Discrete public goods with incomplete information

FUNDAÇÃO GETULIO VARGAS

EPGE

Escola de Pós-Graduação em Economia

SEMINÁRIOS

DE

PES~QUISA

ECONOMICA

"Discrete Public Goods With

Incomplete Information"

Prof. Flavio M. Menezes

(EPGEIFGV)

LOCAL

Fundação Getulio Vargas

Praia de Botafogo, 190 - 10° andar - Auditório Eugênio Gudin

DATA

05/10/2000 (53 feira)

HORÁRIO

16:00h

Discrete Public Goods with Incomplete

Informa tion

1

Flavio M. Menezes

2

Australian National University and EPGEIFGV

À

avio.menezes@fgv.br

Paulo

K

Monteiro

EPGEIFGV

pklm@fgv.br

Akram Teminli

University of AIabama

atemilui@cba.ua.edu

July 27, 2000

1 We would like to thank Mark Armstrong. J ohn Conley. Bryan EUickson, Charles Kahn. Bart Lipman. George Mailath, Paul Pecorino, Todd Sandler, Harris Schlesinger and Hyun Song Shin for helpful discussions and comments. Menezes and Monteiro gratefiJlly acknowledge the i.nancial support ofCNPq. Temimi ac-knowledges the i,nancial support ofthe College ofBusiness Administration at the University of Alabama.

Abstract

We analyze simultaneous discrete public good games with incomplete infor-mation and continuous contributions. To use the tenninology of Admati and Perry (1991), we consider contribution and subscription games. In the for-mer, contributions are not refunded ifthe project is not completed, while in the latter they are. For the special case where provision by a single player is possible we show the existence of an equihbrium in both contnbution and subscription games where a player decides to provide the good by himself. For the case where is not feasible for a single player to provide the good by himself: we show that there exist equilibria of the subscription game where each participant pays the same amount. Moreover, using the technical ap-paratus from Myerson (1981) we show that neither the subscription nor the contribution games admit ex-post eÁ cient equibbria. hl addition. we provide a suÁ cient condition for êontributing zero 'to be the unique equihbrium of the contnbution game with n players.

JEL ClassÍ(,cation:D79, D89, H89

1

Introduction

The literature on private provision of public goods can be divided into two broad categories.1 The i,rst branch of the literature focuses on the provi-sion of continuous publie goods. Papers include Warr (1982, 1983), Comes and Sandler (1984), Bergstrom, Blume and Varian (1986), Andreoni (1988), Gradstein, Nitzan and Slutsky (1994) and others. A standard resuh in this literature is that public goods are underprovided by voluntary contnbutions due to free riding behavior.2 One might conjecture that the govemment can solve this underprovision problem by providing some ofthe good and l,nanc-ing it by imposl,nanc-ing taxes on contnbutors. However, Warr (1982) and Roberts (1984), in two inÀ uential papers, show that govemment contnbutions result in a dollar to dollar reduction in private contnbutions if the tax on con-tributors does not change the set of contnbuting individuais. Bergstrom, Blume and Varian (1986) show that the crowding out e% ect is only partial if one allows for the taxes that pay for the govemment contribution to be also collected from non contnbutors. Hence the literature suggests that un-deiprovision is a robust conclusion if the public good leveI is endogenously determined by voluntary contnbutions.3

The seeond strand ofthe literature foeuses on discrete public goods where a i, xed leve Iof a public good is provided if enough contributions are collected to cover its cost c. Otherwise, the good is not provided. Typical examples include building a bridge, a hbrary ofa certain size, public radio fund raising to l,nance a certaill programo hl all of these examples, if enough money is raised to eover the cost of the publie good, then the good is provided,

1Jn this paper we foeus on voluntary private provisiono There is an extensive mech-anism design literature that treats publie provisiono Papers in this literature develop eÁ cient meehanisms for the provision of publie goods, e.g .• Gradstein (1994). Maskin (1999), d'Aspremont and Varet (1979. 1982). Palfrey and Srivastava (1986) and Comelli (l996).

2The free riding problem becomes worse in the presence of incomplete information. Gradstein (1992) eonsiders a dynamic model ofprivate provision for a continuous public good with incomplete intormation with the restrictioll that players either contribute zero or all exogenous positive amount. He concludes that in additioll to the standard under-provision results, ineÁ ciellcy occurs because ofa delay in contnbutions. This ineÁ cieney does not disappear as the population becomes large.

otherwise the good is not provided.

Palfrey and Rosenthal (1984) developed the (,rst modem treatment of private provision of discrete public goods. More speci(,cally they anaIyze contribution and subscription games 1to use the tenninology of Admati and Perry (1991) 1for a discrete public good under complete infonnation where players' strategies are restricted to either contnbute zero or an exogenous positive amount. In a contribution game, contributions are not refunded if the sum of the contnbutions does not cover the cost of the public good c, while in a subscription game players get their money back ifthe project is not completed. An example with the main features ofa subscription game can be given as follows. The Wisconsin Govemor has recently pledged $27 million in state bonds to (,nance a new $72 million basketball arena on the condition that the rest ofthe money be raised by private donations (Andreon~ 1998). That is, the Govemor will provide $27 million as long as the remaining $45 million is raised, otherwise the 0% er is cancelled. Other exampIes can be found in the fund-raising literature. ExampIes ofcontnbution games include public radio and TV fimd-raising e%orts where contnbutors do not get their money back ifthe program is not provided. Other examples of contributions garnes are situations where contributions take the fonn ofphysicallabor, in this case volunteers cannot recover their e3_{i4ort ifthe project is not compIeted.}

h1 contrast with the standard underprovision resuIt for continuolls public goods, Palfrey and Rosenthal (1984) obtain the startling concIusion that eÁ cient provision of a discrete public good is a possibIe outcome of both contribution and subscription games even though ineÁ cient equihbria also exist. Bagnoli and Lipman (1989) extend Palfrey and Rosenthal model by allowing individuais to make continuous contnbutions. They show that the set of undominated perfect equilibrium outcomes of the subscription game is not only eÁ cient but coincides with the core ofthe economy. Theyalso show that with a dynamic version ofthe subscription game, it is possible to obtain eÁ cient outcornes even if the leveI of the public good is not binary as Iong as the nurnber of units of the public good is countabIe.4 It follows

4 Admati and Perry analyzed the role of commitment in the provision ofpublic goods.

More speci(.cally, they examined two-pIayer contnbution and subscription games with complete and pertect information where pIayers ahemate in making pledges or contribu-ti011S until the public project is completed. They show that Witll linear costs, a necessary

equilib-that the general conclusion from the theoretical literature is equilib-that private provision ofdiscrete public goods with complete information can be eÁ cient. Moreover, eÁ cient provision is a possible outcome for both subscription and contribution games.

An important question that has not been addressed in the literature on voluntary provision of discrete public goods is to what extent these results generalize to a model where players have incomplete information about other p1ayers' valuation of the public good. Moreover, there is both casual evi-dence from the fund-raising literature ofthe superiority ofsubscription games and experimental evidence from Bagnoli and McKee (1991) and Cadsby and Maynes (1999) that a refund increases the chance ofproviding the good. More specij, cally, Cadsby and Maynes consider a discrete public good experiment. They provide experimental evidence showing that provision is encouraged in a subscription game vis-a-vis a contribution game. They also provide evi-dence that a high c discourages provision in the contnbution game but not in the subscription game. Our results wil1 conl. nn these i,ndings.

In this paper, we examine contribution games and subscription garnes for a discrete public good in the presence of incomplete information. We l.rst aTialyze provision games where it is feasible for a p1ayer to provide the public good by hinlself. Common examples inc1ude opening a window in a hot room, rescuing an injured person in a traÁ c accident, etc. We show that when the cost ofprovision eis not prohibitively high as to prevent a single p1ayer from providing the good, there always exists an equihbrium where a p1ayer with a suÁ ciently large valuation provides the good in both the subscription and the contnbution game.

We also examine contribution and subscription garnes when contnbutions or pledges from more than one individual are necessary in order for the good to be provided. We will show that there exist equal sharing equihbria in the subscription game, i.e., equihbria in which each p1ayer contributes the same share of the total cost of providing the public good.. We also ap-ply the mechanims design techniques developed in Myerson (1981) to show that neither the contribution game nor the subscription game admit ex-post eÁ cient equilibria. Final1y we will provide a suÁ cient condition for 'con-tributing zero' to be the unique equilibrium ofthe contribution garne. This conl.rms the evidence from the experirnentalliterature ofthe superiority of subscription games over contnbution games.

This remainder ofthe paper provides more details on all ofthese issues. In Section 2, the speci<. cs of the model are presented. Section 3 provides an example that illustrates the main resuhs of the paper. In Section 4, the case where a single individual can provide the good by himself is ana-lyzed. A cost-sharing type equilibrium for the subscription game is identÍ{, ed Section 6 examines the ex-post eÁ ciency of contnbution and subscription games while in Section 7 we demonstrate that the contribution game can be extremely ineÁ cient. Section 8 illustrates a di%erential equations approach to i,nd equihbrium strategies in games with two players. Our main results are summarized in Section 9.

2

The Model

We now present the formal model There are N ~ 2 individuaIs with private independent values for a discrete public good. That is, each individual i,

i

=

1, ... , N knows his own va lue Vi but only the distnbution ofhis opponents' values. Values are assumed to be independently drawn from a continuous distnbution F. The cost ofproviding the public good is c.

With the exception of Section 4, we assume throughout the paper that an individual cannot provide the good by himself. h1 this case, F (c) = 1 and thus the distnbution has a bounded support. As a normalization we suppose, without loss of generality, the support to be contained in the interval [0,1].

We will examine both subscription and contribution games. h1 a sub-scription game the money contributed by players is retumed if the sum of the pledges is less than c. In a contribution game the money is not retumed even if the good is not provided. Whatever the method to elicit donations is, if dona tions add up to more than c, the additional money is not retumed to the contnbutors.

More specÍ{, cal1y, we analyze one-shot sinlUltaneous move contnbution and subscription games for a discrete public good in the presence ofincom-plete information about preferences.5 Our model aIso relaxes the binary

tnbution restriction imposed in the literature.6 _{While there are important}

instances where binary contnbutions are relevant due to transaction costs, in general players can give any amount ofmoney they desire (alumni dona-tion, donations to a hbrary, etc.). Moreover, in a continuous contributions framework. individuaIs make contributions that best match their preferences as opposed to a discrete contnbution model Cadsby and Maynes (1999) provide experimental evidence showing that allowing continuous rather than binary à Il-or-nothing 'contnbutions facilitates provisiono

As it is standard in many games of incomplete infonnation with ex-ante symmetric players lsuch as in industrial organization and auctions lour focus is mainly on syn1metric equilibrium. Nevertheless, we provide ineÁ-ciency resuhs that hold for both symmetric and asymmetric equilibria; we will appeal to the revenue equivalence theorem to show that for any equilib-rium the probability ofprovision ofthe public good given that it is eÁ cient to do so is less than one in both contnbution and subscription games. We aIso provide suÁ cient conditions under which contnbuting zero is the unique equihbrium ofthe contnbution game.

The multiplicity of equibbria that characterizes these games has two sources. Firstly, this is an instance ofa coordination game. With two players who cannot provide the good by themselves, the coordination nature ofthe game is ele ar. With more than two players, then we have to account for all possible combinations ofplayers who can provide the good as a group. This adds a new layer of complexity to standard coordination games. Secondly, the incompleteness of in forma tion and the assumption that player can con-tnbute or pledge any values they like aIso creates additional coordination issues beyond the standard coordination problem. It is the combination of these two features that makes it diÁ cult to obtain uniqueness of equihb-rium. In particular, as we will explain in the context ofthe example below, well-known arguments to obtain uniqueness, such as the iterated deletion of dominated strategies as in Bagnoli and Lipman (1989) (with complete infor-mation) and Milgrom and Roberts (1990) (with incomplete infonnation) for the case ofsupemlOdular games, do not apply in this context.

