An efficient and fair solution for communication graph games 1

Size: px

Start display at page:

Download "An efficient and fair solution for communication graph games 1"

Jeremy Jenkins
5 years ago
Views:

1 A efficiet ad fair solutio for commuicatio graph games 1 Reé va de Brik 2 Aa Khmelitskaya 3 Gerard va der Laa 4 March 11, This research was supported by NWO (The Netherlads Orgaizatio for Scietific Research) grat NL-RF The research of Aa Khmelitskaya was also partially supported by CRT Foudatio ad the Uiversity of Easter Piedmot at Alessadria through the Lagrage Project J.R. va de Brik, Departmet of Ecoometrics ad Tiberge Istitute, VU Uiversity, De Boelelaa 1105, 1081 HV Amsterdam, The Netherlads. jrbrik@feweb.vu.l 3 A.B.Khmelitskaya,SPbIstituteforEcoomicsadMathematics Russia Academy of Scieces, Tchaikovsky st. 1, St.Petersburg, Russia, a.khmelitskaya@math.utwete.l 4 G. va der Laa, Departmet of Ecoometrics ad Tiberge Istitute, VU Uiversity, De Boelelaa 1105, 1081 HV Amsterdam, The Netherlads. glaa@feweb.vu.l

2 Abstract We itroduce a efficiet solutio for games with commuicatio graph structures ad show that it is characterized by efficiecy, fairess ad a ew axiom called compoet balacedess. This latter axiom compares for every compoet i the commuicatio graph the total payoff to the players of this compoet i the game itself to the total payoff of these players whe applyig the solutio to the subgame iduced by this compoet. Keywords: TU game, commuicatio graph, Myerso value, fairess, efficiecy. JEL code: C71

3 1 Itroductio A situatio i which a fiite set of players ca obtai certai payoffs by cooperatio ca be described by a cooperative game with trasferable utility, or simply a TU-game, beig a pair cosistig of a fiite set of players ad a characteristic fuctio o the collectio of all coalitios of players, that assigs a worth to each coalitio of players. I this ote we cosider TU-games with limited cooperatio possibilities, represeted by a udirected commuicatio graph, as itroduced by Myerso [6]. The odes i the graph represet the players ad the edges represet the commuicatio liks betwee the players. Players ca oly cooperate if they are coected. This yields a so-called (commuicatio) graph game, give by a triple cosistig of a fiite set of players, a characteristic fuctio ad a commuicatio graph. A (sigle-valued) solutio for commuicatio graph games is a mappig that assigs to every commuicatio graph game a payoff vector. The best-kow solutio for commuicatio graph games is the Myerso value [6], which is obtaied as the Shapley value of a restricted game, ad is characterized by compoet efficiecy ad fairess. Compoet efficiecy states that for each compoet of the commuicatio graph the total payoff to the players of the compoet is equal to the worth of that compoet i the characteristic fuctio. Fairess says that deletig a lik betwee two players yields for both players the same chage i payoff. Aother sigle-valued solutio cocept, the so-called positio value, is itroduced i Meesse [5] ad developed i Borm, Owe ad Tijs [1]. Slikker [9] axiomatizes the positio value usig compoet efficiecy ad balaced total threats. For cycle-free commuicatio graph games, Herigs, va der Laa ad Talma [3] itroduced the so-called Average Tree solutio, characterized by compoet efficiecy ad compoet fairess, the latter axiom statig that deletig a lik betwee two players i a cycle-free graph game yields the same average chage i payoff i the two compoets that result from deletig the lik. All these solutios satisfy compoet efficiecy. Therefore, efficiecy is oly guarateed whe the graph is coected ad thus cotais the player set itself as its uique compoet. I cotrast to the reasoig that a set of players ca oly realise its worth whe they are coected, ad thus evetually the players i each compoet distribute the worth of that compoet amog each other, i some situatios efficiecy is obtaied, eve whe the commuicatio graph is ot coected. As a example, cosider a research fud that has a amout of moey available to distribute amogst idividual researchers. Every researcher that submits a proposal takes part i the distributio of the available budget, so writig a idividual proposal is the oly requiremet for a researcher to get access to the fud. However, the board of the research fud has the policy to stimulate iterdiscipliary research ad therefore joit proposals get priority. Researchers ca secure some part of the 1

