Random Priority A Probabilistic Resolution of the Tragedy of the Commons

Random Priority: A Probabilistic Resolution of the Tragedy of the Commons Hervé Crès* H.E.C School of Management Hervé Moulin** Duke University Preliminary Draft*** May 1998 * HEC School of Management, 78351, Jouy-En-Josas Cedex, France, cres@gwsmtp.hec.fr ** Department of Economics, Box 90097, Duke University, Durham NC 27708, moulin@econ.duke.edu *** Stimulating conversations with Scott Shenker, are gratefully acknowledged Special thanks to Anna Bogomolnaia who contributed Proposition 9, and Yan Yu for some numerical simulation Random Priority: A Probabilistic Resolution of the Tragedy of the Commons A Simple Queuing Model Imagine a group of agents who would like to receive one unit of service Think of patients looking for a medical service, of customers at the post office or in a shop, etc There is a single server who needs one unit of time (say five minutes) to serve any one customer Thus, the actual service rendered may vary from one customer to the next, but the length of service is homogeneous across customers Agents differ in their willingness to wait for the service and are otherwise identical (we assume that other costs of service, e.g., monetary, are not under the control of the mechanism designer) Say that an agent is of type q if she is willing to wait at most q units of time: she prefers not to show up at all rather than waiting ( q 1) periods for service, but she prefers to wait q periods (or less) and get service rather than not showing up For simplicity we assume risk neutrality: whenever the decision to "stay in" involves a random waiting time, she will stay if the expected waiting time is at most q and leave otherwise Our final simplifying assumption is that each agent takes a binary decision, at date t 0 , to "stay in" and wait until service is provided according to whatever queuing protocol is in place, or "opt out," get no service and no wait That is we rule out the option of staying in the line for some time and leaving at a later stage before obtaining service How should we organize the queuing protocol in order to maximize efficiency and strategic simplicity of individual decision making? Efficiency in this model results from the familiar equality of marginal cost (here, waiting time) and marginal utility (here, willingness to wait): the total wait incurred when q agents are served is q ( q  1) / (because the agent served first gets to wait only one period, the one served second waits for two periods, etc.) hence marginal cost of the q-th unit is q; therefore maximization of total surplus (sum of willingness to wait of agents who get service, minus total wait) requires to serve the qe agents with highest willingness to wait, where qe is the largest number q such that there are q or more agents of type q or more We consider two simple and natural mechanisms (protocols, in the terminology of the queuing literature), in which each agent must take a binary decision (in or out) and that lead to transparent strategic equilibrium behavior Consequently, neither of these mechanisms implements full efficiency as just described, and we wish to compare their respective inefficiencies In both protocols the order in which agents are served is randomly drawn with uniform probability on all orderings In the first mechanism, average delay, the agents must decide at date t 0 to stay or leave before learning their place in the line: if x agents decide to stay, they each face an expected wait ( x 1) / (namely the average delay per agent) Say that in equilibrium qa agents decide to stay Each of them has a willingness to wait not lower than ( qa 1) / and all agents who choose to leave have it not higher than ( qa  ) / ; in turn this means that the equilibrium is found at the intersection of the demand curve and average delay curve.1 Because average cost is below marginal cost, this equilibrium entails inefficient overproduction, an example of the familiar tragedy of the commons In the second protocol, random priority, the order in which agents will be served is drawn at t 0 and revealed to the agents before they decide to stay or leave The strategic behavior here is especially simple.2 The first agent in the line stays iff his willingness to wait exceeds one period; the second in line faces a waiting time of two if the previous agent chose to stay and of one if he left: he chooses to stay or leave accordingly; the third agent in line faces a wait of periods if both agents before him in the line chose to stay, of periods if only one stayed, and of one period if they both left; and so on Our first goal in this paper is to compare the welfare performance of the two protocols, average delay and random priority The two protocols represent, in a stylized model, the familiar choice between an organized queue where everyone receives a number (random priority) and an unorganized one where the server randomly picks agents from the waiting crowd.3 Intuition and casual experience suggest that random priority is a less inefficient mechanism The formal The possibility of multiple Nash equilibria complicates the strategic analysis a bit; see Section for details The resulting undeterminacy disappears, however, in the continuous limit: see Section Each player has a dominant strategy so that, barring indifferences, there is a unique Nash equilibrium outcome The main simplifying assumption is the one-shot binary decision-making at t 0 ; agents not have the option to hang out in the queue for a while It is not hard to come up with examples where this assumption is plausible model confirms this For instance, consider the particular case of an infinitely elastic demand, namely all agents have identical willingness to wait, say 10 periods at most In average delay, either 20 or 19 agents will stay, with an average wait around 10 and almost no surplus in equilibrium By contrast, random priority is fully efficient, with the first 10 agents in the random ordering staying and