3

An

Example

Before turning to an anaJysis ofthe mode~ it is instructive to consider a sim-pIe example where it is possible to make the point that the eÁ cient resuhs in the discrete public goods literature are not robust to the introduction of incompIete information. This example incorporates the simplest structure for which this lack of robustness is evident and for which we obtain a strik-ing distinction between predicted outcomes in subscription and contnbution games. We begin with the complete information case.

3.1

Complete InfoTI11ation

Assume that players 1 and 2 have valuations for the public good V1 and V2 in [0,1] and that are common knowledge. Let the cost ofprovision ofthe public good c be such that 1

<

c

<

2 and V1 +V2 ~ c. That is, a single player cannot provide the good by himselfand it is socially eÁ cient to provide the good. If

we denote by bi , i = 1,2, player i's pledging strategy in a subscription game,

then we can characterize the set of (pure strategy) equilibria as follows:

(ü) (O, O);

The equilibria described in (i) are eÁ cient, while the equilibria descnbed in (ü) and (m) are not.7 _{Moreover, the equilibria described in (ü) and (m)}

are not strict as best responses are not unique.8

Note that the equilIbria ofthe contribution game can be descnbed by (i)

and (ü) above. Given that individuais in a contnbution game never receive their contnbutions back, the strategies prol.les described in (m) are no longer equilIbria. UnIike in the case ofthe subscription game, the equilibrium strat-egy (O, O) is now a strict equilibrium. That is, contnbuting zero is the unique best response when the other player contributes zero.

This example suggests that under complete information the distinction between contribution and subscription game both in terms of eÁ ciency and

7111e equihbriurn described in (ü) is knOWIl as the strong free riding equihbriurn.

in terms of expected outcomes is unclear. We now consider the incomplete information case.

3.2

Incomplete Infonnation

We now assume that VI and V2 are independent draws from the uniform [0,1] distnbution. Moreover, Player i = 1,2 knows his own valuation but only the distnbution ofhis opponent's valuation. Let us suppose that 1

<

c

<

2. The subscription game has many symmetric equihbria and the contnbution game has on1y the strong free riding equilibria. Let us see some subscription game eq uihbria (, rs t:

1. (O, O);

D. b*(v) = { 0, if V

<

~

c if

>

c

2' V - 2

{

2c-I

+

v

lli. b*(v) = - 6 - 2'

0,

if 2c-I

<

V

< 1

3 -

-otherwise.

That (O, O) is still an equilibrium in the incomplete information follows trivially from the fact that a single player cannot provide the good by him-self. Now let's check that the strategy pro(,le descnbed in (ü) is indeed an equilibrium. Suppose Player 2 follows b*(.). IfPlayer 1 's value is less than ~, then 1 's best reply is to pledge any number less than ~. hl particular, pledging zero is a best response. IfPlayer 1 's value is greater than equal to ~, then Player 1 's best response is to pledge ~. Pledging less than ~ Player 1 obtains zero prol. ts; by pledging more than ~, Player 1 does not increase the probability that the good is provided but decreases prol. ts conditional on provISIono

The equilibrium pledging strategies descnbed in (iü) can be explained as follows. A player pledges the equivalent to the expected value ofthe other player being lower than his 0\\11, conditional on the interval [2C

ã

I ,

1],

that is,

on the relevant interval where pledges are less than or equal to the valuations for the public good, i.e. b(v) ~ v. A formal derivation is left to the appendix.

is not supermodular: Consider two strategies b1 = O and b2 = c and note

that f(b_{2 ,} b_{2 ) -} f(b_{2 ,} b1 ) - [f(bI, b2 ) - f(bI, b2 )] = (v - c) - (v - O) = -v

<

O. Of course this multiplicity of equilibria can potentially raise diÁ culties in questions ofwelfare. Nevertheless, we wilI show later that any equilibria ofthe subscription game is ineÁ cient. More precisely we wilI show that the probability ofprovision ofthe public good given that it is eÁ cient to provide is (unifomlly)less than one in any equilibrium.9 Moreover, we wilI provide a suÁ cient condition under which the welfare comparison between subscription and contribution games is straightforward las the only equilibrium ofthe contribution game is to contnbute zero.

Suppose (bI, b2 ) is an equilibrium of the contribution game. It is

imme-diate that b_{2 (V2)} ~ V2. Let us i,nd the best response of Player 1 to b_2• If Player 1 bids x ~ O his expected utility is

If x

=I

O and c - x ~ 1 then

cp

(x) = -x

<

O. Now suppose c - x

<

1. We have

cp

(x) ~ VP(V2 ~ C - x) - x = v(l - (c - x)) - x = v(l - c)

+

(v - l)x

<

O.

Thus x = O is the best response. (Note that we are not limiting ourselves to symmetric pure strategy equilibrium). Therefore, despite the multiplic-ity or equilibria of the subscription game. we are still able to make welfare comparisons. More precisely, the contribution game is Iess eÁ cient than the subscription game in this example. This result contrasts greatly with the case where players have complete information and both games have eÁ cient equilibria.

4

P

rovis ion

by

a

S ingle P

la

ye

r

In this section, we analyze contribution and subscription games when valua-tions for the public good can take values larger than the cost. The following theorem shows that there always exists a symmetric equilibrium where a player with a suÁ ciently high valuation provides the good by himself.

