Allocating scarce resources: what money can't buy

Allocating scarce resources: what money can’t buy Daniele Condorelli University College London [email protected] This version: December 2007; First Version: November 2005

DRAFT - PLEASE DO NOT QUOTE OR DISTRIBUTE La principale finesse est de ne vouloir point du tout user de finess Descartes In this paper I study the allocation of scarce and indivisible goods (or burdens) in cases where monetary transactions are unfeasible or prohibited. More precisely, I investigate the design of ex-ante Pareto optimal mechanisms for the allocation of m identical indivisible objects to n > m heterogenous risk-neutral agents with a private value and unitary demand for the goods, in environments where transfer payments are not allowed but agents are able to signal their value through some other costly activity (queuing, advertising, exerting effort, etc.). If money was available, the optimal mechanism would achieve the first best. It would allocate the goods efficiently, trough a second price auction and then redistribute the payments back to the agents (see d’Aspremont and Gerard-Varet (1979)). However, in a setting without transferable utility, the designer of the optimal mechanism faces a trade-off between ex-post efficiency and cost minimization. Increasing the efficiency of the allocation requires that agents are screened according to their private valuation for the goods, but eliciting private information comes at a cost. For example, suppose that the designer knows that two agents have values v1 = x and v2 = y, with x > y > 0, but does not know who is who. If he (directly or indirectly) impose a cost who gets the good (at least) equal to y, then only 1 will have an interest to claim it, and the surplus will be x − y. 1

On the other hand, allocating the good at random, provides an expected x surplus of x+y 2 , which is greater than x − y whenever y > 3 . Adopting the mechanism design approach, I’m able to compute the optimal mechanism and show exactly how the optimal degree of screening and the welfare costs of incomplete information depend on the exogenous variables of the problem. In brief, the optimal mechanism is constructed as follows. First, the designer computes for each agent participating in the allocation a priority function. This function is monotone, assigns a priority level for each of the agent’s possible values, and depends on the common knowledge distribution, Fi , from which everyone (including the designer) knows that the value of i has been draw. After, the designer requests each agent to report his priority level and assigns the goods to those that exhibit the highest priority levels. Finally, he needs to make sure that everyone has an incentive to report truthfully. Therefore, he must arrange the process in such a way that those who have been awarded the goods, sustain the cost exactly sufficient to prove that they deserved the award. For the reader familiar with the literature on optimal mechanism design, I note that the priority function corresponds to the ‘ironed’ inverse hazard rate of the distribution of values. Therefore, a monotone increasing hazard rate for agent i implies that the priority function will be flat and that the unique priority level will correspond to the expectation of his private value, E[vi ]. While a monotone decreasing hazard rate implies that the priority function will be identical to the inverse hazard rate itself (see Myerson (1981)).1 The main motivation for my research comes from the issue of allocating those goods (or burdens) for which the use of markets is traditionally forbidden. A non exhaustive list of such problems includes: the allocation of scarce medical resources, procreation, adoption and custody rights, exemption from military service in wartime, demobilization of the army, admission in selective colleges, distribution of humanitarian aids in case of an emergency or permits to migrate, allocation of the right to vote and other civil 1

While in optimal auction literature, the goods go to those bidders with the highest marginal revenue, in our case what matters is the marginal surplus (see Bulow and Roberts (1989))

