Strategy-proof divisible public goods. for quasi-linear preferences.

Transcription

1 Strategy-proof divisible public goods for quasi-linear preferences. Pasha Andreyanov et al. (Preliminary Draft) October 27, 2016 Abstract When agents are risk-neutral and costs are linear, allocating a divisible public good is equivalent to allocating a binary public good with uncertainty. We show that an ex post IC, IR and BB mechanism 1 is equivalent to a lottery over two types of mechanisms. One is a posted price, while the other is an ex-post IC, IR and BS mechanism 2 with residual claimants 3. The latter, in general, can not be decomposed into lotteries over similar deterministic mechanisms, and can not be discarded as inefficient. Lottery decomposition provides immediate results for two special cases of mechanisms: binary and anonymous. We study welfare in these families of mechanisms and, where possible, deliver general ranking results. 1 ex post incentive compatible, individually rational and budget balanced 2 ex post incentive compatible, individually rational and budget surplus generating 3 a group of agents that participate solely by absorbing the surplus 1

2 Contents 1 Introduction 3 2 Setting 7 3 Analysis Randomized mechanisms Deterministic mechanisms Anonymity 23 5 Conclusion 25 2

3 1 Introduction Consider a problem of building a divisible public good, in a quasi-linear setting, when both costs and preferences are linear. The assumption of constant marginal costs is not ubiquitous, but for many economic environments we can say that it is, at least for a range of production levels, close to reality. Same with preferences: using three copies of the public good at the same time might, as an approximation, bring three times higher utility as that of a single copy. For such environments therefore producing a divisible good is equivalent to producing a binary good with uncertainty, because agents are risk-neutral. We pursue the goal of characterizing a family of mechanisms satisfying a number of desirable requirements (such as ex post IC, IR, BS) in terms of a lottery over a collection of mechanisms, as simple as possible, and from the same class. Such lottery decomposition will allow us to analyze the structure of the family which will give us a better understanding of how to design mechanisms for divisible public goods. For example, if a mechanism is a lottery over k deterministic mechanisms with equal probabilities, then we can think of it as k copies of the public good being built, each through one of k separate deterministic mechanisms. The literature on strategy proof implementation of public goods is rich. After the famous negative result by [Arrow] was obtained for abstract preferences and choice functions, researchers decided to focus on more restricted domains. Depending on the choice of the preference domain, we can sort the literature into three big strains. The first strain mainly works with convex preferences. Several important 3

4 results were obtained in [Zho91], [Mou94], [BJ93], and [Ser99] in the context of public goods, when the costs are also convex. The dominating point of view is that anonymous cost sharing schemes are the only reasonable mechanisms for divisible public goods. The second strain works in the quasi-linear domain. Here, the [Cla71], [Gro73], [GL75], [GL77], [Hol79], [Led06] made important contributions to characterizing dominant strategy mechanisms for public projects. However, there was no satisfactory solution found for the problem of balancing the budget ex-post. Finally, in the most restrictive domain of quasi-linear preferences that satisfy single crossing, the most progress was done. The relevant literature spans from fundamental implementation questions [JMtVMZ06], [BM05], [BS12], [BS14], [EC02] to auction design [SOME REFERENCES] and bilateral trade [MS83], [HR87], [CP16]. In the working paper [KS16] the authors characterize binary dominant-strategy mechanisms for public goods with budget balance Following the tradition started in [MS83], we model agents as having onedimensional valuations, that they report to the designer in a direct revelation mechanism. The public good is modeled to have one dimension, that corresponds to different levels or different probabilities of production. The preferences are linear, that is, agent is risk neutral, and the costs are linear as well. As a solution concept we pick ex-post IC, IR and BB mechanisms, that is mechanisms that satisfy the ex post incentive compatibility, individual 4 SHOULD ADD SOME REFS ON [BAYESIAN IMPLEMENTATION], [ASYMPTOTIC IM- PLEMENTATION], [RESIDUAL CLAIMANTS IN CONTRACTS] 4