4 fud by submittig joit proposals. For istace, suppose that the budget of the fud is 12 ad there are three researchers, amed A, B ad C. A idividual proposal just gives access to the fud, but does ot secure ay amout of moey. O the other had, A e B ca secure themselves a grat of 3 whe writig a joit proposal, A ad C a grat of 2 ad B ad C a grat of 4. However, C does ot commuicate with the others, so the oly feasible coalitio is A ad B ad the commuicatio graph cosists of two compoets: the coalitio of A ad B that ca secure themselves 3, ad the sigleto aget C that ca oly secure itself 0. Accordig to the Myerso value the total amout of moey grated to the researchers is oly 3, but i this situatio the board of the research fud will grat 3 to A ad B ad will the distribute the remaiig 9 to the researchers. Although the commuicatio graph is ot coected, the full budget of 12 is still available to the coalitio of all players. So, this requires a value satisfyig efficiecy. Recetly, also Casajus [2] argued by some motivatig example that i some situatios it seems reasoable to require efficiecy, eve whe the commuicatio graph is ot coected ad thus has multiple compoets. He itroduced a solutio for commuicatio graph games that is characterized by efficiecy, equivalece (meaig that the total payoff i case of the complete graph is equal to the total payoff i case of the empty graph), compoet mergig (meaig that mergig the compoets players ito a sigle player does ot affect the total payoff to the compoet) ad a modified versio of Myerso s fairess. I this ote we itroduce a ew solutio for commuicatio graph games that, besides efficiecy ad Myerso s fairess, satisfies a ew axiom called compoet balacedess. This compoet balacedess axiom compares for every compoet i the commuicatio graph the total payoff to the players of this compoet i the game itself to the total payoff of this compoet whe applyig the solutio to the subgame iduced by this compoet. It also ca be see as weak versio of compoet efficiecy. The ew solutio equals the Shapley value whe the graph is coected ad is equal to the equal surplus divisio whe the graph is empty. This ote is orgaized as follows. Basic defiitios ad otatio are itroduced i Sectio 2. The compoet balacedess axiom, the ew solutio ad its characterizatio are give i Sectio 3. At the ed of that sectio we retur to the research fud example described above ad compare our solutio with several others. 2 Prelimiaries A situatio i which a fiite set of players ca obtai certai payoffs by cooperatig ca be described by a cooperative game with trasferable utility, or simply a TU-game, beig a pair N,v, where N IN is a fiite set of 2 players ad v:2 N IR is a characteristic 2

5 fuctioon suchthatv( ) = 0. ForaycoalitioS N, v(s)istheworthofcoalitios, i.e., the members of coalitio S ca obtai a total payoff of v(s) by agreeig to cooperate. We deote the set of all characteristic fuctios o give player set N by G N. Although N is ot fixed, evertheless for simplicity of otatio ad if o ambiguity appears, we write v istead of N,v. For give N, the subgame of a game v G N with respect to a player set T N, T, is the game v T G T defied as v T (S) = v(s), for all S T. We deote the cardiality of a give set A by A, alog with lower case letters like = N, c = C, c = C ad so o. For K IN, we deote IR K as the k-dimesioal vector space which elemets x IR K have compoets x i, i K. For game v G N, a vector x IR N may be cosidered as a payoff vector assigig a payoff x i to each player i N. A sigle-valued solutio, called a value, is a mappig ξ that assigs for every N ad every v G N a payoff vector ξ(v) IR N. A value ξ is efficiet if i N ξ i(v) = v(n) for every v G N ad N IN. The best-kow efficiet value is the Shapley value [8], give by Sh i (v) = {S N i S} ( s)!(s 1)! (v(s) v(s \{i})), for all i N.! For N IN, a commuicatio structure o N is specified by a commuicatio graph N,Γ with Γ Γ N = {{i,j} i,j N, i j}, i.e., Γ is a collectio of (uordered) pairs of odes (players), where a pair {i,j} represets a lik betwee players i,j N, ad N,Γ N is the complete graph o N. Agai, for simplicity of otatio ad if o ambiguity appears, we write graph Γ istead of N,Γ. Let L N deote the set of all commuicatio graphs o N. A pair v,γ G N L N costitutes a game with (commuicatio) graph structure or simply a graph game o N. For give N, the subgraph of a graph Γ L N with respect to set T N, T, is the graph Γ T L T defied by Γ T = {{i,j} Γ i,j T}. For a graph Γ, a sequece of differet odes (i 1,...,i k ), k 2, is a path from i 1 to i k, if for all h = 1,...,k 1, {i h,i h+1 } Γ. A graph Γ o a player set N is coected, if for ay two odes i N there exists a path i Γ from oe ode to the other. For give graph Γ o N, we say that the player set S N is coected, if the subgraph Γ S is coected. For graph Γ o player set N ad S N, a subset T S is a compoet of S if (i) Γ T is coected, ad (ii) for every i S \ T, the subgraph Γ T {i} is ot coected. For Γ o N ad S N, we deote by S/Γ the set of all compoets of S, ad by (S/Γ) i the compoet of S cotaiig i S. Notice that S/Γ is a partitio of S. A sigle-valued solutio for commuicatio graph games, a graph game value, is a mappig ξ that for every N IN ad every v,γ G N L N assigs a payoff vector ξ(v,γ) IR N. A well-kow graph game value is the Myerso value. I Myerso [6] it is assumed that i a commuicatio graph game v, Γ oly coected coalitios are able to cooperate ad to realise their worths. A o-coected coalitio S ca oly realise the sum 3

