Private Provision of Discrete Public Goods - Semantic Scholar

Private Provision of Discrete Public Goods M. Makris∗ University of Exeter, CMPO - University of Bristol, EMOP - AUEB 21/09/04

Abstract We investigate the private provision of a discrete public good. When the size and decomposition of the group are certain, such a problem has, in general, multiple equilibria. Yet, if it is commonly believed that some individuals may be altruists and that the size of the group is a Poisson random variable then equilibrium is unique. If the decomposition uncertainty is very small and the expected group-size sufficiently large then the unique equilibrium is in symmetric mixed strategies. When players’ beliefs about group-size are on average correct, this equilibrium is shown to be very close to the corresponding equilibrium under certainty. This provides a theoretical justification for ignoring the pivotal provision and the zero contributions outcomes, in favour of the symmetric mixed-strategy equilibrium, in sufficiently large discrete public good provision problems.

JEL Classification: C72, D70, H41, D89 Keywords: Discrete Public Goods, Altruism, Poisson Games, Equilibrium Selection.

0∗ Address:

M.Makris, Department of Economics, University of Exeter, Streatham Court, Rennes Drive, Exeter EX4

4PU, UK, Tel: ++44 (0)1392 264486, email: [email protected]. I am indebted to Dieter Balkenborg, Clare Leaver and participants in the University of Exeter Internal Departmental Seminar Series and UCL for comments on an earlier version of this paper. The usual disclaimer applies.

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1

Introduction

One of the usual justifications of government intervention is the provision of public goods, as private provision is often claimed to be inefficient - especially, in a large economy.1 However, for a large class of public goods - namely, for discrete public goods - this may not be true, as provision can be an equilibrium outcome regardless of the group-size. Discrete public goods are provided if and only if a certain amount of contributions is reached; that is, such goods are characterised by a threshold technology. Examples of such problems are: putting up a streetlight, collecting money for an office coffee club, staffing of a wild life refuge, setting up a lobby given the political environment, signing a petition for a referendum to be held e.t.c. In situations of private provision of discrete public goods it is well known that, when the number of potential contributors and their preferences are common knowledge, multiplicity of pure strategy Nash equilibria is the norm. Typically, if the private cost of contribution is strictly less than the marginal willingness to pay, and at least two but not more than all the existing individuals are required to contribute for the public good to be provided , then there is an inefficient equilibrium outcome where no contributions are made towards financing the cost of provision. At the same time, however, there is also an efficient equilibrium outcome where just enough contributions are made so that the public good is provided (see, for instance, Palfrey and Rosenthall (1984)). Importantly, the latter is a strict equilibrium and hence robust to refinements. These, different, in terms of provision, outcomes are equilibria for any group-size, and regardless of whether unused contributions are returned or wasted or kept by a collector. Also, if contributions are not returned when provision fails, these are the only pure-strategy Nash equilibria and the ‘zero contributions’ is also a strict equilibrium. Consequently, provision of discrete public goods involves an equilibrium selection problem. Part of the existing work tackles equilibrium selection either via strategy-revision processes that are subject to noise such as Harsanyian-style payoff disturbances2 or by means of the ‘global games’ approach, where it is assumed that players do not know the underlying state and receive privately observed and noisy signals of the actual game they are playing.3 This strand of research emphasises that depending on the fundamentals of the problem the selected play corresponds to one of the pure-strategy Nash equilibria. An alternative approach puts forward a simple appeal to symmetry. Dixit and Olson (2000), for instance, investigate a problem where unused contributions are returned, and focus on the symmetric 1 See,

for instance, Gradstein (1998). On a related issue, Olson (1965) has conjectured that collective action is more

likely to fail in large groups than in small groups. 2 See

Myatt and Wallace (2004). For equilibrium-selection models where the noise stems from mistakes/mutations see

Young (1993, 1998), Kandori, et. al. (1993). 3 See

Myatt and Wallace (2002), for a game with two players when unused conributions are not returned. For surveys

of the global games literature see Morris and Shin (2003) and Myatt et. al. (2002).

2

equilibria due to “the difficulty of the coordination that is required ... when a subset of a larger group is designated to do one thing and the rest another”. In their provision game, as well as in a game where unused contributions are not returned, the only other symmetric equilibrium, apart from the ‘zero contributions’ one, is a mixed-strategy one.4 This appeal to symmetry has been criticised by Myatt and Wallace (2004) in that it “simply ducks the problem of equilibrium selection: the “difficulty of coordination” reflects the existence of the problem, and does not imply a symmetric equilibrium”. One could also argue that asymmetric equilibria, though theoretically sound given the postulated assumption of common knowledge of preferences and group-size, do not seem realistically very plausible when the expected number of stakeholders is large. The reason is that in such, theoretical, outcome(s), identical individuals perceive very different probabilities that they are pivotal in securing the provision of the public good.5 However, common intuition would suggest that this is implausible in situations where the group-size is very large. That is, common intuition might seem to be in conflict with theoretical predictions. This work contributes to the discussion of provision of discrete public goods by providing a theoretical justification for selecting the symmetric mixed strategy equilibrium in sufficiently large games. Following the ‘global games’ approach, we investigate a problem where potential contributors face uncertainty about the game they are actually in; specifically, there is uncertainty about the groupsize. However, and in contrast to the ‘global games’ paradigm, stakeholders face the same uncertainty over the underlying state. In more detail, to capture population uncertainty we choose to model the voluntary-contribution game as a Poisson game (Myerson (1998, 2000)), and hence assume that the total number of players is drawn from a Poisson distribution with a given mean. Postulating population uncertainty, and specifically that the group-size is a Poisson random variable (PRV), can be motivated in a number of ways. First, when the number of potential contributors is very large, the standard assumption that every player takes every other player’s behaviour as given and known when contemplating her best response seems somewhat implausible. For instance, in large societies, it may be prohibitively expensive to collect the necessary information for who all the stakeholders are - in which case the implausibility, we highlighted above, of the asymmetric equilibria becomes even more obvious. As Myerson notes, in large games “it is unrealistic to assume that every player knows 4 In

this equilibrium, the typical individual contribution rate and the probability of provision decrease with the group-

size, a fact which fares well with the conjecture of Olson (1965). The symmetric mixed-strategy equilibrium is also the focus of Hindriks and Pancs (2002). 5A

similar criticism appears also in the literature on costly-voting games, where it has been shown (Palfrey and

Rosenthal (1983)) that there is mulitplicity of equilibria: both a low-turnout and a high-turnout equilibria exist as long as the size of the electorate and the types of all voters are common knowledge. Nevertheless, the high-turnout equilibria are quite counter-intuitive and implausible in a large electorate: they rely on there being no uncertainty about how many votes one of the two alternatives will receive (see e.g. Palfrey and Rosenthal (1985)) and, thereby, on identical individuals perceiving very differrent probabilities that they are pivotal in determining the election outcome (see e.g. Myerson (1998)).

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all the other players in the game; instead, a more realistic model should admit some uncertainty about the number of players in the game” (Myerson (2000) pp.7). Thus, we follow Myerson’s suggestion and allow for population uncertainty in investigating provision of discrete public goods. Specifically, given the convenient properties associated with the Poisson distribution (see Myerson (1998)), we model the voluntary-provision problem as a Poisson game. As a complementary justification for this modelling choice, suppose that the identity of every stakeholder is indeed common knowledge but also that contributions must be collected by a given time. Standard theory suggests that each agent will decide on her action by taking the number of collected contributions as given. However, potential stakeholders may be ill, postmen may be on strike or computer networks may be down; in short, accidents will happen, albeit with a small, but strictly positive, probability. As a result, in a large environment, stakeholders should actually view the number of players in the contribution game as a Poisson random variable. Despite the fact that the uncertainty about the group-size is common between stakeholders, potential contributors are still involved in an asymmetric information game, as, following Myatt and Wallace (2004), there is also uncertainty over the decomposition of the group. Specifically, we postulate that some players may have contributing as a weakly dominant strategy due to deriving sufficient satisfaction from the action of contributing itself. That is, we postulate that some stakeholders may be ‘warm-glow altruists’.6 An example of the relevance of such altruism is the Olympic Games volunteers.7 We show that if the likelihood of altruism is very small and the expected group-size is sufficiently large, there is a unique equilibrium which is in symmetric mixed strategies. If the players’ common beliefs about the group-size are on average correct - that is if the expected population equals the actual group-size - this equilibrium has the following important properties. First, the individual contribution rate is decreasing with the expected size and converges to zero. Second, the expected number of contributors is strictly positive and independent of the game-size. Third, the distance between the individual contribution rate and the symmetric mixed-strategy equilibrium of the unperturbed game is decreasing with the expected group-size and converges to zero; the same is also true for the expected number of contributors. The first property implies that any pivotal provision outcome (where m players always contribute) is not robust to population and decomposition uncertainty, and could thereby be ignored. The second property implies that the unique equilibrium in symmetric mixed-strategies in the perturbed game can be distinguished in a non-trivial manner from the ‘zero contributions outcome’ in the unperturbed game, and so the latter outcome could also be ignored. The third property implies, in conjuction with the previous implications, that one could select the symmetric mixed-strategy equilibrium, over the pivotal provision and zero contributions outcomes, of the contribution game with no uncertainty. 6 See

Andreoni (1988). See also Palfrey and Rosenthall (1988) and Hindriks and Pancs (2002).

7 See

for instance http://www.athens2004.com /en/Volunteers/indexpage.

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This is in striking contrast to the equilibrium selection by means of noisy strategy-revision processes or the ‘global games approach’. Before embarking on our analysis, we should also emphasise the following. First, there is no a priori reason to always treat a Poisson game as a special case of global games. The reason is that, as Myerson (1998) has shown, any game of asymmetric information with fixed and commonly known number of players can result from a game with population uncertainty. Second, as it will be made clear shortly after, our explanation of equilibrium uniqueness is analytically simpler than the existing theoretical explanations. The organisation of the paper is as follows. Next Section describes the model. Sections 3 investigates equilibria under the assumption of population and decomposition certainty. Section 4 investigates equilibria under the assumption of population and decomposition uncertainty, and Section 5 concludes.

2

The Model

We consider an economy with N ≥ 2 citizens. There are two types of citizens: an ‘egoist’, t = e, and an ‘altruist’, t = a. The probability of a citizen being ‘altruist’ is ε ∈ [0, 1). Types are statistically independent. Each citizen has an endowment of y units of the private good and derives utility from the consumption of a discrete public good g ∈ {0, 1}. Fixed per-capita contributions φ ∈ (0, y] are required towards the provision of the public good g. The citizens must decide whether to contribute or not. The cost of provision, i.e. of g = 1, is k. The income net of any contributions is denoted with ω. We denote the choice of each citizen by c ∈ {0, 1} ≡ C, where c = 1 denotes the decision to contribute. Moreover, we denote the resulting action profile by the vector x = {xc }c∈C and the set of possible action profiles by Z. Here, xc denotes the number of citizens who have chosen action c, and x determines whether the public good is provided or not. Note that provision takes place if and only if x1 ≥ k/φ ≡ κ > 0. That is, κ is the minimum number of contributors that secure provision. Given that κ may not be an integer, define with m the minimum integer such that m ≥ κ. Note that κ > 0 implies that m ≥ 1. Observe also that m < κ + 1. It follows that from an agent’s point of view g = g(c, x− ) with   1 if x1 ≥ m g(c, x− ) = ,  0 if x < m 1

where x1 = x− 1 + c and the superscript

−

denotes the actions of other citizens.

In general the net income of a stakeholder depends on the regime. In this paper we consider voluntary provision of the public good under the institutional arrangement that all unused contributions

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are distributed equally back to the citizens who have contributed.8 Under this rule,9 the net income ω is equal to y − φ(c, x− ), where   φ−    φ(c, x− ) =    

(x− 1 +1)φ−k x− 1 +1

=

k x− 1 +1

if c = 1 and x− 1 +1≥m

0 if c = 1 and x− 1 +1<m

.

(1)

0 if c = 0

Our results are qualitatively robust to allowing for unused contributions not be returned (regardless of the provision outcome or only when provision fails).10 Preferences are assumed without loss of generality to be quasi-linear. In particular, utility is

assumed to be equal to ω + v(g, c, t). For an ‘egoist’ v(g, 1, e) = v(g, 0, e) = u(g). That is, an ‘egoist’ does not derive any direct pleasure from contributing. For an ‘altruist’, on the other hand, we have that v(0, 1, a) > v(0, 0, a), v(1, 1, a) − φ > v(0, 0, a) and v(1, 1, a) > v(1, 0, a). Thus an ‘altruist’ is contributing, regardless of x− and group-size, simply because he derives sufficiently high direct pleasure from contributing. Finally, assume without loss of generality that u(0) = 0 and u(1) = 1; that is, the willingness-to-pay for the provision of the public good on the part of each ‘egoist’ is normalised to 1. We refer hereafter to ‘egoists’ as agents. When an agent contemplates his contribution decision he has to form beliefs about the behaviour of the other stakeholders. Denote with Pr(x− | I) the probability an agent attaches on the event x− occurring given the information available to him I. The optimisation problem each agent is faced with is # " X ¤ £ ¤ £ − − − − Pr(x | I) u(g(c, x )) + y − φ(c, x ) . max E U (c, x ) | I ≡ c∈C

(2)

x− ∈Z −

Note that for any x− such that x1 − < m − 1 the public good is not produced, while for any x− such that x1 − ≥ m the public good is produced, whether an agent contributes or not. We then have in a straightforward manner that for any given population N the net expected benefit from not contributing is given by B(I) = k

N −m−1 X i=−1

Pr(m + i | I) − Pr(m − 1 | I). m+i+1

(3)

To understand the above equation, note that the typical agent faces an expected benefit from not contributing via saving on payment. The agent will need to pay when the public good is provided with 8 Such

an institutionmal arrangement is equivalent to an institution where benefactors could committ ex ante to making

a contribution and thus make only the necessary payment ex post. Examples of such a mechanism abound. To cite one, a group called ‘Access’ called on the residents of Kingsdown, Bristol, UK to pledge a willingness to pay towards a campaign against parking restrictions. 9 The

full-refund rule, under population and decopmposition cetrainty, has also been analysed by Dixit and Olson

(2000). We assume a full-refund rule because we want to isolate the incentive to free-ride out of ‘greed’ (a non-refund or a refund-rule might deter agents from contributing out of ‘fear’ of loosing a contribution). See also Palfrey and Rosenthal (1984, 1988) and Dixit and Olson (2000). 10 Notice

that a non-refund rule would apply if contributions were physical.

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the agent contributing. In this case, the payment is equal to

k x− +1

with x− ∈ {m − 1, ..., N − 1}. Thus

the first term above represents the expected benefit from not contributing. The typical agent faces also an expected cost of not contributing via the lost opportunity to secure g = 1. This loss happens when the agent is pivotal, in the sense that without his contribution total collections fall short of the cost of provision but with his contribution public good can be provided. Thus the second term above represents the expected cost from not contributing. We refer hereafter to Pr(m − 1 | I) as the pivotal probability: the probability that an agent is critical for the provision of the public good. We also assume that: Assumption A1:

k m

< 1.

This assumption ensures that if an agent is certain that he is critical for the provision of the public good then he is willing to contribute.11 Finally, note the following. When m ≥ 2 and ε = 0, if every agent is not contributing we have that no agent is critical for the provision of the public good. Thus, if m ≥ 2 and ε = 0 the group is not ‘privileged’. If, however, m = 1 and ε = 0 then, due to k < m, an agent who is certain that no other contributions have been made can, and does have an incentive to contribute in order to, ensure the provision of the public good; that is, the group is privileged. Note also that if N ≥ m then, due to A1, we have N > k. So, provision of public good maximises the group’s net surplus from consumption. In the following Section we find the equilibria under the assumption that N is common knowledge and ε = 0.

3

Population Certainty and Complete Information

When N is common knowledge we refer to it as the population or game or group size. With N being common knowledge and ε = 0 we have that I = {N, σ − } where σ − = {σ 1 , ...σi−1 , σ i+1 , ...σ N } and σ i is the probability that player i contributes in the event that he is an ‘egoist’. We deploy the following assumption.12 Assumption A2: If N is common knowledge then N ≥ m + 1. This assumption ensures an interesting problem, given that k < m, for if N < m provision would be non-feasible and if m = N then c = 1 would weakly dominate c = 0. It will prove useful in what follows to define the following. First denote with ri the probability 11 If

Assumption A1 was violated then, for any m ≥ 1 and N ≥ 2, c = 0 would weakly dominate c = 1.

12 An

alternative complete information model could be one with exactly (1 − ε)N ‘egoists’ and εN ‘altruists’ in the

society. We would then have directly that if there were at least m ‘altruists’ then the public good would be provided and each agent in the society would not contribute. If on the other hand ‘altruists’ were less or equal to m − 1 agents would face a non-trivial voluntary-provision problem. In a society with ‘altruists’ being strictly less than m one should define m∗ = m − Nε ≥ 1 and N ∗ = N (1 − ε) ≥ m∗ + 1, and substitute m∗ for m amd N ∗ for N in the main text.

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that an agent i attaches on any randomly selected player, from the N − 1 other agents, contributing. Given these beliefs the probability of having z contributors in the group of N − 1 other players is given by p(z; N − 1, ri ), where p(.; M, v) is the binomial distribution with size and likelihood parameters M and v ∈ (0, 1), respectively. Also, p(N −1; N −1, 1) = 1, p(z; N −1, 1) = 0 if z < N −1, p(0; N −1, 0) = 1, P −m−1 p(m+i;N −1,¯r) = p(m − 1; N − 1, r¯). and p(z; N − 1, 0) = 0 if z > 0. Define r¯ as the solution of k N i=−1 m+i+1 This solution r¯ is the probability any randomly selected player from the group of N − 1 other players contributes, that if an agent perceives then he is indifferent between contributing or not. Define also with r(N, m) the function which is implicitly defined by r(N, m) = s∗ (N, m)/(1 + j=i Q (N − (m − 1 + j)) si Pi=N−m m j=1 ∗ ∗ s (N, m)) with s (N, m) being the solution with respect to s of i=1 [ m+i ] j=i Q (m − 1 + j) j=1

= (1 − (k/m))/(k/m). Note that the left-hand side of this equation is equal to zero if s = 0. It is also strictly increasing with s. Thus there is a unique solution s∗ (N, m) > 0 to the above equation for any given N and m. Accordingly, r(N, m) ∈ (0, 1). Clearly, also, the left-hand side of this equation is strictly

increasing with N for any N ≥ m + 1. So, s∗ (N, m), and thereby r(N, m), is strictly decreasing in N for any N ≥ m + 1. We then have that: Lemma: If m = 1 then r¯ = r∗ ≡ r(N, m) and if m ≥ 2 then r¯ ∈ {0, r∗ }. Also if 1 ≥ ri > r∗ then the agent is strictly better off if he refrains from contributing, whilst if 0 < ri < r∗ then the agent is strictly better off if he contributes. Finally, if ri = 0 and m = 1 the agent is strictly better off if he contributes. Proof. Note that due to A2 we have that N − 1 > m − 1. So if ri = 1 we have that i + 1)]p(m + i; N − 1, 1) =

k N

PN −m−1 i=−1

[k/(m +

> 0 = p(m − 1; N − 1, 1). Thus, if ri = 1 then B(I) > 0 and the agent is

strictly better off if he refrains from contributing, and hence r¯ < 1. Note also that due to A1 we have in a straightforward manner that if m = 1 and ri = 0 then PN−m−1 k [k/(m + i + 1)]p(m + i; N − 1, 0) = m < 1 = p(m − 1; N − 1, 0). Thus, if ri = 0 then B(I) < 0 i=−1

and the agent is strictly better off if he contributes. So, if m = 1 then r¯ > 0. PN −m−1 Clearly, also, if m ≥ 2 then r¯ = 0 is a solution of i=−1 [k/(m + i + 1)]p(m + i; N − 1, r¯) = p(m − 1; N − 1, r¯). However, if m ≥ 2 we may as well have a solution of B(I) = 0 such that r¯ > 0.

We have in a straightforward manner that if r¯ > 0 then r¯ = r∗ . Thus, if m = 1 we have r¯ = r∗ and if m ≥ 2 we have r¯ ∈ {0, r∗ }. Finally, given that

Pi=N −m i=1

m [ m+i ]

(r/(1 − r))i

j=i Q

(N − (m − 1 + j))

j=1

j=i Q

is strictly increasing with r,

(m − 1 + j)

j=1

and is equal to zero if r = 0, we have that if 0 < ri < r∗ then B(I) < 0 and the agent is strictly better

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off if he contributes, while if ri > r∗ then B(I) > 0 and he is strictly better off if he refrains from contributing. Define with χ the number of ‘egoists’ who contribute with probability one, and let superscripts denoting equilibrium values. We then have: Proposition 1 (See also Dixit and Olson (2000)) χo = m is a pure strategy equilibrium, for any m ≥ 1. If m = 1 then χo = m is the unique pure strategy equilibrium. If m ≥ 2 the only other pure strategy equilibria are χo ∈ [0, ..., m − 2]. Moreover, χo = m is a strict equilibrium. Proof. See Appendix A. Clearly, then, there is multiplicity of pure-strategy equilibria if m ≥ 2. The public good might be provided and it might not be provided, with the probability of provision being indeterminate. In particular, the expected number of contributors belongs to the set {0, m}, if m = 2, and {0, 1, ..., m − 2, m}, if m ≥ 3. These sets are independent of the size of the game, as long as N ≥ m + 1.13 Turning now to the case of symmetric mixed-strategy equilibria, i.e. with all agents contributing with probability σo ∈ (0, 1), we have: Proposition 2 (See also Dixit and Olson (2000)) The unique equilibrium in symmetric mixed strategies is σ o = r∗ ∈ (0, 1), for any m ≥ 1. Proof.

Straightforward given the definition of r∗ , given that in a mixed-strategy equilibrium agents

must be indifferent between contributing or not, and given that, when ε = 0, in a symmetric mixedstrategy equilibrium we have rio = σ o ∈ (0, 1) for any i = 1, ..., N. Thus a symmetric mixed-strategy equilibrium also exists. Note that under this equilibrium the expected number of contributors is N σo . Note also that this equilibrium mixed strategy σ o is strictly decreasing with N. Thus the probability that a randomly selected agent will contribute, σ o , is strictly decreasing with N. This in turn implies that the effect of group-size on the expected number  of contributors is ambiguous. Moreover, the probability of public good provision is (1 − r∗ )N

i=N P

i=m



Clearly, the effect of group-size on the probability of provision is also ambiguous. 13 Notice

N i

 [r∗ /(1 − r∗ )]i .

that in any given equilibrium with pivotal provision the likelihood that a randomly selected agent is contributing

is m/N, which is strictly decreasing with the size of the game. Moreover, the expected number of contributors is m. Note also that the outcome of provision relies on identical agents perceiving very different probabilities that they are pivotal in securing the provision of the public good: the contributors perceive pivotal probability equal to one and the noncontributors perceive pivotal probability equal to zero. In other words, from the point of view of a contributor the likelihood that a randomly selected agent from the set of N − 1 other agents is contributing is (m − 1)/(N − 1), while from the point of view of a non-contributor the likelihood that a randomly selected agent from the set of N − 1 other agents is contributing is m/(N − 1). Finally note that as N becomes very large both these probabilities converge to zero, and yet m agents will always be found to contribute. For these reasons the pure strategy equilibrium with pivotal contributions, though it is theoretically sound given the postulated assumptions, looks quite an implausible outcome in large societies. See also Myerson (1998) pp. 21-22 for related arguments concerning voting games.

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Notice that the pivotal equilibria which imply public good provision with certainty {χo = m} are robust to any (known) refinement. The reason is that these are strict (or strong): every agent holding Nash conjectures strictly prefers his equilibrium strategy to any other alternative strategy.14 Accordingly, there is no a priori theoretical reason for ignoring the pivotal equilibrium and focusing instead on the symmetric equilibrium in mixed strategies. In addition, the symmetric mixed-strategy equilibrium is very close, at the limit, to the zerocontributions equilibrium. To see this, note, given that r∗ is strictly decreasing with N, that the binomial distribution with parameters N and σ o = r∗ can be approximated by a Poisson distribution with parameter λ = N r∗ , when N is large. We then have that Proposition 3 For sufficiently large group-size, the probability of a randomly selected player contributing decreases with group-size, and in the limit as N → ∞ it tends to zero. Also the expected number of contributions decreases with the group-size. Finally the probability of provision decreases with the group-size. Proof. See Appendix A. That is, if agents randomise then the expected number of contributors and the probability of provision decreases with the group-size, and in the limit they tend to zero. Thus, in the limit the public good will almost certainly not be provided. In the next Section, we provide a theoretical argument for the selection of the symmetric mixedstrategies equilibrium r∗ over the pure-strategies equilibria.15

4

Equilibrium in the Poisson Contribution Game when ε > 0

Hereafter, we model the social interaction as a Poisson game in the presence of random altruism.16 In particular, the number of actual players is a PRV with mean n. We will refer to n as the population or game size. Introducing population uncertainty implies that players can no longer assign a strategy to 14 That

the outcome with pivotal provision is the unique pure-strategy Undominated Perfect Equilibrium is, for instance,

the result in Bagnoli and Lipman (1989). 15 Notice

that there may be other equilibria as well in the voluntary-provision game we focus on here. These would

involve asymmetric strategies with some players randomising. However, we choose to ignore these equilibria, as it is also the practice in the literature. 16 Note

that introducing only small uncertainty over the decomposition of the population, in terms of ‘warm glow’

altruism, will not result in the non-emergence of asymmetric equilibria, and will also fail to select a unique equilibrium. See Appendix C for more details. Also, introducing only population uncertainty, in that the grouip-size is a Poisson random variable, will fail to select a unique equilibrium if more than one contributions are necessary for public good provision. See Appendix D. Finally, if m = 1 then, as it will become clear shortly after, the equilibria of the Poisson game with ε = 0 are very close to the corresponding ones with ε ' 0, for any finite group-size.

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other individual players, simply because they are not aware of who they all are. Instead, we describe strategic behaviour in terms of a distributional strategy (see Myerson (2000)). Such a strategy, τ n , is defined as any probability distribution over the set C×{a, e} such that the marginal distribution on {a, e} is equal to {ε, 1 − ε} in a game of size n. That is τ n (1, a) + τ n (0, α) = ε and τ n (1, e) + τ n (0, e) = 1 − ε. Note that τ n can be interpreted as the ‘beliefs’ players choose to hold that a randomly sampled player will have type t and choose action c in a game of size n.17 As Myerson (1998) puts it “...going to a model of population uncertainty requires us specify a probability distribution over actions for each type of player, rather than for each individual player. In effect, population uncertainty forces us to treat players symmetrically in our game-theoretic analysis.” Accordingly, moving to a contribution game with population uncertainty ensures that the provision outcome will not be characterised by an asymmetric equilibrium. In other words, “all players of the same type must have the same predicted behaviour in equilibrium, because the type of a player includes all behaviorally-relevant attributes of the player that are recognised by others”, Myerson (1998), pp. 16. When players behave according to τ , the number of players of any type that choose any action c is a PRV with mean nτ (c) and hence the expected action profile in a game of size n is nτ ≡ {nτ (c)}c∈C , where τ (c) = τ (c, a) + τ (c, e). To derive expected payoffs for each action, given nτ , we make use of two special features of Poisson games. First, that the number of players choosing c is independent of the number of players who choose all other actions (see Myerson (1998)).18 Second, that any player in a Poisson game attaches the same probability that there are d individuals of type t in the game with him with the probability that the external game theorist would attach on the event that there are d individuals of type t in the whole game. This ‘environmental equivalence’ property implies that “from the perspective of any player of any type, the number of other players (not including himself) who choose action c is also an independent Poisson random variable with the same mean nτ (c)”, Myerson (1998), pp 16. Accordingly, the optimisation problem a type t individual faces is # " X ¤ £ ¤ £ t P (x− | nτ ) v(g(c, x− ), c, t) + y − φ(c, x− ) , max E U (c, x− ) | nτ ≡ c∈C

with

(4)

x− ∈Z −

" # Y e−nτ (c) (nτ (c))x− (c) P (x | nτ ) = . x− (c)! −

c∈C

In essence, then, each player resolves her contribution decision by formulating ‘beliefs’ τ over the likely behaviour of each potential other player and then calculating which action maximises her expected utility given the resulting expected action profile nτ . Following Myerson (2000), we establish 17 For

notational convenience, we supress hereafter the dependence of the strategy τ on the size of the game n whenever

there is no danger of confusion. 18 Without

population uncertainty this property could not be satisfied since the total number of players who choose a

certain action must be equal to the known N, see Myerson (1998) pp.9.

11

an equilibrium if and only if τ (c, S(c, nτ )) = τ o (c) ∀ c ∈ C, where S(c, nτ ) ⊆ {a, e} denotes the set of all types for whom action c maximises (4) given nτ . That is, if “all the probability of choosing action c comes from types for whom c is an optimal action, when everyone else is expected to behave according to this distributional strategy” (Myerson (2000), pp. 11). Given that an ‘altruist’, i.e t = a, has a strictly dominant strategy to contribute, τ o is equilibrium if and only if τ o (1) = (1 − ε)σ o + ε = 1 − τ o (2), where σ o = {σ o (c)}c∈C denotes the equilibrium probability that the typical ‘egoist’ contributes. Also, we focus hereafter on the problem of an agent/‘egoist’. It follows then directly that in a Poisson game I ={σ, ε, n} and the net expected benefit from not contributing is equal to B(I) = k

∞ X f (m + i | nτ ) − f (m − 1 | nτ ), m+i+1 i=−1

(5)

where f (| λ) denotes the Poisson distribution with mean λ. We can now turn to the characterisation of equilibrium. To start with, note that if τ (1) > 0 Pi=∞ m [nτ (1)]i ¯ − (1 − (k/m))/(k/m). Define λ∗ by then B(I) is proportional to ∆(nτ (1)) = i=1 m+i j=i−1 Q (m + j) j=0

¯ ∗ ) = 0. Note that λ∗ is the strictly positive expected number of other contributors which makes an ∆(λ ‘egoist’ indifferent between contributing or not, given that the number of other contributors is a PRV. Note also that an ‘egoist’ is indifferent between contributing or not when the expected number of other contributors is zero if and only if m ≥ 2. Clearly then λ∗ is the counterpart of N r∗ . Regarding pure-strategy equilibria, we have Proposition 4 For any given n (a) τ o (1) = 1 is an equilibrium if and only if n ≤ λ∗ , (b) τ o (1) = ε

(i.e. σo = 0) is an equilibrium if and only if n ≥ λ∗ /ε.