the others leaving at once Yet the model also shows that average delay is not always inferior, welfarewise, to random priority A simple example where the comparison goes strongly the other way has 10 agents, of whom are willing to wait 1.1 (expected) periods, while the last agents' willingness to wait is 2.1 Using the average delay protocol, only the "efficient" agent stays because no other agent is willing to pay even the average delay for two customers; that is, the unique Nash equilibrium is fully efficient Contrast this with the expected outcome of random priority, where times out of  1)  ( 21  2) 0.2 , 10, two agents stay (one of the first and the last one) for a surplus of (11 well below the efficient surplus 2.1-1=1.1 See Section for more discussion of this example Random priority leads to inefficiencies of a different kind than average delay: in the latter, some inefficient agents are served with probability one, while in the former all inefficient agents (willing to wait one period at least) are served with some positive probability Our goal in this paper is to compare these two kinds of inefficiencies in a general model of free access to a commons Our qualified conclusion is that the random priority mechanism not only has better incentives properties than average delay (especially when the number of agents is small), it is also is generally less inefficient In order to explore systematically the welfare consequences of the two mechanisms, average delay and random priority, we shall need to describe the (probabilistic) allocation generated by the latter mechanism This turns out to be a hard combinatorial question In order to help in this comparison, we introduce a third mechanism, directly inspired from serial cost sharing (Moulin and Shenker [1992]) of which it is a probabilistic version The probabilistic serial mechanism is very closely related to random priority; in fact its (expected) outcome is Pareto superior to that of random priority, but the welfare difference is small, and vanishes in the limit model with a continuum of agents Before overviewing our results in Section 3, we generalize in Section the queuing story to a systematic model of the tragedy of the commons, hence opening a much broader range of potential applications Three Mechanisms of Free Access to the Commons A "commons" is a production process of which a given set of agents are the legitimate potential users Decentralized utilization of the commons means that each individual agent decides, independently and selfishly, to join or not in the production process (either by demanding some output or by contributing some input, or both) When the returns of the technologymarginal product or marginal costvary, decentralized utilization of the commons brings about an inefficient outcome, a difficulty already noted by Aristotle and dubbed a "tragedy" by Hardin [1968] In the standard model of the tragedy, the returns decrease (marginal cost increase) which results in overproduction at the decentralized equilibrium outcome (see Moulin and Watts [1995] for a general statement, and references therein about the tragedy) In the queuing model of Section the commons is the server, and decentralized utilization takes the "cost sharing" format namely each agent demands or unit of output (service), and the total cost (=delay) of serving the q "users" (i.e., the agents who demand 1) shared among them In the example, C( q ) q( q 1) / so that returns are decreasing An equally simple model is the dual "output sharing" game, where each agent contributes or unit of input (e.g labor) and total output F ( y ) produced when y agents work must be shared among these workers This second model is standard fare to discuss the exploitation of natural resources, such as fisheries (Gordon [1954], Levahri and Mirman [1975]), forests, oil reserves (Dasgupta and Heal [1979]) etc Input represents, then, the fishing, logging or pumping effort and output is the total catch To fix ideas, we work from now on in the cost sharing format, yet this choice entails no loss of generality All our results and formulas are routinely adapted to the dual model of output sharing (see Section 11) For instance our leading story in Section becomes a farming story A potato field is divided in many lots with unequal yield The technology consists of assigning one worker to one lot for one unit of time (a day of work) All workers are equally skilled and, at the end of the day, they have extracted all the potatoes in the lot to which they have been assigned Each agent can supply or unit of labor (can only work a full day or not at all) Disutility of labor is independent of the lot to which an agent is assigned, but varies across agents In the average output mechanism, agents must decide at am whether to stay (and provide one unit of labor) or leave; if y is total supply of labor, these y workers are assigned to the y most productive lots and total output F ( y ) is equally divided among all workers (alternatively: each worker keeps the yield of his lot, but assignment is random and revealed after the agents have decided to stay or leave) In the random priority mechanism, a priority ordering is drawn at random (with uniform probability on every ordering) after which agents choose sequentially whether or not to stay; the first in the line is offered the best lot, and in general the next agent in the line is offered the best nonassigned lot, with the understanding that she keeps the yield of that lot The three mechanisms presented below offer three ways of governing the commons (in the terminology of Oström [1990]) in the free access regime, namely without imposing any caps on production (all agents are free to actively participate in the production process) and without requiring monetary transfers to, or from, the users from, or to, the nonusers In each one of the three scenarios, the mechanisms orders the agents (all potential users) at random, and the agent who decides to buy the q-th unit pays the marginal