The Communication Complexity of Coalition Formation among Autonomous Agents

Transcription

1 The Commuicatio Complexity of Coalitio Formatio amog Autoomous Agets Ariel D. Procaccia Jeffrey S. Roseschei School of Egieerig ad Computer Sciece The Hebrew Uiversity of Jerusalem Jerusalem, Israel {arielpro, ABSTRACT It is self-evidet that i umerous Multiaget settigs, selfish agets stad to beefit from cooperatig by formig coalitios. Nevertheless, egotiatig a stable distributio of the payoff amog agets may prove challegig. The issue of coalitio formatio has bee ivestigated extesively i the field of cooperative -perso game theory, but util recetly little attetio has bee give to the complicatios that arise whe the players are software agets. The bouded ratioality of such agets has motivated researchers to study the computatioal complexity of the aforemetioed problems. I this paper, we examie the commuicatio complexity of coalitio formatio, i a eviromet where each of the agets kows oly its ow iitial resources ad utility fuctio. Specifically, we give a tight Θ() boud o the commuicatio complexity of the followig solutio cocepts i urestricted games: Shapley value, the ucleolus ad the modified ucleolus, equal excess theory, ad the core. Moreover, we show that i some ituitively appealig restricted games the commuicatio complexity is costat, suggestig that it is possible to achieve subliear complexity by costraiig the eviromet or choosig a suitable solutio cocept. Categories ad Subject Descriptors F. [Theory of Computatio]: Aalysis of Algorithms ad Problem Complexity; I..11 [Artificial Itelligece]: Distributed Artificial Itelligece Multiaget Systems Geeral Terms Theory Keywords Commuicatio complexity, Coalitio formatio Permissio to make digital or hard copies of all or part of this work for persoal or classroom use is grated without fee provided that copies are ot made or distributed for profit or commercial advatage ad that copies bear this otice ad the full citatio o the first page. To copy otherwise, to republish, to post o servers or to redistribute to lists, requires prior specific permissio ad/or a fee. AAMAS 06 May , Hakodate, Hokkaido, Japa. Copyright 006 ACM /06/ $ INTRODUCTION I a eviromet teemig with autoomous agets, it is oly atural for agets to cooperate i order to achieve goals, eve uder the assumptio that the agets are selfish. Cooperatio is realized through the formatio of coalitios: the members of each coalitio share their resources, ad ultimately divide their payoff. I the secod half of the 0th cetury, some of the most promiet researchers i game theory have studied cooperative games; the focus of most research was determiig which coalitios would form i a give game, ad how coalitios should divide their payoff amog their members. The trick is to divide the payoff i a way that keeps the coalitio structure stable: the agets should be motivated to remai loyal to their respective coalitios, istead of deviatig ad formig ew coalitios that might guaratee them a higher expected payoff. Differet otios of stable solutios have bee proposed. The strogest, amed the core of the game, is sometimes empty. Other (weaker) solutios have differet desirable properties. I the past few years, a ew layer has bee added to the problem. The players i our cooperative game are software agets, which are limited i various (practical) ways. This complicatio has motivated researchers to study the computatioal complexity of differet solutio cocepts for cooperative games [7, 1,, 4]. Aother issue, that has so far received little attetio i multiaget systems research, is the commuicatio complexity of cooperative games i multiaget eviromets. I the multiparty commuicatio complexity model, each of the players (agets 1 ) holds some part of the iput, ad the players wish to joitly compute some fuctio o the iput. I this model, we assume the players have ulimited computatioal power; we are oly iterested i the worst-case umber of bits of iformatio they must pass amog themselves. Although the study of commuicatio complexity is ot atural i the cotext of some problems, it seems especially appropriate i the cotext of multiaget systems i geeral (see for example [3]), ad cooperative games i particular. Shehory ad Kraus [8] aalyzed the computatioal ad commuicatio complexity of two algorithms for payoff di- 1 Throughout the paper, we use the terms players ad agets iterchageably. As metioed above, computatioal complexity issues are also importat (perhaps eve more so) i multiaget systems. These issues are ot take ito accout i the commuicatio complexity model, but are studied separately. 505

