INTERNATIONAL ECONOMIC REVIEW
Vol. 44, No. 3, August 2003
COMMONS WITH INCREASING MARGINAL COSTS:
RANDOM PRIORITY VERSUS AVERAGE COST∗
BY HERVÉ CRÈS AND HERVÉ MOULIN1
HEC Paris and Delta, Paris, France; Department of Economics,
Rice University, Houston
Indivisible units are produced with increasing marginal costs. Under average
cost, each user pays average cost. Under random priority, users are randomly
ordered (without bias) and successively offered to buy at the true marginal cost.
Both average cost (AC) and random priority (RP) inefficiently overproduce. RP
tends to overproduce less, but which game collects more surplus depends much on
the demand configuration. We show that a key to compare the welfare properties
of the two mechanisms is the crowding factor, i.e., the number of potential users
over the number of units of output users can afford: The more crowded the
commons, the more RP outperforms AC. In the quadratic cost case, beyond
the threshold value of 2.4 for the crowding factor, RP strongly outperforms AC;
beneath it AC only mildly outperforms RP. Thus the RP mechanism manages
crowded commons better than AC.
1.
INTRODUCTION
1.1. The Average Cost and Random Priority Mechanisms. Managing a commons with decreasing returns provides, historically at least, the seminal illustration
of the trade-off between private incentives and collective welfare. Hardin (1968)
coined the term “tragedy of the commons” in the context of a malthusian model
of growth, and the concept was developed in great detail by the literature on
exhaustible resources. Examples include fisheries (Gordon, 1954; Lehvari and
Mirman, 1980), forests and water resources (Oström, 1991), and mineral riches
(Dasgupta and Heal, 1979).
The management of queues in congested networks such as the Internet, where
the users have the option to leave the queue (balking), is another case in point.
Marginal waiting cost in a queue increases linearly with the length of the queue;
therefore, in the noncooperative equilibrium, balking is inefficiently low (Naor,
1969; Mendelson, 1985; Demers et al., 1990; Shenker, 1995). How to design a queuing protocol to minimize such inefficiencies is an important and difficult question
on which our model throws some light.
Thus a simple model of the commons involves a large set of potential users
(the agents) and a technology producing indivisible units of a homogeneous good,
∗
Manuscript received January 2001; revised February 2002.
Stimulating conversations with Anna Bogomolnaia and Scott Shenker are gratefully acknowledged. Special thanks to Yan Yu, Vianney Dequiedt, and Liqan Wang, who developed the numerical
computations. Moulin’s work is supported by the NSF, under grant SBR9809316. Please address correspondence to: Hervé Crès, Groupe HEC, School of Management, 78351 Jouy-en Josas, France. E-mail:
[email protected].
1
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CRÈS AND MOULIN
which we call a service, at an increasing marginal cost: cq is the cost of producing the
qth unit and we assume 0 < c1 < c2 < · · · < cq < cq+1 < · · · . Each agent consumes
at most one unit of service; agent i is willing to pay ui , ui ≥ 0, for service.
We compare the welfare performance of two canonical mechanisms, dubbed
average cost (AC) and random priority (RP). Both mechanisms are simple to
describe and easy to implement; their incentives and normative properties makes
them canonical in a sense that we explain now.
The average cost mechanism can be described in two equivalent2 ways: as the
one-shot game where each agent i decides to demand service
or not, xi = 1 or 0,
whereupon all active agents (xi = 1) are served and pay ac( x), and all inactive
agents (xi = 0) get no service and pay nothing; or as the revelation game where
agent i reports ui , the mechanism computes the intersection p̄ = ac(q̄) of the
average cost and reported demand curves, charges p̄ for service to all (active)
agents reporting a willingness to pay no smaller than p̄, and charges nothing to
all other (inactive) agents. The AC direct revelation mechanism is strategyproof,
namely, truthtelling is a dominant strategy. Given the large number of agents, this
is equivalent to the property that, when the agents report truthfully, the resulting
allocation is nonenvious.3 In turn, this explains why the two versions of the AC
game are strategically equivalent. Average cost is clearly the best4 strategyproof
and budget-balanced mechanism where an inactive agent is neither subsidized nor
taxed (his net transfer is zero).
Think now of our model as an assignment problem with a large number of
objects, where each agent must be assigned to at most one object, and object
q, q = 1, 2, . . . , means “service at price cq .” The average cost mechanism assigns
the first q̄ objects to the active agents and uses side payments to ensure that they
all end up paying ac(q̄); the inactive agents are assigned no object.
Instead of side payments, consider lotteries, allocating objects randomly. Much
like AC is the compelling assignment method when side payments are available
but lotteries are not, random priority is compelling when lotteries are available
but side payments are not. The RP mechanism draws at random and with uniform
probability an ordering of the agents, and lets them pick, in that order, the best
remaining object, or no object. Thus the agent ranked first buys service at price c1 ,
if ui > c1 , and declines (gets no object) if ui ≤ c1 ; the agent ranked t is similarly
offered service at price cq+1 , where q is the number of agents ranked before him
who did buy service, 0 ≤ q ≤ t − 1. Thus in every realization of the ordering, every
agent faces a price for service that is independent from his or her own message:
therefore the induced revelation game is strategyproof.5
2
The equivalence is a consequence of the assumption that the number of potential users is large.
Any two active agents are charged the same price for service, any two inactive agents receive the
same net transfer, and the marginal agent is indifferent between being active or inactive.
4 Consider a mechanism with these properties, and fix a demand function: If there is at least one
active agent, then the intersection of the average cost and reported demand curves partitions the
agents into active and inactive. Therefore, our mechanism is either AC or the null mechanism (all
agents inactive) in this particular economy.