cost of producing this very unit The only intervention of the mechanism bears on the choice of the probability distribution over the various orderings, and on whether or not the agents are informed about the ordering before choosing to be active or not We contend that these features justify the terminology of "free access," because the mechanism has no coercive power (as would be the case if a cap on total production level is imposed) and does not monitor entry (as would be necessary if, for instance, inactive agents receive a monetary transfer from active ones: think of the competitive equilibrium with equal incomes mechanism described in Moulin [1995] Chapter and elsewhere).4 We now present the general model and our three mechanisms A typical agent i is willing to pay ui , ui 0 , for one (indivisible) unit of a certain good (or service), the same good for every agent The technology is described by the increasing sequence cq ,  c1  c2  , where cq is the marginal cost of producing the q-th unit thus total cost of q units is c1  c2 cq The Average Cost mechanism (AC) is played as a one shot game where each agent chooses to buy the good or not and where total cost is equally divided among all the buyers This is the most familiar model of the tragedy of the commons (although its output sharing version is even more familiar) As noted earlier, when the agents are risk neutral we may interpret the role of the mechanism as that of drawing at random an ordering of all the buyers (all agents who chose to buy) and charging to each buyer the marginal cost corresponding to his ranking in that ordering Oström [1990] discusses a variety of actual instances where the governance of the commons requires such coercive power and monitoring ability from the mechanism The important feature is that an agent must decide to buy or not before learning where he stands in the line, and cannot refuse the price offered to him afterwards.5 The Random Priority mechanism (RP) is played as a sequential game where first, Nature chooses at random (with uniform probability) an ordering of the agents and offers them to buy the good at the successive marginal costs: the agent ranked first is offered the good at price c1 ; if agent i is ranked k, and if exactly q agents among those ranked before i did buy, then agent i is offered the price cq1 Note that the RP game is a plausible description of the way certain commons are utilized in the free access regime Think of our agents as walking around randomly in the forest looking for fruits Someone will be lucky enough to find the fruit hanging lowest and pick it if the fruit is worth this person's effort to reach for that low branch; the next luckiest agent will find the next lowest hanging fruit and decide similarly whether or not to spend the effort to get it; and son on The above story is of some relevance to R&D competition, where input is research effort and output is (the present value of) a patent: think of all teams as supplying the same research effort, and of the lucky one (first in line) as the team who discovers the first and most profitable patent, and so on However, in problems of search involving some random discoveries it is hard to motivate the one shot decision problem (do I climb this tree to get this fruit or not?) as opposed to a sequential process where an agent may choose to search for several periods Our motivation in this paper is exclusively normative in the sense of mechanism design: if both options, to charge average cost or to charge marginal price in some random order, are available, which one should we recommend on the grounds of efficiency and perhaps, equity? This normative viewpoint is especially fruitful in the design of queuing protocols (see Section 1) such as those used in the internet (Demers et al [1990]), job scheduling and the management of congested roads (Gelenbe and Mitrani [1980]) See also the discussion of average cost versus incremental cost policies in Spulber [1994].6 It is easy to check that if the agent can refuse the price offered to him, everyone will chose to "buy" (with no commitment) in the first round and the mechanism will be precisely Random Priority Spulber [1994] discusses a case involving power utilities: at issue is the allocation of the costs of new investments necessary to serve new customers; this cost can be rolled over in the general budget (average cost sharing) or imputed solely to the beneficiaries of the investment (incremental cost sharing) He also argues in favor of the latter on incentives and efficiency grounds The viewpoint motivates the introduction of our third mechanism, a probabilistic version of serial cost sharing (Moulin and Shenker [1992], [1994]) The easiest way to describe the Probabilistic Serial mechanism (PS) is as an allocation of probabilistic "shares" describing with what probability a given agent will be able to buy the good and at what price Consider an agent of type q this means that his willingness to pay is above cq and no larger than cq1 Denote by mk the number of agents of type k or more The idea of PS is that these mk agents are equally entitled to purchasing the good at price ck Hence our agent of type q has an option to buy the good at price c1 with probability probability mq If m1 m1 , at price c2 with probability m2 , , at price cq with  m1q 1 , our agent will exercise all his options; if m1  m1q  , he will exercise only the best ones, that is to say all such options up to the highest index k such that m1  m1k 1 , and this fraction of the ( k 1) -th option bringing his total probability of service to one See Section for details The Probabilistic Serial mechanism is closely related to Random Priority, as demonstrated by the results of Section reviewed in the next section They are both natural but from different viewpoints RP is implemented by the simplest random ordering process, but the probabilistic allocation that it generates (with what probability does an agent of type q buy the good at price cq ?) is hard