Discrete public goods with incomplete information

(1)

- - - . - . . - . - - - - .... - - , - - --'"'"7"" ｟Ｎ｟Ｍ｟Ｎｾ＠ｾｙＳＺＮＱｾ＠ 0 _ _ _ _ _ _ _ _ _ _ _ U ' ' •• ' _ .... ;,. . . : :

. . ｾｆｕｎｄａￇￃｏ＠

. " GETUUO VARGAS . .- - .. I <"':;:\"\.: I. " . '

\I)/OS'

.:.o, EPGE

Escola de Pós-Graduação em Economia

Dl:

_{- ... " (,};)1:1..;

):U· i1S"\':

_{I .}

ｌｃＨＩｾｏｩｖｲ＠

rCA

"Discrete Public

Goods

WÍ.th

Incolllplete

Inforlnation"

J

Prof. Paulo Klinger Monteiro

(EPGE/FGV)

LOCAL

Fundação Getulio Vargas

Praia de Botafogo, 190 - 9° andar - Sala 931

DATA

21/10/99 (Y feira)

HORÁRIO

16:00h

Coordenação: Prof. Pedro CavaJcanti Gomes Ferreira Email: Cerreira@fgv.br·.(021)536-9250

\'

f

ti

(2)

Flavio

1-1.

:\Ienezes·

Australian National University and EPGE/FGV

Paulo K. :\Ionteiro

EPGE/FGV

Akram Temimi

U niversity of .Alabama

Discrete Public Goods with Incomplete Information

t

Octo ber 7. 1999

Abstract

We analyze simultaneous discrete public good games wi.th incom-plete information and continuous contributions. To use the terminol-ogy of Admati and Perry (1991). we consider comribution and sub-scription games. In the former. comrioutions are :1ot rcfunded if the project is not completed. while in thp. iatter they are. For the special case whp.re provision by a single player is possible we show the exis-tence of an equilibrium in Doth cootribution 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 any equilibriwn of both games is inefficient. WE also provide a sufficient condition for "contributing zero" to be the unique equilibrium of the contribution garoe with n players and characterize e<!uilibria of the subscription game involving positive contributions .

• COTn!3pOndence to: F. Menezes, Department of Economics, Faculty of EcoJlOlDicaaod

Commerce, Australian National University, Canberra. AGT. 0200. Australia.

tWe would like to thank John Conley, Charles Kahn, Bart Lipman, Paul Pecorino, 'lbdd Sandler and Hum Schlesinger for helpful comments. Menezes and Monteiro gudefull)

acknowledge the financiai support of CNPq. Temimi acknowiedges the financiai suppori of the College of Business Administration at the University of Alabama.

1

(3)

(.

1

JEL Classification:D79, D89, H89

Key Words: private provisiOri of public gooos: contribution and

sub-scription ｧ｡ｭｾＺ＠ incomplete information.

Introduction

The literature on private provision of public goods can be divided into two

broad categories.1 _{The first branch of the literature fqcuses on the}

provi-sion of continuous public 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 result in trus

literature is that public goods are underprovided by voluntary contributions

due to fIee riding behavior.2 _{One might conjecture that the government ean}

solve this underprovision problem by providing some of the good and finane-ing it by imposine taxes on contributors. However. Warr (1982) and Roberts (1984L in two influential papers. sho\\' rhat gO\'errunent contributions result

in a dollar to Jollar reduction in privare contributions if the tax on

con-rributors does not change the set of contributing individuaIs. Bergstrom, Blume and Varian (1986) show that the crowding out effect is only partia! if one allows for the ta'(es that pay for the government contribution to be

also collected fIom non contributors. Hence the literature suggests that

un-derprovision is a robust conclusion if the public good leveI is endogenously

Jeterrnined bv \'oluntar:-' contriburiollii. J

i In ｾｨｩｳ＠ paper. we focus on private provisiono There is ao extensive mechanism design !iterature that treats public provisiono Papers in rhis literature develop efficient mecha-nisms for the provision of public gooci.:;. These mechanisms are in general complex aod require a central autbúrity to implement tbem aod bence would be best decsribed as mecha-nisms for p:lblic provision of public goods. Gradstein (1994) e.xamines efficient mecbamecha-nisms for discrete public goods. Other papers tbat treat public provision include Maskin (1977), d'Aspremont and Varet (1979, 1982), Palirey aod Srivastava (1986). Coroelli (1996) ex-amines ao optimal mecbanisTO for a monopolist which produces ao excludable good th.at has large fixed costs.

2The free riding problem becomes worse in the presence of incomplete information. Gradstein (1992) considers a dynamic model of private provision for a continuous public good with incomplete information with the restriction that players either contribute zero or an exogenous positive amount. He concludes that in addition to tbe standard under-provision results, inefficiency occurs because of a delay in contributions. This inefficiency does not disappear as tbe population becomes large.

3 Andreoni (1998) analyzes tbe role of seed money in tbe presence of nonconvexities

in tbe production of the public good. He shoW'S that a small amount of seed money can

(4)

ｾＬ＠

The second strand of the literatlU"e rocuses on discrete public goods where a fixed leveI of a public good is provided if enough contributions are collected to cover its cost c, Ot.herwise. the good is not provided. Typical examples inc!uàe buiIding a bridge, a libra!}' of a certain size, public radio fund raising

to finance a certain programo In alI of these exampIe.s, if enough money

is raised to cover the cost of the public good. then the good is provided, otherwise the good is not provided.

Palfrey and Rosenthal (1984) developed thB ｾｳｴ＠ modem treatment of

private provision of discrete public goods, :-'lore specifically they analyze

contribution and subscription games - LO use rhe terminology of Admati and

Perry (1991) - for a discrere public good under complete information where

players: strategies are restricted to either contribute zero or an exogenous

positive amount. In a contribution game. contributions are not refunded if

t he sum of the contri butions does not cover t he cost of the public good c,

whiIe in a subscription game players get their money back if the project is not completed . .-\n example with the main features of a subscription game can be

Ｇｾｩ｜Ｇｰｮ＠ ;1"'; ridl\)\\·:" The \\'i:-;collsin C\.)\'("n1ür il;\S I'l'('l':,:h' pleclged 827 million

in state oOIlcls to nnance a Ile\V :3-;'2 million oasket bati arena on the condition

that the rest of the money be raised by private donations (Andreoni, 1998).

That is, the Governor \ViII provide S27 million as long as the remaining $45 million is raised. otherwise the offer is cancelled. \Jther exampIes can be found in the fund-raising literature. ExampIes of contribution games include public radio anci TV fund-raising efforts where comributors do not get their

Illonev back if r:he program is nor prm·ided. ｏｾｨ･ｲ＠ examples of contributions

games are situations \vhere contributions take the form of physicaIlabor, in

(bis case volunteers cannot recover their errort if the project is not completed, [n CO!1trast \Vith the standard underprovision result for continuous public

goods. Pal.frey and Rosenthal (1984) obtain the startling conclusion that

efficient provision of a discrete public good is a possible outcome of both

contribution and subscription games even though inefficient equilibria also

existo Bagnoli and Lipman (1989) extend Pal.frey and Rosenthal model by

allowing individuals to make continuous contributions. They show that the set of undominated perfect equilibrium outcome.s of the subscription game is not only efficient but coincides with the core of the economy. They also

show that with a dynamic version of the subscription game, it is possible to

obtain efficient outcomes even if the leveI of the public good is not binary

generate a substantial amount of voluntary contributions.

(5)

as long as the number of units of the public good is countable.-! It follows rhat the general conclusion from rhe theoretical lirerature is that private

provision of discrete public good.s with complete information can be efficient .

.\Ioreover, efficient provision is a possible outcome for both subscription and contribution games.

A,n important quesrion that has not been addressed in the literature on

'.-cIllntar.y provision of discrete public goods is to wnat extent these results

generalize to a model where players have incomple,te information about other

players: \'aluation of the public good. )'Ioreover. there' is both casual

evi-dence from the fund-raising literarure of t he superiorin' of subscription games

and experimental e\'idence frorn Bagnoli anà .\Ichee I i.991) and Cadsby and

), Iaynes (1999) that a refund increases the chance of providing the good. More

specifically, Cadsby and .\laynes consider a discrete public good experimento

They provide experimental evidence showing rhat provision is encouraged. in

a subscription game vis-a-vis â. contribution game. They also provide

evi-dence that a high c discourages provision in the cortriblltion game but not ::: rile :::lli);:;cTipr;Ull C!dllW Ou1' F,':"ulrs \\'iil cr:·m1nll ｾ＾ＺＺＺＺＺ･＠ rindings.