5 In the revelation game, each agent reports u and the mechanism charges c
i
q+1 for service to agent
i if ui > cq+1 , and denies service if ui ≤ cq+1 . Note that strategyproofness of RP holds for any finite
number of agents, whereas AC is strategyproof only when the number of agents is large.
3
RANDOM PRIORITY VERSUS AVERAGE COST
1099
Budget balance and the property that an inactive agent gets no cash transfer,
positive or negative, are built in the definition of RP. Conversely, when the number
of agents is large, RP is the only random assignment mechanism meeting the
following three properties: strategyproofness, equal treatment of two agents with
equal willingness to pay for service, and an ordinal version of efficiency relying
only on the ranking of ui and cq , for all i, q (Bogomolnaia and Moulin, 2002).
Last but not the least, when the number of agents is large the (ex ante) random
assignment achieved by the equilibrium of RP is easy to describe: the qth object
is equally split among all agents willing to pay cq or more, i.e., each such agent
receives object q with probability d(c1q ) , where d is the demand function (Crès and
Moulin, 2001).
The AC mechanism is an accurate model of the commons when equality of
the returns to all users is a technological constraint, as in the case of fisheries
or when agents pump water form a common well and the pressure must be the
same for all. On the other hand, the RP mechanism is a plausible stylized description of certain commons with decreasing marginal returns. Think of our agents as
walking around randomly in the forest looking for mushrooms or 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 so on. Another example is 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 that discovers the first and most profitable patent, and
so on.
Our motivation however is mainly normative: if both options (to charge average
cost or to charge marginal price under random ordering) are available, which
one should we recommend? Our answer bears on distributive properties—which
agents receive more surplus in what mechanism?—and on efficiency properties—
which mechanism induces more overproduction, and which one collects more
surplus?
1.2. A Scheduling Example and the Punchline. The two mechanisms AC and
RP are especially relevant to the following simple model of scheduling, that we
will use as our preferred illustration throughout the article.
A server processes (deterministically) one agent per period. The disutility for
waiting is one per period, for every agent. Thus agent i, with willingness to wait ui ,
gets utility ui − t for being served at time t and −t for leaving the queue without
service at time t. All agents are risk neutral. They all show up at time t = 0 and
must decide to stay and wait for service, or to balk. Their decision depends on
their own willingness to wait, on that of other participants, and on the scheduling
protocol.
The first protocol is the unorganized queue: the server picks at random and
with uniform probability one agent in the remaining queue. Thus if q agents decide at time t = 0 to stay in the queue, each one of them faces an expected wait
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CRÈS AND MOULIN
(1 + 2 + · · · + q)/q = (q + 1)/2, and the corresponding game is precisely the AC
mechanism for the technology cq = q.6
The second protocol is the organized queue where the server draws at time t = 0,
randomly and without bias, a scheduling ordering. The agents decide to stay or to
balk after learning their number (whereas in the previous protocol, they take this
decision, in effect, before learning their number). Thus the agent ranked first is
offered service at the waiting price c1 = 1, the agent ranked second is offered the
price c2 = 2 if the first decided to stay, or c1 = 1 if the first balked at t = 0, and so
on. The game is precisely the RP mechanism for the technology cq = q.
Both the organized and unorganized scheduling protocols are commonly observed. The organized one is the familiar “first-in, first-out” method, of which
the analysis is simplified by the assumption that all participants arrive, in random
order, at time t = 0. Most queues among real people (at the post office, in shops,
etc.) are “organized,” and the actual distribution of numbered tickets is common
practice when the wait is expected to be long.7 On the other hand, many “mechanical” queues follow the unorganized pattern, as when we try to get a line from a
busy phone system, access the Internet, and so on.8
Consider the case of a perfectly homogeneous (infinitely elastic) demand,
namely all agents have the same willingness to wait u. Efficiency commands
to serve exactly qE agents, where qE is the largest integer smaller than u. Assume for simplicity u = qE + 23 . Under AC (unorganized protocol), a queue of
length 2qE will form at t = 0 : Indeed, 2qE is the largest integer x such that
ac(x) = (x + 1)/2 ≤ u. Therefore, everyone expects9 to wait qE + 12 , and the overall surplus collected is q3E . Contrast this with the fully efficient outcome of RP
(organized protocol), where exactly qE agents decide to stay at t = 0 : An agent
drawing a number larger than qE observes that qE other agents have decided
to stay and balks. Not only does RP in this example collect the efficient surplus
1
q (q + 13 ), whereas AC dissipates almost all of it; the RP outcome is also (ex
2 E E
ante) Pareto superior to the AC outcome.
The next configuration of the demand is one where, symmetrically, the AC
outcome is fully efficient whereas RP leads to overproduction, and dissipates
most of the surplus. The “waiting technology” is still cq = q and the n agents are
of two types: agent 1 is willing to wait u1 = 2 + ε, whereas agents 2, . . . , n have
ui = 1 + ε, where ε is such that 0 < ε < 12 . In the equilibrium of AC, all agents
but agent 1 balk at t = 0 (given that agent 1 stays, their expected wait is no less
than ac(2) = 1.5) and the efficient outcome qe = 1 ensues, with associated surplus
u1 − c1 = 1 + ε. On the other hand under RP, when agent i, i ≥ 2, gets the highest
priority she stays in the queue, after which all other agents balk, except agent 1: The
Once an agent finds it worthwhile to stay for the first period—as ui ≥ (q + 1)/2—he will have no
incentive to leave later because the expected wait decreases in each period.