2 visio i a multiaget eviromet. The eviromet itroduced by Shehory ad Kraus iduces a partitioig of the iformatio about the game amog the agets a fact that makes this eviromet a obvious cadidate for a study of commuicatio complexity. I this paper, we aalyze the commuicatio complexity of computig the payoff, i differet solutio cocepts, of a arbitrary player. This restricted problem (as opposed to computig the payoffs for all the players) is importat i its ow right: it is reasoable that a aget would like to kow a priori its expected payoff from a game, i order to decide whether to participate i the game at all. 3 It seems possible that, for some solutio cocepts or specific types of games, it may be sufficiet for a aget to elicit iformatio oly from a small subset of the other agets i order to compute its payoff. We focus o two categories of solutio cocepts: Sigleto Solutios: Shapley value, the ucleolus (ad modified ucleolus), ad equal excess theory. Solutios that are possibly empty: the core. For such solutio cocepts, oe ca establish hardess of suitable decisio questios, such as determiig whether the payoff of a certai aget is greater tha 0, or determiig whether the solutio set is empty. Decidig these problems is clearly easier tha computig a aget s payoff i a solutio (the solutio i the case of sigleto solutios). Studyig divisio schemes that do ot fall ito oe of the two categories is less straightforward, sice there are o apparet decisio problems that are easier tha computig the payoff i a solutio. For example, oe could ask whether a give payoff cofiguratio is a solutio, but eve if this problem is hard, it may still be easy to geerate just oe solutio (out of a possibly large solutio set). The isistece o havig such decisio problems will become apparet i Sectio. We ivestigate a eviromet i which there are agets, each holdig a costat amout of iformatio. We show a tight boud of Θ() o the commuicatio complexity of computig the payoff for a arbitrary player uder all solutio cocepts metioed above. Furthermore, we show that i restricted (but commo) types of cooperative games, the commuicatio complexity of fidig the payoff of a player i a solutio is 1. The rest of the paper is orgaized as follows. I Sectio we explai the basics of cooperative games ad commuicatio complexity. I Sectio 3 we prove our results, ad discuss the commuicatio complexity of restricted games. I Sectio 4 we propose directios for future research.. PRELIMINARIES.1 Cooperative Games We start by explaiig the basic ideas of cooperative - perso games i characteristic form, ad defiig restricted types of games that will be eeded later o. We ext specify the differet solutio cocepts. Last, we describe our eviromet ad coect this represetatio with games i characteristic form. 3 This becomes eve more explicit whe the aget has to pay so as to joi the game. I the ext few paragraphs, we follow Chapter of [5]. A cooperative -perso game i characteristic form with side paymets is a pair (N; v), where N = {1,,..., } is a set of players, ad v is the characteristic fuctio, which assigs a real umber v(s) to each S N. v(s) is the value of S: the payoff the players i S ca obtai by cooperatig. It always holds that v( ) = 0. The collectio of payoffs to the players is expressed as a payoff vector: x = {x 1, x,..., x }. A coalitio structure is a partitio of N, of the form φ = {S 1, S,..., S r}, which specifies how the players i N divide themselves ito coalitios. A Payoff Cofiguratio is a pair ( x; φ) = {x 1, x,..., x ; S 1, S,..., S r}, where x is a payoff vector ad φ is a coalitio structure, such that: j [r] : x(s j) = def X i S j x i = v(s j). We shall refer to the coalitio of all players as the grad coalitio. Example 1. Cosider the followig 3-perso game kow as Odd Ma Out : v({1}) = v({}) = v({3}) = v(n) = 0; v({1, }) = 4; v({1, 3}) = 5; v({, 3}) = 6. The payoff cofiguratio (,, 0; {1, }, {3}) would be obtaied if players 1 ad formed a coalitio ad split their payoff equally. A game is superadditive if: S, T N s.t. S T = : v(s T) v(s) + v(t). Superadditivity is a reasoable assumptio i may games, sice the uio S T of two coalitios may, i the worst case, act as two separate coalitios ad receive the payoff v(s) + v(t). We ext discuss special types of games. I weighted majority games, the players are assiged weights w 1, w,..., w, ad a criterio umber q is specified, such that: ( 1 P i S S N : v(s) = wi q 0 P i S wi < q Such a game is represeted by the followig shorthad otatio: [q; w 1, w,..., w ]. Lemma 1. Let [q; w 1, w,..., w ] be a weighted majority game, where for all i, w i {0, 1}. If q / + 1, the the game is superadditive. Proof. Let S, T N such that S T =. If S < q ad T < q, the clearly v(s T) v(s) + v(t). Otherwise, assume without loss of geerality that S q. Sice q / + 1 ad S T =, it holds that T < q; it follows that v(t) = 0. Therefore, v(s T) = 1 = v(s) + v(t). A differet type of restricted game is a apex game. I such games, the oly coalitios that have ozero value are those that cotai a distiguished player called the apex 506

