The Generalized Random Priority Mechanism with Budgets∗ Tadashi Hashimoto† January 12, 2016 Abstract This paper studies allocation problems with and without monetary transfers, such as combinatorial auctions, school choice, and course allocation. Interdependent values and multidimensional signals are allowed. Despite known negative results, a mechanism exists that is feasible, ex post individually rational, ex post incentive compatible, and asymptotically both efficient and envy-free. This mechanism is a special case of the generalized random priority mechanism (GRP), which always satisfies the first three properties. The asymptotic properties follow as a corollary of the main theorem: GRP approximates virtually any infinite-market mechanism in large finite markets. Keywords: ex post incentive compatibility; random allocation; random priority; large market; rational expectations equilibrium; information aggregation.

∗

I am grateful to Eric Budish, John Hatfield, Johannes H¨orner, Matt Jackson, Michihiro Kandori,

Phuong Le, Huiyu Li, Paul Milgrom, Takeshi Murooka, Muriel Niederle, Andrew Postlewaite, John Roberts, Marzena Rostek, and Ilya Segal for insightful comments. I am especially indebted to Mike Ostrovsky, Andy Skrzypacz, Bob Wilson, and Fuhito Kojima for their guidance and encouragement. This research is supported by the SIEPR Leonard W. Ely and Shirley R. Ely Graduate Student Fund Fellowship through a grant to the Stanford Institute for Economic Policy Research. † Department of Economics, Yeshiva University, 245 Lexington Avenue, New York, NY 10016. E-mail: [email protected].

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1

Introduction

This paper studies allocation problems with indivisible goods, including combinatorial auctions, school choice, and course allocation. We introduce a generalized random priority mechanism (GRP) to approximately achieve desirable properties such as efficiency and envy-freeness in large but finite markets, while exactly maintaining feasibility, ex post individual rationality, and most important, ex post incentive compatibility. This mechanism is applicable to cases with and without monetary transfers and can accommodate interdependent valuations and multidimensional signals. The strong requirement, ex post incentive compatibility, eliminates the possibility of profitable misreporting even at the ex post stage when agents observe all the truthfully reported private information. This incentive compatibility robustly induces truthful reports from agents regardless of any posterior distributions or higher order beliefs agents form; the converse is also true (Bergemann and Morris, 2005). Ex post incentive compatibility is considered an appropriate counterpart of dominant-strategy incentive compatibility (a.k.a. strategy-proofness) in environments with interdependent values. Indeed, these two concepts coincide when values are private. Several studies demonstrate difficulties associated with ex post incentive compatibility in markets with finitely many agents. Generically, no ex post incentive compatible mechanisms attain efficiency with interdependent values and multidimensional signals (Maskin, 1992; Jehiel and Moldovanu, 2001; Jehiel et al., 2006; Che et al., 2015).1 If monetary transfers are unavailable, strategy-proof mechanisms become necessarily inefficient under certain fairness conditions even when values are private (Zhou, 1990; Bogomolnaia and Moulin, 2001). On the other hand, efficiency can coexist with ex post incentive compatibility in infinite markets with a continuum of agents. An efficient solution, competitive equilibrium, is interpretable as an infinite-market mechanism that calculates demand 1

The impossibility theorems of Che et al. (2015) allow single-dimensional signals in an environ-

ment without monetary transfers.

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correspondences and then market-clearing prices given reported private information. In this market with a continuum population, each individual has no informational or allocational effect on the entire market and hence cannot manipulate equilibrium prices. Thus, agents truthfully report their information to let the mechanism choose the optimal consumption from the unchangeable budget set; this is incentive compatibility. This paper bridges the gap between these two different markets with seemingly contradicting natures. Our main finding is as follows: An infinite-market mechanism can be asymptotically approximated by a finite-market mechanism that keeps feasibility, ex post individual rationality, and ex post incentive compatibility. That is, even in finite economies, approximate efficiency is always achievable together with incentive compatibility despite the impossibility theorems. Throughout the paper, we employ GRP as an approximating mechanism in finite markets. GRP has additional flexibility in choice sets compared to the random priority mechanism (RP, a.k.a. random serial dictatorship; see Abdulkadiro˘glu and S¨onmez (1998)). RP sequentially assigns objects to agents from the top of a randomly generated priority. The most prioritized agent receives the best choice, the second prioritized receives the best choice from the remaining objects, and so on. This mechanism is clearly feasible, ex post individually rational, and ex post incentive compatible in the case of private values. GRP is designed to inherit these same properties, while also accommodating the aforementioned flexibility at the same time. Like RP, GRP randomly prioritizes agents and then sequentially assigns consumption bundles to them. Before the sequential assignment starts, GRP determines a budget set, the set of available choices, for each agent. The mechanism then sequentially chooses an optimal choice for each agent, subject to two constraints: (i) the choice comes from the budget set and (ii)

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the resulting allocation is feasible in a certain strong sense. The two constraints are constructed not to be manipulable so that GRP is incentive compatible. The feasibility of GRP, guaranteed by the sequential application of the feasibility constraint, is one of the main departures from the existing literature. This paper maintain that feasibility is an obligatory requirement, although some papers allow slightly infeasible allocations (C´ordoba and Hammond, 1998; Kovalenkov, 2002; Budish, 2011; Nguyen et al., 2015).

1.1

Application 1: Combinatorial Auctions

We apply the approximation technique to two assignment problems. Our first application is combinatorial auctions, including auctions with homogeneous goods or single-unit demands as special cases. By approximating the full-information competitive equilibrium (a.k.a. fully-revealing rational expectations equilibrium) in infinite markets, we obtain the following result: A finite-market mechanism exists that is feasible, ex post individually rational, ex post incentive compatible, asymptotically surplus maximizing, and asymptotically envy-free in large finite markets. That is, full efficiency is nearly achievable even with ex post incentive compatibility despite the aforementioned negative findings (Maskin, 1992; Jehiel and Moldovanu, 2001; Jehiel et al., 2006). All these works study environments with interdependent values and multidimensional signals. According to the first two works, full efficiency is generically unattainable with Bayesian incentive compatible mechanisms. Jehiel et al. (2006) claims a stronger result with ex post incentive compatibility; generically, any ex post incentive compatible mechanism is constant almost everywhere as a function of reports in their environments with two alternatives. As Bikhchandani (2006) points out, this generic constancy fails when goods are private, but it has remained unclear how efficient an ex post incentive compatible mechanism can be.

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We need virtually no assumptions on preferences to obtain the result above. In particular, preferences may violate concavity, gross substitutability, and the multidimensional version of the single-crossing condition (as in Bikhchandani, 2006). Instead, we need certain continuity and non-atomicity conditions on utility functions as functions of signals, but we can allow any sort of complementarity in utility functions as functions of a consumption bundle. In this sense, the present paper studies a larger preference domain compared to the literature on finite-economy competitive equilibrium relying on the gross substitute condition (Gul and Stacchetti, 1999; Ausubel, 2006). The approximating mechanism also asymptotically achieves competitive-equilibrium prices and resulting revenues. This feature is desirable in light of revenue considerations in the auction literature. The Vickery auction seems satisfactory for its efficiency and strategy-proofness, but it sometimes generates unacceptably low revenues regardless of how valuable auctioned items are (Ausubel and Milgrom, 2002). This revenue issue motivates Day and Milgrom (2008) to introduce core-selecting package auctions, arguing “[a]uctions that select core allocations with respect to reported values generate competitive levels of sales revenues at equilibrium.” This paper implicitly employs a more straightforward definition of competitiveness with a strong incentive property: The revenues in competitive equilibria are approximately achievable with an ex post incentive compatible mechanism. Note that core-selecting package auctions are efficient but not fully incentive compatible.

1.2

Application 2: One-Sided Matching

Our method also applies to allocation problems without monetary transfers. Agents may or may not demand multiple units. Examples of such problems thus include both school choice and course allocation. We approximate an extension of the pseudomarket mechanism introduced by Hylland and Zeckhauser (1979). The authors define competitive equilibria in hypothetical markets where agents with single-unit demands 5

purchase lotteries of goods using pseudo-money. We extend their equilibrium concept (hereafter, the HZ equilibrium) to the case with a continuum of agents who may have multi-unit demands, establishing the following: A finite-market mechanism exists that is feasible, ex post individually rational, ex post incentive compatible, asymptotically efficient, and asymptotically envy-free in large finite markets. The approximation method overcomes a technical difficulty that arises in extending the HZ equilibrium to multi-unit allocation problems. The HZ equilibrium with single-unit demands relies on the Birkhoff–von Neumann theorem for its foundation to find a joint distribution consistent with the individual lotteries, or marginal distributions, that the equilibrium specifies. This theorem does not globally extend to multi-unit demand, although Budish et al. (2013) partially extend the HZ equilibrium to multi-unit demand with a restricted preference domain that precludes certain complementarities. As previously mentioned, we do not impose such restrictions on preferences in the present paper. In addition to Budish et al. (2013), a number of papers propose mechanisms for ¨ multi-unit assignment problems. S¨onmez and Unver (2010) propose a mechanism that incorporates ideas of sealed-bid auctions and the deferred acceptance algorithm, but their mechanism may be inefficient. Budish (2011) proposes a pseudo-market mechanism with desirable properties in efficiency and fairness, allowing infeasible allocations. Kojima (2009) and Peivandi (2013) extend the probabilistic serial mechanism (Bogomolnaia and Moulin, 2001) to the case with multi-unit demands. Both the original and its extensions are efficient and envy-free in the ordinal sense, while the present paper studies cardinal properties. These proposed mechanisms are practically tractable, but none of them are comparable to the HZ equilibrium with single-unit demand. Independently of the present paper, Peivandi (2013) also proposes a mechanism that approximately achieves HZ allocations with private values. His mechanism

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is exactly feasible, envy-free, and ex post individually rational, as well as approximately efficient and strategy-proof. The present paper’s mechanism is exactly ex post incentive compatible but only approximately envy-free. Even when agents have single-unit demands and private values, it has been unclear how efficient strategy-proof mechanisms can be with satisfactory fairness properties. This question arises from the impossibility theorem by Zhou (1990), which states that exact efficiency is never achievable with a strategy-proof mechanism with a weak fairness condition, even weaker than equal treatment of equals. Positive results are known in the context of ordinal properties: efficiency and envy-freeness are asymptotically achievable in the ordinal sense by strategy-proof mechanisms, in particular RP (Che and Kojima, 2010; Liu and Pycia, 2012). This paper instead studies cardinal properties.

1.3

Related Literature on Large Markets

The present study complements the work by Azevedo and Budish (2012) that also studies incentives and approximation in finite and infinite markets. The authors start from a finite-market mechanism that is exactly envy-free or has a sequence of approximate Bayesian equilibria for all possible priors. They then construct an infinitemarket incentive compatible mechanism that is consistent with the given finite-market outcomes. Unlike Azevedo and Budish (2012), the present paper studies exact (ex post) incentive compatibility in finite markets. Of note, the present study constructs finite-market mechanisms from an infinite-market one, their opposite. The present paper also contributes to the literature on information aggregation in large auctions with interdependent values (e.g., Pesendorfer and Swinkels, 1997, 2000; Kremer, 2002; Reny and Perry, 2006; Matsushima, 2008; Kojima and Yamashita, 2014). These works study single-dimensional or finitely many signals, except that Pesendorfer and Swinkels (2000) allow two-dimensional ones. In the present paper, signals may have any finite number of dimensions. 7

Several papers study incentives in competitive equilibrium with large populations. Roberts and Postlewaite (1976) and Jackson and Manelli (1997) study incentives in the competitive equilibrium mechanism where agents report their demand. C´ordoba and Hammond (1998) and Kovalenkov (2002) propose (approximately) strategy-proof mechanisms that converge to competitive equilibrium outcomes. C´ordoba and Hammond’s mechanism is not fully but asymptotically strategy-proof, while Kovalenkov’s is not fully but asymptotically feasible.2 In contrast, the present paper proposes a mechanism that is always both feasible and ex post incentive compatible regardless of market size. Gul and Postlewaite (1992) and McLean and Postlewaite (2002, 2003, 2004, 2015) study approximation problems with interdependent values and Bayesian incentive compatibility. Gul and Postlewaite (1992) and McLean and Postlewaite (2002, 2003) construct mechanisms with ex post individual rationality and approximate efficiency in pure exchange economies with asymmetric information.3 McLean and Postlewaite (2004, 2015) achieve the approximation of the private-value Vickrey–Clarke–Groves mechanism with exact Bayesian and approximate ex post incentive compatibility. All of the above papers assume finitely many states and types, which preclude subtle difficulties in approximation. The present paper allows continuously many multidimensional states and types with a stronger notion of incentive compatibility. Segal (2003) studies optimal auctions with an unknown distribution that generates i.i.d. signals. He shows that optimal revenues with the unknown distribution and those with known distributions converge. In his model, goods are homogeneous and agents have single-unit demand and (correlated) private values, whereas the present paper allows heterogeneous goods, multi-unit demands, and interdependent values. 2

C´ ordoba and Hammond (1998) also propose two alternative mechanisms: a feasible, strategy-

proof one with asymptotic individual rationality and an individually rational, strategy-proof one with asymptotic feasibility. 3 McLean and Postlewaite (2002) also has an approximation theorem with inefficient target allocations.

