CENTRONE F

MEO, CLAUDIA

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Value allocations in economies with coalition structure Francesca Centrone Claudia MeoDipartimento di Scienze Economiche e MetodiQuantitativi, Università del Piemonte OrientaleDipartimento di Scienze Economiche e MetodiQuantitativi, Università del Piemonte OrientaleAbstractWe embody a notion of stability for coalition structures by Hart and Kurz (1983) into theframework of general equilibrium, by generalizing the classical value allocation notion(Shapley, 1969) to situations where: (a) agents organize themselves voluntarily into coalitionstructures; (b) the process of coalition formation is treated as endogenous. To this end weintroduce the definition of stable coalition structure value allocation and provide, understandard hypotheses, a preliminary existence result for the three--player case in an exchangeeconomy.Citation: Centrone, Francesca and Claudia Meo, (2008) "Value allocations in economies with coalition structure." EconomicsBulletin, Vol. 3, No. 49 pp. 1-8Submitted: July 23, 2008.  Accepted: August 25, 2008.URL: http://economicsbulletin.vanderbilt.edu/2008/volume3/EB-08C70034A.pdf1 IntroductionIn a previous work (Centrone and Meo, 2008), we faced the issue of endogenouscoalition formation in the framework of games without side payments: pre-cisely, we defined a new solution concept, the stable Coalition Structure (CS)valuation, based on Owen’s extension (1977) of the Shapley value for gameswith side payments to situations where players are organized in coalitions (theCoalition Structure value), and on a notion of stability by Hart and Kurz(1983) based on the strong Nash equilibria of a “game among coalitions”andon the CS value. We provided an existence result for the three-player case,where the restrictive setting is caused by the non-existence of stable coalitionstructures in the general case.The aim and main contribution of this note is to introduce the previous ele-ments into the framework of a general equilibrium model by defining a solutionconcept for an exchange economy which seeks (a) to evaluate each agent’s po-sition in the market by taking into account how his own bargaining opportu-nities and gains attainable through cooperation vary within different coalitionstructures and (b) to select a collectively stable outcome. To this end we pro-pose the notion of stable CS value allocation, which parallels and extends theclassical notion of value allocation (Shapley, 1969) to account for endogenousformation of coalitions, with the CS value substituting for the Shapley valueand the CS stable valuation for the classical Shapley one. As a byproduct ofour main Theorem 1 we get an existence result for stable CS value allocations(Corollary 1) in an exchange economy with three players. We point out thatKrasa, Temimi and Yannelis (2003) were the first to introduce the CS valueallocations in the context of differential information economies; the extra twistof our approach is represented by the fact that, through the stable CS valueallocation, agents are able to compare their prospects in the various coalitionstructures and choose the most advantageous one.2 Basic notionsWe will adopt the following notation. Given n ∈ N, Rn denotes the n-dimensional Euclidean space and Rn+ its nonnegative orthant. We will usethe symbol ≥ to order vectors in Rn with the standard interpretation: x ≥y ⇐⇒ xi ≥ yi, for every i = 1, . . . , n. For a set S ⊆ {1, . . . , n}, piS denotesthe projection defined from Rn onto R|S| which maps each x ∈ Rn onto the1element xS with coordinates indexed by elements of S.Let N = {1, . . . , n} be a non-empty finite set of players. Each subset S ⊂ Nis referred to as a coalition. We denote the collection of coalitions, i.e., theset of all subsets of N , by P(N).A game with side payments in characteristic form is a pair (N, ν) whereν : P(N) −→ R is a real-valued function such that ν(∅) = 0, called the char-acteristic function of the game.The real number ν(S) may be interpreted as the total “utility” that membersof S can divide among themselves when they agree to a contract about coop-eration and act