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rights, decisions to recruit or fire employees.2 In all these allocation problems, the candidates may possess private information that is essential in order to implement an efficient allocation, but that is costly to find out. Indeed, my results may contribute to explain why scarce medical resource and admission to college, where screening on observable characteristics is possible, are allocated through a priority system;3 why drafters in wartime are selected by lottery; 4 and why tickets for artistic events and many public services are rationed using queues.5 For an example, consider distributing a limited number of very costly tests for a rare but dangerous disease, that, if discovered in advance, can be successfully treated. Furthermore, suppose that the occurrence of the disease is highly correlated with the lifestyle (e.g. drinking behavior) and other individual but observable characteristics (e.g age), in such a way that the potential individual benefits of taking the test depend on a mixture of public and private information. To make things more concrete assume that a population of n individuals is divided in two equally large age groups (low-age and high-age) and that there are m tests to allocate, with n/2 < m < n. Say that the representative person i in the high-age class has a higher probability of contracting the disease vi , conditional on drinking a certain amount of alcohol per week a ∈ [0, 1], when compared to agent j in the low-age group. Therefore assume that vi = 1 + ai for agents with high age, while vj = aj for low-age agents. Note that the probability of having contracted the disease stands here for the benefit of taking the test. Consider the case where everyone knows the units of alcohol consumed to be uniformly distributed in both populations. Then the optimal mechanism allocates a test to all members of the high-age group class while run a lottery to distribute the remaining tests, within low-age persons. Finally, suppose instead that units of alcohol in the low-age group are 2

For a list of such institution and a description of their functioning, see Elster (1992) and (1989), Calabresi and Bobbit (1978), and Walzer (1983) 3 As in the case for the priority system for kidneys 4 As an example of an alternative system, it has been suggested that the need for military personnel can be met by requiring all young citizens to spend some time in public service, where the amount of time required in each form of service would indicate the relative desirability of each to participate in war (see “Topics: Civilian Service Instead of the Draft” in New York Times, Feb. 15, 1969, p. 28). 5 See Nichols, Smolensky and Tideman (1971) for a paper on this idea.

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distributed according to an exponential distribution with λ = 1. Then the optimal mechanism still assigns tests to all high-age group members, but can allocate the remaining tests on a first come first served based (assuming that this lead to the formation of a queue). The paper which adopts an similar model, and whose results more closely related to mine, is McAfee and McMillan (1992). McAfee and McMillan (1992) study the design of optimal cartels at auctions, under the hypothesis that bidders do not engage in side-payments. They show that a weak cartel cannot do better than having all bidders to bid the reserve price, expecting the auctioneer to randomize the good within them. My results generalize those of McAfee and McMillan (1992) in several directions, as I consider asymmetric bidders and many goods, plus I do not impose restriction on the hazard rate of the distribution of values. I want to emphasize that the extension to the asymmetric case is especially rich in insights. Bagwell and Ramey (1994) study costly advertising as a signal of product quality. In their paper they show that firms that are heterogeneous in the quality of their product may be better off by not having advertising at all, even though consumers will not be able to distinguish the good-quality product from the bad-quality one. Our results provide a more general framework for the study of costly signalling advertising. Finally, Taylor, Tsui and Zhu (2003) and Koh, Yang and Zhu (2006) compare the performance of waiting line auctions and lotteries under specific distribution of values. They have shown a tendency of lotteries to Pareto dominate waiting-line auction. My results generalize these papers by showing exactly under which conditions lottery are optimal compared to other mechanisms without transfers. The paper is organized as follows. The next two sections presents the model and the results. Section 4 discusses some possible applications and Section 5 concludes the paper.

1 1.1

The Model Agents and preferences

Let N = {1, ..., n} represent the set of agents. There are m < n goods to allocate. Each player demands one good only, and has a private valuation vi ∈ V ≡ [0, u) for it (u ∈ 1. Each agent is identified by his observable characteristics, which are common knowledge. Observable characteristics corresponds to the belief that a particular agents induce on the other agents and on the designer, about his private value. Therefore, it is a strictly increasing and continuous distribution function Fi , with density fi , and support in V . We assume that the value of one agent is not informative about the values of other agents, i.e. that Fi are distributions for stochastically independent random variables.

