¡k

_jt

k

)u

k

(t; 0) > U

k

(¹

jt

k

)

( 7)

so t hat agent

k

blocks

¹

at t he int er im st age as soon as t her e exist s

t

k

2 T

k

sat isfying ( 7) .

¹

is an incent ive compat ible coar se cor e mechanism i¤

¹

is incent i ve compat ible and no coalit ion

S

has an incent ive compat ible coar se ob j ect ion t o

¹

. I n par t icular,

¹

is incent ive int er im e¢ cient , namely t her e is no incent i ve compat ible mechanism

º = º

K such t hat ( 6) is sat is…ed for

S = K

, and

¹

is

int er im i ndivi dually r at i onal, namel y

U

k

(¹

jt

k

)

¸

X

t_¡k

q(t

_¡k

_jt

k

)u

k

(t; 0)

8k 2 K; t

k

2 T

k

I n a similar way as in sect ion 3, we de…ne t he incent ive compat ible coar se cor e as t he set of (i nt er im) payo¤ s ( of t he for m

[(w

k

(t

k

))

tk2Tk

]

k2K

2 R

N_,

N =

P

_k

jT

k

j

) t o incent ive compat ible coar se cor e mechanisms. A s point ed

out in [27] and [13], t her e is no inclusion r elat ionship bet ween t he ex ant e incent ive compat ible cor e and t he i ncent ive compat ible coar se cor e; indeed, int er im individual r at ionalit y obviously implies ex ant e individual r at ionalit y, but t he implicat ion goes t he ot her way r ound for e¢ ciency ( see [15]) .

W ilson [29] showed t hat , in t he absence of incent ive const r aint s, t he coar se cor e of a well-behaved exchange economy is non-empt y, as t he st andar d cor e of an appr opr iat ely de…ned balanced N T U cooper at i ve game wit h

N

player s

(k; t

k

)

,

k

2 K

,

t

k

2 T

k ( see also [13]) . Vohr a [27] ext ended his ar gument t o

coarse cor e as in [27] and [13]. Fr om a concept ual point of view, t he concept of t he coarse cor e seems mor e at t r act ive when no st at e of nat ur e has zer o probabilit y.

exchange economies wit h linear ut ilit y funct ions. We pr oceed i n t he same way her e and est ablish t hat t he incent ive compat ible coar se cor e is t he cor e

C(V

_I¤

)

of a char act er ist ic funct ion

V

¤

I. T he player s in t his game ar e, as in W ilson

[29], t he t ypes of t he or iginal model; we call t hem “ auxiliar y pl ayer s” ; t he only vi abl e coal it ions ar e of t he for m

f(k; t

k

)

g

for some

k

2 K

and some t ype

t

k

2 T

k, or of t he for m

f(k; t

k

) : k

2 S; t

k

2 T

k

g

for some or iginal coal it ion

S

_{µ K}

,

_{jSj ¸ 2}

. T he int er pr et at ion is clear : at t he int er im st age, a coal it ion consist s eit her of a si ngl e agent who knows his t ype or of sever al agent s who have t r ivial common knowledge ( as a consequence of our assumpt ions) and t hus consider all t heir t ypes as possible. T he viable coali t ions of t wo agent s or mor e will t hus be ident i …ed wit h t he or igi nal coalit ions

S

µ K

,

jSj ¸ 2

wit h a lit t le abuse of language.

T he char act er ist ic funct ion

V

¤

I is now easily de…ned on viable coalit ions.

For ever y

(k; t

k

)

,

V

I¤

(

f(k; t

k

)

g)

is t he set of vect or payo¤s

v

2 R

N in which

agent

k

of t ype

t

kget s at most his individually rat ional level

P

t_¡k

q(t

¡k

jt

k

)u

k

(t; 0)

.

For ever y coalit ion

S

_{µ K}

,

_{jSj ¸ 2}

,

V

¤

I

(S)

is t he set of vect or payo¤s

w

2 R

N

for which t her e exist s an incent ive compat ible mechanism

¹

_S for coalit ion

S

such t hat

w

k

(t

k

)

· U

k

(¹

jt

k

)

for ever y

k

2 S

and

t

k

2 T

k. F inal ly, as usual ,

let us de…ne

V

¤

I on ar bit r ar y coalit ions by t aking t he super addit ive cover .

T his amount s t o allowing t he auxili ar y player s

(k; t

k

)

of a coalit ion t o use

an ar bit r ar y incent ive compat ible mat ching mechanism if all

(k; t

0_k

)

,

t

0_k

2 T

k,

are also in t he coalit ion and t o leaving unmat ched t he ot her auxiliar y play-er s.

V

_I¤ is a well-de…ned N T U game, but clear ly, t he ar gument showing t hat

V

_A¤ de…nes a T U game does not ext end her e ( see [11] and [13] for r elat ed comment s) .

T he core of t he int er im assignment game

V

_I¤is t he set of all vect or payo¤s

w = [(w

k

(t

k

))

tk2Tk

]

k2K

2 V

I¤

(K)

which cannot be impr oved upon by any

( viable) coalit ion. We deduce t he non-empt iness of

C(V

¤

I

)

fr om Scar f ’s [24]

t heor em.

P r op osi t i on 2 T he ( NT U) assi gnment game de…ned by

V

¤

I i s balanced, so

that the i ncenti ve compati ble coar se core

C(V

¤

I

)

i s non-empty.

P r o of : L et

_B

be a balanced family of viabl e coalit ions i n t he auxiliar y

N

player game de…ned above, wit h associat ed weight s

¸

B,

B

2 B

. We must

show t hat

\

B2B

V

I¤

(B)

µ V

I¤

(K)

. A ssume t hat , for some

k

and

t

k, t he

sin-glet on

_{f(k; t}

_k

)

_g

belongs t o

_B

; t hen by balancedness, all singlet ons

_{f(k; t}

0_k

)

_g

,

t

0

k

2 T

k, must also be in

B

, all wit h same weight , say

¸

k. We can t hus

by mer ging al l

_B

’s element s

_{f(k; t}

k

)

g

,

t

k

2 T

k, int o a single coal it ion, wi t h

weight

¸

k ( and keepi ng unchanged t he coali t i ons of at least t wo player s and

t heir weight s, since t hese al ready corr espond t o or iginal pl ayer s) . Since by de…nit ion of

V

¤

I, for ever y

k

2 K

,

V

I¤

(

f(k; t

k

) : t

k

2 T

k

g) = \

tk

V

I¤

(

f(k; t

k

)

g)

,

we also have

\

B2B

V

I¤

(B) =

\

S2S

V

I¤

(S)

. We can t hus pur sue t he r easoning

in t er ms of t he or iginal player s, but consider ing vect or payo¤s indexed by t he t ypes.

L et

w = [(w

k

(t

k

))

tk2Tk

]

k2K

2 \

S2S

V

I¤

(S)

: for ever y

S

2 S

, t here exist s

an incent ive compat ible mechanism

¹

_S achieving

w

. L et us de…ne

¹

as in lemma 1.

¹

is an incent ive compat ible mechanism for t he gr and coali t ion, such t hat

U

k

(¹

jt

k

)

¸ w

k

(t

k

)

for ever y

k

,

t

k. H ence

w

2 V

I¤

(K)

.

Q.E.D . R em ar k s:

( i) T he pr evious analysis shows t hat under t he assumpt ion t hat all t ype vect or s have posit ive pr obabilit y, t he r elevant char act er ist ic funct ion at t he int er im st age can be de…ned over t he or iginal coali t i ons

S

_{µ K}

, by consid-er ing as achievable all vect or payo¤s

[(w

k

(t

k

))

tk2Tk

]

k2S t hat r esult fr om an

incent ive compat ible mechanism. Coal it ion

S

bl ocks a mechanism

¹

if t her e is an incent ive compat ible mechanism

º

S such t hat (6) hol ds. Of cour se, if

S

is a singlet on, t he incent ive compat ibil it y condit ions ar e vacuous so t hat a single agent blocks any mechanism t hat is not int er im individually r at io-nal. T he const r uct ion of t he auxiliar y game is only necessar y t o use Scar f ’s t heor em, which is not for mulat ed for vect or payo¤s.