as a unit.A game without side payments is a pair (N, V ), where V is a correspon-dence from P(N) into Rn satisfying the following properties:1. V (S) 6= ∅, for all S ∈ P(N);2. if x, y ∈ Rn, S ∈ P(N) and xi ≥ yi for all i ∈ S, then x ∈ V (S) impliesthat y ∈ V (S) (comprehensiveness);3. V (N) is convex;4. V (S) is closed, for all S ∈ P(N);5. piS(V (S))×piT (V (T )) ⊂ piS∪T (V (S∪T )) for all disjoint S and T in P(N)(superadditivity).For each S ∈ P(N), V (S) can be interpreted as the set of feasible payoffvectors which players in that coalition can assure for themselves by actingcooperatively.A game without side payments (N, V ) is compactly generated if there existsa compact set H ⊆ Rn such that:V (N) = {x ∈ Rn : ∃ y ∈ H such that y ≥ x}.The Coalition Structure (CS) value, first introduced by Owen (1977) and fur-ther developed by Hart and Kurz (1983), generalizes the Shapley value tosituations where players can voluntarily organize themselves into coalitions.The simplest form of organization is considered, that of a partition over theplayers set: given a game (N, ν), a coalition structure is a finite partitionB = {B1, . . . , Bm} of N .The CS value has the same properties of the Shapley value (in particular, it is2efficient and individually rational) and is given for each player, by some aver-aging of the player’s expected contributions to the coalitions he is a memberof. This expected contribution takes into consideration the coalition structureB in the following way.An order on N is consistent with B if, for every k = 1, . . . ,m and everyi, j ∈ Bk, all elements of N between i and j also belong to Bk. There are(m!b1!b2! . . . bm!)−1 such consistent orders (each assumed to be equally likely),where bk = |Bk| , k = 1, . . . ,m.In analogy with the Shapley value, the CS value of a game (N, ν) given thecoalition structure B is the unique payoff vector which assigns to player i thequantity:φi(ν,B) = 1m!b1! · · · bm!∑piB[ν(P (piB, i) ∪ {i})− ν(P (piB, i))] ,where the sum is taken over all orders piB consistent with the coalition struc-ture B and, for each piB, P (piB, i) denotes the coalition of players precedingplayer i in the order piB.Since the quantity φi(ν,B) can be interpreted as the expected utility of playeri in participating in the game ν when players are organized in coalitions ac-cording to B, each player is then able to compare his prospects in the variouscoalition structures. Based on this idea, Hart and Kurz (1983) have intro-duced a non-cooperative game of coalition formation whose solutions identifyin a complete endogenous way the “stable” coalition structures; they presenttwo models of stability, each based on the strong Nash equilibria of an appro-priate game and differing in the reaction of the other players when a coalitionbreaks apart.In Model ∆, players announce a coalition they would like to belong to; theplayers which announce the same list will result in a coalition.Model ∆: The noncooperative game ∆ is described by:1. The set of players is N = {1, . . . , n};2. For each i ∈ N , the set of strategies of i is Σi = {S ⊂ N : i ∈ S};3. For each n-tuple of strategies σ = (S1, . . . , Sn) ∈ Σ1 × · · · × Σn andeach i ∈ N , the payoff to i is φi(ν,B(δ)σ ), where B(δ)σ is the coalitionstructure which forms according to the criterion explained above, thatis: Bδσ = {T ⊂ N : i, j ∈ T if and only if Si = Sj}.3In Model Γ, players announce a coalition they would like to belong to. Acoalition forms if and only if all members make the same proposal.Model Γ: The noncooperative game Γ is described by:1. The set of players is N = {1, . . . , n};2. For each i ∈ N , the set of strategies of i is Σi = {S ⊂ N : i ∈ S};3. For each n-tuple of strategies σ = (S1, . . . , Sn) ∈ Σ1 × · · · × Σn andeach i ∈ N , the payoff to i is φi(ν,B(γ)σ ), where B(γ)σ is the coalitionstructure which forms according to the criterion explained above, that isB(γ)σ = {T σi : i ∈ N}, where:T σi ={Si, if Sj = Si for all j ∈ Si,{i}, otherwiseConsidered a coalition structure B and a player i ∈ N , let SBi be the elementof B to which i belongs and set σB = (SBi )i∈N . If the players choose σB, thenin both ∆ and Γ the resulting