1.2

Implementable outcomes

The designer has to set up an allocation mechanism. An allocation mechanism is any game, played by the agents under incomplete information, whose outcomes (terminal nodes) are identified by (i) a probability distriP bution over the n agents (i.e p1 , ...pn , with pi ≤ 1 and i pi ≤ n), indicating the probability with which each of them will gets a good, and (ii) a vector of non negative costs (c1 , ..., cn ) sustained by the agents. We can imagine that taking actions in the mechanism is costly and that, given the combination of actions taken by all agents, a probability distribution over the goods obtains.6 We are assuming here, that the designer is able to impose any cost on the agents, as designing the game, implies between other things, designing the payoffs. While in the context of auctions, the idea that the auctioneer can require any payment to bidders is more natural, it may appear that some justification is needed for this assumption. By noting that the designer may ask agents to stay in queue for how much he thinks is useful, we leave the discussion on practical implementation to section 4. Clearly, we are only interested in equilibrium outcomes of the games. When a mechanism has more than one equilibrium we will assume that agents will end up playing the one that is preferred by the designer (i.e. the 6

Even if the designer may decide to set pi > 1, this is clearly never optimal and therefore we assume from the start that pi ≤ 1

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one that maximize his objective function). This seems reasonable given that the designer has the ability to communicate with agents. Furthermore, we require that each agent is willing to participate in the mechanism. If an agent does not participate, he does not get any good, but does not sustain any cost either and is utility is equal to zero. Therefore an individually rational mechanism must guarantee an expected payoff of at least zero to all agents. Finally, we require that the outcome is permutation symmetric. This essentially means that, in setting up the game, the designer can only discriminate agents on the basis of their observable characteristics. We call implementable any outcome, which is a permutation symmetric, feasible, and individually rational equilibrium of an allocation mechanism.

1.3

The Revelation Principle

The task of searching for an optimal allocation mechanisms may appear daunting. In order to find for an optimal allocation mechanism we would need to proceed by finding out which symmetric and individually rational outcomes can indeed arise as some equilibrium of some allocation mechanism. Then we could pick an optimal implementable outcome within this set. However, the problem with this approach is that potential mechanisms are countless and the problem, (if we are unlucky, i.e. when the first best is not implementable) may easily become untractable. However, by the revelation principle (see, for example, Myerson (1991)) we can restrict attention, without loss of generality, to truthful equilibria of direct mechanisms. A direct mechanism is one where agents communicate their private information to the designer (e.g. agent i choose si ∈ V ) and the designer allocate the goods and costs based on all reports (i.e. a vector s ∈ V n ). Therefore, agents’only action in this game is to report information about the value of the good to the designer. A truthful equilibrium is one equilibrium where every type of every bidders reveals truthfully his information (i.e. reporting s = v is an equilibrium). The revelation principle states that for any equilibrium outcome of any allocation mechanism, there exist a truthful equilibrium of a direct mechanism which achieves the same outcome.7 Therefore, in searching for an optimal mechanism we can restrict attention to outcomes that can be im7

Take any equilibrium of any game and transform it in a direct mechanism where the same outcome as in the original mechanism is assigned to exactly the same terminal node.

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plemented as truthful equilibria of direct mechanisms. Finally, note that there is a one to one correspondence between outcomes that are equilibrium of some allocation mechanisms and direct allocation mechanisms that have a truthful equilibrium.

1.4

Admissible direct mechanisms

We can define direct allocation mechanisms as indicating an outcome as a function of the profile of reports. Definition: A direct allocation mechanism, hp, ci, is a set of real valued functions {pi : T → IR ; ci : T → IR}ni=1 satisfying the following restrictions: pi : T → [0, 1] i = 1, ..., n

(1)

+

(2)

ci : T → IR i = 1, ..., n i=1 X pi (t) ≤ 1 ∀t ∈ T

(3)

n

Here, (1) and (2) restrict the allocation rule for the goods to be a probability measure for the m goods over the number of agents in every state of private information (remember that we are, without loss of generality, restricting each agent to get one good at most). We also require the designer to use a mechanism where equals are treated equally. Therefore we require that p and c are both permutation symmetric, at least for those agent that share the same observable characteristics. This means that ∀i, j such that ∀v ∈ V Fi (v) = Fj (v) it is true that ∀x, y ∈ V n with xi = yj , xj = yi and ∀k 6= i, j xk = yk we have: pi (x) = pj (y) , ci (x) = cj (y)

(4)

We are interested in mechanisms that have an equilibrium where everyone reveals correctly the private information. However, the designer cannot force the agents to reveal correctly their information. In playing mechanism hp, ci the (ex-post) utility to player i from announcing si when its true value is vi , while all other players announce v−i is equal to vi pi (si , v−i )−ci (si , v−i ).

