Accepted Manuscript. Games of threats. Elon Kohlberg, Abraham Neyman S (17) Reference: YGAME Received date: 30 August 2017

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1 Accepted Manuscrpt Games of threats Elon Kohlberg, Abraham Neyman PII: S (7) DOI: eference: YGAME 277 To appear n: Games and Economc Behavor eceved date: 30 August 207 Please cte ths artcle n press as: Kohlberg, E., Neyman, A. Games of threats. Games Econ. Behav. (207), Ths s a PDF fle of an unedted manuscrpt that has been accepted for publcaton. As a servce to our customers we are provdng ths early verson of the manuscrpt. The manuscrpt wll undergo copyedtng, typesettng, and revew of the resultng proof before t s publshed n ts fnal form. Please note that durng the producton process errors may be dscovered whch could affect the content, and all legal dsclamers that apply to the journal pertan.

2 GAMES OF THEATS ELON KOHLBEG AND ABAHAM NEYMAN + Abstract. Agameofthreatsonafntesetofplayers,N, s a functon d that assgns a real number to any coalton, S N, such that d(s) = d(n \S). A game of threats s not necessarly a coaltonal game as t may fal to satsfy the condton d( ) =0. We show that analogs of the classc Shapley axoms for coaltonal games determne a unque value for games of threats. Ths value assgns to each player an average of d(s) acrossall the coaltons that nclude the player. Games of threats arse naturally n value theory for strategc games, and may have applcatons n other branches of game theory.. Introducton The Shapley value s the most wdely studed soluton concept of cooperatve game theory. It s defned on coaltonal games, whch are the standard objects of the theory. A coaltonal game on a fnte set of players, N, s a functon v that assgns a real number to any subset ( coalton ), S N, such that v( ) = 0. The amount v(s) may be nterpreted as the worth of S,.e., what the players belongng to S can jontly get by coordnatng ther efforts. A value s a functon that assgns to each coaltonal game a vector of payoffs, one for each player, that reflects the a pror evaluaton of each player s poston n the game. In hs celebrated paper [4] Shapley proposed four desrable propertes ( axoms ) and proved the remarkable result that there exsts a unque functon satsfyng these propertes. Ths functon, the Shapley value, can be descrbed as follows. The value of player s an average of the margnal contrbutons, v(s ) v(s), of player, where the average s taken over all the orderngs of N, wths denotng the subset of players that precede n the orderng. A game of threats s a functon d that assgns a real number to any coalton, S N, such that d(s) = d(n \ S). The amount d(s) may be nterpreted as the threat power of the coalton S,.e., the maxmal dfference between the total amounts that the players belongng to S and the players belongng to N\S receve, when the players n S coordnate ther efforts to maxmze ths dfference and the players n N \S coordnate ther efforts to mnmze t. Games of threats arse naturally n value theory for strategc games [2]. Date: October 23, 207. Harvard Busness School, Harvard Unversty; ekohlberg@hbs.edu. + Insttute of Mathematcs, and the Federmann Center for the Study of atonalty, The Hebrew Unversty of Jerusalem, Gvat am, Jerusalem 9904, Israel; aneyman@math.huj.ac.l.

3 2 ELON KOHLBEG AND ABAHAM NEYMAN There, the condton d(s) = d(n \S) s a consequence of the mnmax theorem; see the Appendx. Now, games of threats need not be coaltonal games, as they may fal to satsfy the condton d( ) = 0. Thus, f we wsh to obtan an a pror evaluaton for games of threats then we must develop a value theory for such games. Ths paper does that. We show that there s a unque functon, from games of threats to n-dmensonal payoff vectors, that satsfes the analogs of Shapley s four axoms; and that ths functon can be descrbed as follows. The value of a player s the average of the threat powers, d(s), of the coaltons that nclude the player. Specfcally, f d,k denotes the average of d(s) overallk-player coaltons that nclude, then the value of player s the average of d,k over k =, 2,...,n. We take three approaches: a dervaton of the results from classc Shapley value theory for coaltonal games, a drect dervaton based on the formula, and a drect dervaton based on the random-order approach. Ths last dervaton establshes a formula for the Shapley value of games of threats that s analogous to the formula for coaltonal games: the value of player s the average of d(s ), where the average s taken over all the orderngs of N, wths denotng the subset of players that precede n the orderng. We end ths ntroducton by notng that the exstence and unqueness of the Shapley value for games of threats s an essental component of the proof of the exstence and unqueness of a value for strategc games [2]. 2. Games of Threats - Defnton A coaltonal game of threats s a par (N,d), where N = {,...,n} s a fnte set of players. d: 2 N s a functon such that d(s) = d(n\s) for all S N. Example: N = {, 2, 3}. d( ) =, d() = d(2) =, d(3) = 0, d(, 2) = 0, d(, 3) = d(2, 3) =, d(, 2, 3) =. Denote by D(N) the set of all coaltonal games of threats. By choosng, for every S N, ethers or N\S, we can descrbe any d D(N) bymeansof2 n numbers, thereby dentfyng D(N) wth 2n. One convenent choce s (d(s)) S. Wth ths choce, the above example s descrbed as follows. d() =, d(, 2) = 0, d(, 3) =, d(, 2, 3) =. 3. The Shapley Value Let ψ : D(N) n be a map that assocates wth each game of threats an allocaton of payoffs to the players. Followng Shapley [4], we consder the followng axoms. For all games of threats (N,d), (N,e), for all players, j, andforallrealnumbersα, β (and usng the notaton ψ d for ψ(d)()), the followng propertes hold:

4 GAMES OF THEATS 3 Effcency N ψ d = d(n). Lnearty ψ(αd + βe) =αψd + βψe. Symmetry ψ (d) =ψ j (d) fand j are nterchangeable n d (.e., f d(s ) = d(s j) S N\{, j}). Null player ψ d =0f s a null player n d (.e., f d(s ) =d(s) S N). Defnton. Amapψ : D(N) n satsfyng the above axoms s called a value. Theorem. There exsts a unque value for games of threats. follows: () ψ d = n d,k, k= It may be descrbed as where d,k denotes the average of d(s) over all k-player coaltons that nclude. In the example, ψ = ψ 2 = 3 ( ) = 6, ψ 3 = = 2 3. Note: Formula () allocates to each player a weghted average of the d(s) overthe coaltons S that nclude that player. The weght s the same for all coaltons of the same sze but dfferent for coaltons of a dfferent sze. Specfcally, for each k =,...,n,the total weght of n s dvded among the ( n k ) coaltons of sze k that nclude : (2) ψ d = n k= ( n k ) d(s) = n S: S S =k S: S ( n )d(s). S Note: Henceforth we shall refer to the map of equaton as the Shapley value for games of threats. 4. Dervaton from classc Shapley value theory Let V(N) :={v : 2 N, v( ) =0} be the set of standard coaltonal games on N. It can be dentfed wth a subspace of 2n of dmenson 2 n. Let K : V(N) D(N) andl: D(N) V(N) be defned by and (Kv)(S) =v(s) v(n\s) (Ld)(S) = 2 d(s)+ 2 d(n). In the Shapley-value lterature, such players are called substtutes rather than nterchangeable. We adopt the latter term n order to avod potental confusons that may arse from the standard meanng of the term substtutes n economcs.

5 4 ELON KOHLBEG AND ABAHAM NEYMAN Note that the map L s lnear, effcent (.e., L(d)(N) =d(n)), symmetrc (.e., f and j are nterchangeable n d D(N) then and j are nterchangeable n Ld V(N)), and preserves null players (.e., f s a null player n d D(N) then s a null player n Ld V(N)). Thus, f ϕ s a value on V(N), then ϕ L s a value on D(N). Note that K L s the dentty on D(N); therefore n partcular K s surjectve. It follows that, f ψ and ψ 2 are two dfferent values on D(N), then ψ K and ψ 2 K are two dfferent values on V(N). Thus f the value on V(N) s unque then the value on D(N) s unque. As the Shapley value ϕ s the unque value on V(N), ϕ L s the unque value on D(N). Ths completes the proof of the exstence and unqueness of a value on D(N). In order to show that the map ψ of Theorem s, n fact, the Shapley value on D(N), t suffces to show that ψ = ϕ L. ecall that the Shapley value ϕ on V(N) s defned as follows. If v V(N) then (3) ϕ v := (v(p ) v(p )), where the summaton s over the orderngs of the set N and where P denotes the subset of those j N that precede n the orderng. Let d be a game of threats. Then, (Ld)(P P 2 d(p ) (Ld)(P ) = 2 d(p ) 2 d(p ). The complement of P s, where denotes the reverse orderng of. As d s a game of threats, we have )= 2 d(p ). Therefore, ϕ (Ld) = ( 2 d(p )+ d(p )). 2 Snce the set of reverse orderngs s the same as the set of orderngs, ϕ (Ld) = d(p ). But (4) Thus d(p ) = n k= (n )! ϕ (Ld) = n whch completes the proof that ψ = ϕ L. : P =k d,k, k= d(p ) = n d,k. k=