coalition structure is B.Definition 1. The coalition structure B is δ − stable (γ − stable) in thegame (N, ν) if σB is a strong equilibrium of ∆ (Γ, respectively); that is, ifthere exists no nonempty T ⊂ N and no σ̂i ∈ Σi for all i ∈ T , such thatφi(ν, B̂) > φi(ν,B) for all i ∈ T , where B̂ corresponds to ((σ̂i)i∈T , (σBj )j∈N\T ).Remark 1. In general the existence of stable coalition structures is not guar-anteed due to the fact that strong equilibria fail to exist in a broad class ofsituations; nonetheless Hart and Kurz (1984) have proven that all three playersgames have coalition structures that are both δ-stable and γ-stable.3 Stable CS value allocations for pure exchangeeconomiesThe aim of this section is to introduce an “optimal” mechanism of resourceallocation for an exchange economy, based on a cooperative solution concept.We recall the following definition (Centrone and Meo, 2008).Definition 2. Given a game without side payments (N, V ), a stable coali-tion structure valuation (stable CS valuation, henceforth) is a pair (λ∗, ξ∗)such that:41. λ∗ ∈ Σ;2. ξ∗ is feasible for the grand coalition, that is ξ∗ ∈ V (N);3. λ∗i ξ∗i = φi (νλ∗ ,Bsλ∗) , ∀ i ∈ N,where νλ∗ is the game with side payments derived from (N, V ) by allowingtransfers of utilities at the rates λ∗, i.e.:νλ∗(S) = sup{∑i∈S λ∗i ξi : ξ ∈ V (S)}if S 6= ∅0 if S = ∅(1)and Bsλ∗ and φi(νλ∗ ,Bsλ∗) are, respectively, a stable coalition structure for thegame νλ∗ and the CS value for player i corresponding to Bsλ∗.In Centrone and Meo (2008) the following theorem has been proved:Theorem 1. Let (N, V ) be a compactly generated game without side payments,with n = 3. Then a stable CS valuation exists for this game.Consider now a pure exchange economy E consisting of a set N = {1, . . . , n}of economic agents and l commodities.We assume that each agent is able to consume any commodity bundle in Rl+,that is, Rl+ represents the consumption set of each agent i ∈ N . Each agenti ∈ N is characterized by the pair (ei, ui) where ei ∈ Rl+ is his initial endowmentof physical resources and ui is the utility function representing his preferencesor tastes over the various consumption bundles in his consumption set, thatis:ui : Rl+ −→ R.Throughout the rest we will assume that∑ni=1 ei > 0.We denote a list of n consumption vectors of Rl+ by x = (x1, . . . , xn) and callit an allocation of the economy E.An allocation x = (x1, . . . , xn) is feasible if it is a non-wasteful redistributionof the total endowment∑ni=1 ei among the n agents in the economy, that is:n∑i=1xi =n∑i=1ei.For a coalition S ∈ P(N), the allocation x = (x1, . . . , xn) is called S-feasibleif∑i∈S xi =∑i∈S ei.5Given an exchange economy E, the game without side payments (N, Vu) asso-ciated with it is defined in a very natural way as follows:Vu : P(N) −→ RnVu(S) ={ξ ∈ Rn : there exists an S-feasible allocation x = (x1, . . . , xn)such that ξi ≤ ui(xi),∀ i ∈ S}, ∀ S ∈ P(N).(2)The interpretation is clear: for each coalition S, we consider the utility levelsthat may arise, given each agent’s utility function, subject to the constraintthat no coalition can consume more than its total initial endowment.Based on this strict link between general equilibrium and cooperative gametheory, we propose the following definition.Definition 3. Given the exchange economy E, a stable CS value alloca-tion is an allocation x∗ = (x∗1, . . . , x∗n) such that:i. x∗ is feasible;ii. there exists λ∗ ∈ Σ such that (λ∗, ui(x∗i )) is a stable CS valuation of thegame without side payments (N, Vu) associated with the economy E anddefined by (2).Thus, x ∗ is a stable CS value allocation if it is feasible and:λ∗iui(x∗i ) = φi(νuλ∗ ,Bsλ∗), ∀ i ∈ N,where νuλ∗ is the game with side payments associated with (N, Vu).The interpretation is the following: the stable CS value allocation yields toeach agent in the economy a “utility level” which is equal to the sum of theagent’s expected marginal contributions to all coalitions he is a member of,when all the agents are organized in a stable coalition structure.Remark 2. If the stable coalition structure we find in correspondence of theutility weights λ∗ is the trivial one, i.e. Bsλ∗ = {1 | 2 | 3}, then it is easily seenthat the notion of stable CS value allocation coincides with the traditionalnotion of value allocation (Shapley, 1969).We are now able to prove the following existence result:Corollary 1. Let E be an exchange economy with three agents.If the utility functions ui are concave and continuous for each i ∈ N , then61. a stable CS value allocation x∗=(x∗1, x∗2, x∗3) exists;2. if λ∗ = (λ∗1, λ∗2, λ∗3) are weights associated with x∗, then x∗ is constrainedPareto-optimal, that is∑i∈N λ∗iui(x∗i ) = νλ(N) ;3. x∗ is constrained individually rational, that is λiui(xi) ≥ λiui(ei) forevery i ∈ N .Proof. The assumptions of continuity and concavity of the utility functionsguarantee that (N, Vu) is a game without side payments.The existence of a stable CS value allocation for the exchange economy Efollows from Theorem 1 after noticing that the continuity of the ui’s assuresthat the subset of R3 defined by:H = {(u1(x1), u2(x2), u3(x3)) : x = (x1, x2, x3) is a feasible allocation for E}is compact and therefore (N, Vu) is compactly generated.The constrained Pareto optimality and the constrained individual rationalityof the stable CS value allocations are respectively straightforward consequencesof the efficiency and individual rationality of the CS value.4 Conclusions and future workThis short note wants to represent a first preliminary step towards the gener-alization of the value allocation notion (Shapley, 1969) to a context where anendogenous process of coalition formation is introduced into a pure bargainingsituation among economic agents. Such a step is realized primarily by themerging of our previously introduced notion of stable CS valuation (Centroneand Meo, 2008) with the classical definition of value allocation: we introducea new cooperative solution concept for a pure exchange economy model, thestable CS value allocation, for which we provide an existence result in thethree–agent setting.Many open issues still remain to be addressed in order to try to extend ourresult. The most important one concerns the existence of stable coalitionstructures for games with more than three players in order to obtain existenceresults for stable CS value allocations in economic models more general thanthe one we have considered here. Since all the trouble with the existence resultstem from the fact that Hart and Kurz characterize stable coalition structureby means of strong equilibria, which fails to exist in a broad class of situations,7a possible way to obviate the problem is by using a Nash equilibrium notionand thus reinterpreting the non-cooperative coalition formation game substi-tuting fuzzy coalitions for the traditional ones (the existence of Nash equilibriais indeed guaranteed in mixed strategies which call for such a substitution).Moreover, it would be of interest to investigate if one can provide non-cooperativefoundations for stable CS value allocation in order to both explain the dynam-ics of reaching equilibrium outcomes and select among different stable coalitionstructures. This will be the subject of future work.Acknowledgements. The authors gratefully acknowledge the support byResearch Grant n. 2006132713-005 (Ministero dell’Universita` e della Ricerca).References1. Centrone, F., Meo C. (2008) Coalition formation in games without sidepayments. Mathematical Social Sciences, forthcoming.2. Hart, S., Kurz M. (1983) Endogenous formation of coalitions. Econo-metrica 51, 1295–1313.3. Hart, S., Kurz M. (1984) Stable coalition structures. In: M.T. Holler(Ed.), Coalition and Collective Action. Physica-Verlag, Vienna, pp. 235–258.4. Krasa, S., Temimi, A., Yannelis N.C. (2003) Coalition structure valuesin differential information economies: is unity a strength?. Journal ofMathematical Economics 39, 51–62.5. Owen, G. (1977) Values of games with a priori unions. In: R. Hennand O. Moeschlin (Eds.), Essays in Mathematical Economics and GameTheory. Springer-Verlag, pp. 76–88.6. Shapley, L.S. (1969) La De´cision: Agre´gation et Dynamique des Ordresde Pre´fe´rences, Paris: E´ditions du Centre National de la Recherche Sci-entifique.8

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