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Assuming that all opponents will play truthfully (by revealing correctly their private information), expected utility at the interim stage (i.e. when the agents decide which value to report to the designer) is: Ui (vi , si ) = vi Ev−i [pi (si , v−i )] − Ev−i [ci (si , v−i )] We say that a direct allocation mechanism hp, ci is incentive compatible and individually rational if and only if, for every player, truthful revelation of the private type is a Bayesian equilibrium of the game, and the corresponding outcome generates an equilibrium payoff, Uip (vi ), greater than zero for all agents of all values:8 ∀i ∈ N ∀vi ∈ V Uip (vi ) ≡ Ui (vi , vi ) = max Ui (vi , si ) ≥ 0 si ∈V

(5)

Definition: A direct allocation mechanism hp, ci is admissible if it is symmetric, incentive compatible and individually rational. The force of the revelation principle is that, not only we now restrict the search for implementing mechanisms within direct mechanisms only, but we can characterize those outcome that arise as truthful equilibrium in a very simple way. In fact, we have the following. Lemma 1. Write Pi (vi ) = Ev−i [pi (vi , v−i )] and Ci (vi ) = Ev−i [ci (vi , v−i )]. A direct allocation mechanism hp, ci is incentive compatible and individually rational if and only if, for all i: ∀v, v ∗ ∈ V : v ≥ v ∗ Pi (v) ≥ Pi (v ∗ ) Z v ∀v ∈ V Ci (v) = vPi (v) − Pi (x)dx

(6a) (6b)

0

Proof: The result is standard and we will only sketch the proof. Uip (v) is the maximum of a family of convex functions and so it is convex. The convexity of Uip (v) implies that Pi (v) is a non decreasing function. A convex function is absolutely continuous and so is it is differentiable almost everywhere in the interior of its domain. According to (?) we must have P 0 (v)v − C 0 (v) = 0. d Uip (v) = P (v). Integrating back we obtain: Uip (t) = Uip (0) + Hence dv Rv 0 P (x)dx To conclude the proof remember that Up (t) = tP (t) − M (t) and that Uip (0) = 0 because it cannot be positive nor negative. 8

We may require also ex-post individual rationality (IR). It will be clear that the optimal direct mechanism can can always be constructed in such a way to guarantee ex-post IR

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1.5

Objective function

The designer’s task is to identify an optimal mechanism. More precisely, we are looking for an implementable outcome (with the corresponding mechanism) that maximizes the sum of the agents’ ex-ante expected utilities. This outcome will ex-ante Pareto optimal, but it will not be the only one. Formally, we look for an implementable outcome that maximizes: ( n ) X Ev vi pi (v) − ci (v) (7) i=1

The choice of a this specific Pareto optimal agreement is standard in the mechanism design literature and can be justified by appeal to the idea of the original position, often fictionally employed in social choice theories. In particular, this would be the mechanism that utilitarian agents would choice under a veil of ignorance over their identity.9 It is important to observe that, by substituting payments in terms of the allocation rule inside the objective functions (according to Lemma 1) and integrating by parts, the designer’s problem can be reduced to maximize in p, subject to feasibility (1), (2) and (3), symmetry (4), and incentive compatibility (6) the following function: " Ev

n X i=1

2

1 − Fi (vi ) pi (v) fi (vi )

#

Results

Now we are ready to for the main result. As explained, the key blocks of an optimal mechanisms are the priority functions. Before proceeding, we introduce some useful definitions that will allow us to use the ironing technique, as developed in Myerson (1981). Definition: Let Hi (x) =

Rx

1−q 0 fi (F −1 (q)) dq i

and Gi (x) = conv hHi (xi), where

conv stands for the convex hull of the function10 (i.e. Gi (x) is the highest 9

This implies that all possible observable characteristic occurs always with equal probability. This is very strong. However, it is straightforward to modify the objective function and the results to span the entire ex-ante Pareto frontier. For more on this on the next section 10 See Rockafellar (1970) (Part IV, section 17)

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d convex function such that Gi (x) ≥ Hi (x) ∀x). Let gi (x) = dx Gi (x) (where gi (x) is not defined we extend it using its right (or left) derivative). We define the priority function λi for agent i as:

λi (v) = gi (Fi (v)) Remark: Whenever the inverse hazard rate

1−Fi (vi ) fi (vi )

is monotone decreas-

1−Fi (vi ) fi (vi ) .

ing, then λ(v) = Whenever, instead, the inverse hazard rate is monotone increasing we have that λ(v) = H(1) = E[vi ]. The next proposition formally states that, given the observed distributions of values, in the optimal mechanism, the m agents with the highest priority levels obtain the goods and are required to sustain a cost which is equivalent to the minimum value they could have and still obtain the good under the allocation rule. In case of ties, if there are not sufficient goods available to accommodate everyone with the same priority level, the remaining goods are randomized with equal probability. Proposition 2. First, we put agents in ordered sets according to their ranking in terms of the priority level. Define the sets Mx (y) recursively as follows (set M0 (v) ≡ ∅):  M1 (v) ≡

 i ∈ N : λi (vi ) = max λi (vi ) i∈N

    [ Mx+1 (v) ≡ i ∈ N \ Mi : λi (vi ) = max λi (vi )   N \∪i≤x Mi i≤x

Now, define Ij (v) as the set of agents who have the highest priority levels, up to those included in the set Mj (v): I(j) = {i ∈

[

Mi (v)}

0≤i≤j

Let |X| generally denote cardinality of a set X. Then, choose the highest natural number s such that |Is (v)| ≤ m (s = 0 is also possible if M1 (v) > m). Of course, |Is+1 (v)| > m. Finally, define g = |Is (v)|, k = m − g, and r = |Is+1 (v) − Is (v)| = |Ms+1 (v)|. We have that, a mechanism hp, mi maximize (7) in the set of admissible mechanisms if and only if ∀i ∈ N, ∀v ∈ V n : 10

   1   pi (v) = k/r    0

if i ∈ Is (v) if i ∈ Ms+1 (v) otherwise vi

Z mi (v) = pi (v)vi −

pi (xi , v−i )dxi 0

Proof: According to Lemma 1, by substituting payments in terms of the allocation rule inside the objective function, integrating by parts, and setting ∀i Mi (0) = 0 (as al these constants eneter negatively in the objective function), the designer’s problem can be reduced to the following (subject to symmetry): " max

pi :V n →[0,1] i=1,...,n

subject to:

n X

Et

n X i=1

1 − Fi (vi ) pi (v) fi (ti )

#

pi (v) ≤ m ∀v ∈ V n

i=1

Pi (vi ) ≥ Pi (vi∗ ) ∀i ∈ N, ∀vi , vi∗ ∈ V : vi ≥ vi∗ Now, recalling the definitions given we can sum and subtract gi (Fi (ti )) inside the objective function: Ev

( n X i=1

=

n X i=1

 Ev

) 1 − Fi (vi ) = − gi (Fi (vi )) + gi (Fi (vi )) pi (v) fi (vi ) 



 1 − Fi (vi ) pi (v)gi (Fi (vi )) + pi (v) − gi (Fi (vi )) fi (vi )

Consider now the second term of this expression for every i: 



 1 − Fi (vi ) Ev pi (v) − gi (Fi (vi )) = fi (vi )   Z ui 1 − Fi (vi ) = p˜i (vi ) − gi (Fi (vi )) f (vi )dti = fi (vi ) 0 Z ui ui = |Pi (vi ) [Hi (Fi (vi ) − Gi (Fi (vi ))]|0 − [Hi (Fi (vi ) − Gi (Fi (vi ))] dPi (vi ) 0

(8)