7 There are examples of unorganized (AC) queues among real people: think of a packed bar where
the bartender cannot keep track of “who is next.”
8 Some mechanical queues are better described as following the organized pattern (RP): think of a
customer assistance line where a machine announces the waiting time until the next available agent.
9 Half of the agents actually wait longer than u, but it is rational for them to enter the queue at
t = 0, and to stay after each unlucky draw, because past waiting is a sunk cost.
6
RANDOM PRIORITY VERSUS AVERAGE COST
1101
result is inefficient overproduction (an inefficiently long queue) with probability
n−1
; the (ex ante) equilibrium surplus is less than n1 + 2ε, namely a vanishingly
n
small fraction of the efficient surplus if n is large enough and ε small enough. Note
that agents 2, . . . , n all prefer the RP outcome to the AC outcome; hence the latter
is not Pareto superior to the former.
We summarize the main findings of the article in the light of these two examples. Both mechanisms, in general, inefficiently overproduce but RP never overproduces by more than 100% (for any technology and any configuration of the
demand), whereas overproduction in AC is potentially unbounded: Proposition
4. The RP outcome may be Pareto superior to the AC outcome (as in the homogeneous demand example), but the converse (AC outcome Pareto superior to RP
outcome) is impossible: Proposition 3.
Other results comparing the welfare performance of AC and RP rely on the familiar assumption that the demand function can be approximated by a continuous
downward sloping function: This assumption rules out the second example above.
The concavity (resp. convexity) of the function 1/d implies that AC overproduces
less (resp. more) than RP, and we argue that the latter is more plausible for the
usual functional forms of demand functions: Propositions 5–7.
Next we define a numerical index, that we call the crowding factor, and that
is easily computed from the demand and marginal cost functions. The crowding
factor is the ratio of the potential demand (the number of agents who would like
to be served if service is free) over the maximal production (the largest willingness
to pay, i.e., the largest q such that cq ≤ maxi ui ). Thus it is a measure of the level
of congestion of the system.
We show that the relative welfare performance of the two mechanisms is critically affected by this factor. Specifically, we show that the more crowded the commons, the more the RP mechanism outperforms the AC one. In general, when
the crowding factor is very large, the latter collects no surplus whereas the former
collects a positive fraction of the efficient surplus (Proposition 8). In the canonical
model with quadratic costs (as in the scheduling example), the crowding factor is
an exact measure of the relative performance of our two mechanisms. See Sections
3, 4, and Subsection 5.1 for precise statements.
1.3. Organization of the Article. We define the model in Section 2. We have
a continuum of agents, each of which wants at most one unit of service, and
the profile of utilities (willingness to pay for service) is described by a demand
function p −→ d( p) ≡ the number of agents willing to pay p or more. Assuming a
continuum of players raises some technical difficulties (e.g., for the noncooperative
game), but on the other hand it yields a simple description of the equilibrium
outcomes of AC and RP, and simple formulas for measuring their inefficiency,
both in terms of overproduction and of surplus loss: Proposition 1.
Section 3 is devoted to the important special case where the demand and
marginal cost functions are both linear. The only relevant parameter in this case is
the crowding factor: It determines entirely the overproduction and surplus loss—
relative to the efficient benchmark—of our two mechanisms. Beyond the crowding
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CRÈS AND MOULIN
factor 2.4, we find that RP strongly outperforms AC; beneath this critical level,
AC mildly outperforms RP: Proposition 2.
In Section 4, we go back to general demand functions and discuss the distributive
consequences of the two mechanisms. A lower tail of the demand always prefers
RP to AC whereas the agents preferring AC to RP form a—possibly empty—
upper tail of the demand: Proposition 3. In particular, if the demand is flat (identical agents), the RP outcome is fully efficient whereas the AC outcome entirely
dissipates the surplus. Some convexity properties of the demand give an edge to
one mechanism over the other, irrespective of the cost function: to AC over RP
if 1/d is concave (Proposition 5) and to RP over AC if 1/d is convex (Proposition
6). Finally, Proposition 8 explains that RP asymptotically outperforms AC when
the crowding factor goes to infinity as a result of replicating the demand function
while keeping the technology fixed.
We conclude in the last section 5. Before discussing some variations and possible
extensions of our model (Subsection 5.2), we mention in Subsection 5.1 a discrete
version of our model (see Crès and Moulin, 1999), in which all definitions and
results proposed in the present paper have a counterpart. The interest of the discrete model is twofold. First, it provides a rigorous foundation for the continuous
model, by viewing the latter as the limit of a natural sequence of discrete models.
Second, the discrete model allows us to make a statistical argument in favor of
RP over AC. We assume quadratic costs and parameterize the demand space in a
natural way. We show that with many agents, for almost all choices of the demand
function, the pattern of relative overproductions and surplus losses under AC and
RP is precisely the same as in the linear demand case of Section 3: Proposition 9.
This in turn gives a broader support to our finding that 2.4 is the critical value of
the crowding factor beyond which we expect RP to strongly outperform AC.
2.
THE MODEL
An economy consists of marginal cost and demand functions, denoted mc and d.
The function mc is defined for all x, x ≥ 0, strictly increasing and continuous. We
assume mc(0) ≥ 0 and mc(∞) = ∞. We denote by C the cost function and by ac
the average cost function. The demand function d( p) is defined for all nonnegative
prices p; d is nonincreasing and continuous; moreover, for the sake of simplicity,
we assume there is a price P above which nobody wants to buy the good: d(P) = 0.
Denote X0 = d(mc(0)) the size of the potential demand; and set mc(X1 ) = P. If
mc(0) is positive, the part of the demand functions for 0 ≤ p < mc(0) (the agents
willing to pay less than mc(0) for service) plays no role in either the AC or RP
mechanisms: These agents never buy service. Therefore, we can assume mc(0) = 0
without loss of generality.
The crowding factor of the economy (d, mc) is γ = X0 / X1 . Its role appears in
the next sections.
The efficient quantity qE is at the intersection of the marginal cost and inverse
demand curves, i.e., it is the unique solution of d(mc(qE )) = qE (see Figure 1). The
efficient surplus σ E is the sum of the consumer surplus at the “competitive” price
mc(qE ) and of the competitive profit:
RANDOM PRIORITY VERSUS AVERAGE COST
1103
FIGURE 1
THE CROWDING FACTOR
(1)
X1 / X0
σ E = D(mc(qE )) + qE · mc(qE ) − C(qE )
where D( p) is the consumer surplus at price p:
P
D( p) =
p
P
d(y)dy = −
˙
(y − p)d(y)dy
p
The average cost mechanism is the noncooperative game (with a continuum of
players) where each agent chooses to buy service or not, xi = 0 or 1. An agent
who does not
buy pays nothing. All agents who do buy pay the average cost ac(q),
where q = x is the total number of agents who buy.
In equilibrium the quantity qAC produced is where the average cost and inverse
demand curves intersect, i.e., it solves the equation:
d(ac(qAC )) = qAC
Our assumptions on d and mc guarantee a unique solution. An agent with valuation
(willingness to pay for service) above qAC buys service and pays ac(qAC ), and one
with valuation below qAC does not buy. The AC equilibrium surplus is simply the
consumer surplus at price ac(qAC ):
(2)
σAC = D(ac(qAC ))
The random priority mechanism works as follows. Nature draws an ordering of all
agents, with uniform probability on all orderings; agent are successively offered
service at an increasing price (defined inductively) and
can accept or refuse, xi = 0
or 1; the price offered to agent i is mc(q), with q = x, where the sum bears over
all agents preceding i in the ordering in question.
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CRÈS AND MOULIN
The inductive definition of the above integral is a technical issue that we choose
to ignore at this point. The discussion of the RP equilibrium outcome below is
similarly heuristic. In Subsection 5.1, we mention a more rigorous approach (Crès
and Moulin, 1999), where the RP and AC games with a continuum of agents are
viewed as the limit of a sequence of discretized games with finitely many agents.
Think of the RP mechanism as allocating infinitesimally small units δ of the
good. After a total of q units have been allocated, the demand above p = mc(q)
has “shrunk” by a factor λ(q), 0 ≤ λ(q) ≤ 1, because some of these agents are
already served. The shrinking factor is uniform because all agents above p are
treated symmetrically with respect to the allocation of the q units. The shrunk
demand is thus
dq ( p ) = λ(q) · d( p )
for all
p ≥ p = mc(q)
The evolution of λ follows a differential equation. Let δ be a small increment of the
quantity offered, and be the corresponding price increase, = ṁc(q) · δ, where
ṁc is the derivative of mc. Then
dq+δ ( p + ) = dq ( p + ) − δ
where we neglect a second-order term due to the fact that some of δ is allocated
to the agents in [ p, p + ]. Combining the two equations above
λ(q + δ) · d( p + ) = λ(q) · d( p + ) − δ =⇒ λ(q + δ) − λ(q)
=−
δ
δ
=−
d( p + )
d(mc(q))
where in the right-hand side equality we again neglect a second-order term. Define
φ(q) =
0
q
dt
d(mc(t))
with the convention φ(q) = ∞ whenever q ≥ mc−1 (P)
Now the differential equation governing λ and the initial condition λ(0) = 1 yield
λ(q) = 1 − φ(q) as long as φ(q) ≤ 1
Two cases may arise. We may reach a level qRP such that φ(qRP ) = 1, in which
case all agents in d(mc(qRP )) have already been served when qRP units have been
allocated. In this case, the RP mechanism stops because no agent will accept an
offer any longer. If, on the other hand we reach qRP = mc−1 (P) with φ(qRP ) ≤ 1,
then there is no agent left in d(mc(qRP )) and the mechanism stops as well. In
both cases we have dqRP ( p) ≡ 0 whenever p ≥ mc(qRP ) so that qRP is the quantity
produced in equilibrium.
Going back to the definition of λ, we consider an agent i with willingness to pay
p. For any q such that mc(q) ≤ p, agent “p” buys the good at price mc(q) or less
with probability 1 − λ(q). Hence his total probability of service is
1105
RANDOM PRIORITY VERSUS AVERAGE COST
xp =
1
if
mc−1 ( p) > qRP
φ(mc−1 ( p))
if
mc−1 ( p) ≤ qRP
(Note that if qRP = mc−1 (P), the upper case is vacuous.) And his expected cost
share is
b
mc(t)
yp =
dt where b = inf mc−1 ( p), qRP
0 d(mc(t))
Finally, we can compute the surplus collected by RP in equilibrium:
σRP =
0
∞
∞
˙ p) | dp = −
(x p · p − yp ) | d(
=−
0
qRP
0
∞
˙
( p − mc(t))d( p)dp
mc(t)
b
0
p − mc(t)
˙ p)dp
dt d(
d(mc(t))
dt
d(mc(t))
Recall that D( p) is the consumer surplus at p; now we get the formula
qRP
σRP =
(3)
0
D(mc(t))
dt
d(mc(t))
We summarize the above discussion as follows.