6 of the worths of its compoets i S/Γ. This yields the restricted game v Γ G N defied by v Γ (S) = v(t), for all S N. T S/Γ The the Myerso value for commuicatio graph games is the graph game value µ that assigs to every commuicatio graph game v, Γ the Shapley value of its restricted game v Γ, i.e., µ(v,γ) = Sh i (v Γ ) for all v,γ G N L N ad every N IN. It is well-kow that the Myerso value is the uique graph game value that is compoet efficiet ad satisfies the so-called Myerso fairess axiom. The aim of this paper is to itroduce a efficiet ad fair solutio for commuicatio graph games. To coclude the itroductio sectio we recall defiitios of efficiecy, compoet efficiecy ad fairess. A graph game value ξ is - efficiet if for every graph game v,γ o ay player set N, i N ξ i(v,γ) = v(n); - compoet efficiet if for every graph game v,γ o ay player set N, for every C N/Γ, i C ξ i(v,γ) = v(c); - fair if for every graph game v,γ o ay player set N, for every {h,k} Γ, ξ h (v,γ) ξ h (v,γ hk ) = ξ k (v,γ) ξ k (v,γ hk ), where Γ hk = Γ\{{h,k}}. 3 Efficiecy, fairess ad compoet balacedess I this ote we look for a graph game value that is characterized by efficiecy, fairess ad a ew axiom that we refer to as compoet balacedess. Compoet balacedess (CB) For every graph game v,γ o ay player set N, for every compoet C N/Γ, it holds i C (ξ i(v,γ) ξ i (v C,Γ C )) = c ( ) i N ξi (v,γ) ξ i (v (N/Γ)i,Γ (N/Γ)i. First, ote that this axiom oly states a requiremet o the payoffs whe the collectio of compoets N/Γ cotais at least two elemets, otherwise the requiremet reduces to a idetity. Further, otice that the games v C,Γ C ad v (N/Γ)i,Γ (N/Γ)i are defied o the reduced player sets C, respectively, (N/Γ) i. For a compoet C N/Γ, the axiom compares the payoffs that the players of C receive i the game itself to the payoffs that 4