3 (ad are of size at least ), or the remaiig 1 players (called the base) are icluded. Costat-sum apex games have the weighted majority represetatio: [ 1; 1,..., 1,, 1,..., 1], where is the weight of the apex player. Similarly to apex games, i veto games a distiguished player is a ecessary (but ot sufficiet) member of ay coalitio with ozero value. The geeral homogeeous weighted majority represetatio of the veto game i which the veto player eeds oly oe ally is: [; 1,..., 1, 1, 1,..., 1], where 1 is the weight of the veto player..1.1 Solutio Cocepts I this part of the paper, we follow Chapters 3 ad 6 of [5], as well as [9]. Over the years, may differet solutios to characteristic fuctio games with side paymets have bee proposed. These solutios differ i their otio of stability: give a coalitio structure, the payoff divisio should be such that agets are ot motivated to deviate, thus breakig dow coalitios. For example, if v({i}) > 0, ay payoff cofiguratio where x i = 0 could ot be stable, sice player i would prefer to receive the payoff he could get by himself The Core. The core C of a game (N; v) is the followig set of payoff cofiguratios: C = {( x, φ) : S N, x(s) v(s)}. Less formally, the core is the set of payoff cofiguratios that leave o coalitio i a positio to improve the payoffs to all of its members. The core is the strogest of all solutio cocepts; i fact, it is so strog that i some cases the core is empty. I such a case, at least oe coalitio will be dissatisfied with ay payoff cofiguratio Shapley Value. The Shapley value is a payoff divisio scheme that is characterized axiomatically, ad hece satisfies some importat desiderata. Player i is called a dummy if v(s {i}) = v(s) for all coalitios S that do ot iclude i; players i ad j are iterchageable if v((s {i}) {j}) = v(s) for every coalitio S that icludes i but ot j. The axioms of the Shapley value are: Symmetry: If i ad j are iterchageable, the x i = x j. Dummies: If i is a dummy, the x i = v({i}). Additivity: For ay two games (N; v) ad (N; w), x i i (N; v + w) equals x i i (N; v) plus x i i (N; w). It is well kow that the Shapley value, defied by: x i = X S N ( N S )!( S 1)! (v S v S {i} ), N! is the uique payoff divisio scheme that satisfies the axioms listed above The Nucleolus ad the Modified Nucleolus. The excess of a coalitio S with respect to the payoff vector x is: e(s, x) = v(s) x(s). Give a payoff cofiguratio ( x; φ), a excess e(s, x) ca be costructed for ay coalitio S; there are such coalitios. Let θ( x) be a vector of legth, whose compoets are all possible excesses, sorted i o-icreasig order: θ( x) = θ 1( x), θ ( x),..., θ ( x) = e(s 1, x), e(s, x),..., e(s, x), where for all i < j: e(s i, x) e(s j, x). θ( x) is said to be lexicographically greater tha θ( y), deoted θ( x) θ( y), if there exists a iteger q [ ] such that θ p( x) = θ p( y) for all p < q, ad θ q(x) > θ q(y). If θ( x) is ot lexicographically greater tha θ( y), we write θ( x) θ( y). Example. Cosider the followig 3-perso game: v({1}) = v({}) = v({3}) = 0; v(n) = 105 v({1, }) = 90; v({1, 3}) = 80; v({, 3}) = 70. Defie a payoff cofiguratio ( x, φ) = (15, 55, 35; N). The 3 excesses with respect to x are (0, 0, 15, 55, 35, 0, 30, 0) for coalitios (, N, {1}, {}, {3}, {1, }, {1, 3}, {, 3}), respectively. Therefore, θ( x) = (30, 0, 0, 0, 15, 0, 35, 55). Similarly, for ( y; φ) = (45, 15, 45; N) we have: θ( y) = (30, 10, 0, 0, 10, 15, 45, 45). It follows that θ( x) θ( y). The ucleolus of a game is the set of all payoff cofiguratios for which the sorted vector of excesses is lexicographically miimal: {( x; φ) : θ( x) θ( y) for all y, give φ}. The modified ucleolus is defied idetically, except that istead of a vector θ with excesses, a sorted vector with excesses is costructed, with compoets e(s, x) for all coalitios S such that 1 < S <. The ucleolus is uique for each coalitio structure i ay characteristic fuctio game. It is also kow that the modified ucleolus for the grad coalitio cosists of a uique payoff vector Equal Excess. Equal excess theory yields a payoff vector that is the result of a bargaiig process. At each stage, a player lodges a claim for a share of the value of each coalitio of which he is a member. As a startig poit, each player expects a equal share of the value of each such coalitio. More formally, the bargaiig process cosists of discrete rouds. I roud r, player i has a expectatio of the payoff he will obtai from coalitio S of which he is a member; this expectatio is deoted E r (i, S). Let A r (i, S) = max T S [E r (i, T)]; this is player i s highest expectatio from the alterative coalitios to S. Player i s expectatio for roud r + 1 is created by: E r+1 (i, S) = A r (i, S) + v(s) P j S Ar (j, S). S 507

4 Example 3. Cosider the followig 3-perso game: v({1}) = v({}) = v({3}) = v(n) = 0; v({1, }) = 90; v({1, 3}) = 80; v({, 3}) = 70. I roud 0, we have: E 0 (1, {1, }) = E 0 (, {1, }) = 45, E 0 (1, {1, 3}) = E 0 (3, {1, 3}) = 40, ad fially E 0 (, {, 3}) = E 0 (3, {, 3}) = 35. If the game termiates at roud 0 ad the coalitio structure is {{1, }, {3}}, for istace, the the payoffs would be (45, 45, 0). If egotiatio cotiues to roud 1, the for example: A 0 (1, {1, }) = 40, A 0 (, {1, }) = 35, ad thus: E 1 (1, {1, }) = = For the same coalitio structure as before, player 1 s payoff after a roud of egotiatio would be If the bargaiig process eds after m rouds, we say it is the solutio geerated by m-roud equal excess. As far as we kow, for all games examied, the sequece of expectatios coverges to a asymptotic value, but there is o proof of covergece. Assumig such a limit always exists, we say it is the solutio geerated by -roud equal excess..1. Eviromet Descriptio We follow the presetatio of the eviromet i [8] (with some simplificatios). A exact descriptio of the eviromet is importat for our purposes, sice it iduces a distributio of the iformatio about the game (or iput) amog the agets. Our eviromet cosists of autoomous agets 1,,...,, with tasks to fulfill. Each aget has a give amout of the resources l 1,..., l s, which are required i order to deal with tasks (we deote by Q the set of all possible vectors of quatities of resources). A aget receives a payoff for fulfillig tasks. These cocepts ca be formalized as a payoff fuctio U i : Q R +, which gives the payoff of aget i for some arbitrary resources. We assume that resources ca be traded amog agets. We also assume that payoff ca be trasferred from oe aget to aother (usig moey, or some other divisible desirable commodity). 4 The agets may try to tackle the tasks aloe, but may also prefer to form coalitios, thus poolig (ad redistributig) their resources. If S is a coalitio, we say that its value is v(s) if v(s) = P i S Ui (q i ), where q is the vector of resources after optimal redistributio i the coalitio. It is importat to recogize that such a eviromet ca be represeted usig a characteristic fuctio game, ad sometimes vice versa; hece the cocepts from the previous subsectios ca be used. However, the represetatio of the game proposed i this subsectio is more realistic. I essece, the iformatio about the game is distributed amog the agets; each aget holds a costat amout of iformatio, amely its resources ad payoff fuctio. Example 4. Cosider the weighted majority game defied by [q; w 1, w,..., w ]. There is oly oe resource, of 4 Such a trasfer of payoff is called a side paymet. Recall that we assumed side paymets are possible whe we described characteristic fuctio games i Sectio.1. which each aget is give the iitial quatity w i; the utility fuctio for all agets is: ( U i 1 z q (z) = 0 z < q I such games, the private iformatio aget i has its resources ad productio fuctio essetially defies its weight w i (which ca be deduced from its resources), ad the value of q (which ca be deduced from the commo utility fuctio, ad is therefore commo kowledge). We do ot assume that the eviromet is superadditive (although this is implied by the above defiitios), ad our results also apply to eviromets that do ot have this property. However, ote that i a superadditive game, we ca assume that the grad coalitio forms at some stage, ad cocer ourselves oly with the payoff divisio (ad ot the coalitio structure).. Commuicatio Complexity I this subsectio, we preset the multiparty commuicatio complexity model with which we shall deal. A recommeded, more detailed overview of commuicatio complexity theory appears i [6]. There are several ways to geeralize the two-party commuicatio complexity model (itroduced by Yao i [10]) to a multiparty settig. I our model, 5 player i (i N) holds a iput z i {0, 1} k. The players wish to compute together a fuctio f : ({0, 1} k ) A. They commuicate via a public blackboard all the players ca see ay bit a player seds. We are oly iterested i the amout of commuicatio amog the players, ad therefore we allow the players to have ulimited computatioal power. A determiistic protocol P is a biary tree where each iteral ode v is labeled by a fuctio g i,v{0, 1} k {0, 1}, for a sigle i N, ad each leaf is labeled with a A. The value of the protocol P o iput (z 1, z,..., z ) is the label of the leaf reached by startig from the root, ad walkig o the tree: at each iteral ode v labeled by g i,v walkig left if g i,v(z i) = 0, ad right if g i,v(z i) = 1. The cost of the Protocol P o iput (z 1, z,..., z ) is the legth of the path take o this iput. The cost of the protocol P is the height of the tree. The determiistic commuicatio complexity of a fuctio f is the miimum cost of P, over all protocols P that compute f. Ituitively, every iteral ode v labeled by g i,v is associated with a bit set by player i: 0 if g i,v(z i) = 0, ad 1 otherwise. At each poit i the protocol, the curret ode i the tree is determied by the previous bits set by the players. I this way, players take ito accout the bits commuicated so far. It is also possible to cosider the odetermiistic commuicatio complexity of f. A all-powerful prover is tryig to covice the players that f(z 1, z,..., z ) = a 0. If it is ideed true that f(z 1, z,..., z ) = a 0, the the prover should be able to covice the players (by postig the aswer ad a proof to the blackboard). However, if this is ot the case, the players should be able to detect the lie, regardless of what the prover says. It is obvious that a lower boud o the 5 Aother iterestig model is the umber o the forehead model, where each player holds a bit that all other players ca see, but he caot. 508

5 odetermiistic commuicatio complexity of a problem is also a lower boud o the determiistic commuicatio complexity of the problem. Furthermore, it is also a lower boud o the radomized commuicatio complexity of the problem. Remark 1. May problems i the multiparty model have a odetermiistic commuicatio complexity of o(). For example, each player holds oe bit, ad f = max(z 1, z,..., z ). If f(z 1, z,..., z ) = 1, the prover ca post to the blackboard a idex i of the player with z i = 1 (log bits); this player checks whether ideed z i = 1. There are several techiques for obtaiig lower bouds o commuicatio complexity. The most popular techique is the foolig set. Defiitio 1. Let f : ({0, 1} k ) {0, 1}. A subset H of ({0, 1} k ) is called a foolig set (for f) if there exists a value f 0 {0, 1} such that: For every (z 1, z,..., z ) H, f(z 1, z,..., z ) = f 0. For every two distict vectors (z 1 1, z 1,..., z 1 ), (z 1, z,..., z ) H, there exist r 1, r,..., r {1, } such that f(z r 1 1, z r,..., z r ) = 1 f 0. Less formally, by mixig the coordiates of ay two vectors i the foolig set, we ca obtai a iput vector whose value uder f is 1 f 0. It is kow that the existece of a foolig set of size m for f etails a lower boud of logm o the odetermiistic commuicatio complexity of f [6]. Remark. I the cotext of cooperative games, f is the fuctio which, give the resources ad utility fuctio for all the players ad a distiguished player, outputs that player s payoff i some solutio, accordig to a fixed solutio cocept. However, whe provig our lower bouds, we will deal with boolea fuctios 6 (so that we ca use the foolig set techique). 3. RESULTS This sectio is devoted to provig a tight boud of Θ() o the commuicatio complexity of computig a player s expected payoff i the followig solutio cocepts (i this order): Shapley value, the ucleolus ad modified ucleolus, m-roud equal excess ad -roud equal excess, ad the core. 3.1 Upper Boud Obtaiig a upper boud o the commuicatio complexity of ay solutio cocept i our eviromet is immediate: 6 Determiig whether a give player s payoff is greater tha 0, or determiig whether the solutio set is oempty. This stregthes our results, sice these decisio problems are weaker tha computig payoff. Propositio 1. The determiistic commuicatio complexity of ay solutio cocept (eve whe computig the payoff of all players) is O(). Proof. Recall that each aget holds a costat amout of iformatio. Therefore, all agets ca commuicate their etire part of the iput (resources ad utility fuctio), ad the compute a solutio. 3. Lower Bouds The followig lemma will soo be essetial i the proof of Lemma 3. Lemma. log` +1 = Ω(). Proof. Without loss of geerality, assume is eve (for a odd small chages are required, but the proof is similar). Observe that!! + 1 = ( + 1)!( 1)! = ( 1) (/ + ) (/ 1) (/ ) 1. We associate each factor i the deomiator which is greater tha or equal to /4+1, with a factor i the umerator that is exactly twice as large. For example, /4 + 1 is associated with / +. We have /4 1 such pairs, each with a ratio of. The other /4 factors i the umerator are all greater tha the other /4 factors i the deomiator. Therefore, we have: ` +1 /4 1, ad hece: log! + 1 log /4 1 = /4 1 = Ω(). The ext defiitio ad lemma are a part of the proof of Lemma 4. They yield a somewhat roudabout proof for Lemma 4, which is meat to provide ituitio for the correctess of some of the mai propositios. Defiitio. The majority fuctio, deoted by maj, returs 1 if at least + 1 players have the bit 1, ad 0 otherwise. Let E be the set of iput vectors such that the umber of oes is at most + 1, ad deote by maj E the majority fuctio restricted to the games i E. Lemma 3. The odetermiistic commuicatio complexity of maj E is Ω(). Proof. We exhibit a foolig set of size `, where = + 1; the result follows from Lemma. The foolig set cosists of all biary vectors of legth with exactly oes. Clearly, for ay vector there is a majority of oes, ad thus it remais to show that for ay two vectors w 1 ad w i the foolig set, we ca create a vector w where w i = wi 1 or w i = wi for all i N, i such a way that maj( w) = 0. Ideed, for ay two such vectors, there must be i 0 N such that wi 1 0 = 1 but wi 0 = 0. Let w i = wi 1 for all i i 0, ad w i0 = wi 0 = 0. w has exactly 1 oes, as required. Lemma 4. Let G be the set of weighted-majority games [q; w 1, w,..., w ] where q = 1 +, such that wi {0, 1} for all i, w 1 = 1, ad #{i : i w i = 1}

6 Assume that some sigleto solutio cocept satisfies the followig property: g G, maj E (w, w 3,..., w ) = 1 = x 1(g) > 0 for the grad coalitio. The the odetermiistic commuicatio complexity of computig the payoff of a arbitrary aget i this solutio cocept is Ω() (eve to decide whether the payoff of a give aget is greater tha 0). Proof. Notice that all games i G are superadditive by Lemma 1, ad thus we ca assume the grad coalitio forms. There is a obvious reductio from maj E with 1 players to games i G: give a iput w, w 3,..., w for maj E, complete the vector of weights with w 1 = 1, ad set q = +1, where = Assumig property (1) holds, the maj(w, w 3,..., w ) = 1 if ad oly if x 1(g) > 0: oly if follows from the fact that if maj(w, w 3,..., w ) = 0, the P i wi <, ad so the value of all coalitios is 0. We have that if the odetermiistic commuicatio complexity of decidig whether x 1 > 0 is f() for some fuctio f, the the odetermiistic commuicatio complexity of maj E with 1 players is at most f(). From Lemma 3, f() = Ω( 1) = Ω(). We ow prove our lower bouds. We start with the solutio cocepts that correspod to a sigleto set of stable payoff cofiguratios. These proofs rely o Lemma 4. Propositio. The odetermiistic commuicatio complexity of computig the payoff of a arbitrary aget accordig to the Shapley value is Ω() (eve to decide whether the payoff of a give aget is greater tha 0). Proof. Let G be the set of weighted-majority games [q; w 1, w,..., w ] where q = 1 +, such that wi {0, 1} for all i, w 1 = 1, ad #{i : i w i = 1} We show that the Shapley value has property (1); this is sufficiet to complete the proof by Lemma 4. Fix a game from G, ad let S = {i : w i = 1} (this coalitio icludes player 1). If we assume that maj E (w, w 3,..., w ) = 1, the S = q; it follows that x 1 > 0, sice: x 1 = X S N ( N S )!( S 1)! (v S v S {1} ) N! ( N S )!( S 1)! (v S v S N! {1}) ( q)!(q 1)! = (1 0)! > 0. I fact, sice all q players with w i = 1 are iterchageable, ad the rest are ull players, we have from the axioms that characterize the Shapley value that x 1 = 1 q. Propositio 3. The odetermiistic commuicatio complexity of computig the payoff of a arbitrary aget accordig to the ucleolus 7 ad the modified ucleolus is Ω() (eve to decide whether the payoff of a give aget is greater tha 0). 7 Recall that the ucleolus is a sigleto for a give coalitio structure. Here we ivestigate the ucleolus of the grad coalitio. (1) Proof. Let G be the set of weighted-majority games [q; w 1, w,..., w ] where q = 1 +, such that wi {0, 1} for all i, w 1 = 1, ad #{i : i w i = 1} We show that the ucleolus has property (1); this is sufficiet to complete the proof by Lemma 4. Fix a game from G, ad let x be the ucleolus of the grad coalitio. We assume that maj(w,..., w ) = 1; it follows that #{i : w i = 1} = q. We wish to show that x 1 > 0. Cosider the payoff divisio x, where x i = 1 for all i such q that w i = 1, ad x i = 0 otherwise. e(r, x ) = 0 for coalitios R that cotai all players with w i = 1, ad coalitios that do ot cotai ay of these players. Moreover, for ay other coalitio, e(r, x ) < 0: if the coalitio cotais k < q players with w i = 1, the e(r, x ) = k. q Now, assume x 1 = 0. If there exists i such that w i = 0 but x i > 0, the the coalitio S = {i : w i = 1} must have e(s, x) > 0, sice v(s ) = 1 but x(s ) < 1. But this meas θ( x) θ( x ), ad thus we ca assume the payoff is distributed oly amog the players with w i = 1. e(r, x) = 0 for coalitios R that cotai all players with w i = 1, ad all coalitios that do ot cotai ay of these players. Additioally, it holds that e({1}, x) = 0. The umber of zeros i θ( x) is greater tha the umber of zeros i θ( x ), ad thus θ( x) θ( x ); this is a cotradictio to the assumptio that x is the ucleolus. For the modified ucleolus we have to slightly chage the ed of the proof, because the excess of coalitios of size 1 is o loger cosidered. However, observe that if x 1 = 0 (ad we still have that x i = 0 for all players with weight 0), there must be a player i 0 with w i = 1 ad x i 1 ; q 1 hece, e({1, i 0}, x) 1. It holds that the umber of 0 s q 1 i θ( x) ad θ( x ) is