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2

Illustrating Example

This section illustrates the main idea of the paper with an auction example. There are m identical indivisible items and n (= 2m) agents. Each agent i receives private signal si = (αi , βi ) ∈ R2 . Each agent i receives the expected utility of uni (zi , ti |sn ) = zi vin (sn ) − ti when the agent obtains an item with probability zi with payment t in expectation. Here, sn = (s1 , . . . , sn ). Agents never consume multiple items. Agent i’s valuation n

vin (sn ) = αi +

1X βj n j=1

on an item consists of a private-value component αi and a common-value component Pn 1 j=1 βj . The summand βi = θ + δi consists of common state θ and idiosyncratic n noise δi and no agent directly observes θ or δi . The random variables αi , δi , and θ are all independently and uniformly distributed on [0, 1]. In this environment, no ex post incentive compatible mechanism achieves full efficiency. To see this, first observe that efficient allocation must give an object to the agents with αi in the top half. In other words, agent i wins an item if αi is n med = (αj )j6=i . If lower, agent i loses for sure. , the median of α−i higher than α−i

Second, in an ex post incentive compatible mechanism, agent i’s payment does not depend on si given zi and sn−i . Otherwise, after learning sn−i , agent i wishes to switch to a message that gives a cheaper payment with zi unchanged, violation of ex post incentive compatibility (Chung and Ely, 2002). Thus, we can write agent i’s payment as a function pi (zi ; sn−i ). Let pi (s−i ) = pi (1; sn−i ) − pi (0; sn−i ) denote the difference in the payments when zi = 1 and 0. To complete the argument, consider two values, (1)

(2)

med med med αi = α−i + ε and αi = α−i − ε, slightly above and below α−i . Ex post incentive P P (1) (2) compatibility requires αi + n1 j βj ≥ pi (sn−i ) and αi + n1 j βj ≤ pi (s−i ). By

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taking the limit ε → 0, " # n X X 1 1 βi med med + pi (sn−i ) = α−i + βj = α−i βj + . n j=1 n j6=i n This is a contradiction because pi (sn−i ) cannot depend on agent i’s signal si , in particular its second element βi . In contrast, full efficiency is achievable in a hypothetical economy with a continuum of agents by implementing full-information competitive equilibrium. As the market size n becomes infinity large, the valuation vin (sn ) on an item converges to v ∞ (αi , θ) = αi + θ + 1/2. Also, the empirical distribution of αi converges to the uniform distribution on [0, 1] with no aggregate uncertainty.4 Since the valuations of the agents are uniformly distributed over [θ + 1/2, θ + 3/2], its median p∞ (θ) = θ + 1 clears the market as a unique full-information equilibrium price. That is, the agent with a valuation in the top half purchase one item and thus the total demand is half the population, equal to the total supply. This allocation maximizes the surplus and thus is efficient. The full revelation of information might sound questionable, but with the infinitely large population, the following simple direct mechanism implements the above fullrevelation outcome in an incentive compatible manner. From truthful reports, the mechanism estimates the value of the state θ from the fact that the mean of βi equals θ + 1/2. The estimate is almost surely accurate because of the infinite samples. The ˆ of an item. Agents mechanism announces the estimated state θˆ and the price p∞ (θ) with αi > 1/2 purchase an item, knowing their valuations v ∞ (αi , θ) = αi + θ + 1/2 and the price p∞ (θ) = θ + 1. This procedure induces truthful reports, even at the ex 4

There are measure-theoretical difficulties in modeling a continuum of i.i.d. random variables

(Feldman and Gilles, 1985; Judd, 1985). To avoid such complications, this paper mechanically defines infinite-market mechanisms without explicitly modeling information structures for a continuum population. Instead, we unambiguously define finite markets and utilize infinite-market mechanisms as mathematical objects to be approximated. This approach suffices and works for our purpose even though infinite-market mechanisms do not properly define a game.

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post stage, because no individual agent can manipulate the estimation. This argument is approximately valid in large but finite economies. The infiniteeconomy competitive equilibrium above can be approximated by the following mechanism, which generalizes the idea of the random priority mechanism: (i) Each agent i reports sˆi = (ˆ αi , βˆi ) ∈ S ≡ [0, 1] × [0, 2] simultaneously. (ii) A priority  over the agents is randomly chosen with an equal probability. (iii) Agent i receives an object and pays pni (ˆ sn−i ) = if

1 1 Xˆ βj + n − 1 j6=i 2

(a) vi (ˆ sn ) > pi (ˆ sn−i ), and X (b) 1 ≤ m − max 1{vjn (s0i , sˆn−i ) > pnj (s0i , sˆn−{i,j} )}. 0 si ∈S

ji

Otherwise, agent i does not receive or pay anything. Here, 1{· · · } denotes an indicator function. This mechanism is built upon three ideas. The first idea is the estimation of the competitive equilibrium price. As n → ∞, the price pni asymptotically and uniformly converges to the competitive price p∞ . However, these approximately market-clearing prices pni do not necessarily clear the market. We thus need some rationing rule, in this case the random priority; this is the second idea. As the final idea, in order to induce truthful reports, the mechanism needs to minimize the room for manipulation. For this purpose, this mechanism does not allow agents to manipulate their choice set. That is, both the price and the feasibility constraint, (b), are independent of the report from agent i. Agents thus truthfully report their signals; otherwise, the mechanism may choose a suboptimal choice based on wrongly calculated valuations. In other words, agents are optimally obedient as if they were price takers in competitive markets where they cannot manipulate their budget sets. We explain the idea behind step (iii) in more detail. The price pi (ˆ s−i ) estimates the full-information market-clearing price that does not involve agent i’s signal to preserve 11

incentive compatibility. Condition (a) simply ensures that it is individually rational sn−i ). Condition (b) ensures feasibility, for agent i to buy an object at the price pni (ˆ while preserving ex post incentive compatibility at the same time. Apparently, the feasibility constraint can be simplified to the following more straightforward condition: 1≤m−

X

1{vjn (ˆ sn ) > pnj (ˆ sn−j )}.

ji

This condition, however, may allow agents to profitably misreport their signals, demonstrated by the following simple example with m = 1. Suppose s1 = s2 = (0.6, 0.4). With the truthful reports, the more prioritized agent wins the object because v1 (s1 , s2 ) = v2 (s1 , s2 ) = 1 are greater than p1 (s2 ) = p2 (s1 ) = 0.9. Agent 1 can profitably deviate to report sˆ1 = (0, 2), preventing agent 2 from winning because v2 (ˆ s1 , s2 ) = 1.3 < 2 = p2 (ˆ s1 ). In contrast, this misreport does not alter agent 1’s estimated valuation v1 (ˆ s1 , s2 ) = 1 or personalized price p1 (s2 ) = 0.9. Consequently, agent 1 wins the item for sure and receives twice the utility of the truthful report. This generalized random priority mechanism is asymptotically efficient because all the prices and valuations appearing in step (iii) uniformly converge to their infinitemarket counterparts. All of the prices appearing in (iii) are uniformly close enough Pˆ from pn (sn ) = n1 βj + 12 , which almost surely converges to p∞ (θ). That is, max0 pnj (s0i , sn−{i,j} ) − p∞ (θ) ≤ max0 pnj (s0i , sn−{i,j} ) − pn (sn ) + pn (sn ) − p∞ (θ) i,j,si i,j,si " # X 1 1 ≤ max0 · βj + βk + pn (sn ) − p∞ (θ) i,j,si n n(n − 1) k6=j almost surely converges to 0. Similarly, the valuation vj (s0i , s−i ) also uniformly converges to v ∞ (αj |θ). These two facts suggest that condition (a) also uniformly converges to the infinite-market assignment rule, v ∞ (αi |θ) > p∞ (θ). Furthermore, the feasibility condition (b) virtually disappears as m goes to ∞. Even when agent i is

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at the bottom 1% in the priority ordering, the following is very likely to occur: X 1 1{vjn (s0i , sn−i ) > pnj (s0i , sn−{i,j} )} · max m s0i ∈S ji X 2 1{vjn (s0i , sn−i ) > pnj (s0i , sn−{i,j} )} ≤ · max 0 s ∈S n i ji Z 1 1 1{v ∞ (α|θ) > p∞ (θ)} dα = 2 · 0.99 · = 0.99. → 2 · 0.99 2 0 That is, the right-hand side of condition (b) goes to infinity with probability close to 1. We see more formal arguments in subsequent sections, with the general model presented in the next section.

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The Model

3.1

Finite Markets

A sequence of finite markets is indexed by the number of agents, n ∈ {2, 3, 4, . . .}. Each agent i ∈ {1, . . . , n} receives a signal si ∈ S ≡ [0, 1]dS , independently and identically distributed with density f (si |θ) conditional on state θ ∈ Θ ≡ [0, 1]dΘ .5 The density f (s|θ) is positive and continuous on S × Θ, and any two different states θ, θ0 ∈ Θ are statistically distinguishable, i.e., f (s|θ) 6= f (s|θ0 ) for some s ∈ S. The prior on θ is given by a positive continuous density f (θ). The conditional density of θ given sn = (s1 , . . . , sn ) is denoted by f (θ|sn ); explicitly, n

f (sn |θ)f (θ) , f (sn |θ0 )f (θ0 )dθ0 Θ

f (θ|s ) = R 5

More generally, S and Θ can be compact subsets of Euclidean spaces such that (i) S has a

nonempty interior whose closure coincides with S and (ii) either Θ has a positive Lebesgue measure or Θ is a finite set. The function f (θ) is a continuous density when Θ has a positive measure. When Θ is finite, f (θ) is the probability of θ. In either case, f (θ) > 0 for all θ ∈ Θ. The results and proofs extend to the general case without fundamental modification.

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where f (sn |θ) =

Qn

i=1

f (si |θ).

There are L types of indivisible goods. The per-capita supply of type-` goods is q` ∈ (0, ∞], meaning that the total supply of good ` is bnq` c, where b·c stands for the floor.6 The set of all possible consumption bundles is X ⊆ {0, 1, 2, . . . , x¯}L , where a positive integer x¯ signifies the maximum consumption level. The set X contains the zero vector, 0. Agents have a common quasi-linear expected utility function u∞ (xi , ti |si , θ) = v ∞ (xi |si , θ) − ti , where xi ∈ X is a consumption bundle and ti ∈ R is the amount of payment. The function v ∞ (xi |si , θ) is continuous with respect to si and θ. Assumption 1. The set of si ∈ S such that v ∞ (xi |si , θ) − v ∞ (x0i |si , θ) = a has Lebesgue measure 0 for all θ ∈ Θ, distinct xi , x0i ∈ X, and a ∈ R. In other words, an agent almost surely strictly prefers one of the two different choices, (xi , ti ) and (x0i , t0i ), when the true state θ is revealed. This assumption is satisfied, for example, if hxi ,x0i ,θ (si ) = v ∞ (xi |si , θ) − v ∞ (x0i |si , θ) has a continuous first-order derivative that is nonzero at every si ∈ S whenever xi 6= x0i . The welfare of agents is evaluated conditional on signals sn . Define uni (xi , ti |sn ) and vin (xi |sn ) as the expected values of u∞ (xi , ti |si , θ) and v ∞ (xi |si , θ), respectively, conditional on sn . That is, vin (xi |sn )

Z =

v ∞ (xi |si , θ)f (θ|sn )dθ

Θ

and uni (xi , ti |sn ) = vin (xi |sn ) − ti . These functions, u∞ (·|si , θ), v ∞ (·|si , θ), uni (·|sn ), and vin (·|sn ), naturally extend to lottery spaces as expected utility functions. 6

The proportional supply can be slightly weakened to the condition that q`n /n converges to a

positive number. Here, q`n signifies the supply of type-` goods with n agents.