( ii) U nlike t he cor e of a mat ching game wit h complet e infor mat ion, and as t he ex ant e incent ive compat ible cor e, t he incent i ve compat ible coarse cor e is smaller t han t he set of int eri m payo¤s which cannot be blocked ( in t he sense of ( 6) and ( 7) ) by any single agent nor by any pai r

(i; j)

_{2 I £ J}

(see example 1 in sect ion 5) .

5

E x am p l es

I n t hi s sect ion, we mot i vat e t he basic model and t he sol ut ion concept of sec-t ions 2 and 3 on simple housing mar kesec-t s and we illussec-t r asec-t e sec-t wo di¤er ences bet ween mat ching games wit h complet e and incomplet e infor mat ion. Ex-ample 1 consist s of a simple auct i on ( one sell er ) wit h independent pri vat e values, in which ob j ect ions by coalit ions of more t han t wo agent s mat t er .

T he lat t er pr oper t y i s also t r ue i n t he second example, but i s less sur pr ising t her e, because values ar e common and cor r elat ed. T he goal of t his example is t o show t hat t he grand coalit ion deals bet t er wit h incent ive compat ibilit y t han small coalit ions. F inally, in example 3, random mechani sms improve t he char act er ist ic funct i on under incomplet e infor mat ion.

I n t he sequel, for

i

_{2 I}

and

j

_{2 J}

,

_¡u

_i

(t; j) = ¼

i

(t; j)

is int er pret ed as

t he mini mum pr ice at which seller

i

is willing t o sell his house t o buyer

j

when t he st at e of infor mat ion is

t

and

u

j

(t; i)

is int er pr et ed as t he maximum

pr ice t hat buyer

j

i s willing t o pay for sel ler

i

’s house. T he ut ilit y of being unmat ched is

0

for ever y t r ader

k

.

E x am p l e 1

A gent

1

, t he seller , has no pr ivat e infor mat ion and a null r eser vat ion pr ice; agent s

2

and

3

, t he pot ent ial buyer s, have independent pr ivat e values unifor mly dist r ibut ed over

f0; 1g

.9 T he basic par amet er s of t he assignment game ar e t hus:

I =

_f1g

,

J =

_{f2; 3g}

,

T

j

=

f0; 1g

,

j = 2; 3

,

t

2 and

t

3 ar e

i.i.d. unifor mly over

_{f0; 1g}

,

u

1

(t; 2) = u

1

(t; 3) = 0

,

u

j

(t; 1) = u

j

(t

j

; 1) = t

j,

j = 2; 3

.

We shal l show t hat in t his example,

v

¤_A

(

_{f1; 2g) = v}

_A¤

(

_{f1; 3g) =}

1

2

whil e

v

_A¤

(

_{f1; 2; 3g) =}

3

4

H ence a feasi ble vect or payo¤ like

(

1₂

; 0; 0)

cannot be bl ocked by any t wo agent coal it ion but can be blocked by t he gr and coalit ion.

Since t he seller is not submi t t ed t o incent ive const r aint s, one can r est r ict on t r ansfer s summing up t o

0

wit hout loss of gener alit y. I n par t icul ar , in a seller -buyer coalit ion

_{f1; jg}

,

j = 2

or

3

, a mechanism

(x; m)

can be de…ned by t he pr obabilit y of t r ade

x(t

j

)

and t he expect ed t r ansfer

m(t

j

)

fr om buyer

j

t o t he seller ,

t

j

= 0; 1

. T he t ot al expect ed gai ns fr om mechanism

(x; m)

ar e 1

2

x(1)

·

1

2. H ence

v

¤A

(

f1; 2g) = v

A¤

(

f1; 3g) ·

12. On t he ot her hand, 1

2 can be

9_{T he example cor responds t o t he simplest possible discr et e model. Everyt hi ng goes}

t hr ough if for inst ance, t he pair _{f0; 1g} is r eplaced by t he int erval [0; 1]. I n t hi s case,

v¤

achieved wit h a ( const ant ) mechanism sel ling always t he ob j ect at t he pr ice

1 2.

