Serial cost sharing of an excludable public good
available in multiple units1
Yan Yu
Department of Economics
Hong Kong University of Science & Technology
Clear Water Bay, Kowloon
Hong Kong
[email protected]
December 2006
JEL Classification: H41, C72, D71
1I
thank Hervé Moulin for helpful comments. This paper was presented at the Second World
Congress of the Game Theory Society, Marseille, France. I thank the participants for comments. I
thank two referees for their detailed comments.
Abstract
This paper characterizes serial mechanisms for the cost sharing of an excludable public good
that is available in multiple units. We show that serial mechanisms are the only voluntary
mechanisms that satisfy coalition strategy-proofness and consumer sovereignty.
1
Introduction
This paper characterizes serial cost sharing for an excludable public good when it is available
in multiple units. A pure public good is both non-rival and non-excludable, i.e., an agent’s
consumption of the public good does not reduce the amount of the good available to others
and it is impossible to exclude any agent from consuming the public good. An excludable
public good is non-rival but excludable. Examples of excludable public goods include all
kinds of public facilities, such as libraries, highways, parks, and museums, so long as the
facilities are not congested. Almost all information products are excludable public goods,
such as TV programs, software, music CDs, electronic newspapers, e-books. Many excludable
public goods can be provided in multiple units. For example, there are basic, standard, and
premium services of Cable TV; there are amateur and professional versions of software; and
there are regular and gold memberships at country clubs.
On the provision of a pure public good, Moulin (1994) identifies the conservative equal
costs mechanism as the only anonymous, voluntary, and coalition strategy-proof mechanism.
Under the conservative equal costs mechanism, each agent reports his demand, the lowest
demand of the public good is produced, and costs are shared equally among all agents. A
mechanism is coalition strategy-proof if no group of agents can gain by coordinating deviation
from revealing their true preferences. Coalition strategy-proof is also called group strategyproof in the literature. Strategy-proof is a weaker condition than coalition strategy-proof.
A mechanism is strategy-proof if every agent has a dominant strategy to reveal his true
preference. Serizawa (1996) characterizes the set of strategy-proof and voluntary mechanisms
for a pure public good. Ohseto (2000) shows that on the provision of a binary public good,
1
by admitting exclusions, it is always possible to construct Pareto superior cost sharing rules
to the cost sharing rules when exclusion is impossible.
On the provision of an excludable and divisible public good, Moulin (1994) proposes
serial cost-sharing. If we order the agents by increasing demands, yi , the serial cost sharing
formula divides the cost, C(y1 ), equally among all agents, then it divides the incremental
cost, C (y2 ) − C(y1 ), equally among all agents except agent 1; then it divides the incremental
cost, C(y3 ) − C(y2 ), among all agents who consume no less than y3 , and so on. When the
cost of producing the public good is convex, the demand game, in which each agent chooses
a demand, yi , and cost shares are computed by the serial formula, has a unique strong equilibrium outcome for every profile of convex preferences. In a direct serial mechanism, each
agent reports his preference and the mechanism assigns to each reported preference profile
the unique strong equilibrium outcome of the demand game. The direct serial mechanism is
anonymous, voluntary, and coalition strategy-proof.
Moulin (1994, Theorem 3) proves, for the special case of two agents, that any anonymous,
voluntary, and coalition strategy-proof mechanism that meets non-imposition is Pareto dominated by the direct serial mechanism. Non-imposition requires that, for any amount, y, of
the public good within production capacity, there exists a preference profile such that the
mechanism assigns every agent consuming y amount of the public good. Moulin (1994) announces that the theorem holds for n agents and a binary public good. Deb and Razzolini
(1999), Ohseto (2000), and Moulin and Shenker (2001) provide similar characterization results for the serial mechanism of a binary public good. Moulin (1994) also conjectures that
the theorem holds for divisible goods as well. However, Olszewski (2004) shows that Moulin’s
conjecture does not hold and the conjecture holds when the non-imposition condition is re2
placed by a continuity condition on the mechanism. A mechanism is continuous if the final
allocation (or the agents’ final utility) is continuous in the agents’ preference profiles. More
specifically, Olszewski (2004) proves that the serial mechanism Pareto dominates all continuous mechanisms that are anonymous, voluntary, and coalition strategy-proof for divisible
public goods and n agents.
We examine the provision of an excludable public good available in integer units. This
setting contains the binary public good model as a special case. It is topologically simpler
than the divisible public good model but also applicable to real life scenarios. Moulin’s
conjecture does not hold when the public good is available in integer units, unlike the binary
case. This can be shown by adjusting the counter example given in Olszewski (2004, Example
2) slightly by restricting agents’ consumption of the public good to integer units. Here, we
provide a characterization of the serial mechanism. We show that serial mechanisms are the
only anonymous, voluntary, and coalition strategy-proof mechanisms that meet consumer
sovereignty. Consumer sovereignty requires that, for any level of the public good within
production capacity, each agent has a report (of preference) to guarantee him that level of
consumption regardless of other agents’ reports of preferences. This is a condition in the spirit
of freedom of choice. Our characterization result shows that consumer sovereignty can replace
Pareto dominance in characterizing the serial mechanisms. This means that guaranteeing
agents the freedom to choose their consumption levels in fact improves efficiency in voluntary
and coalition strategy-proof cost sharing of excludable public goods.
Section 2 presents the model and definitions. Section 3 states and proves the characterization results. Section 4 concludes the paper.
3
2
The Model
There are one private good, money, and one excludable public good. The public good is
available in y + 1 levels (units): Y = {0, 1, 2, ..., y}. The production capacity, y, is finite. Let
C(y) denote the cost (in the amount of the private good) to produce y units of the public
good. C(0) = 0. C(y) is strictly increasing and convex.
There are n agents. Let N = {1, 2, ..., n} denote the set of agents. The agents’ utilities are
quasi-linear: Ui (yi , xi ) = Wi (yi ) − xi , where yi is agent i’s consumption of the public good,
xi is agent i’s payment for the public good. Agent i’s utility is assumed to be finite, nondecreasing, and concave in yi . Let wi = (wi (1), wi (2), ..., wi (y)) denote agent i’s willingness
to pay vector for each unit of the public good up to the production capacity. Wi (yi ) is agent
i’s total willingness to pay for yi units of the public good: Wi (yi ) =
k=yi
k=1
wi (k), where the
marginal benefit wi (k) is non-increasing in k and wi (k) ≥ 0 for all i, k. Let D stand for the
set of all possible willingness to pay for the public good with the above-mentioned properties.
