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, [email protected]
** Department of Economics, Box 90097, Duke University, Durham NC 27708,
[email protected]
*** 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
1. 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) / 2 (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.
2
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) / 2 (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) / 2 and all agents who choose to leave have it not higher than ( qa  2) / 2 ; 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 3
periods if both agents before him in the line chose to stay, of 2 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
1
The possibility of multiple Nash equilibria complicates the strategic analysis a bit; see Section 5 for details. The
resulting undeterminacy disappears, however, in the continuous limit: see Section 6.
2
Each player has a dominant strategy so that, barring indifferences, there is a unique Nash equilibrium outcome.
3
The main simplifying assumption is the one-shot binary decision-making at t  0 ; agents do 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.
3
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, 9
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 9 times out of
.  1)  ( 21
.  2)  0.2 ,
10, two agents stay (one of the first 9 and the last one) for a surplus of (11
well below the efficient surplus 2.1-1=1.1. See Section 5 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 2 the queuing story to a
systematic model of the tragedy of the commons, hence opening a much broader range of
potential applications.
4
2. 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 1 the commons is the server, and decentralized utilization
takes the "cost sharing" format namely each agent demands 0 or 1 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) / 2 so that returns are decreasing.
An equally simple model is the dual "output sharing" game, where each agent contributes 0
or 1 unit of input (e.g. labor) and total output F ( y ) produced when y agents do 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 1 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 0 or 1 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 8 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
5
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 5 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 , 0  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.
4
Oström [1990] discusses a variety of actual instances where the governance of the commons requires such coercive
power and monitoring ability from the mechanism.
6
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
5
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.
6
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.
7
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
1
mq
. If
1
m1
1
m1
, at price c2 with probability
1
m2
,, at price cq with
 m1q  1 , our agent will exercise all his options; if
1
m1
 m1q  1 , he
will exercise only the best ones, that is to say all such options up to the highest index k such that
1
m1
 m1k  1 , and this fraction of the ( k 1) -th option bringing his total probability of service
to one. See Section 4 for details.
The Probabilistic Serial mechanism is closely related to Random Priority, as demonstrated by
the results of Section 4 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 4 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.
8
3. Overview of the Results
In Section 4 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 1 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 5 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 1 or more, whereas AC gives a positive surplus share to a much smaller upper end
of the demand.
Section 5 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 6 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
9
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 2 provides an explicit formula describing this
common limit.
This formula is used in Section 7 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 8 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 9 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.
4. 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
0  c1 , cq  cq1 for q  1,2,
10
A mechanism is a game form determining a feasible allocation of this economy. The two
mechanisms RP and PS are probabilistic: the allocation determined by the game form is a
random variable. Because we assume risk neutrality, all we need to know about agent i's