¯ Proof. If τ (1) = 1 then B(I) is proportional to ∆(n). Suppose that λ∗ < n. Then for τ (1) = 1 we have ¯ ∆(n) > 0. Thus, deviating to non-contributing is profitable and τ (1) = 1 cannot be an equilibrium. ¯ ≤ 0. Thus if λ∗ ≥ n then deviating Suppose now that λ∗ ≥ n. Clearly for τ (1) = 1 we have that ∆(n) to non-contributing is not profitable, and τ (1) = 1 is an equilibrium. ¯ Consider now Part (b). Note that if τ (1) = ε then B(I) is proportional to ∆(nε). Suppose that ¯ ≥ 0. Thus, deviating to contributing is not profitable, and λ∗ ≤ nε. Then for τ (1) = ε we have ∆(nε)

¯ < 0. τ (1) = ε is an equilibrium. Suppose now that λ∗ > nε. Clearly for τ (1) = ε we have that ∆(nε) Thus, deviating to contributing is profitable, and τ (1) = ε cannot be an equilibrium. Thus, under the population and decomposition uncertainty we focus on, if the game is sufficiently large - i.e. n ≥ λ∗ /ε - the unique pure-strategy equilibrium is zero contributions by every ‘egoist’. 12

The intuition is straightforward: if the population size is large enough and every agent is expected to contribute then the probability that a contributor is pivotal, relative to the probability that the public good is provided without his contribution, is very small. No equilibria in pure-strategies is the prediction when the game is of an intermediate size - i.e. λ∗ /ε > n > λ∗ . The intuition is straightforward. On the one hand, as above, if the population size is large enough (n > λ∗ ) and every agent is expected to contribute then the probability that a contributor is pivotal, relative to the probability that the public good is provided without his contribution, is very small. So, everyone contributing cannot be an equilibrium. On the other hand, if λ∗ /ε > n and no ‘egoist’ is expected to contribute then the expected number of ‘altruists’, nε, is small and the public good is very likely not to be provided in the absence of further contributions. So, contributing is the best response; everyone abstaining from contributing cannot be an equilibrium. Furthermore if n ≤ λ∗ then the unique pure-strategy equilibrium is that everyone contributes. The size of the game is small enough to ensure in expected terms that all contributions are needed for the public good to be provided. Turning to the case of mixed-strategy equilibria, i.e. with τ o (1) ∈ (ε, 1), we have: Proposition 5 For any given n there is a mixed-strategy equilibrium τ o (1) ∈ (ε, 1) if and only if n > λ∗ > nε and τ o (1) = λ∗ /n. Proof.

¯ ∗ ) = 0. Suppose that λ∗ < n and λ∗ > nε. Then Recall that λ∗ > 0 is the solution of ∆(λ

¯ (1)) < 0. Thus, τ (1) ∈ (ε, λ∗ /n) cannot be an equilibrium. In for any τ (1) ∈ (ε, λ∗ /n) we have ∆(nτ

¯ (1)) > 0. Thus τ (1) ∈ (λ∗ /n, 1] cannot be an addition, for any τ (1) ∈ (λ∗ /n, 1] we have that ∆(nτ

¯ (1)) = 0 and thus τ (1) = λ∗ /n is an equilibrium in mixed equilibrium. Finally, if τ (1) = λ∗ /n then ∆(nτ strategies. ¯ (1)) > Suppose that λ∗ ≤ nε. Then for any τ (1) ∈ (ε, 1) we have that τ (1) > λ∗ /n and hence ∆(nτ 0. Thus there is no equilibrium in mixed strategies. Suppose now that λ∗ ≥ n. Clearly for any τ (1) ∈ (ε, 1) we have that τ (1) < λ∗ /n and hence

¯ ∆(nτ (1)) < 0. Thus if λ∗ ≥ n there is no equilibrium in mixed strategies.

So, we have that there is a mixed-strategy equilibrium if and only if the expected size of the game is of an intermediate level - i.e. n ∈ (λ∗ , λ∗ /ε). If the expected population size is either small or large then it is not possible to find a randomisation that makes agents indifferent between contributing or not. Focusing on the case of n ∈ (λ∗ , λ∗ /ε) we have, due to λ∗ being independent of group-size, that the equilibrium mixed-strategy is strictly decreasing with population size. Also, the expected number ∞ P [λ∗ ]i of contributors is equal to λ∗ and the probability of public good provision is equal to exp[−λ∗ ] (i)! , i=m

which are both independent of the size of the game n.

Summarising, we have (a) if n ≤ λ∗ the unique equilibrium is τ o (1) = 1, (b) if n > λ∗ and

n < λ∗ /ε the unique equilibrium is such that τ o (1) = λ∗ /n (c) if n ≥ λ∗ /ε the unique equilibrium 13

is τ o (1) = ε. Thus, the contribution game under the population and decomposition uncertainty we postulate here has a unique equilibrium, regardless of the group-size and the number of contributions needed for provision. In addition, it follows directly from the above that if the group size is low enough and/or ε is low enough and/or m is large enough19 - i.e. if n < λ∗ /ε - then the probability of a randomly selected player contributing, τ o (1), the expected number of contributors, nτ o (1), and the probability of public good provision are independent of ε. If on the other hand n ≥ λ∗ /ε then no ‘egoist’ is contributing and thereby provision depends positively on ε.20 We now ask: what is the equilibrium if the decomposition noise is very small, i.e. if ε → 0? We have, in a straightforward manner, from the above Proposition that, for very small frequency of altruism, there is no equilibrium with zero contributions by every agent. Specifically, if the expected number of stakeholders is sufficiently small, n ≤ λ∗ , then every agent is a contributor. If, on the other hand, the expected group-size is sufficiently large, n > λ∗ , then every agent randomises according to λ∗ /n.21 That is, if the Poisson contribution game is sufficiently large and the frequency of altruism is very small then the unique equilibrium is in symmetric mixed strategies. The above discussion will enable us to develop an argument in favour of selecting the symmetric mixed-strategy equilibrium r∗ as the outcome of sufficiently large contribution game under population and decomposition certainty. In more detail, fix a game of size ν under certainty, i.e. with ε = 0 and N = ν. Suppose also that this game is large enough, i.e. ν > λ∗ . Call this game Γ. Consider now its perturbation with ε ' 0, and Pr(N ) = f (N | nτ ) where n = ν. That is, suppose that players commonly believe that there is an infinitesimal frequency of altruism, that the number of actual players is a Poisson random variable, and that players’ common beliefs about the group-size are on average correct. In this perturbed game Γ(ε, f ), the unique equilibrium individual contribution rate λ∗ /ν is decreasing with the group-size and converges to zero. This implies that any outcome of the unperturbed game Γ with some players contributing with certainty, like the pivotal provision outcome (where m players always contribute), is not robust to population and decomposition uncertainty, and could thereby be ignored. Define now λν ≡ (ν − 1)r∗ . After using the approximation of the Binomial by the Poisson i−(m−1) P − [λ∗ ]i−(m−1) } 1 (m − 1)!{[λν ] [ ] distribution, we have, for sufficiently large games, that i=ν−1 i=m 1+i i! Pi=∞ 1 (m − 1)![λ∗ ]i−(m−1) 22 = i=ν [ 1+i ] . The right hand-side is strictly positive and hence λν > λ∗ , for any i! finite group-size. As λ∗ is independent of group-size, we also have that the distance λν −λ∗ is decreasing 19 Note

that λ∗ is strictly increasing in m.