Let w = (w1 , ..., wn ) denote the profile of n agents’ willingness to pay. For any T ⊆ N, let
(wT , w−T ) denote the profile of willingness to pay whose ith component is wi if i ∈ T and wi
if i ∈
/ T.
Let Z denote the set of all feasible outcomes: Z = {((y1 , x1 ); ...; (yn , xn )) : yi ∈ Y, xi ≥ 0
for all i, and
i xi
= C(maxi yi )}. A mechanism, f, associates to every willingness to pay
profile, w ∈ DN , a feasible outcome f (w) ≡ (y(w), x(w)) ∈ Z.
Notice that by this definition, a mechanism must be budget-balanced and that no agent
receives any subsidy.
4
We denote by S(f(w)) the corresponding vector of utility levels:
Si (fi (w)) = Ui (yi (w), xi (w)) for all i ∈ N.
When it is not confusing, we use S(w) for S(f(w)).
Definition 1: Mechanism f satisfies Equal Treatment of Equals (ETE) if, whenever
two agents have identical willingness to pay, they receive the same utility level:
wi = wj ⇒ Si (w) = Sj (w) for any w ∈ DN .
Definition 2: Mechanism f satisfies No Exploitation (NoEx) if, whenever agent i’s
consumption of the public good is zero, his cost share is also zero:
yi (w) = 0 ⇒ xi (w) = 0 for all i ∈ N, w ∈ DN .
Definition 3: Mechanism f satisfies Consumer Sovereignty (ConSov) if, for all
i ∈ N, yi ∈ Y, there exists wy∗i ∈ D such that yi (wy∗i , w−i ) = yi for all w−i ∈ DN/i .
Definition 4: Mechanism f is Coalition Strategy-Proof (CSP) if, for any coalition
T ⊆ N, any profile w ∈ DN , and any wT ∈ DT , we have:
{Si (wT , w−T ) > Si (w) for some i ∈ T } ⇒ {Sj (wT , w−T ) < Sj (w) for some j ∈ T }.
Definition 5: Mechanism f is Strategy-Proof (SP) if, for any agent i ∈ N , any profile
5
w ∈ DN , and any wi ∈ D, we have:
Si (w) ≥ Si (wi , w−i ) where Si is based on wi .
ConSov requires that it is possible for each agent to consume any amount of the public
good regardless of other agents’ preferences. ConSov, together with NoEx and SP, implies
voluntary participation, since an agent can report w0∗ to get (0, 0) for certain.
A cost sharing pattern, g, is a function associating to an allocation of the public good,
y ∈ Y N , a distribution of the cost, x = (x1 , ...xn ), xi ≥ 0, such that budget is balanced:
i
xi = C(maxi yi ). Denote the set of all such cost sharing patterns G.
For any y ∈ Y N , let m denote the number of distinct consumption levels of the public
good in y and let y 1 , ..., y m denote those consumption levels in an ascending order. Let N 1
denote the set of agents who consume y 1 units of the public good and n1 the number of
agents in N 1 . Define N 2 , ..., N m and n2 , ..., nm accordingly.
Definition 6: The serial pattern, g s , is a cost sharing pattern such that for any
y ∈ Y N , for every agent i ∈ N 1 , the cost share, gis (y), is
cost share, gis (y), is
C(y 1 )
n
+
C(y2 )−C(y1 )
n−n1
C(y1 )
n
+ ... +
+
C(y 2 )−C(y1 )
; ...;
n−n1
C(y1 )
;
n
for every agent i ∈ N 2 , the
for every agent i ∈ N m , the cost share, gis (y), is
C(ym )−C(ym−1 )
m−1 j .
n− j=1
n
In other words, the serial pattern, g s , assigns the cost of any y ∈ Y N according the
following formula:
6
gis (y) =















C(y1 )
n
C(y1 )
n
if i ∈ N 1
+
k
j=2
C(yj
)−C(yj−1 )
m
l
l=j n
if i ∈ N k , k ∈ {2, ..., m}.
Given a cost sharing pattern, g ∈ G, and a willingness to pay profile, w ∈ DN , a demand
game, d(w, g), is the normal form game with a strategy set {0, 1, ..., y} for every agent and
where given a strategy profile y ∈ Y N , agent i’s consumption vector is (yi , gi (y)) and his net
utility is
yi
k=1
wi (k) − gi (y). In other words, in a demand game, each agent simply states
his demand for the public good and then his cost share is determined by all agents’ demands
according to the cost sharing pattern.
A strong Nash equilibrium is a Nash equilibrium where no group of agents can gain
by joint deviation. Moulin (1994, Lemma 1) shows that the demand game corresponding to
the serial pattern always has a unique (at least in terms of utilities) strong Nash equilibrium.
This is still true when the public good is available in integer units.
Given w ∈ DN , we calculate a strong Nash equilibrium as follows:
Stage 1: the (possible) production and cost allocation of the first unit of the
public good.
Search for the largest integer t, 1 ≤ t ≤ n, such that #{i | wi (1) ≥
N} = t, where #{i | wi (1) ≥
C(1)
,i
t
C(1)
,i
t
∈
∈ N} denotes the number of agents whose
willingness to pay of the first unit of the public good is no less than
C(1)
.
t
If such a t doesn’t exist, the public good is not provided and each agent gets (0, 0)
as the final allocation. (This ends the calculation of the strong Nash equilibrium.)
7
Otherwise, let t1 = max{t | #{i | wi (1) ≥
and let N(1) = {i | wi (1) ≥
C(1)
,i
t1
C(1)
,i
t
∈ N} = t, t ∈ {1, 2, ..., n}}
∈ N }. The first unit of the public good is
produced; every agent in N(1) consumes it and pays
C(1)
,
t1
while every agent in
N − N (1) gets (0, 0) as his final allocation in the strong Nash equilibrium. All
agents in N (1) go to Stage 2 for possible update of their allocations.