5 rationality and budget surplus constraints for every profile of valuations. There are two natural candidates that fit our description. One is a (deterministic) posted price. In this mechanism the designer offers each agent to pay a non-negotiable price for the public good, and if all agents agree then the public good is built and the cost is covered by the payments. This mechanism gives incentives for the agent to pay the price if and only if it does not exceed his valuation. The other is a bit more tricky. First of all, it requires an already working ex post IC, IR and BS mechanism. The surplus generated is then absorbed by a group of agents that are called residual claimants. They do not participate in the original mechanism in any way, but they split the surplus among themselves in proportions that they can not choose. Our analysis is very close in spirit to that of bilateral trade in [HR87]. In fact, for two agents with values in [0, 1] 2 and costs c = 1 the problem of allocating a public good is, up to a change in variables, equivalent to the problem of bilateral trade. The answer is, straightforwardly, a lottery over posted prices. Our approach is more general though, for the following reason. In [HR87] two special cases: discrete and smooth allocation functions, are studied separately. In our paper we provide general results in the domain of right-continuous allocation functions, that nests both discrete and smooth functions. An alternative approach to deal with the same problem was taken in [CP16], where a stronger version of the IR constraint was invoked. Our main result is Theorem 1, that states that in the public goods setting, an ex post IC, IR, BB mechanism can be decomposed into a lottery over posted prices and (not necessarily deterministic) ex post IC, IR, BS mechanisms with residual claimants. Theorem 2 argues that further decomposition of ex post 5

6 IC, IR, BS mechanisms in a similar fashion is impossible unless a certain geometric property is satisfied. Moreover, Example 1 shows that ex post IC, IR, BS mechanisms, generally, do not admit a decomposition into deterministic mechanisms from the same class. Our first application of the lottery decomposition concerns deterministic ex post IC, IR and BB mechanisms. It trivially follows from Theorem 1, that such mechanism has to be either a posted price, or an ex post IC, IR, BS mechanism with residual claimants. Using a geometric approach, we show that posted prices are the only allocative undominated in this class of mechanisms. This is, however, not true for non-binary allocation functions, as shown in Example 2. Things turn out quite differently for Pareto dominance, as it is a more restrictive concept. We show that every deterministic ex-post IC, IR, BB mechanism, that is also non-wasteful 5, is Pareto undominated. 6 Our second application concerns anonymous mechanisms. It follows from Theorem 1 that the family of anonymous ex-post IC, IR, BB mechanisms is a lottery over agents indices followed by a lottery over posted prices and ex post IC, IR, BS mechanisms with residual claimants. The whole family of such mechanisms is quite hard to imagine, but we document a rich subset of relatively simple ones, see randomized residual claimants in Example 3. This result stands in sharp contrast with Theorem 3 in [Ser99], that predicts the anonymous posted prices to be the only ones. The difference lies in our assumption of linearity of the cost function. Randomized residual claimants are mechanisms, that are mathematically 5 the allocation function attains its maximum at some point 6 I M NOT SURE ABOUT PARETO AND NON-BINARY YET 6

7 equivalent to any (the simpler the better) ex post IC, IR, BS mechanism among a subset of agents with residual claimants, with a preliminary lottery over agents indices. These mechanisms are, in general, not dominated by posted prices neither allocatively, nor Pareto. Implicit in their construction is a transfer of wealth from highest to lowest types, so that they have a much more equal distribution of surplus than, say, anonymous posted prices. In a world where direct redistribution of wealth is not possible, such mechanisms might be of use. 7 The paper is organized in the following way. Section 2 describes the setting and core definitions. Section 3 first studies randomized mechanisms, then deterministic ones. Section 4 studies anonymous mechanisms. Section 5 concludes. 2 Setting Consider n agents who wish to purchase a pure (non-excludable and nonrivalrous) public good. The good is divisible, and the maximum level of production is normalized to unity. Let each agent have a private valuation v i of the maximum level of public good. Denote v = {v i } n i=1 - the profile of valuations of all agents from the type space V = i V i, where we set each individual type space V i equal to the segment [0, 1] for simplicity. Let the utility of agent i from consuming x amount of the public good, and 7 MAYBE I SHOULD PUSH THIS A BIT FURTHER 7