L et us t ur n t o t he grand coalit ion and consider t he feasible, incent i ve compat ible mechanism induced by a second pr ice auct ion, namely

0 1

0 no sale sell t o 3 at pr ice 0

1 sell t o 2 at pr ice 0 sell t o 2 or 3 wit h pr obabilit y 1₂ at pr ice 1

wher e t he r ows ( r esp., columns) cor r espond t o t he r epor t ed t ypes of agent

2

( r esp.,

3

) . T he t ot al expect ed gains fr om t r ade ar e 3₄, which is t he maxi-mum expect ed value t hat can be achieved, even in t he absence of incent i ve const r aint s.

One can also check t hat no seller -buyer coalit i on can bl ock t he pr evi-ous mechanism at t he i nt er i m st age. I ndeed, t his mechanism is classically ( i.e., wit hout incent ive const r ai nt s) ex ant e e¢ cient and int er im individually r at ional.10

E x am p l e 2

T he seller s, agent s

1

and

2

, bot h know, at t he int eri m st age, whet her t he quali t y of t heir own house i s high or low:

T

1

=

fh

1

; l

1

g

,

T

2

=

fh

2

; l

2

g

.

T he pr obabi lit y dist r ibut ion over t he seller s’ t ypes is

q(h

1

; h

2

) = q(l

1

; l

2

) =

3₈,

q(h

1

; l

2

) = q(l

1

; h

2

) =

1₈. T he pot ent ial buyer s, agent s

3

and

4

, have no pri vat e

infor mat ion. T he r eser vat ion pr ice of a seller for a high (r esp., low) qualit y house is

¼

h ( r esp.,

¼

l) , while for a buyer , t he r eser vat i on pr ices ar e

u

h and

u

l. We assume t hat

u

l

< ¼

l

< ¼

h

< u

h and t hat ₂1

(u

l

+ u

h

) <

1₂

(¼

l

+ ¼

h

)

( e.g.,

0, 9, 12, 20) .11

10

I f t he buyers’ t ypes ar e unifor mly dist ribut ed over [0; 1], coalit ion _{f1; jg}, j = 2; 3, can achieve t he vect or payo¤ (1

4; (tj¡12)I(tj¸ 12))wit h an incent ive compat ible, int erim

individually r at ional, mechanism. Fur t her mor e, one can check t hat 1₄ is t he maxi mum t he sell er can expect from a mechanism wit h t hese pr opert i es in a t wo-agent coal it ion. I n t he gr and coal it ion, t he second pr ice auct ion mechanism yields t he expect ed payo¤s(1

3; t2 2 2; t2 3 2)

and t hus st ricly impr oves all agent s’ payo¤s.

11_{For a concret e example, t hink of t he t er mit es invading some r egions of Fr ance. T he}

qualit ies of houses in t he same neighbor hood are highly corr elat ed. M any owners do not know whet her t hei r house is infect ed or not , but wil l go t hrough a t est in case of pot ent ial sale. H ence, t here is an ex ant e st age, befor e t he seller s know t he qualit y of t heir houses.One might ar gue t hat t he qualit y of a house is ver i…able in t his example but , given t he r isk of false cert i…cat es, an incent ive mechanism looks safer .

Since t he buyer s ar e not submit t ed t o incent ive const r ai nt s, we will, wit h-out loss of gener alit y, focus on t r ansfers summing up t o

0

t hr oughout t he example.

L et us …rst consider a seller -buyer coali t i on

_{fi; jg}

,

i

_{2 f1; 2g}

,

j

_{2 f3; 4g}

. T he seller has t hen t wo equipr obable t ypes, which we denot e as

h

and

l

. We face a simple, di scr et e ver sion of M yer son [19]’s “ lemon pr oblem” . A mecha-nism for a seller -buyer coalit ion consi st s of t he pr obabilit y of t r ade

x

h ( resp.,

x

l) when t he sel ler repor t s t ype

h

( r esp.,

l

) and t he cor r esponding expect ed

t r ansfer s

m

h,

m

l fr om t he buyer t o t he seller . B y eliminat ing t he t r ansfer s

fr om t he incent ive const r aint s ( see, e.g., [17] or [19]) , t he opt imizat ion pr ob-lem of a seller -buyer coalit ion is