9This probability is equal to 1/2 in the equilibrium descnbed in (ü) and equal to 2/3

Theorem 1 Suppose F : [0,00) --+ ]R is a continuous distnbution. Suppose there are N ~ 2 p1ayers for a project with cost c

>

O and that F (c)

<

1. Then there exists an o:

>

O where o: solves o:F (o:)N-l = c such that

b (v) =

{O

~

v:::; a c if v>o:

is an equilibrium strategy for both contribution and subscription games.

Proof. Suppose p1ayers n = 2, ... ,N p1ay according to b(·). Let us l,nd the best response ofP1ayer 1. Ifhis value is v

>

O and his contnbution is x ~ O, his expected payo~ in the contribution game is given by

9 (x) = v Pr (x

+

b ( V2)

+ ... +

b (VN) ~ c) - x

Since b(V2)

+ ... +

b(VN) is either O or not less than c, it follows that if x

<

cthen g(x):::; g(O). Thus max{g(x);x~O} = max{g(O),g(c)} =

max

{v

(1 -

F (o:)N-l) , V - c} . Hence if

v

<

F(Ct)N i = 0:, the best

contri-bution is x = O. If v

>

o: the best contribution is x = c. And if v = o: the player is indi3;" erent between x

=

O and x

=

c. For the subscription game, the expected surplus 1S given by

1

(x) = (v - x) Pr (x

+

b ( V2)

+ ... +

b (v N) ~ c)

The proofis identical to the contnbution game since 1(0) = g(O) and I(c) =

g(C).10 •

Theorem 1 implies thatwhen the costofprovision is notprohibitivelyhigh as to prevent a single player from providing the good, there always exists an equihbrium where a player with a suÁ ciently large valuation provides the good by himself.

5

The Subscription Game

In this section we ana lyze the subscription game where a single player cannot provide the good by himself. This is the case only if F(c) = 1. Thus the

10The solution is not unique in general For example if N = 2, F (x) = x, x E [0,1), then if O < c < l/e another equihbrium strategy for the contribution game is b(x) =

distnbution F has a bounded support and without 10ss of generality we suppose the support is the interval

[0,1].

Players' values are detennined by independent draws from a continuous distribution F :

[0,1]

- t

lR.

The cost

ofthe public good is c E [m, m + 1) where m ~ 1 i.e. m + 1 is the minimum number ofplayers needed to provide the public good.

Ifplayers i = 2,3, ... ,N follow the function b(·) and player 1 with valu-ation v contnbutes x ~ O, his expected surplus is

</> (x) - (v - x) Pr (x

+

b ( V2)

+ ... +

b (v N)

>

c) (1 )

- (v-x)Pr(b(v2)+ ... +b(VN)~C-X)

We see from the above that if N

>

2, to \vrite </> (x) as a function ofthe dis-tnbution ofindividual valuations, F, detailed infonnation on b(·) is needed; it does not suÁ ce to know that b(·) is increasing in the private valuation. Hence the standard technique that is used to characterize equilibrium in-volving increasing strategies in games ofincomplete infonnation cannot be used in this problem when N

>

2. Moreover, it would be fruitless to try to guess all the fimctional fonns that equihbria might take. Thus, it would be extremely diÁ cult to (, nd all the equihbria ofthe subscription game.

5.l

Cost Sharing Equilibria

In this subsection, we show directly that there exists an equilibrium with cut-OY4 strategies where the good is provided if m

+

1 p1ayers have a value greater than a cel1ain cut-o% value.

Proposition 1 Supposc that p1ayers's valuations are distnbuted in the

[0,1]

interval. S uppose the cost ofthe public good is c E

[m,

m+ 1) and the number of p1ayers N ~ m

+ 1. The following strategy is a symmetric equihbrium

strategy ofthe subscription game:

b (v) = {

°

c

m+l

if v:S; aj

if a:S;v:S;1. (2)

where a is such that

CíV-l (1 - F (a))m F (a)N-l-m c

a

-2:~:~ CRr-l (1 - F (a))h F (a)N-l-h - m

+

1·

(3)

d Ch n!

Proof. We (. rst prove the existence of ao Del ne

h(a) = a--~~~--~~--~~-=~~ C'N-l (1 - F (a))m F (a)N-l-m

2:~:r! C~_l (1 - F (a))h F (a)N-l-h aC'N_l

N-l o

C'N-l

+

2:

C~_l (1 - F (a))h-m F (a)m-h

h=m+l

Note that

lim h (a)

= O

and lim h (a)

=

CC~-l

= 1

a-+O a-+l N-l

Since h is a continuous function, by the intennediate value theorern there exists an a such that h (a) = m~l E (0,1) o

We now prove that b (o) is an equihbriurno Suppose v

<

1 and pIayer 1 pledges x ;::: O and players n = 2, o o o ,N foIlow b (vn ) o The expected utility

ofp1ayer 1 is given by

",(x)

~

(v - x)Pr

(t,b(V

n

):2:

c - x) .

The range of

2::=2

b (vn ) is

{O,

m~l'

':~l'

o o o ,

(~~lr

} .

If _{c - x E m+l' m+l'} (.E... U+l)Cj th _{e n}

Thus the optirnal pledge x* ~ v is such that

* { (j

+

1)

c o

>

I}

x E c- 0J - o

m+1'

-Thus

x* = c - (j*

+

1) c E [O 1) JO

*

>

-l.

-Therefore j* E {m,m - I} since ';~1

2::

2m~1

2::

1. Finallywe have j* = m-1

if and only if 4> (m~J

2::

4> (O) or equivalently if and only if

or

Since

Pr

(tb(Vn)

=

me)

=

C;;_1

(1- F (a))m F (a)N-l-m

n=2 m

+

1

and

Pr

(t,b(Vn)

~

m"": 1)

=

~

CJv-l

(1-

F (a»h F (a)N-l-h

we conclude that 4> (m~l)

2::

4> (O) ifand 0111y ifv

2::

a . •

Remark 1 If every player is pivotaL i.e. if N = m+ 1, then the equilibrium (2) is given by:

( ) _ {O

if v

<

m~1

b v - _c_ if _c_<v<1.

m+l m+l -

-Thus each pIayer considers the cost equally divided among the pIayers and contnbutes ifand only ifhis value is at least his share.