allocation is the probability x i that he/she will be served, and his/her expected payment yi .
The Random Priority mechanism
The n stages game where Nature draws agents successively without replacement and with
uniform probability (equivalently, an ordering of the agents is drawn at random, with equal
probability on all orderings). The agent drawn in the first stage is offered the good at price c1 and
chooses between taking the offer or declining it (in both cases, this agent leaves the game). The
second agent in line is offered price c2 if the first agent did buy at c1 , or price c1 if the first agent
declined. And so on: the agent drawn at stage q is offered the price cq1 , where q is the number
of agents drawn before him who did buy.
The strategic analysis of this game is transparent. It is a dominant strategy for an agent to
"buy truthfully" (i.e., buy if and only if ui  cq ); barring indifferences, the dominant strategy
equilibrium is the unique Nash equilibrium, and is also a strong equilibrium (i.e., it resists
coalitional deviations). Even if indifferences are allowed, the above equilibrium remains the
essentially unique strong equilibrium of the game. It is also Pareto superior to any other Nash
equilibrium. Note that these strategic properties are independent of agents' preferences toward
risk.7
The canonical equilibrium is determined by the altruistic tie-breaking rule: whenever
indifferent between buying or not, an agent does not buy. We shall maintain this assumption
throughout the paper; removing it would complicate the analysis of the margin without bringing
any new insight. As no confusion may arise, we simply call this equilibrium "the" RP
equilibrium.
The altruistic equilibrium determines a social choice function associating to any utility
profile the corresponding probabilistic allocation. Viewed as a direct revelation game, this social
choice function is nonmanipulable, even when any coalition of agents can jointly misreport (this
is the coalition strategy-proofness property, see, e.g., Moulin [1996]).
7
See Moulin [1996] for a proof of these claims in the nonrandom framework; a straightforward argument showing
that randomization has no effect on these strategic properties is omitted.
11
The next step toward evaluating the welfare properties of the RP mechanism is to compute
the RP social choice function, namely the vector of probabilistic individual allocations ( xi , yi ) .
Clearly, all that matters to compute agent i's allocation is his type, namely the position of his
utility ui relative to the increasing sequence of marginal costs. We shall say that agent i is of
type q if cq  ui  cq1 : this means that i will buy up to the q-th highest price. A profile of types is
a vector ( n1,, nQ ) where nq is the number of agents of type q (thus nq is zero or a positive
integer) and where q varies from 1 to Q. Given a utility profile ui , we choose Q to be the highest
type in this profile; thus when we write a profile of types ( n1,, nQ ) , we always assume that
nQ  0 .
Given a profile of types ( n1,, nQ ) we wish to compute the probability  k ,q that, in the RP
equilibrium, an agent of type q buys the good at price ck . We can assume 1  k  q because
 k ,q  0 whenever q  k . Note that this probability is not defined whenever nq  0 . Next we
observe that  k ,q   k ,q whenever k  q, q (and nq , nq are both nonzero). Indeed,  k ,q only
depends upon the (random) outcome of the first k stages of the game, and, up to that stage, an
agent of type q and one of type q behave in exactly the same way. Therefore we set  k   k ,q
to be the probability that, in the RP equilibrium, an agent of type k or more buys the good at
price ck . Note that  k is defined for all k  1,, Q because nQ  0 . The RP social choice
function is now:
if agent i is of the type q: xi  k 1  k
q
,
yi  k 1 ck   k
q
(1)
The computation of the numbers  k is difficult. We only provide a recursive formula in
Appendix 1. This allows explicit computations for reasonably small values of n. A couple of
such examples are given after the definition of the PS mechanism, to which we now turn.
We offer two equivalent definitions of the mechanism, first as a direct revelation game,
namely a social choice function, second as a probabilistic demand game.
Fix an arbitrary utility profile with associated profile of types ( n1,, nQ ) . To define the
(probabilistic) allocation selected by the PS mechanism (social choice function), we consider the
largest number q* , 1  q*  Q such that
12

