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Bargaining in committees as an extension of Nash's bargaining theory
Journal of Economic Theory 132 (2007) 291 – 305 www.elsevier.com/locate/jet
Bargaining in committees as an extension of Nash’s bargaining theory Annick Laruellea , Federico Valencianob,∗ a Departamento de Fundamentos del Análisis Económico, Universidad de Alicante, Campus de San Vicente,
E-03071 Alicante, Spain b Departamento de Economía Aplicada IV, Universidad del País Vasco, Avenida Lehendakari Aguirre, 83,
E-48015 Bilbao, Spain Received 28 January 2005; final version received 3 May 2005 Available online 20 July 2005
Abstract This paper addresses the following issue: if a set of agents bargain on a set of feasible alternatives ‘in the shadow’ of a voting rule, that is, any agreement can be enforced if a ‘winning coalition’ supports it, what general agreements are likely to arise? In other words: what influence can the voting rule used to settle (possibly nonunanimous) agreements have on the outcome of consensus? We model the situation as an extension of the Nash bargaining problem in which an arbitrary voting rule replaces unanimity. In this setting a natural extension of Nash’s solution is characterized. © 2005 Elsevier Inc. All rights reserved. JEL classification: C7; D7 Keywords: Bargaining; Voting; Committees
1. Introduction Nash [25] proposes and characterizes axiomatically a cooperative ‘solution’ to the bargaining problem, in the spirit of von Neumann and Morgenstern’s [33] notion of ‘value’ of a zero-sum two-person game, as a rational expectation of the ‘amount of satisfaction’ or expected utility payoff of a ‘highly rational’ player engaging in a bargaining situation ∗ Corresponding author.
E-mail addresses: [email protected] (A. Laruelle), [email protected] (F. Valenciano). 0022-0531/$ - see front matter © 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.jet.2005.05.004
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with another rational player. In Shapley [30] a similar notion of ‘value’ for transferable utility (TU) games is also proposed and characterized. Still in the same spirit, here we are interested in understanding the influence that the voting rule used to settle agreements by a set of bargainers can have on the outcome of negotiations. More precisely, if a set of agents bargain on a set of feasible alternatives ‘in the shadow’ of a voting rule, that is, any agreement can be enforced if a ‘winning coalition’ supports it, what general agreements are likely to arise? Or, put into Nash’s classical terms, what agreements can a rational agent expect to arise when faced with the prospect of engaging in such a situation? The importance of the issue is clear in many contexts. In contrast with the case of a ‘take-it-or-leave-it’ committee, only entitled to accept or reject proposals submitted to it, without the capacity to modify them, it is often the case in a committee that uses a voting rule to make decisions that the final vote is merely the formal settlement of a bargaining process in which the issue to be voted upon has been adjusted to gain the acceptance of all members. This is what we call a ‘bargaining committee’. 1 It seems intuitively obvious that in such committees the voting rule by means of which agreements are settled conditions the outcome of negotiations. With this motivation in mind and in the spirit common to the classical results of von Neumann–Morgenstern, of Nash and of Shapley alluded to above, in this paper we explore an extension of Nash’s [25] model of a bargaining problem. We see Nash’s original model as consisting of two ingredients, a set of (two) players with von Neumann–Morgenstern [33] preferences over a set of feasible agreements, and a voting procedure (unanimity) to settle agreements. Thus, the kind of situation we are interested in can be described by a natural generalization of this model (and its traditional extension to n players), by considering arbitrary voting rules 2 instead of unanimity. In this two-ingredient setting, on grounds similar to those in Nash [25] or Shapley [30], axioms such as efficiency, anonymity, independence of irrelevant alternatives, invariance w.r.t. positive affine transformations and null-player can be adapted, retaining their meaning and motivation. Assuming these conditions entails a significant narrowing of the class of admissible answers or ‘solutions’ which is identified. It is shown that in fact all that remains is to settle the choice (within narrow limits) of an answer for the same issue in the particular case in which the configuration of preferences is TU-like. The possibility of singling out a choice is accomplished by integrating Nash’s and Shapley–Dubey’s characterizations [30,6] into one which goes further beyond both, yielding a surprising solution to the more complex problem under consideration. The rest of the paper is organized as follows: in Section 2 the model is precisely formulated. In Section 3 the natural extension of Nash’s and some of Shapley’s axioms in the more general framework considered here are provided. Section 4 contains the characterizing results, along with the relationships with Nash’s solution and the Shapley value. The meaning of the results is discussed in Section 5 along with some lines of further research. Then Appendix contains the proofs of the characterizing theorems.
1 The distinction between a ‘take-it-or-leave-it’ committee and a ‘bargaining’ committee is similar to the distinction made by Baron and Ferejohn [3] between a closed and an open rule in a legislature. 2 A setting including both a voting rule and a set of feasible agreements has been considered by some authors
but, as far as we know, with different purposes and within a completely different approach concerned with social choice issues (see [27] for a recent overview).
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2. Bargaining in the shadow of a voting rule We are concerned with a situation in which a set of agents bargain on a set of feasible alternatives ‘in the shadow’ of a voting procedure. That is, any agreement can be enforced if a ‘winning coalition’ supports it. The set N = {1, . . . , n} labels the seats of the decision procedure by means of which agreements are to be settled. As only yes/no voting is considered, a vote configuration can be represented by the set of ‘yes’-voters. So, any S ⊆ N represents the result of a vote in which players occupying seats in S voted ‘yes’ and those in N \S voted ‘no’. An N-voting rule is specified by a set W ⊆ 2N of winning vote configurations such that (i) N ∈ W ; (ii) ∅ ∈ / W ; (iii) If S ∈ W , then T ∈ W for any T containing S; and (iv) If S ∈ W , then N\S ∈ / W . W denotes the set of all such N-voting rules. For voting rule W, M(W ) denotes the set of minimal winning configurations, i.e., those that do not contain any other winning configuration. For any S ∈ M(W ) (S = N ), WS∗ denotes the voting rule that results from W by eliminating S from the set of winning configurations, that is, WS∗ := W \ {S}. For any permutation :N → N , W denotes the voting rule W := {(S) : S ∈ W }. A voting rule W is symmetric if W = W , for any permutation . The n players who use an N-voting rule are labelled by the seats in N that they occupy, and we refer to the subset of players denoted by a subset S ⊆ N as coalition S. Thus, depending on the context any S ⊆ N will be referred to either as a vote configuration (of seats) or as a coalition (of players). We will also speak of winning coalitions for a given N-rule, with an obvious meaning. A seat/player i ∈ N is said to be a null seat/player in W if, for any coalition S, S ∈ W if and only if S\ {i} ∈ W . We assume that a set of n (N-labelled) players makes decisions by means of rule W in the following sense. They can reach any alternative within a set, as well as any lottery over them, as long as: (i) a winning coalition supports it, and (ii) no player is imposed upon an agreement worse than the status quo, where all players will remain if no winning coalition supports any agreement. It is also assumed that every player has expected utility (von Neumann and Morgenstern [33] (vNM)) preferences over this set, so that the situation can be summarized à la Nash in utility terms by a feasible set of utility vectors D, together with the particular vector d associated with the disagreement or status quo, as a summary of the situation concerning the players’ decision. Accepting this simplification, the situation can be summarized by a pair (B, W ), where B = (D, d) is a classical n-person bargaining problem, and W is the N-voting rule to enforce agreements. In accordance with this interpretation we assume that Dd := {x ∈ D : x d} 3 is bounded, and D is a closed, convex and comprehensive (i.e., x y ∈ D ⇒ x ∈ D) set containing d, such that there exists some x ∈ D s.t. x > d. B denotes the set of all such bargaining problems. For any permutation : N → N, B := ((D), (d)) will denote the bargaining problem that results from B by -permutation of its coordinates, so that for any x ∈ R N , (x) denotes the vector in R N s.t. (x)(i) = xi . A bargaining problem B is symmetric if B = B, for any permutation . Thus, in this setting we are concerned with pairs (B, W ) ∈ B × W, each of which, consistent with the interpretation that accompanied its introduction, will be referred to as a bargaining problem B under rule W, or for short just a problem (B, W ). 3 We will write for any x, y ∈ R N , x y (x < y) if x y (x < y ) for all i = 1, . . . , n. i i i i
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Before proceeding with the search for a ‘value’ or a ‘solution’ in this setting, let us first see how the model fits into the general nontransferable utility (NTU) games framework and how the classical bargaining problems and simple TU games fit into this model. Let pr S : R N → R S denote the natural S-projection for S ⊆ N , and let x S := pr S (x) for any x ∈ R N . On obvious grounds we can associate an NTU game (N, V(B,W ) ) with every (B, W ) ∈ B×W, by associating with each coalition S the closed, convex and comprehensive subset of R S containing all feasible utility vectors for S, given by if S ∈ W, pr S x ∈ D : x N\S = d N\S V(B,W ) (S) := pr S (ch(d)) if S ∈ / W, where B = (D, d), and ch(d) denotes the comprehensive hull of {d} . The n-person classical bargaining problem corresponds to the case in which W is the unanimity rule W = {N }, with N as the only winning coalition. While when the bargaining ingredient in the model is the (normalized) TU-bargaining problem := (, 0), where := generalized x ∈ R N : i∈N xi 1 , the associated NTU game V(,W ) is equivalent to the simple TU game 4 representing the rule vW , given by 1 if S ∈ W, vW (S) := 0 if S ∈ / W. Thus, the family of associated NTU games (N, V(B,W ) ) : (B, W ) ∈ B × W properly contain all classical bargaining problems and all simple superadditive games. 5 Nevertheless, in spite of the possibility of embedding the family of problems under consideration into a subclass of NTU games, we will deal with the model in terms of its constituent elements, B and W.
3. Rationality conditions for a ‘value’ or a ‘solution’ In the class of situations described by this model the question addressed by Nash [25] and Shapley [30] in their respective cases can also be addressed: What is the ‘amount of satisfaction’ or utility that a rational player can expect from such a situation, or, in classical terms, what is the value for any player of the prospect of engaging in a situation such as this? We will proceed as in the two seminal papers by stating reasonable conditions that will narrow the possible options. Thus, we will impose some conditions on a map : B × W → R N , for vector (B, W ) ∈ R N to be considerable as a rational agreement, or better as a reasonable expectation of utility levels of a general agreement in a bargaining situation B under voting rule W. To begin with, in view of the interpretation in terms of the underlying situation stated in Section 2, we build the requirements of being feasible and no worse than the status quo for any player into the very notion of solution, that is, we impose 4 Note that this model allows for a neat distinction between voting rules and their associated simple TU games.