In this pa.pel'. \\'e examine contribution games anc subscription games for

a discrete public good in the presence of incomplete information. We first analyze provision games where ir is feasible for a player t0 provi de the public good by himself. Common examples include opening a \I,'indow in a hot room, rescuing an injured person in a traffic ar.cident. etc. \Ve show that when the

cost of prm'ision c is nor prohibitiwlv high as to prp':ent :t single player from

pro\'iding rhe good. rhere ｩＱＱｷ｡ｾＧｳ＠ exisrs (l,n AquilibriL.:lTI where a player with

a sufficiemIv large \'aluation provides the gooà in bor 11 the subscription and

r he contri burion game. .\ roreover. \\'e sho\'v' r har r he rree riding problem can

becúme moresevere or less severe as the population oecomes larger.

\Ve also examine contribution and suoscriprion games when contributions

1.-\dmati anei Perry analyzed the role of commirment in d:e provision of public goods. :'Iore specificaliy, they examined tWl?p!ayer contribution and subscription games with complete and perfect information where p!ayers alternate in making pledges or contribu-tions until the public project is complp.ted. They show that v';rh linear costs, a necessary and sufficient condition for completion of the project under a contribution game is that each player would complete the project immediately

b:.

himself if he was the only player. \Ve provi de a surlicient condition and '\\'1th two or more players for "contributing zero" to be the unique equillbrium of the contribution game. For the subscription game, they show

that ali socially desirable projects are completed in equilibrium. fu contrast, we show that with incomplete inforrnation the probability of provision conditional on being efficient to

(6)

ar pledges fram mare than ane individual are necessary in order far the good

to be pravided. \Ve wiII characterize same s!mmetric equilibrium of the

5ubscriptian game and demollstrate that alI equilibria are inefficient, that

is. the prabability of provision given that it is etficient to do so is less than

une. TIús inefficiency stems from the difficulties arising in coordinating to ()vercome the free-rider problem in the presence of inc.:omplete information. We will also pravide a 5urlicient condition for 'conrributing zero: to be the

unique equilibrium of the cantribution game. ｔｨ･ｾ･ｦｯｲ･Ｌ＠ we show that under

same eircumstances the contribution game is never mote. efficient than the

subscription game. This eonfirms the evidenee fram the experimentalliter-ature of the superiority of subscriptian games O\'er contribution games. In cantrast to the mechanism design literature, our approach of characterizing equilibrium behavior in existing mechanisms enables us to provi de results which ean be tested in a laboratory environment ar by using empirical data. TIús remainder of the paper provides more details on alI of these issues. Tn Seetion 2. the speeifics of the madel are presenred. Section 3 provides an

"\:;in:ple r11ilf ilh:::rr;HP:: rhe main rl'::ulb ,:!j' lhe ｰＺ｜ｰｴｾＺＭＮ＠ b Sr:,etion 4, the case

where a single individual can proviàe the good b:" illrn.self is analyzed. The

general analysis of subscription games is eontaineà in Seetion 5. Section 6

examines the eantribution game. Seetion I illustrates a àifierential equations

approach to find equilibriurn strategies in games wirh two players. Our mairt results are summarized in Section 8.

2 The lVIodel

\\-e now present the formal moeie!. There are X

2

:2 individuals with private

independent va1ues for a discrete publie good. That is. eaeh individu31 i,

i = L ... , ｾｖ＠ knows his CVln value l.!j but only the clistribution of his opponents'

\Cllues. Values are assumed to be independently drawn from a continuous

distribution

F.

The cost of providing the public gaod is c.

\Vith the exception of Section 4, we assume throughout the paper that an

individual eannot provided the good by himself. In trus case,

F

(c) = 1 and

thus the distribution has a bounded support. As a normalization we suppose,

without 10ss of generality, the support to be contained in the interval

(0,11.

\Ve wiil examine both subscription and contribution games. In a

sub-5eription game the money contributed by players is returned if the sum of

the pledges is less than c. In a contribution game the money is not returned.

(7)

:.

even if the good is not provided. \Vhatever the method to elicit donations is. if donations add up to more than c. rhe additional monev is not returned to the contributors.

ｾｊｯｲ･＠ specifically, we analyze one-shot simultaneous move contribution and subscription games for a cliscrete public good in the presence of

incom-plete information .:tbollt preferences.5 _{Our model ê.l.so relaxes the}_binarJ

con-tribution restriction imposed in the li terature. li \\ rule there are important

instances where binary contributions are relevanç due to transaction costs, in general players can give any amount of money they dffiire (alumni dona-tion. donations to a lihrary, etc.). \foreover. in a continuous contributions

framework. individuais make contriburions rhar ｾ･Ｕｴ＠ match their preferences

as opposed to a discrete contribution model. Cad<;by and Maynes (1999) provide experimental evidence showing that ailowing continuous rather than binary "all-or-nothing" contributions facilitares pro,-ision .

. -\5 it is standard in many games of incomplete inIormation with ex-ante

5vmmetric plavers - :',uch as in industrial ｯｲｾ｡ｮｩｺＨｬｲｩｲＺｊｬｬ＠ and auctions - our

I·'.)("lb _ｪＮｾ＠ Illailll\' '.'11 .'\"lllmerric !'ql.lilibrilllll. Ｎ｜ﾷｴＺＧ｜ＢｅＺＧｲｾ＠ :.:,·!ess. \',e provide ineffi":

ciency results thar hold for both symmetric and ＼Ｎｾｹｭｭ･ｴｲｩ｣＠ equilibria; we

show that for any equiLibrium the probability of pro\ision of the public good

given that it is efficient todo so is less rhan one ;.n both contribution and subscription games and it is frequently equal to zero in contribution games. \Ve also provi de sufficient conditions under which contributing zero is the

un.ique equilibrillm of the contribution game.

The multiplicity of equilibria rhar characrerize::: these games has two .:3ources. Firstly. this is an instance of a coordination game. \Vith two pIayers

',vho cannot provide the good by ｾｨ･ｭｳ･ｬｶ･ＵＮ＠ Lhe coordination nature of the

game is c1ear. \Vith more than rwo players. rhen we have to account for ali

possible combinarions of players who can prO\ide rhe good as a group. This adds a new layer of complexity to standard coordination games. Secondly,

5It is important to note thatthere are several papers that introduce in complete

in-formation in one form or another but do not address the aoove questions. Palfreyand Rosenthal (1988) analyze the provision of a discrete public good when individuals have inc.omplete infúrmation about the degree of altruism of other players under the restriction rhat players are only allowed to make discrete contributions. :'\itzan and Romano (1990) show that when the cost of the disc:ete public good is uncertain to the players, then efficiency is no longer obtained.

6Bagnoli and Lipman (1989) al.so considers continuous contributions. However their model is with complete information.

(8)

the incompleteness of informarion and the assumption that player can con-tribute or pledge anv \'alues they Iike also creates additional coordination issues beyond the sta!1dard coordination probIem It is the combination of

thE:Se two features r hat makes it diflicult to obtain uniqueness of equililr

r:um, In particular, él5 \ve \Vill explain in rhe comext of the example below,

well-kr.own argumems to obtain uniquene5s, .3uch a.s the iterated deletion of

,Jominuted strategies a..:; in Bagnoli and Li pman I 1969) (v,ri th complete

infor-mation) and \[ilgrom and Roberts (1090) (\\'irh 'incomplete information) for

rhe case of supermodular games. do nor apply in rrus contexto

. 3

An Example

Before turning to an analysis of the mode!. ir is instructive to consider a

sim-pIe example where ir is possible to make rhe poim rhat the efficient results

in rhe ciiscrere public ｾｯｯ､ｳ＠ lirerature are not rObl:.5r to rhe introduction of

::lcnrnnll'r(' i llf."lrtlU I i"l!. T:lis "x;\lllplc :::(';.'1':" ,[':, - .... i hc :-implest srructure

:',)1' \\'ilici! rlli::i Lll'k I.!! l"J[)l.brneS5 i.-; t:',,'it'ir.m :"J..! r'L':' ·.' .. hich ｜｜ＧＨｾ＠ ohtain a

strik-ing distincrion between predicted outcomes in ｳｵ｢ｳ｣ｲｩｾｴｩｯｮ＠ and contribution

games, "'e begin v;irh lhe complete informarioll ca.:::e.