v

_A¤

(

_{fi; jg) = max[}

1

2

x

h

(u

h

¡ ¼

h

) +

1

2

x

l

(u

l

¡ ¼

l

)] s:t: 0

· x

h

· x

l

· 1

so t hat , under our assumpt ions,

v

¤_A

(

_{fi; jg) = 0 i 2 f1; 2g; j 2 f3; 4g}

L et us set

g

h

= u

h

¡ ¼

h

> 0

. Obser ve t hat , in absence of incent i ve

const r aint s,

v

A

(

fi; jg) =

1₂

g

h. T r ade is indeed bene…ci al in st at e

h

, but t he

incent ive compat ibilit y condit ions pr event r evelat ion of infor mat ion fr om t he seller .

L et us t ur n t o t he gr and coalit ion. F ir st best e¢ ciency r equir es t o sel l t he high qualit y houses, and only t hose, at ever y st at e of nat ur e. H ence,

v

A

(K) = g

h

We will const ruct an incent ive compat ible mechanism achievi ng

g

h as sum of

expect ed payo¤s, so t hat

v

¤_A

(K) = g

h

Since

v

A

(S)

¸ v

A¤

(S)

for ever y

S

, t his wil l also show t hat

C(v

A

)

µ C(v

A¤

)

and pr ovide a simple pr ocedur e t o const r uct expect ed payo¤s in t he ex ant e incent ive compat ible cor e.12

12_{T he procedur e can be appli ed t o a large class of assignment pr oblems (see, e.g., [2],}

Consider a mat ching mechani sm in which only high qualit y houses ar e sold, e.g.,

x

2 f0; 1g

I£J _{descr ibed by}

x(h

1

; h

2

) =

µ

1

0

0

1

¶

x(h

1

; l

2

) =

µ

1 0

0 0

¶

x(l

1

; h

2

) =

µ

0

0

0

1

¶

x(l

1

; l

2

) =

µ

0 0

0 0

¶

T he sum of expect ed payo¤s fr om

x

i s

g

h. Obviously,

x

is not incent i ve

compat ible, but one can const r uct t r ansfer s

m

such t hat

(x; m)

is incent i ve compat ible. For inst ance, t he t r ansfer s

m

1 t o t he …rst seller must sat isfy

¡¼

h

+

3

4

m

1

(h

1

; h

2

) +

1

4

m

1

(h

1

; l

2

)

¸

3

4

m

1

(l

1

; h

2

) +

1

4

m

1

(l

1

; l

2

)

1

4

m

1

(l

1

; h

2

) +

3

4

m

1

(l

1

; l

2

)

¸ ¡¼

l

+

1

4

m

1

(h

1

; h

2

) +

3

4

m

1

(h

1

; l

2

)

A possible sol ut ion is

m

1

(h

1

; h

2

) =

3¼h₂¡¼l

+

g₄h

m

1

(h

1

; l

2

) =

3¼l¡¼₂ h

+

g₄h

m

1

(l

1

; h

2

) =

g₄h

m

1

(l

1

; l

2

) =

g₄h

T he t r ansfer s

m

2 t o t he second seller can be chosen in a si milar way. I n

order t o balance t he t r ansfers, one can simpl y set

m

3

=

¡m

1,

m

4

=

¡m

2.

T he mechanism

(x; m)

t hus associat es buyer

3

(r esp.,

4

) wit h seller

1

( resp.,

2

) but sale only t akes place if t he sell er ’s house is of high qualit y.

(x; m)

yields t he expect ed payo¤ gh

4 t o each t r ader. T he mechanism re‡ect s t hat

sale pr ices ar e in‡uenced by t he pr esence of l ow qualit y it ems; t he t r ansfer s in t he low st at e should be int er pr et ed as a fee t hat t he pot ent ial buyer s pay t o get infor mat ion and avoid a bad decision. M any ot her mechanisms achieving ex post e¢ ciency can be const r uct ed. I n par t icular , as in [8], it is possible t o design t he t r ansfer s in such a way t hat t he mechanism i s int er im individually r at ional for t he seller s, who fully ext r act t he sur plus13.