6

Ex-post EÁ ciency

eÁ ciency (the probability ofprovision given that it is eÁ cient to provide the public good) goes to zero. They also claim, although they do not provide a proot: that even for (,nite n that there are no ex-post eÁ cient mechanisms to provide the public good within the framework that we established above. For the case ofn = 2. the ineÁ ciency follows from applying the well known result ofMyerson and Satterthwaite (1983) with the interpretation that player 1 is a buyer with valuation VI distnbuted in the interval

[0,1]

and player

2

is a

seller with valuation c - V2, v2 E

[0,1].

The idea is that player 2 will pay c to provide but receives a transfer from pIayer 1 in order to cover part ofthe cost ofproduction. For n

>

2. it is more diÁ cult to infer the result direct1y. However. the same principIe that is applicable in Myerson and Satterthwaite is also applicable here, namely, the revenue equivalence theorem ofMyerson (1981). We use such theorem not only to show ex-post ineÁ ciency ofboth contnbution and subscription games but also to compute an upperbound for the ineÁ ciency.

6 .. 1

Subscription and contnbution games m the

Jan-guage of mechanisms.

Suppose biddcr i is type Vi

E

Ti =

[0,1], 1

~ i ~ n. Dei,ne T =

I1i=ITi.

lfv E Twe write V-i = (VI, ... ,Vi-I,Vi+l,'" ,vn ). And the expectation

operator E-i denotes the expectation among the variables V-i. We dei, ne a directmechanism as a function (y,x): T ---+ {O, l}x1Rn. The good is provided

if y (v) = 1 and Xi (V) is player i's payment. Thus his expected utility. ir his type is Vi is CPi (Vi) = E-i [ViY (V) - Xi (V)]. Using Myerson's revelation principIe we only consider mechanisms that are individually rational and incentive compa tible. Tha t is

CPi (Vi)

>

°

(IR)

CPi(Vi)

>

E-dviY(S,V-i)-xds,V-i)],'tSETi. (lC)

We may rewrite (IC) as

CPi (a) - CPi (b) ~ (a - b) E-i [y (b, V-i)] , 'ta, b E Ti.

Using the same argument as in Myerson(l981) we conclude that

The voluntary participation constraint is satisi, ed if and only if <PiO

>

O. Bidder i expected payment is therefore

We could prove in the same way as Myerson's the following

Lemma 1 The direct mechanism (y,x) satisfy (IC) and (IR) ifand only if

(5), <PiO

>

O, and u --+ E-i

[y

(u, V-i)] is non-decreasing for every i.

6.2

EÁ cient mechanisms are 110t budget balanced.

We need two dei, nitions:

Dei,nition 1 The mcchanism (y,x) is eÁcientify(v) = 1 ifand onlyif

E~l VI ~ c.

Dei, nition 2 The mechanism (y, x) is budget balanced if for every V E T,

tX1

(v)

~

O and y(v)

(tX1

(v) - c)

~

O.

In this subsection we prove that an eÁ cient mechanism for public provi-sion cannot be budgct balanced. Although this is a Imown result, a simple and direct proof is not available. Note that subscription and contnbution games are budget balanced.and, therefore. we can conclude that they are not eÁ cient.

Proposition 2 Suppose (y,x) is an eÁ cient mechanism. Then

Proof. We need to estimate Pr (l:~=l XI (v) ~ cll:_l VI ~ c). Thus

Pr

(~X

(v)

>

cl

~

v

>

C)

<

Pr

(l:~=l

XI (v)

~

c)

<

E

[l:~=l

XI (v)] .

f:::.

I - ~ I - - Pr (l:I VI ~ c) - c Pr (l:I VI ~ c)

We now proceed to (,l1d an upperbound for E [l:~=l XI (v)] . Since

J;i

E-i

[y

(u, V-i)] du =

E-i

JoVi

[y

(u, V-i)] du using (5) the expected paymellt is E [Xi (v)] = E [ViY

(v)]-<PiO - E

(J;i

Y (u, V-i) du) . Thus

Ifthe mechanism is eÁcient then Y(U,V_i) = 1

<=>

u ~ C-l:j#iVj'

Ifc-l:j#i Vj

>

Vi then

JoVi

y (u, V-i) du

=

O. lf c-l:j#i Vj ~ Vi then

JoVi

y (u, V-i) du =

Vi

~

(c -l:#i Vj) + . Substituting this in (6):

Therefore

Proof. Del. ne X = 'P ({I, ... ,n}). For every v E [0,1]" de(,ne H (v) =

{i:5

n;c- L:j#iVj

2::

O} and hv = #H(v). Ifhv

~

1 we may rewrite

t ( c - LVj)+ - L (c- LVj) =hvc-hv L Vj-(hv-l) L Vi=

i=l j#i iEH(v) i'/=i jrtH(v) iEH(v)

hvc - L Vj - (hv - 1) t Vi = C - . L Vj

+

(hv - 1)

(c -

t

Vi)

:5

c.

jrtH(v)

1=1

JrtH(v) 1=1

The last inequality is strict with positive probability. Thus sinee

L E [c. XL:i=l

VI~cXH(V)=H]

HEX

we (,nish the proof. •

Remark 2 lt is true that

The proofis inllTIcdiate using E

[J;i

[y

(u, V-i)] du] = E

[lffv~)i)y

(v)

f

(v) dv] .

6.3

The probability of provision of budget balanced

mechanisll1s.