PROPOSITION 1. At the RP equilibrium outcome the quantity qRP is produced
with probability one, where
qRP = X1
(4)
qRP solves
qRP
0
if
dt
d(mc(t))
=1
X1
0
dt
d(mc(t))
≤1
otherwise
q̄ dt
, where
An agent willing to pay p = mc(q) buys service with probability 0 d(mc(t))
q̄ mc(t)
q̄ = min{q, qRP } and his expected payment is 0 d(mc(t)) dt. Total surplus collected
at the RP equilibrium is given by (3).
Our goal in the next two sections is to compare qE , qAC , and qRP as well as
σ E , σAC , and σRP . We start by the canonical example where both the demand and
the marginal cost functions are linear.
3.
THE CASE OF QUADRATIC COSTS AND LINEAR DEMANDS
In this section, we assume that marginal costs increase linearly, mc(x) = a · x
for all x > 0 where a is a positive constant.
Consider first the case of a flat—infinitely elastic—demand function: d( p) = X0
for p < P, = 0 for p > P, where X0 is “large,” namely X0 > 2P/a. The demand
is perfectly homogeneous, all agents are willing to pay P for service. Under the
AC mechanism, we have a full “tragedy of the commons,” namely, inefficient
1106
CRÈS AND MOULIN
FIGURE 2
AC EQUILIBRIUM WITH ZERO SURPLUS
overproduction, qAC = 2P/a = 2qE , up to the point where the average cost equals
the common valuation P and so the AC equilibrium dissipates all surplus: σAC = 0
(see Figure 2).
We can use Proposition 1 to compute the RP equilibrium, provided we account
for the discontinuity in d and D. But the direct argument is obvious: The first qE
agents in the random ordering buy the service, and the subsequent agents decline.
Thus the RP equilibrium is first best efficient: qE = qRP and σ E = σRP . The RP
outcome is preferred by every agent to the AC outcome: Everyone gets a fair
share of the efficient surplus under RP and no surplus at all under AC.10
We turn to the case of a linearly decreasing demand function, namely
γ
 · (P − p)
d( p) = a

0
(5)
for
0≤ p≤ P
for
P≤p
In this economy, depicted in Figure 3, we have X0 = γ · P/a and X1 = P/a so that
γ is the crowding factor defined in the preceding section. Applying Proposition 1
delivers easily:
PROPOSITION 2. In the economy with linear marginal cost mc(x) = a · x and
linear demand (5), we have
qE =
P γ
a γ +1
qRP =
P
(1 − e−γ )
a
qAC =
P 2γ
a γ +2
and
σRP
γ +1
(1 − e−2γ )
=
σE
2γ
σAC
4(γ + 1)
=
σE
(γ + 2)2
The straightforward proof is omitted.
10
Note that this conclusion is valid for any marginal cost function.
RANDOM PRIORITY VERSUS AVERAGE COST
1107
FIGURE 3
OVERPRODUCTIONS IN THE AC EQUILIBRIUM
Thus the relative overproductions qAC /qE and qRP /qE , as well as the relative
surpluses σAC /σ E and σRP /σ E only depend upon the crowding factor γ . Figures 4
and 5 depict the overproductions and surpluses when γ varies between 0 and +∞.
The first observation is that qRP ≤ qAC at all levels of γ : The RP equilibrium
overproduces less than AC. On the other hand, AC collects more surplus than RP
when γ ≤ 2.3994 and vice versa:
γ < γ ∗ =⇒ σAC > σRP
γ > γ ∗ =⇒ σAC < σRP
where γ ∗ = 2.3994 is the solution of eγ = γγ +2
.
−2
Figures 4 and 5 also make clear that the relative advantage of RP and AC
is monotonic in γ . First the ratio qAC /qRP increases with γ . Next we note that,
although when γ < 2.4 AC collects only a bit more relative surplus than RP
(σAC /σRP never exceeds 1.03), the ratio σAC /σRP decreases for γ > 1.5 and goes
to zero as γ goes to infinity, at the rate of 8/γ .
Finally, we note that as γ goes to infinity, the RP equilibrium collects only
one half of the efficient surplus: Thus we cannot think of the flat demand case—
discussed at the beginning of this section—as the limit of the γ -linear economies.
Another difference is that, for any γ , the agents with valuation close enough to P
prefer AC to RP. At the AC equilibrium, they get service at the following price:
pAC = ac(qAC ) = P
γ
γ +2
At the RP equilibrium, the price they pay is given by Proposition 1:
qRP
pRP =
0
a·t
γ − 1 + e−γ
dt = P
d(a · t)
γ
One checks easily that pRP ≥ pAC for all γ . Recall that if the demand is flat, all
agents strictly prefer RP to AC.
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CRÈS AND MOULIN
FIGURE 4
COMPARING OVERPRODUCTION IN AC AND RP
4.
WELFARE COMPARISONS IN THE GENERAL CASE
We go back to the general model of Section 2 and compare first the distributive
consequences of our two mechanisms. In the AC equilibrium outcome, all the
“efficient” agents (i.e., the qE agents with the largest willingness to pay) are served
with probability one. The RP equilibrium allocation, on the other hand, guarantees
service only to those agents willing to pay mc(qRP ) or more, and this does not
include all efficient agents because, typically, qRP is strictly larger than qE .
The AC allocation offers no service at all (and zero surplus) to agents with low
utility (i.e., below ac(qAC )), whereas the RP allocation gives a positive probability
FIGURE 5
COMPARING SURPLUS LOSSES IN AC AND RP
1109
RANDOM PRIORITY VERSUS AVERAGE COST
of service to any agent with valuation greater than mc(0) = 0. This is the source of
inefficiency in the equilibrium of RP: efficiency—interpreted as surplus maximization rather than Pareto optimality—forbids serving with any positive probability
an agent with valuation smaller than mc(qE ).
The patterns “low valuation agents prefer RP to AC” and “high valuation agents
prefer AC to RP” are general, with one important qualification: There may be no
one preferring AC.
PROPOSITION 3.
0 ⇐⇒ P > 0;
Consider an economy (d, mc) with a positive surplus: σ E >
(a) there is a valuation p̃, 0 < p̃ ≤ P, such that every agent with valuation in
]0, p̃[ strictly prefers the RP equilibrium to the AC one;
(b) if there is at least one agent that does not strictly prefer the RP equilibrium
to the AC one (resp. that strictly prefers AC over RP), the set of those agents
forms an interval [ p̃, P] (resp. an interval ] p̃, P]);
(c) there are economies where every agent strictly prefers the RP equilibrium
to the AC one.
PROOF. Statement (a) holds with p̃ = ac(qAC ): See the discussion immediately
q
dt
preceding the proposition. For statement (b), assume that qRP solves 0 RP d(mc(t))
=
1 (Proposition 1). Write pAC = ac(qAC ) and observe pAC ≤ pE ≤ mc(qRP ) (both
AC and RP overproduce). By Proposition 1 an agent p such that pAC ≤ p ≤
mc(qRP ) and p = mc(q), weakly prefers AC over RP if and only if
p − pAC ≥
0
q
dt
d(mc(t))
q
p−
0
mc(t)
dt
d(mc(t))
Taking the derivative with respect to q on both sides (recall p = mc(q)), we find
q 1
·
·
·
mc (q) on the left side and ( 0 d(mc)
) mc (q) ≤mc (q) on the right side. Therefore,
the set of agents p in [0, mc(qRP )] weakly preferring AC to RP is empty or is
an interval [ p̃, mc(qRP )]. In the latter case, all agents p, p ≥ mc(qRP ), share this
preference: They get exactly the same allocation as the agent mc(qRP ), whether
the mechanism is AC or RP. The proof in the case qRP = X1 is similar. Statement
(c) is established by the case of an infinitely elastic demand discussed above. The next four propositions compare the performance of AC and RP in terms
of their relative overproduction and surplus loss, just like in Proposition 2.
PROPOSITION 4.
(a) Both the RP and AC equilibrium outcomes inefficiently overproduce: qE ≤
qAC , qRP . The overproduction under RP is at most 100%: qRP ≤ 2qE .
(b) If RP overproduces more than AC, it also collects less surplus:
qAC ≤ qRP =⇒ σAC ≥ σRP
qAC < qRP =⇒ σAC > σRP
1110
CRÈS AND MOULIN
PROOF. That AC overproduces is well known. To prove that RP overproduces,
and by at most 100%, use the computation of qRP in Proposition 1 and the following
inequalities:
1≥
qRP
qE
qE
dt
dt
≤
= 1 ⇒ qE ≤ qRP
d(mc(t))
d(mc(qE ))
0
0
qRP
dt
dt
qRP − qE
⇒ qRP ≤ 2qE
≥
=
d(mc(t))
d(mc(qE ))
qE
qE
qE
This proves statement (a). To prove statement (b), we combine the inequality:
for all t, 0 ≤ t ≤ X1 : D( pAC ) − D(mc(t)) ≥ d(mc(t))(mc(t) − pAC )
a consequence of the convexity of D; and:
qRP
σAC − σRP = D( pAC ) −
0
D(mc(t))
dt ≥
d(mc(t))
qRP
0
D( pAC ) − D(mc(t))
dt
d(mc(t))
a consequence of Proposition 1. They imply
σAC − σRP ≥ C(qRP ) − qRP pAC = qRP (ac(qRP ) − ac(qAC ))
The desired conclusion follows from the monotonicity of ac.
In the linear economies of Section 3, the overproduction of the AC equilibrium
is at most 100% (it is at most 30% at the RP equilibrium: See Figure 4). However,
the cap of 100% on the overproduction under RP (statement (a)) is independent
of both the demand and marginal cost functions. It is easy to construct examples
where the overproduction under AC is arbitrarily large.11
The linear economies of Section 3 provide examples where qAC > qRP and σAC
may be larger or smaller than σRP . Statement (b) tells us that the only other
possible configuration is when AC overproduces less and collects more surplus.
Our next result provides a family of examples.
PROPOSITION 5. If 1/d is a concave (resp. strictly concave) function on the interval [0, mc(qAC )] then qAC ≤ qRP (resp. qAC < qRP ).
PROOF. Fix mc and d. Apply Jensen’s inequality to this function and the following integral:
1
qAC
qAC
0
1
1
dt ≤
d(mc(t))
d
1
qAC
qAC
mc(t) dt
=
0
1
d(ac(qAC ))
By definition, qAC is the solution of d[ac(q)] = q so the above inequality gives
11
Take mc sharply concave and d flat or nearly flat. We omit the details.
RANDOM PRIORITY VERSUS AVERAGE COST
0
qAC
1111
1
dt ≤ 1
d(mc(t))
implying qAC ≤ qRP by Proposition 1.
PROPOSITION 6. If 1/d is a convex (resp. strictly convex) function on [0, P[ then
qAC ≥ qRP (resp. qAC > qRP ).
PROOF. From Proposition 1, qRP is not larger than X1 , therefore if qAC ≥ X1
there is nothing to prove. Assume now qAC < X1 ⇐⇒ mc(qAC ) < P. From Proposition 1 and Jensen inequality for the function 1/d on the interval [0, mc(qAC )], we
have
0
qAC
1
dt ≥ 1 ≥
d(mc(t))
and the desired conclusion qAC ≥ qRP .
qRP
0
1
dt
d(mc(t))
Consider the demand function derived froma single (representative) consumer
ρ
with CES utility function u(x1 , x2 , . . . , x ) = ( j=0 α j x j )1/ρ (see Mas-Colell et al.,
1995 or Chung, 1994), where ρ is between −∞ (Leontief utility function), and 1
(linear utility); recall that ρ = 0 corresponds to a Cobb–Douglas utility function.
1
One computes easily d( p) = W( p + βp 1−ρ )−1 where p is the price of good 1 (other
prices are fixed), W and β are constant. Therefore, 1/d is concave (Proposition
5) if 0 < ρ < 1, and convex (Proposition 6) if ρ < 0. For a Cobb–Douglas utility,
overproduction is the same in both mechanisms, but AC collects a higher surplus.
Note that in order to compute the surplus, we must truncate d above some large
enough finite price P; however, the surplus difference σAC − σRP does not depend
on P.
In general, it is easy to come up with demand functions d vanishing at a finite
price P and such that 1/d is convex: This is explained in the next paragraph. By
contrast, if the function d vanishes at P, then 1/d has a vertical asymptote at P,
and hence cannot be concave.
Recall that a function is called Log-concave if Log f is concave. Then we have
d concave =⇒ d Log-concave =⇒ 1/d convex. Consider a demand function of the
form d( p) = f (α − βp), 0 ≤ p ≤ α/β, where f is increasing and f (0) = 0. Then
1/d is convex if 1/ f is convex. If this holds true, and mc(qAC ) ≤ P = α/β, we
deduce from Proposition 6 that qAC ≥ qRP (and qAC > qRP if 1/ f is strictly convex).
An example is the demand d( p) = (α − βp)ν for any positive exponent ν, because
the function f (x) = x ν is Log-concave.
Our next result offers a family of demand functions for which we guarantee the
pattern qAC ≥ qRP and σAC ≥ σRP .
PROPOSITION 7. If the function D/d is affine on the interval [0, mc(qAC )] then
1/d is convex and we have qAC ≥ qRP and σAC ≥ σRP .
1112
CRÈS AND MOULIN
PROOF. First we check that 1/d is convex. Set f = 1/d. If f is twice differentiable, its convexity obtains easily by differentiating D/d = D · f twice:
˙ D · f = −d · f + D · f˙ = −1 + D · f˙
and
¨ d · f˙
0 = D · f = −d · f˙ + D · f¨ =⇒ f¨ =
≥0
D
The easy argument in the case where f is not twice differentiable is omitted.
As 1/d is convex, Proposition 6 implies qAC ≥ qRP . We combine this inequality
with the fact that an affine function commutes with the integral operation:
qAC
D
D(ac(qAC ))
D
σAC
1
mc(t) dt =
(mc(t)) dt =
=
qAC 0
d
d qAC 0
d(ac(qAC ))
qAC
qAC
qRP
D
D
=⇒ σAC =
(mc(t)) dt ≥
(mc(t)) dt = σRP .
d
d
0
0
1
qAC
Consider the demand function d( p) = (α − βp)ν , with ν > 0. The function D/d is
affine on the interval [0, P] and only there. Therefore Proposition 7 only applies
to a marginal cost function small enough such that mc(qAC ) ≤ P. For instance in
the case of a linear demand (ν = 1) and a linear mc function as in Section 3, this
inequality holds if and only if the crowding factor is at most 2.
The last result shows that for infinitely crowded commons, the RP equilibrium systematically outperforms the AC one. We fix an arbitrary economy E(1) =
(d, mc) and we replicate the demand to λ · d, for some λ ≥ 1, without replicating
the marginal cost function: The crowding factor of the economy E(λ) = (λ · d, mc)
is thus γ (λ) = λ · γ (1). We compare the two mechanisms as λ goes to infinity.
It is easy to check on the formulas of Section 2 (e.g., (2)) that qE (λ), qAC (λ),
and qRP (λ) are increasing in λ. Moreover when λ goes to infinity, the proportions
of the efficient surplus generated by RP and AC converge. The latter converges
to zero: For infinitely crowded commons, AC collects almost nothing of the efficient surplus. The former converges toward a strictly positive ratio (of course
smaller than 1), which means that RP always collects a significant proportion of
the efficient surplus, even for infinitely crowded commons. Formally,
PROPOSITION 8.
lim qAC (λ) ≥ lim qRP (λ) = lim qE (λ) = X1
λ→∞
λ→∞
λ→∞
and
σAC
lim
(λ) = 0
λ→∞ σ E
and
X1 D(mc(t))
dt
σRP
0
d(mc(t))
lim
(λ) =
λ→∞ σ E
P − ac(X1 )
PROOF. The first assertion, lim qRP = X1 , follows easily from Proposition
1. Moreover, for λ infinite, (λd)−1 (t) → P for all t; therefore, we get
RANDOM PRIORITY VERSUS AVERAGE COST
1113
straightforwardly lim qE = X1 . Finally, the limit value of qAC when λ → ∞ satisfies: lim qAC = ac−1 (P).
As a consequence limλ→∞ D[ac(qAC )] = 0, which implies that
q σAC /σ E converges to zero. The last equation follows from the fact that q1E 0 E [(λd)−1 (x) −
mc(x)]dx tends toward P − ac(X1 ).
5.
CONCLUDING COMMENTS
5.1. A Founding Discrete Model: Statistical Comparison of RP and AC. The
extended version of the present paper (Crès and Moulin, 1999) discusses our
model when the set, N , of agents is finite. This allows for a rigorous definition of
the strategic games AC and RP. All results of Sections 3 and 4 have a counterpart in
the discrete model. Moreover, the (continuous) model of the present article can be
viewed as the limit of the discrete model when the size, n, of the population tends
to infinity. Thus the results in the discrete model are the mathematical justification
of those of the present article.