11

The last line follows from integration by parts and remembering the definitions given in the text. Consider now that the first term of (9) is equal to zero: in fact, Hi (0) = Gi (0) and Hi (1) = Gi (1), because Gi (x) is the convex hull of the continuous function Hi (x) and thus they coincide at endpoints. The continuity Hi (x) follows from assuming continuous fi (x). Summing up, the objective function becomes: n X i=1

Ev [pi (v)gi (Fi (vi ))] −

n Z X i=1

ui

[Hi (Fi (vi ) − Gi (Fi (vi ))] dPi (vi )

0

Let’s assume for a moment that the mechanism in the statement of the proposition is admissible. It is easy to see that our candidate hp, mi maximize the first sum as it put all probability on players for which gi (Fi (vi )) is maximal. We will show next that the second term is equal to zero for the mechanism hp, mi. In fact, the second term must be always non negative, as ∀x ∈ [0, 1] Hi (x) ≥ Gi (x). That it is equal to zero in our case follows because Gi (x) is the convex hull of Hi (x) and so, whenever Hi (x) > Gi (x), then Gi (x) must be a straight line below Hi (x). This in turn implies that d d dxx Gi (x) = dx gi (x) = 0 and so gi (Fi (vi )) will be a constant in a neighborhood of ti such that x = Fi (vi ). So, according to our hp, mi, P (vi ) must be a constant as well in the neighborhood of vi whenever Hi (Fi (vi )) > Gi (Gi (vi )). To conclude the proof we need to show that hp, mi is admissible. That hp, mi is feasible follows immediately from its definition. To prove incentive compatibility observe first that gi (Fi (pi )) is monotone increasing: in fact, ∀v−i p(v) is increasing in vi , which implies that Pi (ti ) = Ev−i [pi (v)] is also increasing (as gi () is the derivative of a convex function). Then observe that payments are determined according to (8). Remark: Because the expected cost only depends on the other’s values, the optimal mechanism is implementable in dominant strategies. Therefore, agents will have a dominant strategy to reveal truthfully their value. Remark: It must be said that the mechanism is unique only up to a transformation of the cost allocation c that does not change the expected cost C. However, for other cost structures (when the expected cost also depend on the other’s values), the optimal mechanism is not implementable in dominant strategies. 12

Remark: The optimal mechanism is ex-post individually rational. Therefore, agents will always be better off (also ex-post) by participating in it. Remark: The optimal mechanism is not in general detail free, as knowledge of the distribution function is required by the designer. Therefore, observable characteristics are always relevant to the determination of the optimal mechanism Remark: There are other ex-ante efficient if we depart from symmetry and we put more weight on some agents. In fact the proposition would be simply modified by putting weights on the priority functions of individual agents. Before discussing how the optimal mechanism looks like in some special cases, let me introduce some further definitions and a simple lemma. Definition: A random allocation is a mechanism ha, ci such that ∀ i ∈ N, P ∀ v ∈ V Pi (v) = ai , with 0 ≤ ai ≤ 1 and ai ≤ m, and Ci (v) = 0. A random allocation, which is trivially an admissible mechanism, allocates the object randomly between the different players, regardless of individual values.11 The following lemma shows that if a direct allocation mechanism impose no cost to agents in equilibrium, than it must be a random allocation. Lemma 3. Given any admissible mechanism hp, ci if ∀i ∈ N, ∀v ∈ V Ci (v) = 0, then the mechanism is a random allocation. Proof: Suppose that, for some S and ∀i ∈ S we have that ∀v Ci (v) = 0. Then, incentive compatibility implies that ∀v Pi0 (v)v − Ci0 (v) = 0. Therefore ∀v Pi (v) = pi for some pi ∈ IR. Finally, note that if the mechanism is feasible then it must be that ∀i 0 ≤ pi ≤ 1. Remark: It follows from Lemma 2 that any (maybe non direct) allocation mechanism generates only equilibrium outcomes whose payoff is independent of the private values, whenever no agents engages in a costly activity. Remark: It follows from Lemma 2 that if no costly activity can be exerted by the agents to signal their values, the only admissible mechanisms are random allocations. 11