equal, but the ext smallest coordiate i θ( x) is at least 1, while i θ( x q 1 ) it is. Hece, q θ( x) θ( x ) a cotradictio to our assumptio that x is the modified ucleolus of the grad coalitio. Propositio 4. The odetermiistic commuicatio complexity of computig the payoff of a arbitrary aget accordig to m-roud ad -roud equal excess is Ω() (eve to decide whether the payoff of a give aget is greater tha 0). Proof. Let G be the set of weighted-majority games [q; w 1, w,..., w ] where q = 1 +, such that wi {0, 1} for all i, w 1 = 1, ad #{i : i w i = 1} We show that m-roud equal excess has property (1); this is sufficiet to complete the proof by Lemma 4. Fix a game from G, ad let S = {i : w i = 1}; we assume that S = q. Observe that for all rouds r ad players i S : E r (i, S ) = 1/q, by the symmetry of all players i S. Moreover, clearly it holds (agai, by the symmetry of the players i S ) that for all rouds r, i S ad T N: E r (i, T) 1/q. Therefore, r, i S, T N s.t. S T : A r (i, N) = 1/q. () For all players i / S it holds that: T N : E 0 (i, T) E 0 (i, T ) = 1 q + 1, (3) where T = S {i}. I subsequet rouds r, we claim that: i / S, T N : E r (i, T) 1 q + 1. (4) 510

7 This ca be prove by iductio: the base is give by equatio (3). For the iductio step, we have that A r (i, T) 1 q+1 from the assumptio. Ay coalitio T with o-zero value of which player i is a member also cotais S. For such coalitios: E r+1 (i, T) = A r (i, T) + v(s) P j T Ar (j, T) T 1 q P j S A r (j, T) P j T S A r (j, T) T () = 1 P q + 1 j T S A r (j, T) T 1 q + 1 Cosequetly, for all rouds r: E r+1 (1, N) = A r (1, N) + v(n) P j N Ar (j, N) () = 1 q + 1 P j S A r (j, N) P j N S A r (j, N) () = 1 P q j N S A r (j, N) (4) 1 q q 1 q + 1 > 0. For m-roud equal excess it holds that x 1 = E m (1, N); this completes the proof. Observe that we have from the proof that eve for m =, E (1, N) > 0. We ow wish to aalyze the commuicatio complexity of payoff divisio accordig to the core. The core is ot ecessarily a sigleto, but may be empty. Clearly, determiig whether the core is empty is easier tha computig the payoff of a player i some payoff cofiguratio that is i the core. Propositio 5. The odetermiistic commuicatio complexity of computig the payoff of a arbitrary aget accordig to the core is Ω() (eve to decide whether the core is empty). Proof. We exhibit a foolig set of size `, where = + 1; the result follows from Lemma. All the iputs i the foolig set correspod to weighted majority games with q = 1 = ; thus, a iput i the foolig set ca be fully represeted by the vector of the agets weights. The vectors w 1, w,..., w i the foolig set have 1 i exactly coordiates, ad 0 i the other coordiates. There are ` such vectors. Fix a game w i the set; we wish to show the core is empty. Sice the game is superadditive, 8 it is sufficiet to show that there is o stable payoff divisio for the grad coalitio. Ideed, let x be a payoff vector, 8 The proof of superadditivity is similar to Lemma 1, ad relies o the fact that if v(t) = 1, T cotais at least q players with w i = 1, ad so N T cotais at most oe such player. ad let i 0 = argmax i {x i : w i = 1}. The coalitio S of q players i with w i = 1 ad i i 0 beefits by deviatig, sice v(s) = 1 = x(s)+x(n S) > x(s). It follows that x caot be i the core. We still eed to show that for ay two vectors w 1 ad w i the foolig set, we ca create a vector w where w i = w 1 i or w i = w i for all i N, i such a way that the core is o-empty. For ay two such vectors, there must be i 0 N such that w 1 i 0 = 1 but w i 0 = 0. Let w i = w 1 i for all i i 0, ad w i0 = w i 0 = 0. w has 1 i exactly q coordiates. It is clear that the followig payoff distributio is i the core: x i = 1/q for all i such that w i = 1, ad x i = 0 for all other players. 3.3 Restricted Games I Sectio, we defied some restricted types of games. I this subsectio, we shall look more closely at apex ad veto games. These two types of games have atural, realworld iterpretatios: i a apex game, the apex player may be described as a limited moopolist (who requires oly oe ally), while veto games are similar to such political bodies as the Uited Natios security coucil. Apex ad veto games are particularly iterestig i the cotext of commuicatio complexity, sice a great deal of iformatio about the game ca be commuicated by simply amig the apex or veto player. Propositio 6. I costat-sum apex games, the determiistic commuicatio complexity of computig the payoff of a arbitrary aget i ay solutio cocept is 1. Proof. Recall that costat-sum apex games have the weighted majority represetatio [ 1; 1,..., 1,, 1,..., 1], where is the weight of the apex aget, ad suppose w.l.o.g. that the goal is to compute the payoff of aget 1. It is sufficiet that aget 1 simply aouce (usig oe bit) whether it is