Although conceptually different (the specification of a voting rule does not involve its users’ preferences), both sets are obviously isomorphic. 5 A TU game v : 2N → R is superadditive if v(S ∪ T ) v(S) + v(T ), for all disjoint S, T ⊆ N.
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as prerequisites: (B, W ) ∈ D (feasibility), and (B, W ) d (individual rationality), if B = (D, d). In addition to this, we will use the following conditions, all of them natural adaptations of Nash’s and Shapley’s characterizing properties: 1. Efficiency (Eff): For all (B, W ) ∈ B × W, there is no x ∈ D, s.t. x > (B, W ). 2. Anonymity (An): For all (B, W ) ∈ B × W, and any permutation :N → N , and any i ∈ N , (i) ((B, W )) = i (B, W ), where (B, W ) := (B, W ). 3. Independence of irrelevant alternatives (IIA): Let B, B ∈ B, with B = (D, d) and B = (D , d ), such that d = d, D ⊆ D and (B, W ) ∈ D ; then, (B , W ) = (B, W ), for any W ∈ W. 4. Invariance w.r.t. positive affine transformations (IAT): For all (B, W ) ∈ B × W, and all N and ∈ R N , ∈ R++ ( ∗ B + , W ) = ∗ (B, W ) + , where ∗ B + = ( ∗ D + , ∗ d + ), denoting ∗ x := (1 x1 , . . . , n xn ), and ∗ D + := { ∗ x + : x ∈ D}. 5. Null player (NP): For all (B, W ) ∈ B × W, if i ∈ N is a null player in W, then i (B, W ) = di . 5*. Null player* (NP*): For all (B, W ) ∈ B × W, i (B, W ) = di if and only if i ∈ N is a null player in W . The readers can see for themselves the precise correspondence of these conditions with some of Nash’s and Shapley’s axioms. But it is worth noting some subtle differences arising in this setup. Observe that Eff, IIA and IAT (adaptations of Nash’s axioms) concern the feasible set, while An (adapted from Nash’s and Shapley’s anonymity 6 ) and NP (from Shapley’s system), as well as NP*, concern both. We omit the arguments in support of each of conditions 1–5, which can be found in Nash’s and Shapley’s papers. 7 It may be worth remarking, though, that An entails a consistent relabelling of voters in B and seats in W. Note also that it seems natural in our setting to set null players’ expectations to the status quo level, given their null capacity to influence the outcome given the voting rule according to which final agreements are enforced. Alternatively, the stronger version of null-player NP* requires in addition that nonnull players expectations are strictly greater than the status quo level, which also seems reasonable given that these players are not irrelevant according to the voting rule. The next two conditions impose alternative constraints on the solution for TU configurations of preferences (i.e., when B = ). 6. Transfer (T): For any two rules W, W ∈ W, and all S ∈ M(W ) ∩ M(W ) (S = N ): (, W ) − (, WS∗ ) = (, W ) − (, WS ∗ ).
(1)
6*. Symmetric gain-loss (SymGL): For any voting rule W ∈ W, and all S ∈ M(W ) (S = N ): i (, W ) − i (, WS∗ ) = j (, W ) − j (, WS∗ ), for any two voters i, j ∈ S, and any two voters i, j ∈ N \ S. 6 In Nash [26] anonymity replaces symmetry. We prefer this simpler condition to the slightly weaker symmetry in Nash [25], which can also be adapted to this setting 7 For a careful discussion of Nash’s axioms, see, e.g., chapter 1 in Binmore [5].
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These conditions permit to exploit the lattice structure of the set of simple games (or voting rules), as additivity permits to exploit the vector space structure of the domain of TU games in Shapley’s [30] characterization. Dubey [6] uses T to characterize the Shapley– Shubik index. 8 Laruelle and Valenciano [18] replace T by SymGL (weaker than T in the presence of anonymity) to characterize Shapley–Shubik and Banzhaf [2] indices. In words, T postulates that the effect of eliminating a minimal winning configuration from the set of winning configurations is the same whatever the voting rule, while SymGL requires that the effect of eliminating a (minimal) winning configuration from the list that specifies the voting rule is equal on any two voters belonging (not belonging) to it. Remark 1. Observe also that, fixing B = , conditions 1, 2, and 5 when restricted to (, ·) : W → R N become, respectively, their homonymous efficiency, anonymity and null-player for a value on the domain of simple TU games, while 3 and 4 become empty requirements. If, instead, W is fixed to be the unanimity rule W = {N } (or any symmetric voting rule), then conditions 1–4 restricted to (·, W ) : B → R N become Nash’s homonymous characterizing conditions in the classical one-ingredient bargaining setting, while 5 becomes an empty condition (there are no null-players when W is symmetric), and 5* becomes Roth’s [28] strict individual rationality condition. 9
4. Characterizations In this section we show how assuming some of the above conditions restricts the class of admissible ‘solutions’ or ‘values’ : B × W → R N . Denote by Nash(B) the Nash [25] bargaining solution of an n-person bargaining problem B = (D, d), that is, (xi − di ), Nash(B) = arg max x∈Dd
i∈N
and by Nashw (B) the w-weighted asymmetric Nash bargaining solution [14] of the same problem for a vector of nonnegative weights w = (wi )i∈N , (xi − di )wi . Nashw (B) = arg max x∈Dd
i∈N
In fact, if any of the weights is zero, the w-weighted Nash solution may not be unique under the conditions assumed in the bargaining problems in B. This difficulty can be overcome in two ways: either by assuming a condition of ‘nonlevelness’ on the feasible set, or, as we do here, by imposing the disagreement ‘payoff’ for players whose weight is zero. 8 Formulation
(1)
of
this
condition
is
equivalent
(see
[18])
to
the
more
traditional:
(, W ) + (, W ) = (, W ∪ W ) + (, W ∩ W ). 9 Roth [28] replaces efficiency by this condition in Nash’s system to characterize the Nash bargaining solution.
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That is, redefining Nashw i (B)
:=
argi max
x∈Dd
j ∈J (xj
di
− dj )wj
if i ∈ J, if i ∈ N \ J,
(2)
where J = {i ∈ N : wi > 0} . In what follows whenever we write Nashw (B) we refer always to definition (2). The following result shows in which way conditions 2–4 and 5* considered in the previous section drastically restrict the possible answer to the question raised. Theorem 1. A solution/value : B ×W → R N satisfies anonymity (An), independence of irrelevant alternatives (IIA), invariance w.r.t. affine transformations (IAT) and null player* (NP*), if and only if (B, W ) = Nash(W ) (B),
(3)
for some map : W → R N that satisfies anonymity and null player*. Moreover, = (, ·), where (W ) = (W )/ i∈N i (W ). Proof. See the appendix.
Thus formula (3) can be rewritten as (B, W ) = Nash(,W ) (B).
(4)
Note that efficiency has not been assumed on , but as is mentioned in the proof, this condition is implied by IIA, IAT and NP* by an argument similar to that in Roth [28]. In particular (, ·) is also efficient. Therefore, any map (, ·) : W → R N that satisfies efficiency, anonymity and null player would fit in formula (4) and yield a solution (B, W ) that satisfies the four rationality conditions. In other words assuming An, IIA, IAT and NP*, the solution, given by (4), will be unique as soon as (, ·) is specified. As stated in Section 2, in the case of the TUbargaining problem , for any voting rule W, the associated NTU game V(,W ) is equivalent to the simple TU game vW s.t. vW (S) = 1 if and only if S ∈ W . Thus, the solution is so far determined up to the choice of a value on the domain of simple superadditive games satisfying ‘efficiency’, ‘anonymity’ and null player*. These conditions on (, ·) immediately bring to mind the Shapley [30] value, or more specifically in the context of simple games, the Shapley–Shubik [32] index. 10 Denote by Sh(v) the Shapley [30] value of a TU game v, given by Shi (v) =
(n − s)!(s − 1)! (v(S) − v(S\i)), n!
S:S⊆N
10 But there are other alternatives, for instance, the normalization of any semivalue [7] meets these conditions, as well as some ‘power indices,’ such as the Holler–Packel [13] index.
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and by Sh(W ) the Shapley–Shubik [32] index of a voting rule W, i.e., the Shapley value of the associated simple game vW . We have the following result. Proposition 1. Let : B × W → R N be a solution/value that satisfies Eff, An, NP and T; then, for any voting rule W ∈ W, (, W ) = Sh(W ). Proof. For any W ∈ W, as stated in Section 2, the associated NTU game V(,W ) is equivalent to the simple TU game vW s.t. vW (S) = 1 if and only if S ∈ W . And, as is well known, efficiency, anonymity, null player and transfer characterize the Shapley (–Shubik) value in the domain of simple (superadditive or not) games ([6]; see also [18]). It can then be easily verified that in our setting conditions Eff, An (recall that is symmetric), NP and T become their homonymous for (, ·) : W → R N . Thus, (, W ) = Sh(vW ) = Sh(W ) for any voting rule. Then as an easy corollary of Theorem 1 and Proposition 1 we have: Theorem 2. There exists a unique solution/value : B×W → R N that satisfies anonymity (An), independence of irrelevant alternatives (IIA), invariance w.r.t. affine transformations (IAT), null player* (NP*) and transfer (T), and it is given by (B, W ) = NashSh(W ) (B). Proof. See the appendix.
