Cores of Cooperative Games in Information Theory

Hinda wi Publishing Corporation EURASIP Journal on Wir eless Communications and Networking V olume 2008, Article ID 318704, 12 pages doi:10.1155/2008/318704 Research Ar ticle C ores of C ooper ativ e Games in Information Theory Mokshay Madiman Depart ment of Statistic s, Y ale University , 24 Hillhouse A venue, New Haven, CT 06511, USA Correspondence should be addressed to Moksha y Madiman, mokshay .madiman@yale.edu Rec eived 2 September 2007; Revised 18 December 2007; Ac cepted 3 Mar ch 2008 Rec ommended by Liang-Liang Xie Cores of cooperative games are ubiquitous in infor mation theory and ar ise most frequently in the characterization of fundamental limits in various scenarios inv olving multiple users. Examples include classical settings in network information theor y such as Slepian-W olf source coding and multiple access channels, classical settings in statistics such as robust hyp othesis testing, and new settings at the intersection of networking and statistics such as distributed estimation problems for sensor networks. Cooperative game theory allows one to understand aspects of all these problems from a fr esh and unifying perspective that treats users as pla yers in a g ame, sometimes leading to new insig hts. At the hear t of these analyses are fundamental dualities that hav e been long studied in the context of cooperative games; for informat ion theoretic purposes, these are dualities b etween information inequalities on the one hand and properties of r ate, capacit y , or other resource allocation regions on the other . Copyright © 2008 Moksha y Madiman. This is an open access article distributed under the Creative C ommons Attribution License, which permits unrestricted use, distr ibution, and repr oduction in any medium, provided the original work is properly cited. 1. INTRODUCTION A central problem in information theor y is the determina- tion of rate regions in data compression problems and that of capacity reg ions in communication problems. Althoug h single-letter characterizations of these regions were g iv en for lossless data compression of one source and for communica- tion from one transmitter to one receiv er by Shannon him- self, more elaborate scenarios in volving data compression from man y correlated sources or communication between a network of users remain of great theoretical and prac tical interest, with many key problems remaining open. In these multiuser scenarios, rate and capacit y regions are subsets of some Euclidean space whose dimension depends on the number of users. The search for an “ optimal” rate point is no longer tr i vial, even if the rate reg ion is known, because of the fact that there is no natural total ordering on points of Euclidean space. Indeed, it is important to ask in the first place what optimality means in the multiuser conte xt— ty pical cr it eria for optimality , depending on the scenario of interest, would deriv e from c onsider ations of fairness, net e ffi ciency , extraneous costs, or r obust ness to various kinds of network failures. Our primar y goal in this paper is to point out that notions from cooperativ e game theor y arise in a ver y natural way in connection w ith the study of rate and capacity regions for several impor tant problems. Examples of these problems include Slepian-W olf sourc e coding, multiple access chan- nels, and certain distr ibuted estimation problems for sensor networks. U sing notions from cooperative game theory , certain properties of the rate regions follow from appropriate information inequalities. In the case of Slepian-W olf coding and multiple access channels, these results are ver y well known; perhaps some of the interpretations are unusual, but the experts w ill not find them sur pr ising. In the case of the distributed estimation setting , the results are recent and the interpretation is new . W e supplement the analysis of these rate regions by pointing out that the classical capacit y-based theor y of composite hypothesis testing pioneered by Huber and Str assen also has a game-theoretic interpretation, but in terms of games with an uncountable infinit y of players. Since most of our results concern new interpretations of known facts, we label them as T ranslations. The paper is organized as follows. In Section 2 , some basic facts from the theory of cooperative games are reviewed. Section 3 treats using the g ame-theor etic frame- work the distributed compression problem solv ed by Slepian and W olf. The extreme points of the Slepian-W olf rate region are inter pr eted in terms of robustness to certain kinds of net- work failures, and allocations of r at es to users that are “fair” or “tolerable” are also discussed. Section 4 considers var ious classes of multiple access channels. An interesting sp ecial 2 EURASIP Journal on Wir eless Communications and Networking case is the Gaussian multiple access channel, where the game associated with the standar d setting has sig nificantly nicer structure than the game studied by La and Anantharam [ 1 ] associated w ith an arbitr arily var ying setting . Section 5 describes a model for distr ibuted estimation using sensor networks and studies a game associated w ith allocation of risks for this model. Sectio n 6 looks at various games invol v- ing the entropies and entropy powers of sums. These do not seem to have an operational interpretation but are related to recently developed infor mation inequalities. Section 7 discusses connections of the game-theoretic fr amew ork with the theor y of robust hypothesis testing. Finally , Section 8 contains some concluding remarks. 2. A REVIEW OF COOPERA TIVE GAME THEORY The theor y of cooperativ e games is classical in the economics and game theor y literature and has been extensively de v el- oped. The basic setting of such a game consists of n players, who can form arbitr ary coalitions s ⊂ [ n ], where [ n ] denotes the set { 1, 2, ... , n } of players. A g ame is specified by the set [ n ] of play ers, and a value function v :2 [ n ] → R + , where R + is the nonnegative real numbers, and it is always assumed that v ( φ ) = 0. The value of a coalition s is equal to v ( s ). W e wil l usually interpret the cooperative game (in its standard form) as the setting for a cost allocation problem. Suppose that player i contributes an amount of t i . Since the game is assumed to in volve (linearly) transferable utility , the cumulativ e cost to the play ers in the coalition s is simply ! i ∈ s t i . Since each coalition must pay its due of v ( s ), the individual costs t i must satisfy ! i ∈ s t i ≥ v ( s ) for e v er y s ⊂ [ n ]. This set of cost vectors, namely A ( v ) = " t ∈ R n + : # i ∈ s t i ≥ v ( s ) for each s ⊂ [ n ] $ , (1) is the set of aspirations of the game, in the sense that this set defines what the players can aspire to. The goal of the game is to minimize social cost, that is, the total sum of the costs ! i ∈ [ n ] t i . Clearly this minimum is achieved when ! i ∈ [ n ] t i = v ([ n ]). This leads to the definition of the cor e of a g ame. Definition 1. The core of a game v is the set of aspir ation vectors t ∈ A ( v ) such that ! i ∈ [ n ] t i = v ([ n ]). One may think of the core of an arbit rary game as the intersection of the set of aspirations A ( v ) and the “ e ffi ciency hyperplane”: F ( v ) = " t ∈ R n : # i ∈ [ n ] t i = v ([ n ]) $ . (2) The core can be equivalently defined as the set of undominated imputations; see, for example, Owen ’ s book [ 2 ] for this approach, and a proof of the equivalence. In this paper , we w ill not c onsider the question of where the value function of a game comes from but rather tak e the value func tion as given and study the corresponding g ame using str uctural results from g ame theor y . How ever , in the original economic interpretation, one should think of v ( s ) as the amount of utility that the members of s can obtain from the game whatever the remaining players may do . Then, one can interpret t i as the payo ff to the i th player and v ( s ) as the minimum net payo ff to the members of the coalition s that they wil l accept. This g i ves the aspiration set a slig htly di ff erent interpretation. Indeed, the aspir ation set can be thought of as the set of pay o ff vectors to players that no coalition would block as being inadequate. For the purposes of this paper , one may think of a cooperative game either in terms of payo ff s as discussed in this par agraph or in terms of cost al location as described earlier . A pathbreaking result in the theor y of tr ansferable utilit y games was the Bondareva-Shapley theorem characterizing whether the core of the game is empty . First, we need to define the notion of a balanced game. Definition 2. Given a collection C of subsets of [ n ], a function α : C → R + is a fractional partit ion if for each i ∈ [ n ], we hav e ! s ∈ C : i ∈ S α ( s ) = 1. A game is balanced if v ([ n ]) ≥ # s ∈ C α ( s ) v ( s ) (3) for any fractional par tition α for any collection C . A ctually , to check that a game is balanced, one does not need to show the inequalit y ( 3 ) for all fr actional par titions for all collections C . It is su ffi cient to check ( 3 ) for “minimal balanced collections ” (and these collections tur n out to yield a unique frac tional par tition). Details may be found, for example, in Owen [ 2 ]. W e now state the Bondareva-Shapley theorem [ 3 , 4 ]. F act 1. The core of a g ame is nonempty if and only if the game is balanced. Proof. Consider the linear program: Maximize # s ⊂ [ n ] α ( s ) v ( s ), subject to α ( s ) ≥ 0 for each s ⊂ [ n ], # s ⊂ [ n ], s % j α ( s ) = 1 for each j ∈ [ n ] . (4) The dual problem is easily obtained Minimize # j ∈ [ n ] t j , subject to # j ∈ s t j ≥ v ( s ) for each s ⊂ [ n ] . (5) If p ∗ and d ∗ denote the primal and dual optimal values, duality theor y tells us that p ∗ = d ∗ . Also, the game being balanced means p ∗ ≤ v ([ n ]), while the cor e being nonempt y means that d ∗ ≤ v ([ n ]). (Note that by setting α ( s ) = 0 for some subsets s ⊂ [ n ], frac tional par titions using ar bitrar y collections of sets can be thoug ht of as fr actional par titions using the full power set 2 [ n ] .) Thus, the game having a nonempty core is equivalent to its being balanced. Moksha y Madiman 3 An important class of games is that of con vex games. Definition 3. A game is convex if v ( s ∪ t )+ v ( s ∩ t ) ≥ v ( s )+ v ( t ) (6) for any sets s and t . (In this case, the set func tion v is also said to be super modular .) The connection b etween con vexity and balancedness goes back to Shapley . F act 2. A con vex game is balanced and has nonempt y core; the con verse need not hold. Proof. Shapley [ 5 ] showed that con vex games have nonempty core, hence they must be balanced by Fact 1 . A direct proof by induction of the fac t that con vexity implies fr actional superadditivit y inequalities (which include balancedness) is given in [ 6 ]. Incidentally , Maschler et al. [ 7 ] (cf., Edmonds [ 8 ]) noticed that the dimension of the cor e of a conv ex game was determined by the decomposability of the game, which is a measure of how much “additivity” (as opposed to the kind of superadditiv ity imposed by con vexity) there is in the value function of the game. There are var ious alternative characterizations of con vex games that are of interest. Fo r any game v and any ordering (permutation) σ = ( i 1 , ... , i n ) on [ n ], the marginal wor th vector m σ ( v ) ∈ R n is defined by m σ i k ( v ) = v ( { i 1 , ... , i k } ) − v ( { i 1 , ... , i k − 1 } ) (7) for each k> 1, and m σ i 1 ( v ) = v ( { i 1 } ). The con vex hull of all the marginal vectors is called the W eber s et . W eber [ 9 ] showed that the W e ber set of any game contains its cor e. The Shapley-Ichiishi theorem [ 5 , 10 ] says that the W e ber set is identical to the core if and only if the game is conv ex. In particular , the extreme points of the core of a conv ex game are precisely the marginal vectors. This characterization of con vex g ames is obv iously useful from an optimization point of view , as studied deeply by Edmonds [ 8 ] in the closely related theor y of poly- matroids. Indeed, polymat r oids (str ictly speaking , contra- polymatroids) may simply be thoug ht of as the aspiration sets of conv ex games. Not e that in the presence of the conv ex- ity condition, the assumption that v takes only nonnegative values is equivalent to the nondecreasing condition v ( s ) ≤ v ( t ) if s ⊂ t . Since a linear program is solv ed at extreme points, the results of Edmonds (stated in the language of polymatroids) and Shapley (stated in the language of conv ex games) imply that any linear func tion defined on the cor e of a conv ex game (or the dominant face of a poly matr oid) must be extremized at a marginal vector . Edmonds [ 8 ] uses this to develop greedy methods for such optimization problems. Historically speaking, the two parallel theories of poly matr oids and conv ex games were developed around the same time in the mid-1960s w ith awareness of and stimulated by each other (as e videnced by a foot note in [ 5 ]); howev er , in information theor y , this paral lelism does not seem to be par t of the folklore, and the g ame interpretation of rate or capacity regions has only been used to the author’ s knowledge in the important paper of La and Anantharam [ 1 ]. The Shapley value of a game v is the centroid of the marginal vectors: φ [ v ] = 1 n ! # σ ∈ S n m σ , (8) where S n is the sy mmetric group consisting of all permuta- tions. As shown by Shapley [ 11 ], its components are given by φ i [ v ] = # s % i ( | s | − 1)!( n − | s | )! n ! % v ( s ) − υ ( s \{ i } ) & , (9) and it is the unique vector satisfy ing the following axioms: (a) φ lies in the e ffi ciency hyper plane F ( v ), (b) it is invariant under permutation of players, and (c) if u and v are two games, then φ [ u + v ] = φ [ u ]+ φ [ v ]. Clearly , the Shapley value g i ves one possible formalization of the notion of a “fair allocation ” to the play ers in the game. F act 3. For a con vex game, the Shapley value is in the cor e. Proof. As pointed out by Shapley [ 5 ], this simply follows from the representation of the Shapley value as a conv ex combination of marginal vectors and the fact that the core of a con vex game contains its W eber set. For a cooperative game, conv exit y is quite a strong property . It implies, in par ticular , both that the game is exact and that it has a large core; we descr ibe these notions below . If ! i ∈ s y i ≥ v ( s ) for each s , does there exist x in the core such that x ≤ y (component-wise)? If so, the core is said to be large . Sharkey [ 12 ] show ed that not all balanced games hav e large cores, and that not all games with large cores are con vex. H owev er , [ 12 ] also showed the following fact. F act 4. A conv ex game has a large core. A game with value funct ion v i s said to be ex act if for every set s ⊂ [ n ], there exists a cost vector t in the cor e of the game such that # i ∈ s t i = v ( s ) . (10) Since for any point in the core, the net cost to the members of s is at least v ( s ), a g ame is exact if and only if v ( s ) = min " # i ∈ s t i : t is in the cor e of v $ . (11) The exact ness and large core properties are not comparable (counter examples can be found in [ 12 ] and Biswas et al. [ 13 ]). Ho wever , Schmeidler [ 14 ] showed the following fact. F act 5. A conv ex game is exact. 4 EURASIP Journal on Wir eless Communications and Networking Inter esting ly , Rabie [ 15 ] showed that the Shaple y value of an exact game need not be in its core. One may define, in an exactly complementary way to the abov e development, cooperativ e games that deal with resource allocat ion rather than cost allocation. The set of aspirations for a resourc e allocation game is A ( v ) = " t ∈ R n + : # i ∈ s t i ≤ v ( s ) for each s ⊂ [ n ] $ , (12) and the core is the intersection of this set w ith the e ffi - ciency hyper plane F ( v ) defined in ( 2 ), which represents the maximum achievable resource for the gr and coalition of all play ers, and thus a public good. A resource allocation game is conca ve if v ( s ∪ t )+ v ( s ∩ t ) ≤ v ( s )+ v ( t ) (13) for any sets s and t . The conca vit y of a game can be thoug ht of as the “ decreasing marginal returns” propert y of the value function, which is well motivated by economics. One can easily formulate equivalent versions of Facts 1 , 2 , 3 , 4 , and 5 for resourc e allocation games. For instance, the analogue of F act 1 is that the cor e of a resource al location game is nonempty if and only if υ ([ n ]) ≤ # S ∈ C α ( s ) υ ( s ) (14) for each fractional par tition α for any collection of subsets C (we cal l this propert y also balancedness, with some slight abuse of terminolog y). This follows from the fact that the duality used to prov e Fact 1 remains unchanged if we simultaneously change the sig ns of { t i } and υ , and r everse relevant inequalities. N otions from cooperativ e game theory also appear in the more recently de v eloped theory of combinatorial auc tions. In combinatorial auction theor y , the interpretation is slig htly di ff erent, but it remains an economic interpretation, and so we discuss it briefly to prepare the g round for some additional insights that we w ill obtain fro m it. Consider a resourc e allocation g ame v :2 [ n ] → R , w her e [ n ] indexes the items available on auction. Think of v ( s ) as the amount that a bidder in an auction is w illing to pay for the par t icular bundle of items indexed by s . In desig ning the rules of an auction, one has to take into account al l the recei ved bids, represented by a number of such set func tions or “valuations” v . The auction design then determines how to make an allocation of items to bidders, and computational concerns often play a major role. W e wish to hig hlight a fact that has emerged from combinatorial auction theor y; first we need a definition introduced by Lehmann et al. [ 16 ]. Definition 4. A set function v is additive if there exist non- negative real numbers t 1 , ... , t n such that v ( s ) = ! i ∈ s t i for each s ⊂ [ n ]. A set function v is X OS, if there are additive value functions v 1 , ... , v M for some positive integer M such that v ( s ) = max j ∈ [ M ] v j ( s ) . (15) The terminolog y X OS emerged as an abbrev iation for “X OR of OR of singletons” and was motivated by the need to represent value functions e ffi ciently (without storing all 2 n − 1 values) in the computer science literature. Feige [ 17 ] prov es the following fact, by a modification of the argument for the Bondareva-Shapley theorem. F act 6. A game has an X OS value function if and only if the game is balanced. By analog y with the definition of exactness for cost allocation g ames, a resource allocation game is exact if and only if v ( s ) = max " # i ∈ s t i : t is in the core of v $ . (16) In other words, for an exact game, the additive value fun- ctions in the X OS representation of the game can be taken to be those corresponding to the elements of the core (if we allow maximizing o ver a potentially infinit e set of additive value functions). Some of the concepts elaborated in this section can be extended to games with infinitely many players, althoug h many new technicalities arise. Indeed, there is a whole theory of so-called “ nonatomic games” in the economics literature. This is briefly alluded to in S e c t i o n 7 , where we discuss an example of an infinite game. 3. THE SLEPIAN-WOLF GAME The S lepian-W olf problem refers to the problem of loss- lessly compr essing data from two correlated sources in a dist ributed manner . Let p ( x 1 , ... , x n ) denote the joint probability mass function of the sources ( X 1 , ... , X n ) = X [ n ] , which take values in discrete alphabets. When the sources are coded in a centralized manner , an y rate R>H ( X [ n ] ) (in bits per sy mbol) is su ffi cient, w here H denotes the joint entropy , that is, H ( X [ n ] ) = E [ − log p ( x 1 , ... , x n )]. W hat r at es are achievable w hen the sources must be coded separ at ely? This problem was solv ed for i.i.d sources by Slepian and W olf [ 18 ] and extended to jointly ergodic sources using a binning argument by Cover [ 19 ]. F act 7. Correlated sources ( X 1 , ... , X n ) can be describ e d separately at rates ( R 1 , ... , R n ) and reco vered w ith arbitrar ily low er r or probability by a common decoder if and only if # i ∈ s R i ≥ H ' X s | X s c ( = : v SW ( s ) (17) for each s ⊂ [ n ]. In other words, the Slepian-W olf rate region is the set of aspirations of the cooperative game v SW , which we call the Slepian-W olf game. A key consequence is that using only knowledge of the joint distribution of the data, one can achieve a compression rate equal to the joint entropy of the users (i.e., there is no loss