3.1 Complete Information

.-\:,sume rhar plan:'l'S 1 ;!!ld ｾ＠ haw' \'aIuariolls

r,::,'

;i:c· ;)ubIic good 1:[ and 1!',? in

.IU; and tbar are COllilllOll knowiedge, Ler rhe cosr ()i' provisioIl of the public

'Zood c be such rhat 1 < c < '2 and t'! -"-1"2

2:

c. Thar i5, a single player cannot

provide the gaod bv hirnself and it 15 socially pmciem to provi de the good. If

i\'e denote by bi , i = 1. 2, player 1'5 pIedging straregy in a subscription game,

[hen we can characterize the set of (pure strateg:.') equilibria as follows:

(ii) (O, O):

The equilibria described in (i) are efficient. while the equilibria described

in (ii) and (iii) are noto j \Ioreover. the equilibria described in (ii) and (iii)

(9)

:-,

- .. ----. -.. - .. ---.--- -"-" -

ｾ｟ＮｾＮＭ

-- .. - ---,--.. ,- .,

Ｎ｟ＭｾＮＮ＠

, .. ""I

are not strict a.s best responses are nor unique . .:l

)Iote that the equilibria of rhe conrrioution gam2 can be described by (i)

aúd (ii) above. Given that indi\'iduals in ê. conrriblltion game never receive

rheir contriburions back. the srraregies profiles descróed in (iii) are no Ionger equilibria. Cnlike in rhe case of the subscriprion game. rhe equilibrium

strat-･ｾｹ＠ (O. O) is now a srricr equilibriwn. T!lar is. conrribut.:ing zero is the unique besr response when rhe orher plaver comrioutes zero.

This example sllggesrs that under complete ﾷＺｾｩｯｲｭ｡ｴｩｯｮ＠ the distinction

between contribution and subscription game both in terins of efficiency and

in rerrns of expected Ollrcomes is llnc1ear. \\'e no\\' consider the incomplete

information case.

3.2 Incomplete Information

\Ve now assume ｲｨ｡ｴｬｾｬ＠ andv'2 are independem draws [rom the uniform

[0,1)

disrriburion. \loreover. Plaver Z = 1.2 kno\\'s hi::; 0\'.11 \'aluarion but only the

,ii:-:uiburiol! (,I hi:-; I 'PpUl!<::llt'S \'Ji:l;Hiull. L'.:t : :.' ＮＧｌＺ［ｾＬＡｪｬｾ［ＺＺ［ＧＺｾ＠ til,H L

<

c < 2. The

.;ubscripriOll ｾ｡ｬｬｬ･＠ h,l'S many symmerric c-quiiiória ,mel tlle cumribution game

has on1:v rhe strong free riding equilibria. Ler 11S ＺＳｾ＠ :::ome subscription game

equilibria hrsr:

i. (O, O):

{

(I. if /' <: -_:

ii. IJ' ( /' j

=

ｾＬ＠

ir /.

2:

ｾ＠

.. ..

I'f _-J-:2c-! _..2::<'" l.' <

(nherwise.

That (O, O) i5 still an equiJibriwn in the incomplete information follows

trivial1y from the fact that a singIe player cannor provi de lhe 'good by

him-ｾ･ｬｦＮ＠ｾｯｷ＠ let's check that the srrategy profile descrioed in (ii) is indeed an

equilibriwn, Suppose Player :2 follows b" (.), If Pla,ver 1 's value is Iess than

ｾＮ＠ the!l 1 's best reply is to pledge any number less than ｾＮ＠ In particular,

pledging zero is a best response, If PIayer l's \'alue is greater than equal to

}, then PIayer l's best response is to pledge ｾＬ＠ Pledging less than ｾ＠ PIayer

! Bagnoli and Liprnan (1989) show that the inefficienr equilibria i!1 (ii) and (iii) can be

(10)

ｾＮＩ＠

",

""

:..,

1 obtains zero profits: by pledging more than ｾＮ＠ Player 1 does not increase

rhe probability that the good is provided but decreases profits conditional on proVISIOn.

The equilibrium pledging strategies described in (iii) can be explained as

Follows. A player pledges the equivalent to the expected value of the other

player being lower tha,n rus own. conditional on lhe iaterval

[2c;l,

1],

that is,

vn the relevant interval where pledges are iess rhan or equal to the valuations

for the public good. i,e. 6('/.1)

S

V. A formal derivar,ion is left to the appendix.

Note that the iterated elimination of dominated strategies does not bite

here. In OlU setting, in a first round of elirnination Player i elirninates

strate-gies 6i ('Vi) > /li· The elimination process stops r here! The type of bounds

obtained in ｾｉｩｬｧｲｯｭ＠ and Roberts( 1990) are not possible here since the game

is not supermodular: Consider two srrategies 61

=

O and 62

=

c and note

[hat 1(62,62 ) - /(62 , 6d - [/(bl , 62 ) - /(61 , 62 )] = '(v - c) -

(v

-

O) =

-v <

O.

Of COlUse this multiplicity of equilibria can potentially raise difficulties

in questions of welfare. ｾ･ｶ･ｲｴｨ･ｬ･ｳｳＮ＠ "'e wiU :::how later that any equilibria

"r' rhe subscripri,jp ｾ｡ｭ｣＠ is inetficic:m. :Ü.Jre preu:::ei'.· 'Vi; \\'i11 show that the

probability of provision of the public good given r har it is efficient to provi de

is (uniformly)less than one in any ･ｱｵｩＱｩ｢ｲｩｵｭＮｾ＠ :\Ioreover, we wiII provide a

sufficient condition wlder \vruch the \velfare comparison between subscription

and contribution games is srraightforward - <'..5 the onJy equilibrium of the

contribution game is to contribute zero.

\Ve now retlUn to rhc example and élnalyze rile cf)ntribution game. This

example satisfles rhis sutficiem condition rhat \\"iH be specified later.

How-f!Ver, ir may be instructive to show directly rhat (0.0) is the unique

equi-ｾｩ｢ｲｩｬｬｲｮＮ＠ Suppose (61.6"2 j is an -:quilibrium of rhe contribution game. It is

ｾｭｭ･､ｩ｡ｴ･＠ that b2 (/.12) :::; u2' Let us find the best response of Player 1 to b2 •

If Player 1 bids x

2:

O rus expected urility' is

If x

=f.

O and c - x

2:

1 then cp (x) = -x

<

O. !\"ow suppose c - x

<

1. We have

o (x)

S

VP(V2

2:

c - x) - x

=

v(1 - (c - x)) - I

=

v(1 - c)

+

(v - l)x ｾ＠ O.

9This probability is equal to 1/2 in the equilibrium described in (ii) and equal to 2/3 ln the equilibrium described in (iii).

(11)

;.,

Thus x =

o

is the best response. (:'-Iote that \ve are not lirniting ourselvE'S

to symmetric pure strategy equilibriurn J. Therefore, despite the

multiplic-ity of equilibria of the subscription game. we are still able to make welfare comparisons . .\Iore precisely, the contribution game is less efficient than the subscription game in this example. This result contrasts greatly with the case \Vhere players ha\'e complete informarion and both games have efficient equilibria.

4 Provision by a Single Player

In this section. we analyzp. contribution anà subscription games when valua-tions for the public good can take values larger than the cost. The following

rheorem shows that there always exists a symmetric equilibrium where a

player with a sufficiently high valuation provides the good by himself.

Theorem 1 .C:uppo.çp F : :0. x: - ?: i" " i:O!l:;:: .. j ·18 distrihlLtion. Suppose ·he,.e .!lE ,\'

2:

:2 piuljeTS for ([ pro)ect ＧｾＧｉ｛ｨ＠ｲＺｲＩＮｾ｛＠ l.· . / O and that F (c)

<

1.

Then there exists an a

>

O where o: solves aF (Q:) :,' -1

=

c such that

ó(l') = .

{

O

1I

. C Ij

I' ::; o

c>o

'8 an equibbri.um .c.trateqy for both contlloution ｣ＺＺＬＺＬＧｾｕ｢ｓｃｔｩｰｴｩｯｮ＠ games.

Proof. Suppose players TI

=

2 ... V play accordin!S to b (-). Let us find the

best response of Plaver 1. If his v; lue is L'

2:

I) anà his contribution is x

2:

O,

his expect.ed payoff in the contribution game is giwn by

g(x)

=

vPr(x+ 6(V2)"'" +b(L'S)

2:

c) - x

Since b (V2)

+ .,. +

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

I

<

c theng(x)::; g(O). Thus max{g(x);x

2:

O}

= max{g(O),g(c)} =

max

{v

(1 -

F (a),v-l) ,V - c} . Rence if l'

<

F(a;:::-l =

a,

the best

contri-bution is x = O. If v

>

a the best contribution is x

=

c, And if v

=

a the

player is indifferent between x = O and x = c. For the subscription game, the

expected surplus is given by

f

(x)

=

(11 - x) Pr (x

+

b ( V2)

+ ...

+ b ( V N)

2:

c)

(12)

r,

\

-...

The proof is identical to the contribution game since 1(0) = 9(0) and

I(c)

= !}(C),IU •

Remark 1 This equilibrium ls not ex-post ejji.cieT'..i, Although efficiency

re-I]',úres that the public good be pro1..'ided u'henel'er the :3um 01 individual's valu-atiol1s exceed c, the good is provided onllj

li

ar ieast one mdividual's valuation t;'Iceeds Q: which is greater than c, J/oreoL'er u'e u'lii ÚI.OW later that any

equi-libril.lm is inejficient,

Theorem 1 implies r hat when r he C05t c;f pro\'!5iGn is nor

ｰｲｯｨｩ｢ｩｴｩｷｾｬｾＺ＠ high as to pre\'ent a 5ingle player f:-,::-m providing the good, rhere always exists an equilibrium where a player \\ith a sufficiently large caluation provides the good by rurnself.

Bliss

and NalebuH (1984) analyze a war of attririon game where the public good is provic!ed by a single inàividual. They show that in this dynamic t'ramewnrk the free rider problem is :ess imnonam .,.::: - he poplllation becomes

ＺＬｩｲｾｬＮＡｲ＠ ;111<1 pr,!\'i, i(' ( I ｊｦＡＨｩｩｴｩＨｪＡｬｾ＠ Ill:Clr:,r \\'Ilic!: <:,,:' r::',< '(::-;t 1.-; ilchieved as the p')plliatiOIl approaches innnity, Hence, a llatunlÍ qUf5tion is whether in our framework. r he probability r hat

c

he good gets pro\'ided increases or decreases as r he population becomes larger. The an.swer ro r

rus

qllestion depends on rlle ･ｬ｡ｳＨｩ｣ｩｲｾＧ＠ oi' rhe disrribllrion funcrion, T,.J ｾ･･＠ rili5, first note that Q:N is

increasing in N and that rhe probabilitv char rhe good is provided is

.

,- :

f'

,. :

1 - FI' Q \' ) '. = 1 - ' "', '"

, . 1:1..,"

Taking the derivative of crus \\'ith respect to O", \\'P. conclllde that the

..leri\'ative is positive if

F';_7

_{, °VI}v 'o,,,,, < 1 anci ne2:ati\'e _-

ir

this term is o!ITeater r han one, That is, free riding can becorne less se\'ere 01' more severe as the

population becornes larger. There is no presumption r hat it will get worse or rhat it wili get better. It depends on the distribution function, For example, if F is uniform, rhen the elasticity is equal to one and the probability that the good gets ｰｾｯｶｩ､･､＠ does not change as N increases,

iOThe solution is not unique in general. For example if.\' = 2,F(x)

=

x,x E [0,1], rhen if O < c < l/e another equilibrium strategy for rhe contribution game is

o

(x)

=

max {c +-k log (x) lO} where k is such that k'" = e-C, There are two solutions: k, < l/e <

(13)

r ..

5 The Subscription Game

In crus ser.::tion \ve analyze rhe subscription game where a single player cannot provide the good by rlÍmself. This is the case onh'

ir

F( c) = 1. Thus the distribution F has a bounded support anà ｜｜Ｇｩｾｨｯｵｴ＠ loss of generality we :::l.lppose rhe support is rhe imen'al

[O,

r:.

Pla\'ers' \';.tiues are determined by

ｾｮ､･ｰ｣ｮ､･ｮｲ＠ drJ.\\'s frnm a cominuol.ls ciistrioution ç ,

:0, lJ -,.

iR. The cost

ＰｾＧ＠ the public ｾｯｯ､＠ is c ｾ＠

:m, m

-

1) \\'here m

2:

1, i.'=. ";'l -+ 1 is the minimum numoer of players needed to provi de rhe public gooà,

If players i = 2,3, .. , , .:V follow rhe function b (.', 2nd play'er 1 with valu-Jrion L' contributes x

2:

O, his expecred surpius is

o (x) = ( 1.' - r) Pd x

+

b ＨｬＮＧｾ＠ j , .. , ..:.... 1j ( c'N)

2

c) ₍₁₎

= (u - r) Pr (b ( L'2) '7" ... ,. b (./.'.\' I

2

r: - x)

We ｳ･ｾ＠ from the above rhat if X > :2, to \\Tite o (r'! àS a function of the

dis-aibution of individual ｜Ｇｾｬｬｬ｡ｲｩｯｬＱＵＬ＠ F. derailed inforn:2..t;on on b(-) is needed:

.: l:Ul.''::; 111.'1 . . ｾｬｬｭｮＧ＠ !', Kl;Cj\\' ｩｩＡｾｬｲ＠ :"., ｩＮｾ＠ｩｉＺＨＧＺＧｻＺＺＭｬＮＺＺＬｩＡｬｾ＠ ::, '!le pri\,;lte valuation,

Ht'nce rhe s[c1llJard rC'chniqu2 tilar is u.:::eà ;-0 char'lcr::rize equilibrium in-\'olving increasing strategies in games of incomplete :!1formation cannot be l.lsed in this problem when ｾｖ＠ > 2, :'Ioreover. ir wowd be fruitless to try to guess all rhe funcrional forllls rhar equilibria migllr ＺｩＧ｜ｾ･Ｌ＠ T!lus, it would be extremely difficult to find all the equilibria of the subs.:ription game,

In this sC'r.tion. \\'P ::ho\\' dirprr!\' rllar ｲｾＡｦｾｲ･＠ＨＧＺＮＺｩＺＭＬＢｾ＠ ;H! ｴｾｱｬｬｩｬｩ｢ｲｩｵｭ＠ with

,·ut-<jff ｾｪｲｲ｡ｴ･ｧｩ･ｾ＠ \\'i!prr ilw gCIIJà i::i p[lAirieri if rn - ｾ＠ｾｬ｡Ｇｹ･ｲｳ＠ have a ",alue grearer rhan a certain cut-oh' \,;:llue, \\'e theI! silo\\' ｲＺＺＮＺｾｲ＠ for any' equilibrium,

ｾｨｅＺ＠ probabilit.v thar rhe good gers pro\'icied ｾｩｙ･ｮ＠ rh,:u ir is etficient to so is

ｾ･ｳｳ＠ t han one,

Proposition

2

Suppose that p!.ayers·s caiuations <U'F. rJistributed in the

[0,1]

imervaL, Suppose lhe cost of th,e public !.lOod is c E [m. rn

+

1) and the number of players IV

2

m

+

1, The following strategy ls a symmeb'ic equilibrium strategy of the subscription yame:

wnere a is .5uch that

( O

｢ＨｶＩ］ｾ＠ｾ＠

l m+i

if l' ｾ＠ a: if

a:::;

I.' ｾ＠ 1.

C':J_l (1 - F(a))m F(a),V-l-m

12

(2)

c

(3)

(14)

d C'h n! an n = hl( -h'l' , n ),

Proof. \Ve first prove the existence af a, Define

h (a) C

N_

I (1 - F (a))m F (a).v-l-m

｡ＭＭｾｾｾＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭｾｾ＠

ｌｾｶ］Ｍｾ＠ｃＮｾＧＭｉ＠ (l - F (a)t F (a"V--I-h aC,r:J_i

=

'v-I

C.r:J_l

+

I:

ｃＮｾＭＱ＠ (1 - F (a))h-m F,(a)m-h h=m+l

.\'at.e that

_em

lim h (a) = O and lim h (a)

=

e:

-1

=

1

a-O a-I .V-l

Since h is a caminuous function. by the intermediate value theorem there

exists an a such that h (a) =

m:1

E (0.1).