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A deterministic allocation (xn , tn ) is an element of X n × Rn . It is feasible if x1 + · · · + xn ≤ bnqc and transfer-free if t1 = · · · = tn = 0. A random allocation is an element of ∆(X n × Rn ). Here, ∆(T ) denotes the space of lotteries (i.e., Borel probability measures) on T endowed with the topology of weak convergence. The expected utility from a random allocation λn is simply written as uni (λn |sn ). n n n A (finite-market) mechanism g = {g n }∞ n=2 consists of mappings g : S → ∆(X ×

Rn ). A mechanism is feasible if g n (sn ) is feasible for all n and sn . A mechanism is transfer-free if g n (sn ) is transfer-free for all n and sn . A mechanism is ex post incentive compatible if

uni g n (sn ); sn ≥ uni g n (ˆ si , sn−i ); sn for all n, i, sn , and sˆi . A mechanism is ex post individually rational if uni (g n (sn ); sn ) ≥ 0 for all n, i, and sn . Probability Space and Asymptotic Properties To avoid ambiguity in probabilistic statements, we clarify that the underlying probQ ability space is (S ∞ × Θ, F, P), where S ∞ = ∞ i=1 S. The conditional density f (si |θ) naturally induces the conditional probability measure Pθ on S ∞ and then the unconditional one, P. The σ-algebra F is the completion of the Borel σ-algebra with respect to P. We are ready to define asymptotic properties. Given a mechanism {g n }, the size of envy at sn is measured by

 max uni gjn (sn ); sn − uni gin (sn ); sn ∨ 0, i,j

where x ∨ y = max{x, y} and gjn (sn ) is the marginal distribution of (xj , tj ) associated with the joint distribution g n (sn ). The mechanism is asymptotically envy-free if the size of envy, as a function of sn , almost surely converges to 0 as n → ∞. Efficiency is measured by surplus when monetary transfers are available. The 15

maximally achievable total surplus (per capita) is n n X 1X n n vi (xi |s ) subject to xi ≤ nq S (s ) = max xn ∈X n n i=1 i=1 n

n

at sn . The size of surplus loss at sn is defined as n

1X n n n n S (s ) − v (g (s )|s ), n i=1 i n

n

given a finite-market mechanism {g n }. The mechanism is asymptotically surplusmaximizing if the size of surplus loss almost surely converges to 0. We finally define approximate efficiency when monetary transfers are unavailable. For ε ∈ [0, 1], a random allocation λn ∈ ∆(X n ) is ε-Pareto efficient at sn if there exists a (potentially infeasible) random allocation λn∗ ∈ ∆(X n ) such that (1 − ε)vi (λn∗ |sn ) ≤ vi (λn |sn ) for all i and no feasible random allocation λn∗∗ ∈ ∆(X n ) Pareto-dominates λn∗ , i.e., vi (λn∗∗ |sn ) ≥ vi (λn∗ |sn ) for all i and this inequality is strict for some i. Given a mechanism {g n }, the size of Pareto inefficiency at sn is the infimum of ε ∈ [0, 1] such that g n (sn ) is ε-Pareto efficient. The mechanism {g n } is asymptotically Pareto efficient if the size of Pareto inefficiency almost surely converges to 0.

3.2

Infinite Markets

We now define mechanisms and related concepts for the corresponding markets with a continuum of infinitely many agents. Such a market has no aggregate uncertainty conditional on state θ. The conditional density of the agents with signal s relative to the whole population is f (s|θ), to make infinite markets consistent with finite markets. Agents receive a private signal s ∈ S, as in finite markets, but the distribution of s has no aggregate uncertainty given state θ ∈ Θ.

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In infinite markets, a (random) allocation is a Lebesgue-measurable mapping from S to ∆(X). An allocation z ∞ is feasible at state θ if Z E[z ∞ (s)]f (s|θ) ds ≤ q, S

where E[z] =

P

x∈X

x · z(x) for lotteries z ∈ ∆(X).7

In this paper, an (infinite-market) mechanism is defined as a pair g ∞ = (z ∞ , t∞ ) : S ×Θ → ∆(X)×R of Lebesgue-measurable mappings. That is, an agent with signal s receives lottery z ∞ (s|θ) and pays t∞ (s|θ) when the state is θ. A mechanism is feasible if z ∞ (·|θ) is feasible at any state θ. A mechanism is transfer-free if t∞ (s|θ) = 0 for all s ∈ S and θ ∈ Θ. A mechanism is ex post incentive compatible if

u∞ g ∞ (s; θ) s, θ ≥ u∞ g ∞ (ˆ s; θ) s, θ for all s, sˆ ∈ S, and θ ∈ Θ. A mechanism is ex post individually rational if u∞ (g ∞ (s; θ)|s, θ) ≥ 0 for all s ∈ S and θ ∈ Θ. A mechanism has bounded subsidies if inf s,θ t∞ (s; θ) is finite. The concept of budget sets plays a central role in this paper. A budget set B is a finite set of pairs (z, t) ∈ ∆(X) × R such that (0, 0) ∈ B. Let B denote the set of budget sets. An (infinite-market) budget rule is a mapping B ∞ : Θ → B. A budget rule B ∞ implements a mechanism g ∞ = (z ∞ , t∞ ) if (z ∞ (s; θ), t∞ (s; θ)) ∈ arg max u∞ (z, t|s, θ) (z,t)∈B ∞ (θ)

for all s ∈ S and θ ∈ Θ. When B ∞ implements g ∞ , we simply state that (g ∞ , B ∞ ) is an (infinite-market) mechanism. Note that (g ∞ , B ∞ ) is automatically ex post incentive compatible and individually rational. 7

This definition of feasibility implicitly assumes that the group of agents with a particular signal

s is also a continuum. This interpretation makes E[z(s)] not only the expected value but also the total consumption within this group. The total consumption of the whole market is now expressed R as the double integrals S E[z(s)]f (s|θ) ds.

17

In the auction example in the previous section, we employ the Walrasian budget set ∞ Bp(θ) = {(x, t) : x · p(θ) − t = 0}

as the infinite-market budget set. A more general example follows: Example 1. Consider a deterministic mechanism g ∞ = (x∞ , t∞ ). For each θ, suppose that g ∞ (·|θ) constitutes a full-information competitive equilibrium, together with some price vector p(θ) ∈ [0, ∞)L in the following sense: (i) Market clearing:

R

x∞ (s; θ)f (s|θ)ds ≤ q and p` (θ) = 0 if the inequality is strict

in the `-th dimension. (ii) Optimality: (x∞ (s; θ), t∞ (s; θ)) is a solution of the following utility maximization ∞ . problem: max u(x, t|s, θ) subject to (x, t) ∈ Bp(θ)

The first part of the market clearing condition is identical to the feasibility. The ∞ implements the mechanism g ∞ . See optimality condition implies that B ∞ (θ) = Bp(θ)

Section 6 for more details.

4

Generalized Random Priority Mechanism

This section introduces the generalized random priority mechanism (GRP). This mechanism is always feasible and ex post incentive compatible after properly configured. Although it rarely achieves full efficiency, GRP may asymptotically achieve efficiency and other desiderata when associated budget sets are appropriately designed. GRP generalizes the random priority mechanism (RP). This original mechanism is defined by the following procedure: (1) RP randomly chooses a priority  over the agents with equal probability. (2) Agents simultaneously report their preferences.

18

(3) Based on the reported preferences, the most prioritized agent receives the best choice from the available bundles. Next, the second prioritized agent obtains the most preferred bundle from the remaining items. This procedure continues until the bottom of the priority ranking. With private values, the above rule embraces two important features generalized in GRP. First, RP effectively lets agents choose their favorites in their turns. Hence, agents have no incentive to misreport their preferences to avoid a suboptimal choice being selected. Second, RP feasibly allocates objects at any stage of its sequential assignment procedure. Consequently, the final allocation is also always feasible. Built upon these two observations, GRP generalizes the RP algorithm by flexibly controlling available choices as follows: (1) GRP randomly and independently chooses a priority  over the agents and randomization devices rn = (r1 , . . . , rn ) ∈ (0, 1)n . Each priority occurs with equal probability. The randomization devices are uniformly drawn. (2) Each agent i ∈ {1, . . . , n} simultaneously submits a report sˆi ∈ S. (3) The mechanism sequentially determines (xi (, ri ; sˆn ), ti (, ri ; sˆn )) from the top of the priority ordering  as follows:8 (a) GRP finds a solution yi (, ri ; sˆn ) = (zi (, ri ; sˆn ), ti (, ri ; sˆn )) of the following problem: max uni (zi , ti |ˆ sn ) subject to (i) (zi , ti ) ∈ Bin (ˆ sn−i ), and zi ,ti

n (ii) zi ∈ Cin (, ri ; sˆn−i ),

with some budget constraint Bin (ˆ sn−i ) ∈ B and feasibility constraint n Cin (, ri ; sˆn−i ) ⊆ ∆(X). If this problem has multiple solutions, the mecha-

nism uses a deterministic predetermined rule to select (zi , ti ) from them.9 8 9

Here, ri = (rj )ji and ri = (rj )ji . As a default, we let GRP select the largest solution with respect to the lexicographic order

19

(b) GRP determines an outcome xi (, ri ; sˆn ) consistently with the lottery zi (, ri ; sˆn ) by using i’s randomization device ri :
xi (, ri ; sˆn ) = xL ri ; zi (, ri ; sˆn ) . Here, xL (ri ; z) = x(k) if ri ∈ [Rk−1 (z), Rk (z)), Rk (z) =

P

0

k0 ≤k

zi (x(k ) ), and

(x(1) , . . . , x(|X|) ) is the lexicograhic order of X.10 GRP becomes RP when values are private and the constraints are given by P Bin (sn−i ) = X × {0} and Cin (sni ) = {xi ∈ X : xi + ji xj (; snj ) ≤ nq}. Although this feasibility constraint is straightforward, it needs to be strengthened in order to accommodate interdependent values, as discussed in Section 2. To handle interdependent values, we employ a stronger feasibility constraint that is hereafter used throughout the paper. To formulate the constraint, consider the unconstrained optimum: yi∗ (sn ) = (zi∗ (sn ), t∗i (sn )) ∈

arg max uni (zi , ti |sn ).

(zi ,ti )∈Bin (sn −i )

We call zi∗ (sn ) unconstrained demand. Correspondingly, define x∗i (ri ; sn ) = xL (ri ; zi∗ (sn )) as the counterpart of xi (, ri ; sn ). Henceforth, we focus on the following feasibility constraint: (i) Agent i may choose anything from the budget set if ¯ + max x 0 si ∈S

X

x∗j (rj ; s0i , sn−i ) ≤ nq,

(1)

ji

¯ is the L-dimensional vector (¯ where x x, . . . , x¯). on ∆(X) × R by identifying (z, t) ∈ ∆(X) × R with a vector (z(x(1) ), . . . , z(x(|X|) ), t). The choice of a tie-breaking rule is mostly irrelevant to the results. One exception is the measurability; see Appendix A.1. A more important exception appears in the proof of Theorem 5, where agents need to choose a cheapest choice from the optimal ones. 10 For example, if (zi (0, 0), zi (0, 1), zi (1, 0), zi (1, 1)) = (0.1, 0.4, 0.3, 0.2), then xL (ri , z) is (0, 0) if ri ∈ (0, 0.1), (0, 1) if ri ∈ [0.1, 0.5), (1, 0) if ∈ [0.5, 0.8), and (1, 1) if ∈ (0.8, 1).

20

(ii) Otherwise, agent i must choose (0, 0), no consumption with no payment. The inequality (1) strengthens the exact feasibility constraint, xi +

P

ji

xj ≤ nq,

incorporating worst-case considerations in several directions. First, agent i’s con¯ of consumption. Thus, no matsumption bundle xi is replaced by the upperbound x ter what bundle is selected as xi , the total demand never exceeds the total supply. Second, x∗i (ri ; sn ) is at least as large as the actual consumption bundle xi (, ri ; sn ), either 0 or identical to x∗i (ri ; sn ). Finally, and most important, GRP ensures the feasibility inequality holds for every possible report from agent i by taking the maximum over s0i . This treatment is to make the feasibility constraint independent of sˆi , as in the auction example in Section 2. This strong feasibility constraint guarantees that the total quantity assigned never exceeds the total supply nq. Also, this mechanism is ex post incentive compatible. Both the budget and feasibility constraints are independent of si as in the auction example. Unable to manipulate either constraint, agents can distort only their utility functions that the mechanism uses to select the optimal choices. That is, misreporting may lead to a suboptimal choice with the two constraints totally unchanged. Therefore, Z

1 X n n n n ui gGRP (s ) s = uni yi (, ri ; sn ) sn drn n!  Z

1 X n ≥ uni yi (, ri ; s0i , sn−i ) sn drn = uni gGRP (s0i , sn−i ) sn . n!  n This is the definition of ex post incentive compatibility. Here, gGRP denotes GRP.

