Discussion Papers In Economics
And Business

Coordination and free-riding problems in the provision ofmultiple public goods

Ai Takeuchi and Erika Seki

Discussion Paper 19-15-Rev.

June 2020

Graduate School of Economics
Osaka University, Toyonaka, Osaka 560-0043, JAPAN.

Coordination and free-riding problems in the provision ofmultiple public goods1
Ai Takeuchi

Erika Seki

College of Economics, Ritsumeikan
University

Graduate School of Economics, Osaka
University

5 June 2020
Abstract
This study considers the twin problems of free riding and coordination failure prevailing in theprovision of multiple public goods with diminishing marginal returns in which the payoff-sum
maximising Pareto optimal outcome requires less-than-full contributions by group members. We
examine theoretically and experimentally whether the provision of information on the demand for
public goods helps overcome these problems and improves efficiency. We construct a game of two
public goods, each with an upper bound on effective contributions. Theoretical analysis predicts
that this information improves efficiency as it prompts efficiency concerned individuals to matchthe upper bound of each public good in equilibrium. The experimental results show countervailing
effects of demand information, i.e., it improves coordination but deteriorates the free-riding problem.
(JEL: C72, C91, C92, H41)
Keywords: Charity, Free riding, Coordination, Multiple public goods, Laboratory experiment,
Information.

Acknowledgments: We are thankful for the helpful comments of David Flath, Takao Kusakawa, Charles
Noussair, Judd Kessler, Dongryul Lee, and the participants of the 2017 Asian-Pacific ESA Conference andthe 2018 WEAI Annual conference. The experiment was approved by the Ritsumeikan University Ethics
Review Committee for Research Involving Human Participants. [Fundings] This research was supportedby The Seimeikai Foundation during FY 2016, and JSPS KAKENHI Grant Number 19K01565.
E-mail: ai-tak@fc.ritsumei.ac.jp (Takeuchi); erika_seki@econ.osaka-u.ac.jp (Seki)
1.

Declarations of interest: The authors declare that they have no conflict of interest.

2

1. Introduction
This study considers the twin problems of free riding and coordination failure
prevailing in the provision of multiple public goods with diminishing marginal returns.
We examine theoretically and experimentally whether the provision of information on
the demand for public goods helps overcome these problems and enhances efficiency.
Most studies on voluntary contribution mechanisms (VCM) have examined the freeriding problem of a single linear public good (see, e.g., Ledyard, 1995 and Croson,
2010 for survey). However, many social services share characteristics of multiple
public goods with diminishing marginal returns in which the payoff-sum maximising
Pareto optimal outcome requires less-than-full contributions by group members. The
feature of non-linearity makes them vulnerable not only to the free-riding problem
but also to coordination failure.1 In our study, we give two motivating examples ofcharitable giving.2
Blood banks, for example, are an important public health service based primarily
on human blood donated by volunteers. For an effective operation of blood banks it isessential to match supply and demand in any season of the year, say the summer andwinter seasons. Accordingly, the operations of blood banks in each seasons can beconsidered as a different public goods, thus making this situation a game of multiple
public goods game. Blood components are perishable and donors must observe aninterval period between consecutive blood donations (between 4 to 16 weeks). 3
Because of these aspects, an excess supply of blood at a given period preempts supply
in the subsequent period, causing wastage in the previous period and a shortage ofdonated blood in the latter.
Charitable action, such as post-disaster volunteering, is another example of public
services requiring a coordination among volunteers. Often multiple locations are inneed of volunteer support that can be considered as multiple public goods provided
by on-site volunteers. Coordination problems arise when some popular locations
attract too many volunteers, causing an oversupply in those communities, while other
1. Compared to the large experimental literatures on the linear public goods game, experiments on public
goods with diminishing returns are limited (see Laury and Holt, 2008 for literature in Economics andKameda, Tsukasaki, Hastie, and Berg, 2011 for literature in Social Psychology). Experiments on multiple
public goods are few but increasing. Corazzini, Cotton, and Valbonesi (2015) study a case involving
multiple threshold public goods. Blackwell and McKee (2003) and Fellner and Lünser (2014) analyze
a case involving local and global public goods. Bernasconi, Corazzini, Kube, and Maréchal (2009) andEngl, Riedl, and Weber (2017) analyze a case involving two identical public goods with one receiving
institutional support to promote contributions. Except for Corazzini et al. (2015), these studies adopted
linear public goods models and did not address coordination problems.
2. In conformity with the existing experimental literature on donations, we use the public goods game tomodel charitable giving. Vesterlund (2015), in her survey on experimental literature on charitable giving,
explains the relevance of the public goods game for examining charitable giving.
3. Blood products are manufactured for each blood type and have different shelf lives—4 days forplatelets, 21 days for red blood cells and whole blood, and 1 year for plasma (Ministry of Health, Labourand Welfare, 2019). In the case of excess supply, there is always a risk of some blood components reaching
their expiration dates before use.

3

locations are left with few or no volunteers. Such misallocation of volunteers hampers
efficient provision of a much needed support for the disaster affected residents.
Similar coordination issues may also arise in connection with charitable donations on
crowdfunding websites: Some projects attract funding far in excess of their operational
capacities, while other projects are left underfunded, compelling them to remain below
their minimum operational levels (Corazzini et al., 2015).
The above examples share two common features. First, the efficient provisionof multiple public goods requires the coordination of contributors. Second, a lackof information about the required amount of contributions, i.e., the demand forthem, makes coordination difficult. For instance, the absence of information about
a seasonally varying scarceness of blood makes it difficult for potential donors tocoordinate the timing of their donations with the demand across time periods. Without
sufficient information about the availability of alternative volunteers, some of them
may become redundant.
To overcome coordination and free-riding problems and thereby improve theefficiency of social service provision, it is essential to ensure that the necessary
information is available. Providing information about the demand for blood donations
across time and the needs of volunteers across locations may solve these twin
problems. In view of fast-progressing information technologies, such as smartphone
apps and data streaming, the forecasting of intertemporal demand for blood donations
across the seasons and the provision of real-time information about alternative
locations in need of volunteers can be made readily available. Thus, our focus is on
answering the following question: Does the provision of information about the relativeneed of contributions (i.e., demand for each public good) promote coordination,
reduce free riding, and thus improve the efficient provision of multiple public goods?
In general, the following features permeating the social services give rise to social
dilemma situations with free riding and coordination failure: i) the contribution toeach public good exhibits diminishing marginal returns, ii) there are multiple public
goods, and iii) the contributions, once made, are not transferable between the public
goods. When these three features coexist, as in the motivating examples, the efficiency
loss of over-contribution to one public good has a larger consequence in the sense thatan over-contribution to one leads to an insufficient contribution to the other. It may
therefore be advantageous for society to make efficient use of the altruistic/pro-social
behavior of individuals, coordinating them to allocate their contributions to different
public goods. In this study, we investigate whether provision of fuller information on
the demand for each public good facilitates coordination of contributions to multiple
public goods, enabling potential contributors to respond to relative scarcities andthereby improving efficiency. 4
4. In a similar vein, Slonim, Wang, and Garbarino (2014) propose a donor registry as a central
clearinghouse system. The registry collects information about past blood donors, including theirconvenient donation times and locations. The call center contacts registered people for blood donations. A
field experiment with the Australian Red Cross Blood Service reveals that registered past donors are more
likely to donate in response to calls for donation and that the registry system helps blood banks to facilitate