\Ve nO\v prove that b (,) is an equilibrium, Sl.Ippose l'

<

1 and player 1

ｩＩｬ･､ｾ･ＺＺｩ＠ .L'

2:

I) and pla\'el's 11 = 2. , , , .. \- foilaw r;: i'", . The expected utilityaf

piayer 1 is given by

ó(x) = (v - r) Pr

(f.

bit,")

2:

c -

x) .

n=2

I ' , '

"

.

{

- "c . \-. ,., " .

,

I.le rar:o<Ye oí / ' ., b (/'n) IS O. -"-1' ---, .. , , , -'-'-- > '

｟ｾｮ］｟＠ n1- tn-l .'71-J. )

If c - x E (.L, (j-,-l 1C] then

m ... 1 m ... l j,

o (x) = (u - x) Pr

Ｈｾ｢＠

(v_n)

2:

(j

ｾ＠

1)

c')

:s

o

(r:

_

....,;.(J_.

+_l)_C)

L m-t-1 , m + 1

n=2 "

Ihus the optimal pledge x·

:s

v is such that

Thus

x· E

{c _

(j

+

1)

c: ).

>

-I} .

. m+

l '

-• - _ (P

+

1) c [O 1) .•

>

-1

x -c E , ,J _ .

m+

1

(15)

Thereiore j' E

{mo

m - I} since _{m+l -}ｾ｣＠＾ＺＲｾ＠ _m-l_-

>

1. Finallv we haveJ'· _"

=

m-l

if al1d only if CJ ＨｭｾｬＩ＠ｾ＠ o (O) or equivalently if and only if

or

l.' Pr

(t

b ( u_{n )}= me)

ｾ＠

c Pr

(t

Í) ( V_{n )'}

ê

me).

n=2 m

+

1 m -+- 1 n=2 m

+

1

Since

and

we conclude that (j)

(,n:J

ｾ＠ 0(0) if and only if I'

2:

11,.

Remark 2 The above equilibrium is not ex-post efJieient. In this

equilib-rium.

t/

the !/ood is pT01'ided then the "um

0/

［ＬｾＮＺ＠ ,J:aljer 's ualuations is at ifast (m

+

1) a > c,

Remark 3 li cuery player Is ｰｩｶｯｾｊｌＮ＠ i. t, II.Y = m - 1. then the equiiibriurn

i 2) is gwen by:

b(V)={

ｾ＠

m+l

lf L'

<

_c_

m+l lf _c_o

<

L'

<

1

m ... l - - '

Thus eaeh piayer eonsiders the cost equally divided among the piayers and cantributp..s il and anly if his vaiue is at lea.st his share. Effieieney requires that the good is provided whenever

L

1\ ｾ＠ C whereas the above equilibrium requires each Vi to be greater ar equal to m:l'

A natural question is whether the subscription game admits anyequilibria

rhat are efficient. \Ve show below that every equilibrium of the subscription garoe is inefficient. vVe need two lemmas to prove the inefficiency resulto

(16)

,.,

Lemma 3 Suppose piayer j,::? ::; j

S

n pLalJs strategy 6j (-) and that b (.) is

tl 6est response for piayer l. Then

t' > ,,'

=

Pr

(t

"J

(Uj J

2:

c - b ( ,.')

2:

Pr

Ｈｾ＠

bj (I)j)

2:

c - b

(V')) ,

(4)

Proof. Define H (C-i j

=

_--J-...ｾｮ｟Ｉ＠ DJ' (I'.) ,b

=

o

i ('i ano 6'

=

b (u'). Then bv the

,J' 'IJ

definition of b (-) we have

(1'-b)PrlH(I'_l) ｾｲＺＭｨ￭ｬＧＩＩ＠

>

(1'-;;',Pr'I-!(I'_l) ｾｲＺＭ｢ＨｶＧＩＩ＠ and

(U' - 1/) Pr (H ( I.' _

d

ｾ＠ c - b ( v') )

>

(l" - bi Pr ( H (v _ 1)

2:

c - 6 ( V)) .

A,ddin!S both inequalities and simplifving we nave

(I' - ,,') (PdH(I'-:) ｾ＠ c - 6(1')) - Pr(H(,'-: ｾ＠ c - b(1.")))

2:

00

•

Lemma 4 Suppose !7 !j such that Pr

ＨＧ［Ｇ｟Ｗ］ｾ＠

.:;.'

I I:.

2:

c - !J ("[o))

>

00 Then

ffJr nmj I'

2:

,.'::::!7 ICP' I!rlt'eo,"')

2:

1>(,.',. ｔｾｵＮＧ＠ O"I? optimal response6(0)

IS non-decreasing for any signal v greater than some signal

v

that gives a

/Jositivp. proba6ildl! of qetúnq the public {iOod.

Proof. Suppose /' > i" > !7. Then frorn i -1:) él 00\"(:' 0.':0 have

PdH(u_tJ ｾ＠ c-bit'i) > PrIHí/'_:;

2:

c-I)(7)')) > Pr(Hí('_:J2:r:-I)(V)l>Oo

:'\ow note that this and the nonnegativity of the expected utility implies that

b

=

6 ( v)

S

v and 6'

=

6 ( v')

S

v' o If b > u' t hen 6 > 6'. And if b sv':

(v'-6')Pr(H(v_d 2:c-b')

>

(v'-b)PrIH(v_d 2:c-b)

>

(v' - ó) Pr (

H

(V-I)

2:

c - 6') o

In the last inequality above we used (4) againo Therefore v' - 6'

2:

v' - 6 and

b

2:

b'o •

(17)

,- - --, ＭＭｾｾＭＭＭｾ］ｾＭ］ＭＭｾＭ］］ｾＭＭＭＭ］］ＭＭＭＭＭＭＬ］ＭＭＮＮＮＮＢＮＮＮＮＮＮＮＮＮＬＮＭＭＮＮＢＮＮＮＮＮＢＮＭＮＮＮＮＮＮＮＮＢＮＮＮＮＮＭＮＭ｟ＬＮＮＬＮＮＮＮＭＭＭＭＭＭ］ＢＬＬｾＭＭＭＭ

,.

Remark 4 The Last tu'o Lemmas show that aLthougn an equiLibrium strategy

may not be non-decreaszng it is non-decreasmg on/y :>Jr any vaLuation greater thcln a vaLuation that y/eLds a positive proàabilit!1 oj ,:,';taining the good. Also

ó(v) < I.' in this Cclse.

FinalIy \ve can pro\'f: rhar alI equilibria {)f the ｾｬｬｄｳ｣ｲｩｰｴｩｯｮ＠ game are in-eíficient ,

Theorem 5 (Subscription game inefficiency) ｾＧ［ｬｊｰｰｯｳ･＠ the distribution

F l1as a densüy ｦＨｾＧＩ＠ >

n.

I ｾ＠ (0.1), Then the .:'robabiLity that the good

i,'; provúied gwen Ihat it i8 efficient to do ,50 iS uruior'nllJ Less than one in the

,-ubscnption game.

Proof. Fix a (. _, such rhat _It.f.

< (

< 1. \\"p, want to :3how rhat there is a

R

<

1

.:;uch thar for any equili brium (61 _{('). , , , .}_ón

( , )). r he conditional probability

I ｾ＠

Pr

(Li,.

l i ' . , ' ( '

>

\)=1 - ,

\\"íC: nore rhar 1/ (I') is a solution of

\.

r"l

11..

rnax (/, - L) Pr

(L'

/.:r' ;'/'.;)

2:: ,'-.::') ,

..:.: ,·11

-- \ )=1 .

(5)

Suppose first char for some l S n, Pr

ＨｾＮＧ［ｲｴ＠

,jJ I i,' 2 r; - i/ (;)) = O. Then

I

" 1 f ' ') h . /" . r p (/ ｾ＠ 1 • .. ... I l ( ),) O li

ir IS cear rom (-t t' at Ir I'; S ｾＬ＠ r .é...-j:;:l ＺＺｾＧ＠ ! /'.'! :..:::: (' - U I:i, = as we .