Stage k : k = {2, 3, ..., y}, the (possible) production and cost allocation of the
kth unit of the public good.
Search for the largest integer t, 1 ≤ t ≤ n, such that #{i | wi (k) ≥
N} = t, where #{i | wi (k) ≥
C(k)−C(k−1)
,i
t
C(k)−C(k−1)
,i
t
∈
∈ N} denotes the number of agents
whose willingness to pay for the kth unit of the public good is no less than
C(k)−C(k−1)
.
t
If such a t doesn’t exist, the kth unit is not provided and every agent in N (k − 1)
gets (k − 1,
s=k−1
s=1
C(s)−C(s−1)
)
ts
as his final allocation. (This ends the calculation
of the strong Nash equilibrium)
Otherwise, let tk = max{t | #{i | wi (k) ≥
and let N(k) = {i | wi (k) ≥
C(k)−C(k−1)
}.
tk
C(k)−C(k−1)
}
t
= t, t ∈ {1, 2, ..., n}}
The kth unit of the public good is
produced; every agent in N (k) consumes it and pays an additional
while every agent in N (k) − N (k − 1) gets (k − 1,
s=k−1
s=1
C(s)−C(s−1)
)
ts
C(k)−C(k−1)
,
tk
as his final
allocation in the strong Nash equilibrium. All agents in N (k) go to Stage k + 1
for possible update of their allocations, if k < y. If k = y, every agent in N(y)
gets (y,
s=y
s=1
C(s)−C(s−1)
)
ts
as his final allocation.
Since the demand game corresponding to the serial pattern always has a unique strong
8
Nash equilibrium in terms of utilities, if there exist multiple strong Nash equilibria, they are
all equivalent to the one defined above in terms of utilities.
Definition 7: A serial mechanism (or serial cost sharing rule) is a mechanism that
associates to every profile, w ∈ DN , a strong Nash equilibrium outcome of the demand game,
d(w, g s ), corresponding to the serial pattern.
3
Characterization Result
Theorem: The serial mechanisms are the only mechanisms that satisfy ETE, NoEx, ConSov,
and CSP.
It’s clear that the serial mechanism satisfies ETE and NoEx. Serial mechanism is coalition
strategy-proof when the public good is available in integer units as pointed out by Moulin
(1994, page 315). Now we show that the serial mechanism satisfies ConSov. According to
the method to calculate the strong Nash equilibrium, we can easily verify the following:
By reporting that wi = (0, .., 0 ), agent i gets (0, 0) regardless of other agents’ preferences.
y times
By reporting that wi = (C(1) + , 0, 0, ..., 0), agent i consumes the first unit of the public
good (t1 = 1 and N (1) = {i}, if no larger t1 exists) and only the first unit of the public good
for certain (since wi (2) = 0, i ∈
/ N (2)), regardless of other agents’s preferences.
By reporting that wi = (C(k) + , C(k) + , ...C (k) + ,0, 0, ..., 0), agent i consumes ex
k times
y−k times
actly the first k units of the public good for certain, regardless of other agents’ preferences.
Hence, the serial mechanism satisfies ConSov.
The remainder of this section develops some interesting intermediate results in proving
the theorem. Recall the notation: fi (w) ≡ (yi (w), xi (w)).
9
According to ConSov, for any yi ∈ Y, agent i has a willingness to pay vector, wy∗i , such
that, by reporting this willingness to pay, agent i gets yi for certain, i.e., yi (wy∗i , w−i ) = yi
for any w−i ∈ DN/i . Note that there may exist multiple willingness to pay vectors which
can guarantee that agent i consume yi for certain. wy∗i denotes any such willingness to pay
vector.
Lemma 1: If f satisfies ConSov and CSP, then S(wy∗i (w) , w−i ) = S(w) for all i ∈ N, and
all w−i ∈ DN/i .
Proof : Let yi = yi (w). According to the definition of wy∗i , we have yi (wy∗i , w−i ) =
yi . CSP implies strategy-proofness (SP), i.e., no agent should be able to gain
by unilaterally misreporting his willingness to pay. According to SP, we have
fi (wy∗i , w−i ) = fi (w)1 . Suppose S(wy∗i , w−i ) = S(w). Then there exists an agent j
such that either Sj (wy∗i , w−i ) > Sj (w) or Sj (wy∗i , w−i ) < Sj (w). If Sj (wy∗i , w−i ) >
Sj (w), then when the agents’ willingness to pay profile is (wy∗i , w−i ), the coalition
{i, j} can successfully gain by letting agent i report wi as his willingness to pay,
since agent i gets the same allocation while agent j is strictly better off. Therefore, if f is CSP, Sj (wy∗i , w−i ) > Sj (w) cannot hold. Similarly, Sj (wy∗i , w−i ) <
Sj (w) cannot hold either. Hence, we have S(wy∗i , w−i ) = S(w).
Lemma 2: Let fy (wT, w−T ) = (yT, y−T ). If f satisfies ConSov and CSP, then S(wy∗T , w−T ) =
S(w) for all T ⊆ N.
1
If not, then either xi (wy∗i , w−i ) > xi (w) or xi (wy∗i , w−i ) < xi (w). Suppose xi (wy∗i , w−i ) > xi (w), then,
when the agents’ willingness to pay profile is (wy∗i , w−i ), agent i is better off by reporting wi . This is
contradictary to SP. Hence, xi (wy∗i , w−i ) > xi (w) can not hold if f is strategy-proof. Similarly, xi (wy∗i , w−i ) <
xi (w) is also impossible.
10
Proof : Without loss of generality, assume that T = {1, 2, 3, ..., t}. By Lemma 1,
S(wy∗1 , w−1 ) = S(w). We first show that S(wy∗1 , wy∗2 , w−1,2 ) = S(w).
(i) If y2 (wy∗1 , w−1 ) = y2 .
By Lemma 1, S(wy∗1 , wy∗2 , w−1,2 ) = S(w).
(ii) If y2 (wy∗1 , w−1 ) = y2 and S(wy∗1 , wy∗2 , w−1,2 ) = S(w).
By Lemma 1, S(w1 , wy∗2 , w−1,2 ) = S(w). If S1 (wy∗1 , wy∗2 , w−1,2 ) < S1 (w), when
agents’ preferences are (wy∗1 , wy∗2 , w−1,2 ), agent 1 can gain by reporting w1 . Contradictory to strategy-proofness. Hence S1 (wy∗1 , wy∗2 , w−1,2 ) ≥ S1 (w).
If S1 (wy∗1 , wy∗2 , w−1,2 ) > S1 (w), when agents’ preferences are (w1 , wy∗2 , w−1,2 ), agent
1 can gain by reporting wy∗1 . Contradictory to strategy-proofness. Hence,
S1 (wy∗1 , wy∗2 , w−1,2 ) = S1 (w) = S1 (w1 , wy∗2 , w−1,2 ).
(1)
If S2 (wy∗1 , wy∗2 , w−1,2 ) > S2 (w), by the definition of CSP, S1 (wy∗1 , wy∗2 , w−1,2 ) <
S1 (w). Contradictory to (1).
If S2 (wy∗1 , wy∗2 , w−1,2 ) < S2 (w), by the definition of CSP, S1 (wy∗1 , wy∗2 , w−1,2 ) >
S1 (w). Contradictory to (1).
Hence, S2 (wy∗1 , wy∗2 , w−1,2 ) = S2 (w) = S2 (w1 , wy∗2 , w−1,2 ).
If there exists any i = 1, 2 such that Si (wy∗1 , wy∗2 , w−1,2 ) > Si (w), when agents’
preferences are w, the coalition (1, 2, i) can make Pareto improvement by reporting (wy∗1 , wy∗2 , wi ). Contradictory to CSP. Similarly, Si (wy∗1 , wy∗2 , w−1,2 ) < Si (w) is
also contradictory to CSP.
11
Therefore, S(wy∗1 , wy∗2 , w−1,2 ) = S(w).
Applying the above analysis consecutively to agent 3, 4, ..., t, we have S(wy∗T , w−T ) =
S(w).
A mechanism is called semi-constant cost sharing if, in the range of f , the same
allocation of the public good always implies the same cost allocation, i.e.,
y(w) = y(w ) ⇒ x(w) = x(w ) for all w, w ∈ DN .
Proposition 1: If a mechanism, f, satisfies ConSov and CSP, then f is semi-constant
cost sharing.
Proof 2 : Take any w, w ∈ DN such that y(w) = y(w ) = (y1 , ...yn ).
By Lemma 2, we have S(wy∗1 , ..., wy∗n ) = S(w). Similarly, we have S(wy∗1 , ..., wy∗n ) =
S(w ). Hence S(w) = S(w ). Therefore, if y(w) = y(w ), it must be true that
x(w) = x(w ), i.e., f is semi-constant cost-sharing.
If f is semi-constant cost-sharing, then there exists a corresponding cost sharing pattern.
Let g f denote the cost sharing pattern corresponding to f. One way to define g f is that, for
all y ∈ Y N , let g f (y) = x(wy∗1 , ..., wy∗n ).
Given a willingness to pay profile, w ∈ DN , and a semi-constant cost sharing mechanism,
f, the corresponding demand game is the demand game d(w, g f ).
Lemma 3: If f satisfies ConSov and CSP, then for any w ∈ DN , f (w) must be a strong
Nash equilibrium outcome of the corresponding demand game, d(w, g f ).
2
Proof of Proposition 1 and Lemma 3 is similar to that of Lemma 1 in Moulin (1999). It is reproduced
here for consistency of notation and completeness of the paper.
12
Proof : Suppose for some w ∈ DN , f (w) is not a strong Nash equilibrium outcome of d(w, g f ). Let y(w) = (y1 , ..., yn ) = y. Then, there exists a group of
agents, T ⊆ N, and a choice of the public goods, yT = yT ∈ Y T , such that, based
on w, Si (yi , gi (yT , y−T )) ≥ Si (yi , gi (y)) for all i ∈ T and the inequality holds
strictly for at least one agent in T.
By Lemma 2, we have S(f (wT , wy∗−T )) = S(f (w)). However, when every agent,
j ∈ N − T, has a willingness to pay, wy∗j , the coalition T , whose willingness to pay
profile is wT , can gain by reporting wy∗ for all i ∈ T and getting (yi , gi (yi , y−i ))
i
for all i ∈ T. This contradicts CSP. Therefore, Lemma 3 must hold.
Based on Lemma 3, it must be true that, for any willingness to pay profile, w ∈ DN ,
the corresponding demand game, d(w, g f ), must have a strong Nash equilibrium. This point
plays an important role in the proof of our characterization result.
Mechanism f is semi-equal cost sharing if any two agents who have the same consumption of the public good also have the same cost share, i.e., for any w ∈ DN , yi (w) = yj (w)
implies xi (w) = xj (w).
Proposition 2: If f satisfies ETE, ConSov, and CSP, then f must be semi-equal cost
sharing.
Proof: According to Proposition 1, f has a corresponding cost sharing pattern, g f . Suppose that f is not semi-equal cost sharing, then there exists a willingness to pay profile w ∈ DN , such that for some i, j ∈ N, yi (w) = yj (w) = k and
xi (w) = xj (w). So in g f , gif (k, k, y−i,j (w)) = gjf (k, k, y−i,j (w)). Let wi = wj ∈ D
such that the willingness to pay for the kth unit is greater than C(k) and the
13
willingness to pay for the (k + 1)th unit is 0. Note that by design, if f satisfies ConSov and CSP, by reporting wi , agent i will get at least k units of the
public good. We check the case3 where wi = wy∗i =k and wj = wy∗j =k . In any
strong Nash equilibrium of the demand game d((wi , wj , wy∗−i,j ), g f ), agent i and
j’s demand are both k, unless agent i and j can get additional units for free. By
ETE, Si (f (wi , wj , wy∗−i,j )) = Sj (f (wi , wj , wy∗−i,j )). Therefore, yi (wi , wj , wy∗−i,j ) ≥
k, yj (wi , wj , wy∗−i,j ) ≥ k, and xi (wi , wj , wy∗−i,j ) = xj (wi , wj , wy∗−i,j ).
We show that xi (wi , wj , wy∗−i,j ) = xi (wy∗i =k , wy∗j =k , wy∗−i,j ) by three claims.
∗
).
We claim that (a) xi (wi , wj , wy∗−i,j ) = xi (wy∗i =k , wj , w−i,j
∗
Proof of the claim: If xi (wi , wj , wy∗−i,j ) > xi (wy∗i =k , wj , w−i,j
), when agents’
preference profile is (wi , wj , wy∗−i,j ), agent i can gain by reporting wy∗i =k . Contra∗
dictory to SP. Similarly, xi (wi , wj , wy∗−i,j ) < xi (wy∗i =k , wj , w−i,j
) is also impossible.
∗
We also claim that (b) xj (wy∗i =k , wj , w−i,j
) = xj (wy∗i =k , wy∗j =k , wy∗−i,j ).
∗
) > xj (wy∗i =k , wy∗j =k , wy∗−i,j ), when
Proof of the claim: If xj (wy∗i =k , wj , w−i,j
∗
agents’ preference profile is (wy∗i =k , wj , w−i,j
), agent j can gain by reporting wy∗j =k .
∗
Contradictory to SP. Similarly, xj (wy∗i =k , wj , w−i,j
) < xj (wy∗i =k , wy∗j =k , wy∗−i,j ) is
∗
also impossible. Note that yj (wy∗i =k , wj , w−i,j
) ≥ k.
∗
Next, we claim that (c) xi (wy∗i =k , wj , w−i,j
) = xi (wy∗i =k , wy∗j =k , wy∗−i,j ).
∗
Proof of the claim: If xi (wy∗i =k , wj , w−i,j
) > xi (wy∗i =k , wy∗j =k , wy∗−i,j ), when
∗
agents’ preference profile is (wy∗i =k , wj , w−i,j
), the coalition {i, j} can gain by
reporting (wy∗i =k , wy∗j =k ). (By claim (b), according to wj , agent j is indifferent
3
Results for the other cases where either wi = wy∗i =k or / and wj = wy∗j =k follows directly from the proof.
14
∗
between fj (wy∗i =k , wj , w−i,j
) and fj (wy∗i =k , wy∗j =k , wy∗−i,j ). ) Contradictory to CSP.
∗
Similarly, xi (wy∗i =k , wj , w−i,j
) > xi (wy∗i =k , wy∗j =k , wy∗−i,j ) is also impossible. There∗
fore, xi (wy∗i =k , wj , w−i,j
) = xi (wy∗i =k , wy∗j =k , wy∗−i,j ). Combining claim (a) and (c), we have xi (wi , wj , wy∗−i,j ) = xi (wy∗i =k , wy∗j =k , wy∗−i,j ) =
gif (k, k, y−i,j ). Symmetrically, we can establish xj (wi , wj , wy∗−i,j ) = xj (wy∗i =k , wy∗j =k , wy∗−i,j ) =
gjf (k, k, y−i,j ). Since gif (k, k, y−i,j ) = gjf (k, k, y−i,j ), xi (wi , wj , wy∗−i,j ) = xj (wi , wj , wy∗−i,j ).
Contradictory to ETE. Proposition 2 is established. Proposition 2 says that agents who consume the same level of the public good have the
same cost share. We are left to show that this cost share is determined in the serial pattern.
The remaining proof of the theorem is provided in the Appendix.
Remark: Olszewski (2004, Theorem 2) says that a continuous mechanism is coalition
strategy-proof, individually rational, and satisfies the ETE property if and only if it is a type
of serial mechanism with constrained range. These mechanisms are rather complicated to
define. See Olszewski (2004) for precise definitions. Roughly speaking, a serial mechanism
with constrained range is based on the serial pattern but each agent’s demand is restrained
within a certain range depending on the ranking of his demand and a prefixed sequence of
ranges. The serial mechanism is the only mechanism that meets the consumer sovereignty
among the mechanisms characterized in Olszewski (2004) Theorem 2. Therefore, based on
Olszewski (2004), we can show that, for a divisible public good, the serial mechanisms are the
only continuous mechanisms that are coalition strategy-proof, and satisfy the ETE, NoEx,
and ConSov properties. It is not clear whether the continuity of mechanisms is necessary
for generalizing our theorem to a divisible public good. Note that individual rationality is
15
stronger than No Exploitation. Individual rationality implies that when an agent consumes
no public good, his cost share is zero, i.e., NoEx. On the other hand, NoEx, ConSov, and
SP together imply individual rationality. Note also that the Equal Treatment of Equals
property defined in Olszewski (2004) requires that agents with identical preference orderings
have the same allocation: consume and contribute the same amount. The ETE property in
Olszewski (2004) is stronger than the ETE property defined in this paper. ETE defined in
this paper only requires that agents with same preferences end up with same utilities.
Remark: Our Theorem is not implied by Olszewski (2004). Continuity (or left-continuity,
right-continuity) plays no role in our characterization result.
Remark: Our characterization result is tight. The following examples show that there
are other mechanisms satisfying any three of the four properties: ETE, NoEx, ConSov, and
CSP. In the following examples, the public good is binary with a production cost of 10 and
there are only two agents.
Example 1: Define a mechanism such that:
(i) If w1 ≥ 4, w2 ≥ 6, then f 1 = (1, 4), f 2 = (1, 6);
(ii) if w1 < 4, w2 ≥ 10, then f 1 = (0, 0), f 2 = (1, 10);
(iii) if w1 ≥ 10, w2 < 6, then f 1 = (1, 10), f 2 = (0, 0);
(iv) otherwise, f 1 = f 2 = (0, 0).
It is clear that f 1 satisfies NoEx, ConSov, and CSP, but not ETE. Note that in Example
1, 2, and 4, each agent can guarantee himself to consume the public good by reporting
wi = 10 and guarantee himself not to consume the public good by reporting wi = 0.
Example 2: Define a mechanism such that:
(i) If there is an agent i, i ∈ {1, 2} such that wi > 5, while wj = 0 for the other
16
agent, then f i = (1, 5) and f j = (0, 5).
(ii) If there is an agent i, i ∈ {1, 2} such that wi > 5, while wj > 0 for the other
agent, then f 1 = f 2 = (1, 5).
(iii) If max{w1 , w2 } ≤ 5, f 1 = f 2 = (0, 0).
The above mechanism satisfies ETE, ConSov, and CSP, but not NoEx.
Example 3: Define a mechanism such that the public good is provided and each agent
pays 5 only when both agents’ willingness to pay is no less than 5, otherwise, the public
good is not provided and each agent pays zero. This mechanism satisfies ETE, NoEx, and
CSP, but not ConSov.
Example 4: Define a mechanism such that:
(i) if min{w1 , w2 } > 5, f 1 = f 2 = (1, 5);
(ii) if there is an agent i, i ∈ {1, 2} such that wi ≤ 5,
(a) if wj ≥ 10, f i = (0, 0) and f j = (1, 10)
(b) if wj < 10, f i = f j = (0, 0).
This mechanism satisfies ETE, NoEx, and ConSov, but not CSP. (When w1 = 7 and
w2 = 5 , the mechanism assigns (0, 0) to both agents. The coalition {1, 2} can make Pareto
improvement by letting agent 2 report w2 > 5. ) In fact, it is strategy-proof, but not coalition
strategy-proof.
Remark: By definition, cost-sharing rules are budget balanced. Note that budget balance is not used in the proof of Lemma 1, 2, 3, and Proposition 1, 2. So any mechanism
(including those that are not budget balanced, i.e., mechanism may collect more than enough
money to cover the cost) that satisfies ETE, ConSov, and CSP must have a corresponding
cost sharing pattern and is semi-constant cost sharing. But budget balance is needed for the
17
characterization result. If budget balance is relaxed, for any production technology C(y),
we can inflate the cost to be 110% of C(y) and the serial mechanisms corresponding to this
inflated costs satisfy ETE, NoEx, ConSov, and CSP.
4
Conclusion
This paper characterizes serial mechanisms for the provision of an excludable public good
that is available in multiple units. We show that serial mechanisms are the only mechanisms that meet ETE, NoEx, ConSov, and CSP. Previous characterization results on serial
mechanism are mostly for a binary public good, with the exception of Olszewski (2004), who
characterizes the serial mechanism for a divisible public good within the class of continuous
mechanisms. All those characterization results use Pareto dominance, i.e., all other mechanisms with similar properties are Pareto dominated by the serial mechanisms. Comparing
our characterization result with previous ones, we see that serial mechanisms stand out not
only in terms of efficiency but also in terms of freedom of choices given to agents. To that
extent, efficiency and freedom of choices go hand in hand.
Appendix: Proof of the Theorem
Two allocations of the public good, y, y ∈ Y N , are distinct if one is not a permutation
of the other, i.e., y = (y1 , ..., yn ) = (yΠ(1)
, ..., yΠ(n)
) for any permutation Π : {1, ..., n} →
{1, ..., n}. First, we order all possible distinct allocations of the public good.
Let (y 1 , ..., y 1 , ..., y m , ...y m ) denote a distinct allocation where m is the number of distinct
n1
nm
consumption levels in the allocation and y 1 < y 2 < ... < y m are those consumption levels.
Let N k denote the set of agents whose consumption is y k and let nk denote the number of
18
agents in N k .
We order all possible distinct allocations lexicographically according to (y 1 , ..., y 1 , ..., y m , ...y m )
n1
nm
in ascending order. The beginning of this sequence is:
(0, ..., 0), (0, ..., 0, 1), (0, ..., 0, 2), ..., (0, ..., 0, y ),
n
n−1
n−1
n−1
(0, ..., 0, 1, 1), (0, ..., 0, 1, 2), ..., (0, ..., 0, 1, y),
n−2
n−2
n−2
(0, ..., 0, 1, 1, 1), (0, ..., 0, 1, 1, 2), ...,
n−3
n−3
According to Proposition 2 and NoEx, it is clear that, for the first y + 1 allocations,
costs must be shared according to the serial pattern. According to the serial pattern, g s ,
the cost share for every agent in N k is
C(y1 )
n
+
C(y 2 )−C(y1 )
n−n1
+ ... +
C(yk )−C(y k−1 )
.
j
n− k−1
j=1 n
In addition to
"same consumption same cost", the serial pattern requires that the cost of each unit of the
public good must be shared equally among all agents who consume that unit of the public
good. We are left to show that there should not be any cross subsidy from agents of different
consumption levels of the public good.
We first establish a Lemma. For any agent i in N k , define agent i’s marginal cost share
to be the increase in his cost share when all agents in N k demand the y k th unit of the public
good, while holding other agents’ consumption constant. For any T ⊆ N, let gis (yT − 1, y−T )
denote agent i’s cost share according to the serial pattern when agents in T consume 1 unit
less than in y and the consumption of other agents remains unchanged.
Lemma 4: According to the serial cost sharing pattern, for any agent i ∈ N k , his
marginal cost share does not change when all agents in N j demand one unit less, i.e.,
gis (y) − gis (yN k − 1, y−N k ) = gis (yN k , yN j − 1, y−N k −N j ) − gis (yN k − 1, yN j − 1, y−N k −N j ).
19
Proof of Lemma 4: According to the serial cost sharing pattern, costs are
shared on unit-by-unit basis: first we distribute cost of producing the first unit
equally among all agents who consume the first unit, then we distribute the cost
of producing the second unit equally among all agents who consume the second
unit,..., and so on until no agent consumes any more unit of the public good.
Each agent’s total cost share is simply the summation of his cost share for each
unit up to the last unit he consumes.
If y k < y j , when all agents in N j change their consumptions from y j to y j − 1,
it does not affect the cost distribution of the first y k units of the public good.
Therefore, for any agent in N k , his marginal cost share does not change.
If y k > y j , when all agents in N j change their consumptions from y j to y j − 1,
it only affects the cost distribution of the y j th unit. When the public good consumption is (yN k , yN j , y−N k −N j ), if all agents in N j consume one unit less, the
cost share of the y j th unit increases from
C(y j )−C(yj −1)
n− jl=1 nl
−
C(yj )−C(yj −1)
,
l
n− j−1
l=1 n
C(yj )−C(yj −1)
l
n− j−1
l=1 n
to
C(yj )−C(yj −1)
.
n− jl=1 nl
Let x =
then for any i ∈ N k , gis (yN k , yN j − 1, y−N k −N j )−
gis (y) = x. When the public good consumption is (yN k − 1, y−N k ), if all
agents in N j consume one unit less, the cost share of the y j th unit still increases from
C(y j )−C(yj −1)
j−1 l
n− l=1
n
to
C(yj )−C(y j −1)
.
n− jl=1 nl
Therefore, for any i ∈ N k , gis (yN k −
1, yN j − 1, y−N k −N j )− gis (yN k − 1, y−N k ) = x. Hence, for any i ∈ N k , gis (y) −
gis (yN k − 1, y−N k ) = gis (yN k , yN j − 1, y−N k −N j ) − gis (yN k − 1, yN j − 1, y−N k −N j ). We now prove the theorem by induction. More specifically, we prove the following claim.
Claim: If the costs are shared according to the serial pattern for all allocations before
20
the Ith allocation in the sequence, then the costs must be shared according to the serial
pattern for the Ith allocation.
Proof of the Claim: It is clear that the claim holds for I ≤ y + 1, i.e.,
from allocation (0, ..., 0) to (0, ..., 0, y). When I > y + 1, suppose that the claim
n
n−1
holds for all allocations before the Ith allocation, but the claim does not hold
for the Ith allocation. Let y denote the Ith allocation. Then, there exist i ∈ N
such that gif (y) < gis (y) and j ∈ N such that gjf (y) > gjs (y) . Define (N 1 , ..., N m )
based on the Ith allocation, y. According to Proposition 2, all agents in N k have
the same cost share. Let l denote the smallest k such that the cost share for
agents in N k is less than their serial cost share, i.e., l = min{k| gif (y) < gis (y) for
all i ∈ N k }. Let h denote the smallest k such that the cost share for agents in
N k is more (higher) than their serial cost share, h = min{k| gif (y) > gis (y) for all
i ∈ N k }.
We construct a willingness to pay profile, w, as follows.
•
For every agent i in N l , wi (k) > C(y l − 1) − C(y l − 2) for all k ≤
y l − 1, and gif (y) − gis (yN l − 1, y−N l ) < wi (y l ) < gis (y) − gis (yN l − 1, y−N l ), and
wi (y l + 1) = 0.4
•
For every agent j in N h , wj (k) > C(y h − 1) − C(y h − 2) for all k ≤
y h − 1, and gjs (y) − gjs (yN h − 1, y−N h ) < wj (y h ) < gjf (y) − gjs (yN h − 1, y−N h ), and
wj (y h + 1) = 0.5
4
Note that the allocation, (yN l − 1, y−N l ) is before the Ith allocation in the sequence. Hence, g f (yN l −
1, y−N l ) = g s (yN l − 1, y−N l ).
5
Note that the allocation, (yN h − 1, y−N h ) is before the Ith allocation in the sequence. Hence, gf (yN h −
1, y−N h ) = g s (yN h − 1, y−N h ). Every agent j in N h would demand y h − 1 in d(w, gf ) but would demand y h
(at least) in d(w, g s ).
21
•
For every agent o in N r where r = l, h and r ∈ {1, ..., m}, wo = wy∗r ,
i.e., in the demand game d(w, g f ), agent o would demand y r regardless of the
other agents’ demands6 .
Note that, throughout the proof, (N 1 , ..., N m ) is defined based on the Ith
allocation, y.
Lemma 3 proves that f (w) must be a strong Nash equilibrium outcome of
d(w, g f ). We prove the claim in two steps. In Step 1, we show that in d(w, g f )
there is no strong Nash equilibrium in which either (1) agents in N l demand no
more than y l or (2) agents in N h demand no more than y h . In Step 2, we show
that even if there is a strong Nash equilibrium in which agents in N l demand
more than y l and agents in N h demand more than y h , it is impossible that f (w)
is that strong Nash equilibrium outcome.
Step 1: In d(w, g f ), there is no strong Nash equilibrium in which either (1)
agents in N l demand no more than y l or (2) agents in N h demand no more than
yh.
Note first that based on the constructed preference profile w, every agent
o, o ∈ N r , r = l, h, would always demand y r . Regardless of the other agents’
demands, agents in N l would demand at least y l − 1 units of the public good,
and agents in N h would demand at least y h − 1 units of the public good.
6
The existence of such wy∗r is guaranteed by the ConSov property of f. It’s hard to define clearly what
kind of willingness to pay vector wy∗r should have without knowing gf . A willingness to pay vector such that
wo (yr ) > C(y r ) and wo (y r + 1) = 0 will guarantee agent o to consume at least y r , but may not guarantee
consuming exactly y r . Such a willingness to pay can guarantee agent o to consume exactly y r under the
serial mechanisms.
22
When some or all agents in N l demand y l − 1, the best reply7 of agents in N h
is to demand y h because by the induction assumption the relevant portion of g f
is the same as the serial pattern.
When agents in N h demand y h , the best reply8 of agents in N l is to demand
y l because by construction gif (y) − gis (yN l − 1, y−N l ) < wi (y l ) for all i ∈ N l .
When agents in N l demand y l , the (joint) best reply of agents in N h is to
demand y h − 1, because by construction wj (y h ) < gjf (y) − gjs (yN h − 1, y−N h ) for
all i ∈ N h .
When agents in N h demand y h − 1, the (joint) best reply of agents in N l is to
demand y l −1, because by the induction assumption, g f (yN h −1, y−N h ) = g s (yN h −
1, y−N h ) and g f (yN h − 1, yN l − 1, y−N h −N l ) = g s (yN h − 1, yN l − 1, y−N h −N l ), also
by construction, for all i ∈ N l , wi (y l ) < gis (y) − gis (yN l − 1, y−N l ) = gis (yN l , yN h −
1, y−N l −N h ) −gis (yN l −1, yN h −1, y−N l −N h ). The last equality follows from Lemma
4.
The two best replies, of agents in N l and N h respectively, never meet when
either agents in N l demand no more than y l or agents in N h demand no more
than N h .
Step 2: It is impossible that f(w) is a strong Nash equilibrium outcome of
d(w, g f ) such that agents in N l get more than y l and agents in N h get more than
yh.
7
When agents in N l demand yl − 1, agents in N h may have more than one best replies only if when
agents in N h demand more than yh , their cost share remain the same as when they demand y h .
8
When agents in N h demand yh , agents in N l may have more than one best replies only if when they
demand more than y l , their cost share is the same as when they demand y l .
23
We first prove a monotonicity result.
Lemma 5: (Consumption Monotonicity) If f satisfies ConSov and CSP,
then in the corresponding cost sharing pattern g f , no agent can consume more
and pay less, i.e., for all y−i ∈ Y N/i and yi > yi ∈ Y, gif (yi , y−i ) ≥ gif (yi , y−i ).
Proof of Lemma 5:
Suppose there exist y−i ∈ Y N/i and yi > yi ,
such that gif (yi , y−i ) < gif (yi , y−i ). According to ConSov, there exists wy∗i ∈ Y ,
such that agent i gets yi for certain by reporting it as his willingness to pay.
If gif (yi , y−i ) < gif (yi , y−i ), agent i would gain by reporting wy∗ when his true
i
willingness to pay is wy∗i and other agents’ willingness to pay is wy∗−i . This violates
CSP.9 l+
h+
Suppose that there is a strong Nash equilibrium, (yN
l , yN h , y−N l −N h ), in which
agents in N l demand y l+ > y l and agents in N h demand y h+ > y h and f y (w) =
l+
h+
10
(yN
. Based on w, every agent, o ∈ N r , r = l, h, gets y r for
l , yN h , y−N l −N h )
certain. Since wi (k) = 0 for all k > y l and all i ∈ N l , demanding y l+ can be one
of the best replies only when agent i’s cost share is the same as when demanding
f
l+
h+
h+
l
y l , i.e., gif (yN
l , yN h , y−N l −N h ) = gi (yN l , yN h , y−N l −N h ) for all i ∈ N .
l+
h+
h+
f
We now show that g f (yN
l , yN h , y−N l −N h ) = g (yN l , yN h , y−N l −N h ). Since f (w) =
l+
h+
l+
h+
f
∗
((yN
l , yN h , y−N l −N h ), g (yN l , yN h , y−N l −N h )), by Lemma 2, S(w) = S(wN h = wy h+ , w−N h ),
(where (wN h = wy∗h+ , w−N h ) denote the willingness to pay profile in which every
9
Notice that if we have strict consumption monotonicity, i.e., an agent must pay more to consume more,
there is no need to prove Step 2 because demanding more than yl (y h ) is never a best reply for agents in
N l (N h ). The rest of the proof is to eliminate the special case where according to gf , agents may consume
more without paying more.
10
The proof of the case where only part of agents in N l demand more than yl and part of agents in N h
demand more than y h is the same as this case, but more tedious notation-wise. It is omitted here.
24
agent in N h reports the preference to get y h+ for certain, while all other agents’
preferences are the same as in the constructed w). If for any k ∈ N − N l ,
f
l+
h+
h+
gkf (yN
l , yN h , y−N l −N h ) > gk (yN l , yN h , y−N l −N h ), when the agents’ preference pro-
file is (wN h = wy∗h+ , w−N h ), the coalition (N l ∪ k) can gain by letting agents in N l
f
l+
h+
h+
report wy∗l . If for any k ∈ N − N l , gkf (yN
l , yN h , y−N l −N h ) < gk (yN l , yN h , y−N l −N h ),
when the agents’ preference profile is (wN h = wy∗h+ , wN l = wy∗l , w−N h −N l ), the
coalition (N l ∪ k) can gain by letting agents in N l report wy∗l+ . Therefore,
l+
h+
h+
f
g f (yN
l , yN h , y−N l −N h ) = g (yN l , yN h , y−N l −N h ).
Based on Consumption Monotonicity, we must have gjf (yN l , yN h , y−N l −N h ) ≤
h+
h
gjf (yN l , yN
h , y−N l −N h ) for all j ∈ N .
l+
h+
l+
f
Similarly, we must have: g f (yN
l , yN h , y−N l −N h ) = g (yN l , yN h , y−N l −N h ) and
l+
l
gif (yN l , yN h , y−N l −N h ) ≤ gif (yN
l , yN h , y−N l −N h ) for all i ∈ N .
Combining those inequalities, we get:
l+
h+
h
gjf (yN l , yN h , y−N l −N h ) ≤ gjf (yN
l , yN h , y−N l −N h ) for all j ∈ N ,
l+
h+
l
gif (yN l , yN h , y−N l −N h ) ≤ gif (yN
l , yN h , y−N l −N h ) for all i ∈ N .
Suppose either of the above two inequalities holds strictly. Then the coalition
N l + N h can gain by having agents in N l report wy∗l and agents in N h report
wy∗h , since wi (k) = 0 for all k > y l , i ∈ N l and wj (k) = 0 for all k > y h , j ∈ N h .
25
Therefore, we must have the following:
l+
h+
h
gjf (yN l , yN h , y−N l −N h ) = gjf (yN
l , yN h , y−N l −N h ) for all j ∈ N ,
l+
h+
l
gif (yN l , yN h , y−N l −N h ) = gif (yN
l , yN h , y−N l −N h ) for all i ∈ N .
Note that every agent, j, in N h is indifferent between (y h , gjf (y)) and
l+
h+
l+
h+
h
(y h+ , gjf (yN
l , yN h , y−N l −N h )). If y(w) = (yN l , yN h , y−N l −N h ), then agents in N
have incentive to report wy∗h −1 as their willingness to pay to get y h − 1 for certain
because y(wy∗h −1 , w−N h ) = (yN l − 1, yN h − 1, y−N l −N h )11 . This contradicts CSP.
Therefore, the claim holds. By induction, the cost must be shared according to the serial pattern for y ∈ Y N .
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11
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r
26
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27