13_{A dding (resp., subt ract ing)} gh

4 t o (resp., fr om) all previous t r ansfer s gives t he sur

-plus t o t he seller s (r esp., buyers). A ll t hese mechanisms, including t he lat t er , are int erim individually r at ional for t he seller s. A not her mechanism wit h t he same proper t ies is

m1(h1; h2) =3uh₂¡¼l m1(h1; l2) = 3¼l¡u₂ h

L et us end t he analysis of t he example by showing t hat t he expect ed payo¤ fr om

(x; m)

, namely,

(

gh 4

;

gh 4

;

gh 4

;

gh

4

)

, belongs t o

C(v

¤A

)

. We have evaluat ed

v

A

(

fi; jg)

,

i

2 f1; 2g

,

j

2 f3; 4g

and

v

A

(K)

. T o complet e t he descr ipt ion of

v

A, we comput e t hat

v

A

(

fi; 3; 4g) =

g

h

2

i = 1; 2

v

A

(

f1; 2; jg) =

5g

h

8

j = 3; 4

H ence,

(

gh 4

;

gh 4

;

gh 4

;

gh

4

)

2 C(v

A

)

. I t foll ows fr om our pr evious r emarks t hat

(

gh 4

;

gh 4

;

gh 4

;

gh

4

)

2 C(v

A¤

)

. T he same r easoni ng applies t o

(

gh

2

;

gh

2

; 0; 0)

.

A s announced above, t his example shows t hat it may be bet t er for t he agent s ( i n t he sense of gener at ing a higher sum of expect ed payo¤s) t o st ay t oget her at t he ex ant e ( or int er im) st age in or der t o exchange infor mat ion wi t hi n the grand coali ti on. For inst ance, t his enables t he agent s t o exploit t he possible cor r el at ion bet ween t ypes and t o achieve …rst best e¢ ciency t hr ough full r evelat ion.

E x am p l e 3

T his example will con…rm t he advant age of r andom mechanisms, by illus-t r aillus-t ing illus-t haillus-t

v

_A¤

(K)

can be st r ict ly lar ger t han t he maxi mum sum of payo¤s t hat can be achieved wit h an incent ive compat ible det er mi ni sti c mat ching mechanism ( t o which we will r efer as

v

¤

AD

(K)

) . T he fr amewor k will consist

of one sell er ( agent

1

) and one buyer ( agent

2

) , wit h i ndependent types

t

1

and

t

2, but common values: at t he st at e of infor mat ion

(t

1

; t

2

)

, t he r eser

va-t ion pr ice of va-t rader

1

( r esp.,

2

) is

¼

1

(t

1

; t

2

)

( r esp.,

u

2

(t

1

; t

2

)

) . We will fur t her

assume t hat each t r ader only has t wo equipr obabl e t ypes (

T

k

=

ft

k1

; t

k2

g

,

q(t

k1

) = q(t

k2

) =

1₂,

k = 1; 2

) .

T his seems one of t he simplest models in whi ch one can expect t o have

v

_AD¤

(K) < v

¤_A

(K)

. For inst ance, under independent pr ivat e values, …rst best e¢ ciency can be achieved in an incent ive compat ible way, so t hat

v

¤

A

(K) =

v

A

(K)

( see, e.g., [3] and [4] and t he concluding r emar ks below) and t hus

v

_AD¤

(K) = v

_A¤

(K)

in t his case. I t is a r emar kable fact t hat

v

¤_AD

(K)

and

v

¤

A

(K)

also coincide in seller -buyer models wit h common values in which

only one t r ader has pr ivat e infor mat ion, as it can be checked by adapt ing M yer son [19]’s resul t s on t he “ lemon pr oblem” of t o our discr et e fr amewor k. L et us complet e our example by assuming t he following possible r eser va-t ion pr ices

¼

1

(t

1

; t

2

)

,

u

2

(t

1

; t

2

)

:

t

21

t

22

t

11

6; 2 2; 3

t

12

7; 6 0; 4

Since t he t r ader s’ t ypes ar e independent , one can r est r ict on mechanisms in which t he t r ansfer s sum up t o

0

wit hout loss of gener alit y14_{. Fur t her mor e, by}

eliminat ing t he t r ansfer s fr om t he incent ive compat ibilit y condit ions ( see e.g., Johnson, Pr at t and Zeckhauser [17]) , we can wr it e t he opt imizat i on pr oblem of t he seller -buyer coali t i on as

v

_A¤

(

f1; 2g) = maxf¡4®

11

+ ®

12

¡ ®

21

+ 4®

22

g

s.t .