8 the corresponding cost of production c(x), both take a linear form: u i (v i, x, t i ) = v i x t i, c(x) = cx, where t i is the monetary transfer from agent i to the designer, and c [0, n] is the marginal cost of production. Because these preferences are quasi-linear and risk-neutral, we can think of x as allocation of a fraction of the public good, as well as the probability of allocating the full amount. Since the cost is linear, these two interpretations become equivalent from the designer s point of view. We focus on direct revelation mechanisms, that is one-shot normal form games where the message space is the type space of the agent (as opposed to an arbitrary extensive form game with arbitrary message spaces). Definition 1. A mechanism is a tuple of functions (ϕ, {τ i } n i=1 ), where 1. ϕ : V [0, 1] is the expected allocation function 2. τ i : V R is the expected transfer from agent i Note that expectation is taken conditional on the realization of v. We pick the desirable properties to be ex-post incentive compatibility, individual rationality and budget surplus (or balance). Definition 2. A mechanism (ϕ, {τ i } n i=1 ) is: 1. ex-post IC (incentive compatible) if u i (v i, ϕ(v), τ i (v)) u i (ˆv i, ϕ(v), τ i (v)) for all v V, v i V i, where v = (v i, v i ) is the true profile of valuations. 2. ex-post IR (individually rational) if u i (v i, ϕ(v), τ i (v)) 0 for all v V, where v = (v i, v i ) is the true profile of valuations. 8

9 3. ex-post BS (running budget surplus) if i τ i(v) 0 4. ex-post BB (running budget balance) if i τ i(v) = 0 We define the expected equilibrium utility of agent i, associated with the mechanism, that he gets when all agents report their truthful types. Definition 3. Expected equilibrium utility U i : V R associated with a mechanism (ϕ, {τ i } n i=1 ) is defined by formula: U i(v) = u i (v i, ϕ(v), τ i (v)), where v = (v i, v i ). Finally, by revelation principle [SOME REFERENCE HERE], we can establish an equivalence relation on the set of all (not just direct) mechanisms, whenever their direct revelation counterparts coincide. Definition 4. Two (indirect) mechanisms are equivalent, if their associated expected allocation function and transfers coincide. For example, a mechanism that allocates for free 1/2 of the public good with probability 1, and a mechanism that allocates for free 0 or 1 levels of public good with equal probability are equivalent, because they have the same expected allocation function and transfers. This completes the description of the setting and the minimal mechanism design toolbox. But before starting our analysis, we would like to define two special classes of mechanisms. Definition 5. A mechanism (ϕ, {τ i } n i=1 ) is a posted price if its allocation function takes the form: ϕ(v) = 1[v i p i, i], 9

10 where prices p i are nonnegative. We call it non-deficit if pi c, and we call it zero-surplus, if pi = c. Let I be a subset of indices (we call these agents residual claimants) and denote ṽ I (v) as vector v where all coordinates with indexes from I are substituted with zero. Definition 6. A mechanism (ϕ, {τ i } n i=1 ) is an ex-post IC,IR,BS with residual claimants if for some subset of the indices I: 1. (ϕ, {τ i } n i=1 ) do not depend on coordinates from I, 2. ( ϕ, { τ i } n i=1 ) is an ex post IC, IR and BS mechanism, where ϕ(v) = ϕ(ṽ I (v)), τ i (v) = τ i (ṽ I (v)) We call it non-deficit if τ i 0 and we call it zero-surplus if τ i = 0. It is important that the division of surplus among the residual claimants does not depend on the reports, which follows from {τ i } not depending on the coordinates from I. 3 Analysis We start our analysis with two important lemmas. First lemma - is a classic result, that characterizes an ex post incentive compatible mechanism (ϕ, {t i } n i=1 ) together with the associated equilibrium utilities {U i } n i=1 in our quasi-linear setting. This lemma states that all the local IC constraints of agent i can be replaced with a simple geometric condition 10