Thus fram Fubini's theorem ir fo11o\\'s rhar

Pr

(t

fi (Vj)

2:

c.

t

l'j

2:

c)'

= Pr

(t

f)J (rj)

2:

c,

t

Vj 2 C. Vi

>

ＬｾＩ＠

$

)=1 )=1 )=1 )-1

Pr

(t

I_'j

2

c, Vi

>

ｾＩ＠

:::;

ｴｾｾｾ＠

Pr

(t

t';

2:

c, I',

>

ｾＩ｜＠

<

Pr

(t

Vj

ｾ＠

c) .

)=1 )=1 )-1

Suppose now that for every i :::; n. Pr

(L)=,

IY (".' i

2:

c - bi

ＨｾＩＩ＠

>

O. We now divide the proof in two cases.

(18)

Case 1: If I:7=1 bJ (()

<

c. Define

x

=

{v

E

[O.

ç"t :

t

L',;

2 c} .

)=1

(6)

It is clear that Pr (X)

>

O and since for every /'.' ｾ＠ｾＭ ir is true that

bi

(Vj)

ｾ＠

max {bJ (;) : IY I()} = /y' (() then >7=1 /)J It')j

<

c if (CI:···: u_{n )}EX.

There-fore

Pr

(t

tJ

(Vj)

2 c.

t

/'j

2 c)

=

)=1 j=l

Pr

(t

ó' (';)

2

c.

t

Vj

2

c. v

ｾ＠

X )

<

Pr

(t

L')

2

c. t'

ri.

x)

)=1

- Pr

(tu,

2 c)

-

Pr(

XI

" ｾ＠ -ｾ＠

<

Pr

(I

t

L',:

2 c)' .

)=1

Case 2: l:]=1 bJ (Ç)

2:

c.

In this case if Vj

2 (

for alI j o!= i rhen

tJ

(l'j)

2

f-) (() for all j

=I

i. Thus

Therefore

Pr

Ｈｾ＠

(ri (Vj)

2

c - ()

2

Pr (t'j

2 (,

j

i=

i) = (1 - F (()

r-I

=: ,. )?=I

Hence

u - b

(vi

2

(v - b ( v

I I

Pr

(f;

oi

(Vj)

2

c - b ( v ) )

>

(U-

O

Pr(f;iJi(V;)2

C-()

2

(v-()'y.

(19)

ｉｾＭｾｾＧｾｾＭＧ＠ .. _ _ Ｎ］Ｍｾ］ＮＭｾＭ］ＭＬ］ＭＭｾ］］］］ＭＭ］ＭｾｾｾＭｾＭ］＠ .. - = - ; . . Ｍ］Ｍ］Ｍ］ＭＭ］ＮＭｾｾＭ］ＭＭ］ＭＭ］＠ .. =-=--=.-=-:.=-= .. _= .. ｟］ＮＭＭ］ＮＭＭｾＭ］ＭＭｾＮＭ］Ｎ｟］ＭｾＭ .,-.,-=--= .. "" .•. =.-.=--=-=-.--=-=.-.=-.-=-.==-=-"""-=--=.=--=-. .,,-. -=--=._=._,..,-, .. =" ...

'"

Thus

I' (1 - ｾ＠ .. )

+

ｾＭ＼＠

2:

h ( L') .

Define Y =

{u

E [O, 1]" : f} (tlj)

:s

I:j, for all

j} .

.

-\n application of Fubini's

theorem shows that

Lfu E Y and

)''''7=1

bJ (1'j)

2:

c then

I:7=i

l.'j

2:

c. Thus

Pr

(t

bJ (Vj)

2:

c,

t

t'j

ｾ＠

c) = Pr

(v

ro-

t

f} (Uj)

2:

c,)

)=1

<

Pr

(t

rnin (

v) , Vj

(1 -

ｾＩ＠

+

-,ç}

2:

c)

<

Pr

(t

Vj

2:

c) .

Thus if \\'e rake

R

=

max

we finish the proof.

•

Pr

(2:7=1

min

{L"J'

;')

(1 - -.j - -.(}

2:

C),

Pr

(2::7=1

/.'j

2:

c, /.'

ｾ＠

X) ,

maX1sisn Pr

(2:7=t

I')

2:

c. 1', >

ｾＩ＠

,

(7)

Remark 5 The hypathesis of positive denszty ar eren the existence of a

den-sity for the distribution F can be avoided if R de.fined in (7) is less than 1.

6 The Contribution Carne

In this section we analyze the contribution game i,e. where players make

contributions that are not refunded if the threshold is not met, Players:

values are determined by independent draws from a continuous distribution

F:

[0,1]

--+ lR. The cost of the public good is c,

(20)

If players i = 2.3 ... V folIow rhe funcrion b (.} and player 1 with

valu-ation v contributes x

2:

O. his expected surpius is

u(x) ',prrx..,..b(I'·2)-·· . ..,..OICsi 2:c)-x

L'Pr(6(1.'2)

+ ... -

OIL'S)

2:

c - x) - x

The following rheorem shows rhar rhe coordinarion problem in the contri-bution game is so se vere that if the cosr of the p.liblic good is slightly above rhe aggregate mean of the \·aluations. rhen rhe unique. equilibrium of the

contribution garne is for each player to contribure zero no matter what his

\'alue is.

Denote bv _... J1 r he mean and b\' _.. Ｈｪｾ＠ t he \'ariance of t he distribution

F.

.")

Define ｾＨＮｶ＠ as the root c of N (7"2 - (N - 1) c ＨＮｾ＠ - 11) - = O. Define a!so RN

=

max

h's,

J.l (;V - 1) .J.. 1

+

av'"\; - l} .

TheorE:m 6 (Strong inefficiency) Suppose that nialjers' valuations are

dis-','1hufprj in Ihl:

:n.

1: .

nrr:rud (/n(i .'Ii!' nlUI/Ofr _{"/ :":':}_.·ore:_i.'i _.\".

!f

_the_{cost is}

, ::::

n.\'.

rlll'lI (lIe /l/lU!UI: t;1!llIiibr!u:n oi :n.; ＬＨＩｮｲｲＺ｛ｬｩｬＢｾＬｮ＠ tjlLTTle is IY

==

O.

Froof. Suppose l' < 1 is Plaver's i valuarion. If (b! ..... brt) is an equilibrium,

L'S expected urilit.v Erom contributing X is

Since IY í I'j) :::; I') we have

'jIX) :::; I:Pr

ＨｾＯＧｊ＠

2:

C -

,1:)

-x:::; Pr

(I:

I')

2

C

-.r)

-.1: =: g(x), (8)

Ｉｾｉ＠｜ＩｾＬ＠

the inequality being strict if Pr

(2::

_i;éi Ui

2:

c -

x)

>

O. Thus it suffices to

prove r hat if 1

2:

x 2: ｉｾ＠ then 9 (x)

S

O. Since in r hi5 case the optimal bid

O

S

6' (v)

<

_Ｌｾ＠ and therefore ｉＺｾＺｬ＠

bi

(ui)

<

c for e\'ery (Vi, ... , UN) . Hence

01 (Vi) = O,

Suppose therefore that x

2:

ＺｾＬ＠ \Ve then have

Pr

(2:

Vj

2:

c -

x)

jf:i

Pr

Ｈｾ＠

(Vi - j.1.j

2:

Ｈｾ＠

-

x - (IV - 1)

J1.)

ＩｾＱ＠

(21)

,';'\

. 'l

Ir 10110ws (har

<

=

-

.

-E

I ()'

}=, ; I'; - f-l)

r :

; c - ,r - , .Y - 1) f-l :

-(.Y - 110'2

ｾ＠ c - .L' - , ,\" - 1 i

f-ll-(.Y - 1) O':l . ' .

g(xl-:;: ,-.r=:n(x),

(c - 1" -

\'.v -

1;

j.1.)-The function h

C)

is strictIv convex, Sore rhar h (1) = ＨＮｶＨＭｬＩｬＷｾ＠ 2 - 1

<

O

. c-l- ｎＭｬＩｾＩ＠ - '

.-\lso

,v

(N - 1)

ｃｾＬ＠

-

j.1.)