7 these players receive i the subgame o C. Cosiderig two compoets C,C N/Γ, this axiom implies that 1 c (ξ i (v,γ) ξ i (v C,Γ C )) = 1 c i C i C (ξ i (v,γ) ξ i (v C,Γ C )), meaig that cosiderig oly the players i compoet C, the chage i the average payoff of the players i this compoet is the same as the chage i the average payoff of the players i ay other compoet C resultig from cosiderig oly the players i that compoet C. We refer to this axiom as compoet balacedess because it has some flavour of the balaced cotributios 1 property of Myerso [7], but i terms of the average chage of payoffs i compoets. 2 Compoet balacedess also ca be see as a weak versio of compoet efficiecy sice every graph game value that satisfies compoet efficiecy satisfies compoet balacedess. This follows straightforward sice compoet efficiecy implies that i C ξ i(v,γ) = i C ξ i(v C,Γ C ) = v(c), for all C N/Γ. As metioed, we will show that there is a uique graph game value that satisfies efficiecy, fairess ad compoet balacedess. This solutio is obtaied by takig the Shapley value of a slight modificatio of the restricted game v Γ. If we wat to obtai a efficiet graph game value as the Shapley value of some restricted game, the at least the worth of the grad coalitio N i the restricted game must be v(n). It turs out that this modificatio is sufficiet to obtai the uique graph game value satisfyig efficiecy, fairess ad compoet balacedess. So, for a player set N IN ad v,γ G N L N, we defie v Γ G N by 3 v Γ (S) = { v Γ (S), S N, v(n), S = N, ad cosider the graph game value ψ give by ψ(v,γ) = Sh i (v Γ ) for all v,γ G N L N ad every N IN. We the have the followig theorem. 1 Balaced cotributios for commuicatio graph games states that isolatig a player, say i, i the commuicatio structure has the same effect o the payoffs of aother player, say j, as the effect o the payoff of i as a result of isolatig player j, i.e., for every graph game v,γ ad i,j N, it holds that ξ i (v,γ) ξ i (v,γ\γ j ) = ξ j (v,γ) ξ j (v,γ\γ i ), where Γ h = {{i,j} Γ h {i,j}}. 2 It is worth remarkig that compoet balacedess also has the flavor of several other kow types of axioms such as cosistecy (lookig at reduced games, but ot sayig that the payoffs of remaiig players do ot chage) or compoet fairess (comparig average chages of payoffs i a compoet, but ot after lik deletio). 3 So, v Γ is the Myerso restricted game, except that v Γ (N) = v(n) istead of H N/Γ v(h). 5

8 Theorem 3.1 The graph game value ψ is efficiet, fair ad satisfies compoet balacedess. Proof. Sice v Γ (N) = v(n), efficiecy follows by efficiecy of the Shapley value. So, we oly have to show fairess ad compoet balacedess. By defiitio we have that v Γ = v Γ +w, where w G N is give by { 0, S N, w(s) = v(s) v Γ (S), S = N, i.e., game v Γ is obtaied by addig v(n) v Γ (N) times the uaimity game 4 of N to the Myerso restricted game v Γ. From this ad the additivity ad symmetry properties of the Shapley value it follows that ψ i (v,γ) = Sh i (v Γ ) = Sh i (v Γ )+Sh i (w) = µ i (v,γ)+ v(n) vγ (N), (3.1) where the last equality follows by defiitio of µ ad w. Hece, ψ i (v,γ) ψ i (v,γ ij ) = µ i (v,γ)+ v(n) vγ (N) = µ i (v,γ) µ i (v,γ ij ) vγ (N) v Γ ij (N) ( ) µ i (v,γ ij )+ v(n) vγ ij (N) = µ j (v,γ) µ j (v,γ ij ) vγ (N) v Γ ij = ψ j (v,γ) ψ j (v,γ ij ), where the third equality follows by fairess of µ. Hece, ψ satisfies fairess. To show compoet balacedess, by (3.1) we obtai for every C N/Γ that ψ i (v,γ) = µ i (v,γ)+ c ( v(n) v Γ (N) ). i C i C Further, i C µ i(v,γ) = v(c) because of compoet efficiecy of the Myerso value, ad the total payoff that ψ assigs to the players i C i the subgame v C,Γ C is equal to v(c) because of the efficiecy of ψ itself. Thus, with (3.1) (ψ i (v,γ) ψ i (v C,Γ C )) = v(c)+ c ( v(n) v Γ (N) ) v(c) = c ( v(n) v Γ (N) ). i C Also, by efficiecy of ψ we have i N ψ i(v (N/Γ)i,Γ (N/Γ)i ) = H N/Γ i H ψ i(v H,Γ H ) = H N/Γv(H), ad thus ( ψi (v,γ) ψ i (v (N/Γ)i,Γ (N/Γ)i ) ) = v(n) v(h) = v(n) v Γ (N). i N H N/Γ 4 It is well kow [8] that the collectio of uaimity games {u T }T N, defied as u T (S) = 1, if T S, T ad u T (S) = 0 otherwise, from a basis i G N. 6

9 Hece, ψ satisfies compoet balacedess. Note that (3.1) gives a alterative defiitio of the graph game value ψ assigig to every graph game its Myerso value ad distributig the differece betwee the worth of the grad coalitio N ad the sum of the worths of all compoets equally over all players. I this sese the solutio ψ ca be see as combiig elemets of the Shapley value ad equal divisio solutio. 5 The ext theorem characterizes the graph game value ψ. Theorem 3.2 There is a uique graph game value ξ satisfyig efficiecy, fairess ad compoet balacedess. Proof. By Theorem 3.1 we oly eed to show uiqueess. We first cosider the case that Γ is the empty graph. The N/Γ = {{i} i N}, i.e., every ode is a sigleto compoet, ad compoet balacedess requires for every i N that ξ i (v,γ) ξ i (v {i},γ {i} ) = 1 ( ξj (v,γ) ξ j (v {j},γ {j} ) ). (3.2) j N By efficiecy j N ξ j(v,γ) = v(n) ad also for every j N, ξ j (v {j},γ {j} ) = v({j}). Hece, whe Γ is the empty graph, the by (3.2) the payoffs ( ξ i (v,γ) = v({i})+ 1 v(n) ) v({j}), i N, (3.3) j N are uiquely determied. We ow proceed by iductio similar as i [6], but replacig compoet efficiecy by efficiecy ad compoet balacedess. Cosider graph game v,γ G N L N, ad suppose that we determied the payoffs for every v,γ K N G K L K with Γ < Γ. Efficiecy requires that ξ i (v,γ) = v(n). i N (3.4) Further, for a compoet C N/Γ, compoet balacedess implies that i C (ξ ( ) i(v,γ) ξ i (v C,Γ C )) i N ξi (v,γ) ξ i (v (N/Γ)i,Γ (N/Γ)i =. (3.5) c 5 This idea is similar to Kamijo [4] who itroduced a solutio for games i coalitio structure, i.e., the player set is partitioed ito uios, that allocates to every player its Shapley value i the game restricted to its ow uio ad distributes the Shapley value of its uio i the (quotiet) game betwee the uios equally amog the players i each uio. Cosiderig the associated commuicatio graph, beig the graph where there is a lik betwee ay pair of players i the same uio ad o liks betwee players i differet uios, the uios are exactly the compoets i that commuicatio graph. 7

10 Whe C = N, this is a idetity ad the equatio is redudat. Otherwise, efficiecy requires that for every H N/Γ, ξ i (v H,Γ H ) = v(h). (3.6) i H Usig the equatios (3.4) ad (3.6), equatio (3.5) reduces to ξ i (v,γ) v(c) = c v(n) v(h). (3.7) i C H N/Γ Let m = N/Γ be the umber of compoets. Sice summig up the equatios (3.5) over all compoets yields a idetity, the umber of idepedet equatios (3.7) is m 1. So, equatios (3.4) ad (3.7) yield together m idepedet liear equatios. Next, let Γ be a spaig subforest of Γ, i.e., Γ Γ with Γ = m, ad N/Γ = N/Γ (both forests have the same collectio of compoets). Note that for every lik {i,j} Γ it holds that N/Γ ij > N/Γ (deletig ay lik from Γ icreases the umber of compoets). For every lik {i,j} Γ, by fairess it holds that ξ i (v,γ) ξ i (v,γ ij ) = ξ j (v,γ) ξ j (v,γ ij ). (3.8) Sice Γ ij = Γ 1, for every {i,j} Γ all values ξ h (v,γ ij ), h N, have bee determied by the iductio hypothesis. The (3.8) yields m liearly idepedet equatios. So, together the system of equatios (3.4), (3.7) ad (3.8) yield 1+(m 1)+ ( m) = liearly idepedet equatios i ukow payoffs ξ i (v,γ), i N, ad so all payoffs ξ i (v,γ), i N, are uiquely determied. Noticefromequatio(3.3)thatthesolutioξ dividestheexcessv(n) j N v({j}) equally amog the players whe the graph is empty, ad thus yields the equal surplus divisio solutio. O the other had, whe the graph is complete the solutio ξ gives the Shapley value of v. For the example give i the itroductio we have v({i}) = 0, i = A,B,C, v({a,b}) = 3, v({a,c}) = 2, v({b,c}) = 4, v(n) = 12 ad Γ = {{A,B}}. So N/Γ = {{A,B},{C}}. The Shapley value of v is efficiet ad yields Sh(v) = (3 1 2,41 2,4), ad the Myerso value is compoet efficiet ad yields µ(v,γ) = ( 3, 3,0). The ew solutio is efficiet ad yields ψ(v,γ) = (4 1 2,41,3). Of course, efficiecy requires that the total budget of 12 is allocated. Sice all sigleto worths are zero, fairess implies that the lik betwee A ad B gives them a equal payoff i this example 6. So, we oly eed 6 This follows sice, as metioed before, efficiecy ad compoet balacedess imply equal surplus divisio (every player gets its sigleto worth ad the remaider is equally distributed over all players) for the empty graph. 8

11 to determie the shares i the total budget of C compared to A ad B together. This is doe by compoet balacedess which requires that the total payoff of A ad B together mius their worth (beig equal to 3) is twice the differece betwee the payoff of C ad its worth, implyig that A ad B together get a fractio 2 3 ad C gets a fractio 1 3 of the surplus 12 3 = 9. As ca be see i this example, our solutio favors cooperatio amog players sice the stad aloe player C gets less tha oe third of the budget. The outcome of Casajus value for this example is (3 1 4,41 4,41 ), ad thus the stad aloe player 2 C gets more tha oe third of the budget. This occurs because this solutio favors (o cooperative) stad aloe players. Fially, we show logical idepedece of the axioms of Theorem 3.2. First, the Myerso value is fair ad satisfies compoet balacedess (there is zero excess to divide amog the compoets), but is ot efficiet. Secod, the equal divisio solutio give by ED i (v,γ) = v(n), i N, v,γ N GN L N, N IN, is efficiet ad fair, but is ot compoet balaced. Third, the compoet-wise equal divisio solutio give by CED i (v,γ) = v((n/γ) i) (N/Γ) i + 1 efficiet ad compoet balaced, but ot fair. (v(n) H N/Γ v(h) ), i N, v,γ G N L N, N IN, is Refereces [1] Borm, P., Owe, G., Tijs, S. (1992), O the positio value for commuicatio situatios, SIAM Joural o Discrete Mathemathics 5, [2] Casajus, A. (2007), A efficiet value for TU games with a cooperatio structure, Workig paper, Uiversität Leipzig, Germay. [3] Herigs, P.J.J., va der Laa, G., Talma, A.J.J. (2008), The average tree solutio for cycle-free graph games, Games ad Ecoomic Behavior 62, [4] Kamijo, Y. (2009), A two-step Shapley value for cooperative games with coalitio structures, Iteratioal Game Theory Review, 11, [5] Meesse, R. (1988), Commuicatio games. Master Thesis, Uiversity of Nijmege. [6] Myerso, R.B. (1977), Graphs ad cooperatio i games, Mathematics of Operatios Research 2, [7] Myerso, R.B. (1980), Coferece structures ad fair allocatio rules, Iteratioal Joural of Game Theory 9,

12 [8] Shapley, L.S. (1953), A value for -perso games, i: Tucker AW, Kuh HW (eds.) Cotributios to the theory of games II, Priceto Uiversity Press, Priceto, NJ, pp [9] Slikker, M. (2005), A characterizatio of the positio value, Iteratioal Joural of Game Theory 33,

A New Constructive Proof of Graham's Theorem and More New Classes of Functionally Complete Functions

A New Constructive Proof of Graham's Theorem and More New Classes of Functionally Complete Functions A New Costructive Proof of Graham's Theorem ad More New Classes of Fuctioally Complete Fuctios Azhou Yag, Ph.D. Zhu-qi Lu, Ph.D. Abstract A -valued two-variable truth fuctio is called fuctioally complete,