In the model with a finite set of agents, a demand is identified to a profile of types.
Each agent i has a willingness to pay ui for service; agent i is said to be of type q
when cq < ui ≤ cq+1 . The profile of types is the sequence (n0 , n1 , n2 , . . . , n Q) (such
Q
that q=0
nq = n), where nq is the number of agents of type q. The crowding factor
is γ = n/Q, where Q is the largest integer q such that ui ≥ cq for some i ∈ N .
The discrete model allows for a statistical comparison of RP and AC in
economies with quadratic costs and a large number of agents. This comparison
implies that the results of Section 3 apply to a much larger class of demand configurations. If we randomize over demand functions (i.e., profiles of types), for almost
all choices of the demand function the pattern of relative overproductions and
surplus losses under AC and RP is precisely the same as in the linear demand case
of Section 3! This is, of course, an asymptotic result that holds when the number
of agents is large.
The randomization is natural: all n agents are dispatched within Q types at
most12 (some types can be empty, even the last one, Q). We call Bose–Einstein
statistics (BE statistics in short), the uniform distribution over all these possible
profiles of types.13
12 The number of overall possible profiles of types is the number of ways one can distribute n balls
into Q urns:
Q+ n− 1
.
Q− 1
13 The Bose–Einstein statistics over profiles of types results from a simple urn model, known as
the Polya–Eggenberger urn schemes. The randomly ordered n agents select, in sequence, a ball in
an urn containing Q balls, one for each type. After each draw, the agent puts back the ball into the
urn together with one other of the same type. Consequently, all agents modify the urn (which at the
end of the process contains Q + n balls) and a type which has already been drawn is more likely to be
drawn once more. Hence the Bose–Einstein statistics assumes some homogeneity of preferences; more
homogeneity at least than if the urn were not modified (this last case corresponds to the Maxwell–
Boltzmann statistics, which obviously generates an almost uniform profile of types if n Q; then
Proposition 9 straightforwardly holds true).
1114
CRÈS AND MOULIN
The asymptotic BE statistics is the limit of this probability distribution when the
number n of agents, as well as the number Q = n/γ of units distributed (where γ is
fixed), goes to infinity. Then the following convergence result obtains: The variance
(n)
(n)
of the random variable xM = qM /Q (where M = E or RP or AC) converges to
zero asymptotically. Hence its limit xM behaves according to a Dirac measure with
(n)
probability 1 on the asymptotic limit of the mean of the random variable qM /Q.
PROPOSITION 9. Asymptotically under the BE statistics, with probability one, the
efficient proportion xE of the Q units that should be distributed, and the proportion
xRP (resp. xAC ) of the Q units that is distributed by RP (resp., by AC) under the
assumption of quadratic costs are
xE =
γ
γ +1
xRP = 1 − e−γ
xAC =
2γ
γ +2
Moreover, asymptotically under the BE statistics, with probability one, the ratio of
the efficient surpluses secured by RP and AC are
σRP
γ +1
=
(1 − e−2γ )
σE
2γ
σAC
4(γ + 1)
=
σE
(γ + 2)2
See Crès and Moulin (1999) for a proof.
Our finding that 2.4 is the critical value of the crowding factor beyond which
we expect RP to strongly outperform AC (Proposition 2) remains true in the
statistical model just discussed.
5.2. Variations and Extensions. 1. Does RP outperform AC as a simple mechanism to manage the commons?
The answer is unambiguously yes when the distribution of types (willingness
to pay) is uniform or nearly uniform (see the discussion of Section 3) or from a
statistical point of view by taking demand functions “at random” (see discussion
of Subsection 5.1); and Proposition 8 establishes that the answer is yes whenever
the commons are sufficiently crowded.
The most appealing feature of these results is that the crowding factor is determined by the intercepts of the demand function with the two axes, namely, the
number of potential users of the commons and the largest willingness to pay: The
details of the demand between these two points do not matter. Thus the crowding
factor ought to be easy to estimate empirically.
On the other hand, two important assumptions limit the scope of our analysis:
Marginal costs are increasing and agents are risk neutral. For instance, the RP
mechanism collapses under decreasing marginal costs and uniform demand, if
the first (highest) marginal cost exceeds the common willingness to pay. And risk
aversion works against RP but has no effect on AC when the latter is implemented
via side payments (Subsection 1.1) instead of lotteries (Subsection 1.2).
2. The robustness of our results could be tested in several natural variations
of the model. One of them is the dual “output sharing” game, where each agent
contributes 0 or 1 units of input (e.g., labor) and total output F(y) produced
RANDOM PRIORITY VERSUS AVERAGE COST
1115
when y agents do work must be shared among all 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 the fishing, logging, or pumping effort
and output is the total catch. It is easy to adapt our results to the output sharing
context.
Another natural variation of our model is to allow for variable demands: Agents
can buy more than one unit of the good. The RP mechanism generalizes as follows: Units of output with increasing marginal costs are successively offered to an
agent selected randomly and without bias from the pool of active agents—where
an agent is active as long as he did not refuse any such offer. When the pace
of discretization (the size of each indivisible unit) goes to zero, this mechanism
converges to the serial cost sharing mechanism analyzed by Moulin and Shenker
(1992). The welfare comparison of the serial and average cost sharing mechanisms is still a widely open subject: See, however, Moulin and Shenker (1994) and
chapter 6 in Moulin (1995).
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