from the incapability to communicat e). H owev er , this is not automatic from the characterization of the rate Moksha y Madiman 5 region above; one needs to check that the Slepian-W olf game is balanced. The balancedness of the Slepian-W olf game is precisely the content of the lower bound in the following inequality of Madiman and T etali [ 6 ]: for any fractional partition α using C , # S ∈ C α ( s ) H ' X s | X s c ( ≤ H ' X [ n ] ( ≤ # S ∈ C α ( s ) H ( X s ) . (18) This weak fractional form of the joint ent r opy inequalities in [ 6 ] coupled with Fact 1 pro ves that the joint entrop y is an achievable sum rate even for distr ibuted compression. In fact, the Slepian-W olf game is much nicer . T ranslation 1. The Slepian-W olf game is a con vex game. Proof. T o sho w that the Slepian-W olf g ame is con vex, we need to show that v SW ( s ) = H ( X s | X s c ) is supermo dular . This fact was first explicitly pointed out by Fujishige [ 20 ]. By apply ing Fact 2 , the core is nonempty since the game is con vex, which means that there exists a rate point satisfying # i ∈ [ n ] R i = v SW ([ n ]) = H ' X [ n ] ( . (19) This recove rs the fact that a sum rate of H ( X [ n ] ) is achievable. N ote that, combined with Fact 1 , this observation in turn gives an immediate proof of the inequality ( 18 ). W e now look at how robust this situation is to network degradation because some users drop out. First note that by Fact 5 , the Slepian-W olf game is exact. Henc e, for an y subset s of users, there exists a vector R = ( R 1 , ... , R n ) that is sum- rate optimal for the g rand coalition of all users, which is also sum-rate optimal for the users in s , that is, ! i ∈ s R i = v SW ( s ). Ho wever , in gener al, it is not possible to find a rate vector that is simultaneously sum-rate optimal for multiple proper subsets of users. Below , we obser v e that finding such a rate vector is possible if the subsets of interest arise from users potentially dro pping out in a certain order . T ranslation 2 (Robust Slepian-W olf coding). Suppose the users can only drop out in a cer tain order , which w ithout loss of generality we can take to be the natural decreasing order on [ n ] (i.e., we assume that the first user to potentially drop out would be user n , followed by user n − 1, etc.). Then, there exists a rate p oint for Slepian-W olf coding which is feasible and optimal irrespective of the number of users that have dropped out. Proof. The solution to this problem is related to a modified Slepian-W olf game, g i ven by the utility function: v SW ( s ) = H ' X s | X s c \ > s ( , (20) where > s ={ i ∈ [ n ]: i > j for e v er y j ∈ s } . Indeed, if this game is shown to have a nonempty core, then there exists a rate point which is simultaneously in the Slepian-W olf r ate region of every [ k ], for k ∈ [ n ]. Ho wever , the nonemptiness of the core is equivalent to the balancedness of v SW , which follows from the inequality H ' X [ n ] ( ≥ # S ∈ C α ( s ) H ' X s | X s c \ > s ( , (21) where α is any frac tional par tition using C , which was prov ed by Madiman and T etali [ 6 ]. T o see that the core of this modified game actually contains an optimal point (i.e., a point in the core of the subgame corresponding to the first k users) for each k , simply note that the marg inal vector corresponding to the natur al order on [ n ] gives a constructive example. The main idea here is known in the literature, although not interpreted or proved in this fashion. Indeed, other interpretations and uses of the extreme points of the Slepian- W olf rate region are discussed, for example, in Coleman et al. [ 21 ], Cristescu et al. [ 22 ], and Ramamoor th y [ 23 ]. It is interesting to interpret some of the game-theoretic facts descr ibed in Section 2 for the Slepian-W olf game. This is par ticularly useful w hen there is no natural ordering on the set of players, but rather our goal is to identify a permutation- invariant (and more gener ally , a “fair”) rate point. By Fact 3 , we hav e the following tr anslation. T ranslation 3. The Shaple y value of the Slepian-W olf game satisfies the following properties. (a) It is in the core of the Slepian-W olf g ame, and hence is sum-r ate optimal. (b) It is a fair allocation of compression r at es to users because it is permutation-invariant. (c) Suppose an additional set of n sources, independent of the first n , is int r oduced. Suppose the Shapley values of the Slepian-W olf games for the first set of sources is φ 1 , and for the second set of sources is φ 2 . If each source from the first set is paired with a distinct source from the second set, then the Shapley value for the Slepian-W olf game played by the set of pairs is φ 1 + φ 2 . (In other words, the “fair” allocation for the pair can be “fairly” split up among the partners in the pair .) It is per tinent to note, moreov er , that implementing Slepian-W olf coding at an y point in the core is practically implementable. While it has been noticed for some time that one can e ffi ciently construc t codebooks that nearly achieve the r ates at an extreme point of the core, Coleman et al. [ 21 ], building on work of Rimoldi and Urbank e [ 24 ] in the multiple access channel setting, show a practical appr o ach to e ffi cient coding for any r ate point in the core (based on view ing any such rate point as an extreme point of the core of a Slepian-W olf game for a larger set of sources). Fact 4 says that the Slepian-W olf game has a large core, which may be interpreted as follows. T ranslation 4. Suppose, for each i , T i is the maximum compression rate that user i is wil ling to tolerate. A tolerance vector T = ( T 1 , ... , T n ) is said to be feasible if # i ∈ s T i ≥ v SW ( s ) (22) for each s ⊂ [ n ]. Then, for any feasible tolerance vector T , it is always possible to find a r ate point R = ( R 1 , ... , R n ) in 6 EURASIP Journal on Wir eless Communications and Networking the cor e so that R i ≤ T i (i.e., the r at e point is tolerable to all users). 