(5)
The same solution can be characterized assuming only the weaker NP if Eff is also assumed. That is, we have the following alternative characterization: 11 Theorem 3. There exists a unique solution/value : B × W → R N that satisfies Eff, An, IIA, IAT, NP and T, and it is given by (5). Note that (5) yields for any symmetric voting rule W (B, W ) = Nash(B), while for any rule W it yields (, W ) = Sh(W ). Moreover, as pointed out in Remark 1, NP and T become empty requirements when W is any fixed symmetric rule, while NP* becomes Roth’s [28] strict individual rationality. Thus, the characterizing axioms in Theorems 1 and 2 (resp., Theorem 3) when restricted to (·, W ) : B → R N for any fixed symmetric rule, become Roth’s (resp., Nash’s) axiomatic system. On the other hand, as conditions IIA and IAT become empty requirements when fixing B = , Proposition 1 can also be rephrased as follows: the characterizing axioms in 11 We omit the proof that is an easy adaptation of the proof of Theorem 1 and Lemma 3.
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Theorem 3 when restricted to (, ·) : W → R N become Shapley–Dubey’s characterizing system of the Shapley value (or Shapley–Shubik index) in W. But observe that the same is not true for Theorem 2 (as IIA and IAT lose their bite if B = , efficiency cannot be derived from strict individual rationality alone). But a similar replacement of efficiency by strict individual rationality for the Shapley value is possible if the domain includes and all its positive affine transformations. In other words, Theorem 2 integrates Roth’s and Shapley–Dubey’s characterizations into one, and similarly Theorem 3 integrates Nash’s and Shapley–Dubey’s characterizations into one. But both theorems go further beyond these previous characterizations, yielding a surprising solution to the more complex problem under consideration given by (5). Finally, in Proposition 1, and in Theorems 2 and 3, transfer (T) can be replaced by the weaker (in the presence of anonymity) condition of SymGL. Thus, we have still another two characterizations of (5), whose proof, based on Laruelle and Valenciano [18] characterization of the Shapley–Shubik index, we omit. Theorem 4. (i) There exists a unique solution/value : B × W → R N that satisfies An, IIA, IAT, NP* and SymGL, and it is given by (5). (ii) The same is true if satisfies Eff, An, IIA, IAT, NP and SymGL. In Theorem 4, part (i) (resp., (ii)) integrates Roth’s (resp., Nash’s) and Laruelle– Valenciano’s characterizations into one. Remark 2. The following solutions show the independence of the axioms used in Theorem 2, in Theorem 3 and in Theorem 4 (and a fortiori in Theorem 1). (B, W ) = d satisfies all axioms in Theorems 2 and 4(i) except NP*, and all axioms in Theorems 3 and 4(ii) except Eff. (B, W ) = NashSh (W ) (B), where Sh is any nonsymmetric -weighted Shapley value [16], satisfies all axioms in Theorems 2–4 except An. (B, W ) = Nash(B) satisfies all axioms in Theorems 3 and 4(ii) except NP. (B, W ) = d + t¯ Sh(W ), where t¯ = max {t ∈ R : d + t Sh(W ) ∈ D}, satisfies all axioms in Theorems 2–4 except IAT. (B, W ) = d + t¯ ((m − d) ∗ Sh(W )), where t¯ = max {t : d + t ((m − d) ∗ Sh(W )) ∈ D} with m ∈ R N s.t. mi = max pr i (Dd ), satisfies all axioms in Theorems 2–4 except IIA (note that this family yields Kalai–Smorodinsky [17] solution as the particular case (·, W ) when W is symmetric). (B, W ) = NashBz(W ) (B), where Bz is the Banzhaf [2] index, satisfies all axioms in Theorems 2 and 3 except T, and all except SymGL in Theorem 4.
5. Conclusion Theorem 1 extends the Nash bargaining solution and provides an endogenous justification of the ‘weights’ in Kalai’s [14] asymmetric Nash solutions in the complete information
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context. 12 Kalai’s solutions emerge when symmetry is dropped in Nash’s system, or still assuming symmetry in an adequately ‘replicated’ problem [14], as if each player negotiated on behalf of a number of players, as is usually the underlying situation when a nonsymmetric rule is used to make decisions. Binmore [5, p. 78], justifies the asymmetric Nash solutions as reflecting the different ‘bargaining power’ of the players “determined by the strategic advantages conferred on players by the circumstances under which they bargain”, and uses the term ‘bargaining power’ to refer to the players’ weights. 13 In our case these ‘circumstances’ consist of the voting rule that governs negotiations. Thus, the extension provided by Theorem 1 re-opens the old power indices’ ‘contest’ in a richer and more interesting setting than the traditional one of bare simple games. Now they (i.e., efficient, anonymous values for simple games satisfying the null player condition) compete to adequately represent the ‘bargaining power’ (in a precise and relevant sense) that a voting rule confers to its users in a bargaining committee. Theorems 2–4 single out the Shapley–Shubik index by integrating two axiomatic systems into a single, consistent one, which when restricted to classical bargaining problems becomes either Roth’s or Nash’s system, and when restricted to simple games becomes a characterizing system (Dubey–Shapley’s or Laruelle–Valenciano’s) of the Shapley value. As a result, the unique solution emerging turns out to be consistent with both Nash’s solution and Shapley’s value. Since Nash’s and Shapley’s classical results, several attempts to obtain a satisfactory ‘solution’ or a general notion of ‘value’ in the more general nontransferable utility (NTU) context have been made by different authors. Harsanyi [8,9], Shapley [31], Kalai and Samet [15], Maschler and Owen [22,23] (see also Hart and Mas-Colell [12]) have provided different proposals. 14 As classical bargaining problems are two-person NTU games, and TU games are a particular case of the NTU model, the way to proceed in all cases is to look for a notion of value for NTU games that coincides with Nash’s solution and Shapley’s value when restricted to bargaining problems and TU games, respectively. As is well known these extensions differ, and there are no clear grounds for claiming superiority for any one of them over the others. Curiously enough, in general, no solution in the NTU literature coincides with the one obtained here on axiomatic grounds in the class of NTU games associated with the class of two-ingredient models considered by us. 15 This seems to corroborate the excessive abstraction and consequent lack of intuitive basis of the bare NTU model: unless you put something else in it does not provide sufficient sure ground for a solution. So far, axiomatic characterizations in the NTU domain have come (if at all) only after the proposals of solutions satisfying the aforementioned double consistency with Nash and Shapley (e.g., [1,11]). 12 See Harsanyi and Selten [10] for the incomplete information case. 13 This interpretation is consistent with the outcome of Rubinstein’s [29] alternating offers model within the
noncooperative approach (see [4]). 14 See McLean [24] for a recent review. 15 The solution obtained depends on the bargaining problem and the players’ weights (given by the Shapley– Shubik index of the voting rule). Consequently, the bargaining problem can be modified in many ways so that the feasible set for some coalition(s) changes without the solution changing, unlike other solutions of the associated NTU game. Nevertheless, as Sergiu Hart pointed out, the solution characterized here can be seen as the Shapley NTU value-like of an NTU game by associating the whole feasible set with every winning coalition, but this means an ad hoc extension of the Shapley NTU value notion and of the very notion of the NTU game.
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But here integration has been accomplished directly at the axiomatic level with surprising results. Nevertheless, the credit of this particular solution as characterized by Theorems 2 and 3 (resp., 4) depends on the compellingness of the transfer (resp., symmetric gain–loss) condition, which is not obvious. 16 In this respect we think that the two-ingredient model of a bargaining committee, in spite of containing more meat than bare NTU games, is still rather minimalistic. This will possibly be obvious from a noncooperative point of view. Here we have used a cooperative approach, in which the details of the way in which players interact are ignored, or kept in a black box, the outcome of which is partially characterized (Theorem 1) or singled out (Theorems 2–4) assuming ideal rationality conditions à la Nash. But if in addition to these two ingredients a bargaining protocol is specified, a noncooperative model will possibly be very sensitive to the specified protocol. In other words, the equilibrium outcome will depend on the protocol. For this reason we find both the crisp (Theorems 2–4) and noncrisp (Theorem 1) characterizations provided interesting. Several lines of further research can be suggested. First, and possibly most important, the ‘Nash program’ challenge: the noncooperative foundation of the ‘cooperative’ solutions characterized. It will be interesting to investigate the noncooperative foundation of the solution characterized in Theorems 2–4, and that of the family of solutions characterized in Theorem 1. Rubinstein [29] and Binmore [4], along with the works in the wake of Baron and Ferejohn [3], seem the natural term of reference as a starting point. A different line of work is open if the results in this paper are taken as a new starting point for addressing the normative issue about the ‘fair’ voting rule. 17 That is, if the members of a committee in which bargaining is the usual practice represent groups of individuals of different size, what voting rule is adequate?
Acknowledgments We want to thank the audiences of the seminars at the Center for Rationality at the Hebrew University of Jerusalem and at the CORE in the Université catholique de Louvain, where preliminary versions of this paper were presented. We thank Robert J. Aumann, Marco Mariotti, Michael Maschler, Jean-François Mertens, Bezalel Peleg and Walter Trockel for their comments. We are especially indebted to Ken Binmore, Sergiu Hart, Abraham Neyman and William Thomson for their insightful comments. Of course, the usual disclaimer applies. We also want to thank David Galloway, whose inspiring interview on 23.06.02 generated some of the ideas in this work, and the EU Commission, which financed the interview under the project “Searching & Teaching the EU Institutions”. This research has been supported by the Spanish Ministerio de Ciencia y Tecnología under projects BEC200308182 and SEJ2004-08011/ECON, by the Generalitat Valenciana (Grupo 3086) and the 16 In Laruelle and Valenciano [19] we explore the extension of the results presented here to a wider domain including random voting rules. In this setting it is shown that there are two conditions (each of which is compelling in different informational environments) which are compatible for TU bargaining problems and yield T. 17 Some authors, e.g., Mariotti [21,20], favor a normative interpretation of the Nash bargaining solution. A similar interpretation is possible for the results presented here.
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IVIE, and by the Universidad del País Vasco under project UPV00031.321-H-14872/2002. The first author also acknowledges financial support from the Spanish M.C.T. under the Ramón y Cajal Program. Part of this paper was written while the second author was visiting the Department of Economic Analysis at the University of Alicante, whose hospitality is gratefully acknowledged.