11 - monotonicity of the allocation function ϕ(, v i ), and the evolution of his transfers τ i (, v i ) and utilities U i (, v i ) is pinned down by it s shape. Second lemma - is a direct implication of the first lemma, applied to the public goods setting with budget surplus and individual rationality. Lemma 1. A mechanism (ϕ, {τ i } n i=1 ) with equilibrium utilities {U i} n i=1 is expost IC if and only if the following three conditions hold: 1. ϕ(v) is nondecreasing in every coordinate 2. τ i (v i, v i) τ i (v i, v i) = v i zdϕ(z, v i ) 3. U i (v i, v i) U i (v i, v i) = v i ϕ(z, v i )dz v i v i for all v i, v i and v in the support. Proof. see Appendix. Because in the ex-post IC mechanism transfers τ i can be restored from the allocation function ϕ and utilities U i, the tuple (ϕ, {U i } n i=1 ) carries the same information as (ϕ, {τ i } n i=1 ), so we can use both in a characterization of a mechanism. Lemma 2. A mechanism (ϕ, {U i } n i=1 ) is ex post IC, IR and BS if and only if the following three conditions hold: 1. ϕ(v) is nondecreasing and is equal to zero if v i < c. 2. U i (v) is nondecreasing in v i and (a) U i (v) 0 if v i = 0 (b) U i (v) = 0 if v i c j i v j 11

12 (c) if v i c and v i > 0 then U i (v) = U i (max(0, c j i vi v j ), v i ) + max(0,c ϕ(ξ, v i )dξ j i v j) 3. ϕ(v)( v i c) U i (v), with equality in case of BB. for all v in the support. Proof. see Appendix. Starting from this point we re going to assume that the allocation function is right-continuous (on top of just non-decreasing). Assumption 1. The allocation function ϕ(v) is right-continuos on V. This assumption is important for our results, yet we argue that it is just a necessary and convenient normalization of the functional space. We give three facts in favor of our choice: 1. Right-continuous allocation functions contain both smooth and binary allocation functions, so we don t need to treat them separately. 2. Right-continuity removes the ambiguity in the description of mechanisms on threshold levels of valuations. For instance, in a posted price we would like the public good to be built with certainty, when agents valuations coincide with the offered prices. 3. An allocation function of a simple lottery over posted prices is in fact the CDF of that lottery, if we assume it to be right-continuous. Generally speaking, right-continuity does not seem to be restrictive at all, because it only affects the mechanism (ϕ, {U i } n i=1 ) on a set of measure zero. 12

13 3.1 Randomized mechanisms In this section we will attempt to decompose ex-post IC, IR, BS mechanisms into a lottery over the ones we already know, like posted prices and ex post IC, IR, BS with residual claimants. It is worthy to mention that after the allocation function ϕ(v) is fixed, the only freedom for the designer comes from the boundary levels of utilities U i (0, v i ), and only if j i v j > c. These boundary utilities are not completely arbitrary but should be also consistent with the boundary allocation function ϕ(0, v i ). For our purposes, it is convenient to first abstract from the boundary utilities and focus on the interior of the allocation function. Definition 7. A mechanism (ϕ, {τ i } n i=1 ) with associated utilities {U i} n i=1 has zero boundary utility, if U i (0, v i ) = 0 for all i and v V. The following lemma will give us a simple geometric test to tell whether an allocation function corresponds to a lottery over posted prices, when the boundary utilities are set to zero. Lemma 3. An ex post IC, IR, BS mechanism with zero boundary utility is equivalent to the lottery over non-deficit posted prices iff the rectangular condition holds, and a lottery over zero surplus posted prices iff it holds with equality. Proof. see Appendix. Lemma 4. The allocation function ϕ(v) of the lottery over non-deficit posted prices is also the CDF of that lottery. 13