The above argument implicitly assumes that yi (, ri ; sn ) is a measurable function of rn . To guarantee the measurability, each correspondence Bin needs to be Borel measurable as a mapping from S n−1 to the space B of budget sets, endowed with the Hausdorff metric  dH (B, B 0 ) = max max min ky − y 0 k∞ , max min ky − y 0 k∞ . 0 0 0 0 y∈B y ∈B

y ∈B y∈B

In words, dH (B, B 0 ) ≤ ε if and only if any element of B has an ε-close element in B 0 , and vice versa. Here, k·k∞ denotes the uniform norm: k(z, t)k∞ = max{maxx∈X z(x), t}. 21

When each Bin is Borel measurable, yi (, ri ; sn ) is Lebesgue measurable (Appendix A.1). Now it is natural to define a finite-market budget rule as a collection {Bin }i,n of Borel measurable mappings Bin : S n−1 → B. Theorem 1 summarizes the above arguments, adding an additional desideratum, ex post individual rationality. This property immediately follows from (0, 0) ∈ Bin (sn ). Theorem 1. With any finite-market budget rule, GRP is feasible, ex post incentive compatible, and ex post individually rational. Furthermore, yi (, rn ; sn ) is a Lebesguemeasurable function of rn and sn . Proof. Appendix A.1 proves the measurability.

5

Main Results: Approximation Theorems

We now let GRP approximate infinite-market mechanisms (g ∞ , B ∞ ), as in the auction example in Section 2. Recall in the example that finite-market prices pni (sn−i ) approximate the infinite-market equilibrium price p∞ (θ). We generalize this price convergence by interpreting the convergence as finite-market budget sets Bin (sn−i ) = {(0, 0), (1, pn−i (sn−i ))} converging to the infinite-market one B ∞ (θ) = {(0, 0), (1, p∞ (θ))}. In the current general framework, we can construct such convergent budget sets when B ∞ (·) is continuous. As in the auction example, we first estimate θ from sn−i . Any well-behaving estimation method works for this purpose; for concreteness, we choose the maximum likelihood estimator (MLE):11 θMLE (sn−i ) ∈ arg max f (sn−i |θ). θ∈Θ

We then define Bin (sn−i ) by replacing θ with θMLE (sn−i ) from B ∞ (θ):
Bin (sn−i ) = B ∞ θMLE (sn−i ) .

(2)

This finite-market budget set (almost surely) converges to the infinite-market counterpart B ∞ (θ) due to the consistency of MLE as well as the continuity of B ∞ (·). 11

When there are multiple maximizers, we choose lexicographically the largest solution.

22

Here, the continuity is crucial. If B ∞ is discontinuous at θ, even if θMLE is very close to θ, the resulting budget set B ∞ (θMLE ) may totally differ from the target B ∞ (θ). GRP approximates the infinite-market target (g ∞ , B ∞ ) with the budget rule (2). The following theorem presents selected important properties of the approximating GRP. Denote the finite- and infinite-market expected revenues per capita, respectively, by n Z 1X n Π (s ) = ti dgGRP (xn , tn |sn ), n i=1 Z ∞ Π (θ) = t∞ (s|θ)f (s|θ)ds. n

n

Theorem 2. Let (g ∞ , B ∞ ) be a feasible infinite-market mechanism. Suppose B ∞ : Θ → B is continuous.12 Then, GRP with the finite-market budget rule (2) satisfies the following: n (sn )|sn ) − u∞ (g ∞ (si |θ)|si , θ) almost surely converges to 0. (i) maxi∈{1,...,n} uni (gGRP (ii) Πn (sn ) − Π∞ (θ) almost surely converges to 0. (iii) GRP is asymptotically envy-free. If g ∞ is transfer-free, then so is GRP. Proof. See Appendix A.2 for the proof and more results. The proof reduces to the fact that the inequality in the feasibility constraint asymptotically holds in large finite markets. Recall that the feasibility inequality, ¯ + max x 0 si ∈S

X

x∗j (rj ; s0i , sn−i ) ≤ nq,

ji

contains the operation of maximization. Although we need to handle this operation and difficulties associated with it throughout the proof, we temporarily disregard this problem to illustrate na¨ıve but basic ideas of the proof. 12

The continuity as a mapping is equivalent to satisfying both upper- and lower-hemicontinuity

as a correspondence (Aliprantis and Border, 2007, Theorem 17.15).

23

We outline how to achieve the above inequality—without the max operator. In the proof, we first verify that utility functions and budget sets converge to the infinitemarket counterparts; that is, optimization problems become virtually identical to that in the infinite market when the market size is large. GRP thus chooses, as unconstrained demand zi∗ (sn ), lotteries similar to the infinite-market optimum z ∞ (si , θ) for P most agents. Consequently, the average expected demand n1 nj=1 E[zj∗ (sn )] converges R to the infinite-market average demand E[z ∞ (s, θ)]f (s|θ) ds ≤ q. In particular, n

1X E[zj∗ (sn )] ≤ 1.001q n j=1

(3)

occurs with probability almost equal to 1 in large markets. To see how this inequality works, consider an unfortunate agent i whose priority is, for example, at P the bottom 1%. By the weak law of large numbers, n1 ji x∗j (rj ; sn ) is close to P P P 0.99 · n1 nj=1 E[zj∗ (sn )]. In particular, n1 ji x∗j (rj ; sn ) ≤ 0.991 · n1 nj=1 E[zj∗ (sn )] occurs with probability nearly equal to 1. This inequality, combined with (3), implies the desired feasibility: ( ) X ¯+ ¯ − 0.991 · 1.001nq ≥ 0.008nq − x ¯, nq − x x∗j (, rn ; sn ) ≥ nq − x ji

where the last difference is positive when n is large enough. That is, even if an agent has extremely low priority, the feasibility inequality—without the max operator—is likely to hold in large markets. To handle the operation of maximization, we aim to replace x∗j (rj ; sn ) with n ∗ 0 n x∗∗ j (rj ; s ) = maxi6=j, s0i xj (rj ; si , s−i ) in the feasibility inequality without fundamen-

tally modifying the above arguments.13 Our objective now is to obtain the stronger P n ¯ + ji x∗∗ inequality x j (rj ; s ) ≤ nq. To begin with, we incorporate a similar operation in calculating errors between finite- and infinite-market objects. For example, 13

The definition of x∗∗ j slightly differs in the actual proof to avoid technical arguments on mea-

surability, but the two definitions stem from the same idea.

24

we evaluate the error size of MLE as εnMLE (sn , θ) =

sup

i, j6=i, s0j ∈S

MLE 0 n

θ (sj , s−{i,j} ) − θ .

With errors thus defined, MLE, utility functions, and budget sets robustly converge to the corresponding limits in the infinite market. The supremum operation does not obstruct the convergence: each individual signal has only a vanishingly small informational influence in markets with excessively many samples. Now we can innocuously replace x∗j with its upper bound x∗∗ j . First observe that the robust individual error of optimal choices, εnj,z (sn , θ) =

sup kz ∗ (s0i , sn−i ) − z ∞ (sj , θ)k∞ ,

i6=j, s0i ∈S

is vanishingly small for most agents (more precisely,

1 n

Pn

n j=1 εj,z

robust convergence of utility functions and budget sets.

→as 0) due to the

When εnj,z is infinitesi-

mally small, the corresponding agent’s consumption bundle x∗j (rj ; sn ) = xL (rj ; zj∗ (sn )) hardly changes even after replacing si with any s0i . In particular, x∗j (rj ; sn ) is virtually n identical to its upper bound x∗∗ j (rj ; s ). Correspondingly, their expected values, Z 1 Z 1 ∗ n ∗ n ∗∗ n E[zj (s )] = xj (rj ; s ) drj and Ij = x∗∗ j (rj ; s ) drj , 0

0

also take almost the same values. Therefore, we can replace x∗j and E[zj∗ (sn )] with ∗∗ their upper bounds, x∗∗ j and Ij , without significantly changing any value or argument.

Approximation Without Continuity It is certainly restrictive that the approximation method above requires the continuity. To relax this requirement, we introduce an additional trick: discretization of the state space Θ. We partition this space into finitely many subspaces Θ1 , . . . , ΘK . Each cell Θk has a representative element θk∗ that serves as a proxy of all the other elements θk ∈ Θk . That is, we employ B∗∞ (θk ) = B ∞ (θk∗ ) for θk ∈ Θk 25

(4)

as the budget set for state θk . The new budget rule B∗∞ becomes continuous enough— continuous almost everywhere—when the partition satisfies mild regularity conditions.14 In the actual construction, we gradually make the partition finer as n increases to reduce the size of errors due to the discretization. We prepare a sequence of ∞ partitions {Pm }∞ m=0 and construct Bm (·) as in (4). The partition to be used depends

on n: we employ the m-th partition Pm only when n is large enough to achieve desirable approximation outcomes with Pm . After constructing a partitioning rule m(n), we define Bin by
∞ Bin (sn−i ) = Bm(n) θMLE (sn−i ) , identical to the budget rule (2) with continuity, except the subscript m(n). Theorem 3. Let (g ∞ , B ∞ ) be a feasible infinite-market mechanism with bounded subsidies. There exist a finite-market budget rule {Bin } and sequence {θnP } of mappings θnP : Θ → Θ such that GRP with {Bin } satisfies the following: (i) θ˜P ≡ θnP (θ) almost surely converges to θ. n (sn )|sn )−u∞ (g ∞ (si |θ˜P )|si , θ˜P ) almost surely converges to 0. (ii) maxi∈{1,...,n} uni (gGRP (iii) Πn (sn ) − Π∞ (θ˜P ) almost surely converges to 0. (iv) GRP is asymptotically envy-free. If g ∞ is transfer-free, then so is GRP. Proof. See Appendix A.3 for the proof and more results. The use of slightly inaccurate states raises two potential concerns: infeasibility 14

This budget rule B∗∞ (·) is continuous in Θ◦k , the interior of each cell Θk . Thus, a sufficient

condition for almost everywhere continuity is that for all k, (i) the closure of Θ◦k is identical to that of Θk and (ii) the boundary of Θ◦k has Lebesgue measure 0. These two conditions are easily satisfied in the proof’s construction where open balls U1 , . . . , UK ⊆ Θ generate Θ1 , . . . , ΘK by S Θk = Uk \ `
26

and the inaccuracy itself. As for infeasibility, we can arbitrarily reduce the size of infeasibility in infinite markets by partitioning the state space fine enough: the aggregate demand is a continuous function of the inaccuracy. As a result, we can also make large finite markets almost feasible even though the infeasibility may not completely disappear. Recall that GRP translates a small size of excess demand into a small probability of rationing and hence a small reduction of utility. That is, small initial infeasibility in infinite markets just slightly reduces the welfare of agents through the random rationing. The second, more fundamental concern is that the small inaccuracy in states could lead to unintended outcomes. This concern may or may not be valid, depending on what to approximate. In other words, this issue stays outside of the approximation methodology and will be resolved, positively or negatively, only by investigating the properties of what to be approximated. In the next section, we will see two positive examples where the inaccuracy in states does not obstruct our objectives.

6 6.1

Applications Combinatorial Auctions

As discussed in the introduction, it is difficult to achieve efficiency with interdependent values. In particular, all the ex post incentive compatible mechanisms must be almost constant in generic cases (Jehiel et al., 2006). Although the constancy result is known to dissipate with private goods (Bikhchandani, 2006), it remains unclear to what extent ex post incentive compatible mechanisms can be efficient. Our approximation method provides an answer: The size of surplus loss, per capita, almost surely disappears as the market size goes to infinity. Theorem 4. A finite-market mechanism exists that is feasible, ex post incentive compatible, and ex post individually rational, asymptotically envy-free, and asymptotically surplus-maximizing. 27

Proof. This theorem is a corollary of Theorem 3 and Proposition 3 below. The target of approximation is the full-information competitive equilibrium. The following definition is slightly stronger than that of Azevedo et al. (2013), in that consumption is always deterministic in the definition below. They prove that their equilibria, and hence ours, always maximize the total surplus. Formally, surplusmaximizing allocation in infinite markets is defined as an allocation x∗ : S → X such R that v ∞ (x∗ (si )|si , θ)f (si |θ) dsi = S ∞ (θ), where Z ∞ S (θ) = sup v ∞ (x(si )|si , θ)f (si |θ) dsi . feasible allocation x:S→X

Definition 1. A (full-information) competitive equilibrium (x∗ , p∗ ) at state θ consists of an allocation x∗ : S → X and a vector p∗ = (p∗1 , . . . , p∗L ) ∈ [0, ∞)L such that (i) x∗ (s) ∈ arg maxx∈X {v(x) − p · x}. R (ii) x∗ (s)f (s|θ) ds ≤ q and p∗` = 0 if the inequality is strict in the `-th dimension. Proposition 1. A competitive equilibrium exists at any state θ. Proof. See Appendix A.4. Proposition 2 (Azevedo et al. (2013)). The allocation of any competitive equilibrium is surplus-maximizing. We employ the second approach, discretization, in approximating competitive equilibria. Hence, we need not know about properties of the equilibrium price correspondence. It is true that the equilibrium price correspondence is upper hemicontinuous and thus that the budget set is continuous in the region where an equilibrium is unique. However, the points of discontinuity may possess a positive measure, which prevents immediate application of the first approximation method. There remains a subtle issue that maximum surplus could be discontinuous between finite- and infinite markets. In particular, the maximum surplus in finite markets might be significantly greater than that in infinite markets. This possibility would nullify the approximation approach, but fortunately, this is not the case. 28

Proposition 3. lim supn→∞ S n (sn ) = S ∞ (θ) almost surely. Proof. See Appendix A.5. Theorem 4 now follows from the second approximation theorem, Theorem 3. Let n SGRP (sn ) be the average surplus the approximating mechanism generates at sn . Then P orem 3 claims that |SGRP (sn ) − S ∞ (θm(n) (θ))| converges to 0. Since S ∞ (·) is conP n tinuous, S ∞ (θm(n) (θ)) goes to S ∞ (θ). Therefore, SGRP (sn ) and S n (sn ) have the n same limit and hence the surplus loss, S n (sn ) − SGRP (sn ), needs to converge to 0,

almost surely.