Appendix The basic idea of the proof of Theorem 1 is similar to that of Nash in the proof of his classical result and Kalai’s extension, but the possibility of null players needs to be handled with care. To make the proof simpler we establish three lemmas which will be of use. In Lemma A.3 and in the proof of the theorem we use the following notation: for any set C ⊆ R N , and J ⊆ N, C J denotes the set C J := {x ∈ C : xi = 0 for all i ∈ N \ J } . We also denote by sup(W ) the support of rule W, that is, the set of nonnull players. We leave to the reader the simple proof of the following lemma which will be of use: Lemma A.1. For any vector of nonnegative weights w ∈ R N (w = 0) Nashw () = w, where w := w/ i∈N wi denotes the normalization of w. In particular, if i∈N wi = 1, then Nashw () = w. Lemma A.2. Let : B × W → R N be a solution/value that satisfies IIA; then, for any two problems in B, B = (D, d) and B = (D , d ), such that d = d and Dd = Dd , and any W ∈ W, (B, W ) = (B , W ) holds. Proof. Just note that for such problems ch(Dd ) = ch(Dd ), and this set is contained in D and in D . By the individual rationality condition embodied in the solution concept, (B, W ) and (B , W ) must lie on Dd ⊆ ch(Dd ). Then, by IIA we have (B, W ) = (B , W ) = ((ch(Dd ), d), W ). Lemma A.3. Let : B × W → R N be a solution/value that satisfies IIA, IAT and NP*, then, for any (B, W ) ∈ B × W with B = (D, d) such that d = 0, and D J = J forJ = sup(W ), (B, W ) = (, W ) holds. Proof. We first prove a special case. Let W ∈ W, and J = sup(W ). For any ∈ (0, 1), let T J, : R N → R N the linear map given by xi if i ∈ J, J, Ti (x) := xi if i ∈ N \ J. Then, for any ∈ (0, 1), by IAT, as (, W ) = 0 if and only if i ∈ J by NP*, we have (T J, (), W ) = T J, ((, W )) = (, W ).
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By Lemma A.2, the same conclusion holds by replacing T J, () by the problem T+J, () := N ), that is, (T J, (), W ) = (, W ). (T+J, (), 0), where T+J, () := ch(T J, () ∩ R+ + Now let B = (D, d) ∈ B such that d = 0, and D J = J for J = sup(W ). For N , and con ∈ (0, 1) sufficiently small, T+J, () ⊆ D. By NP*, (B, W ) ∈ J ∩ R+ J, sequently (B, W ) ∈ T+ (). Then by IIA it must be that (B, W ) = (T+J, (), W ) = (, W ). Proof of Theorem 1. (⇐): Let : W → R N be any map satisfying anonymity and null player*. For any (B, W ) ∈ B × W, Nash(W ) (B) exists by the compactness of Dd , whose convexity makes it unique under definition (2). It is then easy to see that the solution (B, W ) := Nash(W ) (B) satisfies An, IIA, IAT and NP*.Note in particular that, by Lemma A.1, for any W, (, W ) = Nash(W ) () = (W )/ i∈N i (W ) = (W ). (⇒): Let : B × W → R N be a value or solution that satisfies An, IIA, IAT and NP*. First note that := (, ·) : W → R N then satisfies anonymity, null-player*, and also efficiency; this last condition is implied by IIA, IAT and NP*, by an argument similar to that in Roth [28]. Now let any problem (B, W ), with B = (D, d). Without loss of generality in view of IAT we can assume that d = 0, and in view of Lemma A.2 we can assume that D = ch(Dd ). Denote x ∗ := Nash(,W ) (B), which exists and is unique under definition (2), and J := sup(W ). By (2) and NP*, and the conditions on D, it is immediate that xi∗ = 0 if and only if i ∈ J . Now let p ∈ R N the vector ⎧ ⎨ i (, W ) if i ∈ J, pi := xi∗ ⎩ 0 if i ∈ N \ J ; and for any k > 0, let Bp,k = (Dp,k , 0) the problem in which Dp,k = x ∈ R N : px 1, xi k (∀i ∈ N \ J ) . Note that x ∗ ∈ Dp,k (px ∗ = i∈J i (, W ) = 1, and xi∗ = 0 < k for all i ∈ N \ J , by the efficiency and the null player* conditions satisfied by (, ·)), and that Bp,k is a result of transforming the problem Bk = (Dk , 0), in which N xi 1, xi k (∀i ∈ N \ J ) , Dk = x ∈ R : i∈J
by means of the linear map T : R N → R N ⎧ ⎨ 1 x if i ∈ J, Ti (x) := pi i ⎩x if i ∈ N \ J, i
that is, Bp,k = T (Bk ). As (Dk )J = J , by Lemma A.3, (Bk , W ) = (, W ). Thus, by IAT (Bp,k , W ) = (T (Bk ), W ) = T ((Bk , W )) = T ((, W )) = x ∗ .
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Finally, for k sufficiently large D ⊆ Dp,k ; thus, again by IIA, we conclude that (B, W ) = (Bp,k , W ) = x ∗ . Proof of Theorem 2. (Existence) Let (B, W ) := NashSh(W ) (B). By Lemma A.1, (, W ) = NashSh(W ) () = Sh(W ). Thus, (, ·) = Sh, which satisfies anonymity, and null player* (as is well known Shi (W ) = 0 if and only if i is a null player in W). Therefore, by Theorem 1, (B, W ) = NashSh(W ) (B) satisfies An, IIA, IAT, and NP*. Finally, as the Shapley value satisfies transfer, satisfies T. (Uniqueness) If : B × W → R N satisfies An, IIA, IAT and NP*, by Theorem 1, it must be that (B, W ) = Nash(,W ) (B), where (, ·) : W → R N satisfies efficiency, anonymity and null player. Then as IIA, IAT and NP* imply Eff and NP, and T is assumed, by Proposition 1, we have (, W ) = Sh(W ). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24]
R.J. Aumann, An axiomatization of the non-transferable utility value, Econometrica 53 (1985) 599–612. J.F. Banzhaf, Weighted voting doesn’t work: a mathematical analysis, Rutgers Law Rev. 19 (1965) 317–343. D.P. Baron, J.A. Ferejohn, Bargaining in legislatures, Amer. Polit. Sci. Rev. 83 (1989) 1181–1206. K. Binmore, Nash bargaining theory III, in: K. Binmore, P. Dasgupta (Eds.), The Economics of Bargaining, Blackwell, Oxford, 1987, pp. 61–76. K. Binmore, Game Theory and the Social Contract II, Just Playing, MIT Press, Cambridge, 1998. P. Dubey, On the uniqueness of the Shapley value, Int. J. Game Theory 4 (1975) 131–139. P. Dubey, A. Neyman, R.J. Weber, Value theory without efficiency, Math. Oper. Res. 6 (1981) 122–128. J.C. Harsanyi, A bargaining model for the cooperative N-person game, Ann. Math. Stud. 40 (1959) 325–355. J.C. Harsanyi, A simplified bargaining model for the N -person cooperative game, Int. Econ. Rev. 4 (1963) 194–220. J.C. Harsanyi, R. Selten, A generalized Nash bargaining solution for two person bargaining games with incomplete information, Manage. Sci. 18 (1972) 80–106. S. Hart, An axiomatization of Harsanyi’s nontransferable utility solution, Econometrica 53 (1985) 1295– 1313. S. Hart, A. Mas-Colell, Bargaining and value, Econometrica 64 (1996) 357–380. M.J. Holler, E.W. Packel, Power, luck and the right index, J. Econ. 43 (1983) 21–29. E. Kalai, Nonsymmetric Nash solutions and replications of 2-person bargaining, Int. J. Game Theory 6 (1977) 129–133. E. Kalai, D. Samet, Monotonic solutions to general cooperative games, Econometrica 53 (1985) 307–327. E. Kalai, D. Samet, On weighted shapley values, Int. J. Game Theory 16 (1987) 205–222. E. Kalai, M. Smorodinsky, Other solutions to Nash’s bargaining problem, Econometrica 43 (1975) 513–518. A. Laruelle, F. Valenciano, Shapley–Shubik and Banzhaf indices revisited, Math. Oper. Res. 26 (2001) 89–104. A. Laruelle, F. Valenciano, Bargaining, voting, and value, IVIE discussion paper WP-AD 2004-17. M. Mariotti, Fair bargains: distributive justice and Nash bargaining theory, Rev. Econ. Stud. 66 (1999) 733–741. M. Mariotti, Maximal symmetry and the Nash solution, Soc. Choice Welfare 17 (2000) 45–53. M. Maschler, G. Owen, The consistent Shapley value for hyperplane games, Int. J. Game Theory 18 (1989) 389–407. M. Maschler, G. Owen, The consistent Shapley value for games without side payments, in: R. Selten (Ed.), Rational Interaction, Springer, Berlin, 1992, pp. 5–12. R.P. McLean, Values of non-transferable utility games, in: R.J. Aumann, S. Hart (Eds.), Handbook of Game Theory with Economic Applications, vol. 3, Elsevier Science Publishers, North-Holland, Amsterdam, 2002, pp. 2077–2120.
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[25] J.F. Nash, The bargaining problem, Econometrica 18 (1950) 155–162. [26] J.F. Nash, Two-person cooperative games, Econometrica 21 (1953) 128–140. [27] B. Peleg, Game-theoretic analysis of voting in committees, in: K.J. Arrow, A.K. Sen, K. Suzumura (Eds.), Handbook of Social Choice and Welfare, vol. 1, Elsevier Science Publishers, North-Holland, Amsterdam, 2002, pp. 395–423. [28] A.E. Roth, Individual rationality and Nash’s solution to the bargaining problem, Math. Oper. Res. 2 (1977) 64–65. [29] A. Rubinstein, Perfect equilibrium in a bargaining model, Econometrica 50 (1982) 97–109. [30] L.S. Shapley, A value for N -person games, Ann. Math. Stud. 28 (1953) 307–317. [31] L.S. Shapley, Utility comparison and the theory of games, in: La Décision: Agrégation et Dynamique des Ordres de Préférence, Paris, Edition du CNRS, 1969, pp. 251–263. [32] L.S. Shapley, M. Shubik, A method for evaluating the distribution of power in a committee system, Amer. Polit. Sci. Rev. 48 (1954) 787–792. [33] J. von Neumann, O. Morgenstern, Theory of Games and Economic Behavior, Princeton University Press, Princeton, 1944 (1947, 1953).
