The Myerson value in terms of the link agent form: a technical note

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1 The Myerson value n terms of the lnk agent form: a techncal note André Casajus (September 2008, ths verson: October 1, 2008, 18:16) Abstract We represent the Myerson (1977) value n terms of the value ntroduced by Vázquez-Brage, García-Jurado and Carreras (1996) appled to the lnk agent form (Casajus, 2007) accompaned by some natural coalton structure. Journal of Economc Lterature Classfcaton Number: C71. Key Words: Myerson value, Owen value, lnk agent form. Char of Economcs and Informaton Systems, HHL Lepzg Graduate School of Management, Jahnallee 59, D Lepzg, Germany. Professur für Mkroökonomk, Wrtschaftswssenschaftlche Fakultät, Unverstät Lepzg, PF , D Lepzg, Germany. e-mal: casajus@wfa.un-lepzg.de 1

2 2 1. Introducton Recently, Casajus (2007) characterzes the poston value (Meessen, 1988; Borm, Owen and Tjs, 1992) for TU games wth a cooperaton structure (undrected graph on the player set) n terms of the Myerson value (Myerson, 1977) of some natural modfcaton of the orgnal game the lnk agent form. In ths note, we show that the Myerson value can be obtaned from the lnk agent form by addng some natural coalton structure (partton of the player set) and applyng the value ntroduced by Vázquez-Brage et al. (1996) for TU games that come wth both a cooperaton structure and a coalton structure. Theplanofthsnotesasfollows: Bascdefntons and notaton are gven n second secton. The thrd secton contans our result. 2. Basc defntons and notaton A (TU) game s a par (N,v) consstng of a non-empty and fnte set of players N and the coalton functon v V (N) : f :2 N R v ( ) 0 ª. Subsets of N are called coaltons, and v (K) s called the worth of coalton K. Avalue s an operator ϕ that assgns payoff vectors to all games, ϕ (N,v) R N. An order of a set N s a bjecton σ : N {1,..., N } wth the nterpretaton that s the σ ()th player n σ. The set of these orders s denoted by Σ (N). The set of players not after n σ s denoted by K {j : σ (j) σ ()}. The margnal contrbuton of n σ s defned as MC v :v (K ) v (K \{}). The Shapley value Sh (Shapley, 1953) s defned by Sh (N,v) : Σ (N) 1 MC v, N. (1) For K N, we denote by ϕ K (N,v, ) the sum P K ϕ (N,v, ). A coalton structure on N s a partton P 2 N of N; a CS-game s a game together wth a coalton structure, (N,v,P). The elements of P are referred to as components; P () denotes the component contanng player.a CS-value s an operator ϕ that assgns payoff vectors to all CS-games, ϕ (N,v, P) R N. For any coalton structure P on N, Σ (N,P) :{σ Σ (N) P P, j P : σ () σ (j) < P } (2)

3 3 s the set of orders on N compatble wth P. The Owen value (Owen, 1977) s gven by Ow (N,v,P) : Σ (N,P) 1 MC v, N. (3) σ Σ(N,P) A cooperaton structure on N s an undrected graph (N,L), L L N : {λ N λ 2}. A typcal element, lnk, ofl s wrtten as λ. Gven any graph (N,L),Nsplts nto (maxmal connected) components whch consttute the partton C (N,L) of N; C (N,L) C (N,L) denotes the component contanng N. The restrcton of L to K N s denoted by L K, L K : {λ L λ K}. (4) A CO-game (communcaton stuaton) (N,v,L) s a game (N,v) together wth a cooperaton structure (N,L); a CO-value s an operator ϕ that assgns payoff vectors to all CO-games, ϕ (N,v,L) R N.The Myerson value μ (Myerson, 1977) s defned by μ (N,v,H) :Sh N,v L, v L (K) : v (S), K N. (5) S C(K,L K ) For any CO-game G (N,v,L), Casajus (2007) defnes ts lnk agent form LAF (G) N,v, as follows: N N N (), N () :{(, λ) λ L } (6a) o L N(), o : {īj j L}, īj : {(, j), (j, j)} (6b) N v K v N K, N K : N N () K 6 ª, K N(6c) Vázquez-Brage et al. (1996) combne the Myerson value and the Owen value nto a soluton concept,, for games (N,v) that come wth both a coalton structure (N,P) and a cooperaton structure (N,L),(N,v,L,P) R N. Ths -value s defned by (N,v,L,P) Ow N,v L, P. (7) 3. Man result Consder a communcaton stuaton Γ (N,v,L), tslnkagentformγ LAF (Γ) N,v,,and the coalton structure P Γ N () N ª.