6 GAMES OF THEATS 5 emark: The map K s lnear, effcent, symmetrc, and preserves null players. Thus, f ψ s a value on D(N) thenψ K s a value on V(N). In partcular, the exstence of a value on D(N) mples the exstence of a value on V(N)). Young [6] showed that the exstence and unqueness theorem for the Shapley value n V(N) remans vald when the axoms of lnearty and null player are replaced by an axom of margnalty, whch requres that the value of a player n a game v depend only on the player s margnal contrbutons, v(s ) v(s). Now, the map L preserves margnalty (.e., f the margnal contrbutons of player are the same n two games d,d 2 D(N) then the margnal contrbutons are the same n Ld,Ld 2 V(N)); therefore the same argument as above mples the followng. Corollary. The Shapley value s the unque mappng ψ : D(N) n satsfyng the axoms of effcency, symmetry, and margnalty. emark: ThemapK also satsfes margnalty. Thus the exstence of a value satsfyng Young s axoms on D(N) mples the exstence of such a value on V(N)). We end ths secton by notng that the results for games of threats are also vald for constant-sum games. Let C(N) :={v : 2 N, v(s) +v(n\s) =v(n) for all S N} be the set of constant-sum coaltonal games on N. It can be vewed as a subspace of 2n of dmenson 2 n. Let K : V(N) C(N) andl : C(N) V(N) be defned by and (K v)(s) = (v(s) v(n\s)+v(n)) 2 (L c)(s) =c(s). Then both L and K are lnear, effcent, symmetrc, preserve null players, and preserve margnalty, and K L s the dentty on C(N). Ths mples the exstence and unqueness of a value on C(N) n exactly the same way that the parallel statement for D(N) mpled the exstence and unqueness of a value on D(N). Corollary 2. The Shapley value s the unque mappng ψ : C(N) n satsfyng the axoms of lnearty, effcency, symmetry, and null player, as well as the unque such mappng satsfyng the axoms of effcency, symmetry, and margnalty. 5. Drect dervaton from the formula of Theorem Defnton 2. Let T N, T. The unanmty game, u T D(N), s defned by

7 6 ELON KOHLBEG AND ABAHAM NEYMAN T f S T, u T (S) = T f S N\T, 0 otherwse. Proposton 2. The unanmty games span D(N). Proof. It s suffcent to show that the 2 n games (u T ) T are lnearly ndependent. Suppose, then, that a j u Tj = 0, where T T j for j and not all the a j are zero. Snce, for j, T T j and T T j {}, nether set s contaned n the other s complement and therefore { Tj f T (5) u Tj (T )= T j, 0 otherwse. Among the T j for whch the coeffcent a j s non-zero choose one, say T, wth a mnmum number of players. Then for any j>, T T j and therefore, by (5), u Tj (T ) = 0. Thus 0= a j u Tj (T )=a u T (T )=a T 0, a contradcton. Proof of Theorem. In the unanmty game u T,all/ T are null players and all T are nterchangeable. It follows that any map that satsfes the effcency, symmetry, and null player axoms, s unquely determned on u T : { for T, (6) ψ u T = 0 for T. If the map also satsfes lnearty then by Proposton 2 t s determned on all of D(N). Ths establshes unqueness. It s easy to verfy that the map ψ defned n () satsfes lnearty and symmetry. To verfy that t satsfes the null player axom, proceed as follows. Consder the basc condton d(s) = d(n \S). As S ranges over all sets of sze k that nclude, N\S ranges over all sets of sze n k that do not nclude. Averagng over all these sets, we have (7) d,k = d,n k, where d,k denotes the average of d(s) overallk-player sets that do not nclude. Now, f s a null player then d(s) =d(s ) for any set S that does not nclude. Takng the average over all such sets S of sze n k, wehaved,n k = d,n k+, whch combned wth (7) yelds d,k = d,n k+. Summng over k =,...,n and dvdng by n yelds ψ d = ψ d.thusψ d =0. It remans to prove effcency, namely, that (8) ψ d = d(n). =

8 GAMES OF THEATS 7 Let D k denote the average of the d(s) overallk-player coaltons. Snce d,k s the average of all the d(s) wth S and S = k, t follows by symmetry that the average of d,k over =,...,n s D k.thus ψ d = = = n d,k = k= k= n d,k = = D k. Note that there s just one n-player coalton, namely, N, sod N = d(n). Thus, to prove (8), we must show that n D k =0. But ths follows from the fact that k= D k = D n k for k =,...,n, whch n turn follows from the basc condton d(s) = d(n\s), as seen by notng that as we take an average of the left sde over all the sets of sze k, the rght sde s averaged over all the sets of sze n k. Note: The theorem remans vald when the axom of lnearty s replaced by the weaker axom of addtvty. The proof s the same, wth the addtonal observaton that effcency, symmetry, and the null player axoms determne the value not only on the unanmty games but also on any multple, cu T, of such games. k= 6. Drect dervaton by the andom Order Approach In ths secton we provde an alternatve formulaton of Theorem that s analogous to Shapley s classc formulaton. Proposton 3. There s a unque value for games of threats. follows: (9) ψ d = d(p ), It may be descrbed as where the summaton s over the possble orderngs of the set N and where P denotes the subset of those j N that precede n the orderng. d(p Proof. Equaton (4) establshes that ) = n n k= d,k. Therefore, by Theorem, the map ψ defned n (9) s the unque Shapley value for games of threats. In the rest of ths secton we present a drect proof of the proposton, wthout relyng on Theorem. The proof of unqueness s the same as before. Frst, observe that the unanmty games span all games of threats and that the axoms unquely determne the value for the unanmty games; then apply lnearty.

9 8 ELON KOHLBEG AND ABAHAM NEYMAN Next, we must prove that the map ψ of equaton (9) satsfes Shapley s four axoms. Lnearty and symmetry are easy to verfy. To prove effcency, proceed as follows. Let Sj be the set of the frst j elements n the orderng. Then, for every j, d(s j )=D j. As shown earler, D j + D n j =0for j<n. Therefore, ψ d = d(p ) = d(p ) = = = = d(sj )= D j = D n = d(n). j= Next, we must prove that the map ψ of equaton (9) satsfes Shapley s four axoms. Lnearty and symmetry are easy to verfy. To prove effcency, proceed as follows. Let Sj be the set of the frst j elements n the orderng. Then, for every j, d(s j )=D j. As shown earler, D j + D n j =0for j<n. Therefore, ψ d = d(p ) = (0) d(p ) = = = = () d(sj )= D j = D n = d(n). j= It remans to verfy the null player axom. If s a null player then d(p ) = d(p )= d(p ). As ranges over all the orderngs, so does. Therefore, ψ d = whch mples that ψ d =0. d(p ) = j= j= d(p ) = ψ d, Appendx A. The value of strategc games A strategc game s a trple G =(N, A, g), where N = {,...,n} s a fnte set of players, A s the fnte set of player s pure strateges, and g =(g ) N, where g : S A s player s payoff functon. Denote by X S the probablty dstrbutons on S A ; these are the correlated strateges of the players n S. Let G G(N). Defne the threat power of coalton S as follows: 2 (2) (δg)(s) := max x X S mn y X N\S g (x, y) g (x, y). S S 2 Expressons of the form max or mn over the empty set are gnored.

10 GAMES OF THEATS 9 By the mnmax theorem, (δg)(s) = (δg)(n\s) for any S N. Thus δg s a game of threats. We defne the value, γg, of the strategc game G by takng the Shapley value of the game of threats δg. In [2] t s proved that γ s the unque functon that satsfes fve propertes that are desrable n a map provdng an a pror evaluaton of the poston of each player n a strategc game. Four of these propertes are analogs of the Shapley axoms for the value of cooperatve games. Formula () then mples the followng: the value of a player n an n-person strategc game G s an average of the threat powers, (δg)(s), of the subsets of whch the player s a member. Specfcally, f δ,k denotes the average of (δg)(s) overallk-player coaltons that nclude, then the value of player s the average of δ,k over k =, 2,...,n. emark: The game of threats assocated wth a strategc game G, as defned n (2), s dfferent from von Neumann and Morgenstern s orgnal coaltonal game, whch s defned as follows: (vg)(s) := max mn g (x, y). x X S y X N\S As was ponted out by von Neumann and Morgenstern [5], Shapley [3], Harsany [], and others, ths defnton has some defcences. In the context of the Shapley value, the shft to defnton (2), whch was frst proposed by Harsany [], has proven successful n addressng these defcences. We wonder whether, n the context of other cooperatve soluton concepts, e.g., the stable set or the core, the defcences could smlarly be addressed by a shft to games of threats or to other varants of the von Neumann Morgenstern coaltonal game. eferences [] Harsany, J. (963), A Smplfed Barganng Model for the n-person Cooperatve Game, Internatonal Economc evew, 4, [2] Kohlberg, E. and A. Neyman (207), Cooperatve Strategc Games, Federmann Center for the Study of atonalty dscusson paper 706. [3] Shapley, L. (95), Notes on the n-person Game II: the Value of an n-person Game, AND research memorandum ATI [4] Shapley, L. (953), A Value for n-person Games. In Kuhn, H.W. and Tucker, A.W. (eds.), Contrbutons to the Theory of Games, Annals of Mathematcs Studes, 28, pp [5] von Neumann, J. and O. Morgenstern (953), Theory of Games and Economc Behavor, thrd edton, Prnceton Unversty Press. [6] Young H.P. (985), Monotonc Solutons of Cooperatve Games, Internatonal Journal of Game Theory, 4, S