We now foeus on budget balanced mechanisms. Sinee they are neeessari1y ineÁ cient it is important to know how ineÁ eient they are. We will rneasure ineÁ ciency by the probability o fprovision given that it is social1y desirable to provide the good. More precisely, if(y, x) is a budget balanced rnechanisrn we want to obtain an upper bound for Pr (y (v) = 1\

2:7=1

VI ~ c). Note that the

Proposition 3 If (x, y) is a budget balanced mechanism then

Proof. The following chain ofillequalities prove the proposition .

•

Example late

Suppose n = 2 and the unifonn distnbution. We have to

calcu-Suppose 2

>

c

>

1. Then

fa+b>c2

(a

+

b -1)+ dadb = 2

f

LI f La

(a

+

b -1) dbda =

2 J.l _c-I (_1 + ₂ 1a₂ 2 - 1c₂ 2

+

c) da - _i -₃ 3c2

+

4c +

~C3.

₃ And Pr (a

+

b

>

_{_} c) =

(2_,.,2 -4-3c2+4c+ 2c3 _{22 1}

~. Thus Pr(y(v) = 11vl +V2

>

c):5 1 ~ 1 = 3

c; .

We now l.nd

• 2 C

a lower bound for lhe supremum eA ciency. Consider the following symmetric equilibrium 11:

b*(v) = - 6 -

+

2' .1 - 3 - _ V - , {

2c-l v 'f 2c-l

< <

1

0, otherw1Se.

W _{e nee to compute} d Pr({(v}'v2);b*(VI)+b*(V2)~C}) _{P ( { ( ) } ) '} Notethat r VI, V2 ; VI

+

V2 ~ c

Pr ({(VI, V2); b*(VI)

+

b*(V2) ~ C })

_ H

4

_~

_2C)'

₊

Cc;

_{1 -}

_{(c _}

_{1»)(1 _}

_c;

_{1 )}

_ i _ ic

+

~C2

=

~

(c _ 2)2

3 3 3 3

Moreover Pr ({(VI, V2); VI

+

V2 ~ c}) = ~ (2 - C)2. Therefore,

Pr (b*(VI)

+

b*(V2)

~

CIVI

+

V2

~

c) =

~.

From the cost sharing equilibria described in the previous section we know that the probability of provision given that is eÁ cient to provide the good, ahhough less than one, is positive in the subscription game. In the next section we demonstrate that the probability of provision in a contnbution ga-me is equal to zero when the cost ofprovision is slightly above the mean ofthe distribution ofvalues.

,

7

The Contribution Game Strong IneA ciency

We now prove a much strong result for the contnbution game. Namely that the probability of provision is zero if the cost c is high enough. In terms ofmechanisms the contribution game is a mechanism (y,x) such that Xi (V)

does not depel1d on V-i' An all-pay auction is a contnbution mechanism in

this sensc.

The following thcorcm shows that the coordination problem in the contri-bution game is so severe that ifthe cost ofthe public good is slightlyabove the aggregate mean of the valuatiol1s, tben the unique equilibrium of the cOl1tributiol1 game is for each p1ayer to contribute zero no matter what his value is.

Del. ne I.L = E

[VI]'

Proof. The contribution game is the game in which the mechanism Xi (V) =

Xi (Vi) for every V-i. Thus from (5) we have

Xi (Vi) ::::; ViE-i [y (V)]

_lVi

E-i [y (U,V_i)] du::::; E_iy (1, V-i) .

Thus E~=l Xi (Vi) ::::; E~=l E_iy (1, V-i)' Since provision neveroccurs ifEI VI

<

C, E_iy (V)

<

Pr (EI#i VI

~

C - Vi) . Therefore the following inequality is

true:

tXi (Vi)

<

t P r (LVI

~

c- Vi) ::::; t P r (LVI

~

C-I)

= nPr (LVI

~

C-I).

i=l i=l I#i i=l l#i 1#1

Thus

n E

[EI#l VI]

(n - 1) J.L

~ Xi (Vi) ::::; n

=

n

<

C

~ c-I c-I

i=l

ending the proof. •

Remark 3 If Cn = ~

+

~Jl

+

4n (n - 1) J.L it is true that ~ -+ J.L if n -+ 00. Thus if cost per capita is higher than the average the strong free riding

equihbrium is the unique equilibrium ofthe contnbution game.

Remark 4 Del. ne (j2 as the distribution variance. A more precise inequal-ity for the strong ineÁ ciency lower bound is given by the positive root of

(c - 1 - (n - 1) J.L)2 C = n (n - 1) (j2. We omit the proof.

Example 2 If F (x)

=

x then J.L

=

!

and (j

=

vb.

Then ifn

=

30, c ~ 17.5

If n = 100, c ~ 54.39.

The above result implies that for a wide range ofthe cost ofthe public good there is extreme underprovision when refunds are not allowed.

Remark 5 For small n we can obtain sharper bounds on Cn. For example if n = 2, and the distribution fimction satis(.es F(x) ~ x,x E [0,1] then the free riding equilibria is the unique equilibria of the contnbution game. We omit the proof.

8

The di% erential equation approach to (, nd

e quilibriu m s tra te gie s

The theory ofsingle-object auctiollS with private independent values provides us with a method to l. nd the (symmetric) equihbrium strategy that is very natural. Usually oue supposes that bidders i = 2, ... ,N bid accordingly to the strictly increasing strategy b (.) and i,nds Bidder 1 's best reply x by means ofthe (. rst order condition for expected utility maximization. Then in equilibrium x = b (v) and we obtain a di% e re ntia 1 equation for b (.). Let us try

to mimic this approach to i., nd equihbria for the contnbution and subscription games. Recall from the discussioll following equation (1) that if N

>

2, detailed infonnatlon on b(·) is needed in order to derive the distnbution of L:j#i b(vj). Let us therefore suppose N

=

2. We suppose also that

f

=

F'

>

O exists and is continuous everywhere. First we examine the subscription garoe.

8.1

The subscription game di% erential equation.