Bargaining in committees as an extension of Nash’s bargaining theory Annick Laruellea , Federico Valencianob,∗ a Departamento de Fundamentos del Análisis Económico, Universidad de Alicante, Campus de San Vicente,
E-03071 Alicante, Spain b Departamento de Economía Aplicada IV, Universidad del País Vasco, Avenida Lehendakari Aguirre, 83,
E-48015 Bilbao, Spain Received 28 January 2005; final version received 3 May 2005 Available online 20 July 2005
Abstract This paper addresses the following issue: if a set of agents bargain on a set of feasible alternatives ‘in the shadow’ of a voting rule, that is, any agreement can be enforced if a ‘winning coalition’ supports it, what general agreements are likely to arise? In other words: what influence can the voting rule used to settle (possibly nonunanimous) agreements have on the outcome of consensus? We model the situation as an extension of the Nash bargaining problem in which an arbitrary voting rule replaces unanimity. In this setting a natural extension of Nash’s solution is characterized. © 2005 Elsevier Inc. All rights reserved. JEL classification: C7; D7 Keywords: Bargaining; Voting; Committees
1. Introduction Nash [25] proposes and characterizes axiomatically a cooperative ‘solution’ to the bargaining problem, in the spirit of von Neumann and Morgenstern’s [33] notion of ‘value’ of a zero-sum two-person game, as a rational expectation of the ‘amount of satisfaction’ or expected utility payoff of a ‘highly rational’ player engaging in a bargaining situation ∗ Corresponding author.
E-mail addresses: [email protected] (A. Laruelle), [email protected] (F. Valenciano). 0022-0531/$ - see front matter © 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.jet.2005.05.004
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with another rational player. In Shapley [30] a similar notion of ‘value’ for transferable utility (TU) games is also proposed and characterized. Still in the same spirit, here we are interested in understanding the influence that the voting rule used to settle agreements by a set of bargainers can have on the outcome of negotiations. More precisely, if a set of agents bargain on a set of feasible alternatives ‘in the shadow’ of a voting rule, that is, any agreement can be enforced if a ‘winning coalition’ supports it, what general agreements are likely to arise? Or, put into Nash’s classical terms, what agreements can a rational agent expect to arise when faced with the prospect of engaging in such a situation? The importance of the issue is clear in many contexts. In contrast with the case of a ‘take-it-or-leave-it’ committee, only entitled to accept or reject proposals submitted to it, without the capacity to modify them, it is often the case in a committee that uses a voting rule to make decisions that the final vote is merely the formal settlement of a bargaining process in which the issue to be voted upon has been adjusted to gain the acceptance of all members. This is what we call a ‘bargaining committee’. 1 It seems intuitively obvious that in such committees the voting rule by means of which agreements are settled conditions the outcome of negotiations. With this motivation in mind and in the spirit common to the classical results of von Neumann–Morgenstern, of Nash and of Shapley alluded to above, in this paper we explore an extension of Nash’s [25] model of a bargaining problem. We see Nash’s original model as consisting of two ingredients, a set of (two) players with von Neumann–Morgenstern [33] preferences over a set of feasible agreements, and a voting procedure (unanimity) to settle agreements. Thus, the kind of situation we are interested in can be described by a natural generalization of this model (and its traditional extension to n players), by considering arbitrary voting rules 2 instead of unanimity. In this two-ingredient setting, on grounds similar to those in Nash [25] or Shapley [30], axioms such as efficiency, anonymity, independence of irrelevant alternatives, invariance w.r.t. positive affine transformations and null-player can be adapted, retaining their meaning and motivation. Assuming these conditions entails a significant narrowing of the class of admissible answers or ‘solutions’ which is identified. It is shown that in fact all that remains is to settle the choice (within narrow limits) of an answer for the same issue in the particular case in which the configuration of preferences is TU-like. The possibility of singling out a choice is accomplished by integrating Nash’s and Shapley–Dubey’s characterizations [30,6] into one which goes further beyond both, yielding a surprising solution to the more complex problem under consideration. The rest of the paper is organized as follows: in Section 2 the model is precisely formulated. In Section 3 the natural extension of Nash’s and some of Shapley’s axioms in the more general framework considered here are provided. Section 4 contains the characterizing results, along with the relationships with Nash’s solution and the Shapley value. The meaning of the results is discussed in Section 5 along with some lines of further research. Then Appendix contains the proofs of the characterizing theorems.
1 The distinction between a ‘take-it-or-leave-it’ committee and a ‘bargaining’ committee is similar to the distinction made by Baron and Ferejohn [3] between a closed and an open rule in a legislature. 2 A setting including both a voting rule and a set of feasible agreements has been considered by some authors
but, as far as we know, with different purposes and within a completely different approach concerned with social choice issues (see [27] for a recent overview).
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2. Bargaining in the shadow of a voting rule We are concerned with a situation in which a set of agents bargain on a set of feasible alternatives ‘in the shadow’ of a voting procedure. That is, any agreement can be enforced if a ‘winning coalition’ supports it. The set N = {1, . . . , n} labels the seats of the decision procedure by means of which agreements are to be settled. As only yes/no voting is considered, a vote configuration can be represented by the set of ‘yes’-voters. So, any S ⊆ N represents the result of a vote in which players occupying seats in S voted ‘yes’ and those in N \S voted ‘no’. An N-voting rule is specified by a set W ⊆ 2N of winning vote configurations such that (i) N ∈ W ; (ii) ∅ ∈ / W ; (iii) If S ∈ W , then T ∈ W for any T containing S; and (iv) If S ∈ W , then N\S ∈ / W . W denotes the set of all such N-voting rules. For voting rule W, M(W ) denotes the set of minimal winning configurations, i.e., those that do not contain any other winning configuration. For any S ∈ M(W ) (S = N ), WS∗ denotes the voting rule that results from W by eliminating S from the set of winning configurations, that is, WS∗ := W \ {S}. For any permutation :N → N , W denotes the voting rule W := {(S) : S ∈ W }. A voting rule W is symmetric if W = W , for any permutation . The n players who use an N-voting rule are labelled by the seats in N that they occupy, and we refer to the subset of players denoted by a subset S ⊆ N as coalition S. Thus, depending on the context any S ⊆ N will be referred to either as a vote configuration (of seats) or as a coalition (of players). We will also speak of winning coalitions for a given N-rule, with an obvious meaning. A seat/player i ∈ N is said to be a null seat/player in W if, for any coalition S, S ∈ W if and only if S\ {i} ∈ W . We assume that a set of n (N-labelled) players makes decisions by means of rule W in the following sense. They can reach any alternative within a set, as well as any lottery over them, as long as: (i) a winning coalition supports it, and (ii) no player is imposed upon an agreement worse than the status quo, where all players will remain if no winning coalition supports any agreement. It is also assumed that every player has expected utility (von Neumann and Morgenstern [33] (vNM)) preferences over this set, so that the situation can be summarized à la Nash in utility terms by a feasible set of utility vectors D, together with the particular vector d associated with the disagreement or status quo, as a summary of the situation concerning the players’ decision. Accepting this simplification, the situation can be summarized by a pair (B, W ), where B = (D, d) is a classical n-person bargaining problem, and W is the N-voting rule to enforce agreements. In accordance with this interpretation we assume that Dd := {x ∈ D : x d} 3 is bounded, and D is a closed, convex and comprehensive (i.e., x y ∈ D ⇒ x ∈ D) set containing d, such that there exists some x ∈ D s.t. x > d. B denotes the set of all such bargaining problems. For any permutation : N → N, B := ((D), (d)) will denote the bargaining problem that results from B by -permutation of its coordinates, so that for any x ∈ R N , (x) denotes the vector in R N s.t. (x)(i) = xi . A bargaining problem B is symmetric if B = B, for any permutation . Thus, in this setting we are concerned with pairs (B, W ) ∈ B × W, each of which, consistent with the interpretation that accompanied its introduction, will be referred to as a bargaining problem B under rule W, or for short just a problem (B, W ). 3 We will write for any x, y ∈ R N , x y (x < y) if x y (x < y ) for all i = 1, . . . , n. i i i i
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Before proceeding with the search for a ‘value’ or a ‘solution’ in this setting, let us first see how the model fits into the general nontransferable utility (NTU) games framework and how the classical bargaining problems and simple TU games fit into this model. Let pr S : R N → R S denote the natural S-projection for S ⊆ N , and let x S := pr S (x) for any x ∈ R N . On obvious grounds we can associate an NTU game (N, V(B,W ) ) with every (B, W ) ∈ B×W, by associating with each coalition S the closed, convex and comprehensive subset of R S containing all feasible utility vectors for S, given by if S ∈ W, pr S x ∈ D : x N\S = d N\S V(B,W ) (S) := pr S (ch(d)) if S ∈ / W, where B = (D, d), and ch(d) denotes the comprehensive hull of {d} . The n-person classical bargaining problem corresponds to the case in which W is the unanimity rule W = {N }, with N as the only winning coalition. While when the bargaining ingredient in the model is the (normalized) TU-bargaining problem := (, 0), where := generalized x ∈ R N : i∈N xi 1 , the associated NTU game V(,W ) is equivalent to the simple TU game 4 representing the rule vW , given by 1 if S ∈ W, vW (S) := 0 if S ∈ / W. Thus, the family of associated NTU games (N, V(B,W ) ) : (B, W ) ∈ B × W properly contain all classical bargaining problems and all simple superadditive games. 5 Nevertheless, in spite of the possibility of embedding the family of problems under consideration into a subclass of NTU games, we will deal with the model in terms of its constituent elements, B and W.
3. Rationality conditions for a ‘value’ or a ‘solution’ In the class of situations described by this model the question addressed by Nash [25] and Shapley [30] in their respective cases can also be addressed: What is the ‘amount of satisfaction’ or utility that a rational player can expect from such a situation, or, in classical terms, what is the value for any player of the prospect of engaging in a situation such as this? We will proceed as in the two seminal papers by stating reasonable conditions that will narrow the possible options. Thus, we will impose some conditions on a map : B × W → R N , for vector (B, W ) ∈ R N to be considerable as a rational agreement, or better as a reasonable expectation of utility levels of a general agreement in a bargaining situation B under voting rule W. To begin with, in view of the interpretation in terms of the underlying situation stated in Section 2, we build the requirements of being feasible and no worse than the status quo for any player into the very notion of solution, that is, we impose 4 Note that this model allows for a neat distinction between voting rules and their associated simple TU games.