14 Proof. This is actually obvious. From now on we can focus mainly on the part of the domain, where production is not inefficient: D = {v V : v i c} The rectangular condition mentioned in Lemma 3 is a discrete analog of a nonnegative mixed derivative of the allocation function. Definition 8. An allocation function ϕ(v) satisfies the rectangular condition in a convex domain D V, if for nonnegative a, b, c: (n = 2) for any rectangle with vertices (x + a, y + b), (x + a, y), (x, y + b) and (x, y =) in D, such that a, b are positive: ϕ(x + a, y + b) ϕ(x + a, y) ϕ(x, y + b) + ϕ(x, y) 0 (n = 3) for any parallelepiped with vertices (x, y, z), (x+a, y, z), (x, y+b, z), (x, y, z+ c),... and (x + a, y + b, z + c) in D: ϕ(x, y, z) ϕ(x + a, y, z) ϕ(x, y + b, z) ϕ(x, y, z + c)+ +ϕ(x + a, y + b, z) + ϕ(x + a, y, z + c) + ϕ(x, y + b, z + x) ϕ(x + a, y + b, z + c) 0 (n > 3) analogously. The rectangular condition is satisfied with equality, if the RHS is zero. 14

15 The next Lemma connects the rectangular condition to the assumption of ex post budget balancedness. Lemma 5. An ex-post IC, IR, BB mechanism has the rectangular condition satisfied with equality. Proof. From Lemma 2 it follows that ϕ(v)( v i c) U i (v) where U i (v) = U i (max(0, c j i vi v j ), v i ) + max(0,c ϕ(ξ, v i )dξ j i v j) In a special case, when the allocation function is smooth it is easy to check that the integral equations above imply that the mixed n-th derivative is equal to zero: n v 1... v 2 ϕ(v) = 0 which implies that the rectangular condition holds. For non-smooth functions, the proof is a bit harder. It requires using generalized functions [SOME REFERENCES HERE] and showing (see Lemma 5.1) that the differential equation above holds in weak form:... ϕ(v)ψ(v)dv = 0 for all smooth test functions ψ(v) with support in D. Lemma 5.2 then shows that under right-continuity the equation above implies that the rectangular 15

16 condition holds with equality. Finally, all the effort required to derive the rectangular condition will now pay off in the following powerful result. We shall demonstrate how to, quite mechanically, decompose an ex-post IC, IR, BB mechanism into the lottery over mechanisms from the same class. Theorem 1. An ex post IC, IR, BB mechanism is equivalent to a lottery over two families of mechanisms: 1. zero-surplus posted prices 2. zero-surplus ex post IC, IR, BS mechanisms with residual claimants. Proof. First of all, by Lemma 5, an ex post IC, IR, BB mechanism has the rectangular condition satisfied with equality. We will now show that it is possible to extract layers of expected allocation function and expected equilibrium utilities, corresponding to zero-surplus ex post IC, IR, BS mechanisms with residual claimants, while preserving the rectangular condition. Consider n = 2 and c (0, 1) for exposition. Let ϕ(v) be the allocation function and U 1 (v), U 2 (v) be equilibrium utilities, where v [0, 1] 2. Define the first mechanism for one agent (ϕ 1, U 1 ) with surplus S 1 as: ϕ 1 (v 1 ) = ϕ(v 1, 0), U 1 (v 1 ) = v1 c ϕ 1 (ξ)dξ, S 1 (v 1 ) = ϕ 1 (v 1 )(v 1 c) U 1 (v 1 ). Define the second mechanism for one agent (ϕ 2, U 2 ) with surplus S 2 as: ϕ 2 (v 2 ) = ϕ(0, v 2 ), U 2 (v 2 ) = v2 c ϕ 2 (ξ)dξ, S 2 (v 2 ) = ϕ 2 (v 2 )(v 2 c) U 2 (v 2 ) 16