4. MUL TIPLE AC CESS CHANNELS AND GAMES A multiple access channel (MA C) refers to a channel between multiple independent senders (the data sent by the i th sender is t ypically denoted X i ) and one receiv er (the recei ved data is t ypically denoted Y ). The channel charac ter istics, defined for each t ransmission by a probability tr ansition p ( y | x 1 , ... , x n ), is assumed to be known. W e wil l fur ther restrict our discussion to the case of memor yless channels, where each t r ansmission is assumed to occur independently accor ding to the channel transition probability . E ven within the class of memor yless multiple access channels, there are se veral notable special cases of interest. The first is the disc rete memor yless multiple a ccess channel (DM-MA C), where all r ando m variables tak e values in possibly di ff erent finite alphabets, but the channel transition matrix is otherwise unrestricted. The second is the Gaussian memor yless multiple access channel (G-MAC); here each sender has a power constraint P i , and the noise introduced to the superposition of the data from the sources is additive Gaussian noise with variance N . In other words, Y = # i ∈ [ n ] X i + Z , (23) where X i are the independent sources, and Z is a mean- zero , var ianc e N is normal indep endent of the sources. N ote that although the power constraints are an additional wr inkle to the problem compared to the DM-MA C, the G-MA C is in a sense more special because of the strong assumption; it makes on the nature of the channel. A third interesting special case is the P oisson memor y less multiple access channel (P -MAC), which models optical communication from many senders to one receiv er and operates in continuous time. Here, the channel tak es in as inputs data from the n sources in the form of wav eforms X i ( t ), whose peak powers are constrained by some number A ; in other words, for each sender i ,0 ≤ X i ( t ) ≤ A . The output of the channel is a Poisson process of r at e: λ 0 + # i ∈ [ n ] X i ( t ), (24) where the nonnegative constant λ 0 represents the r ate of a homogeneous Poisson process (noise) called the dar k current. For further details, one may consult the references cited below . The capacit y region of the DM-MA C was first found by Ahlswede [ 25 ] (see also Liao [ 26 ] and Slepian and W olf [ 27 ]). Han [ 28 ] de v eloped a clear approach to an even more gener al problem; he used in a fundamental wa y the poly matr oidal properties of ent r opic quantities, and thus it is no surprise that the problem is closely connected to cooperative games. Below I denotes mutual information (see, e.g ., [ 29 ]); for notational con venience, we suppress the dep endenc e of the mutual information on the joint distribution. F act 8. Let P be the class of joint distributions on ( X [ n ] , Y ) for w hich the marginal on X [ n ] is a product distribution, and the conditional dist ribution of Y given X [ n ] is fixed by the channel characteristics. For µ ∈ P , let C µ be the set of capacity vectors ( C 1 , ... , C n ) satisfying # i ∈ s C i ≤ I ' X s ; Y | X s c ( (25) for each s ⊂ [ n ]. Th e capacity region of the n -user DM-MA C is the closure of the conv ex hull of the union ∪ { C µ : µ ∈ P } . This rate reg ion is more complex than the Slepian- W olf r ate region because it is the closed con vex hull of the union of the aspir ation sets of many cooperative games, each corresponding to a product dist ribution on X [ n ] . Y et the analogous result turns out to hold. M ore specifically , even though the di ff erent senders hav e to code in a distributed manner , a sum capacity can be achieved that may be interpreted as the capacity of a single channel from the combined set of sources (coded together). T ranslation 5. The DM-MAC capacit y region is the union of the aspiration sets of a class of concav e games. In particular , a sum capacit y of sup I ( X [ n ] ; Y ) is achievable, w her e the supremum is taken over all joint dist ributions on ( X [ n ] , Y ) that lie in P . Proof. Let Γ denote the set of al l conditional mutual information vectors (in the Euclidean space of dimension 2 n ) corresponding to the discrete dist ributions on ( X [ n ] , Y ) that lie in P . More precisely , corresponding to any joint distribution in P is a point γ ∈ Γ defined by γ ( s ) = I ' X s ; Y | X s c ( (26) for each s ⊂ [ n ]. Han [ 28 ] showed that for any joint distribution in P , γ ( s ) is a submodular set function. In other words, each point γ ∈ Γ defines a concav e game. As shown in [ 28 ], the DM-MAC capacity region may also be charac t erized as the union of the aspir ation sets of games from Γ ∗ , where Γ ∗ is the closure of the c on vex hull of Γ . It remains to check that each point in Γ ∗ corresponds to a conca ve game, and this follows from the easily verifiable facts that a conv ex combination of conca ve games is concave, and that a limit of conca ve games is concav e. For the second assertion, note that for any γ ∈ Γ ∗ ,a sum capacity of γ ([ n ]) is achievable by F act 2 (applied to resourc e al location games). Combining this w ith the above characterization of the capacity region and the fact that γ ([ n ]) = I ( X [ n ] ; Y ) for γ ∈ Γ completes the argument. W e now take up the G-MA C. The additive nature of the G-MA C is reflected in a simpler g ame-theor etic descr iption of its capacity region. F act 9. The capacit y region of the n -user G-MAC is the set of capacity allocations ( C 1 , ... , C n ) that satisfy # i ∈ s C i ≤ C ) ! i ∈ s P i N * = : v g ( s ) (27) Moksha y Madiman 7 for each s ⊂ [ n ], where C ( x ) = (1 / 2)log(1 + x ). In other words, the capacity reg ion of the G-MAC is the aspiration set of the g ame defined by v g , which we may call the G-MAC game. T ranslation 6. The G-MAC game is a concav e game. In particular , its core is nonempty , and a sum capacit y of C ( ! i ∈ [ n ] P i /N ) is achievable. As in the previous sec tion, we ma y ask whether this is robust to network deg radation in the form of users dro pping out, at least in some order; the answer is obtained in an exactly analogous fashion. T ranslation 7 (Robust coding for the G-MAC). Suppose the senders can only drop out in a certain order , which without loss of generality we can tak e to be the natural decreasing order on [ n ] (i.e., we assume that the first user to potentially drop out would be sender n , followed by sender n − 1, etc.). Then, there exists a capacit y allocation to senders for the G-MA C which is feasible and optimal ir respecti ve of the number of users that have dropped out. Furthermore, just as for the Slepian-W olf game, Fact 4 has an interpretation in terms of tolerance vectors analogous to T ranslation 4 . W hen there is no natural ordering of senders, Fact 3 suggests that the Shaple y value is a good choice of capacit y allocation for the G-MA C game. Practical implementation of an ar bitrar y capacit y allocation point in the core is discussed by Rimoldi and U rbanke [ 24 ] and Y eh [ 30 ]. W hile the ground for the study of the geometr y of the G- MA C capacity region using the theor y of poly matr oids was laid by Han, such a study and its implications were fur ther developed, and in the more general setting of fading that allows the modeling of wireless channels, by T se and Hanly [ 31 ] (see also [ 30 ]). Clearly statements like T ranslation 7 can be carr ied over to the more general setting of fading channels by building on the obser vations made in [ 31 ]. La and Ananthar am [ 1 ] provide an eleg ant analysis of capacity al location for a di ff erent Gaussian MA C model using cooperativ e game theoretic ideas. W e briefly review their results in the context of the preceding discussion. Consider an Gaussian multiple access channel that is arbitrarily var ying , in the sense that the users are potentially hostile, aware of each others’ codebooks, and are capable of forming “jamming coalitions” . A jamming coalition is a set of users, say s c , who decide not to communicat e but instead get t ogether and jam the channel for the remaining users, who constitut e the communicating coalition s . As before, each user has a power constraint; the i th sender cannot use power greater than P i whether it w ishes to communicate or jam. It is still a Gaussian MAC because the receiv ed signal is the super p osition of the inputs provided by all the senders, plus additiv e Gaussian noise of var ianc e N . In [ 1 ], the value function v LA for the game corresponding to this channel is derived; the value for a coalition s is the capacit y achievable by the users in s e v en when the users in s c coherently combine to ja m the channel. F act 10. The capacity region of the arbitr arily var ying Gaus- sian MAC with potentially hostile senders is the aspiration set of the La-Anantharam game, defined by v LA ( s ): = C + P , s Λ s c + N - , (28) where P s = ! i ∈ s P i , Λ s = [ ! i ∈ s . P i ] 2 , and , s ={ i ∈ s : P i ≥ Λ s c } . N ote that two things have changed relativ e to the naive G- MA C game; the power available for t ransmission (appearing in the numerator of the argument of the C function) is reduced because some senders are rendered incapable of communicating by the jammers, and the noise term (appearing in the denominator) is no longer constant for all coalitions but is augmented by the power of the jammers. This tightening of the aspiration set of the La-Anantharam game versus the G-MA C game causes the concavity property to b e los t . T ranslation 8. The La-Anantharam g ame is not a conca ve game, but it has a nonempt y core. In particular , a sum capacity of C ( ! i ∈ [ n ] P i /N ) is achievable. Proof. La and Anantharam [ 1 ] show that the Shapley value need not lie in the core of their game, but they demonst rate the existence of another distinguished point in the core. By the analogue of Fact 3 for resourc e allocation g ames, the La- Anantharam game cannot be concav e. Although [ 1 ] shows that the Shaple y value may not lie in the core, the y demonstrate the existence of a unique capacity point that satisfies three desirable axioms: (a) e ffi ciency , (b) invariance to permutation, and (c) envy-fr eeness. While the first two are also among the Shapley value axioms, [ 1 ] provides justification for envy-freeness as an appropriate axiom from the point of v iew of applications. W e mention here a natural question that we leave for the reader to ponder : given that the La-Ananthar am game is balanced but not concav e, is it exact? Note that the fact that the Shapley value does not lie in the core is not incompatible with exactness, as shown by Rabie [ 15 ]. Finally , we tur n to the P -MAC. Lapidoth and Shamai [ 32 ] performed a detailed study of this communication problem and showed in particular that the capacit y region w hen all users ha ve the same peak po wer constraint is given as the closed con vex hull of the union of aspiration sets of cer tain games, just as in the case of the DM-MA C. As in that case, one may check that the capacity region is in fac t the union of aspiration sets of a class of conca ve games, and in par ticular , as shown in [ 32 ], the maximum throughput that one may hope for is achievable. Of course, there is much more to the well-developed theor y of multiple access channels than the memor yless scenarios (discrete, Gaussian and Poisson) discussed above. For instance, there is much rece nt work on multiuser channels w ith memor y and also w ith feedback (see, e.g., T atikonda [ 33 ] for a deep treatment of such problems at the intersection of communication and control). W e do not 8 EURASIP Journal on Wir eless Communications and Networking discuss these works further , except to make the obser vation that thing s can change considerably in these more general scenarios. Indeed, it is quite conceivable that the appropriate games for these scenarios are not conv ex or concav e, and it is ev en c onceivable that such games may not be balanced, which may mean that there are unexpected limitations to achieving the sum rate or sum capacit y that one may hope for at first sight. 5. A DISTRIBUTED ESTIMA TION GAME In the nascent theor y of distr ibuted estimation using sensor networks, one w ishes to characterize the fundamental limits of performing statistical tasks such as parameter estimation using a sensor network and apply such charac te rizations to problems of cost or resource allocation. W e discuss one such question for a toy model for dist r ibuted estimation intro- duced by Madiman et al. [ 34 ]. By ignor ing communication, computation, and other constraints, this model allows one to study the centr al question of fundamental statistical limits without obfuscation. The model we consider is as follows. The goal is to estimate a parameter θ , which is some unknown real number . Consider a class of sensors, all of w hich hav e estimating θ as their go al. Howev er , the sensors cannot measure θ directly; they are immersed in a field of sources (that do not dep end on θ and may be considered as producers of noise for the purposes of estimating θ ) . M ore specifically , suppose there are n sources, w ith each source producing a data sample of size M according to some known probability distribution. Let us say that source i generates X i ,1 , ... , X i , M . The class of sensors a vailable corresponds to a class C of subsets of [ n ], w hich indexes the set of sources. Ow ing to the geog raphical placement of the sensors or for other reasons, each sensor only sees certain agg r e gate data; indeed, the sensor corresponding to a subset s ⊂ [ n ], known as the s -sensor , only sees at any given time the sum of θ and the data coming from the sources in the set s . In other words, the s -sensor has a c cess to the obser vations Y s = ( Y s ,1 , Y s ,2 , ... , Y s , M ), where Y s , j = θ + # i ∈ s X i , j . (29) Clearly , θ shows up as a common location par ameter for the observations seen by any sensor . From the obser vations Y s that are available to it, the s -sensor constructs an estimator , θ s ( Y s ) of the unknown parameter θ . The goodness of an estimator is measured by comparing to the “best possible estimator in the worst case ” , that is, b y comparing the r isk of the given estimat or with the minimax risk. If the risk is measured in terms of mean squared error , then the minimax risk achievable by the s - sensor is r M ( s ) = min all estimators , θ s max θ E / , θ s ( Y s ) − θ 0 2 . (30) (For location parameters, Girshick and Savage [ 35 ] showed that there exists an estimator that a c hieves this minimax risk.) The cost measure of interest in this scenario is er r or variance. Suppose w e can give varianc e permissions V i for each source, that is, the s -sensor is only allowed an unbiased estimator w ith var ianc e not more than ! i ∈ s V i , or more generally , an estimator with mean squared risk not more than this number . For the var iance permission vector ( V 1 , ... , V n ) to be feasible with respect to an ar bitrary sensor configuration (i.e., for there to exist an estimator for the s - user w ith worst-case r isk bounded by ! i ∈ s V i , for ever y s ), we need that # i ∈ s V i ≥ r M ( s ) (31) for each s ⊂ [ n ]. Thus, we hav e the following fact. F act 11. The set of feasible var ianc e permission vectors is the aspiration set of the cost al location game v DE ( s ): = r M ( s ), (32) which we call the distributed estimation game. The natural question is the following . Is it possible to allot variance permissions in such a way that there is no wasted total variance, that is, ! i ∈ [ n ] V i = r M ([ n ]), and the allotment is feasible for arbitrar y sensor configurations? The a ffi rmative answer is the content of the following result. T ranslation 9. Assuming that all sources hav e finite variance, the distributed estimation game is balanced. Consequently , there exists a feasible var iance allotment ( V 1 , ... , V n ) to sources in [ n ] such that the [ n ]-sensor cannot waste any of the variance allotted to it. Proof. The main result of Madiman et al. [ 34 ] is the following inequality relating the minimax risks achievable by the s - users from the class C to the minimax risk achievable by the [ n ]-user , that is, one who only sees obser vations of θ corr upt ed by all the sources. U nder the finite variance assumption, for any sample size M ≥ 1, r M ([ n ]) ≥ # s ∈ C β ( s ) r M ( s ) (33) holds for any fractional par tition β using any collection of subsets C . In other words, the game v DE is balanced. Fact 1 now implies that the core is nonempty , that is, a total var ianc e as low as r M ([ n ]) is achievable. T ranslation 9 implies that the optimal sum of variance permissions that can be achieved in a distr ibuted fashion using a sensor network is the same as the best var iance that can be achieved using a single centralized sensor that sees all the sources. Other interesting questions relating to sensor networks can be answered using the inequalit y ( 33 ). For instance, it suggests that using a sensor configuration corresponding to the class C 1 of all singleton sets is better than using a sensor configuration corresponding to the class C 2 of all sets of size 2. W e refer the reader to [ 34 ] for details. An interesting open problem is the determination of w hether this dist ributed estimation game has a large core. Moksha y Madiman 9 6. AN ENTROPY POWER GAME The entropy power of a continuous random vector X is N ( X ) = exp { 2 h ( X ) /d } / 2 π e , where h denotes di ff erential entropy . Ent r opy power plays a key role in several problems of multiuser information theor y , and entropy power inequal- ities have been key to the determination of some capacit y and rate regions. (Such uses of ent r opy power inequalities may be found, e.g ., in Shannon [ 36 ], Berg mans [ 37 ], Ozarow [ 38 ], Costa [ 39 ], and Oohama [ 40 ].) F urthermore, rate regions for several multiuser problems, as discussed already , inv olve subset sum constr aints. Thus, it is concei v able that there exists an inter pr etation of the discussion below in terms of a multiuser communication problem, but we do not know of one. W e make the fol lo w ing conjectur e. Conjecture 1. Le t X 1 , ... , X n be ind e pendent R d -valued ran- dom vectors w ith de nsities and finite covariance mat rices. Suppose the region of interest is the s et of points ( R 1 , ... , R n ) ∈ R n + satisfy ing # j ∈ s R j ≥ N ) # j ∈ s X j * (34) for each s ⊂ [ n ] . Then, the re ex ists a point in this reg ion such that the total sum ! j ∈ [ n ] R j = N ( ! j ∈ [ n ] X i ) . By Fact 1 , the following conjecture, implicitly proposed by Madiman and Barron [ 41 ], is equivalent. Conjecture 2. Le t X 1 , ... , X n be ind e pendent R d -valued ran- dom vectors w ith densit ies and finite covar iance mat rices. For any collection C of subse ts of [ n ] , le t β be a fractional par tition. Then, N ( X 1 + ··· + X n ) ≥ # s ∈ C β ( s ) N ) # j ∈ s X j * . (35) Equality holds if and only if all the X i are normal w ith proportional covar iance mat rices. N ote that Conjecture 2 simply states that the “ entropy power g ame ” defined by v EP ( s ): = N ( ! j ∈ s X j ) is balanced. Define the maximum degree in C as r + = max i ∈ [ n ] r ( i ), where the de gree r ( i ) of i in C is the number of sets in C that contain i . Madiman and Bar r on [ 41 ] showed that Conjecture 2 is true if β ( s ) is replaced by 1 /r + , where r + is the maximum degree in C . W hen every index i has the same degree, β ( s ) = 1 /r + is indeed a fractional par tition. The equi valence of Conjectures 1 and 2 ser v es to underscore the fact that the balancedness inequality of Conjecture 2 may be regarded as a more fundamental pro p- ert y (if true) than the gener alized entropy power inequalities in [ 41 ] and is therefore worthy of attention. The interested reader may also w ish to consult [ 42 ], where we giv e some further ev idence to wards its validity . Of course, if the entropy power g a m e above turns out to be balanced, a natural next question would be w hether it is exact or even con vex. W hile on the topic of games invol ving the entropy of sums, it is wor th mentioning that much more is known about the game with value function: v sum ( s ): = H ) # i ∈ s X i * , (36) where H denotes discrete entropy , and X i are independent discrete r andom var iables. Indeed, as shown by the author in [ 42 ], this game is concave, and in par ticular , has a nonempty core which is the con vex hull of its marginal vectors. For independent continuous random vectors, the set function v ( s ) = h ) # i ∈ s X i * , (37) where h denotes di ff erential entropy , is submodular as in the discrete case. Ho wever , this set function does not define a game; indeed, the appropriate con vention for v ( φ ) is that v ( φ ) = −∞ , since the null set corresponds to looking at the di ff erential entropy of a constant (say , zer o), which is −∞ . Because of the fact that the set funct ion v is not real- valued, the submodularit y of v does not imply that it is even subadditive (and thus υ certainly does not satisfy the inequalities that define balancedness). On the other hand, if X is a continuous random vect or independent of X 1 , ... , X n , and w ith di ff erentiall entropy h ( X ) = 0, then the modified set function v sum ( s ) = h ) X + # s ∈ C X i * (38) is indeed the value function of a balanced cooperative game; see [ 42 ] for details and further discussion. 