4

To incorporate the above mentioned three features underlying a social dilemma
with coordination and a free-riding problem, we construct a game of two public goods,
each with an upper bound on effective contributions. The upper bound implies thatthe marginal return of a contribution to a public good becomes zero when the sumof contributions to this public good goes beyond its upper bound. This is a special
case of a public goods model with a diminishing marginal return of contributions.
Instead of the marginal return gradually decreasing, it remains constant up to thethreshold, dropping to zero beyond it. This specification has three advantages. First,
this game is simple and much easier for subjects to understand when compared to otherspecifications of diminishing marginal returns, such as quadratic or Cobb-Douglas
type utility functions used in the experimental literature (e.g., Laury and Holt, 2008).
Second, it has an interior payoff-sum maximising Pareto Optimal outcome as intended
but maintains the property of free riding as a dominant strategy. This allows us tocompare our results against the existing literature on the linear public goods game.
Finally, the specification of multiple public goods with upper bounds fits well with theexample of blood banks with growing concerns over coordination issues.5
Two alternative information regimes are considered. In the partial information
condition, demands are probabilistically determined and ex ante players only know
the probability distribution of demands for each public good. In the full information
condition, demands are deterministic and players know the exact demands before
making contributions.
Theoretical analysis predicts that information conditions affect the provision of
public goods differently, depending on the assumption of individual preferences. If
each subject aims to maximise own payoff, information conditions will not affectindividual behavior. The underlying reason is that the demands for contributing to eachpublic good do not alter the essential property of the standard public goods game inwhich free riding is a dominant strategy. If, by contrast, each player aims to maximise
the payoff-sum of all players, information provision will increase efficiency since full
information on the demand prompts the payoff-sum maximising players to match thedemand for each public good in equilibrium. However, when only partial informationon the demand is available, it will become impossible to match the demand for eachpublic good. Particularly, when demand follows a uniform distribution, contributions
donor coordination. Slonim et al. (2014) compared the donation behaviors of registered and unregistered
past donors and found that among those contacted 9.7% of registered members came to the donation
venues, while only 5.9% of control subjects came to the venues 4 weeks after receiving the call. Contrary
to their centralized clearinghouse system, we consider a decentralized system that enables independent
decision makers to better coordinate their individual actions, based on additional common information on
the demands met by the blood bank.
5. The model’s applicability may differ between examples. The case of blood donations closely
resembles the model: The marginal benefit is constant until the demand is satisfied and drops to about
zero for contributions beyond because the shelf life of many blood products is short. On the contrary, forthe case of post-disaster volunteers, the change in the marginal benefit from contributions may be smootherand could even become negative (because of congestion due to too many volunteers.

5

to each public good will be about equal. This could cause over or under contributions,depending on the realized value of the upper bounds, thereby lowering efficiency.
To test whether and how information provision affects individual’s behavior, we
conducted a laboratory experiment. Based on theoretical analysis, we hypothesize that,
if there are subjects who are efficiency-concerned, providing full information on the
demand structure will increase efficiency.
The results of our laboratory experiment do not support our hypothesis but reveal
countervailing effects of demand information. We find that full information improves
the coordination of contributions but worsens the free-riding problem, which morethan offsets the positive effects of improved coordination. On the one hand, demand
information directs more contributions to the public good in case of relatively highdemand. On the other hand, demand information tends to lower subjects’ contributions
to the public good in case of low demand. The latter effect can be explainedby strategic uncertainty, making subjects withhold contributions to avoid potential
wastage in view of possibile over-contribution.
The paper is organized as follows. Section 2 presents the model and theoreticalpredictions, followed by Section 3, describing the experimental design andhypotheses. The subsequent sections, 4 and 5, compare the contribution rates
and efficiency under the two information conditions and discuss the possible
interpretations of subjects’ behavior under the full information condition. The final
section, 6, concludes the paper. The Appendix includes full proofs of propositions.

2. The Model
2.1. Basic setup
We consider a model of an n-person society with two public goods, denoted as A andB. Each member i ∈ {1, . . . , n} has a semi-binary choice of whether to contribute toeither public good, A or B, at a cost of c > 0 or not to contribute.6 Thus, the strategy
set of each individual is xi = (xiA , xiB ) ∈ {(1, 0), (0, 1), (0, 0)}.
In this model, considering scenarios in which excessive contributions to a public
good lower efficiency, we assume that each public good k ∈ {A, B} requires a certain
level of contribution d k ∈ N, and any excessive contribution would be wasted.7 In otherwords, each contribution to a public goods yields a constant marginal return β until
the sum of contributions to public good k reaches d k , but, any contribution beyond d kyields a marginal return of zero. The benefit from public good k for each individual at
6. This reflects the infeasibility of individuals to contribute to both public goods at the same time. For
example, regarding blood donations, to donate blood during consecutive periods is prohibited in many
developed countries. As for volunteers, it is not possible for one volunteer to be at two different locations
at the same time.
7. In the blood donation example, d k corresponds to the demand for blood donations in period k. In the
post-disaster volunteer example, d k corresponds to the number of volunteers needed in location k.

6a strategy profile x = (x1 , x2 , ..., xn ) can thus be written asn

Bk (x) = β × min{ ∑ xkj , d k },k ∈ {A, B}.

j=1

The payoffs of individuals are defined, using the marginal cost of contribution cand the benefit from the public goods. Individual i’s payoff at a strategy profile x is
πi (x) = −c × (xiA + xiB ) + BA (x) + BB (x).

(1)

We have the following assumptions on the size of the parameters. First, we assume
that d A + d B ≤ n, which implies that there are sufficient individuals to satisfy thedemand for contributions. Second, as in the standard linear public goods game, we
assume that β < c and nβ > c. β < c reflects the fact that an individual’s marginal
benefit from an individual contribution is less than an individual’s marginal cost.
nβ > c reflects the fact that society’s marginal benefit from an individual’s contribution
is larger than an individual’s marginal cost. The latter implies that it is efficient tocontribute until the upper bound of the effective contribution d k is reached. Due tothese assumptions, this game presents a social dilemma situation, in which individuals
have an incentive to free ride, although it is advantageous for society to have theplayers coordinate and contribute to the upper bounds.
2.2. Different information conditions
This study investigates whether efficiency can be increased by providing information
about the demand for contributions. To investigate the effects of the change in
information provision, we model the baseline situation with less or no information
provision as a game of imperfect information, in which the upper bounds are initiallydetermined by nature (partial information condition). None of the players know therealized value of the upper bounds, but they know the probability distribution inwhich each combination of the upper bound may occur. In contrast, as a result of
being provided with more information, players will have a better understanding of
the needs for each public good. This baseline situation with information provision ismodeled as a game of perfect information, in which the values of the upper bounds
are exogenously given (full information condition).
More precisely, the model of the latter case, the full information condition, refers
to the game explained in Section 2.1 above. The model of the former case, the partial
information condition, shares all basic aspects of the full information condition models
except that we assume that d A and d B are endogenous variables determined by nature.
For simplicity, we assume that each combination of d A and d B would occur with equal
probability.8
8. Let D(n) be the set of all possible combinations of (d A , d B ) for a given number of players n. Thiscondition on probability would impose a discrete probability distribution over D(n) because we assume
that d A , d B ∈ N and d A +d B ≤ n. For example, when n = 3, D(3) = {(1, 1), (1, 2), (2, 1)}, thus each element
would occur with a probability of 1/3.