17 Note that by Lemma 2 each of the two mechanisms is ex post IC, IR, BS and has the rectangular condition satisfied with equality (because they depend on only one of two variables). Finally, pick the remainder mechanism (ϕ 0, {U1 0, U 2 0 }), which is defined as: ϕ 0 (v) = ϕ(v) ϕ 1 (v 1 ) ϕ 2 (v 2 ) U 0 1 (v 1, v 2 ) = U 1 (v 1, v 2 ) S 2 (v 2 ) U 0 2 (v 1, v 2 ) = U 2 (v 1, v 2 ) S 1 (v 1 ) It can be straightforwardly checked (see Appendix for calculations) that the conditions in Lemma 2 are satisfied for this mechanism and it has zero boundary utilities. Therefore, by Lemma 3 it is a lottery over posted prices. The three mechanisms above are the decomposition we were looking for. For n > 2 the algorithm is essentially the same, but it requires more care, because decomposition is not necessarily unique. For example, for n = 3, one can first extract the layers of ϕ and {U i } encoded on the edges of the cube, and only then on the faces of the cube. After removing all mechanisms associated with the lower bounds of support, the remainder is going to be a lottery over posted prices. Unfortunately, a similar result can not be established for ex-post IC, IR, BS mechanisms, because their allocation functions do not necessarily satisfy the rectangular condition, and this is as far as we can get without imposing additional restrictions on the allocation function. The following example shows 17

18 that no reasonable decomposition can be guaranteed for an ex-post IC,IR,BS mechanism. Example 1. Consider the following ex-post IC, IR, BS mechanism. Let there be 2 agents with valuations (x, y) [0, 1] 2 and the cost c = 1. The allocation function ϕ is determined by formula: 1/2, x + y 3/2 ε 1, x + y 3/2 + ε 0 otherwise where ε < 1/2 and utilities and transfers are as in Lemma 1. This mechanism generates budget surplus for all v V (except for point (0,0) where the surplus is exactly zero), but at the same time it can be only decomposed into a lottery over two deterministic ex-post IC,IR mechanisms: 1, x + y 3/2 ε, 0, otherwise 1, x + y 3/2 + ε 0, otherwise The first one generates budget deficit, while the second one generates budget surplus. A lottery with probabilities (1/2,1/2) over these two mechanisms gives the original mechanism that yields budget surplus. We can, nevertheless, establish a subset of IC, IR, BS mechanisms that admit a lottery representation, like in Theorem 1. The are, unsurprisingly, those that have an allocation function satisfying the rectangular condition. Theorem 2. An ex post IC, IR, BS mechanism is equivalent to a lottery over 18

19 two families of mechanisms: 1. non-deficit posted prices 2. non-deficit ex post IC,IR,BS mechanisms with residual claimants. if and only if its allocation function satisfies the rectangular condition. Proof. The proof is essentially the same as in Theorem 1, but this time we re starting with a rectangular condition that may be satisfied with inequality. However, because each of the layers of mechanisms described in Theorem 1 has rectangular condition satisfied with exact equality, removing them preserves the rectangular condition. The remainder mechanism, of course, should be defined to take the surplus generating nature of the grand mechanism into account. If S(v) is the surplus of the original mechanism, the formulas for n = 2 should be adjusted in the following way: ϕ 0 (v) = ϕ(v) ϕ 1 (v 1 ) ϕ 2 (v 2 ) U 0 1 (v 1, v 2 ) = U 1 (v 1, v 2 ) S 2 (v 2 ) S(0, v 2 ) U 0 2 (v 1, v 2 ) = U 2 (v 1, v 2 ) S 1 (v 1 ) S(v 1, 0) It can be verified (see Appendix for calculations) that the remainder mechanism is an ex-post IC, IR, BS mechanism with zero boundary utilities. Because the rectangular condition is still satisfied, it is going to be a lottery over non-deficit posted prices by Lemma 3. This ends our analysis of randomized mechanisms. Lottery decomposition 19