We set   1  1
q*
1
1
k 1
mk
where mk  t  k nt
q*
Q
(2)
1
and define our allocation as follows:
mk
i of type 1: xi 
1
c
, yi  1
m1
m1

i of type k ,1  k  q *: xi  t 1
k
k c
1
, yi  t 1 t
mt
mt
(3)

i of type q*  1,, Q: xi  1 , yi  1
q*
ct
   cq* 1
mt
Note that the lower part of the above formula disappears if q*  Q .
The formula (3) defines a social choice function (from the profile of types to a probabilistic
allocation) that we call the Probabilistic Serial social choice function. Note the analogy with
formula (1). If we denote by  k the probability that, in the PS allocation, an agent of type k or
more buys the good at price ck , we have:
k 
1
if k  q*;  q* 1   ;  k  0 if k  q*  2
mk
(4)
Proposition 1
The PS and RP social choice functions are both coalition strategy-proof.
The statement about RP has been discussed above. In the case of PS, the result is a particular
case of the general nonmanipulability properties of serial cost sharing (Moulin and Shenker
[1992]). Consider the probabilistic extension of the cost function C. If agent i receives the good
~
with probability x i , the total demand x  x1  xn can be served at minimal cost C ( x ) , where
~
C is the convex hull of C, namely the largest convex function (defined for positive real
demands) bounded above by C (defined only for positive integer demands). Because C itself is
convex, its probabilistic extension obtains simply by linear interpolation:
if a  x  a  1, where a is an integer:
~
C ( x )  ( a  1  x ) C( a )  ( x  a ) C( a  1)
(5)
13
Now consider the demand game where each agent i demands to be served with probability x i .
The mechanism selects a probability distribution over all coalitions of agents such that i) each
agent i is indeed served with probability x i and ii) the expected cost is as small as possible given
the service requirement (property i). It is straightforward to check that the cheapest expected
~
cost is given by C .
~
Now, if we use the serial cost sharing formula to share the cost C ( x1  xn ) among the n
users, we obtain a demand game with all the usual properties of the serial cost sharing game.
~
Indeed C is convex and individual preferences over ( xi , yi ) are linear (represented by Von
Neumann-Morgenstern utility functions); so the general results in Moulin and Shenker [1992]
apply.8 It is then easy to check that the equilibrium allocation of the probabilistic serial demand
game is precisely the allocation (3). See Appendix 2 for details.
We emphasize the striking contrast in the way we introduced the two corresponding
mechanisms. Random Priority is implemented by a simple random process, and the social
choice function it implements is hard to compute (see ( ) in Appendix 1). Probabilistic Serial is
given first as a simple social choice function (formulas (3)) that can be realized by a fairly simple
random process described in Appendix 2. This process, however, is based on the entire profile
of types9, hence it is a mere random device to deliver the PS allocation. By contrast, the process
used by RP is universal, namely independent of individual types.
We turn to numerical examples where we compare the RP and PS social choice functions.
First, we note that in any economy where every agent is of type 3 at most, the two allocations
coincide. This follows from statement a) in Theorem 1 below if the efficient output is 3.
Verification of the other cases is straightforward and omitted.
8
~
In fact, these results must be adapted to take into account possible indifferences, as neither C nor the preferences
are strictly convex. The demand game always has a unique strong equilibrium, but indifferences may yield multiple
Nash equilibria. The unique strong equilibrium is selected under our altruistic assumption stating that an agent
refrains from buying when indifferent between buying or not.
9
In fact, it is not necessary to elicit the distributions of types beyond q  1 (see (3)), because the PS allocation does
*
not depend on that part of the distribution. Thus the mechanism could for instance ask successively: whose type is 1
or more? whose type is 2 or more? etc., and stop once the summation 
1
mk
exceeds one.
14
In general, the comparison of the RP and PS allocations amounts to comparing the two
sequences  k ((1)) and  k ((4)).
Example 1:
n  4 ; profile of types (1, 1, 1, 1)
Here we have 4 agents with utilities ui for the good and c1  u1  c2  u2  c3  u3  c4  u4 .
The two vectors  and  are
1 1 3 1
1 1 5
);   ( , , ,0)
4 3 8 24
4 3 12
 ( , , ,
This means that, in both RP and PS, agents 1 and 2 get respectively the allocations
( x1, y1 )  ( 14 , 14 c1 ) ; ( x2 , y2 )  ( 127 , 14 c1  13 c2 )
However agents 3 and 4 both get a strictly better allocation under PS:
RP
PS
agent 3
23 1
( 24
, 4 c1  13 c2  83 c3 )
(1 , 14 c1  13 c2  125 c3 )
agent 4
(1 , 14 c1  13 c2  83 c3  241 c4 )
(1 , 14 c1  13 c2  125 c3 )
The efficient quantity is 2 units, but RP produces 2.79 units on average, versus 2.83 units for PS.
We can easily compute the relative surplus gain from RP to PS. Denote by  r and  s ,
respectively, the total surplus collected by each mechanism:
23
 r  u4  24
u3  127 u2  14 u1  ( c1  c2  43 c3  241 c4 )
 s  u4  u3  127 u2  14 u1  ( c1  c2  10
12 c3 )
   r 1 u3  c4  2c3
 s


s
24
s
One checks that  is maximal for u4  c4 , u2  c2 , u1  c1 and u3  c4 . Hence:

c4  c3
24c4  (9c1  5c2  10c3 )
For instance, if marginal costs increase linearly (quadratic costs, as in the Example of Section 1),
the loss is at most 2.2%.
Example 2: n  6 , profile of types ( 0, 1, 3, 2)
The efficient output level is 3 and we find, again, that the least efficient agents receive the
same allocation in RP and PS:
15
( x1, y1 )  ( 13 , 16 c1  16 c2 )
agent 1:
agents 2, 3, 4: ( x1, yi )  ( 158 , 16 c1  16 c2  15 c3 )
The last two agents are not treated identically by the two mechanisms:
RP
PS
( 109 , 16 c1  16 c2  15 c3  11
(1, 16 c1  16 c2  15 c3  157 c4 )
30 c4 )
agents 5, 6:
The relative surplus loss when switching from PS to RP is no longer very small: here, with
quadratic costs, we find:   10.7% . The expected quantity produced is respectively 3.73 for RP
and 3.93 for PS.
We turn to the general result comparing our two mechanisms. Fix an arbitrary profile of
types ( n1,, nQ ) , and denote by qe the efficient output level, namely the largest integer q such
that q  mq . We also denote by qr and qs , respectively, the expected quantities produced by the
RP and PS allocations.
Theorem 1
a) For all agents of type at most qe , both the RP and the PS allocations coincide, namely:
for i of type q, q  qe : xi  k 1
q
q
1
c
, yi  k 1 k
mk
mk
b) Both allocations overproduce but PS overproduces more. Neither allocation overproduces
by more than 100%.
qe  qr  qs  2qe
c) The PS allocation is Pareto superior to the RP, or they coincide.
Note that the upper bound on qs is tight: for any c and q, there exists a profile of types such that
q is the efficient output and qr  qs is arbitrarily close to 2q. The proof of this claim and of
Theorem 1 is in Appendix 3.
Our last welfare result about the RP and PS social choice functions compares them to a fully
efficient social choice function. Given a utility profile ( ui )iN and the technology
( ck ) k  1, 2,  , the Stand Alone surplus v ( S ) of a coalition S, S  N is the surplus generated
when the agents of S use the commons at their leisure, without any interference by the other
agents:
v( S )  max{ ui  C(| T | )}
T :T  S
iT
16
Proposition 2
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 4.
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 2 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.
5. 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 1 . 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:
xi  0 
xi  1 
yi  0
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( 2) 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
17
multiplicity of equilibria becomes negligible (a precise formulation of this fact is given in
Section 6).
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 1 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 do 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
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
(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
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  1
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 2.
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 3
a) Among all agents of type 1 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 do all agents with higher utility than i; the agent preferring RP to AC form
a nonempty lower tail of the demand.
19
c) Same statement as b), upon replacing RP by PS.
The proof is in Appendix 6. We turn to a few numerical examples.
Example 4: where AC is almost Pareto superior to RP (and PS)
This example has already been introduced in Section 1. 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 1. 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 1 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:
1  u1  2  u2  3  u3  4  u4  5
The efficient output is qe  2 and the efficient surplus is  e  u3  u4  3 . The AC
equilibrium allocation is unique: qa  3 and corresponding surplus  a  u2  u3  u4  6 . Finally
qs  2.83 and
 s  14 u1  127 u2  u3  u4  112 .
Note that agent 1 always prefers PS (or RP) to AC (by Proposition 3). Here agents 3 and 4
prefer AC to PS (and RP), and the preferences of agent 2 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
1  e   s 13

;


e
4
24
e
48
7  s 19


8  a 18
20
(all bounds are exact).
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 2 continued
The profile of types is (0, 1, 3, 2) and we assume quadratic costs, cq  q for all q:
2  u1  3  u2  u3  u4  4  u5  u6  5
The efficient quantity is qe  3 and  e  u4  u5  u6  6 . 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  9 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 3 it is enough
to show that agent 6 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

1  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.
6. 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 ( 0) 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
21
irrelevant to either mechanism (they can be removed without changing anything to the
equilibrium outcomes).
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:
D( p ) 
z

p
d ( t )dt ; A( p )  p  mc 1 ( p )  C( mc 1 ( p )), where mc 1 is the inverse function of mc
Figure 1 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 1(  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
22
qr  mc 1 ( p )
z
z
mc 1 ( p )
if
0
otherwise qr is the solution of
qr
0
1
dt  1
d ( mc( t ))
1
dt  1
d ( mc( t ))
(7)
Theorem 2
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 
z
inf( q, qr )
0
1
dt ,
d ( mc( t ))
yqr 
z
inf( q, qr )
0
mc( t )
dt
d ( mc( t ))
(8)
The corresponding surplus is
r 
z
qr
0
D( mc( t )
dt
d ( mc( t )
(9)
Statement a) is a folk theorem of sorts, its proof is easy. Statement b) is much harder to prove.
See Appendix 7.
Proposition 3 for the discrete model extends straightforwardly to the continuous case. To see
this, denote  r ( p)  p  xqr  yqr , where q  mc 1( p ) , the surplus of an agent with utility p for the
good. The above formulas imply 0  d r ( p ) / dp  1 so that the proof in Appendix 6 extends at
once.
7. Two Simple Examples
We illustrate Theorem 2 in two simple configurations of the demand and marginal cost
function.
Example 5 Linear demand and quadratic costs
Consider a linear demand and a linear marginal cost functions:
mc( q )    q ; d ( p )  max{  p,0}
23
where the constants  ,  ,  are all positive. Define x   , then we have
qa
x 1
 2
qe
x2
qr 1  x