7

2.3. Theoretical prediction
In this section, we analyze the game under the standard assumption of payoff
maximising players. We find that different information conditions do not affectplayers’ behavior because in both conditions players’dominant strategy is to free ride.
However, in laboratory experiments on the public goods game, subjects often choose
contribution levels that yield higher efficiency than that of no contributions, at leastin the first few periods of the game (see, e.g., Ledyard, 1995 and Croson, 2010). As
a reference point, we also analyze the game with an alternative assumption of payoffsum maximising players. In this case, we find it is possible to achieve higher efficiency
in equilibrium under the full information condition than under the partial information
condition.
2.3.1. Payoff maximising players. We start our analysis with the assumption of
payoff maximising players. As stated above, with the assumption of β < c and nβ > c,the decision to contribute or not to each public good has a feature similar to that ofthe prisoner’s dilemma game. This gives us the first proposition wherein the players’
dominant strategy is not to contribute to either public good under the full information
condition.
P ROPOSITION 1. Let i ∈ {1, . . . , n}. In the full information condition, for any(d A , d B ), xi = (0, 0) is the strictly dominant strategy for player i.
Changing the information condition from full to partial does not significantly
change Proposition 1. It changes the calculation of the expected benefit fromcontributing. However, since this is going to be less than or equal to β , it will belower than cost c. Thus, it is also a dominant strategy to not contribute in the partial
information condition.
P ROPOSITION 2. Let i ∈ {1, . . . , n}. In the partial information condition, xi = (0, 0)
is the strictly dominant strategy for player i.
Propositions 1 and 2 imply that if the players maximise own payoffs, thendifferences in information provision would not affect the equilibrium strategies. All
players decide to completely free ride, and no public good is provided.
2.3.2. Payoff-sum maximising players. Next, we analyze the two information
conditions in the case of payoff-sum maximising players. The payoff of player i at
strategy profile x is:nΠi (x) =∑ π j (x)
j=1

n

n

n

= −c × ∑ (xAj + xBj ) + nβ × (min{ ∑ xAj , d A } + min{ ∑ xBj , d B }).
j=1

j=1

j=1

8

This equation shows that if the sum of contributions to public good k is less thandemand d k , then the marginal benefit from contributing to public good k is nβ , whichis larger than cost c. Thus, while the contribution is effective, additional contributions
will increase the payoff sum. However, unlike in the standard linear public goods
game where the payoff-sum is maximised when all players contribute their wholeendowment, the social optimum is obtained when the demand for contributions isjust met in both public goods. The reason is that any contribution beyond the upperbound yields a marginal benefit of zero. This implies the next proposition for the full
information condition.
P ROPOSITION 3. Under the full information condition, when all players maximise
their payoff-sum, the strategy profile x∗ is a Nash equilibrium if and only if the sum ofcontributions to each public good k is equal to its upper bounds d k .
Proposition 3 implies that, under the full information condition when all players
are maximising the payoff-sum of the n players, the Nash equilibrium will be efficientand the payoff-sum will be maximised. This game has multiple equilibria, and hence
there may be a coordination problem as to which equilibrium to play. However, withineach equilibrium the coordination problem associated with the choice of public good
is resolved.
For the partial information condition, however, a similarly efficient result will not
hold because the information about the upper bound is ex ante unknown to players.
Thus, they cannot adjust contributions to match the upper bounds even in equilibrium.
To prove the inefficiency in the partial information condition, we show that the amountcontributed to each public good is going to be almost equal in equilibrium.
L EMMA 1. Let x∗ be a Nash equilibrium of the game under the partial information
condition when all players are maximising the payoff-sum. Then, | ∑nj=1 x∗Aj −n∗B∑ j=1 x j | ∈ {0, 1}.
The main point of this lemma is that the group’s total contribution to each public
good would be about the same in equilibrium. One reason is that the expected marginal
payoff from contributing to public good k decreases with the number of other players
contributing to that public good. Another reason is that all combinations of (d A , d B )have an equal likelihood of occurrence and the expected upper bounds of the two
public goods are ex ante symmetric to the players. Consequently, in equilibrium almost
the same number of players contributes to each public good.9
9. There are two possible cases when the difference would equal 1. The first is a boundary case inwhich the marginal expected payoff from contributing to public good k and k0 is exactly equal whenn∗k0
| ∑nj=1 x∗kj − ∑ j=1 x j | = 1. The second is a special case in which the expected marginal payoff fromcontribution is positive even when everyone has contributed. In this case, if n is an odd number, thedifference equals 1.

9Finally, we show that under the partial information condition, depending on thesize of the parameters of n relative to c and β , the set of Nash equilibria would be
one of two types: In one type of the equilibrium only a subset of players contribute,and in the other type all players contribute. The former type arises when the relative
cost of contribution c/β is sufficiently large because it is not beneficial for all players
to contribute due to probable waste. Let us state this result more precisely in the next
proposition.
P ROPOSITION 4. Let x∗ be a Nash equilibrium of the game under the partial
information condition, in which all players maximise the payoff-sum.10• When n < 4c/β − 1, ∑nj=1 x∗kj ∈ {bt∗ c + 1, dt∗ e} for k ∈ {A, B}, wheres4c(n − 1)1(2n − 1) −+ 1 < (n − 1)/2.
t∗ :=
2
β
0

n
0
∗k
• When n ≥ 4c/β − 1, ∑nj=1 x∗k
j = dn/2e and ∑ j=1 x j = bn/2c for k, k ∈ {A, B}.

Proposition 4 implies that even if the players maximise the payoff-sum, there
may still be ex post inefficiencies in equilibrium under the partial information
condition. There may be over and under contributions with respect to the upperbounds, depending on the realized values of d A and d B because the players cannot
adjust their contributions due to lack of information.
In reality, not all individuals are payoff maximisers or payoff-sum maximisers.
Nonetheless, if there are some payoff-sum maximisers believing there are similar
players, then providing more information on the upper bounds may improve overall
efficiency. Both, Propositions 3 and 4, imply that players would respond to thedemand information under the full information condition but not under the partial
information condition. In order to investigate whether an increase in information
provision increases efficiency and people’s responsiveness to the demand, we examine
the two information conditions in a laboratory experiment.