So suppose Player 2 bids accordingly to b(·) which is strictly increasing. P1ayer 1 's expected surplus given that be has a value v and contnbutes x ~ O is given by

cp

(x) - (v - x) Pr (b ( V2) ~ c - X) _ (V - X) Pr (V2 ~ b-1 (C - X))

_ (v-x)(1-F(b-1_(c-x))).

(7)

(8)

Note that we took the inverse of b (-) to pass from (7) to (8). Player 1 's expected payo% is simply bis surplus if the project is completed times the probability ofcompletion. The i,rst-order condition is

cp'

(x)

= -

(1 -

F

(b-

1 _{(c - x)))}

+

_{(v - x)}

_{f (b-}

1 _{(c - x))}

_(b-

1_{)' (v - x)}

=

o.

1n a symmetric equilibrium x

=

b (v), so

(v - b (v))

f

(b-

1 _{(c - b (v)))}

(b-

1_{)' (c - b (v)) = 1-F}

_(b-

1 _{(c - b (v))). (9)}

II\'iuU I i:.i;A MAHIU Ha~Hl!J.UE SllII1UlISEW

This is not an ordinary di'i4 erential equation since the function b (v) appears inside the argument. We will transfoffi1 (9) in a system of two ordinary di% erential equations. Dei, ne G (v) = b-I _{(c - b (v)) . This implies that}

b(G(v))+b(v) = c (10)

(v - b (v))

f

(G (v))

(b-

I)' (b (G (v))) = I - F (G (v)). (11)

In (11) we used that b (G (v)) = c - b (v). Since b-I (b (w)) = w we have that (b-I

)' (b(w))b' (w) = 1. Thus choosing w = G (v) we get

(b-

I

)' (b (G (v))) =

I/b'

(G (v)).

Substituting this in (11) it follows that

(v - b (v))

f (

G (v)) = b' (G ( ))

I-F(G(v)) v . (12)

Now from (10) applied to v and to G (v) we have that b (G (v))

+

b (v) = c =

b (G2 (v))

+

b (G (v)). Hence b (G2 _(v))

₌

_{b (v) and therefore G (G (v))}

₌

_v.

Therefore from (12) we conc1ude that

b' (v) = (G (v) - b (G (v)))

f

(v) .

1 - F (v) (13)

Di%erentiating (10) we obtain b' (G (v)) G' (v)

+

b' (v) = O. The following theorem descnbes the system of di'i4erential equations that an increasing equihbrium strategy to the subscription game must satisfY.

Theorem 3 The solution ofthe system of di'i4 erential equations below char-acterizes a symmetric equilibrium involving strict1y increasing strategies for the subscription game with two players with values deteffi1ined by independent draws from a distributiol1 F : [0,1] --+ lR, where F is continuously di'i4

eren-tiable and

f

= F'

>

O everywhere.

b'(v) = (G (v) - b (G (v)))f (v),

I - F (v) (14)

G' (v) = _ (1 - F ( G ( v ) )) (G (v) - b ( G (v))) f (v) . (1 - F (v)) (v - b (v)) f (G (v))

8.2

The contribution game

di%

erential equation.

An analysis similar to the analysis made above gives the system ofdi% erential equations that a strictly increasing equilibrium ofthe contnbution game must satisfy.

Theorem 4 The solution ofthe system of di% erential equations below ehar-acterizes a symmetric equilibrium involving strictly increasing strategies for the contribution game with two p1ayers with values determined by indepen-dent draws from a distribution F :

[O, 1]

~ ]R, where F is eontinuously diY4erentiable and

i

= F'

>

O everywhere.

b' (v) = G (v) i (v) , G(v)i(v) G' (v)

=

vi (G (v)) . For example if F is the uniform distnbution,

b' (v)

-G' (v)

-G(v) , G (v)

v

(15)

This gives G(v)

=

~ and b(v)

=

a+klog(v). Naturally b(·) eannot be an equilibrium for aU v since it is nega tive for small v. If O

<

c

<

l/e, b (v)

=

max:

{O,

c

+

k log (v)} where k is sueh that kk

=

e-c, is an equilibrium.

9

Conclusion

We analyzed two mechanisms for private provision of diserete public goods with ineomplete infonnation and eontinuous contributions. The subserip-tion meehanism refunds the money to contnbutors ifthe publie good is not provided whereas the contnbution mechanism does not allow for refunds.

References

[1] Admati, A R. and M. Perry, 1991. Ioint Projects without Commit-ment,'Review of Economic Studies 58, 259-276.

[2] Andreon~ J., 1998, l'oward a Theory of Charitable Fund-Raising: Joumal ofPolitical Economy 106, 1186-1203.

[3] Andreolli, J., 1988, Privately Provided Public Goods in a Large Econ-omy: The Limits of Altruism,' Joumal of Public Ecollomics 35, 57-73.

[4] Bagnoli, M and M. McKee, 1991, Voluntary Contnbution Game: EÁ-cient Private Provision ofPublic goods,'Economic Inquiry 29,351-356.

[5] Bagnoli, M and B. L. Lipman, 1989, Provision ofPublic Goods: Ful1y Implementing the Core through Private Contributions,' Review of Economic Studies 56, 583-602.

[6] Bergstrom, T., L. Blume and H. Varian, 1986, On the Private Provision ofPublic Goods,'J oumal of Public Economics 29, 25-49.

[7] Cadsby, C. B. and E. Maynes, 1999, Voluntary Provision of Public Goods with Continuous Contributions: Experimental Evidence,'Jour-nal ofPublic Economics 71,53-73.

[8] Comes, R. and T. Sandler, 1984, JEasy Riders, Joint Production, and Public Goods,'Economic Joumal 94, 580-598.

[9] Comelli, F., 1996, Optimal Selling Procedures with Fixed Costs,' Joumal ofEconomic Theory 71,1-30.

