Free riders and the optimal prize in public‐good funding lotteries

• Date: 01 September 2020
• DOI: http://doi.org/10.1111/jpet.12460
• Author: Paan Jindapon,Zhe Yang
• Published date: 01 September 2020

J Public Econ Theory. 2020;22:1289–1312. wileyonlinelibrary.com/journal/jpet © 2020 Wiley Periodicals LLC

|

1289

Received: 15 November 2019

|

Accepted: 16 June 2020

DOI: 10.1111/jpet.12460

ORIGINAL ARTICLE

Free riders and the optimal prize in

public‐good funding lotteries

Paan Jindapon

1

|Zhe Yang

2

1

Department of Economics, Finance, and

Legal Studies, University of Alabama,

Tuscaloosa, Alabama

2

Department of Economics, Wofford

College, Spartanburg, South Carolina

Correspondence

Paan Jindapon, Department of Economics,

Finance, and Legal Studies, University of

Alabama, Tuscaloosa, AL.

Email: pjindapon@ua.edu

Abstract

We prove the existence and uniqueness of an equili-

brium in a game where players, whose preferences

exhibit constant absolute risk aversion or constant

relative risk aversion, contribute to a public good via

lottery‐ticket purchases. Contrasting models with

risk neutrality, we show that an equilibrium with a

strictly positive amount of the public good may not

exist without a sufficient number of participants who

are not too risk‐averse. We show that players who are

more risk‐averse purchase fewer lottery tickets and

are more likely to free ride in equilibrium. In fact, it is

possible for free riders to place a larger value on

the public good than do those who contribute. In a

symmetric equilibrium, we show that an upper

bound exists for the amount of the public good,

even though there are infinitely many participants.

Furthermore, we derive a lottery prize that max-

imizes the amount of the public good in a symmetric

equilibrium and find that such a prize always results

in an overprovision of the public good.

1|INTRODUCTION

A lottery is an efficient and popular fundraising mechanism. In 2018, Americans purchased $77.7

billion of lottery tickets in 44 states; 30% of the amount was allocated to fund education and other

social programs.

1

Asiswellknownintheliterature,Morgan's(2000) lottery is more effective in

1

On the basis of the 2018 fiscal year (July 1, 2017–June 30, 2018), see La Fleur's 2018 World Lottery Almanac.

providing a public good than the voluntary contribution mechanism (VCM). While positive ex-

ternalities associated with the public‐good dissuade players from contributing in all public‐good

funding mechanisms, a contribution through lottery purchase causes negative externalities by

reducing other players' probabilities of winning. This advantage of using a lottery to fund a public

good is supported by many experiments in the laboratory and in the field.

2

Whether a lottery mechanism dominates an all‐pay auction depends on the information

structure. Between the two mechanisms, Goeree, Maasland, Onderstal, and Turner (2005)theo-

retically show that an augmented all‐pay auction dominates the lottery mechanism in a private

value setting, while Corazzini et al. (2010) find experimental evidence to support the lottery me-

chanism in a common value setting. In a complete information setting, Damianov and Peeters

(2018) show in theory that an all‐pay auction dominates the lottery mechanism, but in the lab the

number of players determines which mechanism is optimal. Like Damianov and Peeters (2018), we

adopt the complete information setting and show that not only the number of players, but the

chosen prize also determines the effectiveness of the lottery mechanism. In addition to the fact that

this setting is less complex than other information structure, it bears more resemblance to many

lottery games in the real world where all lottery buyers place the same valuation on the jackpot.

Even though many studies have extended Morgan's (2000) framework in various directions,

and alternative schemes have been proposed,

3

very little theoretical analysis on risk attitudes in

Morgan's mechanism is available. Specifically, most researchers adopt Morgan's (2000) as-

sumption of quasilinear utility, so that each player's marginal utility from private consumption

is constant, regardless of his wealth or whether he wins a prize. This functional form leads to an

important theoretical result: The amount of the public good provided in equilibrium increases

with the lottery prize.

4

Thus, it is feasible to raise money in any desired amount, provided the

lottery prize is large enough.

In this paper, we propose a framework that allows for concavity in utility functions with

respect to both private‐and public‐good consumption. Duncan (2002) is the first to examine the

effect of risk attitude on the amount of the public‐good provision in Morgan's lottery. Using a

Cobb–Douglas utility function, Duncan shows that there is an upper bound for the amount of

public good that can be provided in equilibrium. Lange et al. (2007b) provide a more formal

approach to analyzing the effect of risk aversion; they prove the existence and uniqueness of

equilibrium in a lottery mechanism where players exhibit constant absolute risk aversion

(CARA). On the basis of their experiment, they find that public‐good contributions decline in

risk aversion which is consistent with their theoretical prediction. To the best of our knowledge,

there is no other theoretical proof of the existence of equilibrium in Morgan's lottery me-

chanism, given risk‐averse players.

5

2

For example, see Morgan and Sefton (2000), Lange, List, and Price (2007b), Corazzini, Faravelli, and Stanca (2010), and

Carpenter and Matthews (2017).

3

See Goeree et al. (2005), Lange, List, and Price (2007a), Maeda (2008), Corazzini et al. (2010), Kolmar and Wagener

(2012), Faravelli and Stancab (2012,2014), Franke and Leininger (2014), Kolmar and Sisak (2014), Bos (2011,2016),

Damianov and Peeters (2018), and Ghosh and Stong (2018).

4

See Lemma 5 in Morgan (2000). For example, Morgan and Sefton (2000) assume that each player's utility is

u

wG w βG(, )= + , where

β

is a positive constant called the marginal per capita return. It follows that the amount of

public good in equilibrium is

G

=

Rnβ

nβ

e(−1)

(1−)

, where

R

is the lottery prize and

n

the number of players, and that →

∞G

e

as

→∞

R

.

5

In the contest literature where there is no public‐good provision, Cornes and Hartley (2012) provide a general

framework for analyzing an equilibrium given heterogeneous risk‐averse players. Beyond risk aversion, Treich (2010)

and Jindapon and Whaley (2015) explore the effects of downside risk aversion in contests while Kelsey and Melkonyan

(2018) and Stong (2018) focus on ambiguity aversion.

1290

|

JINDAPON AND YANG