3. Experimental Design
The experiment was conducted in January and February 2017 at the experimentalroom CHOCOLA in Ritsumeikan University, Japan. One hundred and twelve subjects
were recruited via posters and leaflets distributed on campus. The proportion ofundergraduate was 92% and the female students accounted for 25.9% of the subjects.
Although we recruited subjects from all departments on campus, about half of themwere from the Economics department. To the best of our knowledge, the last economic
10. Here, we use d·e and b·c to represent operations rounding up and down the decimals to the nearest
whole numbers. For instance, d5/2e = 3 and b5/2c = 2.

10experiment conducted at Ritsumeikan University took place four years prior to ourexperiments. Hence, for most subjects this was their first economic experiment.11
3.1. Experimental treatments and theoretical predictions
In the experiment, subjects repeatedly played a four-person multiple public goods
game with upper bounds for 10 periods in the same group. The cost of contribution c
was 500 points, and the utility from the effective contribution β was 250 points. We
also gave an endowment of 500 points to remove the possibility of negative payoffs.
Thus, subject i’s payoffs in the experiment werenn

πi (x) = 500 − 500 × (xiA + xiB ) + 250 × (min{ ∑ xAj , d A } + min{ ∑ xBj , d B }).
j=1

j=1

The main treatment variable in this experiment was the availability of theinformation on the upper bounds (i.e., demands). With n = 4, the possiblecombinations for the demands (d A , d B ) were {(1, 1), (2, 1), (1, 2), (2, 2), (3, 1), and(1, 3)}. In each period of the game, and separately for each group, one of the six
combinations were randomly chosen by the computer with equal probability. In
the full information treatment, subjects were informed of the realized values of thedemands before making their contribution decision, whereas in the partial information
treatment, subjects made their decision without knowledge of the realized values. A
treatment comparison was conducted in the between-subject design. Table 1 shows
the summary of the treatments and the distribution of realized values of the demands.
With these parameters, the theoretical prediction of payoff-sum maximising
players for the partial information treatment (based on Proposition 4) is that either
1 or 2 subjects would contribute to each treatment. In other words, any strategy
profile x in which (∑nj=1 xAj , ∑nj=1 xBj ) ∈ {(1, 1), (1, 2), (2, 1), (2, 2)} would be a Nashequilibrium.12
3.2. Experimental procedures
When subjects entered the experimental room, they were randomly assigned a seatwith a network-connected computer terminal. To ensure anonymity among subjects,
seats were partitioned so that the computer monitors of the other subjects were
unobservable. At the start of the experiment, the experimenter read out the consent
forms, and subjects were given sufficient time to read the forms before signing.
Subsequently, subjects were shown written instruction on their individual computer
11. We conducted a pre-experiment in January 2017 to test the procedures and the program. Twenty
subjects participated in the pre-experiment but not in the actual experiment.
12. With our parameter values of c = 500 and β = 250, 4c/β − 1 = 7 is larger than n = 4. According
to Proposition 4, in the equilibrium the sum of contributions to each public good should equal bt∗ c + 1 ordt∗ e. Here, t∗ = 1, and hence the former is equal to 2 and the latter equal to 1.

11
TABLE 1. Summary of treatments and realized values of demands

Number of subjects
Number of subjects by session
Sessions (1,3,5)Sessions (2,4,6,7)

Full information
48

Partial information
64(12,20,16)(16,20,16,20)

Frequencies of demand pairs(1,1)(1,2) or (2,1)(2,2)(1,3) or (3,1)

16
43
23
38

28
46
34
52

Note: Since the values of the demands were drawn randomly for each group in each period, the realized
frequency of the demands was counted in terms of the number of groups. The total adds up to the (number ofgroups) × (number of periods), which is 120 in full information and 160 in partial information treatments.

screen, which they were allowed to read at their own pace. After reading them,subjects solved a quiz to ensure they had understood the rules of the game. Subjects’
answers to the quiz questions were checked individually by the experimenters.13
The experimental program started after all subjects had answered all quiz questions
correctly. This ensured that all subjects understood the rules of the experiment.
We also read the answers aloud to assure subjects that they had all received thesame instructions and that the rules of the game were common knowledge. All thematerials, including instruction, quiz, program, and the questionnaire, are available assupplementary material. Throughout the experiment, no deception were used.

Current Round

Current RoundRemaining Time [sec.]: 25

1Your ID is 2.
Necessary amount of investment:
Project A:
1.
Project B:
3.

Please make your investment choice:Remaining Time [sec.]: 25

1Your ID is 2.
Necessary amount of investment:
Project A:
1.
Project B:
3.

Invest in Project A
Invest in Project B
Not Invest

Please make your investment choice:
Invest in Project A
Invest in Project B
Not InvestOK( A ) Full Information

OK( B ) Partial Information
F IGURE 1. Decision making screens

At the beginning of the instruction, subjects were warned about general rules ofthe experiment such as the use of cell phones, or private conversations with otherparticipants. They were then given detailed information on the rules of the game,matching protocols, how to use the computer interface, and the payment method. Since
13. As a result of this procedure, we do not know exactly how much time it took subjects to read the
instruction. We do know, though, that in each session, it took subjects about 30 minutes from the start ofreading the instructions to the end of making corrections to the quiz. Please note that there was a large
individual heterogeneity regarding the length of time it took subjects to complete reading the instruction
and solving the quiz. We report the average of the longest time taken by subjects in each session.

12

they read the instructions at their own pace, they were told that all participants in theroom had received the same instructions and quiz. All materials used in the experimentwere in neutral frames.
The experiment was programmed and conducted using z-tree (Fischbacher, 2007).
The main and only difference in the program between the two treatments lay in thedesign of the decision-making screen. In the full information treatment (see Figure
1(A)), subjects received information about the necessary amount of investment foreach project, whereas in the partial information treatment (see Figure 1(B)), the spacefor realized values was left blank. Except for this difference, all screen interfaces of
the two treatments were identical.14
In both treatments, the program proceeded as follows. At the beginning of thefirst period, subjects were randomly matched into groups of four and stayed in theirgroup throughout the experiment (partner matching protocol). Each period started
with a decision-making screen as shown in Figure 1, informing subjects about theiridentification number, which changed in every period to reduce the reputation effect.
On the same screen, they subsequently made their decision by choosing one of thefollowing three options: “invest in project A,” “invest in project B,” and “not invest.”
After all subjects had made their choice, they proceeded to the end-of-period feedback
information, which was identical in both treatments. The subjects received informationon the following: the choices made by all four group members, the necessary amountof investment for each project, the total amount of investment in each project, thepoints gained from each project, the total points gained from the two projects, thepoints kept by the subject, and the points gained in that period. After the last period
ended, there was a final screen enabling subjects to confirm the results in each period.
After the 10 periods, subjects answered a questionnaire. Most of the questions
were administered using z-tree except for the social value orientations (SVOs)
questions, which were administered on paper due to programming difficulties. 15
Subjects were paid for the points they had earned in the two randomly selectedperiods. After completion of the 10 periods, the experimenter rolled two 10-sided
dices to determine the payment period.16 Subjects received payment of 1 yen for eachpoint earned in the experiment. The show-up fee was 500 yen. The average payment
was 1795 yen (which amounted to about $16 at the time of the experiment) for an
experiment that lasted less than 1.25 hours.
14. For the partial information treatment, we maintain the statement “The necessary amount ofinvestment for this period was as follows. . . ” This was to make subjects aware that the values of thedemands for that period were unknown but already determined.
15. In social psychology, SVOs comprise a measure developed to distinguish individuals by a preference
regarding their respective payoffs relative to the payoff of others. Based on subjects’ responses to thequestions on primary slider measures developed by Murphy, Ackermann, and Handgraaf (2011), we
obtained parameters of SVOs and used them in the analysis in the next section.
16. When two rolled numbers turned out to be the same, we re-rolled one of the dices again to obtain
another number.