 (1  e x )
qe
x
. for all x  0 whereas qa / qe is as high
In particular, qr  qa for all x  0 ; moreover qr / qe  13
as 2 when x becomes large.
Next we have
a
x 1
4
e
( x  2 )2
 r 1 x

 (1  e2 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 2 and 3 show the graphs of these 4 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
1
2
1  e x
, qa   
, qr   
1 x
2 x
x
2
2

1

4
 2 1  e 2 x
e 

, a 

, r 

2 1  x
2  ( 2  x )2
2
2x
qe   
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 9.
Example 6 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 5 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 4 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 5 of the low demand agents.
24
At the AC equilibrium, the quantity produced and the efficiency loss are easily computed
qa 
10

;
 e   a  10( c2  c1 )  (1   )
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 0    23 , the RP mechanism overproduces less and collects more
surplus: qr  qa and  a   r . If, on the other hand,
2
3
   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.
8. 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 4
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 8.
The two examples in the previous section (Examples 5 and 6) show that all three other
configurations of qa versus qr and  a versus  r are possible:
qr  qa and  a   r : Example 5 for 2.4  x  
Example 6 for 0   <
2
3
qr  qa and  r   a : Example 5 for 0  x < 2.4
qa  qr and  r   a : Example 6 for
2
3
  1
Our next results relate convexity properties of the demand function with these comparisons.
25
Proposition 5
If 1 / 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 9.
Combining Propositions 4 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 0    1 and 0  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 6
If 1/d is a convex (resp. strictly convex) function on [0, mc( qa )] , then qr  qa (resp. qr  qa ).
Proof is in Appendix 9.
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  1 / d convex
Examples include a linear demand function: d ( p )    p for 0  p  p   /  . Notice that
we can apply Proposition 6 only if mc( qa )  p ; thus in the case of Example 5, Proposition 6 can
only be invoked for a subset of values of the parameters  and  , where f is increasing and
f ( 0)  0 .
More generally, consider a demand function of the form d ( p )  f (  p ) , 0  p   / 
where f is increasing and f ( 0)  0 . Then 1/d is convex if 1/f is convex. Examples include:
26
f ( x )  a  ( x ) where a  0 and   0 (here f is Logconcave )
f ( x )  a( ex  1) where 0  a, 
(here f is Logconcave )
f ( x )  ax  bx  cx  dx where 0  a, b, c, d
2
f (x) 
3
4
x
where 0    a
ax

Our last result in this section offers a family of demand functions for which we guarantee the
pattern qr  qa and  r   a .
Proposition 7
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 9.
Examples of application include:.
d ( p) 
a
, where   1 and a, b  0
( p  b )
d ( p )  (  p ) , 0  p  p 

, where   0

d ( p )  e  p
9. Replication
Proposition 8 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 1( p )  lim qa  ac 1( p )
lim  e  A( p ) , lim  a  0 , lim  r 
z
mc 1 ( p )
0
D( mc( t ))
dt
d ( mc( t ))
(10)
27
Proposition 8 follows easily from Theorem 2. 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 1 ( p )
0
1
dt   . Then for all  , qr (  ) is defined by
d ( mc( t ))
z
qr (  )
0
1
dt   , an
d ( mc( t ))
increasing function of  , and it converges to mc 1( p ) as  goes to infinity.
Case 2:
z
mc 1 ( p )
0
1
dt   . Then qr (  )  mc 1( 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 
1
2
z
mc 1 ( p )
0
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 do 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 min  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 )
for 0  t  q0
0
for q0  t
(11)
28
(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.
One possibility is to fix q  d ( 0) , 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 9
Assume a quadratic cost function C, and fix the size of the concerned agents q  d ( 0) , 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

2
min r 
 e 2  log( qq )
min
(12)
e
In particular min  a /  e  min  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 5 depicts the two guaranteed shares when qe / q varies between 0 and 1, and shows the
superiority of RP versus AC on account of this particular criterion.
We conjecture that the inequality min  a /  e  min  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 1 , and total output to be shared is
29
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.
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.)
30
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