0

_{· ®}

rs

· 1 r; s = 1; 2

and

®

11

¡ 2®

12

¡ ®

21

+ 2®

22

¸ 0

( 8)

¡®

11

+ ®

12

+ 2®

21

¡ 2®

22

¸ 0

( 9)

wher e

®

rs is t he pr obabilit y of t rade when t he seller ( r esp., buyer ) r epor t s

his

r

th _{( r esp.,}

_s

th _{t ype)}

_{r; s = 1; 2}

_{. ( 8) is t he seller ’ s incent ive compat ibilit y}

const r aint ( aft er eliminat ion of t he t r ansfer s) which expr esses t hat t he sum of his expect ed payo¤s when he always t ells t he t r ut h is larger t han t he sum of his expect ed payo¤s when he always li es. ( 9) has a similar int er pr et at ion for t he buyer .

T he only ex post e¢ ci ent mechani sm in t his example,

® =

0 1

0 1

, i s not

incent ive compat ible for t he buyer . T he mechanism

® =

0

2 3 2 3

1

sat is…es ( 8) and ( 9) - wit h an equalit y - so t hat

v

¤_A

(

f1; 2g) ¸ 4

.15

14_{A s shown in, e.g., [17], if t ypes ar e independent and ¹ = (x; m) is incent ive}

com-pat ible, so ar e (x; m), mk(tk) = Pt¡kq(t¡k)mk(t), and (x;m)e , mek(t) = mk(tk)¡

1 jKj¡1

P

l6=kml(tl).m(t)e is exact ly balanced for ever y t. 15_{I n fact , one can check t hat v¤}

A(f1; 2g) = 4. U nder t he feasibilit y const r ai nt s, t he

ob j ect ive is_{· ®12¡ ®21}+ 4while t he sum of t he incent ive compat ibilit y const raint s yields

We now check t hat

v

¤

AD

(

f1; 2g) < 4

. L et

®

be a det er minist ic mechanism;

if

®

11

= 1

, t he ob j ect ive

¡4®

11

+ ®

12

¡ ®

21

+ 4®

22 is

· 1

; and si milar ly if

®

22

= 0

.

v

AD¤

(

f1; 2g)

is t he maximum of

®

12

¡ ®

21

+ 4

s.t .

®

12,

®

21

2 f0; 1g

,

( 8) and (9) ( wit h

®

11

= 1

and

®

22

= 0

) , which is

3

.

6

C on cl u d i n g r em ar k s

I n t his paper , we have focused on t he ex ante assignment game ( which is de…ned wit hout any ambiguit y) and a possible version of t he int er im assign-ment game, in which agent s do not communicat e unt il t hey ar e in a coali-t ion. W e have escoali-t ablished coali-t hacoali-t coali-t he associacoali-t ed cor es ar e non-empcoali-t y. Ocoali-t her cor e concept s have been pr oposed at t he int er im st age, st ar t ing wit h W ilson [28]’ s …ne core ( de…ned in absence of incent ive const r aint s). T he de…nit ion of an int er im cor e concept t ur ns out t o be quit e delicat e, especially if some communicat ion is allowed at t he coalit ion for mat ion st age ( see, e.g., [9] [10], [15], [13], [20]) and we will not invest i gat e t his pr oblem fur t her her e. B ut t he pr evi ous analysis suggest s t hat t wo-sided mat ching games wit h i ncom-plet e infor mat ion could be an appr opr iat e fr amewor k in which t o appl y or devel op int er im cor e concept s. I ndeed, t he model has a simple, well-behaved st r uct ur e, a number of par t icular cases ar e well-under st ood ( e.g., seller -buyer bar gaini ng, auct ions,...) and t he benchmar k incent ive compat ible coarse cor e is not empt y.