[10] d'Aspremont, C. and L. A Gerard-Varet, 1982, Bayesian Incentive Compatible Belicfs,'Joumal of Mathematical Economics 10(1), 83-103.

[11] d'Aspremont, C. and L. A Gerard-Varet, 1979, Incentives and Incom-plete Information, 'Joumal of Public Economics 11 (1), 25-45.

[13] Gradstein, M., 1992. ~ime Dynamics and Incomplete Information in the Private Provision ofPublic Goods,'Joumal of Political Econ-omy 100(3),1236-1244.

[14] Gradstein, M., S. Nitzan and S. Slutsky, 1994.Neutrality and the Pri-vate Provision of Public Goods with fucomplete Information,'Eco-nomics Letters 46, 69-75.

[15] Mailath, G. and A Postlewaite, 1990, Asymmetric Information Bar-gaining Problems with Many Agents, 'Review ofEconomic Studies 57,351-367.

[16] Maskin, E., 1999, Nash equihbrium and Welfare Optimality,'Review ofEconomic Studies. 66 (1), 23~8.

[17] Milgrom, P. and J. Roberts, 1990, Rationalizability, leaming, and equi-librium in games with strategic complementarities, 'Econometrica 58, 1255-1277.

[18].Myerson, R. B., 1981, Optimal Auction Design,'Mathematics of Operations Research 6, 58-73.

[19] Myerson. R. B. and M. A. Satterthwaite, 1983. EÁ cient Mechanisms for Bilateral Trading, 'J oumal of Economic Theory 29, 265-281.

[20] Nitzan, S. and R. Romano, 1990. Private Provision ofa Discrete Public Good with Uncertain Cost,'Journal ofPublic Economics 42, 357-370.

[21] Palfrey, T. and H. RosenthaL 1988, Private Incentives in Social Dilem-mas: The EY4ects ofIncomplete fuformation and Altruism,'Journal of Public Economics 35, 309-332.

[22] Palfrey, T. and H. RosenthaL 1984, Participation and the Provision ofDiscrete Public Goods: A Strategic Analysis,'Journal of Public Economics 24, 1 71-193.

[23] Palfrey, T. and S. Srivastava, 1986, Nash Implementation Using Un-dominated Strategies,'Mimeo. Camegie Mellon University.

I.

I

I

[25] Warr. P. G .. 1983, .Pareto Optimal Redistnbution and Private Charity,' J oumaI of Public Economics 19, 131-138.

Appendix

The subscription game with two players: The

uni-form dis tribution case

In what fol1ows. we provide some intuition how to solve this problem when we take into account the boundary conditions. For the uniform distnbution on the interval

[0,1],

the unconstrained solution to the system ofdPl4erential equations stated in Section 7.1 is given by the fol1owing pledging function:

b( ) v -_ 2c - 1

+-

v

6 2

Note, however, that in equilibrium a player may not follow b(v) for any v in

[0,1]

as this may lead to pledging more than his value or more than the cos t of the public good. Hence we need to impose the following boundary conditions

b(v)

<

c b(v)

<

v

b(v)

>

°

(b 1)

(b2) (b3)

It tums out that (bl) and (b3) are not binding since c

>

1. Condition (b2) is binding as b(v)

>

v for v

<

2c;1. We now show formally that the fol1owing is a symmetric equilibrium pledging strategy for the subscription game with two players whose values are uniformly distributed on

[0,1]

and 1

<

c

<

2:

{

2c-l

+

~ ir 2c-l

<

V

<

1 b* ( ) - 6 2 , . 3 - - ,

v - O, otherwise .

To verifY that this is an equilibrium, we assume that P1ayer 2 is following the proposed cquiltbrium plcdging stratcgy, we have to (,nd the best response ofplayer 1. We (,rst show that for 2c;1 _~ V _~ 1, _b1(v) = 2c6"1

+

_~ is a best response to b*(V2). Playcr 1 's expectcd surplus, ifhe pledges b, is given by

To l, nd the maximurn of

4>

l rst note that ifO

< c -

b

<

b*

eyl) =

2cã1

then

4>(b) = (v-b)(1_ 2c

ã"1)

~ (v-~)(1_2cãl) = 4>(~). Note thatif

c-b

>

b*

(1)

then

4>

(b) = O. Let us consider now c-b*

(1)

<

b

<

c-b*

eCã1) .

Then we have

4>

(b) - (v - b) Pr ( { V2

~

2c

~

1 ; V2

>

4C:

1 - 2b } ) =

(v _ b)

(1 _ max { 2c

~

1 ,

4C:

1 -

2b } ) .

There are two cases to consider:

1)

2c-l

>

4c+1 - 2b

3 3

In this case 4>(b) = (v - b)

(1- 2cã"1)

<

4>

(~).

2)

2c-l

<

4c+1 - 2b

. 3 - 3

In this case

4>

(b) = (v - b)

(1 - 4ct1

+

2b) . This quadratic function has a unique maximum at b* = 2c;1

+ ~.

Thus b* is the optimal pledge if

b* E

[c -

b

(1) ,c -

b

e

C

ã"I)]

and

2cã1

~ 4Ct1 - 2b*. The last inequality is valid for all v E [0,1]. The l,rst inequality is valid if v E [

2C

ã

1, 1]. Thus

b(v) = 2c;1

+

~ is the best response to b*(V2) if v E [

2cã"l,

1]. To l,nish

let us l, nd the best response for v E [O,

2cã"1).

It is clear from the reasoning in (2) above that the rnaxirnurn of

4>

is not interior. Thus we need on1y to compare 4>

(c -

b*

(1))

=

4>

eC-I)

=

(v

-~)

(1- 4cf

+

22cã"1)

=

O and 4>

(c -

b*

e

C

_ã"I))

_{= 4>}

(C!I)

₌

(v -

C!I)

_{(1 -}

~Cãl)

_<

_{O. Thus if}

v

_{E [O,}

_2cã"1)