13

3.3. Experimental Hypothesis
As mentioned in Section 2.3, if there are some payoff-sum maximising players, theywould try to match the demands in the full information treatment. Since behavior in
the partial information treatment does not vary with demands, there could be over- or
under-contributions with respect to the demands, likely to cause inefficiency. These
theoretical results lead to the following two hypotheses tested in the next section.
H YPOTHESIS 1. In the full information treatment, average contribution rates to a
public good increase with an increase in its demand.
H YPOTHESIS 2. Efficiency is higher in the full information treatment than in thepartial information condition.

4. Results
4.1. Effects of demand information on the aggregate contribution rates
We examine the aggregate propensity to contribute (average contribution rate) toeither public good. Recall that each subject can contribute at most once to one of
the two public goods in each period. Hence, the proportion of subjects’contributions
to either public good in each period yields the per-period aggregate contribution rate.
Figure 2 shows the trend of contribution rates observed in full and partial information
treatments. In both treatments, over 50% of subjects contributed in the first period,
with contribution rates declining over the periods. Averaging across the 10 periods,
the mean contribution rate of the full information treatment (0.348) and that of thepartial information treatment (0.375) were not statistically different.17 Thus, we canstate the following:
R ESULT 1. The provision of information on the demand for each public good has noeffect on the aggregate propensities to contribute.

4.2. Effects of demand levels on contribution rates in the full information
treatment
Result 1 implies that the “availability of information” on the demand does not havea significant impact on the aggregate contribution rates. With reference to Hypothesis
17. Using the decision of individuals to contribute or not in each period as a unit of observation (480
observations for the full information treatment and 640 observations for the partial information treatment),
the p-value of the two-sample test of proportions is 0.351. The Wilcoxon rank-sum test of the comparisonof a group’s average contribution rates across 10 periods also supports no difference between the treatments(p = 0.66).

0

.1

.2

Contribution rates.3 .4 .5 .6 .7

.8

.9

1

14

1

2

3

4

5

full information

Period

6

7

8

9

10

partial information

F IGURE 2. Average contribution rates by treatment

1, we focus on the observations made in the full information treatment and examine
whether and how demand levels affect the average contribution rates.
First, we test whether contribution rates differ with the sum of demands by
evaluating 480 observations in the full information treatment. The first column ofTable 2 reports contribution rates, i.e., the share of individual decisions to contributeto either public good in each period. We note that the greater the summed demand of
the two public goods, the higher is individuals’ propensity to contribute. Odds ratios
confirm this observation: Compared to the baseline case where the total demand istwo, subjects are about 6.5 times more likely to contribute than not when the total
demand is 3, and 8.3 times more likely to contribute than not when the total demandis 4.18 The contribution rates increase with the sum of upper bounds, i.e., the total
demand.
Next, we examine the relations between individual choices regarding which public
good to contribute and demand sizes. Recall that, under the full information treatment,
subjects are informed of the demand in each of the two public goods and decide
whether and to which public good to contribute. Figure 3 visualizes the propensity
to contribute to the public good by demand. When demand pairs are asymmetric, thatis, (2,1), (1,2), (3,1), or (1,3), contribution rates to the public good, for which there is agreater demand, tend to be higher than contribution rates to the public good, for which
there is a smaller demand. Concerning the demand pairs (2,1) and (1,2), the averagecontribution rate to the public good for which there is a larger demand is 0.279,
18. Pairwise Bonferroni adjusted multiple comparisons of differences in odds ratios between total
demands equaling 2 and 3, and 2 and 4 yield z values that are statistically significant (p < 0.01) but
not between total demands of 3 and 4 (p = 0.222).

15
TABLE 2. Odd ratios of contributions by the sum of upper bounds
Summed upper bound(Contribution Rates)

Odds ratio

Standard error

2 (0.078)
3 (0.355)
4 (0.414)
constant

1
6.485
8.334
0.085

3.192
4.030
0.040

Z (p-value)
3.80 (0.000)
4.38 (0.000)-5.30 (0.000)

Notes: Logistic regression with contribution to either public good as dependent variables (=1 ifcontribution is made to either public good, =0 otherwise). The odds of contributions indicate howoften an individual contributes relative to how often an individual does not contribute, that is,Pr(contribute|summed upper bounds)/Pr(not contribute|summed upper bounds). Logistic regression takes thelog of the odds and yields the odds ratios using the summed upper bound = 2 as a base.

with a 95% confidence interval [0.212, 0.346], while the average contribution rate
to the public good for which there is a smaller demand is 0.076, with 95% confidence
interval [0.036, 0.115]. Similarly, for the demand pairs (3,1) and (1,3), the contribution
rates to the public good, for which there is a larger and a smaller demand, are 0.303
with a confidence interval [0.229, 0.376] and 0.092 with a confidence interval [0.046,
0.138], respectively. Facing asymmetric demands, contributors tend to direct theircontributions to a public good with a larger demand. Result 2 summarizes the above
two findings.

0.4
0.35
0.3

0.303
(0.0
(0.037)0.279(0.034)

0.25
0.2
0.15
0.1

0.076
(0.020)0.092(0.023)0.05
0contribute to dk=2 contribute to dk=1
Summed demand = 3

contribute to dk=3 contribute to dk=1
Summed demand = 4

F IGURE 3. Propensities to contribute when demands are asymmetric: mean contribution rates(standard errors in parenthesis) and 95% confidence intervals

