Random Priority A Probabilistic Resolution of the Tragedy of the Commons

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 expect

* HEC School of Management, 78351, Jouy-En-Josas Cedex, France, cres@gwsmtp.hec.fr

** Department of Economics, Box 90097, Duke University, Durham NC 27708,

moulin@econ.duke.edu

*** Stimulating conversations with Scott Shenker, are gratefully acknowledged Special thanks

to Anna Bogomolnaia who contributed Proposition 9, and Yan Yu for some numerical simulation

Random Priority:

A Probabilistic Resolution of the Tragedy of the Commons

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 q e agents with highest willingness to

wait, where q e is the largest number q such that there are q or more agents of type q or more.

We consider two simple and natural mechanisms (protocols, in the terminology of the

queuing literature), in which each agent must take a binary decision (in or out) and that lead to transparent strategic equilibrium behavior Consequently, neither of these mechanisms

implements full efficiency as just described, and we wish to compare their respective

inefficiencies

In both protocols the order in which agents are served is randomly drawn with uniform

probability on all orderings In the first mechanism, average delay, the agents must decide at date t 0 to stay or leave before learning their place in the line: if x agents decide to stay, they

each face an expected wait (x 1 2 (namely the average delay per agent) Say that in ) /

equilibrium q a agents decide to stay Each of them has a willingness to wait not lower than(q a1 2 and all agents who choose to leave have it not higher than () / q a 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 familiarchoice 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.

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 randomordering 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 ofwhom 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

10, two agents stay (one of the first 9 and the last one) for a surplus of ( 11 1 ) (  21 2 )0 2 , 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 ofrandom 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

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 (marginalcost 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 oneworker 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

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 variousorderings, 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

We now present the general model and our three mechanisms A typical agent i is willing to pay u u i, i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 c q,0c1c2  where c, q is

the marginal cost of producing the q-th unit thus total cost of q units is c1c2  c q

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 themechanism 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.

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 c q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 forfruits 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.

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 c q and no larger than c q1 Denote by

m k the number of agents of type k or more The idea of PS is that these m k agents are equally

entitled to purchasing the good at price c k Hence our agent of type q has an option to buy the good at price c1 with probability 1

1

m , at price c2 with probability 1

2

m , , at price c q with probability 1

m q If m11  m1q 1, our agent will exercise all his options; if m11  m1q 1, he

will exercise only the best ones, that is to say all such options up to the highest index k such that

m   m k  , 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 bythe 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

c q ?) 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 probabilisticallocation 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

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 importantlinks between these two allocations:

 They are identical for the most "inefficient" agents (those with type at most q e where q e is the efficient output level)

 Both mechanisms overproduce; but not by more than 100%; their expected output satisfy the following inequalities:

q eq r q s 2q e

 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 mainresult, is a convergence result about the equilibrium outcomes of RP and PS when the continuous

model is viewed as the limit of a sequence of discrete models (with a finite but increasing

number of agents) It turns out that the RP and PS equilibrium outcomes converge to the same limit; in other words the welfare superiority of PS over RP disappears in the limit model with a continuum of agents Moreover, Theorem 2 provides an explicit formula describing this

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 ACbased 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 firsttwo 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 u u i, i 0 When the consumption of

agent i is a random variable, we interpret u i 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 c q the q-th marginal cost: c q C q( ) C q(  1 We assume)

C( )0  and0

0c1 , c q c q1 for q1 2, ,

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 y i

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 andchooses 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 c q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 u i c q); 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 socialchoice 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.

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 ( , )x y i i

Clearly, all that matters to compute agent i's allocation is his type, namely the position of his utility u i relative to the increasing sequence of marginal costs We shall say that agent i is of

type q if c q u i c q1: this means that i will buy up to the q-th highest price A profile of types is

a vector ( , ,n1  n Q) where n q is the number of agents of type q (thus n q is zero or a positive

integer) and where q varies from 1 to Q Given a utility profile u i , we choose Q to be the highest

type in this profile; thus when we write a profile of types ( , ,n1  n Q), we always assume that

n Q0

Given a profile of types ( , ,n1  n Q) we wish to compute the probability k q, that, in the RP

equilibrium, an agent of type q buys the good at price c k We can assume 1  k q because

k q, 0 whenever q k  Note that this probability is not defined whenever n q 0 Next we observe that k q, k q, whenever k q q , (and n n q, q 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 c k Note that k is defined for all k1, , Q because n Q 0 The RP social choice function is now:

if agent is of the type i q: x i  k q k , y i  k q c k k

 1  1  (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  n Q) To define the (probabilistic) allocation selected by the PS mechanism (social choice function), we consider the

largest number q*,1   such that q* Q

1m k m k t k n t

Q k

m y

c m

t t k

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 c k, we have:

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 particularcase 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  x n 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:

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 x~( 1  x n) 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 ( , ) x y i i 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 simplerandom process described in Appendix 2 This process, however, is based on the entire profile oftypes9, 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

m k exceeds one.

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 u i for the good and c1u1c2 u2 c3 u3c4 u4 The two vectors  and  are

4

13

38

124

14

13

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:

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: