Games of Threats. Elon Kohlberg Abraham Neyman. Working Paper

Games of Threats Elon Kohlberg Abraham Neyman Workng Paper 18-023

Games of Threats Elon Kohlberg Harvard Busness School Abraham Neyman The Hebrew Unversty of Jerusalem Workng Paper 18-023 Copyrght 2017 by Elon Kohlberg and Abraham Neyman Workng papers are n draft form. Ths workng paper s dstrbuted for purposes of comment and dscusson only. It may not be reproduced wthout permsson of the copyrght holder. Copes of workng papers are avalable from the author.

GAMES OF THREATS ELON KOHLBERG AND ABRAHAM NEYMAN + Abstract. A game of threats on a fnte set of players, 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 the threat powers, d(s), of the coaltons that nclude the player. 1. 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 [2] 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 random orders of N, wth S denotng the subset of players that precede n the random order. 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 [1]. There, the condton d(s) = d(n \S) s a consequence of the mnmax theorem; see the Appendx. Date: September 19, 2017. Harvard Busness School, Harvard Unversty; ekohlberg@hbs.edu. + Insttute of Mathematcs, and the Federmann Center for the Study of Ratonalty, The Hebrew Unversty of Jerusalem, Gvat Ram, Jerusalem 91904, Israel; aneyman@math.huj.ac.l. 1

2 ELON KOHLBERG AND ABRAHAM NEYMAN 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) over all k-player coaltons that nclude, then the value of player s the average of d,k over k = 1, 2,..., n. We take three approaches: a dervaton of the results from the 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 random orders of N, wth S denotng the subset of players that precede n the random order. We end ths ntroducton by notng that the exstence and unqueness of the Shapley value for games of threats s an essental component n the proof of exstence and unqueness of a value for strategc games [1]. 2. Games of Threats - Defnton A coaltonal game of threats s a par (N, d), where N = {1,..., n} s a fnte set of players. d: 2 N R s a functon such that d(s) = d(n\s) for all S N. Example: N = {1, 2, 3}. d( ) = 1, d(1) = d(2) = 1, d(3) = 0, d(1, 2) = 0, d(1, 3) = d(2, 3) = 1, d(1, 2, 3) = 1. Denote by D(N) the set of all coaltonal games of threats. By choosng, for every S N, ether S or N\S, we can descrbe any d D(N) by means of 2 n 1 numbers, thereby dentfyng D(N) wth R 2n 1. One convenent choce s (d(s)) S 1. Wth ths choce, the above example s descrbed as follows. d(1) = 1, d(1, 2) = 0, d(1, 3) = 1, d(1, 2, 3) = 1. 3. The Shapley Value Let ψ : D(N) R n be a map that assocates wth each game of threats an allocaton of payoffs to the players. Followng Shapley [2], we consder the followng axoms. For all games of threats (N, d), (N, e), for all players, j, and for all real numbers α, β (and usng the notaton ψ d for ψ(d)()), the followng propertes hold: Effcency N ψ d = d(n).

GAMES OF THREATS 3 Lnearty ψ(αd + βe) = αψd + βψe. Symmetry ψ (d) = ψ j (d) f and j are substtutes n d (.e., f d(s ) = d(s j) S N\{, j}). Null player ψ d = 0 f s a null player n d (.e., f d(s ) = d(s) S N). Defnton 1. A map ψ : D(N) R n satsfyng the above axoms s called a value. Theorem 1. There exsts a unque value for games of threats. follows: (1) ψ d = 1 n d,k, It may be descrbed as where d,k denotes the average of the d(s) over all k-player coaltons that nclude. In the example, ψ 1 = ψ 2 = 1 3 ( 1)+ 1 3 1+0 2 + 1 3 1 = 1 6, ψ 3 = 1 3 0+ 1 3 1+ 1 3 1 = 2 3. Note: Formula (1) allocates to each player a weghted average of the d(s) over the 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 = 1,..., n, the total weght of 1 n s dvded among the ( n 1 k 1) coaltons of sze k that nclude : (2) ψ d = 1 n 1 ( n 1 k 1) d(s) = 1 n S: S S =k S: S 1 ( n 1 )d(s). S 1 Note: Henceforth we shall refer to the map of equaton 1 as the Shapley value for games of threats. 4. Dervaton from classc Shapley value theory Let V(N) := {v : 2 N R, v( ) = 0} be the set of standard coaltonal games on N. It can be dentfed wth a subspace of R 2n of dmenson 2 n 1. Let K : V(N) D(N) and L: D(N) V(N) be defned by and (Kv)(S) = v(s) v(n\s) (Ld)(S) = 1 2 d(s) + 1 2 d(n). Note that the map L s lnear, effcent (.e., L(d)(N) = d(n)), symmetrc (.e., f and j are substtutes n d D(N) then and j are substtutes 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)).