16

R ESULT 2. Contribution rates increase with demand: In the full information
treatment, the contribution rate increases, on average, with an increase in the sumof demands. Moreover, subjects direct more contributions to the public good with arelatively high demand than to the public good with a low demand.
4.3. Effects of demand information on individual decisions to contribute
Until now, Result 1 has indicated that the provision of information on demands has
no significant impact on the aggregate contribution rates. However, Result 2 indicates
that, with information on demands, the average individual contribution rates increase
with demand such that more contributions are directed to the public good with a larger
demand. Based on these observations, we investigate whether and how the provisionof information on the demands in the full information treatment affects decisions tocontribute at an individual level.
We regress an individual’s contribution choices against the informational treatment
dummy, its interaction with the sum of demands for the two public goods, and somecharacteristics of the individual. Columns (1), (2), and (3) of Table 3 report the resultsof the ordinary least squares (OLS) specification. The coefficient on information(dummy for the full information treatment) is not significantly different from zeroin specification (1), which echoes Result 1. In specification (2), the coefficient on
the interaction term between information and the sum of demands is positive andsignificant, while the coefficient on information is negative and significant. These
coefficients imply countervailing effects of information; information on the summed
demand increases individuals’ propensities to contribute, while the provision of
information on the demand structure itself depresses contribution rates. As a resultof these effects offsetting each other, there is no effect of information provision
on average (Result 1). At the expected value of the summed demand (3.33), thepoint estimate of the predicted contribution rate is 0.302, in the full information
treatment which is marginally lower than in the partial information treatment (0.329).
Specification (3) includes additional control for subjects’ SVOs. For the range of
SVOs corresponding to the “prosocial” orientation, which is above 22.45 according toMurphy et al. (2011), the observed association between the prosocial value orientationand individual contribution rates is positive. However, for the “individualistic” and“competitive” ranges, the association is negative. Specifications (4), (5), and (6) use
Probit models. Although it is known that probit specifications are likely to introduce
bias in the estimates of the interaction terms, the probit models produce qualitatively
the same results as those with linear specifications. Thus, we state the following.
R ESULT 3. Countervailing effects of information about the demand structure on
contribution rates: On the one hand, the baseline effect of information about thedemand structure on contribution rates is negative—the provision of information alone
lowers the contribution rate. On the other hand, the provision of a demand structure
directs the subject’s contribution toward the public good with a higher demand.

17
TABLE 3. Individual decision to contribute by information on the demand structure (Random effectpanel data analysis: Dependent variable is subject’s contribution decision to either public good)Linear specification

Information

(1)

(2)

(3)

(4)

(5)

(6)

-0.036(0.053)0.506**(0.218)0.034(0.205)-1.744***(0.410)0.478***(0.107)0.543**(0.220)-0.023(0.206)

0.345***(0.046)

0.342***(0.046)-0.484***(0.094)0.126***(0.027)0.145**(0.060)0.007(0.054)-0.006(0.004)0.0003***(0.001)0.291***(0.057)-0.108(0.195)0.144**(0.061)-0.001(0.055)-0.463***(0.094)0.127***(0.027)0.154**(0.060)0.000(0.055)-0.530***(0.170)-0.546***(0.171)-1.828***(0.406)0.475***(0.106)0.514**(0.216)-0.060(0.200)-0.023*(0.014)0.001***(0.004)-0.725***(0.202)1110
0.01841110
0.03891110
0.0731

1110

1110

1110

-635.9109-625.0010-618.7843

Information× (d A + d B )FemaleEconomic major
SVO
SVO2
Constant
Observations
R-sq
Log likelihood

Probit specification

Notes: Figures between brackets are robust standard errors clustered by subjects. Information represents thetreatment dummy, which equals 1 for the full-information treatment. *** Significant at a 1% level, ** 5% level,
and * 10% level.

4.4. Effects of demand information on the efficiency of group outcomes
According to Result 2, in the full information treatment the average contribution rateincreases with an increase in demand, supporting Hypothesis 1. The countervailing
effects of information about demand structures on an individual’s contribution
decision presented in Result 3 are likely to exert ambiguous impacts on the efficienciesof group outcomes. We turn to examine Hypothesis 2, i.e., the relationship between
the provision of information on the demands and efficiencies of outcomes at grouplevel. Figure 4 plots the average efficiency of group outcomes for each treatment.
Efficiency is measured here as a ratio of actually attained group profits to the
maximum attainable group profit, over and above the minimum attainable groupprofit.19 On average, efficiency rates observed in the groups under the full information
19. For example, consider a case where the demand structure is (d A , d B ) = (1, 1). The maximum payoffsum occurs when one member contributes to one and another member to the other public good. The
minimum payoff-sum is reached when all four subjects contribute to the same public good. Suppose thetotal actual group payoff is 2000. Using the parameter settings of the experiment, the maximum andthe minimum attainable group payoff are 3000 and 1000, respectively. The resulting efficiency measureamounts to (2000 − 1000)/(3000 − 1000) = 0.5.

18

0

.1

.2

.3

Efficiency.4 .5 .6

.7

.8

.9

1

and partial information treatments are 0.569 and 0.526, respectively. The overall
efficiency appears to be only marginally higher in the full information treatment.20

1

2

3

4

5

full information

Period

6

7

8

9

10

partial information

F IGURE 4. Average efficiency of group outcomes by treatment

In this game, there are two sources of inefficiency—one caused by insufficient
contributions, the other by excessive contributions. We try to visualize the differencein these sources by classifying each group outcome into one of the following
four categories: “Efficient,” “Free riding,” “Coordination failure,” or “Free riding
and coordination failure.” “Efficient (EF)” refers to the case in which the summed
contribution by group members exactly satisfies the demands of both public goods.
“Free riding (FR)” occurs when the summed contribution to each public good is lessthan or equal to demand and strictly less for at least one public good. “Coordination
failure (CF)” refers to the case in which the summed contribution to at least one
public good exceeds demand, and the excess is sufficient to cover any shortage ofcontributions directed to another public good. “FR and CF” occur when the summed
contribution to only one public good exceeds demand and the excess contribution to
one public good is not sufficient to offset the shortage of the other. As an example,
Figure 5 demonstrates the classification of these categories of outcomes when the
demand structure is (2,1).
Eyeballing the frequency distribution of group outcomes in terms of the efficiency
categories in Table 4, we notice that the provision of information tends to improve
20. Using each group’s data in each period as a unit of observation, a Wilcoxon Rank Sum test of theequality of distribution of unmatched data was performed; the test yields a Mann-Whitney statistic of
1.512 (p=0.1306).

19

Contribute to B

4
3

2
1
0

1

2

3

4

Contribute to A

Free riding (FR)Coordination failure (CF)Free riding and coordination failure (FR & CF)
Efficient outcome

F IGURE 5. Efficiency categories of group outcomes when the demand structure is (d A , d B ) = (2, 1)

TABLE 4. Frequencies and proportions (in parentheses) of group outcomes by efficiency categories
No.
ofobservations

Free riding(FR)

Coordination FR and CF
failure (CF)120

7(0.058)105(0.875)

4(0.033)

4(0.033)160

9(0.056)122(0.762)13(0.081)16(0.3)

full information
partial information

Efficient(EF)outcome efficiencies mainly by reducing coordination failures. Under the full
information treatment, groups’ outcomes are much less likely to be characterized as
CF with a ratio of 0.033 than under the partial information treatment with a ratioof 0.081. It is worth noting that the outcomes of FR and CF are extremely rare
under the full information treatment with a ratio of 0.033, compared to the partial
information treatment case with a ratio of 0.3. FR outcomes occur more frequently
under the full information treatment than under the partial information treatment, andthe frequencies of efficient (EF) outcomes do not differ between the treatments.
R ESULT 4. The coordination-enhancing effect of information on upper bounds:
Provision of information on the upper bounds of effective contribution improves
efficiency by reducing excessive contribution and coordination failures. However, such
efficiency gains are not sufficiently robust to offset the efficiency losses arising from apersistent majority of free-riding outcomes.

20

5. Discussion
At the aggregate level, the result of our experiment shows the neutral effect of
information about the demand (Result 1). At an individual level, the core effectof information provision on the contribution rate is even negative (Result 3).
Recasting these findings on our theoretical predictions put forward in the Experimental
Design section it seems reasonable to conclude that individuals are monetary payoff
maximisers. Thus, the provision of information on the upper bounds does not leadto differences in individual contribution behavior (Propositions 1 and 2). A further
analysis of individual contributions based on the size of summed demands for bothpublic goods reveals the effects of information, driving a higher demand to induce ahigher contribution. Moreover, larger contributions are made to the public good with ahigher demand, when subjects are informed of asymmetric demands between the two
public goods (Result 2).
Our result further shows that the provision of information on the demandof multiple public goods tends to depress subjects’ average contribution rates,
while demand information directs more contributions to the public good, given a
relatively high demand. In a first attempt to explain the puzzling effects of demand
information on individual contribution behavior, we propose a hypothesis about
strategic uncertainty and avoidance of potential wastage generated from excessive
contributions. In our theory discussed in section 3, Experimental Design, we statethat if all individuals are payoff-sum maximisers and are informed of the upperbounds, they would contribute efficiently in equilibrium. Since there are a numberof equilibria, an issue might arise over selecting an equilibrium on which individuals
can coordinate their contributions. Uncertainty about each group member’s action
complicates coordination of such equilibria. In view of the underlying strategic
uncertainty and difficulties of coordination, payoff-sum maximisers may end up
withholding contributions to avoid potential wastage.
Particularly, when demand is low, strategic uncertainty may additionally lower
the contribution of an efficiency-concerned individual, the reason being that thisindividual might perceive the futility of own contributions and hence avoid them.
Under the full information treatment, when the demand structure is (d A , d B ) = (1, 1),almost all outcomes (15 out of 16, 93.7%, of group outcomes) are categorized as
free riding. When the demand structure is (d A , d B ) = (2, 2), free-riding outcomesoccur less often (20 out of 23, 86.9%). By comparison, under the partial information
treatment, corresponding frequencies of free-riding outcomes occur 16 out of 28 times(57.1%) when (d A , d B ) = (1, 1), and 24 out of 29 times (85.2%) when (d A , d B ) =(2, 2). With reference to outcomes under the partial information treatment, the freeriding outcomes are observed more frequently under the full information treatment,
particularly, when demands are low. This evidence corroborates our result that theapparent negative effect of demand information on individual contribution is, at least
partly, associated with strategic uncertainty.

21

6. Conclusion
Efficient provisions of multiple public goods with upper bound demands require
overcoming the twin problems of coordination failure and free riding. We investigate
the role of information in solving these problems and improving efficiency. According
to standard theory, stating that players maximise their payoffs, information conditions
do not affect an individual’s contribution behavior: The dominant strategy of players
is to withhold contributions regardless of information conditions. On the contrary, if
players maximise the payoff-sum of all players, then information provision would
increase efficiency. The reason for this is that when information on demands is madeavailable, payoff-sum maximising players will in equilibrium match the upper boundin each public good. However, when such information is not available, contributions
to each public good will be about the same in equilibrium. This could cause over- andunderachievement of the realized upper bounds, thereby lowering overall efficiency.
Thus, we hypothesize that if there are efficiency-concerned individuals, providing fulldemand information will increase efficiency.
Based on the laboratory experiment, we find that information provision about thedemand structure promotes individual contributions to the public good. However,at the aggregate level, information does not increase average contribution rates or
efficiency. The reason is that the difficulty of interpersonal coordination offsets
the efficiency-enhancing effect of information as individuals tend to withhold
contributions to avoid contribution wastage.
In conclusion, we offer some reflections on the example of a blood bank.
The provision of information on the seasonal demand for blood may not solely
improve the overall efficiency of blood banks. This may be attributable to thecountervailing effects of information on an efficient provision of blood banks. In
other words, information on demand helps blood banks to overcome the interseasonal coordination problem. However, such information provision may worsen
the free-riding problem when potential donors fear over donation. Information
provision for comprehensively improving efficiency may become practicable when it
is complemented by mechanisms to reduce strategic uncertainty and manage aversion
to wasteful donations. Future studies should address these issues, for example,
by using the framework of a sequential voluntary contribution mechanism. Someexperimental evidences associated with a real-time voluntary contribution mechanism
show that providing information about others’ contributions and allowing subjects toadjust contributions upward increases the provision of public goods (Dorsey, 1992;
Kurzban, McCabe, Smith, and Wilson, 2001). As Kurzban et al. (2001) point out, thereal-time public goods game closely mirrors the mechanism discussed in Schelling(1960) (pp.45-46)—dividing a contribution into smaller contributions lowers the riskof the initial contribution, thereby boosting contributions. Similar mechanisms, suchas providing real-time information on donations with a prediction on demand, may
improve the efficiency of blood donations.

22

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23

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Appendix
Proof of Proposition 1
Proof. Let t k (x−i ) := ∑ j6=i xkj denote the sum of contributions to public good k byplayers other than i. The marginal benefit from contributing to public good k is zero
when t k (x−i ) ≥ d k , and is β when t k (x−i ) < d k . In either case, it is lower than thecost of contribution c. Hence, independent of whether the contribution is ineffective
or effective, xik = 0 is a best response to any x−i for k ∈ {A, B}. Thus, xi = (0, 0) is astrictly dominant strategy for all i ∈ N.

Proof of Proposition 2
Proof. Let P(t k (x−i ) < d k ) denote the probability in which the sum of contributions
by players, other than i, is less than d k . This is also the probability in which i’scontribution in public good k would be effective. Since both t k (x−i ) and d k are integers,
P(t k (x−i ) < d k ) ==(n − t k (x−i ) − 1) + (n − t k (x−i ) − 2) + · · · + 1(n − 1) + (n − 2) + · · · + 1
k(n − t (x−i ) − 1)(n − t k (x−i )),n(n − 1)(A.1)
0where the numerator in the first equation counts the possible combinations of (d k , d k ),given that t k (x−i ) < d k , and the denominator counts the number of all possible0
combinations of (d k , d k ). Using this probability, we can calculate the expectedmarginal payoff for player i from contributing in public good k asP(t k (x−i ) < d k ) × (β − c) + 1 − P(t k (x−i ) < d k ) × (−c).
Since β < c, this is clearly negative for any t k (x−i ) ≤ n, which is lower than themarginal payoff from not donating in either public good. Thus, strategy (0, 0) strictly
dominates (1, 0) and (0, 1).