to compute: in Section we apply a recursive algorithm to deliver these probabilities (the algorithm is given in Appendix 1) In contrast, PS yields a natural probabilistic allocation This allocation is easily computed and implemented However the corresponding random process relies on the whole profile of types, hence the PS mechanism must elicit individual types before (randomly) assigning goods and cost shares The Probabilistic Serial mechanism shares the extreme strategic transparency of Random Priority; in particular both mechanisms can be interpreted as direct revelation games (where each agent reports his type and the mechanism implements the corresponding equilibrium allocation) with the property of coalition strategyproofness (no individual agentor group of agentshas an incentive to misreport, or jointly misreport, his type) However, the implementation of Probabilistic Serial requires to elicit the whole profile of types, agent whereas Random Priority simply offers a (random) price to the agents, one at a time In short, Probabilistic Serial is more subtle and requires more monitoring than Random Priority Overview of the Results In Section we examine the strategic and welfare properties of the RP and PS mechanisms We give a recursive formula computing the probabilistic equilibrium allocation of RP (Appendix 1) and compare it with the allocation implemented by PS Theorem uncovers several important links between these two allocations:  They are identical for the most "inefficient" agents (those with type at most qe where qe is the efficient output level)  Both mechanisms overproduce; but not by more than 100%; their expected output satisfy the following inequalities: qe qr qs 2qe  The PS outcome is Pareto superior (or equal) to the RP outcome We show on a few numerical examples that the welfare loss from RP to PS is typically small Finally we show that the Shapley value of the (first best) Stand Alone game is Pareto superior to the PS outcome (Proposition 2) In Section we compare the AC mechanisms with the two mechanisms RP and PS The strategic equilibrium of AC is much less robust than that of RP or PS: in particular it is not robust to coalitional deviations, and we can have multiple equilibria We check that the AC equilibrium may be Pareto inferior to those of RP and PS but the reverse cannot happen: the agents with the lowest utility among those who are ready to pay c1 always prefer RP (or PS) to AC (Proposition 3) Indeed RP and PS spread the surplus among all agents of type or more, whereas AC gives a positive surplus share to a much smaller upper end of the demand Section also offers a series of numerical examples with a small number of agents and of types These support the initial intuition that RP (or PS) brings in general a higher total surplus than AC In the remaining sections to 8, we consider the continuous model with a continuum of infinitesimally small agents, each one with a different utility for one unit of the good Thus the utility profile is represented by a continuous, downward sloping demand function and the technology by a continuous upward sloping marginal cost function Theorem 2, our second main result, is a convergence result about the equilibrium outcomes of RP and PS when the continuous model is viewed as the limit of a sequence of discrete models (with a finite but increasing number of agents) It turns out that the RP and PS equilibrium outcomes converge to the same limit; in other words the welfare superiority of PS over RP disappears in the limit model with a continuum of agents Moreover, Theorem provides an explicit formula describing this common limit This formula is used in Section to compare the respective performance of AC and RP in a number of simple configurations of the demand and marginal cost functions: when both functions are linear (corresponding to quadratic costs and uniform distribution of agents over the range of their utilities); when both functions are simple "step functions" corresponding to a piecewise linear cost (with two pieces) and only two distinct utility levels for all agents In both simple examples, we find that RP performs generally, but not systematically, better than AC In Section we show first that if RP overproduces more than AC, it must collect a smaller share of the efficient surplus (Proposition 4) Then we derive some systematic comparisons of RP and AC based on the convexity properties of the demand function (Propositions 5, 6, 7) In Section we show that upon replicating the demand, the RP mechanism always collects a positive fraction of the efficient surpluses, whereas AC collects a vanishingly small fraction of this surplus (Proposition 8) The final Section 10 gives a last argument in favor of RP against AC based on a fixed cost function and the worst case configuration of the demand function (Proposition 9) Section 11 gathers some concluding comments The Random Priority and Probabilistic Serial Mechanisms We repeat the basic assumptions of our simple production economy, already given in the first two sections The set N of potential users of the technology is finite and of cardinality n The willingness to pay of agent i for the indivisible output is ui , ui 0 When the consumption of agent i is a random variable, we interpret ui as agent i's Von Neumann Morgenstern utility, and assume risk neutrality with respect to monetary payments The cost of producing q units of output is C( q ) and we denote by cq the q-th marginal cost: cq C( q )  C( q  1) We assume C( 0) 0 and  c1 , cq  cq 1 for q 1,2, 10 The utility profile of the PS allocation (as well as that of the RP allocation) is Pareto inferior to the Shapley value of the Stand Alone surplus game Proof is in Appendix The Shapley value spreads the efficient surplus among all agents who can derive some Stand Alone surplus, however small This is similar to the way our two mechanisms operate Proposition says that the connection between the two concepts, the SP allocation and the surplus distribution according to the Shapley value, is tight Shapley and Shubik [1969] argue in favor of the Shapley value in a related model of bilateral trade The Average Cost Mechanism compared to RP and PS The average cost mechanism (AC) The one stage game where each agent chooses to buy the good or not: xi 0 or An agent who does not buy pays nothing An agent who buys pays the average cost Denoting yi the cost share imputed to agent i: x i 0  y i 0 xi 1  yi ac(  xi ) iN where we write ac( q ) C( q ) / q for the average cost function The strategic analysis of this mechanism is straightforward and well known Denote by d the (set valued) demand function: d ( p ) {i  N | p  ui } for all p 0 Let qa be the largest integer such that q | d |( ac( q )) , where | Z| denotes the cardinality of Z Consider the outcome where exactly qa agents in an upper tail of d ( ac( qa )) are served (that is no other agents in d ( ac( qa )) is willing to pay more for the good than any agent in the upper tail) This outcome is a Nash equilibrium There may be other equilibria but the above one is a surplus maximizing equilibrium (verification of these claims is immediate) In the discrete model the multiplicity of equilibria can be severe: for instance, if we have c1  ui  ac( ) for all i  N , there are n equilibria, and in each equilibrium exactly one agent buys the good (there are equilibria in mixed strategies as well) However, when approaching the limit case with a continuum of agents and continuous demand and marginal cost functions, the multiplicity of equilibria becomes negligible (a precise formulation of this fact is given in Section 6) 17 A consequence of the multiplicity of equilibria is envy: if an equilibrium does not serve all the agents in d ( ac( qa )) , the excluded agents would gladly exchange their allocation ( xi 0, yi 0) for the allocation of any one of the active agents ( x j 1, y j ac( qa )) Contrast this with the two mechanisms RP and PS where there is always envy ex post (e.g., in RP, the lucky agent who is served first is envied by all agents of type or more) On the other hand, there is no envy ex ante in the (probabilistic) equilibrium allocation of either mechanism We leave the easy verification of this fact to the reader.10 Thus the strategic properties of the AC mechanism are clearly inferior to those of RP or PS; in particular the AC equilibrium is generally vulnerable to coalitional deviations, e.g., by the "grand" coalition N We turn to the welfare comparison of the equilibrium allocations in AC versus RP or PS All three games yield inefficient equilibrium outcomes but for different reasons Recall that, in the RP (or PS) mechanism, the last user who buys faces the "correct" marginal cost hence her decision to buy is efficient (given the purchasing decisions of agents before her), whereas in the average cost game, the marginal buyer makes an inefficient move, because his willingness to pay is close to average cost, hence inferior to actual marginal cost On the other hand, in the RP game, the "wrong" agent may be offered a lucky deal: if pe is the efficient price the first agent in line may be willing to pay more than the cheapest price c1 but less than pe (i.e., c1  ui  pe ) so that efficiency requires that i not be served Thus, unlike the average cost mechanism, the RP mechanism facing a heterogeneous set of potential users is unable to serve only an upper tail of the demand function; every agent willing to pay more than the cheapest marginal cost c1 will be able to so with a small but positive probability The same conclusion holds for the PS mechanism In the particular case of an homogeneous demand (i.e., all agents have the same willingness to pay), the second type of inefficiency disappears hence the RP equilibrium is fully efficient (and so is the PS equilibrium by Theorem 1) whereas the average cost equilibrium wipes out most of the surplus Example 3: where RP (and PS) is Pareto superior to AC: homogeneous demand 10 This fact also follows from the interpretation of PS as a probabilistic demand game using the serial formula to share costs It is known that the equilibrium of serial cost sharing is nonenvious: Moulin and Shenker [1992] 18 Fix C and a willingness to pay u, common to all n agents The efficient quantity qe is the largest integer such that cqe u (we assume qe n ) The RP (and PS) allocations are fully efficient: each agent is served with probability xi qe / n and his expected payment is yi C( qe ) / n (thus, conditional upon being served, his expected payment is ac( qe ) ) Turning to the AC allocation, let qa be the largest integer such that ac( qa )  u Assuming qa n the AC game has multiple equilibria, in all of which qa (or perhaps qa  1, and u ac( q A  1) ) agents buy at price ac( qa ) Therefore, the RP allocation is Pareto superior to any AC equilibrium allocation if (and only if) qe ( u  ac( qe ))  u  ac( qa ) n (6) The above inequality holds true if n qa , in which case it reduces to u C( qa )  C( qe ) which follows from u  cqe  qa  qe When n is larger than qa , consider the case of quadratic costs ( cq increases linearly) One checks easily that (6) holds true if n  qe  qe2 , for any u such that qe is the efficient quantity Note that in the continuous limit of the discrete model, inequality (6) is always true, because u  ac( qa ) vanishes: see Theorem If the AC equilibrium can be Pareto inferior to the RP and PS allocations, the reverse configuration cannot happen, as shown by our next result In the statement of this result we say for brevity "agent i prefers mechanism X to mechanism Y" to mean that she prefers the corresponding equilibrium allocations at a given utility profile Proposition a) Among all agents of type or more, the one with the smallest utility always prefers RP (and PS) to AC If he is indifferent, the two allocations are equal and fully efficient b) The agents preferring AC to RP form an upper tail of the demand (possibly empty): if i prefers AC to RP, so all agents with higher utility than i; the agent preferring