4 4 Theorem For all N, μ (N,v,L) (,λ) N,v,, P Γ. Proof. Frst observe that any σ Σ N,P Γ unquely nduces some σ N Σ (N) and σ N() Σ N () as follows: If, j N are such that σ (, λ) < σ (j, λ 0 ) for all (, λ) N () and (j, λ 0 ) N (j), thenσ N () < σ N (j); σ N() (, λ) < σ N() (, λ 0 ) ff σ (, λ) < σ (, λ 0 ) for all (, λ), (, λ 0 ) N (). For σ Σ (N) and ρ Σ N (), we set Σ N,P Γ,σ,ρ : σ Σ N,P Γ σ N σ σ N() ρ ª. (8) For Γ (N,v,L), we have (,λ) N,v,, P Γ (7) (3) Ow (,λ) ³ N,v, P Γ Σ N,P Γ 1 Σ N,P Γ 1 (8) Σ N,P Γ 1 Σ N,P Γ 1 Σ N,P Γ 1 Σ N,P Γ 1 σ Σ( N,P Γ ) σ Σ( N,P Γ ) (,λ) (σ) (,λ) (σ) ρ Σ( N()) σ Σ ( N,P Γ,σ,ρ) ρ Σ( N()) σ Σ ( N,P Γ,σ,ρ) ρ Σ( N()) σ Σ ( N,P Γ,σ,ρ) Σ N () Σ N,P Γ,σ,ρ Σ N,P Γ N ()! Q j N,j6 N (j)! Σ (N) Q j N N (j)! (σ N ) (,λ) (σ) ρ Σ( N()) σ Σ ( N,P Γ,σ,ρ) 1

5 5 Σ (N) 1 (1) Sh N,v L (5) μ (N,v,L), where the ffth equaton s shown below, the sxth equaton follows from σ N σ for σ Σ N,P Γ,σ,ρ by (8), and the nnth equaton holds by (2) and (8). The ffth equaton can be seen as follows. Let σ Σ N,P Γ,σ,ρ for N, σ Σ (N) and ρ Σ N ().Wethenhave (,λ) (σ) v K (,λ) (σ) v K (,λ) (σ) \{(, λ)} (8) v (5) (6c) j K (σ N ) j K (σ N ) j K (σ N ) N (j) v N(j), j K (σ N ) N(j), N(j), j K (σ N ) v S N(j), v N S v S N (j) v N S

6 6 (6) S C K (σ N ),L K (σ N ) v (S) S C K (σ N )\{},L K (σ N )\{} v (S) (5) v L (K (σ N )) v L (K (σ N ) \{}) (σ N ). Ths concludes the proof. References Borm, P., Owen, G. and Tjs, S. (1992). On the poston value for communcaton stuatons, SIAM Journal on Dscrete Mathematcs 5: Casajus, A. (2007). The poston value s the Myerson value, n a sense, Internatonal Journal of Game Theory 36(1): Meessen, R. (1988). Communcaton games, Master s thess, Department of Mathematcs, Unversty of Njmegen, the Netherlands. (n Dutch). Myerson, R. B. (1977). Graphs and cooperaton n games, Mathematcs of Operatons Research 2: Owen, G. (1977). Values of games wth a pror unons, n R. Henn and O. Moeschln (eds), Essays n Mathematcal Economcs & Game Theory, Sprnger, Berln et al., pp Shapley, L. S. (1953). A value for n-person games, n H. Kuhn and A. Tucker (eds), Contrbutons to the Theory of Games, Vol. II, Prnceton Unversty Press, Prnceton, pp Vázquez-Brage, M., García-Jurado, I. and Carreras, F. (1996). The Owen value appled to games wth graph-restrcted communcaton, Games and Economc Behavor 12: