We propose a simple mechanism that implements the Ordinal Shapley Value (Pérez-Castrillo and Wettstein [2005]) for economies with three or less agents.Ordinal Shapley Value, implementation, mechanism design

David Pérez-Castrillo

David Wettstein

English

Research Papers in Economics

Implementation of the Ordinal Shapley Value for a
three-agent economy1
David Pérez-Castrillo2
Universitat Autònoma de Barcelona
David Wettstein3
Ben-Gurion University of the Negev
April 2005
1We gratefully acknowledge ﬁnancial support from project BEC 2003-01133, the Generali-
tat de Catalunya (2001 SGR-00162 and Barcelona Economics, CREA), and the Israeli Science
Foundation.
2Corresponding Author: Department of Economics & CODE, Universitat Autònoma de
Barcelona, Ediﬁcio B, 08193 Bellatera (Barcelona), Spain. Email: david.perez@uab.es.
3Department of Economics, Ben-Gurion University of the Negev, Monaster Center for Eco-
nomic Research, Beer-Sheva 84105, Israel. Email: wettstn@bgumail.bgu.ac.il. Ph: 972-8-
6472291, Fax: 972-8-6472941Abstract
We propose a simple mechanism that implements the Ordinal Shapley Value (Pérez-
Castrillo and Wettstein [2005]) for economies with three or less agents.
JEL Classiﬁcation numbers: C72, D50, D63.
Keywords: Ordinal Shapley Value, implementation, mechanism design.1 Introduction
The Ordinal Shapley Value (OSV) is a way to allocate gains realized by cooperation
in general economic environments. It is invariant with respect to the utility representa-
tion of the agents’ preferences and enjoys several desirable properties such as eﬃciency,
monotonicity, anonymity, and individual rationality (see Pérez-Castrillo and Wettstein
[2005]). It provides a reasonable outcome for a large class of environments even where
competitive equilibria or core allocations may fail to exist.
The OSV is a normative solution concept. An alternative approach to the problem of
sharing gains from cooperation consists of proposing mechanisms whose equilibria yield
“good” outcomes.1 In this paper, we propose the use of a bidding mechanism,w h i c h
combines and adapts to exchange economies proposals suggested in Pérez-Castrillo and
Wettstein [2001] and [2002]. Informally, the mechanism proceeds as follows: In stage 1
the agents bid to choose the proposer. Each agent bids by submitting an n-tuple of real
numbers, one number for each agent (including himself). The number submitted by agent
i for an agent j, is a commitment to forego a commodity bundle in case j is chosen as the
proposer. The bids submitted by an agent must sum up to zero. The agent for whom the
aggregate bid (sum of bids submitted for him by all agents including himself) is the highest
is chosen as the proposer. Before moving to stage 2, all the agents (including the proposer)
pay the “bid” (i.e., the promised commodity bundles) they submitted for the proposer. In
stage 2 the proposer oﬀers a feasible allocation of the total initial resources. The oﬀer is
accepted if all the other agents agree. In case of acceptance each agent receives the bundle
suggested for him in this allocation. In the case of rejection all the agents other than the
proposer play the same game again where the new initial endowments incorporate the
allocations paid and received by the end of stage 1.
We prove that the proposed bidding mechanism implements in Subgame Perfect equi-
librium the OSV correspondence for economies with at most three agents.
1See Moore and Repullo [1988] and Maniquet [2003] for papers dealing with implementation in general
environments. Winter [1994], Dasgupta and Chiu [1996] and Vidal-Puga and Bergantiños [2003] deal with
the implementation of the Shapley value in Transferable Utility games.
12 The Ordinal Shapley Value and the Bidding Mech-
anism
We consider a pure exchange economy E consisting of a set N = {1,2,...,n} of agents and
k ≥ 2 commodities. Agent i ∈ N is described by {ºi,w i}, where wi ∈ Rk is the vector of
initial endowments and ºi is a continuous and monotonic preference relation deﬁned over
Rk. We denote by Âi and ∼i the strict preference and indiﬀerence relationships associated
with ºi, and e ≡ (1,...,1) ∈ Rk.
For this economy, Pérez-Castrillo and Wettstein [2005] propose and prove the existence
of a solution concept, called the Ordinal Shapley Value (OSV), the construction of which
relies on the notion of concessions. The use of concessions allows to “measure” the beneﬁts
from cooperation. Concessions are made in terms of the reference bundle e.
Deﬁnition 1 The Ordinal Shapley Value is deﬁned recursively.
(n =1 ) In the case of an economy with one agent with preferences º1 and initial
endowments a1 ∈ Rk, the OSV is given by the initial endowment: OSV(º1,a 1)={a1}.
For n ≥ 2, suppose that the solution has been deﬁned for any economy with (n−1) or
less agents.
(n) In the case of an economy (ºi,a i)i∈N w i t has e tN of n agents, the OSV ((ºi,a i)i∈N)
is the set of eﬃcient allocations (xi)i∈N for which there exists an n−tuple of concession
vectors (ci)i∈N,c i ∈ Rn−1 for all i ∈ N that satisfy:
n.1) for all j ∈ N, there exists y(j) ∈ OSV
¡
(ºi,a i + c
j
ie)i∈N\j
¢
such that xi ∼i y(j)i
for all i ∈ N\j,a n d
n.2)
P
i∈N\j
c
j
i =
P
i∈N\j
ci
j for all j ∈ N.
By deﬁnition, the OSV is eﬃcient.I ti sa l s oconsistent in the sense that any set of
(n − 1)a g e n t si si n d i ﬀerent between keeping their allocation or taking the concessions
made by the remaining agent and reapplying the solution concept to the (n−1) economy
(property n.1). Moreover, to ensure fairness, the concessions balance out (property n.2).
In fact, Pérez-Castrillo and Wettstein [2005] show that the concessions associated with
OSV allocations satisfy the stronger condition that they are symmetric, that is, the
2concession ci
j of agent i to agent j is equal to the concession c
j
i of agent j to i.A l s o ,t h e
matrix of concessions associated with any OSV allocation is unique.
The bidding mechanism to implement the OSV is recursively deﬁned as follows:
If there is only one agent {i}, he receives his initial endowments, so he obtains utility
ui(wi). (If only one player plays, there is no bidding stage.)
Given the rules of the mechanism for at most n − 1 agents, the mechanism for N =
{1,...,n} proceeds as follows:
t =1 : Each agent i ∈ N makes bids bi
j ∈ <,o n ef o re v e r yj ∈ N, with
P
j∈N bi
j =0 .
Hence, at this stage, a strategy for player i is a vector (bi
j)j∈N ∈ Hn, where Hn = n
z ∈ <n|
P
j∈N zj =0
o
.
For each i ∈ N,d e ﬁne the aggregate bid to player i by Bi =
P
j∈N b
j
i.L e t α =
argmaxi(Bi) where an arbitrary tie-breaking rule is used in the case of a non-unique
maximizer. Once the proposer α has been chosen, every player i ∈ N pays bi
αe and
receives (Bα/n) · e.
t =2 : The proposer α oﬀers a feasible allocation (x1,...,x n) ∈ Rkn given the initial
resources (wi)i∈N.
t =3 : The agents other than α, sequentially, either accept or reject the oﬀer. If an
agent rejects it, then the oﬀer is rejected. Otherwise, the oﬀer is accepted.
If the oﬀer is accepted, each agent i receives xi. Therefore, the ﬁnal payoﬀ to an agent
i is ui(xi). On the other hand, if the oﬀer is rejected, all players other than α proceed
t op l a yt h es a m eg a m ew h e r et h es e to fa g e n t si sN\{α} and the initial resources for
these players are (wi −(bi
α − Bα/n)e)i∈N\{α}; while player α is on his own with resources
wα − (bα
α − Bα/n)e.T h eﬁnal payoﬀ to α is uα(wα − bα
αe +( Bα/n)e).T h eﬁnal payoﬀ to
any agent i 6= α is the payoﬀ he obtains in the game played by N\{α}.
3 The implementation for economies with at most
three agents
We start by proving several properties of the OSV allocations for economies with two
agents.
3Lemma 1 (a) For a two-agent economy, both agents are indiﬀerent among the OSV
allocations.
(b) The concession c is the same for every OSV allocation and is a continuous function
of the initial endowments.
(c) Let x ∈ OSV ((ºi,w i)i=1,2) and x0 ∈ OSV ((ºi,w i + λe)i=1,2), λ>0. Then,
xi0 Âi xi for i =1 ,2.
Proof. (a) As shown in Pérez-Castrillo and Wettstein [2004], the OSV for a two-
agent economy consists of the eﬃcient allocations (x1,x 2) such that x1 ∼ w1 + ce and
x2 ∼ w2 + ce for some c ≥ 0. Consider now x,y ∈ OSV ((ºi,a i)i=1,2) and denote by
c = c1
2 = c2
1 and d = d1
2 = d2
1 the concessions associated respectively with x and y.T h e n ,
x1 ∼1 a1+ce, x2 ∼2 a2+ce, y1 ∼1 a1+de,a n dy2 ∼2 a2+de. It is immediate that x1 ≺1 y1
if and only if x2 ≺2 y2.T h ee ﬃciency of both allocations x and y implies x1 ∼1 y1 and
x2 ∼2 y2.
(b) The previous argument also shows that c = d, while the continuity of preferences
implies that the concession varies continuously with the initial endowments.
(c) There exist c and c0 such that: x1 ∼ w1 + ce, x2 ∼ w2 + ce, x01 ∼ w1 +( λ + c0)e
x02 ∼ w2 +(λ+c0)e. The allocation x0 is Pareto eﬃcient in (ºi,w i +λe)i=1,2 whereas x is
feasible, yet not Pareto eﬃcient for the economy (ºi,w i +λe)i=1,2. Hence, it must be the
case that λ + c0 >cand x0i Âi xi for i =1 ,2.
The next theorem shows that the set of Subgame Perfect equilibrium outcomes (SPE)
of the bidding mechanism coincides with the OSV for economies with at most three agents.
Theorem 1 The bidding mechanism implements the Ordinal Shapley Value in Subgame
Perfect Equilibrium in economies with n ≤ 3.
Proof. The proof proceeds by induction.
(a) We ﬁrst prove that every SPE outcome of the bidding mechanism is in the
OSV ((ºi,w i)i∈N).
For n =1the proof is trivial. Note also that for economies with one agent, there is
only one OSV allocation.
Claim 1. In any SPE,any agent i diﬀerent than the proposer α accepts the proposal x
at t =3if xi Â yi for every i ∈ N\{α}, where y ∈ OSV
¡
(ºj,w j − (bj
α − Bα/n)e)j∈N\{α}
¢
.
4Proof: First, by induction, in case of rejection the agents expect to obtain an allocation
in the OSV in the economy without the proposer (and where the concessions have been
added to or substracted from their initial endowment); second, by Lemma 1 (a),a g e n t si n
a two-agent economy are indiﬀerent among OSV allocations (as is the case for a one-agent
economy).
Claim 2. In any SPE o ft h eg a m et h a ts t a r t sa tt =2 , the proposer α proposes
an allocation x that is Pareto eﬃcient and satisﬁes xi ∼ yi for every i ∈ N/{α}, where
y = OSV
¡
(ºj,w j − (bj
α − Bα/n)e)j∈N\{α}
¢
. Moreover, every agent i ∈ N\{α} accepts
any oﬀer x such that xi % yi for every i ∈ N/{α}.
Proof: These are clearly equilibrium strategies for the agents other than the proposer.
As regards the proposer, he cannot gain by switching to another oﬀer that is accepted.