20 is only possible for budget balanced mechanisms, and this decomposition utilizes budget surplus mechanisms. Budget surplus mechanisms, in turn, have a lottery representation if and only if the additional rectangular condition is satisfied. 3.2 Deterministic mechanisms In this section we characterize ex post IC, IR, BS mechanisms that have a binary allocation function, with two different approaches. The first approach is a direct implication of our analysis of randomized mechanisms, which is summarized in Corollary 1. Corollary 1. A deterministic ex post IC, IR, BB mechanism is either a zerosurplus posted price, or a deterministic zero-surplus ex-post IC,IR,BS mechanism with residual claimants. A deterministic ex post IC, IR, BS mechanism with rectangular condition is either a non-deficit posted price, or a deterministic non-deficit ex-post IC, IR, BS mechanism with residual claimants. The second approach is to characterize an ex-post IC, IR, BS mechanism as a certain transformation of a well-known mechanism like posted price. Definition 9. A mechanism (ϕ, {τ i } n i=1 ) is a contraction of a posted price, if the allocation function ϕ(v) does not exceed that of some posted price at all v, and is strictly lower for some v. Lemma 6. Let ϕ be a deterministic allocation function of an ex-post IC, IR, BS mechanism, then it is either a zero-surplus posted price, or a contraction of a posted price. 20

21 Using this simple geometric characterization we can attempt to rank the deterministic mechanisms on an efficiency scale. Let ϕ A (v) and ϕ B (v) be allocation functions associated with mechanisms A and B. Definition 10. Mechanism A ex post allocation dominates B, if ϕ A (v) is at least as high as ϕ B (v) for all v V and is strictly higher for some v V. Theorem 3. In the family of deterministic ex post IC, IR, BS mechanisms the ex post allocative undominated ones are zero-surplus posted prices. Proof. We know from Lemma 6 that a deterministic ex-post IC, IR, BS mechanism is either a zero-surplus posted price or a contraction of one. Two zero-surplus posted prices can not be ranked from the point of view of the ex post allocation function. That is because with budget balance a price for one agent can be lowered only at the expense of the price of some other agent. At the same time, any contraction of a zero-surplus posted price is inferior to that posted price, by definition. QED. Unfortunately, we can not establish a similar result for randomized mechanisms, because not every randomized mechanism is a contraction of a collection of posted prices, see next example. Example 2. Consider the following ex post IC, IR, BS mechanisms. Let there be 2 agents with valuations (x, y) [0, 1] 2 and the cost c = 1. The allocation 21

22 function ϕ is determined by formula: 1, x 3/4, y 3/4 3/5, 1/3 x < 3/4, y 3/4 3/5, 1/3 y < 3/4, x 3/4 0 otherwise where utilities and transfers are as in Lemma 1. This mechanism can be allocative improved only at the expense of the budget surplus. But budget surplus is already equal to zero in the x 3/4, y 3/4 region, so such an improvement is not possible. Consider an alternative ranking criterion. Let U A i (v) and U B i (v) be equilibrium utility functions associated with mechanisms A and B. Definition 11. Mechanism A ex post Pareto dominates B, if U A i (v) is at least as high as U B i (v) for all i and v and is strictly higher for some i and v. Note that absence of allocative ranking precludes Pareto ranking. Since most of deterministic mechanisms are not allocative ranked, it is not a big surprise that a negative result could be formulated for Pareto ranking. In the following theorem we treat budget surplus as a wealth of an additional agent. Theorem 4. In the family of deterministic ex post IC, IR, BS mechanisms, each mechanism is ex post Pareto undominated. Proof. Note that an ex post IC, IR, BS mechanism is either a zero-surplus posted price, or a contraction of one. 22