Although conceptually different (the specification of a voting rule does not involve its users’ preferences), both sets are obviously isomorphic. 5 A TU game v : 2N → R is superadditive if v(S ∪ T ) v(S) + v(T ), for all disjoint S, T ⊆ N.
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as prerequisites: (B, W ) ∈ D (feasibility), and (B, W ) d (individual rationality), if B = (D, d). In addition to this, we will use the following conditions, all of them natural adaptations of Nash’s and Shapley’s characterizing properties: 1. Efficiency (Eff): For all (B, W ) ∈ B × W, there is no x ∈ D, s.t. x > (B, W ). 2. Anonymity (An): For all (B, W ) ∈ B × W, and any permutation :N → N , and any i ∈ N , (i) ((B, W )) = i (B, W ), where (B, W ) := (B, W ). 3. Independence of irrelevant alternatives (IIA): Let B, B ∈ B, with B = (D, d) and B = (D , d ), such that d = d, D ⊆ D and (B, W ) ∈ D ; then, (B , W ) = (B, W ), for any W ∈ W. 4. Invariance w.r.t. positive affine transformations (IAT): For all (B, W ) ∈ B × W, and all N and ∈ R N , ∈ R++ ( ∗ B + , W ) = ∗ (B, W ) + , where ∗ B + = ( ∗ D + , ∗ d + ), denoting ∗ x := (1 x1 , . . . , n xn ), and ∗ D + := { ∗ x + : x ∈ D}. 5. Null player (NP): For all (B, W ) ∈ B × W, if i ∈ N is a null player in W, then i (B, W ) = di . 5*. Null player* (NP*): For all (B, W ) ∈ B × W, i (B, W ) = di if and only if i ∈ N is a null player in W . The readers can see for themselves the precise correspondence of these conditions with some of Nash’s and Shapley’s axioms. But it is worth noting some subtle differences arising in this setup. Observe that Eff, IIA and IAT (adaptations of Nash’s axioms) concern the feasible set, while An (adapted from Nash’s and Shapley’s anonymity 6 ) and NP (from Shapley’s system), as well as NP*, concern both. We omit the arguments in support of each of conditions 1–5, which can be found in Nash’s and Shapley’s papers. 7 It may be worth remarking, though, that An entails a consistent relabelling of voters in B and seats in W. Note also that it seems natural in our setting to set null players’ expectations to the status quo level, given their null capacity to influence the outcome given the voting rule according to which final agreements are enforced. Alternatively, the stronger version of null-player NP* requires in addition that nonnull players expectations are strictly greater than the status quo level, which also seems reasonable given that these players are not irrelevant according to the voting rule. The next two conditions impose alternative constraints on the solution for TU configurations of preferences (i.e., when B = ). 6. Transfer (T): For any two rules W, W ∈ W, and all S ∈ M(W ) ∩ M(W ) (S = N ): (, W ) − (, WS∗ ) = (, W ) − (, WS ∗ ).
(1)
6*. Symmetric gain-loss (SymGL): For any voting rule W ∈ W, and all S ∈ M(W ) (S = N ): i (, W ) − i (, WS∗ ) = j (, W ) − j (, WS∗ ), for any two voters i, j ∈ S, and any two voters i, j ∈ N \ S. 6 In Nash [26] anonymity replaces symmetry. We prefer this simpler condition to the slightly weaker symmetry in Nash [25], which can also be adapted to this setting 7 For a careful discussion of Nash’s axioms, see, e.g., chapter 1 in Binmore [5].
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These conditions permit to exploit the lattice structure of the set of simple games (or voting rules), as additivity permits to exploit the vector space structure of the domain of TU games in Shapley’s [30] characterization. Dubey [6] uses T to characterize the Shapley– Shubik index. 8 Laruelle and Valenciano [18] replace T by SymGL (weaker than T in the presence of anonymity) to characterize Shapley–Shubik and Banzhaf [2] indices. In words, T postulates that the effect of eliminating a minimal winning configuration from the set of winning configurations is the same whatever the voting rule, while SymGL requires that the effect of eliminating a (minimal) winning configuration from the list that specifies the voting rule is equal on any two voters belonging (not belonging) to it. Remark 1. Observe also that, fixing B = , conditions 1, 2, and 5 when restricted to (, ·) : W → R N become, respectively, their homonymous efficiency, anonymity and null-player for a value on the domain of simple TU games, while 3 and 4 become empty requirements. If, instead, W is fixed to be the unanimity rule W = {N } (or any symmetric voting rule), then conditions 1–4 restricted to (·, W ) : B → R N become Nash’s homonymous characterizing conditions in the classical one-ingredient bargaining setting, while 5 becomes an empty condition (there are no null-players when W is symmetric), and 5* becomes Roth’s [28] strict individual rationality condition. 9
4. Characterizations In this section we show how assuming some of the above conditions restricts the class of admissible ‘solutions’ or ‘values’ : B × W → R N . Denote by Nash(B) the Nash [25] bargaining solution of an n-person bargaining problem B = (D, d), that is, (xi − di ), Nash(B) = arg max x∈Dd
i∈N
and by Nashw (B) the w-weighted asymmetric Nash bargaining solution [14] of the same problem for a vector of nonnegative weights w = (wi )i∈N , (xi − di )wi . Nashw (B) = arg max x∈Dd
i∈N
In fact, if any of the weights is zero, the w-weighted Nash solution may not be unique under the conditions assumed in the bargaining problems in B. This difficulty can be overcome in two ways: either by assuming a condition of ‘nonlevelness’ on the feasible set, or, as we do here, by imposing the disagreement ‘payoff’ for players whose weight is zero. 8 Formulation
(1)
of
this
condition
is
equivalent
(see
[18])
to
the
more
traditional:
(, W ) + (, W ) = (, W ∪ W ) + (, W ∩ W ). 9 Roth [28] replaces efficiency by this condition in Nash’s system to characterize the Nash bargaining solution.
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297
That is, redefining Nashw i (B)
:=
argi max
x∈Dd
j ∈J (xj
di
− dj )wj
if i ∈ J, if i ∈ N \ J,
(2)
where J = {i ∈ N : wi > 0} . In what follows whenever we write Nashw (B) we refer always to definition (2). The following result shows in which way conditions 2–4 and 5* considered in the previous section drastically restrict the possible answer to the question raised. Theorem 1. A solution/value : B ×W → R N satisfies anonymity (An), independence of irrelevant alternatives (IIA), invariance w.r.t. affine transformations (IAT) and null player* (NP*), if and only if (B, W ) = Nash(W ) (B),
(3)
for some map : W → R N that satisfies anonymity and null player*. Moreover, = (, ·), where (W ) = (W )/ i∈N i (W ). Proof. See the appendix.
Thus formula (3) can be rewritten as (B, W ) = Nash(,W ) (B).
(4)
Note that efficiency has not been assumed on , but as is mentioned in the proof, this condition is implied by IIA, IAT and NP* by an argument similar to that in Roth [28]. In particular (, ·) is also efficient. Therefore, any map (, ·) : W → R N that satisfies efficiency, anonymity and null player would fit in formula (4) and yield a solution (B, W ) that satisfies the four rationality conditions. In other words assuming An, IIA, IAT and NP*, the solution, given by (4), will be unique as soon as (, ·) is specified. As stated in Section 2, in the case of the TUbargaining problem , for any voting rule W, the associated NTU game V(,W ) is equivalent to the simple TU game vW s.t. vW (S) = 1 if and only if S ∈ W . Thus, the solution is so far determined up to the choice of a value on the domain of simple superadditive games satisfying ‘efficiency’, ‘anonymity’ and null player*. These conditions on (, ·) immediately bring to mind the Shapley [30] value, or more specifically in the context of simple games, the Shapley–Shubik [32] index. 10 Denote by Sh(v) the Shapley [30] value of a TU game v, given by Shi (v) =
(n − s)!(s − 1)! (v(S) − v(S\i)), n!
S:S⊆N
10 But there are other alternatives, for instance, the normalization of any semivalue [7] meets these conditions, as well as some ‘power indices,’ such as the Holler–Packel [13] index.
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and by Sh(W ) the Shapley–Shubik [32] index of a voting rule W, i.e., the Shapley value of the associated simple game vW . We have the following result. Proposition 1. Let : B × W → R N be a solution/value that satisfies Eff, An, NP and T; then, for any voting rule W ∈ W, (, W ) = Sh(W ). Proof. For any W ∈ W, as stated in Section 2, the associated NTU game V(,W ) is equivalent to the simple TU game vW s.t. vW (S) = 1 if and only if S ∈ W . And, as is well known, efficiency, anonymity, null player and transfer characterize the Shapley (–Shubik) value in the domain of simple (superadditive or not) games ([6]; see also [18]). It can then be easily verified that in our setting conditions Eff, An (recall that is symmetric), NP and T become their homonymous for (, ·) : W → R N . Thus, (, W ) = Sh(vW ) = Sh(W ) for any voting rule. Then as an easy corollary of Theorem 1 and Proposition 1 we have: Theorem 2. There exists a unique solution/value : B×W → R N that satisfies anonymity (An), independence of irrelevant alternatives (IIA), invariance w.r.t. affine transformations (IAT), null player* (NP*) and transfer (T), and it is given by (B, W ) = NashSh(W ) (B). Proof. See the appendix.
(5)
The same solution can be characterized assuming only the weaker NP if Eff is also assumed. That is, we have the following alternative characterization: 11 Theorem 3. There exists a unique solution/value : B × W → R N that satisfies Eff, An, IIA, IAT, NP and T, and it is given by (5). Note that (5) yields for any symmetric voting rule W (B, W ) = Nash(B), while for any rule W it yields (, W ) = Sh(W ). Moreover, as pointed out in Remark 1, NP and T become empty requirements when W is any fixed symmetric rule, while NP* becomes Roth’s [28] strict individual rationality. Thus, the characterizing axioms in Theorems 1 and 2 (resp., Theorem 3) when restricted to (·, W ) : B → R N for any fixed symmetric rule, become Roth’s (resp., Nash’s) axiomatic system. On the other hand, as conditions IIA and IAT become empty requirements when fixing B = , Proposition 1 can also be rephrased as follows: the characterizing axioms in 11 We omit the proof that is an easy adaptation of the proof of Theorem 1 and Lemma 3.