6.2

One-Sided Matching

When monetary transfers are unavailable, it is naturally hard to achieve efficiency in an incentive compatible manner. Impossibility arises even with private values: no strategy-proof mechanism is efficient with an axiom weaker than equal treatment of equals (Zhou, 1990). It is also impossible to achieve efficiency with interdependent values and ex post incentive compatibility (Che et al., 2015).15 Despite these negative observations, approximate efficiency is achievable with ex post incentive compatible mechanisms, as in the first application. Theorem 5. A finite-market mechanism exists that is feasible, ex post incentive compatible, ex post individually rational, asymptotically envy-free, and asymptotically Pareto efficient. Proof. See Appendix A.7. We prove this statement by approximating equilibria of markets with pseudomoney. Hylland and Zeckhauser (1979) are the first to introduce such a market 15

Indeed, Che et al. (2015) employ a weaker incentive compatibility appropriate for their ordinal

assignment problem. They also have the triviality result with group ex post incentive compatibility.

29

mechanism with single-unit demand. Their mechanism is exactly both efficient and envy-free but manipulable with a finite population.16 We extend Hylland and Zeckhauser’s mechanism to markets with multi-unit demand and continuously many agents. Consider a pure exchange economy where lotteries are linearly priced with respect to probability shares. Each good ` has a unit price p` , and the price of a lottery z ∈ ∆(X) is p · E[z]. Given a price vector p, an agent with signal s maximizes utility with a unitary budget: max v(z|s, θ) subject to p · E[z] ≤ 1.

(5)

z∈∆(X)

Definition 2. A generalized Hylland-Zeckhauser equilibrium (or simply an HZ equilibrium) at state θ ∈ Θ is a pair (z HZ , pHZ ) that consists of an allocation z HZ : S → ∆(X) L HZ at θ and price vector pHZ = (pHZ 1 , . . . , pL ) ∈ [0, ∞) such that

(i) z HZ (s) is a solution of the problem (5); and R (ii) E[z HZ (s)]f (s|θ) ds ≤ q and pHZ = 0 if the inequality is strict in the `-th ` dimension. Proposition 4. A generalized Hylland-Zeckhauser equilibrium exists at any state θ. Proof. See Appendix A.6. To approximate the HZ equilibrium, we need to reduce the HZ budget set BpHZ = {z ∈ ∆(X) : p·E[z] ≤ 1} to a finite set, or more specifically, the finitely many extreme points of the convex polytope BpHZ . By Assumption 1, almost every agents have strict preferences over these extreme points and never choose a non-extreme point from the original set BpHZ . In approximating HZ equilibrium, we again employ the second approximation approach to avoid trying to find a sufficiently continuous equilibrium path. Instead of directly approximating HZ equilibrium with the true total supply q, we aim to 16

He et al. (2015) show that an HZ equilibrium is asymptotically strategy-proof in regular

economies, as a special case of their more general environment.

30

approximate HZ equilibrium with slightly higher total supply (1 + 1/m)q. We let 1/m gradually decrease in n and go to 0 as n → ∞. The desired asymptotic efficiency follows from two observations. First, it is very P likely that the average demand n1 ni=1 E[zi∗ (sn )] is very close to (1 + 1/m)q and thus more than q. Recall that the unconstrained demand zi∗ (sn ) is the optimal choice from Bin (sn−i ). Second, it is also highly likely that all the agents face the same price vector. ∞ (θ) is locally constant after discretization. Note that the infinite-market budget Bm

Since the individualized estimators θMLE (sn−i ) uniformly converge to θ, all the finite∞ MLE n ∞ market budget sets Bin (sn−i ) = Bm (θ (s−i )) are likely to be identical to Bm (θ).

If both of the above two situations simultaneously occur, then (z1∗ (sn ), . . . , zn∗ (sn )) and the common price vector p constitute, casually speaking, an HZ equilibrium with total supply greater than the actual supply nq. This means that the bundle of unconstrained demand cannot be Parate-dominated by any feasible allocation with the actual total supply nq. The GRP allocation gradually approaches these undominated lotteries, hence the asymptotic efficiency.

A A.1

Proofs Proof of Theorem 1

Borel measurability of yi∗ (sn ) = (zi∗ (sn ), t∗i (sn )) follows from the measurable maximum theorem (Aliprantis and Border, 2007), summarized below with appropriately reduced generality for our purpose. Lemma 1 (Theorems 18.10 and 18.19, Aliprantis and Border (2007)). Let A and T be compact subsets of Euclidean spaces. Let h : A × T → R be a Carath´eodory function; i.e., h(a, t) is continuous in a for all t ∈ T and Borel measurable in t for all a ∈ A. Let A0 be a Borel-measurable mapping from T to the space C(A) of nonempty compact subsets of A, equipped with the Hausdorff metric. Then the following hold:

31

(i) H(t) ≡ maxa∈A0 (t) h(a, t) is Borel measurable. (ii) A∗1 (t) ≡ arg maxa∈A0 (t) h(a, t) ∈ C(A) is Borel measurable. (iii) Let ψ1 , . . . , ψK : A × T → R be Carath´eodory functions. For each k = 1, . . . , K define A∗k+1 (t) = arg maxa∈A∗k (t) ψk (a, t). Then, A∗k+1 (t) is Borel measurable for all k ∈ {1, . . . , K}. In particular, the unique lexicographic maximum of A∗1 (t) is Borel mesurable. Proof. The first two statements are simplified restatements of Theorems 18.10 and 18.19 from Aliprantis and Border (2007). Repeatedly apply the second to prove the third. The measurability of yi∗ implies that Ri,k (sn ) =

P

0

k0 ≤k

zi∗ (x(k ) |sn ) is also Borel

measurable. This fact, in turn, proves that x∗i (ri , sn )

=

|X| X

x(k) · 1{Ri,k−1 (sn ) ≤ ri < Ri,k (sn )}

k=1

is also Borel measurable. Finally, ( yin (, rn , sn ) = yi∗ (sn ) · 1 x¯1L + max 0 si

n X

) 1{j  i}x∗i (ri , si , sn−i ) ≤ q (n)

j=1

P is Lebesgue measurable as a function of , rn , and sn . This is because nj=1 1{j  i}· P x∗i (ri , sn ) is Borel measurable and thus maxs0i nj=1 1{j  i}x∗i (ri , si , sn−i ) is Lebesgue measurable due to the projection theorem (Theorem 18.25 of Aliprantis and Border, 2007).

A.2

Proof of Theorems 2

We introduce a few more notations and definitions. The maximum likelihood estimator (MLE), θMLE (sn ), is defined as lexicographically the largest solution of maxθ∈Θ f (sn |θ). Let Qn be the probability measure that represents the probabilistic structure on (, r1 , . . . , rn ). That is, Qn ({} × U1 × · · · Un ) = 32

1 · mU1 · · · mUn n!

for all  ∈ P n and Borel subsets U1 , . . . , Un ⊆ (0, 1), where m is the Lebesgue measure and P n is the set of priorities (i.e., strict total orders) on {1, . . . , n}. Given a GRP with {B n } specified, we use the following notations as in Section 4: yin (, rn , sn ), zin (, rn , sn ), tni (, rn , sn ), xni (, rn , sn ), yi∗ (sn ), zi∗ (sn ), t∗i (sn ), and P x∗i (sn ). Recall Πni (sn ) = n1 ni=1 t∗i (sn ) as the expected revenue per capita. Also, given an infinite-market mechanism g ∞ = (z ∞ , t∞ ), let Z A(θ) = E[z ∞ (s)]f (s|θ)ds t¯∞ (θ) = Also, recall that Π(θ) =

R

max

(z,t)∈B ∞ (θ)

|t|.

t∞ (s)f (s|θ)ds.

Theorem 2 follows from the following slightly stronger theorem. An infinite-market mechanism is α-feasible if A(θ) ≤ (1 + α)q for all θ. Theorem 6. Let (g ∞ , B ∞ ) be an α-feasible infinite-market mechanism. Suppose B ∞ (·) is Borel measurable and continuous almost everywhere. Then, the generalized random priority with finite-market budget rule Bin (sn−i ) = B ∞ (θMLE (sn−i )) satisfies the following almost surely in P: (i) εnB (sn , θ) = supi, j6=i, s0j ∈S dH (Bin (s0j , sn−{i,j} ), B ∞ (θ)) converges to 0. P (ii) n−1 ni=1 εni,y (sn , θ) converges to 0, where εni,y (sn , θ) = maxj6=i, s0j ∈S kyi∗ (s0j , sn−j ) − y ∞ (si , θ)k∞ . ¯ n (sn ) = maxi Qni (sn ) is at most α, where Qni (sn ) = (iii) The limit superior of Q Qn {(, sn ) : yin (, rn ; sn ) 6= yi∗ (sn )}. (iv) The limit superior of maxi |uni (gBn n (sn ); sn )−u∞ (g ∞ (si ; θ); si , θ)| is at most α(kvk∞ + t¯∞ (θ)).   (v) The limit superior of maxi,j uni (yje (sn ); sn ) − uni (yie (sn ); sn ) is at most α(kvk∞ + t¯∞ (θ)), where yie (sn )

=

(zie (sn ), tei (sn ))

Z = P n ×(0,1)n

33

yi (, rn , sn )dQn (, rn ).

(vi) The limit superior of |Πn (sn ) − Π∞ (θ)| is at most αt¯∞ (θ). Before starting the proof, we show some auxiliary results. A.2.1

Convergence of Utility Functions

Lemma 2. The following almost surely converges to 0: εnv (sn , θ) =

sup x∈X, i, j6=i, s0j ∈S

n vi (x|s0j , sn−j ) − v ∞ (x|si , θ)

To prove Lemma 2, we first investigate properties of the full-information posterior. Let β(·|sn ) denote the posterior distribution of θ given sn , i.e., R Qn f (si |θ)f (θ)dθ 0 0 n β(Θ |s ) = RΘ Qni=1 i=1 f (si |θ)f (θ)dθ Θ for all Borel-measurable Θ0 ⊆ Θ. The following lemma claims that β converges to the true state in a robust manner, as an immediate consequence of Doob’s consistency theorem (see, e.g., van der Vaart, 2000). Lemma 3. For all θ ∈ Θ and a closed set Θ0 ⊆ Θ \ {θ}, β ∗ (Θ0 |sn ) = max β(Θ0 |s0j , sn−j ) 0 j, sj ∈S

almost surely converges to 0 conditional on θ. Proof. Let c = maxsj ,s0j ,θ {f (sj |θ)/f (s0j |θ)}. Then, 0

β(Θ

|ˆ sj , s˜n−j )

R f (ˆ sj , s˜n−j |θ)f (θ)dθ Θ0 R = f (ˆ sj , s˜n−j |θ)f (θ)dθ Θ R sn |θ)f (θ)dθ 2 RΘ0 f (˜ = c2 β(Θ0 |˜ sn ) →as 0 ≤c · n f (˜ s |θ)f (θ)dθ Θ

conditional on θ, where the limit follows from Doob’s consistency theorem. Proof of Lemma 2. Let U (δ, θ) = {θ0 ∈ Θ : kθ − θ0 k < δ} and ε∞ v (si , θ; δ) =

sup kv ∞ (·|si , θ) − v ∞ (·|si , θ0 )k∞ . θ0 ∈U (δ,θ)