7. GAMES IN COMPOSITE HYPOTHESIS TESTING Inter esting ly , similar notions also come up in the study of composite hypothesis testing but in the setting of a cooperative resource allocation g ame for infinitely many users. Let ( Ω , A ) be a P olish space with its Borel σ - lgebra, and let M be the space of pr obabilit y measures on ( Ω , A ). W e may think of Ω as a set of infinitely many “micr oscopic players” , namely ω ∈ Ω . The allowed coalitions of microscopic users are the Borel sets. For our purposes, we specify an infinite cooperative game using a value function v : A → R that satisfies the following conditions: (1) v ( φ ) = 0, and v ( Ω ) = 1, (2) A ⊂ B ⇒ v ( A ) ≤ v ( B ), (3) A n ↑ A ⇒ v ( A n ) ↑ v ( A ), (4) for closed sets F n with F n ↓ F , v ( F n ) ↓ v ( F ). The contin uity conditions are necessary regularit y con- ditions in the context of infinitely many players. The normalization v ( Ω ) = 1 (which is also sometimes imposed in the study of finite g ames) is also useful. In the mathematics literature, a value function satisfying the itemized conditions 10 EURASIP Journal on Wir eless Communications and Networking is called a cap a c ity , while in the economics literature, it is a (0,1)-normalized nonatomic game (usually additional conditions are imposed for the latter). There are many subtle analytical issues that emerge in the study of capacities and nonatomic g ames. W e avoid these and simply mention some infinite analogues of already stated facts. For any capacity v , one may define the family of probability measures P v ={ P ∈ M : P ( A ) ≤ v ( A ) for each A ∈ A } . (39) The se t P v may be thought of as the core of the game v . Indeed, the additiv ity of a measure on disjoint sets is the continuous formulation of the transfer able utility assumption that earlier caused us to consider sums of resourc e allocations, while the restriction to probability measures ensures e ffi ciency , that is, the maximum possible allocation for the full set. Let P be a family of probability measures on ( Ω , A ). The set function v ( A ) = sup P ∈ P P ( A ), A ∈ A , (40) is cal led the upper envelope of P , and it is a capacit y if P is weakly compact. N ote that such upper env elopes are just the analogue of the X OS valuations defined in S e c t i o n 2 for the finite setting. By an extension of Fact 6 to infinite games, one can deduce that the core P v is nonempt y when v is the upper env elope game for a weakly compact family P of probability measures. W e say that the infinite game v is concav e if, for all measurable sets s and t , v ( s ∪ t )+ v ( s ∩ t ) ≤ v ( s )+ v ( t ) . (41) In the mathematics literature, the value func tion of a conca ve infinite g ame is often called a 2-alter nating capac i t y , following Choquet’ s seminal work [ 43 ]. W hen v defines a conca ve infinite game, P v is nonempty ; this is an analog of Fact 2 . Further more, by the analog of Fact 5 , v is an exact game since it is concave, and as in the remarks after Fact 6 , it follows that v is just the upper env elope of P v . Conca ve infinite games v are not just of abst r act interest; the families of probability measures P v that are their cores include impor tant classes of families such as total variation neighborhoods and contamination neighbor hoods, as dis- cussed in the refer ences cited below . A famous, classical result of Huber and Strassen [ 44 ] can be stated in the language of infinite cooperative games. Suppose one w ishes to test b etw een the composite hypothe- ses P u and P v , where u and v define infinite games. The criterion that one w ishes to minimize is the decay rate of the probability of one ty pe of error in the worst case (i.e., for the worst pair of sources in the two classes), given that the error probability of the other kind is kept below some small constant; in other words, one is using the minimax criterion in the Neyman-P earson framework. N ote that the selection of a cr itical region for testing is, in the game language, the selection of a coalition. In the setting of simple hypotheses, the optimal coalition is obtained as the set for w hic h the Radon-Nik o dy m derivative between the two probability measures corresponding to the two hypotheses ex ceeds a threshold. Although ther e is no obv ious notion of Radon- Nik ody m der i vative between two composite hypotheses, [ 44 ] demonstrates that a likelihood ratio test continues to be optimal for testing between composite h ypotheses under some conditions on the games u and v . T ranslation 10. For conca ve infinite g ames u and v , consider the composite h y potheses P u and P v that are their cor es. Then, a minimax Neyman-P earson test between the P u and P v can be constr ucted as the likelihood ratio test between an element of P u and one of P v ; in this case, the representativ e elements minimize the K ullback divergence between the two families. In a certain sense, a con verse statement can also be shown to hold. W e refer to H uber and Str assen [ 44 ] for proofs and to V eeravalli et al. [ 45 ] for context, further results, and applications. Relat ed results for minimax linear smoothing and rate distortion theor y on classes of sources were given by P oor [ 46 , 47 ] and to channel coding w ith model uncertaint y were g iven by Ger aniotis [ 48 , 49 ]. 8. DISCUSSION The gener al approach to using cooperative game theor y to understand rate or capacity regions in volves the following steps. (i) Formulate the region of interest as the aspiration set of a cooperative game. This is frequently the r ight kind of formulation for multiuser problems. (ii) Study the properties of the value function of the g ame, starting with checking if it is balanced, if it is exact, if it has a large core, and ultimately by checking conv exity or concavity . (iii) Interpret the properties of the game that follow from the discov ered properties of the value function. For instance, balancedness implies a nonempt y core, while conv exit y implies a host of results, including nice properties of the Shaple y value. These are structur al results, and their game-theoretic inter pr etation has the potential to provide some additional intuition. There are numerous other papers which make use of cooperativ e game theor y in communications, althoug h with di ff erent emphases and applications in mind. See, for example, van den Nouw eland et al. [ 50 ], Han and Poor [ 51 ], Jiang and Baras [ 52 ], and Y a ¨ ıche et al. [ 53 ]. Ho wever , we hav e pointed out a very fundamental connection between the two fields—arising from the fac t that rate and capacity reg ions are often closely related to the aspiration sets of cooperativ e games. In several ex emplar y scenarios, both classical and relativ ely new, we hav e reinterpreted known results in terms of g ame-theoretic intuition and also pointed out a number of open problems. W e expect that the cooperative game theoretic point of view w ill find utility in other scenarios in network information theory , distributed inferenc e, and robust statistics. ACKNO WLEDGMENTS I am indebted to Rajesh Sundaresan for a detailed discussion that clarified my understanding of some of the literature, and Moksha y Madiman 11 the identification of an error in the first version of this paper . 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