Proof of Proposition 3
Proof.
First, given x−i , when compared to the case of not contributing to either public good,the marginal payoff for player i from contributing to public good k is nβ − c when

24

the contribution is effective, and it is −c when it is ineffective. When the contribution
is effective, i’s contribution increases the monetary payoff of all players by β . Since
there are n players, it increases the utility of payoff-sum maximiser i by nβ at a costof c. When the contribution is ineffective, it creates no extra benefit but still costsc. To summarize, for payoff-sum maximising player, it is better to contribute if thecontribution is effective, and it is better not to contribute if it is ineffective.
Second, we show by contradiction that t k (x∗ ) = d k for k ∈ {A, B} is a necessary
condition for x∗ to be a Nash equilibrium. Assume that x∗ is a Nash equilibrium andthat t k (x∗ ) 6= d k , for some k ∈ {A, B}.
[Case 1: t k (x∗ ) > d k for some public good k] There is over contribution to public goodk, so there is at least one player i, who can refrain from contribution and increase their∗
payoff by c. Thus, xi∗ is not a best response to x−i.
k ∗k[Case 2: t (x ) < d for some public good k] There are two possible sub-cases toconsider: (i) t A (x∗ )+t B (x∗ ) = n, and (ii) t A (x∗ )+t B (x∗ ) < n. When (i), then t k (x∗ ) < d k00
implies that t k (x∗ ) > d k . Then, the argument for Case 1 applies. When (ii), there is atleast one player i who is not contributing to either public good. Since i’s contribution
to k will be effective, it is better to contribute to k. Thus, xi∗ is not a best response to∗.
x−i
Therefore, if x∗ is a Nash equilibrium, t k (x∗ ) = d k .
Finally, we show that t k (x∗ ) = d k for k ∈ {A, B} is a sufficient condition for x∗
to be a Nash equilibrium. We begin by considering the case where a player i is notcontributing at the strategy profile x∗ . If player i deviates and contributes to one of the
public goods, then the marginal payoff of player i would be −c. This is because theupper bound is already met by the contributions of the other players. Thus, xi∗ = (0, 0)∗
for non-contributing players.
is a best response to x−i
Next, we consider the case where a player i is contributing to public good k at
strategy profile x∗ . If player i deviates and contributes to another public good k0 6= k,then the payoff of player i would decrease by nβ > 0. This is because the upperbound for k0 is already met through other players’ contributions. Finally, if player ideviates and does not contribute to either public good, then i’s payoff would decreaseby nβ − c > 0. For both deviations, player i’s payoff decreases, and hence xi∗ is a best∗
response to x−ifor contributing players.
Proof of Lemma 1
Proof. Using the probability in Equation (A.1), we can calculate the expectedmarginal payoff of player i from contributing to public good k compared to notcontributing as
P(t k (x−i ) < d k ) × (nβ − c) + P(t k (x−i ) ≥ d k ) × (−c)
cβ kt (x−i )2 − (2n − 1)t k (x−i ) + (n − 1)(n − ) .
=(n − 1)
β

(A.2)

25

For the payoff-sum maximising players, the values of the expected marginal payoff
can be positive. Hence, an individual’s decision to contribute can be optimal.
One point to note is that because P(t k (x−i ) < d k ) is decreasing in t k (x−i ) for0 ≤ t k (x−i ) ≤ n − 1, the expected marginal payoff for player i from contributing to0public good k also decreases in t k (x−i ). Hence, if t k (x−i ) < t k (x−i ), then it would bebetter to contribute to k than to k0 .
Suppose that x∗ is a Nash equilibrium and |t A (x∗ ) − t B (x∗ )| ≥ 2. We denote thepublic good with a higher contribution as k and the other as k0 . Let i be a contributor
to k in x∗ . Then,
0
0
∗
∗
).
) > t k (x∗ ) = t k (x−it k (x∗ ) − 1 = t k (x−iThis implies that i’s expected marginal payoff would be higher if i contributes to k0 .
Thus, x∗ is not a Nash equilibrium if the difference in the sum of contribution to eachpublic good is more than two.

Proof of Proposition 40

Proof. Without the loss of generality, let us assume that t k (x∗ ) ≤ t k (x∗ ). Using
Equation (A.2), we solve the condition of the player i, under which it is better to∗),contribute to public good k than not contribute. This is equivalent to solving for t k (x−ik ∗satisfying Equation (A.2) ≥ 0. Since Equation (A.2) is a quadratic function of t (x−i ),this condition is satisfied when:s
14c(n − 1)k ∗t (x−i ) ≤ t∗ :=(2n − 1) −
+1
2
β
or

s


4c(n − 1)1k ∗∗t (x−i ) ≥ t :=(2n − 1) ++12
βwhere t ∗ and t∗ are the two solutions for Equation (A.2) = 0.
Given our restrictions on the parameters that c, β > 0 and n ≥ 2, t∗ and t ∗ take realvalues. Additionally, t ∗ turns out to be larger than n/2, ass1
4c(n − 1)n∗t −(2n − 1) − n +
=
+1
2
2
β
s


1
4c(n − 1)(n − 1) +
=
+1
2
β
> 0.
Therefore, in what follows, we consider the conditions for t∗ only.
First, we check the possibility of t∗ < 0. This is a trivial case in which it will neverbe optimal for the player i to contribute to public good k. It can be verified that ifnβ > c, then t∗ > 0. Hence, t∗ > 0 in our game.

26

Next, we examine the condition t∗ ≥ (n − 1)/2. If this condition holds, then,because of Lemma 1, all players would be contributing to one of the two public goodsin equilibrium.
By rearranging the equations, we find that:
s


n−1
1
4c(n − 1)(2n − 1) −+1 ≥t∗ ≥ (n − 1)/2 ⇐⇒2β2s4c(n − 1)⇐⇒ n ≥+1β⇐⇒n2 − 4cn/β + 4c/β − 1 ≥ 0.

This inequality holds when n ≤ 1 or when n ≥ 4c/β −1.21 Since we assume that n ≥ 2,
only the second condition needs to be considered.
We can summarize our results depending on whether the second condition issatisfied.
When n ≥ 4c/β − 1, t∗ ≥ (n − 1)/2, and hence player i chooses to contribute to
public good k. This contribution would continue up to the point where half of the
population contributes to k. Since the number of contributors to k is less than or equal
to that of the other, in the equilibrium, every player would be contributing to one of0
the two public goods. Thus, t k (x∗ ) = bn/2c and t k (x∗ ) = dn/2e.
When n < 4c/β − 1, t∗ < (n − 1)/2. Hence, except for the boundary cases, only
a subset of players would be contributing. In this case, player i would contribute if∗
t k (x−i) ≤ t∗ . Since the players are symmetric, the total number of contributors in eachpublic good would be bt∗ c + 1 = dt∗ e in both public goods if t∗ is not a whole number.
If t∗ is a whole number, then player i would be indifferent between contributing and∗) = t∗ . In this case, the total number of contributors to
not contributing when t k (x−ieach public good would be either dt∗ e or bt∗ c + 1. Thus, t k (x∗ ) ∈ {dt∗ e, bt∗ c + 1} fork ∈ {A, B}.
21.

It can be verified that 1 < 4c/β − 1 when 2c > β , which is satisfied under our assumption that c > β .