23 We already know that zero surplus posted prices are allocative undominated, therefore they are Pareto undominated. Since any contraction of a posted price generates budget surplus, it is not dominated by any posted price. Finally, we need to compare two different contractions. Note that Pareto ranking requires allocative ranking, and two contractions can be only allocative ordered iff one is a contraction of the other. But this means that one that is allocative dominated generates more budget surplus or gives higher boundary utility to the residual claimant, which means that Pareto dominance is not possible. 4 Anonymity In this section we apply our analysis of randomized and deterministic mechanisms to a special setting, when anonymity is required. Definition 12. A mechanism (ϕ, {τ i }) is anonymous, if 1. ϕ(v) = ϕ(p(v)) 2. τ i (v i, v i ) = τ i (v i, p(v i )) for any permutation of coordinates p(.) and any v = (v i, v i ) in the support. Theorem 5. The family of anonymous ex post IC,IR,BB mechanisms is equivalent to all ex post IC,IR,BB mechanisms with a preliminary stage, where the indices are assigned to the agents with equal probability. This characterization is almost tautological, but it gives us an understanding how to generate the space of anonymous ex post IC, IR, BB mechanisms 23

24 for divisible public goods. For exposition we will scale the allocation function such that production levels appear discrete. Consider a setting where three copies of the public good are available for constructions, that it, ϕ(v) can take values from {0, 1, 2, 3}. Let there be three agents with valuations per copy belonging to the segment [0, 2], and the cost of production be c = 2 per copy. Example 3 (Randomized residual claimants). A mechanism (ϕ, {τ i } 3 i=1 ) proceeds in the following way: 1. At stage 1: agents submit their valuations. 2. At stage 2: three separate instances of the same deterministic (0 or 1), anonymous, ex post IC,IR,BS sub-mechanism ( ˆϕ, {ˆτ i } 2 i=1 ) for 2 agents are launched, one per pair of agents, and they are executed with the reports collected in the previous stage. 3. At stage 3: for every instance of the sub-mechanism ( ˆϕ, {ˆτ i } 2 i=1 ) one copy of public good is built, if ˆϕ yields 1 and the respective transfers are made, and all of the generated surplus is absorbed by the agent that did not belong to the active pair. This mechanism is, by construction, anonymous and ex-post IC, IR and BB. Mechanism designer has a lot of freedom in choosing which sub-mechanism to pick. Consider the following two examples: Example 3.1: The ex post IC, IR, BS sub-mechanism is an anonymous zero-surplus posted price for two agents ϕ(x, y) = 1[x 1, y 1] 24

25 [MORE CALCULATIONS, MAYBE PICTURES HERE] Example 3.2: The ex post IC, IR, BS sub-mechanism is an anonymous contraction of the previous example: ϕ(x, y) = 1[x + y 2] [MORE CALCULATIONS, MAYBE PICTURES HERE] Despite the fact that the mechanism is example 3.1 is allocative superior to example 3.2, the randomized residual claimants mechanism using example 3.2 as a sub-mechanism is not Pareto dominated by the one using 3.1, because of the surplus absorbed by the residual claimant in each instance of the submechanism. These examples are in sharp contrast with [Ser99], that tells that equal cost sharing schemes (in our language anonymous posted prices) are the only anonymous mechanisms that are ex post IC, IR and BB. The difference comes from the fact that in our setting costs are linear (as opposed to convex in [Ser99]). If the costs were strictly convex, then the production of one of the copies of the public good would increase the cost of the other copy, thus making the side payments that agents receive implicitly dependent of their reports. 5 Conclusion In the quasi-linear setting with linear costs and preferences, ex-post mechanism design for divisible public goods is significantly more complicated than, for 25

26 example, that of bilateral trade or indivisible public goods. We show lottery decomposition, similar to [HR87] for an ex post IC, IR, BB mechanism in a divisible public goods setting. We demonstrate that traditional posted prices (cost sharing rules) are not the only ones that fit the requirement of strategy proofness and budget balancedness. Even when anonymity is imposed, there exist a rich set of ex post IC, IR, BB mechanisms, with a convoluted recursive structure that uses residual claimants. Finally, we describe a family of relatively transparent anonymous mechanisms, that on top of being ex post IC, IR, BB, aslo have a surprising property of surplus redistribution. In the world where different agents have different intensity of using the public good, an anonymous compensating mechanism that transfers welfare from those who benefit from that public good most to those who benefit least, might be of interest. 26

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