L et us t ur n t o possible ex post pr oper t i es of ex ant e incent ive compat ible cor e solut ions. A number of paper s ( see, e.g., [2], [3], [4], [7], [8], [17]) ident ify assumpt ions ( on beliefs and ut il it y funct ions) which ensur e t he exist ence of mechanisms ( involving exact ly bal anced t r ansfer s) which ar e bot h incent i ve compat ible and ex post e¢ cient . Given t he T U aspect , it is not di¢ cul t t o see t hat t he expect ed payo¤s achieved by such mechanisms ar e in t he ex ant e incent ive compat ible cor e ( t his is illust r at ed on example 2 in t he pr evious sect ion) . I n ot her wor ds, one can easily pr oduce assumpt ions under which t he ex ant e incent ive compat ible core cont ains ex post e¢ cient mechanisms. Such r esult s ar e used in [11] ( wher e an analog of pr oposit ion 1 does not hol d) t o est ablish t he non-empt iness of t he ex ant e incent ive compat ible cor e.

e¢ ciency but not ex post individual r at ionalit y.16 Rot h [22] consider s a much st r onger r equir ement : ex post st abilit y in t he absence of t r ansfer s. H e anal yzes “ mar r iage pr obl ems” ( t ypically wit hout t r ansfer s) in which agent s pr ivat ely know t heir own ut ilit ies ( in our t er minology, “ pr ivat e val ues” ) . A s in t he pr esent paper , he consider s gener al mechanisms ( allowing in par t icular for lot ter i es over mat chings) and applies t he r evelat ion pr inciple in or der t o focus on dir ect r evelat ion mechanisms. H e calls such a mechanism “ st abl e” if it select s a st able mat ching for any st at ed ut ilit ies. To expr ess a similar st abilit y pr oper t y in our fr amewor k, let us de…ne, for ever y

t

_{2 T}

,

v

t as t he

( T U ) mat ching game ( wit h complet e i nfor mat ion) when t he t ypes ar e

t

. A s shown in [26],

v

t is t he super addit ive cover of

v

t

(

fkg) = u

k

(t; 0)

,

k

2 K

,

v

t

(

fi; jg) = u

i

(t; j) + u

j

(t; i)

,

i

2 I

,

j

2 J

. A mat ching mechanism

¹

is

“ st able” in t he sense of Rot h [22] if for ever y

t

_{2 T}

( int er pr et ed as a vect or of r epor t ed t ypes) ,

¹(t)

select s a solut ion in

C(v

t

)

, i.e., if

¹

is ex post st able.

Rot h ar gues t hat t his r equir ement is not “ excessively st r ong” i n a model wi t h pr ivat e values. H owever , he shows t hat incent ive compat ible, ex post st able, mat ching mechanisms do not exist in gener al. A s suggest ed above, in our fr amewor k ( wit h t r ansfer s) , it is not di ¢ cult t o const r uct examples in which incent ive compat ible ex post e¢ cient mechanisms do exist , but none of t hem is ex post individually r at ional.

Rot h [22] also obser ves t hat sever al proper t ies involving st rategy-proofness in mar r iage pr oblems wit h complet e i nfor mat ion have an immediat e count er -par t in mar r iage pr oblems wit h incomplet e infor mat ion but pr ivat e values.17 I n par t icular , t here exist mat ching mechanisms such t hat r evealing one’s t r ue pr efer ences is a dominant st r at egy for one side of t he mar ket . Such r esult s are also valid her e, but do not seem t o be hel pful in analyzing t he incent i ve compat ible cor e unless one side of t he mar ket cont ains only one agent ( as in st andar d auct ions) .

T he pr evious comment s r aise t he quest ion of t he validit y of our r esult s in mat ching mar ket s wher e monet ar y t r ansfer s ar e not possible ( like in Shapley and Scar f [25]) . Our solut ion concept s can be used in t his fr amewor k. Except for t he T U aspect in sect ion 3, all our r esult s hold when all monet ary t r ansfer s are imposed t o be null. T he ( N T U ) ex ant e incent i ve compat ible cor e is t hen non-empt y as a consequence of Scar f [24]’s t heorem.

16

Ex ant e and, in some cases like [7] and [8], int er im individual rat ionalit y can be guarant eed.

17_{A st r at egy-proof mechanism induces a domi nant-strategy incent ive compat ibl e}