20 These 21 This

predictions on the effect of altruism are consistent with those in Hindriks and Pancs (2002)

is also the complete equilibrium characterisation of the Poisson game with m = 1 and ε = 0. See Appendix D

for more details. 22 See

Appendix B.

14

with ν and converges to zero.23 It follows that the distance | r∗ − τ o | is also decreasing with ν and converges to zero.24 Thus, the distance between the individual contribution rate λ∗ /ν in the perturbed game and the symmetric mixed-strategy equilibrium of the unperturbed game r∗ is decreasing with the gamesize ν and converges to zero. This observation however cannot justify, on its own, the selection of the symmetric mixed-strategy equilibrium r∗ over the ‘zero contributions’ equilibrium, χo = 0, in the game Γ. The reason is that the latter equilibrium is also ‘approached’ as ν increases by the unique equilibrium λ∗ /ν of the perturbed game. Nevertheless, note also that νr∗ − λ∗ =

ν ν−1 λν

− λ∗ > λν − λ∗ > 0. Recall that λ∗ is independent

of ν. It follows directly that that the distance νr∗ − λ∗ =

ν ν−1 λν

− λ∗ is decreasing with group-size and

converges to zero. Therefore, the distance between the strictly positive and independent of the game-size expected number of contributors in the perturbed game, λ∗ , and the expected number of contributors in the symmetric mixed-strategy equilibrium in the game Γ, νr∗ , is decreasing with group-size and converges to zero. In other words, while the distance between the expected number of contributors in the perturbed game and the expected number of contributors in the ‘zero contributions’ equilibrium in the unperturbed game is positive and constant, the distance between the expected number of contributors in the perturbed game and the expected number of contributors in the symmetric mixed-strategy equilibrium in the unperturbed game vanishes in the limit. So, in a game of discrete public good provision of size N > λ∗ , one could select the symmetric mixed-strategy equilibrium over both the pivotal equilibria and the zero contributions equilibrium. This is in striking contrast to the equilibrium selection by means of noisy strategy-revision processes or the ‘global games approach’.

5

Conclusions

In this paper we are concerned with equilibrium selection in sufficiently large discrete public good provision games. We develop our argument by investigating the perturbed game with very small frequency of altruism and very small population uncertainty. We show that in the perturbed game, there is a unique equilibrium which is in symmetric mixed strategies. This equilibrium has the following crucial properties. First, the individual contribution rate is decreasing with the expected size and converges to zero. Second, the expected number of contributors is strictly positive and independent of the game23 See

Appendix B.

24 First,

note that λν > λ∗ implies r∗ >

∂((ν−1)r ∗ −ντ o ) τ o . Note now that < 0. So, we have that

∂(r ∗ −τ o ) ∂ν


0, and more contributions will result to higher individual costs k/(m + 1). Clearly χo = m is a strict equilibrium. Note that χo = 0 is trivially an equilibrium if m > 1, since deviations will not alter the level of public good provision, which - in this outcome - is g = 0. Also, note that if m = 1 then χo = 0 is not an equilibrium since a deviation will result in public good provision at an individual cost of k/m = k < 1.

17

If m ≥ 3 then χo ∈ [1, m − 2] are equilibria since deviations will not alter the level of public good provision, which - in these outcomes - is g = 0. Finally, note that χ = m − 1 cannot be an equilibrium since deviations to contributing will result to provision of public good at an individual cost of k/m < 1.

Proposition 3. Proof. The proof follows very closely that of a related result in Hindriks and Pancs (2002). Consider a group of size N 0 ≥ N + 1 and the associated equilibrium mixed strategies σ 0 and σ. We know that (1 − (k/m))/(k/m) =



σ 0m+i−1 (1 − σ 0 )N−1−(m+i−1) 

N0 − 1

 

m+i−1 =  N0 − 1  σ 0m−1 (1 − σ 0 )N−1−(m−1)  m−1   N − 1  σ m+i−1 (1 − σ)N −1−(m+i−1)  m+i−1 Pi=N −m m [ m+i ] . Change of variables j = m + i − 1,   i=1 N −1  σ m−1 (1 − σ)N −1−(m−1)  m−1 and approximation by the Poisson distribution gives for any sufficiently large N > m ≥ 1 that Pj=N 0 −1 m λ0j−(m−1) (m − 1)! Pj=N −1 m λj−(m−1) (m − 1)! [ ] [ j+1 ] = , where λ = (N − 1)σ and j=m j=m j+1 j! j! Pj=N −1 m [λ0j−(m−1) − λj−(m−1) ](m − 1)! Pj=N 0 −1 m λ0j−(m−1) (m − 1)! [ j+1 ] [ j+1 ] = − j=N < λ0 = (N 0 −1)σ 0 . Hence j=m j! j! 0 0. It follows directly that λ < λ and thereby σ 0 < [(N − 1)/(N 0 − 1)]σ < σ. Note that limN →∞ σ = Pi=N 0 −m i=1

m [ m+i ]



limN →∞ [λ/(N − 1)] = 0 since λ0 < λ for any N 0 > N. Thus the probability of a randomly selected player contributing decreases with group size, and in the limit as N → ∞ it tends to zero. Substituting in the above N for N − 1 and N 0 for N 0 − 1 we have again that λN 0 < λN for any N 0 > N, where now λN = σN and λN 0 = σ 0 N 0 . Thus the expected number of contributors decreases with group size. This in turn allows us to approximate the number of contributors by a Poisson distribution f (i | vN ) for which the mean vN is lower in the larger group. Then applying theorem 33.2 in pp. 92 in m−1 m−1 P P f (i | vN ) < f (i | vN 0 ) for any m ≥ 1 and N 0 > N. Hence, the Schmetterer (1974) we have that i=0

i=0

probability of no provision increases with group size.

8

Appendix B

One can easily see from Propositions 2 and 5 that the symmetric mixed-strategy equilibria in the unperPν−m−1 p(m+j;ν−1,r ∗ ) m turbed and perturbed games, Γ and Γ(ε, f ), with n = ν, are given by j=0 [m+j+1] p(m−1;ν−1,r ∗ ) = P∞ f (m+j|ντ o ) m (1 − (k/m))/(k/m) and j=0 [m+j+1] f (m−1|ντ o ) = (1 − (k/m))/(k/m), respectively. Thus, after approximating, for large games, the Binomial distribution p(z; ν − 1, r∗ ) with the Poisson distribution

f (z | λν ) where λν ≡ (ν − 1)r∗ , using the fact that λ∗ = nτ o , and deploying the change of variables 18

P P∞ f (i|λν ) f (i|λ∗ ) m m i = j + m, we have ν−1 i=m [i+1] f (m−1|λν ) = i=m [i+1] f (m−1|λ∗ ) . P∞ m f (i|λ∗ ) Pν−1 Pν−1 m f (i|λν ) f (i|λ∗ ) [ f (m−1|λ − f (m−1|λ So, i=m [i+1] ∗) ] = i=ν [i+1] f (m−1|λ∗ ) , or, equivalently, i=m ν) P∞ m (m−1)! ∗ i−m+1 ∗ i−m+1 } = i=ν [i+1] i! [λ ] . Thus, [λ ]

m (m−1)! {[λν ]i−m+1 − [i+1] i!

∞

ν−1 X

X m! m! {[λν ]i−m+1 − [λ∗ ]i−m+1 } = [λ∗ ]i−m+1 . (i + 1)! (i + 1)! i=m i=ν

(6)

Recall that λ∗ is independent of group-size ν. As the right hand-side of (6) is strictly positive, it follows directly that λν > λ∗ for any finite ν. P∞ f (j|λ∗ ) Pν P f (j|λ∗ ) f (j|λ∗ ) Note that the right hand-side of (6) is equal to ∞ j=ν+1 f (m|λ∗ ) = j=m f (m|λ∗ ) − j=m f (m|λ∗ ) . Pν (j|λ∗ ) Notice that i=m ff(m|λ ∗ ) is strictly increasing with ν. Also, the inverse hazard rate at there being m P∞ m! P∞ f (i|λ∗ ) [λ∗ ]i−m+1 is strictly contributors i=m f (m|λ∗ ) is independent of the group-size. Therefore, i=ν (i+1)! P∞ Pν P∞ m! (j|λ∗ ) f (j|λ∗ ) [λ∗ ]i−m+1 = j=m ff(m|λ decreasing with group-size ν. In fact, due to i=ν (i+1)! ∗) − j=m f (m|λ∗ ) P ∗ i−m+1 m! we also have that ∞ converges to zero. These two observations in conjuction with i=ν (i+1)! [λ ]

the fact that λν > λ∗ for any finite ν, imply that λν is a strictly decreasing function that converges Pν 0 −1 m! i−m+1 0 − [λ∗ ]i−m+1 } = to λ∗ . To see this consider group-size ν 0 > ν and note that i=m (i+1)! {[λν ] Pν 0 −1 m! Pν−1 m! i−m+1 − [λ∗ ]i−m+1 } + i=ν (i+1)! {[λν 0 ]i−m+1 − [λ∗ ]i−m+1 }. In addition, after usi=m (i+1)! {[λν 0 ] Pν 0 −1 m! {[λν 0 ]i−m+1 − [λ∗ ]i−m+1 } < ing the monotonicity of the right hand-side of (6), we have i=m (i+1)! Pν−1 m! Pν−1 m! Pν−1 m! i−m+1 i−m+1 i−m+1 0 −[λ∗ ]i−m+1 }. So i=m −[λ∗ ]i−m+1 } > i=m − i=m (i+1)! {[λν ] (i+1)! {[λν ] (i+1)! {[λν ] 0 P P ν −1 m! ν−1 m! i−m+1 i−m+1 0 − [λ∗ ]i−m+1 }. That is, i=m − [λν 0 ]i−m+1 } > [λ∗ ]i−m+1 } + i=ν (i+1)! {[λν ] (i+1)! {[λν ] Pν 0 −1 m! i−m+1 −[λ∗ ]i−m+1 } > 0, where the latter inequality follows from λν > λ∗ for any finite i=ν (i+1)! {[λν 0 ] P∞ m! [λ∗ ]i−m+1 ν. It follows then directly that λν > λν 0 > λ∗ for any ν 0 > ν. Finally, the fact that i=ν (i+1)! Pν−1 m! converges to zero implies directly from (6) that i=m (i+1)! {[λν ]i−m+1 − [λ∗ ]i−m+1 } also converges to zero. The latter implies from (6), due to λν > λ∗ for any finite ν, that λν also converges to λ∗ .

9

Appendix C

Suppose that N is common knowledge. Interestingly, then, multiplicity of equilibria arises even if we relax the assumption that the type of each individual is common knowledge by postulating a small noise due to random ‘altruism’. Suppose that individuals do not know the payoff relevant decomposition of the group. What they know instead is that the decomposition of the group of size N is a random variable, and described by the likelihood of a player being an ‘altruist’ ε ∈ (0, 1). In other words, it is common knowledge that the number of ‘egoists’ follows a binomial distribution with parameters N and 1− ε. We model this social interaction as a Bayesian game. In such a game an equilibrium strategy profile is σ ˆ o = {(σ o1 , 1), ..., (σ oN , 1)} with σ oi ∈ [0, 1] being the probability of contributing for individual i conditional on her being an ‘egoist’, for any i = 1, ..., N. If σ oi = σ oi0 for any i, i0 we say that we have equilibrium in symmetric strategies. If σ oi 6= σ oi0 for some i, i0 we say that we have equilibrium in 19

asymmetric strategies. In the environment in question each individual i knows his type (and hence if he is ‘altruist’ he contributes with probability one and if he is ‘egoist’ he contributes with probability σ oi ). However, the other players i− , despite knowing who player i is, do not know the type of player i. Nevertheless they do know that i player deploys the strategy σ ˆ oi = (σ oi , 1). We start with equilibria such that σ oi = σ o for any i = 1, ..., N. That is, we focus on symmetric equilibria where each agent deploys the same (type-dependent) strategy. Clearly σ = 1 cannot be an equilibrium. The reason is that if σ = 1 then each agent is certain that the number of other contributors is equal to N − 1, for any ε. Hence, each agent is certain, given A2, that the public good is provided even if he does not contribute and thus has an incentive to deviate to not contributing to save on payment. That is, it must be that σ o < 1. Focus now on symmetric equilibria such that σ o ∈ [0, 1). The analysis is as the one in Section 3 when we considered symmetric mixed strategies, with the difference that here for any individual i we have that rio = (1 − ε)σ o + ε ≡ ro ∈ [ε, 1) for any symmetric strategy σ o ∈ [0, 1) on the part of the ‘egoists’. It follows then that, Proposition 6 (1) There is a symmetric mixed-strategy equilibrium σ o ∈ (0, 1) if and only if r∗ > ε and σ o =

r o −ε 1−ε

with the probability of a randomly selected player contributing, ro , being ro = r∗ . (2)

Finally, σ o = 0 is a symmetric pure-strategy equilibrium if and only if r∗ ≤ ε. Proof.

Part (1): It follows directly from Lemma that in a symmetric mixed-strategy equilibrium

σ o ∈ (0, 1) we have that r∗ = (1 − ε)σ o + ε ≡ ro . So, we have σ o =

r ∗ −ε 1−ε .

Part (2): Follows directly from the definition of r∗ , the fact that in such an equilibrium all agents have no incentive to deviate to contributing, Lemma, and that (1 − ε)σ o + ε is equal to ε ∈ (0, 1) if σ o = 0. Thus, a pure-strategy equilibrium where all agents do not contribute exists if and only if the probability of a citizen being ‘egoist’ is low enough, for given N and m. Equivalently, after recalling the definition of r∗ , a pure-strategy equilibrium where all agents do not contribute exists if and only if N is large enough, for given ε. Also, a symmetric mixed-strategy equilibrium exists if and only if a pure-strategy equilibrium where all agents do not contribute does not exist. Or, a symmetric mixed-strategy equilibrium exists if and only if ε is small enough and/or N is low enough. In particular, since 1 − r∗ ∈ (0, 1), we have that as the likelihood of ‘altruism’ becomes very small, i.e. ε → 0, the only symmetric equilibrium is the mixed-strategies one. We turn to the case of asymmetric equilibria in pure strategies with certain provision. That is, we turn to the case of #{i | σ oi = 1} = m and #{i | σ oi = 0} = N − m .25 If m players contribute, regardless of their type, we have that from the point of view of a non-contributing agent the public good is provided 25 Clearly,

no more than m ‘egoists’ will be contributing, due to free-riding.

20

with certainty, and so there is no reason to deviate. On the other hand, from a contributing agent’s point PN−m−1 −m,ε) − p(0; N − m, ε). So the of view, the expected benefit from not contributing is k i=−1 p(1+i;N m+i+1 PN−m−1 p(1+i;N −m,ε) ≤ p(0; N − m, ε), pivotal outcome is a Bayesian Nash equilibrium if and only if k i=−1 m+i+1 PN −m m p(i;N −m,ε) i.e. if and only if i=1 [ m+i ] p(0;N−m,ε) ≤ (1 − (k/m))/(k/m). The latter is true if and only if 

ε ≤ rˆ∗ , where rˆ∗ is defined by

PN −m

m [ m+i ]

  rˆ∗i (1−ˆ r∗ )N−m−i  

N −m i

    

= (1 − (k/m))/(k/m). Notice   N −m PN−m m ∗i  , where that the left-hand side of the latter equation can be re-written as i=1 [ m+i ]ˆ s  i sˆ∗ ≡ rˆ∗ /(1 − rˆ∗ ). Clearly, then, sˆ∗ > 0 and rˆ∗ ∈ (0, 1). That is, the pivotal outcome is an equilibrium if i=1

(1−ˆ r ∗ )N−m

and only if the frequency of ‘altruism’, ε, is non-greater than the mixed-strategy the non-contributing agents should have deployed to leave contributing agents indifferent between contributing or not, rˆ∗ . Thus, we have shown that Proposition 7 The pivotal outcome #{i | σ oi = 1} = m and #{i | σoi = 0} = N − m is an asymmetric pure-strategy equilibrium if and only if rˆ∗ ≥ ε. It follows then directly from the above discussion that if the likelihood of ‘altruism’ becomes very small, i.e. ε → 0, both the pivotal outcome and the symmetric mixed-strategy r∗ are equilibria. That is, only introducing some noise in the decomposition of the fixed-in-size group does not lead to uniqueness of equilibrium.

10

Appendix D

Suppose that the group-size is a Poisson random variable. Interestingly, then, multiplicity of equilibria arises when ε = 0, unless m = 1. We show this next. Here, τ o = σ o . Consider first pure-strategy equilibria. It follows that: Proposition 8 For any given finite n (a) τ o (1) = 0 is an equilibrium if and only if m ≥ 2, and (b) Pi=∞ m λ∗ i = τ o (1) = 1 is an equilibrium if and only if n ≤ λ∗ with λ∗ > 0 such that i=1 m+i j=i−1 Q (m + j) j=0

(1 − (k/m))/(k/m). Proof.

Part (a) follows directly after noting that if τ (1) = 0 then f (x− 1 = 0 | nτ ) = 1, and hence

B(I) = 0 if m ≥ 2 while B(I) = (k/m) − 1 < 0 if m = 1. Note now that for any given τ with τ (1) > 0 we have that B(I) = exp[−nτ (1)] {

∞ P

x− 1 =m−1

nτ (1)m−1 (m−1)! }.

k x− 1 +1

Part (b) follows from the fact that if τ (1) = 1 then it must be that B(I) ≤ 0 - otherwise

deviations to not contributing would be profitable. In more detail, note that if τ (1) = 1 then B(I)

21

−

[nτ (1)]x1 (x− 1 )!

−

P ¯ is proportional to ∆(n) = i=∞ i=1

ni m m+i j=i−1 Q

(m + j)

− (1 − (k/m))/(k/m). Define λ∗ by the solution of

j=0

¯ ¯ ∗ ) = 0. Note that λ∗ > 0. Suppose that λ∗ < n. Then for τ (1) = 1 we have ∆(n) > 0. Thus τ (1) = 1 ∆(λ ¯ ≤ 0. Thus cannot be an equilibrium. Suppose now that λ∗ ≥ n. Clearly for τ (1) = 1 we have that ∆(n) τ (1) = 1 is an equilibrium if and only if λ∗ ≥ n.

Thus, after noting that λ∗ is independent of n, we have that, under population uncertainty, if

m ≥ 2 and the game is sufficiently large - i.e. n > λ∗ - zero contributions by every agent is the unique pure strategy equilibrium. The intuition is straightforward: if m ≥ 2 the group is not privileged, and if the expected population size is large enough and every agent is expected to contribute then the probability that a contributor is pivotal, relative to the probability that the public good is provided without his contribution, is very small. So, contributors have an incentive to deviate. Uniqueness of pure-strategy equilibria is also the prediction here if m = 1 and n ≤ λ∗ The unique equilibrium in pure strategies is that everyone contributes. The reason is that the group is priviledged, and hence if the expected population is small enough the probability of being pivotal is high enough to sustain unilateral provision. No equilibria in pure-strategies is, on the other hand, the prediction when m = 1 and the game is large enough - i.e. n > λ∗ . The intuition is again straightforward. On the one hand, as above, if the population size is large enough and every agent is expected to contribute then the probability that a contributor is pivotal, relative to the probability that the public good is provided without his contribution, is very small. On the other hand, given A1, if m = 1 then the group is privileged. So, unilaterally providing the public good is optimal when every other player is expected to refrain from contributing. Multiplicity of pure-strategy equilibria is the prediction if m ≥ 2 and n ≤ λ∗ . In particular, zero voluntary contributions and hence no provision is an equilibrium outcome, regardless of the expected group-size. The other possible pure strategy equilibrium outcome, now, is that each agent contributes! The intuition for the latter is simple: if the population size is small enough and every agent is expected to contribute then the probability that a contributor is pivotal, relative to the probability that the public good is provided without his contribution, is large enough to sustain contributions. Turning to the case of mixed-strategy equilibria, i.e. with τ o (1) ∈ (0, 1), we have: Proposition 9 For any given finite n, τ o (1) ∈ (0, 1) is an equilibrium if and only if n > λ∗ and τ o (1) = λ∗ /n. In a mixed strategy equilibrium agents are indifferent between contributing or not - i.e. Pi=∞ m [nτ (1)]i ¯ − B(I) = 0. In more detail, if τ (1) > 0 then B(I) is proportional to ∆(nτ (1)) = i=1 m+i j=i−1 Q (m + j)

Proof.

j=0

(1 − (k/m))/(k/m). Note that the first term is a non-negative valued and strictly increasing function 22

¯ ∗ ) = 0. Suppose that λ∗ < n. of τ (1) for any n > 0. Recall that λ∗ > 0 is the unique solution of ∆(λ ¯ (1)) < 0. Thus, deviations to contributing are profitable, Then for any τ (1) ∈ (0, λ∗ /n) we have ∆(nτ

and τ (1) ∈ (0, λ∗ /n) cannot be an equilibrium. In addition, for any τ (1) ∈ (λ∗ /n, 1) we have that

¯ ∆(nτ (1)) > 0. Thus, deviations to non-contributing are profitable, and τ (1) ∈ (τ ∗ /n, 1) cannot be an

¯ (1)) = 0, and thus τ (1) = λ∗ /n is an equilibrium. Finally, if λ∗ < n and τ (1) = λ∗ /n then ∆(nτ equilibrium in mixed strategies. Suppose now that λ∗ ≥ n. Clearly for any τ (1) ∈ (0, 1) we have that

¯ ∆(nτ (1)) < 0. Thus if λ∗ ≥ n there is no equilibrium in mixed strategies.

So, there is a mixed-strategy equilibrium if and only if the expected size of the game is large enough - i.e. n > λ∗ . Summarising, we have that if m = 1 then the equilibrium is unique. Specifically, if the mean population is small, n ≤ λ∗ , every player in the game contributes, while if otherwise then every player in the game contributes with probability λ∗ /n. If, on the other hand, m ≥ 2, then there is multiplicity of equilibria. In particular, apart from the above equilibria, every stakeholder refraining from contributing is also an equilibrium, regardless of the expected group-size. Thus, only introducing population uncertainty does not resolve the problem of equilibrium multiplicity. It is of some interest to note that, when m = 1, the solution with ε → 0 can be approximated by

the solution with ε = 0. The reason is that if ε → 0 then λ∗ /ε → ∞.

Nevertheless, this is not true if m ≥ 2, since if ε = 0 then τ o (1) = 0 is an equilibrium for any n, while if ε → 0 then for any finite n there is no equilibrium with zero contributions by every agent. The reason is simple. If ε = 0 and m ≥ 2 then if no ‘egoist’ is expected to contribute we have that f (x− 1 = m − 1 | nτ ) = 0, whilst if ε > 0 and no ‘egoist’ is expected to contribute then f (x− 1 = m − 1 | nτ ) > 0. Accordingly, if ε = 0 the typical ‘egoist’ is indifferent between contributing or not, whilst if ε > 0 the incentives of the typical ‘egoist’ depend on the expected size of the game n and the threshold number of contributions m.

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