'2 h

ｃｾＢＩ＠

= ,\"0'2 - ; ,V - 1 c

ｃｾ＠

-

I.L

r -:;:

O.

TllU::i

•

Remark 6 For example il F Is unilo1iTl/Y distrib'utea 017, [O, 1], then IJ. =

t

'111d (J'2 = :!2' Then 1/

.v

= :)0. c ｾ＠ rnax f 17.1:3. 17'.'-':' ｾ＠ = 17.13 the unique

rt/ll.ilibriIUTI ui fhe rOl/iribuliull gume ＡＮｾ＠ lhe ::r:m "(!iti_{,:,rill.ll/..}

[I

IV = ·100 lhe

!ower bound I.') 3.3.95 II ,V = 1000 the iou:er DOllnll .: .512.75. Jloreover il

.\' - x. ｾﾷＮ｜Ｇｲｖ＠ - f L.

The above result irnplies t har for a wide range OI' ; he cost of the public

good. there is extreme underprovision wnen refunàs rue not allowed .

. For small IV we can obtain sharper bounds on Rs· for exampleif N

=

2,

then RN = 1. vVe turn to this in the next proposirion. \Ve show that the

strategy ,ócontributing zero" for both p[ayers is t he only equilibrium for a

general family of distributions .

Proposition 7 Suppose 1

<

c

<

2.

11

the úistribution function satisfies

F(z)

2:

Z, z E (0,1) then the best response to b_{2 (-)} 5uch that b2 (V2) $ v2 is

OI (vr) = O .

(22)

Ｚｾ＠

.

-Proof. Define (J) (x) = vP(b₂(v'2)

2:

c-xl-x. If x > I: then dJ (x) $ v-x

<

O.

Thus to maximize o \ve must have x ::; ｾＧＮ＠ Suppose x

>

O. If x

<

c - 1 then

o(x)

=

7)·O-x = -x

<

O. Suppose now c-I

S

x ::; L'. \Ve have 4>(c-1) =-x

and 6( v) ::; O. Now:

d> (x)

<

I: P (!-'2

2:

C - x) - I = d 1 - F í r: - x)) - x

<

cil-(c-xj)-x=r(1-c)-í,c-1)x<0 .

•

_{Note that the above family of distributions includes well known}

distribu-rions such as the uniforrn a.nd the exponemial distributions. :\Jore generally

any concave distribution satisfies ir.

Remark 7 We couLd prove aLso that there is no mixed strategy equilibrium.

We omit the proof.

The proposirion (i) abm'e is not tme for C\'e!T ,:iistrihution. Inttútively

if the distriburion gives a lligh \\'eight to high \'aiuarions then an equilibriwn

exists. This is the sense of the next proposition.

Proposition 8 Suppose that c

>

1 and that there erists an a E (0,1) such

that F _{ＨＲｾＩ＠} = 1 - a. Then

b

(

/!) =

{O} 1I

if

/. >...f..

_- _'20

ｾＮ＠ < O)c

.Q

is an equiLibrium strategy of th·.;. contribution game u:lth two pLayers.

\Ve omit the proof but apply the result in an example.

Example 1 lf F (x)

=

x'Y, then Q is a solution 01 ＨｾＩＢＧＡ＠

=

a'Y (1 - a). Since

max { a 'Y (I - a) ; a

E

[O,

1]}

=

(1 : ( )

I

(1

ｾ＠

')' ) ,

there is an equiiibrium if c $ 2

(d::;)

I

12'"1'

Thus if')'

=

13, c

=

1.5 wehave

(23)

r.r

The previous theorems shows that for high enough cost (i.e. greater than

RN ) the contribution game efficiency is zero. The following theorem shows

that even for low COSt5. any equilibrium of rhe contribution game is inefficient.

Define Sm (x)

=

Pr

(I:;:

1 L'j

ｾ＠

x). The following theorem shows that

rhe probability that the good is provided given that ir i5 efficient to do so is

hounded away from 1 in any equilibrium of rhe cOlltribution game.

Theorem 9 (Contribution game inefficiency) The efficiency

probabil-ity of the contribution game is at most

Pr ＨｌｾＱ＠ /'! (1 -

S..,_.

i,C - I';))

2:

I:)

I -

--==:..!.--- <

1.

1 -

5

n(c)

Proof. Suppose (61 (, ), ... , 6n (-)) is an equilibrium of the contribution

game. I t is clear that 61 (t') _ｾ＠ _tifor ali v. Player _iwith valuation _Viexpected

utilit:y is

Vi Pr

(2.::

[r1(Vj)

2:

c - b1(fi_{d )} - b1(i\)

2:

i\ Fr(I: lJ1(Vj)

2:

c)

2:

O.

Ji:1 . ; 0=1

Therefore

ｾ＠

_"Pr ( \ " _L- _r:::".,> C - ,-:) _,

} 'FI .

•

- P

ＨｾＧＢ＠

'... /;'-))' hi(-)

2:

I', r

L

t:r I/':} :::::: '.' - ) (/I, ｾ＠ Vi·

)Y"

<

Pr

(t

1.'lll - Sn-li.C - ud)

2:

c)

,=1

n

<

Pr(L t'i

2:

c)

i=l

(24)

(25)

....

differential equations, Define G (L') = 6-1 (c - 6 (v)) , This implies that

6íG(L')} ｾＶｩＧｴＬＩ＠ f' (12)

(1.-6(v))J(e(L'))(6-i_)'(6(eírl)) ₌ _1-F(e(v)), ₍₁₃₎

In d3) we used that 6

(e

(v)) = c - 6 (1.,) , Since b-1 _{1':1(; .. ;))}_{= :....;}_{we have that}

íÍJ-1)' (6 (w)) 6' (:....;) = 1. 'Thus choosing : .... : = G íl') we ger (6-1_)'(6(e(L'l))₌ _L'b'(GJrii,

Substituting rhis in (13) it follO\\is rhat

lI.' - Ú (I') I

fie

(I'i I

----...:...--- = ｾＭＬＧＨｇＨｉＧＺＢ＠

l-F(G(u)) (14)

Xow from (12) applied to v and to

e

(1') we have rhat 6 (C (u))

+

6 (v) = c =

6 (e'2 (v))

+

b ( C ( I.' )) , Hence 6 (e2 (v)) = 6 ( L'.i and r herefore

e

(C ( v )) = v,

Therefore from (1,1) we conclude that

;1 . _

., " I

.-r;'; - l.' G ' i , ..

1 - F: /'1 (15)

Differemiating (12) \\'e obtain 6' (e (/'i) C' (,,) -:- 6'! /.)

=

O, The following

rhcorem describes the sysrem of differemial equarions rhat an increasing equilibrillm srraregy ro rhe sllbscription game Illusr :::arisfy,

Theorem 10 The solution of the sy8tem of dlfJerentlc1 equations below

char-licteri,:es !l ,sljrTl'me;ric o/udibrilLm ｉｮＯＧｏｩＯＮＧｩｮＡｊｾｴｲｩ｣ｴｬｵ＠ ,.' ,:reasing strategies for

:he SUOSCllptiOll gume Icith tl1'O plaljcrs .'nth caiues detérmined by independent

,Jmws from a dlstribution F :

:0, 1; -

?:, l.1:nere F is r:ontinuo1LsLy differen-tlable and

.f

=

F'

> I) ever7jwhere,

(e(!'1 - hiG(i·'!,i "(I')

b' ( 1.') = ' . r ' . ' ,

. 1-r(l'\ (16)

C' (L') = _ ( 1 - F (e ( v) )) (e (u) - b ( G ( v ｾＩ＠ j

J

(

v) ,

(1 - F (L')) {v - 6 (1.,))

f

(G iC.J)

Remark 8 It is important to note that the system (J 6) gives only a candidate

(26)

"

....

7.2 The contribution game differential equation.

An analysis similar to the analysis made above gives the system of differential equations that a strictly increasing equilibrium of the cO!ltribution game must satisfY.

Theorem 11 The soilLtion 01 the system of differential equ.ations below

char-I.Lcterizes a symmetTic equzl-ibrium involzring strictly increasing strategies for the contribution game u-ith two players with ｲ｡ｬｵｾｳ＠ determined by

indepen-dent draws from a distribution F :

[O,IJ -

iR.where· F is continuously

differentiable and f = F'

>

O e'l:erywhere. b'(v)=G{L')f(v),

G'

(v) = _

G

(

v)

f

(v) .

l'/(G(v))

For example if

F

is the uniform distribution.