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Theorem 3 when restricted to (, ·) : W → R N become Shapley–Dubey’s characterizing system of the Shapley value (or Shapley–Shubik index) in W. But observe that the same is not true for Theorem 2 (as IIA and IAT lose their bite if B = , efficiency cannot be derived from strict individual rationality alone). But a similar replacement of efficiency by strict individual rationality for the Shapley value is possible if the domain includes and all its positive affine transformations. In other words, Theorem 2 integrates Roth’s and Shapley–Dubey’s characterizations into one, and similarly Theorem 3 integrates Nash’s and Shapley–Dubey’s characterizations into one. But both theorems go further beyond these previous characterizations, yielding a surprising solution to the more complex problem under consideration given by (5). Finally, in Proposition 1, and in Theorems 2 and 3, transfer (T) can be replaced by the weaker (in the presence of anonymity) condition of SymGL. Thus, we have still another two characterizations of (5), whose proof, based on Laruelle and Valenciano [18] characterization of the Shapley–Shubik index, we omit. Theorem 4. (i) There exists a unique solution/value : B × W → R N that satisfies An, IIA, IAT, NP* and SymGL, and it is given by (5). (ii) The same is true if satisfies Eff, An, IIA, IAT, NP and SymGL. In Theorem 4, part (i) (resp., (ii)) integrates Roth’s (resp., Nash’s) and Laruelle– Valenciano’s characterizations into one. Remark 2. The following solutions show the independence of the axioms used in Theorem 2, in Theorem 3 and in Theorem 4 (and a fortiori in Theorem 1). (B, W ) = d satisfies all axioms in Theorems 2 and 4(i) except NP*, and all axioms in Theorems 3 and 4(ii) except Eff. (B, W ) = NashSh (W ) (B), where Sh is any nonsymmetric -weighted Shapley value [16], satisfies all axioms in Theorems 2–4 except An. (B, W ) = Nash(B) satisfies all axioms in Theorems 3 and 4(ii) except NP. (B, W ) = d + t¯ Sh(W ), where t¯ = max {t ∈ R : d + t Sh(W ) ∈ D}, satisfies all axioms in Theorems 2–4 except IAT. (B, W ) = d + t¯ ((m − d) ∗ Sh(W )), where t¯ = max {t : d + t ((m − d) ∗ Sh(W )) ∈ D} with m ∈ R N s.t. mi = max pr i (Dd ), satisfies all axioms in Theorems 2–4 except IIA (note that this family yields Kalai–Smorodinsky [17] solution as the particular case (·, W ) when W is symmetric). (B, W ) = NashBz(W ) (B), where Bz is the Banzhaf [2] index, satisfies all axioms in Theorems 2 and 3 except T, and all except SymGL in Theorem 4.
5. Conclusion Theorem 1 extends the Nash bargaining solution and provides an endogenous justification of the ‘weights’ in Kalai’s [14] asymmetric Nash solutions in the complete information
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context. 12 Kalai’s solutions emerge when symmetry is dropped in Nash’s system, or still assuming symmetry in an adequately ‘replicated’ problem [14], as if each player negotiated on behalf of a number of players, as is usually the underlying situation when a nonsymmetric rule is used to make decisions. Binmore [5, p. 78], justifies the asymmetric Nash solutions as reflecting the different ‘bargaining power’ of the players “determined by the strategic advantages conferred on players by the circumstances under which they bargain”, and uses the term ‘bargaining power’ to refer to the players’ weights. 13 In our case these ‘circumstances’ consist of the voting rule that governs negotiations. Thus, the extension provided by Theorem 1 re-opens the old power indices’ ‘contest’ in a richer and more interesting setting than the traditional one of bare simple games. Now they (i.e., efficient, anonymous values for simple games satisfying the null player condition) compete to adequately represent the ‘bargaining power’ (in a precise and relevant sense) that a voting rule confers to its users in a bargaining committee. Theorems 2–4 single out the Shapley–Shubik index by integrating two axiomatic systems into a single, consistent one, which when restricted to classical bargaining problems becomes either Roth’s or Nash’s system, and when restricted to simple games becomes a characterizing system (Dubey–Shapley’s or Laruelle–Valenciano’s) of the Shapley value. As a result, the unique solution emerging turns out to be consistent with both Nash’s solution and Shapley’s value. Since Nash’s and Shapley’s classical results, several attempts to obtain a satisfactory ‘solution’ or a general notion of ‘value’ in the more general nontransferable utility (NTU) context have been made by different authors. Harsanyi [8,9], Shapley [31], Kalai and Samet [15], Maschler and Owen [22,23] (see also Hart and Mas-Colell [12]) have provided different proposals. 14 As classical bargaining problems are two-person NTU games, and TU games are a particular case of the NTU model, the way to proceed in all cases is to look for a notion of value for NTU games that coincides with Nash’s solution and Shapley’s value when restricted to bargaining problems and TU games, respectively. As is well known these extensions differ, and there are no clear grounds for claiming superiority for any one of them over the others. Curiously enough, in general, no solution in the NTU literature coincides with the one obtained here on axiomatic grounds in the class of NTU games associated with the class of two-ingredient models considered by us. 15 This seems to corroborate the excessive abstraction and consequent lack of intuitive basis of the bare NTU model: unless you put something else in it does not provide sufficient sure ground for a solution. So far, axiomatic characterizations in the NTU domain have come (if at all) only after the proposals of solutions satisfying the aforementioned double consistency with Nash and Shapley (e.g., [1,11]). 12 See Harsanyi and Selten [10] for the incomplete information case. 13 This interpretation is consistent with the outcome of Rubinstein’s [29] alternating offers model within the
noncooperative approach (see [4]). 14 See McLean [24] for a recent review. 15 The solution obtained depends on the bargaining problem and the players’ weights (given by the Shapley– Shubik index of the voting rule). Consequently, the bargaining problem can be modified in many ways so that the feasible set for some coalition(s) changes without the solution changing, unlike other solutions of the associated NTU game. Nevertheless, as Sergiu Hart pointed out, the solution characterized here can be seen as the Shapley NTU value-like of an NTU game by associating the whole feasible set with every winning coalition, but this means an ad hoc extension of the Shapley NTU value notion and of the very notion of the NTU game.
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But here integration has been accomplished directly at the axiomatic level with surprising results. Nevertheless, the credit of this particular solution as characterized by Theorems 2 and 3 (resp., 4) depends on the compellingness of the transfer (resp., symmetric gain–loss) condition, which is not obvious. 16 In this respect we think that the two-ingredient model of a bargaining committee, in spite of containing more meat than bare NTU games, is still rather minimalistic. This will possibly be obvious from a noncooperative point of view. Here we have used a cooperative approach, in which the details of the way in which players interact are ignored, or kept in a black box, the outcome of which is partially characterized (Theorem 1) or singled out (Theorems 2–4) assuming ideal rationality conditions à la Nash. But if in addition to these two ingredients a bargaining protocol is specified, a noncooperative model will possibly be very sensitive to the specified protocol. In other words, the equilibrium outcome will depend on the protocol. For this reason we find both the crisp (Theorems 2–4) and noncrisp (Theorem 1) characterizations provided interesting. Several lines of further research can be suggested. First, and possibly most important, the ‘Nash program’ challenge: the noncooperative foundation of the ‘cooperative’ solutions characterized. It will be interesting to investigate the noncooperative foundation of the solution characterized in Theorems 2–4, and that of the family of solutions characterized in Theorem 1. Rubinstein [29] and Binmore [4], along with the works in the wake of Baron and Ferejohn [3], seem the natural term of reference as a starting point. A different line of work is open if the results in this paper are taken as a new starting point for addressing the normative issue about the ‘fair’ voting rule. 17 That is, if the members of a committee in which bargaining is the usual practice represent groups of individuals of different size, what voting rule is adequate?
Acknowledgments We want to thank the audiences of the seminars at the Center for Rationality at the Hebrew University of Jerusalem and at the CORE in the Université catholique de Louvain, where preliminary versions of this paper were presented. We thank Robert J. Aumann, Marco Mariotti, Michael Maschler, Jean-François Mertens, Bezalel Peleg and Walter Trockel for their comments. We are especially indebted to Ken Binmore, Sergiu Hart, Abraham Neyman and William Thomson for their insightful comments. Of course, the usual disclaimer applies. We also want to thank David Galloway, whose inspiring interview on 23.06.02 generated some of the ideas in this work, and the EU Commission, which financed the interview under the project “Searching & Teaching the EU Institutions”. This research has been supported by the Spanish Ministerio de Ciencia y Tecnología under projects BEC200308182 and SEJ2004-08011/ECON, by the Generalitat Valenciana (Grupo 3086) and the 16 In Laruelle and Valenciano [19] we explore the extension of the results presented here to a wider domain including random voting rules. In this setting it is shown that there are two conditions (each of which is compelling in different informational environments) which are compatible for TU bargaining problems and yield T. 17 Some authors, e.g., Mariotti [21,20], favor a normative interpretation of the Nash bargaining solution. A similar interpretation is possible for the results presented here.
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IVIE, and by the Universidad del País Vasco under project UPV00031.321-H-14872/2002. The first author also acknowledges financial support from the Spanish M.C.T. under the Ramón y Cajal Program. Part of this paper was written while the second author was visiting the Department of Economic Analysis at the University of Alicante, whose hospitality is gratefully acknowledged.