RP to AC form a nonempty lower tail of the demand c) Same statement as b), upon replacing RP by PS The proof is in Appendix We turn to a few numerical examples 19 Example 4: where AC is almost Pareto superior to RP (and PS) This example has already been introduced in Section The cost function is given and arbitrary The n agents are split in two groups of size ( n  1) and 1: u1 c2   ; for i 2, , n: ui c1    ac( 2) The efficient quantity is 1, to serve agent The AC mechanism has a unique and efficient equilibrium, which yields the profile of surplus ( c2  c1   , 0, , 0) The RP and PS allocations coincide and are easily computed: x1 1, y1 (1  1n )c2  1n c1 xi  1n , yi  1n c1 The corresponding profile of surplus is ( 1n ( c2  c1 )   , 1n  , 1n  ,  , 1n  ) Example continued The profile of types is (1, 1, 1, 1) We have computed earlier the RP and PS allocations Here we assume quadratic costs: cq q for all q Therefore the assumptions on utilities are:  u1   u2   u3   u4  The efficient output is qe 2 and the efficient surplus is  e u3  u4  The AC equilibrium allocation is unique: qa 3 and corresponding surplus  a u2  u3  u4  Finally qs 2.83 and  s  14 u1  127 u2  u3  u4  11 Note that agent always prefers PS (or RP) to AC (by Proposition 3) Here agents and prefer AC to PS (and RP), and the preferences of agent can go either way (he prefers PS/RP if and only if u2 2.6 ) A straightforward computation yields the upper and lower bounds of, successively, the relative efficiency loss of AC, that of PS, and finally the ratio of the PS surplus over the AC surplus 0 e  a 1  e   s 13  ;   e 24 e 48  s 19    a 18 (all bounds are exact) 20 Here the welfare performances of AC and PS are close, and either one of the mechanisms may collect more surplus The relative efficiency loss of PS has less variance than that of AC Example continued The profile of types is (0, 1, 3, 2) and we assume quadratic costs, cq q for all q:  u1   u2 u3 u4   u5 u6 5 The efficient quantity is qe 3 and  e u4  u5  u6  The AC equilibrium is unique and produces qa 5 units, with  a u2  u3  u4  u5  u6  15 Finally the (expected) quantity and produced by PS is qs 393 11  s  13 u1  158 ( u2  u3  u4 )  u5  u6  15 It turns out that the PS allocation always gathers more surplus than the AC equilibrium In fact, the PS allocation is Pareto superior to the AC one To check this, by Proposition it is enough to show that agent pays less in PS Note that she also prefers RP to AC if and only if u6 4 13 Finally one computes the range of relative efficiency losses and relative surpluses: 0125  e  a   s .5 ; 139  e .345 e e   s 178 a To be completed: a systematic comparison of AC and PS/RP for all profiles of types with a small number of agents and small number of types The Continuous Model The cost function is now described by a strictly increasing and continuous marginal cost function mc( q ) defined for all nonnegative (real) q; we assume mc( 0) 0 and mc( )  The demand function d ( p ) is defined for all nonnegative (real) p, is nonincreasing and continuous; moreover lim d ( p ) 0 as p goes to infinity We denote by q  d ( ) the size of the potential demand Note that our assumption mc( 0) 0 is merely a normalization convention and entails no loss of generality If mc( 0) is positive, all agents whose willingness to pay is below mc( 0) are irrelevant to either mechanism (they can be removed without changing anything to the equilibrium outcomes) 21 We denote by D( p ) the user's surplus given a competitive price p and by A( p ) the maximal profit of a competitive firm facing the price p: z  D( p )  d ( t )dt ; A( p )  p mc  1( p )  C( mc  1( p )), where mc  is the inverse function of mc p Figure illustrates these two functions The efficient level of production qe is found at the intersection of the demand and marginal cost functions, and the efficient surplus is  e  D( pe )  A( pe ), where pe mc( qe ) Next we explain how the continuous model can be viewed as the limit of a sequence of increasingly finer discrete approximations Fix an integer n and consider the discrete economy E ( n) with n agents, and the following utility profile and marginal costs: q cq  mc( q ) q 1,2, n q ui  d  ( i ) i 1,2, , n, where d -1 ( q ) sup{ p 0| d ( p ) q} for all q 0 n Consider, for each n, an allocation z ( n ) in the economy E ( n) That is, z ( n ) is the list of probabilistic allocations ( xi ( n ), yi ( n )) to all agents i 1,  , n In the continuous economy, an allocation is a function q  zq ( xq , yq ) , specifying, for every "type" q, a probabilistic allocation We say that the sequence z ( n ) converges to the (continuous) allocation z if z is the pointwise limit of z ( n ) (we omit the details) Our next result, the second main result of this paper, says that all three equilibrium allocations (for the AC, RP and PS mechanisms respectively) converge in the continuous model; moreover the latter two have the same limit We define the critical quantity qr (it turns out to be the expected output in the continuous RP and PS allocations) as follows Denote by p the smallest price such that d ( p ) 0 (note that p  is possible) Then we set qr mc  ( p ) if otherwise qr is the solution of z z mc  1( p ) qr dt  d ( mc( t )) dt 1 d ( mc( t )) (7) Theorem 22 a) The AC equilibrium allocation of the discrete economies converges to the following continuous allocation Let qa be the intersection of the average cost and (inverse) demand functions: d ( ac( qa )) qa We denote pa ac( qa ) The limit AC allocation is zqa ( 0, 0) if q qa ; zqa (1, pa ) if q  qa The corresponding surplus is  a  D( pa ) b) The RP and PS allocations of the discrete economies converge to the same continuous allocation, that we denote z r ( x r , y r ) : xqr  The corresponding surplus is z inf( q, qr ) dt , d ( mc( t )) z yqr  inf( q, qr ) mc( t ) dt d ( mc( t )) z r  qr D( mc( t ) dt d ( mc( t ) (8) (9) Statement a) is a folk theorem of sorts, its proof is easy Statement b) is much harder to prove See Appendix Proposition for the discrete model extends