the apex player; it ca do this by determiig whether its weight is (i other words, checkig that it has resource uits). By the symmetry betwee the o-apex agets, this kowledge is sufficiet to compute the payoff of aget 1. Moreover, may importat veto games have the weighted majority represetatio: [; 1,..., 1, 1, 1,..., 1], where 1 is the weight of the veto player. Obviously, the determiistic commuicatio complexity of payoff divisio i these games is also CONCLUSIONS AND FUTURE WORK I a eviromet with -agets, each holdig a costat amout of iformatio (the aget s utility fuctio ad iitial resources), we have show that the commuicatio complexity of computig the payoff of a arbitrary player is Θ() i the followig solutio cocepts: Shapley value, the ucleolus ad modified ucleolus, equal excess theory, ad the core. Additioally, we have show that i costat-sum apex games ad certai veto games, the determiistic commuicatio complexity is

8 As our upper boud is trivial, clearly the sigificace of the results lies i the lower bouds. Fortuately, a commuicatio complexity lower boud of Ω() is usually ot a obstacle. Nevertheless, whe iterpreted egatively, our results show that solvig cooperative games may be ifeasible i scearios where the commuicatio is severely restricted, or the umber of agets is very large. Our approach has some limitatios. The results i this paper are relevat to coalitio formatio i our specific eviromet model; the bouds o commuicatio complexity may be differet i other models, although the model which was our focus is a ituitive ad well-kow oe. Aother possible criticism of this work is that, eve if a payoff divisio amog the players is kow i advace, i geeral it takes O() commuicatio to broadcast this solutio to the players. Nevertheless, recall that our goal here was to determie the value of a specific player i a give solutio cocept. As we oted i the itroductio, this problem is also importat i its ow right. Our aalysis of restricted games shows that Ω() commuicatio is ot a lower boud for this problem i certai games. There are several appealig directios i which this research ca be exteded. A importat task is to characterize the iterestig games i which payoff divisio has a commuicatio complexity of o() (small O). A related issue is to determie whether there exist reasoable sigleto solutio cocepts with commuicatio complexity of o() i geeral characteristic fuctio games. It may also be the case that the commuicatio complexity may be lowered for certai solutios, but oly i specific games (where other solutios are still hard), or particular eviromets. Our methods of obtaiig lower bouds have preveted us from ivestigatig such importat solutios as the kerel ad the bargaiig set (see [5]). These solutios should also be examied i the future. Aother directio is to augmet our model by addig aother parameter m, which is the amout of iformatio each aget holds (i our eviromet m = O(1), sice we assumed that each aget holds a costat amout of iformatio). We expect that the bouds for may of the solutio cocepts ca be geeralized to Θ( logm), but the same issues we studied here may be eve more explicit i this augmeted model. 5. ACKNOWLEDGMENT This work was partially supported by grat # from the Israel Sciece Foudatio. 6. REFERENCES [1] V. Coitzer ad T. Sadholm. Complexity of determiig oemptiess of the core. I Proceedigs of the Iteratioal Joit Coferece o Artificial Itelligece (IJCAI), pages , 003. [] V. Coitzer ad T. Sadholm. Computig Shapley values, maipulatio value divisio schemes, ad checkig core-membership i multi-issue domais. I Proceedigs of the Natioal Coferece o Artificial Itelligece (AAAI), pages 19 5, 003. [3] V. Coitzer ad T. Sadholm. Commuicatio complexity of commo votig rules. I Proceedigs of the ACM Coferece o Electroic Commerce (ACM-EC), pages 78 87, 005. [4] X. Deg ad C. H. Papadimitriou. O the complexity of cooperative solutio cocepts. Mathematics of Operatios Research, pages 57 66, [5] J. P. Kaha ad A. Rapoport. Theories of Coalitio Formatio. Lawrece Erlbaum Associates, [6] E. Kushilevitz ad N. Nisa. Commuicatio Complexity. Cambridge Uiversity Press, [7] T. Sadholm ad V. Lesser. Coalitios amog computatioally bouded agets. Artificial Itelligece, 94(1):99 137, Special issue o Ecoomic Priciples of Multiaget Systems. [8] O. Shehory ad S. Kraus. Coalitio formatio amog autoomous agets: Strategies ad complexity. I From Reactio to Cogitio, Lecture Notes i Artificial Itelligece Number 957, pages 57 7, [9] T. Sadholm. Distributed ratioal decisio makig. I G. Weiß, editor, Multiaget Systems: A Moder Itroductio to Distributed Artificial Itelligece, chapter 5. MIT Press, [10] A. C. Yao. Some complexity questios related to distributed computig. I Proceedigs of the 11th ACM Symposium o Theory of Computig (STOC), pages 09 13,