34

The latter converges to 0 as δ → 0 since v ∞ is continuous. Let β ∗ (sn , θ; δ) = β ∗ (Θ \ U (δ, θ)|sn ). Then, almost surely, εnv (sn , θ) =

sup x∈X, i, j6=i, s0j ∈S

|vin (x|s0j , sn−j ) − v ∞ (x|si , θ)|

≤ β ∗ (sn , θ; δ) · 2kvk∞ + (1 − β ∗ (sn , θ; δ))ε∞ v (si , θ, δ) → 0 as δ → 0. A.2.2

MLE and Budget Sets

Lemma 4. For all n, θMLE is Borel measurable as a mapping from S n to Θ. Also, εnMLE (sn , θ) almost surely converges to 0. Proof. The Borel-measurability follows from Lemma 1. This lemma also finds a Borel-measurable mapping θWMLE (sn , θ) ∈

arg max

kθˆ − θk

MLE (sn ):i=1,...,n} ˆ θ∈{θ −i

for each n. Since θWMLE (sn , θ) = θMLE (sn−j ) with some j, n

1X 1 1X log f (si |θWMLE (sn , θ)) + log f (sj |θMLE (sn )) log f (si |θMLE (sn )) ≤ n i=1 n i6=j n n

C 1X log f (si |θWMLE (sn , θ)) + ≤ n i=1 n for all n, sn , and θ with a constant C = maxs,θ log f (s|θ) − mins,θ log f (s|θ). Such an approximate MLE is strongly consistent; i.e., for all θ ∈ Θ, θWMLE (sn , θ) →as θ conditional on θ (Perlman (1972), Lemma 1.1). Since θWMLE is measurable, ˜ = lim sup kθWMLE (sn , θ) − θk →as 0 max kθMLE (sn−i ) − θk i

n→∞

without conditioning. Lemma 6 serves as the continuous mapping theorem for this case. Its proof is based on Lemma 5, a probablistic uniform continuity result. 35

Lemma 5. Let X be a compact subset of Rk and (Y, d) a compact metric space. Consider a mapping g : X → Y that is continuous on a Borel subset XC ⊆ X with a positive Lebesgue measure. For all ε > 0, there exists δ > 0 and a compact set X ∗ ⊆ XC such that (i) m(X ∗ ) ≥ m(XC ) − ε and (ii) d(g(x∗ ), g(x)) < ε whenever kx∗ − xk < δ for all x∗ ∈ X ∗ and x ∈ X. Proof. Since the Lebesgue measure is regular, there exists a compact set X ∗ ⊆ XC such that m(X ∗ ) > m(XC ) − ε. For each x ∈ X ∗ , let δx =

1  inf kxk − x0 k : d(g(xk ), g(x0 )) ≥ ε/2 , 2

(6)

which must be positive by the definition of continuity. Let U (δ, x) = {x0 ∈ X : kx − x0 k < δ} be the δ-open neighborhood of x. Since X ∗ is compact, an open covering {U (δx , x)}x∈X ∗ of X ∗ has a finite sub-covering {U (δxk , xk )}K k=1 . Let δ = mink δxk , and suppose that x∗ ∈ X ∗ , x ∈ X, and kx∗ − xk < δ. Since ∗ ∗ {U (δxk , xk )}K k=1 covers X , there exists k such that kx − xk k < δxk . This implies

kx − xk k ≤ kx − x∗ k + kx∗ − xk k < 2δxk , and thus d(g(x∗ ), g(x)) ≤ d(g(x∗ ), g(xk )) + d(g(xk ), g(x)) < ε by (6). Lemma 6. Let B ∞ be a budget rule continuous almost everywhere. Then max0

i, j6=i, sj ∈S


dH B ∞ ◦ θiMLE (s0j , sn−{i,j} ), B ∞ (θ0 ) →as 0.

Proof. Lemma 5 implies εnB,k (sn , θ) =

h   i 1 + ∞ · 1 θ 6∈ Θk + 1 εnMLE (sn , θ) > 1/k k

is greater than the maximum appearing in the statement of this lemma, where Θk ⊆ Θ is a compact subset as in the statement of the lemma. Since it is Lebesgue measurable, 36

so is its infimum εnB (θ, sn ) = inf k εnB,k (θ, sn ). By Lemma 4, εnB,k →as ∞ · 1{θ 6∈ Θk }. This limit almost surely converges to 0 as k. Therefore, εnB,k and hence the maximum in the statement almost surely converge to 0. Lemma 6 is useful due to Lemmata 7 and 8, presented below. Lemma 7 shows that the two very similar budget sets generate almost the same levels of maximum utility. Lemma 8 illustrates a more subtle implication in terms of optimal choices. Lemma 7. Let B, B 0 ∈ B and let v0 : ∆(X) → [−¯ v , v¯] be a linear mapping. Define u0 : ∆(X) × R → R by u0 (z, t) = v0 (z) − t. Then, 0 u0 (y ) ≤ (|X| · kv0 k∞ + 1)dH (B, B 0 ). max u0 (y) − max 0 0 y∈B

y ∈B

Proof. By the definition of the Hausdorff metric, (z ∗ , t∗ ) = y∗ ∈ arg maxy∈B u0 (y) has an element (z∗0 , t0∗ ) = y∗0 ∈ B 0 such that ky ∗ − y∗0 k∞ ≤ dH (B, B 0 ). Hence, " # X max u0 (y) = u0 (y ∗ ) ≤ u0 (y∗0 ) + v0 (x)|z ∗ (x) − z∗0 (x)| + |t∗ − t0∗ | y∈B

x∈X

≤ max u0 (y) + (|X| · kv0 k∞ + 1)dH (B, B 0 ). 0 0 y ∈B

Lemma 8. Let B, B 0 ∈ B. Let v and v 0 be linear mappings from ∆(X) to R. Define u, u0 : ∆(X) × R → R by u(z, t) = v(z) − t and u0 (z, t) = v(z) − t. Suppose there exists y∗ ∈ B such that u(y∗ ) > u(y) + α

(7)

for all y ∈ B \ {y∗ }, where α = 2kv − v 0 k∞ + 2(|X| · kvk∞ + 1)dH (B, B 0 ). Then, ky∗ − y∗0 k∞ ≤ dH (B, B 0 ) for all y∗0 ∈ arg maxy0 ∈B 0 u(y 0 ). Proof. Suppose to the contrary that there exists y∗0 ∈ arg maxy0 ∈B 0 u(y 0 ) such that ky∗ − y∗0 k∞ > dH (B, B 0 ). By the definition of the Hausdorff metric, there exist y and y 0 such that both ky∗ − y 0 k∞ and ky∗0 − yk∞ are at most dH (B, B 0 ). However, u0 (y 0 ) − u0 (y∗0 ) ≥ u(y∗ ) − u(y) − α > 0, 37

contradicting the optimality of y∗0 . In light of Lemma 8, it is important to evaluate how often the condition (7) holds. However, it is unclear if the condition (7) is measurable when both the utility function uni (·|sn ) and budget set Bin (sn−i ) randomly fluctuate. More precisely, we wish to show that δ ∞ (s, θ) = u∞ (y ∞ (s|θ)|s, θ) −

max

y∈B ∞ (θ)\{y ∞ (s|θ)}

u∞ (y|s, θ)

(8)

is Borel measurable. In showing the desired measurability, it is useful to explore topological properties of the metric space (B, dH ) equipped with the Hausdorff metric dH . Define ρ : B → (0, ∞] by   ∞ if B is a singleton ρ(B) =  min{ky − y 0 k∞ : y, y 0 ∈ B, y 6= y} otherwise. Lemma 9. Let Bd = {B ∈ B : ρ(B) ≥ d}. (i) Bd is closed in (B, dH ) for any d ∈ R. (ii) If B, B 0 ∈ B and dH (B, B 0 ) < ρ(B)/2, then there exists an injection ι : B → B 0 such that ky − ι(y)k∞ ≤ dH (B, B 0 ) for all y ∈ B. (iii) There exists a countable set Bd such that (a) each A ∈ Bd is a Borel subset of (B, dH ), S (b) Bd ⊆ Bd , and K(A)

1 (c) for each A ∈ Bd , there exist finitely many continuous mappings yA , . . . , yA K(A)

1 A → ∆(X) × R such that B = {yA (B), . . . , yA

(B)} for all B ∈ A.

Proof. (i) We prove that the complement B\Bd is open. Suppose B ∈ B and ρ(B) < d. Find y1 , y2 ∈ B that give ky1 − y2 k∞ = ρ(B). Let r = (d − ρ(B))/2. If B 0 ∈ B and dH (B, B 0 ) < r, then ky1 − y10 k∞ < r and ky2 − y20 k∞ < r for some y10 , y20 ∈ B 0 by the definition of the Hausdorff metric. Finally, by the triangle inequality, ρ(B 0 ) ≤

38

:

ky10 − y20 k∞ ≤ ky1 − y2 k∞ + ky1 − y10 k∞ + ky2 − y20 k∞ < ρ(B) + 2r = d. Therefore B 0 ∈ B \ Bd . (ii) For each y ∈ B, the budget set B 0 has an element ι(y) such that ky − ι(y)k∞ ≤ d(B, B 0 ). This ι(·) is injective because kι(y) − ι(y 0 )k∞ ≥ ky − y 0 k∞ − ky − ι(y)k∞ − ky 0 − ι(y 0 )k∞ ≥ ρ(B) − 2d(B, B 0 ) > 0 for any distinct y, y 0 ∈ B. (iii) Let C be the space of non-empty compact subsets of ∆(X) × R. Since the metric space (C, dH ) is σ-compact and thus separable, (Bd , dH ) is also separable. Let Qd be a countable dense subset of (Bd , dH ). Let Bd = {U(d/2, B) : B ∈ Qd }, where Ud (d/2, B) = {B 0 ∈ Bd : dH (B, B 0 ) < d/2}. This set Bd is countable, consists only of Borel sets (open balls of a closed subspace Bd ), and covers Bd . We construct the desired continuous mappings yB1 , . . . , yBK for each B = {y 1 , . . . , y K } ∈ Qd . For any B 0 ∈ U(d/2, B), we can find injections as in (ii) in both directions, because ρ(B) ≥ d, ρ(B 0 ) ≥ d, and dH (B, B 0 ) < d/2. In particular, there exists a bijection ιB 0 : B → B 0 such that ky k − ιB 0 (y k )k∞ ≤ dH (B, B 0 ) for all k. Let yBk (B 0 ) = ιB 0 (y k ) for each k. Each yBk (·) is continuous because kyBk (B) − yBk (B 0 )k∞ = ky k − ιB 0 (y k )k∞ ≤ dH (B, B 0 ). A.2.3

Proof of Theorem 6

First, (i) follows from Lemma 6. Also, the first three properties imply the last three. Proof of (i)–(iii) ⇒ (iv)–(vi). First we show (iv). By Lemmata 2 and 7, |uni (yi∗ (sn )|sn ) − u∞ (y ∞ (si , θ)|si , θ)| ≤ |uni (yi∗ (sn )|sn ) − u∞ (yi∗ (sn )|si , θ)| + |u∞ (yi∗ (sn )|si , θ) − u∞ (y ∞ (si , θ)|si , θ)| ≤ εnv (sn , θ) + CεnB (sn , θ),

(9)

39

where C = (|X| · kvk∞ + 1). Let εnI (sn , θ) denote (9). Then, 0 ≤ uni (yi∗ (sn )|sn ) − uni (yie (sn )|sn ) = Qni (sn )uni (yi∗ (sn )|sn ) ¯ n (sn ) [kvk∞ − tn (sn )] ≤Q i ¯ n (sn ) [kvk∞ + t¯∞ (θ) + εnB (sn , θ)] . ≤Q

(10)

We therefore obtain |uni (yie (sn )|sn ) − u∞ (y ∞ (si , θ)|si , θ)| ≤ |uni (yie (sn )|sn ) − uni (yi∗ (sn )|sn )| + |uni (yi∗ (sn )|sn ) − u∞ (y ∞ (si , θ)|si , θ)| ¯ n (sn ) [kvk∞ + t¯∞ (θ) + εnB (sn , θ)] + εnI (sn , θ). ≤Q The limit superior of the last expression is, almost surely, at most α(kvk∞ + t¯∞ (θ)). Hence (iv). The inequality (10) is also useful to show (v). If uni (yje (sn )|sn ) < 0 then there is nothing to prove. If uni (yje (sn )|sn ) ≥ 0, then, using (10), we obtain uni (yje (sn )|sn ) − uni (yie (sn )|sn ) ≤ uni (yj∗ (sn )|sn ) − uni (yie (sn )|sn ) ≤ uni (yi∗ (sn )|sn ) − uni (yie (sn )|sn ) + CεnB (sn , θ) ¯ n (sn ) [kvk∞ + t¯∞ (θ) + εn (sn , θ)] + Cεn (sn , θ). ≤Q B B The limit superior of the last expression is, almost surely, at most α(kvk∞ + t¯∞ (θ)). Finally, we show (vi): n ∞ n 1X 1X n ∗ n 1X ∞ n n ∗ n ∞ t (si |θ) ≤ |ti (s ) − ti (si |θ)| + Q |t (s )| Π (s ) − n n i=1 i n i=1 i i i=1 ∞

1X n n ¯ n (sn ) [t¯∞ (θ)] . ≤ εi,y (s , θ) + Q n i=1 Since n−1

Pn

∞ i=1 ti (si |θ)

→as Π∞ (θ), we obtain lim supn→∞ |Πn (sn )−Π∞ (θ)| ≤ αt¯∞ (θ)

almost surely. Proof of (ii). Define δ ∞ (s, θ) as in (8). To show δ ∞ is Borel measurable, we consider the set Ωa = {(s, θ) : δ ∞ (s, θ) > a} with a fixed a ≥ 0. Note that a < 0 makes Ωa = 40

K(A)

1 S ×Θ clearly Borel measurable because δ ∞ is non-negative. Let Bd and yA , . . . , yA S∞ as in Lemma 9 (iii). Define B = k=1 B1/k and ΩA = S × {θ : B ∞ (θ) ∈ A} for each

A ∈ B. We use {ΩA }A∈B to decompose Ωa as follows: ΩaA = Ωa ∩ ΩA . Each ΩaA is a Borel set because K(A)

ΩaA

=

[ \

j k (B ∞ (θ))|s, θ) ≥ u∞ (yA (B ∞ (θ))|s, θ) + a . (s, θ) ∈ ΩA : u∞ (yA

k=1 j6=k k ` Here, u∞ (yA (B ∞ (θ))|s, θ) is Borel measurable as a function of (s, θ), because u∞ , yA , S and B ∞ are all Borel measurable on their domains. Its union Ωa = A∈B ΩaA is also

a Borel set since B is countable. Therefore, δ ∞ (s, θ) is Borel measurable. To utilize Lemma 8, let Di,k = {(sn , θ) : δ ∞ (si , θ) > 1/k} and Ekn = {(sn , θ) : 2εnv (sn , θ) + 2CεnB (sn , θ) ≤ 1/k}. Lemma 8 then implies 1Ekn · 1Di,k · εni,y (sn , θ) ≤ εnB (sn , θ), where 1Ekn and 1Di,k are indicator functions. Thus, n

n

1X n n 1X εi,y (s , θ) ≤ εnB (sn , θ) + 1Ekn · (1 − 1Di,k )εni,y (sn , θ) n i=1 n i=1 n

+ (1 −

1Ekn )

1X n n · ε (s , θ) n i=1 i,y

 X  n   1 n ≤ + (1 − 1Di,k ) + (1 − 1Ek ) 2 + 2t¯∞ (θ) + εnB (sn , θ) n i=1   →as 2 + 2t¯∞ (θ) pk (θ), (11) εnB (sn , θ)

where pk (θ) is the conditional probability of δ ∞ (si , θ) ≤ 1/k given θ. The last inequality is due to     ∗ 0 n ∞ ∞ n n ∞ ¯ ¯ εni,y (sn , θ) ≤ max ky (s , s )k +ky (s , θ)k ≤ 1+ t (θ)+ε (s , θ) + 1+ t (θ) . ∞ i ∞ i j −i B 0 j6=i, sj ∈S

The limit (11) becomes arbitrarily small as k → ∞ for all θ; therefore (ii). Proof of (iii). We construct a Borel-measurable mapping x∗∗ i that serves as a robust P 0 upperbound for xi . For zi ∈ ∆(X), let Rk (zi ) = k0 ≤k zi (x(k ) ) and let xL (ri ; zi ) =

|X| X

 
x(k) · 1 ri ∈ Rk−1 (zi ), Rk (zi ) .

k=1

41

Recall that x(1) , . . . , x(|X|) are the elements of X sorted in the lexicographic order and that x∗i (ri ; sn ) = xL (ri ; zi∗ (sn )). Now we define x∗∗ i by n ∞ x· x∗∗ i (ri ; s , θ) = xL (ri ; zi (si , θ)) + 2¯

X

 1 |ri − Rk (z ∞ (si , θ))| ≤ |X|εni,y (sn , θ) .

0
This mapping satisfies n ∗ 0 n x∗∗ i (ri ; s , θ) ≥ xi (ri ; sj , s−j )

(12)

for all two distinct i, j ∈ {1, . . . , n}, ri ∈ (0, 1), sn ∈ S n , θ ∈ Θ, and s0j ∈ S. Obviously, ¯ ≥ x∗i when |ri − Rk (z ∞ (si , θ))| ≤ |X|εni,y (sn , θ) with some 0 < k < |X|. x∗∗ ≥ x n (k) Suppose x∗∗ i (ri ; s , θ) = x . This happens only when

Rk−1 (z ∞ (si , θ)) + |X|εni,y (sn , θ) < ri < Rk (z ∞ (si , θ)) − |X|εni,y (sn , θ). The above inequalities imply x∗i (ri ; s0j , sn−j ) = x(k) because Rk−1 (zi∗ (sn )) = Rk−1 (z ∞ (si , θ)) +

X h

i 0 0 zi∗ (x(k ) |sn ) − z ∞ (x(k ) |si , θ)

k0 ≤k−1

≤ Rk−1 (z ∞ (si , θ)) + |X|εni,z (sn , θ) and similarly Rk (z ∞ (si , θ)) − |X|εni,y (sn , θ) ≤ Rk (zi∗ (sn )). Hence (12). To apply Chebyshev’s inequality to x∗∗ i , it is useful to observe n

1 X ∗∗ n η (s , θ) →as A(θ), n i=1 i where Ii∗∗ (sn , θ) =

R1 0

(13)

n ∗∗ n ∗ x∗∗ x · |X|pi with i (ri ; s , θ)dri . Since Ii (s , θ) = E[z (si , θ)] + 2¯

some pi ∈ [0, 2|X|εni,y (sn , θ)],

n

1 X

Ii∗∗ (sn , θ) − A(θ)

n

i=1 ∞

n n

1 X

X

∞ 2 1 ≤ E[z (si , θ)] − A(θ) + 4¯ x|X| · εni,y (sn , θ) →as 0.

n n i=1 i=1 ∞

42

¯ n . Note that y n (, rn ; sn ) = y ∗ (sn ) if x ¯ + Now we are ready to evaluate Q i i P

ji

n ¯/(kq∗ ), x∗∗ i (ri ; s , θ) ≤ nq, due to (12). Let q∗ = min` q` . When n ≥ x ( ) L L X X X n ∗∗ n n Qi ≤ xi,` (ri ; s , θ) > nq` ≤ Q x¯ + πi,` (sn , θ), ji

`=1

where

( n

πi,` (s , θ) = Q

n

`=1

1 X ∗∗ xi,` (ri ; sn , θ) > n ji

  ) 1 1− q` . k

Note that for all m < (1 − α − 3/k)n,   2 m 1 X ∗∗ n · I (s , θ) < 1 − q` n n ji i,` k in the event ( Hkn =

n

1 X ∗∗ n (sn , θ) : I (s , θ) ≤ (1 + α + 1/k)q n i=1 i

) .

An application of Chebyshev’s inequality (Lemma 10, shown below) proves  2 2    x¯ k 3 1 3 1 n · + α+ + πi,` (s , θ) < 1 − α − − k n nq∗2 k n in the event Hkn . Hence, Qni ≤ 1Hkn · Qni + (1 − 1Hkn ) ≤

1Hkn

·

L X

πi,` (sn , θ) + (1 − 1Hkn )

`=1  2 2    3 1 x¯ k 3 1 n + α+ + · + (1 − 1Hkn ). ≤ 1Hk · 1 − α − − k n nq∗2 k n

The last expression, independent of i, almost surely converges to α+3/k as n → ∞ due ¯n to (13). Since k can be arbitrarily large, we obtain the desired result, lim supn→∞ Q ≤ α. This proof is completed by proving the remaining lemma: Lemma 10. Let a > 0 and ϕ1 , . . . , ϕn be Borel-measurable mappings from (0, 1) R1 to [0, a]. Define µi = 0 ϕi (ri )dri . Under the probability measure Qn , for all m ∈ {1, . . . , n} and κ > 0, the probability of m 1X 1 1X ϕj (rj ) > · µj + n ji n n j6=i κ conditional on agent i ranking m-th is less than a2 κ2 /n. 43

Proof. Rewrite the sum as

P

ji

ϕj (rj ) =

Pn

j=1

hj ()ϕj (rj ), where hj () = 1{j  i}.

The conditional covariance of hj and hk is negative when i, j, and k are all distinct:  2  2 m−1 m−1 (m − 1)(m − 2) (n − m)(n − m − 1) · 1− · Cov(hj , hk ) = + (n − 1)(n − 2) n−1 (n − 1)(n − 2) n−1    2(m − 1)(n − m) m−1 m−1 − · 1− (n − 1)(n − 2) n−1 n−1 (m − 1)(n − m) ≤ 0. =− (n − 1)2 (n − 2) Hence, the conditional variance of the sum is ! n X X X a2 + Var hj ϕj ≤ j=1

j6=i

Cov(hj , hk ) · µj µk < na2 .

j,k∈{1,...,n}\{i}

The conditional mean is 1 Xm−1 m 1X · µj < · ·µj . n j6=i n − 1 n n j6=i From these calculations, we obtain the desired conditional probability by Chebyshev’s inequality.

A.3

Proof of Theorem 3

Again, we prove a stronger theorem than the one shown in the main text. We utilize the additional condition, (vii), in the proof of Theorem 5. Theorem 7. Let {αm }∞ m=0 be a sequence of non-negative numbers that converges to ∞ ∞ 0. For each m, let (gm , Bm ) be an αm -feasible infinite-market mechanism. Suppose

that there exists τ∗ ∈ (−∞, 0) such that tm ≥ τ∗ for all m ∈ N, θ ∈ Θ, and (zm , tm ) ∈ ∞ Bm (θ). Then, there exists m(·) : N → N and Borel-measurable mappings θ0P , θ1P , . . . :

Θ → Θ such that GRP with ∞ P Bin (sn−i ) = Bm(n) ◦ θm(n) ◦ θiMLE (sn−i )

satisfies all of the following almost surely in P: 44

(i) There exists N (s∞ , θ) such that εnB (sn , θ) = 0 for all n ≥ N (s∞ , θ), where εnB (sn , θ) = (ii) n−1

Pn

n n i=1 εi,y (s , θ)

max0

i, j6=i, sj ∈S


P ∞ (θ) . ◦ θm(n) dH Bin (s0j , sn−{i,j} ), Bm(n)

converges to 0, where

∗ 0 n

∞ P

.

yi (sj , s−j ) − ym(n) εni,y (sn , θ) = max (s , θ (θ)) i m(n) 0 ∞ j6=i, sj ∈S

¯ n (sn ) converges to 0. (iii) Q ∞ P P (iv) maxi |uni (yie (sn ); sn ) − u∞ (ym(n) (si , θm(n) (θ)); si , θm(n) (θ))| converges to 0.   (v) maxi,j uni (yje (sn ); sn ) − uni (yie (sn ); sn ) converges to 0.

(vi) |Πn (sn ) − Π∞ m(n) ◦ θm(n) (θ)| converges to 0. Furthermore, the above set of conditions may include the following additional condition with any sequence {cm } of positive numbers:17 Pn ∗ n n n 0 P P (vii) kAn (sn ) − A∞ i=1 E[zi (s )] and m(n) (θm(n) (θ); θm(n) (θ))k∞ < cm , where A (s ) = Z ∞ Am (θ; θ0 ) = arg max u∞ (y|s, θ)f (s|θ) ds. ∞ (θ P (θ )) y∈Bm m 0

Proof. The aggregate demand A∞ m (θ; θ0 ) is written as X

A∞ m (θ; θ0 ) =

∞ E[z] · Dm (θ; θ0 , (z, t)),

∞ (θ ) (z,t)∈Bm 0

where ∞ Dm (θ; θ0 , y)

Z =

Y

1{u∞ (y|s, θ) > u∞ (y 0 |s, θ)}f (s|θ)ds.

∞ (θ )\{y} y 0 ∈Bm 0

is the fraction of agents who choose y from the choice set B ∞ (θ0 ).

Note that

∞ Dm (θ; θ0 , y) is continuous in θ because Z Y  ∞ lim∗ Dm (θ; θ0 , y) = lim∗ 1 u∞ (y|s, θ) > u∞ (y 0 |s, θ) · lim∗ f (s|θ) · ds θ→θ

y 0 ∈B ∞ (θ0 )\{y}

θ→θ

θ→θ

∞ ∗ = Dm (θ ; θ0 , y). 17

P Of course, m(n) and {θm } could be different after adding (vii).