straightforwardly to the continuous case To see r r this, denote  r ( p )  p xq  yq , where q  mc  ( p ) , the surplus of an agent with utility p for the good The above formulas imply d r ( p ) / dp 1 so that the proof in Appendix extends at once Two Simple Examples We illustrate Theorem in two simple configurations of the demand and marginal cost function Example Linear demand and quadratic costs Consider a linear demand and a linear marginal cost functions: mc( q )  q ; d ( p ) max{  p,0} where the constants  ,  ,  are all positive Define x  , then we have qa x 1 2  qe x 2 qr  x  (1  e  x ) qe x 23 for all x 0 whereas qa / qe is as high In particular, qr qa for all x  ; moreover qr / qe 13 as when x becomes large Next we have a x 1 4 e ( x  )2  r 1 x  (1  e  x ) e 2x In particular  r /  a .969 for all x ; moreover  r /  e 5 for all x , whereas  a /  e goes to zero when x becomes large Figures and show the graphs of these functions The corollary establishes the unambiguously superior performance of RP over AC in this particular context Its proof is a straightforward application of equations (7), (8), (9) One computes easily  e x qe   , qa   , qr   1 x 2x x 2     e x e   , a   , r   2  x  (  x )2 2 2x In the linear d/linear mc example, an increase in the parameters  ,  of the demand function leaving  /  constant amounts to make the demand flatter while preserving the highest willingness to pay  /  Such a change corresponds to a replication of the utility profile, namely a uniform proportional increase d   d for some parameter  greater than one When x is large enough, this change is favorable to RP in the sense that  r increases while  a decreases This corresponds to a general fact, the subject of Section Example Two types of consumers and two marginal costs The cost function is C( q ) c1 q if q 10 and C( q ) 10c1  c2 ( q  10 ) if q 10 : the marginal cost jumps from c1 to c2 , c2  c1 , at the output level 10 The demand splits in a group of size with high willingness to pay p1 , where c2  p1 , and a very large group of agents with low willingness to pay p2 , where c1  p2  c2 We set p2 c1  (1   )c2 Figure shows the configuration of d and mc: efficiency commands to produce 10 units of output, to serve all high demand agents and a subset of size of the low demand agents At the AC equilibrium, the quantity produced and the efficiency loss are easily computed 10 qa  ;   e   a 10( c2  c1 ) (1   ) 24 It is straightforward to extend the formulas (7), (8), (9) to account for the discontinuities in d and mc One computes: qr 15 and  e   r 5( c2  c1 )  Therefore, two cases arise If   23 , the RP mechanism overproduces less and collects more surplus: qr  qa and  a   r If, on the other hand,   1 , both inequalities are turned around: qa  qr and  r   a In Example 6, the tension between the welfare properties of AC versus RP is clear: the higher  the less overproduction and surplus loss in AC, and the more surplus loss in RP Convexity Properties of the Demand and Welfare Comparisons We give in this section a series of results whereby a certain geometric property of the demand function allows us to derive, irrespective of the marginal cost function, which mechanism from AC or RP overproduces less, and/or which mechanism collects more surplus Before turning to these results, we show first that a certain configuration of the relative size of qa versus qr and  a versus  r is impossible Proposition If the RP equilibrium overproduces more than the AC equilibrium, then it brings a smaller surplus as well: qa  qr   r   a , and qa qr   r  a The proof is in Appendix The two examples in the previous section (Examples and 6) show that all three other configurations of qa versus qr and  a versus  r are possible: qr  qa and  a   r : Example for 2.4  x  Example for  < qr  qa and  r   a : Example for x < 2.4 qa  qr and  r   a : Example for   1 Our next results relate convexity properties of the demand function with these comparisons Proposition 25 If / d is a concave (resp strictly concave) function on the interval [0, mc( qa )] , then qa qr (resp qa  qr ) Proof of this and the next two propositions is in Appendix Combining Propositions and 5, we find that if 1/d is concave on [0, mc( qa )] , then AC overproduces less and collects more surplus An example is the demand function d ( p)  a where    and  a , b ( p  b ) Note that p  for the above demand function If p is finite, the concavity of 1/d cannot hold on the whole interval [0, p ] because 1/d goes to  as p approaches p In general, the concavity of 1/d implies (but is not implied by) the convexity of d This means that the density of agents increases when the willingness to pay decreases Proposition If 1/d is a convex (resp strictly convex) function on [0, mc( qa )] , then qr qa (resp qr  qa ) Proof is in Appendix Unlike in the case of Proposition 5, it is very easy to come up with simple functional forms of the demand for which 1/d is convex Recall that a function f is Logconcave if Logf is concave Then we have d concave  d Logconcave  / d convex Examples include a linear demand function: d ( p )   p for  p  p  /  Notice that we can apply Proposition only if mc( qa )  p ; thus in the case of Example 5, Proposition can only be invoked for a subset of values of the parameters  and  , where f is increasing and f ( ) 0 More generally, consider a demand function of the form d ( p )  f (  p ) ,  p  /  where f is increasing and f ( 0) 0 Then 1/d is convex if 1/f is convex Examples include: f ( x ) a ( x ) where a  and   (here f is Logconcave ) f ( x ) a ( ex  1) where  a,  (here f is Logconcave ) f ( x ) ax  bx  cx  dx where  a , b, c, d x f (x)  where    a a x  26 Our last result in this section offers a family of demand functions for which we guarantee the pattern qr qa and  r  a Proposition If the function D is affine on the interval [0, mc( qa )] then (1/d is convex and) we have d qr qa and  r  a Proof is in Appendix Examples of application include: a , where   and a , b  ( p  b )  d ( p ) (  p ) ,  p  p  , where    d ( p)  d ( p ) e  p Replication Proposition Replication Fix the marginal cost function and a demand function d where the highest willingness to pay p is finite Say that the economy is replicated if we replace d by d where  is a parameter greater than one A replication raises qe , qa , qr as well as  e and  r On the other hand, for  large enough, the surplus  a decreases When  goes to infinity, we have: lim qe lim qr mc  ( p )  lim qa ac  ( p ) lim  e  A( p ) , lim  a 0 , lim  r  z mc  ( p ) D( mc( t )) dt d ( mc( t )) (10) Proposition follows easily from Theorem As  increases the demand  d becomes flatter, which in turn brings the surplus  a to zero As for the statement about qr and  r it is a consequence of (7), (9) Two cases arise Case 1: z mc  ( p ) dt  Then for all  , qr (  ) is defined by d ( mc( t )) z qr (  ) dt  , an d ( mc( t )) increasing function of  , and it converges to mc  ( p ) as  goes to infinity 27 Case 2: z mc  ( p ) dt   Then qr (  )  mc  ( p ) for  large enough d ( mc( t )) Note that when d is linear, formulas (10) imply that the asymptotic RP surplus is one half of total surplus, regardless of the cost function Indeed, one computes: d ( p )  ( p  p ) , all p 0   lim  r  12 z mc  ( p ) D ( p )  12 ( p  p ) d ( p  mc( t ))dt  12 A( p ) 10 Guaranteed Surplus Share Strategyproof mechanisms are precious, whenever the designer has little or no information about the demand; on the other hand, our mechanisms AC, RP, and PS require her to know the technology C Whenever we compare the relative merits of various mechanisms, a very risk averse designer will consider their performance in the worst case scenarios of demand In our model, such a designer would want to compute, for instance, the minimal share of the efficient surplus that a given mechanism collects in the worst configuration of the demand From now on, we fix the cost function C and we want to evaluate the infimum of the ratios  r /  e ,  a /  e over all possible demand functions If we not impose further restrictions on the demand, the answer is obvious: both infimum are zero It is enough to take a flat demand function to show that  a /  e is zero As for the ratio  r /  e , its infimum is also zero, as shown by the following family of demand functions where q and  are two positive parameters d ( mc( t )) q0 e ( q0  t ) 0 for t q0 for q0  t (11) (note the discontinuity of d, that in all rigor should be "smoothed") One checks that the ratio  r /  e becomes vanishingly small when  grows arbitrarily large In order to capture some useful lower bounds, it is necessary to fix some parameters of the demand, so as to rule out the kind of unbounded examples just offered 28 One possibility is to fix q  d ( ) , another is to fix qe By using precisely the same families of demand functions (flat to cancel  a /  e , truncated exponential like (11) to cancel  r /  e in the limit) one shows that the infimum is still zero in both cases However, if we fix both q and qe the minimization of  a /  e and of  r /  e over the family of demand functions such that d ( 0) q and d ( mc( qe )) qe brings about some interesting results Proposition Assume a quadratic cost function C, and fix the size of the concerned agents q  d ( ) , as well as the efficient output level qe The relative share of surplus guaranteed by the average cost and random priority mechanisms (over all conceivable demand functions) are: a q  max{2  ,0} e qe  r   e  log( qqe ) (12) In particular  a /  e   r /  e as long as q qe 0 The proof of this Proposition is due to Anna Bogomolnaia It is given in Appendix 10 Figure depicts the two guaranteed shares when qe / q varies between and 1, and shows the superiority of RP versus AC on account of this particular criterion We conjecture that the inequality  a /  e   r /  e holds for every strictly convex cost function C 11 Concluding Comments a) The dual output sharing model Here the technology is described by a decreasing sequence of (positive) marginal returns rq Each agent can supply zero or one unit of input: yi 0 or , and total output to be shared is r1 rq , where q  yi Each agent i  N has a disutility ei for "working" (contributing one iN unit of input) These numbers generate a nondecreasing supply curve 29 The analog of AC is the Average Return mechanism, sharing total return equally among the active agents The analog of RP consists of ordering (randomly) our agents and offering them to work for the current marginal return Our entire analysis can be adapted to the output sharing model Formulas to be completed b) Open questions Our analysis left open several difficult questions:  in the discrete model, is the configuration qa  qr and  a   r possible? What about the configuration qa  qs and     s ?  can we compute the upper and lower bounds of  r /  a for a given cost function C and d( 0) , but when d varies over all conceivable demands (Section 10)? What about the upper bound of s in the discrete model? r c) Strategyproof probabilistic social choice functions The RP and PS mechanisms offer two instances of probabilistic social choice functions: agents reveal their willingness to pay for the service and receive a probabilistic allocation Both of these social choice functions are nonmanipulable (even by coalitions) The natural next problem is to describe the entire class of all such nonmanipulable social choice functions in the "free access" regime: this could mean that each agent is guaranteed service if he is willing to pay enough, and that participation is voluntary (no agent is ever asked to pay more than his willingness to pay) This class does include inequitable mechanisms, e.g., when we follow a fixed, deterministic, priority What else? d) Extensions of the model These include variable individual demands (each agent chooses a quantity of output to consume), and/or heterogeneous goods (each agent demands a different good, as in Moulin and Shenker [1996], who assume decreasing marginal costs.) 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