6' (rI

G' (1.')

G (1.,).

G (v)

v

(17)

This gives G (u) = ｾ＠ and b (v) = a .;- k log {v) . .\'aturally b (.) cannot be

an equilibrium for ali li since it is negative for small v. If O

<

c

<

l/e,

b (/') = ma.x {O, c-;- k log ( 1.'

J)

where k is such t h,H ｾＺｫ＠ = ,,-':. is an equilibriurn.

8 Conclusion

\VC analyzed t\VO mechanisms for pri\'ate provision of discrete public goods

with incomplete information a.nd continuous contributions. The

subscrip-rion mechanism refunds the money to contributors if the public good is not

provided whereas the contribution mechanism does not allow for refunds. Our analysis showed that, unlike'the mo dei with complete information,

the coordination problem becomes more comple.x and efficient provision is no

longer possible, ｾｬｯｲ･ｯｶ･ｲＬ＠ the two mechanisms lead to completely different

(27)

"

.

-,

.-,

References

[1] Admati. A. R. and ?vI. Perry, 1991. "Joint Projects without

Commit-ment." Review of Economic Studies 58. 259-276.

;2) Andreoni.

J .. 1998. "Toward a Theory of Charirable Fund-Raising,"

Journal of Political Economy 106. 1186-1203.

[3] Andreoni. J .. 1988. "Privately Provided PubLic Goods in a Large

Econ-omy: The Limits of Altruísm." Journal of Public Economia 35,

.j7-73.

H]

Bagnoli. ｾｉＮ＠ and 11. ｾＭｉ｣ｋ･･Ｌ＠ 1991. 'Voluntary Contribution Game:

Effi-cient Private Provision of Public goods." Economic lnquiry 29,

351-356.

)J

Bagnoli. ｾｉＮ＠ and B. L. Lipman. 1989. "Pro\'Ísion of Public Goods: Fully

fmplcmcnring rhe ClJre rrúough Pri\'ate ｃｯｮｲｲｲｾｾｬＱｴｩｯｮｳＮＢ＠ Review of

Econamic Studies 56. 563-602.

[6] Bergstrom. T .. L. Blume and H. Varian. 1986. "00 rhe Private Provision

of Public Goods." Journal of Public Economics 29. 25-49.

[7] Bliss, C. and B. Nalebuff. 1984, "Dragon-Slaying and Ballroom Dancing:

The Private Supply of a Public Good." Journal of Public Economics

ＲＵｾ＠ 1-12.

'S] Cadsby, C. B. and E. :\Iaynes. 1999. "Voluntary Provision of Public

Goods with Continuous Comributions: Experimental Evidence,"

Jour-nal of Public Economics 7L 53-73.

[9] Comes, R. and T. Sandler, 1984,

"Easy

Riders. Joint Production, and

Public Goods," Economic Journal ＹＴｾ＠ 580-598.

[lO] Cornelli, F., 1996, "Optimal Selling Procedures with Fixed Costs,"

Journal af Economic Theory 71, 1-30 .

)IJ

d'Aspremont, C. and L. A. Gerard-Varet, 1982, "Bayesian Incentive

Compatible Beliefs," Journal of Mathematical Economics 10(1),

83-103.

(28)

[12] d·Aspremont.

C.

and L. A. Gerard- \-aret. 1979. "Incentives and

Incom-plete Iruormation.·· Journal of Public Economics 11(1),25-45.

i13]

Gradstein . .\1.. 1994. "Efficient Provision of a Discrete Public Good,"

International Economic Review 35(4), 05/7'-897.

:1-1) Gradstein . .\L. 1992. 'Time Dvnamics and Incomplete Iruormation in the Private Provision of Public Güods.·· J ournal of Political

Econ-omy 100(3). 1:236-12-1-1.

ｾＱＵ｝＠ Gradstein . .\L. S. ｾｩｴｺ｡ｮ＠ and S. Slutskv. 1994. ":'\eutrality and the Pri-vare Provision üf Public Goods with Incomplete Information," Eco-nomics Letters 46. 69-7'·5.

)6] .\Iaskin.

E ..

1977. ":'\ash equilibriurn and \Velfare Optimality," ｾｬｩｭ･ｯＮ＠

) I)

.\-lilgrom. P. and J. Roberts. 1990. ·"Rationalizability. learning, and

equi-librium in games \vith ｳｴｲ｡ｴ･ｾｩ｣＠ compiememariries.·' Econometrica 58.

; ].):,-1 ｾＺＭＭ［ＭＮ＠

)8)

l'iitzan. S. and R. Romano. 1990. "Pri\'are Provision of a Discrete Public

Good with Pncertain Cost." Journal of Public Economics 42,

357-370.

[19] Palfrey. T. and H. Rosenthal. 1988. "Privare Incentives in Social

Dilem-mas: The Effects of Incompiere Informarion and A.ltruism." Journal of

Public Economics 35. 3UCJ-:332.

'"]01 Palfrey, T. and H. Rosemhal. 1984. "Participation alld the Provision

1)[ Discrete Public Goods: A Strategic Anai:vsis.·' .Journal of Public

Economics 24. 1 n-193.

;211 Palfrey, T. and S. Srivastava. 1986. 'Sash Implementation Using

Un-dominated Strategies," .\Iimeo. Carnegie .\Iellon Gniversity.

)2] Roberts, R., (1984) ... :\ Positive .\Iodel of Pri\'are Charity and Wealth

Transfers." Journal of Political Economy 92. 136-148.

[23) \Varr. P. G .. 1983. "Pareto Oprimal Redistribution and Private Charity,"

Journal of Public Economics 19. 131-138.

ｾＲＴｊ＠ \Varr. P. G., 1982. "The Private Provision of a Public Good Independent

of the Distribution of Income," Economics Letters 13, 207-211.

(29)

AppenciL"X

'The subscriprion game \\irh

t\VO

players:

The

lllll-form distribution case

In what follows. we provide some imuition how ro soh-e trus problem when

ＧＮｮｾ＠ take imo accoum r he boundary conàitions. for r he uniform distribution

011 the imerval

[O. li.

the unconstrained 501ution.ro rhe 5"stem of differential

equations stated in 5ecticn 7.1 is gi,'en oy the folfowing,pledging function:

).Tote. however. that in equilibrium a player may not follow b( v) for any

L' in

[O. 1J

as trus may lead to pledging more than his ,'alue or more than the

cost of the public good. Hence we need to impose rhe following boundary

ＬＧＨｾｮｲｬｩｴｪｮｮｾ＠

Lil L'; <. f_', \bll (18)

b( 1') < ;'0 ( b2)

(19)

b(v)

>

_{O. (b31}

₍₂₀₎

It turns out that (b 1) and (b3) are not binding since c

>

l. Condition

￭ｾＩＲＱ＠ is binding as h(/') :> /' for /' < Ｇｾ｣ＺＺＬ＠ \\'e no,," :-:h,)\\" formal1y that the flJllowing is a s\"ll1merric equiiibrium piedging ｾｲｲ｡ｲ･ＲＮＢ｜ﾷ＠ '"'-Jr the subscription

game \Virh two piavers whose ,'alues are lmiformi,' (iisrr:Duted on

[0,1]

and

L < c < 1:

ｾｩ＠ • ( (,) = o _ . I -

-{

ｾ｣ＺＭｩ＠ＭＧＭｾＮ＠ if ｾｲＭｩ＠

<

!'

<

l.

O.otherwIse.

To verify that trus is an equilibrium, we assume that Player 2 is following

lhe proposed equilibrium pledging 5rrategy, \\'C have to find the best response

of player 1. \Ve first show rhat for 2c; I :::;: ,.' :::; l. bd I.' j = Ｒ｣ｾ＠ I

+

7is a best

response to 6'(/)2). Player 1'5 expected surplus. if he pledges b, is given by

o

(b) = (ti - b) Pr (b

+

bO (1.'2) ｾ＠ c) .

To find the maximum of d; first note that if O

<

c - 6

<

b" _{ＨＲｃＺｾ＠} 1) = 2C_3-1then