45

Similarly, the expected revenue Π∞ m (θ; θ0 ) =

P

∞ (θ ) (z,t)∈Bm 0

∞ t · Dm (θ; θ, (z, t)) is contin-

uous in θ. To determine how to discretize the state space, let18  U (δ, θ0 ) = θ ∈ Θ : A∞ (θ; θ0 ) < (1+δ)q, |Π∞ (θ; θ0 )−Π∞ (θ0 ; θ0 )| < δ, and kθ−θ0 k < δ . This set U (δ, θ0 ) is an open neighborhood of θ0 because A∞ (θ; θ0 ) and Π∞ (θ; θ0 ) are continuous in θ. Let αm∗ = αm + 1/m. For each m = 1, 2, . . ., {U (αm∗ , θ)}θ∈Θ 1 constitutes an open covering of the compact set Θ; thus, there exists a finite set {θm , SK(m) K(m) k P . . . , θm } ⊆ Θ such that Θ = k=1 U (αm∗ , θm ). Now we define θm : Θ → Θ by k(θ)

P θm (θ) = θm , where k(θ) is the smallest k such that θ ∈ U (1/m, θkm ).

We can naturally construct an αm∗ -feasible, incentive compatible, infinite-market ∞ ∞ ∞ (θ) = (·) almost surely continuous: Bm∗ with a budget rule Bm∗ mechanism gm∗ ∞ P (s, θ). Thanks to Theorem 6, GRP with Bin (sn−i ) = (θ)) and g(s|θ) = ym B ∞ (θm ∞ Bm ◦ θMLE (sn−i ) almost surely satisfies all the six convergence properties. We can

therefore find Nm and a measurable set Ωm ⊆ S ∞ × Θ such that P(Ωm ) ≥ 1 − 2−m and all of the following hold for all n ≥ Nm and (s∞ , θ) ∈ Ωm : ∞ ` ∞ k (θm )), clearly implying εnB,m∗ (sn , θ) = 0. (θm ), Bm (i) εnB,m∗ (sn , θ) < mink, `6=k dH (Bm P (ii) n−1 ni=1 εni,y,m∗ (sn , θ) < 1/m.

¯ n (sn ) < αm∗ . (iii) Q m∗ e ∞ (iv) maxi |uni (yi,m∗ (sn ); sn ) − u∞ (gm∗ (si ; θ); si , θ)| < αm∗ (kvk∞ + T ) + 1/m.  n e  e (v) maxi,j ui (yj,m∗ (sn ); sn ) − uni (yi,m∗ (sn ); sn ) < αm∗ (kvk∞ + T ) + 1/m.

(vi) |Πn (sn ) − Π∞ m∗ (θ)| < αm∗ T + 1/m. Finally, define m(n) = max({m : n ≥ Nm } ∪ {0}). The almost sure convergence follows by applying the Borel–Cantelli Lemma to {(S ∞ × Θ) \ Ωm }∞ m=0 . P To obtain (vii), add kAnm∗ (sn ) − A(θ; θm (θ))k∞ < cm /2 to the above conditions

and kA(θ; θ0 )−A(θ0 ; θ0 )k∞ < cm /2 to the definition of U (δ, θ0 ). Note that kAnm∗ (sn )− 18

The strict inequality in the L-dimensional space means strict inequality at every dimension.

46

P A(θ; θm (θ))k∞ →as 0 by the strong law of large numbers.

A.4

Proof of Proposition 1

Fix θ ∈ Θ. For all p ∈ [0, ∞)L , the demand correspondence D(p; s, θ) is a singleton for almost every s ∈ S. Thus the aggregate demand correspondence Z A(p) = D(p; s, θ)f (s|θ) ds is indeed a function of p ∈ [0, ∞)L . Furthermore, A(p) is continuous because Z Z lim0 A(p) = lim0 D(p; v)f (s|θ) ds = D(p0 ; v)f (s|θ) ds = A(p0 ) p→p

p→p

by Lebesgue’s dominated convergence theorem. We focus on p ∈ [0, kvk∞ ]L because p` > kvk∞ implies A` (p) = 0. For each p ∈ [0, kvk∞ + x¯]L , define φ(p) = p + A(p) − q.

(14)

The image of φ is a subset of D = [−q ∗ , kvk∞ + x¯]L , where q ∗ = max` q` . If p` ≤ p¯ then φ` (p) ≤ p` + A` (p) ≤ kvk∞ + x¯. Also, φ` (p) ≤ p` ≤ kvk∞ + x¯ when p` > kvk∞ . From the function φ, define a mapping ψ : D → D by ψ(p) = φ(p ∨ 0).19 This mapping is continuous because so is A(p). Therefore, the mapping ψ has a fixed point pF by Brouwer’s fixed point theorem. We prove that p∗ = pF ∨ 0 constitutes a competitive equilibrium together with a consumption function x∗ (v) such that x∗ (v) ∈ D(p∗ ; v). Since p∗ ≥ pF , the market clearing condition is satisfied: Z x∗ (s)f (s|θ) ds = A(p∗ ) = q − p∗ + φ(p∗ ) = q − (p∗ − pF ) ≤ q. 19

Here p0 = p ∨ 0 is the dimension-wise maximum of p and the L-dimensional zero vector. That

is, p0` = max{p` , 0}.

47

The inequality is strict in the `-th dimension only when p∗` > pF` , which in turn implies pF` < 0 and then p∗` = 0. Therefore, the pair (x∗ , p∗ ) comprises a competitive equilibrium at θ.

A.5

Proof of Proposition 3

In proving Proposition 3, we first establish certain upper-hemicontinuity of allocations between finite and infinite markets. Because allocations are differently formulated in the two types of markets, we need to transform finite- and infinite-market allocations into mutually comparable forms, more specifically, joint distributions on X × S. In the infinite market, allocations can be identified with joint distributions and vice versa. For each θ ∈ Θ, let µθ denote the probability measure on S induced by R f (s|θ), i.e., µθ (T ) = T f (s|θ)ds for all Borel subsets T of S. Let Mθ be the set of λ ∈ ∆(X × S) such that the marginal distribution of λ on S coincides with µθ . Lemma 11. Let θ ∈ Θ. For any λ ∈ Mθ , there exists a Borel-measurable mapping z : S → ∆(X) such that Z

Z

h(z(s), s) dµθ

h(x, s)dλ =

(15)

S

S×X

for any bounded Borel-measurable mapping h : X × S → R.20 Conversely, for any Borel-measurable mapping z : S → ∆(X), there exists unique λ ∈ Mθ such that (15) holds. Proof. See Theorems F.1 and 4.20 of Pollard (2002). The next lemma achieves the desired upper-hemicontinuity. Let δx denote the Dirac measure concentrated on x, i.e., the probability measure such that δx ({x}) = 1. 20

Such z is unique up to subsets of S with Lebesgue measure 0. See Theorem F.1 of Pollard

(2002).

48

Lemma 12. Let θ ∈ Θ. Let s1 , s2 , . . . ∈ S and λn ∈ ∆(X n ) for each n ∈ {2, 3, . . .}. P If the empirical measure n1 ni=1 δsi weakly converges to µθ , then there exists a Borelmeasurable mapping z : S → ∆(X) such that Z Z n 1X n lim sup h(xi , si )dλ = h(z(s), s) dµθ n→∞ X n n i=1 S for any bounded Borel-measurable mapping h : X × S → R. Proof. Find a subsequence {λn(m) } of {λn } such that Z Z n(m) n 1 X 1X n(m) lim h(xi , si )dλ = lim sup h(xi , si )dλn . m→∞ X n(m) n(m) n→∞ X n n i=1 i=1 For each n, define κn ∈ ∆(X × S) by n

 1 X  λn κ = xi ⊗ δsi , n i=1 n

where α ⊗ β denotes the product measure of the measures α and β. Recall that xλi

n

is the marginal distribution of λn for xi . The space ∆(S × X), with the topology of weak convergence, is a compact metric space since S × X is compact. Due to the compactness, the sequence {κn(m) } has a converging subsequence {κn(m(k)) } with a limit κ∞ ∈ ∆(S × X). By the definition of weak convergence, Z Z ∞ h(x, s)dκn(m(k)) h(x, s)dκ = lim X×S

k→∞

X×S

Z = lim sup n→∞

Xn

n

1X h(xi , si )dλn n i=1

for any bounded Borel-measurable mapping h. We now can find the desired mapping, z, by Lemma 11. Now we prove Proposition 3. For each sn ∈ S n , let λ∗ (sn ) be a feasible alloP cation that maximizes the surplus ni=1 vin (xi |sn ). Since the empirical measure in Lemma 12 almost surely converges to µθ , we almost surely find an infinite-market ∞ n n allocation z(s ∞ ,θ) whose total surplus is as high as lim supn→∞ S (s ). Therefore,

lim supn→∞ S n (sn ) ≤ S ∞ (θ) almost surely. The opposite direction follows from Theorem 3. 49

A.6

Proof of Proposition 4

Define the excess demand correspondence Z A(p) = E[Z(p; v)]dλ(v) − q. Due to Assumption 1, A(p) is indeed a continuous function. Restrict the domain of A(p) to D0 = [0, 2q ∗ x¯]L and define φ as in (14). The image of φ is then a subset of D = [−q ∗ , (2q ∗ + 1)¯ x]L . The rest of the proof is identical to the proof of Proposition 1.

A.7

Proof of Theorem 5

In this proof, GRP always chooses the cheapest choices among the optimal ones in HZ budget sets. If there are multiple cheapest choices, GRP uses the lexicographically largest one. Lemma 1 still guarantees the measurability of GRP. HZ (·; θ), pHZ For each m ∈ {1, 2, . . .} and θ ∈ Θ, find an HZ equilibrium (zm m (θ))

at state θ with total supply αm q, where αm = 1/m. We then naturally construct ∞ ∞ ∞ HZ an αm -feasible infinite-market mechanism (gm , Bm ) with gm (s; θ) = zm (s; θ) and ∞ (θ) = BpHZ . Recall BpHZ = {z ∈ ∆(X) : p · E[z] ≤ 1}. Bm m (θ)

By Theorem 7, we can configure GRP so that for all ε > 0, there exists N such that the probability of the following event, Ωε,N , is more that 1 − ε: n ≥ N implies P ¯ n (sn , θ) < ε, and (i) εnB (sn , θ) = 0, (ii0 ) n1 ni=1 εni,y (sn , θ) ≤ αm(n) q∗ , (iii) kvk∞ · Q P P (vii) kAn (sn ) − A∞ m(n) (θm(n) (θ); θm(n) (θ))k∞ < αm(n) q∗ . Here, the Roman numerals are

consistent with Theorem 7. Also, recall q∗ = min` q` . n We show that the GRP allocation, gGRP (sn ), is ε-efficient when (s∞ , θ) ∈ Ωε,N

and n ≥ N . To this end, let λn∗ be a (potentially infeasible) allocation whose i-th marginal distribution is the unconstrained demand zi∗ (sn ). For example, λn∗ {xn } = m{rn : xL (ri ; zi∗ (sn )) = xi for all i}. Due to (iii), we already know n ¯ n (sn , θ))un (z ∗ (sn ); sn ) > (1 − ε)un (λn (sn ); sn ). uni (gGRP (sn ); sn ) ≥ (1 − Q ∗ i i i

50

P It remains to show that no feasible allocation can dominate λn∗ . Let θ∗ = θm(n) (θ) ∗ ∗ n and p∗ = pHZ m(n) (θ ) for notational simplicity. First observe that zi (s ) is a (cheapest)

solution of max uni (zi |sn ) subject to zi ∈ BpHZ ∗ zi

∞ ∗ (θ ) = Bin (sn ). Second observe that p∗` = 0 if An` (sn ) < because (i) implies BpHZ ∗ = B

q` . By (vii), ∗ ∗ n n A∞ m(n),` (θ ; θ ) < A` (s ) + αm q∗ < (1 + αm )q` . HZ (·; θ∗ ), p∗ ) is an (infinite-market) HZ equilibrium This implies p∗` = 0 because (zm(n)

with total supply (1 + αm )q. Now we obtain a version of the first welfare theorem. Suppose to the contrary that ˆ Pareto-dominates λn . Let zˆi be the i-th marginal distribution a feasible allocation λ ∗ ˆ Due to the first observation, p∗ · E[ˆ of λ. zi ] ≥ p∗ · E[zi∗ ] for all i and the inequality is strict for some i. Then, p∗ · q ≥

n X

p∗ · E[ˆ zi ] >

i=1

n X

p∗ · E[zi∗ ] = p∗ · An (sn ) = p∗ · (An (sn ) ∨ q) ≥ p∗ · q,

i=1

a contradiction. Here, the last equality follows from the second observation.

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