If he makes an unacceptable oﬀer he obtains the bundle wα − (bα
α − Bα/n)e) which if
preferred to xα would violate the Pareto eﬃciency of the proposal x.
Claim 3. In any SPE, Bi =0for i ∈ N. Moreover, each agent is indiﬀerent about
the identity of the agent who is chosen as the proposer.
Proof: Denote by M the set of agents for whom the aggregate bid is the largest, that
is, M ≡ {i ∈ N|Bi =m a x j∈N Bj}.W e ﬁrst claim that any agent j ∈ N is indiﬀerent
between any agent in M being chosen as the proposer. Indeed, if j would strictly prefer
some particular agent, say i ∈ M to win, agent j would slightly increase his bid to agent
i and decrease his bid to the other agents in M so that agent i is chosen as the proposer
for sure. Following the change, by Lemma 1 (b), agent j would be better oﬀ.
If M = N, Claim 3 is proven. Suppose, by way of contradiction, that M 6= N and
denote by m (<n )t h ec a r d i n a l i t yo fM. Assume, for convenience, 1 ∈ M. We now show
that agent 1 can achieve a better outcome by changing the bids he submitted in stage 1.
Consider the following change in the bids by agent 1: b10
1 = b1
1 −  , b10
i = b1
i − 2  for any
i ∈ M\{1}, b10
j = b1
j +(2m−1)  for a particular j/ ∈ M, and b10
j = b1
j otherwise, with  >0
and small enough. Then, B1 >B 0
1 >B 0
i for all i 6=1 .I n p a r t i c u l a r ,1 is chosen as the
proposer for sure. We claim that 1 is strictly better following this change in bids. To see
this, note ﬁrst that bi
1 did not change for any i 6=1and B0
1 <B 1. Hence, in the economy
with agents N\{1}, after the change in the bids the “initial endowments” change from
(wj−(b
j
1−B1/n)e)j∈N\{1} to (wj−(b
j
1−B0
1/n)e)j∈N\{1}. Given B1 >B 0
1, by Lemma 1 (c),
5all agents in N\{1} are worse oﬀ in the OSV of the second economy than in the OSV
of the former. Hence, at stage 2,a g e n t1 can oﬀer an allocation that is worse oﬀ for all
j ∈ N\{1} and, by Pareto eﬃciency, better oﬀ for himself. Therefore, agent 1 is better
oﬀ after bidding according to b10 than after bidding according to b1.
Claim 4. In any SPE, the oﬀer x made by the proposer at t =2always belongs to
OSV ((ºj,w j)j∈N). Moreover, the agents’ bids at t =1are bi
j = ci
j for all i,j ∈ N, i 6= j,
where c is the matrix of concessions of x, and bi
i = −
P
j∈N\{i} ci
j.
To prove Claim 4,d e n o t eb yx = x(i) ∈ OSV
¡
(ºj,w j − b
j
ie)j∈N\{i}
¢
the proposal that
agent i ∈ N makes if he is chosen as the proposer (we notice that Claim 3 states that
Bi =0 ) .W ea r eg o i n gt op r o v et h a tx ∈ OSV ((ºj,w j)j∈N). First, according to Claim 2,
x i saP a r e t oe ﬃcient allocation. Moreover, the n−t u p l eo fv e c t o r so fb i d s(bi)i∈N satisﬁes:
i) By Claim 3, xk ∼k x(j)k for j ∈ N,k 6= j,w h e r ex(j) ∈ OSV
¡
(ºk,a k − bk
je)k∈N\{j}
¢
,
ii)
P
i∈N\{j}
b
j
i =
P
i∈N\{j}
bi
j for all j ∈ N (by Claim 3, Bj =0 , i.e.,
P
i∈N\{j}
bi
j = −b
j
j;
moreover, the rules of the mechanism impose that
P
i∈N
b
j
i =0 , i.e., −b
j
j =
P
i∈N\{j}
b
j
i).
Therefore, the allocation x is in the set OSV ((ºj,w j)j∈N) taking the matrix of con-
cessions ci
j = bi
j for all i,j ∈ N, i 6= j.
( b )W en o wp r o v ee v e r ya l l o c a t i o nx in the set OSV ((ºi,w i)i∈N) is an SPE outcome
of the bidding mechanism. We denote c the matrix of concessions of x. We propose the
following strategies for the case of n agents:
At t =1 , each agent i, i ∈ N, announces bi
j = ci
j for every j ∈ N\{i} and bi
i =
−
P
j∈N\{i} ci
j.
At t =2 , agent i, if he is the proposer, proposes an allocation z that is Pareto eﬃcient
and satisﬁes that zj ∼ yj for every j ∈ N/{i}, where y ∈ OSV
¡
(ºj,w j − (b
j
i − Bi/n)e)j∈N\{i}
¢
.
(We recall that, according to Lemma 1 (a), in economies with one or two agents either
there is only one OSV allocation or agents are indiﬀerent among the several OSV allo-
cations.)
At t =3 , agent i, if agent j ∈ N/{i} is the proposer, accepts any oﬀer z such that
zi % yi,w h e r ey ∈ OSV
¡
(ºk,w k − (bk
j − Bj/n)e)k∈N\{j}
¢
and rejects it otherwise.
F i r s to fa l l ,w en o t i c et h a ti ft h ea g e n t sm a k et h ep r e v i o u sb i d s ,t h e nt h ea g g r e g a t e
bid to each one is zero. This is a direct consequence of the symmetry of the concessions.
6Second, if they make these bids, the proposal at t =2will certainly be x, given that
x is eﬃcient and guaranties the rest of the agents their OSV of the game without the
proposer and with the proposer’s concessions added to their initial endowment. Hence, if
the agents follow the previous strategies, the ﬁnal outcome is always x.
W ep r o v et h a tt h es t r a t e g i e sa r ei n d e e dSPE strategies. By the induction argument,
what the agents other than the proposer, say agent j, expect after the bids is some
allocation in OSV
¡
(ºk,w k − (bk
j − Bj/n)e)k∈N\{j}
¢
. Therefore, it is easy to check that
the previous strategies are SPE strategies from t =2on. Consider now the strategies at
t =1 . Remember that we have shown that Bi =0for all i ∈ N. If agent i changes his
bid, the proposer will be the agent (or one of the set of agents) to whom i increases his
bid. Denote by α the proposer, and B0
α > 0 the new aggregate bid. If α = i, then the
other agents will face a situation where all their initial endowments increase by the same
amount B0
i/n. By Lemma 1 (c), all these agents are better oﬀ in the new situation, hence
agent i is worse oﬀ. If the new proposer is α 6= i, then the outside option for agent i will
be a situation where all the agents other than i and α will see their initial endowment
increased by B0
α/n while agent i’s initial endowment will decrease by (n − 1)B0
α/n. An
argument similar to that of Lemma 1 (c) s h o w st h a ta g e n ti’s situation is worse oﬀ after
the change. Therefore, deviating is not proﬁtable.
The major diﬃculty with extending this result for any number of agents is intimately
related to the transfer paradox (Safra [1984]). We brieﬂy explain here this diﬃculty.
It is crucial for our result that (as stated in Claim 3) the equilibrium aggregate bids
are zero for every agent. For this result, it must be the case that a proposer can not
gain by increasing the bid for himself and facing at the proposal stage agents with larger
endowments. However, similar to the transfer paradox, an agent can be worse oﬀ in an
OSV allocation when the initial endowments of all agents (including himself) increase. If
this happens, the proposer may ﬁnd it “easier” (less costly in terms of his own welfare)
to make an acceptable proposal to the set of agents with larger endowments.
The mechanism constructed provides a non-cooperative foundation for the OSV for
environments with a small number of agents. It also shows that the concessions underlying
the OSV concept can be interpreted in terms of bids.
74 References
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Implementation of the Ordinal Shapley Value for a three-agent economy

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