Asymptotics for Provisioning Problems with a Large Number of Participants

We consider problems of provisioning a good to a number of participants who are able to communicate information about their private preferences for the good. The good may or may not be a public good. This provisioning is to be done in a manner that is incentive compatible, rational and feasible (in senses to be described). We show that as the number of participants becomes large there is a limiting problem, whose solution takes a simple form. For instance, it is near optimal to set a fixed fee and allow a participant to access the good if he is willing to pay this fee. The advantage of this is that the provisioning policy and fee structure can be easily communicated to the participants. 1 A problem of providing a public good This paper is about problems of provisioning a good amongst a number of participants who are able to communicate information about their private preferences for the good. The good may or may not be a public good. This provisioning is to be done in a manner that is incentive compatible, rational and feasible (in senses that are described below). In Section 2 we show that as the number of participants becomes large there is a limiting problem, whose solution takes a simple form. For instance, it is near optimal to set a fixed fee and allow a participant to access the good only if he is willing to pay this fee. The advantage of this is that the provisioning policy and fee structure can be easily communicated to the participants. In Section 3 we describe some models to which our result can be applied. We prove the main theorem in Section 4 and discuss further directions in Section 5. Appendices A and B contain proofs for specialized and generalized versions of our main theorem. We begin by summarising a model of Hellwig (2003). Consider n agents that bargain about the provision of a public good. A public good, once provided, can be enjoyed by all the agents at once. To provide the good in quantity (or, in some applications, quality) Q costs c(n,Q). Once the good is provided, the net benefit to agent i is θiu(Q)− pi where pi is the payment that the agent makes towards the cost of the good. The preference parameter θi is the realization of a random variable with prior distribution function F on ∗Department of Computer Science Athens University of Economics and Business, Evelpidon Str., Athens 11362, GR †Statistical Laboratory, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WB, UK 1 [0, 1]. While F is known to all agents, the value of θi is known to agent i alone, We suppose that θ1, . . . , θn are independent and identically distributed. Given θ = (θ1, . . . , θn), we ask what Q(θ) should be, and what payments, p1(θ), . . . , pn(θ), the agents should be required to make. An allocation is said to be feasible if the sum of the payments always covers the cost, i.e., c(n,Q(θ)) ≤ n ∑ i=1 pi(θ) (1) for all θ ∈ [0, 1]n. We say an allocation is weakly feasible if the expected sum of the payments covers the expected cost, i.e., ∫ c(n,Q(θ))dF(θ) ≤ n ∑ i=1 ∫ pi(θ)dF(θ) . (2) We suppose that the social planner may, if he wishes, exclude some agents from participating. Let πi(θ) be the probability with which he chooses to include agent i given announced preferences θ. If exclusion is not an option for the planner, then we simply make the restriction πi(θ) = 1 for all θ, i. We also consider the fact that there must be an incentive for each agent to participate. Agent i must expect to have a positive net benefit. This is the condition of individual rationality θi ∫ πi(θi, θ−i)u(Q(θi, θ−i))dFn−1(θ−i)− ∫ πi(θi, θ−i)pi(θi, θ−i)dFn−1(θ−i) ≥ 0 . (3) Here θ−i is the vector of the preference parameters of the n− 1 agents other than agent i. Finally, we have an incentive compatibility condition, that agent i maximizes his expected net benefit by a truthful declaration of θi, i.e., θi ∫ πi(θi, θ−i)u(Q(θi, θ−i))dFn−1(θ−i)− ∫ πi(θi, θ−i)pi(θi, θ−i)dFn−1(θ−i) ≥ θi ∫ πi(θi, θ−i)u(Q(θ̂i, θ−i)dFn−1(θ−i)− ∫ πi(θi, θ−i)pi(θ̂i, θ−i)dFn−1(θ−i) ≥ 0 for all θ̂ ∈ [0, 1]. We can now define Vi(θi) = ∫ πi(θi, θ−i)u(Q(θi, θ−i))dFn−1(θ−i) (4) and Pi(θi) = ∫ πi(θi, θ−i)pi(θi, θ−i)dFn−1(θ−i) . (5) The incentive compatibility condition implies two things. Firstly, we must have [θ̂iVi(θ̂i)− Pi(θ̂i)] + [θ̄iVi(θ̄i)− Pi(θ̄i)] ≥ [θ̂iVi(θ̄i)− Pi(θ̄i)] + [θ̄iVi(θ̂i)− Pi(θ̂i)]