Appendix The basic idea of the proof of Theorem 1 is similar to that of Nash in the proof of his classical result and Kalai’s extension, but the possibility of null players needs to be handled with care. To make the proof simpler we establish three lemmas which will be of use. In Lemma A.3 and in the proof of the theorem we use the following notation: for any set C ⊆ R N , and J ⊆ N, C J denotes the set C J := {x ∈ C : xi = 0 for all i ∈ N \ J } . We also denote by sup(W ) the support of rule W, that is, the set of nonnull players. We leave to the reader the simple proof of the following lemma which will be of use: Lemma A.1. For any vector of nonnegative weights w ∈ R N (w = 0) Nashw () = w, where w := w/ i∈N wi denotes the normalization of w. In particular, if i∈N wi = 1, then Nashw () = w. Lemma A.2. Let : B × W → R N be a solution/value that satisfies IIA; then, for any two problems in B, B = (D, d) and B = (D , d ), such that d = d and Dd = Dd , and any W ∈ W, (B, W ) = (B , W ) holds. Proof. Just note that for such problems ch(Dd ) = ch(Dd ), and this set is contained in D and in D . By the individual rationality condition embodied in the solution concept, (B, W ) and (B , W ) must lie on Dd ⊆ ch(Dd ). Then, by IIA we have (B, W ) = (B , W ) = ((ch(Dd ), d), W ). Lemma A.3. Let : B × W → R N be a solution/value that satisfies IIA, IAT and NP*, then, for any (B, W ) ∈ B × W with B = (D, d) such that d = 0, and D J = J forJ = sup(W ), (B, W ) = (, W ) holds. Proof. We first prove a special case. Let W ∈ W, and J = sup(W ). For any ∈ (0, 1), let T J, : R N → R N the linear map given by xi if i ∈ J, J, Ti (x) := xi if i ∈ N \ J. Then, for any ∈ (0, 1), by IAT, as (, W ) = 0 if and only if i ∈ J by NP*, we have (T J, (), W ) = T J, ((, W )) = (, W ).
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By Lemma A.2, the same conclusion holds by replacing T J, () by the problem T+J, () := N ), that is, (T J, (), W ) = (, W ). (T+J, (), 0), where T+J, () := ch(T J, () ∩ R+ + Now let B = (D, d) ∈ B such that d = 0, and D J = J for J = sup(W ). For N , and con ∈ (0, 1) sufficiently small, T+J, () ⊆ D. By NP*, (B, W ) ∈ J ∩ R+ J, sequently (B, W ) ∈ T+ (). Then by IIA it must be that (B, W ) = (T+J, (), W ) = (, W ). Proof of Theorem 1. (⇐): Let : W → R N be any map satisfying anonymity and null player*. For any (B, W ) ∈ B × W, Nash(W ) (B) exists by the compactness of Dd , whose convexity makes it unique under definition (2). It is then easy to see that the solution (B, W ) := Nash(W ) (B) satisfies An, IIA, IAT and NP*.Note in particular that, by Lemma A.1, for any W, (, W ) = Nash(W ) () = (W )/ i∈N i (W ) = (W ). (⇒): Let : B × W → R N be a value or solution that satisfies An, IIA, IAT and NP*. First note that := (, ·) : W → R N then satisfies anonymity, null-player*, and also efficiency; this last condition is implied by IIA, IAT and NP*, by an argument similar to that in Roth [28]. Now let any problem (B, W ), with B = (D, d). Without loss of generality in view of IAT we can assume that d = 0, and in view of Lemma A.2 we can assume that D = ch(Dd ). Denote x ∗ := Nash(,W ) (B), which exists and is unique under definition (2), and J := sup(W ). By (2) and NP*, and the conditions on D, it is immediate that xi∗ = 0 if and only if i ∈ J . Now let p ∈ R N the vector ⎧ ⎨ i (, W ) if i ∈ J, pi := xi∗ ⎩ 0 if i ∈ N \ J ; and for any k > 0, let Bp,k = (Dp,k , 0) the problem in which Dp,k = x ∈ R N : px 1, xi k (∀i ∈ N \ J ) . Note that x ∗ ∈ Dp,k (px ∗ = i∈J i (, W ) = 1, and xi∗ = 0 < k for all i ∈ N \ J , by the efficiency and the null player* conditions satisfied by (, ·)), and that Bp,k is a result of transforming the problem Bk = (Dk , 0), in which N xi 1, xi k (∀i ∈ N \ J ) , Dk = x ∈ R : i∈J
by means of the linear map T : R N → R N ⎧ ⎨ 1 x if i ∈ J, Ti (x) := pi i ⎩x if i ∈ N \ J, i
that is, Bp,k = T (Bk ). As (Dk )J = J , by Lemma A.3, (Bk , W ) = (, W ). Thus, by IAT (Bp,k , W ) = (T (Bk ), W ) = T ((Bk , W )) = T ((, W )) = x ∗ .
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Finally, for k sufficiently large D ⊆ Dp,k ; thus, again by IIA, we conclude that (B, W ) = (Bp,k , W ) = x ∗ . Proof of Theorem 2. (Existence) Let (B, W ) := NashSh(W ) (B). By Lemma A.1, (, W ) = NashSh(W ) () = Sh(W ). Thus, (, ·) = Sh, which satisfies anonymity, and null player* (as is well known Shi (W ) = 0 if and only if i is a null player in W). Therefore, by Theorem 1, (B, W ) = NashSh(W ) (B) satisfies An, IIA, IAT, and NP*. Finally, as the Shapley value satisfies transfer, satisfies T. (Uniqueness) If : B × W → R N satisfies An, IIA, IAT and NP*, by Theorem 1, it must be that (B, W ) = Nash(,W ) (B), where (, ·) : W → R N satisfies efficiency, anonymity and null player. Then as IIA, IAT and NP* imply Eff and NP, and T is assumed, by Proposition 1, we have (, W ) = Sh(W ). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24]
R.J. Aumann, An axiomatization of the non-transferable utility value, Econometrica 53 (1985) 599–612. J.F. Banzhaf, Weighted voting doesn’t work: a mathematical analysis, Rutgers Law Rev. 19 (1965) 317–343. D.P. Baron, J.A. Ferejohn, Bargaining in legislatures, Amer. Polit. Sci. Rev. 83 (1989) 1181–1206. K. Binmore, Nash bargaining theory III, in: K. Binmore, P. Dasgupta (Eds.), The Economics of Bargaining, Blackwell, Oxford, 1987, pp. 61–76. K. Binmore, Game Theory and the Social Contract II, Just Playing, MIT Press, Cambridge, 1998. P. Dubey, On the uniqueness of the Shapley value, Int. J. Game Theory 4 (1975) 131–139. P. Dubey, A. Neyman, R.J. Weber, Value theory without efficiency, Math. Oper. Res. 6 (1981) 122–128. J.C. Harsanyi, A bargaining model for the cooperative N-person game, Ann. Math. Stud. 40 (1959) 325–355. J.C. Harsanyi, A simplified bargaining model for the N -person cooperative game, Int. Econ. Rev. 4 (1963) 194–220. J.C. Harsanyi, R. Selten, A generalized Nash bargaining solution for two person bargaining games with incomplete information, Manage. Sci. 18 (1972) 80–106. S. Hart, An axiomatization of Harsanyi’s nontransferable utility solution, Econometrica 53 (1985) 1295– 1313. S. Hart, A. Mas-Colell, Bargaining and value, Econometrica 64 (1996) 357–380. M.J. Holler, E.W. Packel, Power, luck and the right index, J. Econ. 43 (1983) 21–29. E. Kalai, Nonsymmetric Nash solutions and replications of 2-person bargaining, Int. J. Game Theory 6 (1977) 129–133. E. Kalai, D. Samet, Monotonic solutions to general cooperative games, Econometrica 53 (1985) 307–327. E. Kalai, D. Samet, On weighted shapley values, Int. J. Game Theory 16 (1987) 205–222. E. Kalai, M. Smorodinsky, Other solutions to Nash’s bargaining problem, Econometrica 43 (1975) 513–518. A. Laruelle, F. Valenciano, Shapley–Shubik and Banzhaf indices revisited, Math. Oper. Res. 26 (2001) 89–104. A. Laruelle, F. Valenciano, Bargaining, voting, and value, IVIE discussion paper WP-AD 2004-17. M. Mariotti, Fair bargains: distributive justice and Nash bargaining theory, Rev. Econ. Stud. 66 (1999) 733–741. M. Mariotti, Maximal symmetry and the Nash solution, Soc. Choice Welfare 17 (2000) 45–53. M. Maschler, G. Owen, The consistent Shapley value for hyperplane games, Int. J. Game Theory 18 (1989) 389–407. M. Maschler, G. Owen, The consistent Shapley value for games without side payments, in: R. Selten (Ed.), Rational Interaction, Springer, Berlin, 1992, pp. 5–12. R.P. McLean, Values of non-transferable utility games, in: R.J. Aumann, S. Hart (Eds.), Handbook of Game Theory with Economic Applications, vol. 3, Elsevier Science Publishers, North-Holland, Amsterdam, 2002, pp. 2077–2120.
A. Laruelle, F. Valenciano / Journal of Economic Theory 132 (2007) 291 – 305
305
[25] J.F. Nash, The bargaining problem, Econometrica 18 (1950) 155–162. [26] J.F. Nash, Two-person cooperative games, Econometrica 21 (1953) 128–140. [27] B. Peleg, Game-theoretic analysis of voting in committees, in: K.J. Arrow, A.K. Sen, K. Suzumura (Eds.), Handbook of Social Choice and Welfare, vol. 1, Elsevier Science Publishers, North-Holland, Amsterdam, 2002, pp. 395–423. [28] A.E. Roth, Individual rationality and Nash’s solution to the bargaining problem, Math. Oper. Res. 2 (1977) 64–65. [29] A. Rubinstein, Perfect equilibrium in a bargaining model, Econometrica 50 (1982) 97–109. [30] L.S. Shapley, A value for N -person games, Ann. Math. Stud. 28 (1953) 307–317. [31] L.S. Shapley, Utility comparison and the theory of games, in: La Décision: Agrégation et Dynamique des Ordres de Préférence, Paris, Edition du CNRS, 1969, pp. 251–263. [32] L.S. Shapley, M. Shubik, A method for evaluating the distribution of power in a committee system, Amer. Polit. Sci. Rev. 48 (1954) 787–792. [33] J. von Neumann, O. Morgenstern, Theory of Games and Economic Behavior, Princeton University Press, Princeton, 1944 (1947, 1953).