The reader may wonder that we have not shown how to pass from P to p

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Remark: It follows from Lemma 2 that the first best allocation (the one that the designer could implement under complete information or using transfers) is not implementable. In fact, in the first best the designer assigns the objects to the highest value agents and no costly screening is necessary. Now, consider a random allocation ha, mi. Take S, a subset of k agent P in N (the set of all agents), and r goods, when i∈S ai = r and ∀i ai = r/k we will speak of a lottery for r goods within S. When there exist a subset of m agents S, with ai = 1 for all i ∈ S, we speak of a priority mechanism. Remember that, because of symmetry, it must be that, whenever Fi ≡ Fj we have ai = aj . The following sets of remark, illustrate how the optimal mechanism look likes in some important special cases. Remark: Consider the so called regular case, where hazard rates are all monotonically increasing. In this case the optimal mechanism is a priority system, and the goods are allocated according to the expected private value as suggested by the observable characteristics of the subjects. When agents are symmetric, the mechanism becomes a lottery Remark: Consider the case where all agents are symmetric hazard rates are all monotonically decreasing. In this case the optimal mechanism is a firstarrived first-served mechanisms, where only those that are able to sustain the higher costs (for example in terms of time) get the goods. If agents are asymmetric, the mechanism will tend to be biased in favor of weak agents, in order to induce high level agents to properly reveal their valuation, even when it is very high. Remark: In general, because hazard rates can be always decreasing only in the case of an unbounded type space, efficient mechanisms will almost always include some pooling of the highest values. The following example illustrate the construction of an optimal mechanism, for the case of non monotonic hazard rates but symmetric agents. Example: Consider the case of two agents, whose values are distributed in the interval [0, 1] according to the following cumulative distribution function: 1 1 F (v) = v 2 . It follows that f (v) = 12 t− 2 , and it is easy to see that the hazard rate is decreasing in the interval [0, 0.25] and increasing elsewhere. 14

The optimal collusive mechanism, obtained using the ironing technique developed in proposition 4, assigns the object to the player with the highest value, if one or both values belong to [0, 0.0625]. The object is randomly assigned between the two players when both values are in [0.0625, 1]. Each agent pays the highest value he could have and still get the object assigned under the optimal mechanism.

3

Applications and practical implementation

[TO BE COMPLETED]

4

Conclusions

[TO BE COMPLETED]

References Bagwell, K., and G. Ramey (1994): “Advertising and Coordination,” Review of Economic Studies, 61(1), 153–171. Bulow, J., and J. Roberts (1989): “The Simple Economics of Optimal Auctions,” Journal of Political Economy, 97(5), 1060–1090. Calabresi, G., and P. Bobbit (1978): Tragic Choices. W. W. Norton & Company, New York. d’Aspremont, C., and L.-A. Gerard-Varet (1979): “Incentives and Incomplete Information,” Journal of Public Economics, 11, 25–46. Elster, J. (1989): Solomonic Judgements. Cambridge University Press, Cambridge, UK. (1992): Local Justice. Russel Sage Foundation, New York. Koh, W., Z. Yang, and L. Zhu (2006): “Lottery rather than waiting-line auction,” Social Choice and Welfare, 27, 289–310. McAfee, P., and J. McMillan (1992): “Bidding Rings,” American Economic Review, 82(3), 579–99.

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Myerson, R. (1981): “Optimal Auction Design,” Mathematics of Operations Research, 6(1), 58–73. (1991): Game Theory. Harvard University Press. Nichols, D., E. Smolensky, and N. Tideman (1971): “Discriminating by Waiting Time in Merit Goods,” American Economic Review, 61, 312– 323. Rockafellar, T. (1970): Convex Analysis. Princeton University Press. Taylor, G., K. Tsui, and L. Zhu (2003): “Lottery or waiting-line auction?,” Journal of Public Economics, 87, 1313–1334. Walzer, M. (1983): Spheres of Justice. Basic Books, New York.

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