4 ELON KOHLBERG AND ABRAHAM NEYMAN 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 1 and ψ 2 are two dfferent values on D(N), then ψ 1 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 1 s, n fact, the Shapley value on D(N), t suffces to show that ψ = ϕ L. Recall that the Shapley value ϕ on V(N) s defned as follows. If v V(N) then (3) ϕ v := 1 n! R (v(p R ) v(p R )), where the summaton s over the n! possble orderngs of the set N and where P R denotes the subset of those j N that precede n the orderng R. Denote S = P R. In the orderng R, the margnal contrbuton of player s v(s) v(s ). In the reverse orderng the margnal contrbuton s (v(n \ S ) (v(n \ S)). Snce the set of reverse orderngs s the same as the set of orderngs, we may replace each summand n the r.h.s. of (3) by the average margnal contrbutons n the orderng and ts reverse. Thus ϕ v = 1 ( 1 ((v(s) v(n \ S)) + (v(n \ S ) v(s ))). n! 2 S R:P R =S But as S ranges over the subsets of N that nclude, so does N \ S. Thus we have ϕ v = 1 (s 1)!(n s)!(v(s) v(n \ S)). n! S Recall that v(s) = (Ld)(S) = 1 2 d(s) + 1 2d(N), and so v(s) v(n \ S) = d(s). Therefore the above equaton can be rewrtten as ϕ v = 1 n (k 1)!(n k)! (n 1)! S, S =k d(s) = 1 n d,k, whch s formula (1). Thus (ϕ L)d = ϕ(ld) = ψd. Remark: The map K s lnear, effcent, symmetrc, and preserves null players. Thus, f ψ s a value on D(N) then ψ 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 [3] 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.,

GAMES OF THREATS 5 f the margnal contrbutons of player are the same n two games d 1, d 2 D(N) then the margnal contrbutons are the same n Ld 1, Ld 2 V(N)); therefore the same argument as above mples the followng. Corollary 1. The Shapley value s the unque mappng ψ : D(N) R n satsfyng the axoms of effcency, symmetry, and margnalty. Remark: The map K 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 R, 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 R 2n of dmenson 2 n 1. Let K : V(N) C(N) and L : C(N) V(N) be defned by and (K v)(s) = 1 (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). Therefore, Corollary 2. The Shapley value s the unque mappng ψ : C(N) R 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 1 Defnton 2. Let T N, T. The unanmty game, u T D(N), s defned by 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 1 games (u T ) T 1 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.

6 ELON KOHLBERG AND ABRAHAM NEYMAN Snce, for j, T T j and T T j {1}, nether set s contaned n the other s complement and therefore { 1 f T T (4) u Tj (T ) = j, 0 otherwse. Among the T j for whch the coeffcent a j s non-zero choose one, say T 1, wth a mnmum number of players. Then for any j > 1, T 1 T j and therefore by (4), u Tj (T 1 ) = 0. Proof of Theorem 1. In the unanmty game u T, all / T are null players and all T are substtutes. It follows that any map that satsfes the effcency, symmetry, and null player axoms, s unquely determned on u T : { 1 (5) ψ u T = T for 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 (1) 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 (6) d,k = d,n k, where d,k denotes the average of d(s) over all k-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, we have d,n k = d,n k+1, whch combned wth (6) yelds d,k = d,(n k+1). Summng over k = 1,..., n and dvdng by n yelds ψ d = ψ d. Thus ψ d = 0. It remans to prove effcency, namely, that (7) ψ d = d(n). =1 Let D k denote the average of the d(s) over all k-player coaltons. Snce d,k s the average of all the d(s) wth S and S = k, t follows that by symmetry the average of d,k over = 1,..., n s D k. Thus ψ d = =1 =1 1 n d,k = 1 n d,k = =1 D k.

GAMES OF THREATS 7 Note that there s just one n-player coalton, namely, N, so d N = d(n). Thus, to prove (7), we must show that But ths follows from the fact that n 1 D k = 0. D k = D n k for k = 1,..., n 1, 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 axom determne the value not only on the unanmty games but also on any multple, cu T, of such games. 6. Drect dervaton by the Random Order Approach In ths secton we provde an alternatve defnton of the Shapley value for games of threats that s analogous to the classc defnton: Proposton 3. (8) ψ d = 1 n! R d(p R ), where the summaton s over the n! possble orderngs of the set N and where P R the subset of those j N that precede n the orderng R. denotes Proof. The equaton says that ψ d s the average of the d(s) over the sets S that nclude, weghted accordng to the frequency of those random orderngs where the players n S appears frst, wth last among them. (9) The number of such orderngs s ( S 1)!(n S )!. Thus ψ d = 1 n! = 1 n! ( S 1)!(n S )! d(s) S: S (k 1)!(n k)! d(s), S: S S =k whch s the same as (2).

8 ELON KOHLBERG AND ABRAHAM NEYMAN Note that equaton (8) may be rewrtten as follows: (10) (11) ψ d = = 1/n! R = 1/n! R d(p R ) d(p R ), where R denotes the reverse orderng of R. To see (10), observe that P R s the complement of P R, hence d(p R ) = d(p R ). To see (11), observe that the map, R R, from an orderng to ts reverse, s a surjectve and njectve map on the space of orderngs. (12) Note further that equaton (9) can be rewrtten as follows: ψ d = 1 S!(n S 1)! d(s) n! = 1 n! S: S n 1 k=0 S: S S =k k!(n k 1)! d(s). Ths follows from equaton (10), whch says that ψ d s mnus the average of the d(s) over the sets S that exclude, weghted accordng to the frequency of those random orderngs where the players n S appear frst, wth mmedately followng them. Next, we provde an alternatve proof of Theorem 1 that s based on the random order approach. Let ψ be the map defned by (8). It s easy to verfy that ψ satsfes lnearty and symmetry. To prove effcency, proceed as follows. If two games satsfy the effcency condton, (7), then so does any lnear combnaton of them. By Proposton 2, then, t s suffcent to prove effcency on the unanmty games. Consder the unanmty game u T and let T. Then u T (P R ) = 1 f (P R ) T ;.e., f s the last to appear among the members of T, and zero otherwse. Clearly, the probablty that n a random orderng s last among T players s 1 T. Thus ψ u T = 1 T. Smlarly, f T, then u T (P R ) = 1 f all members of T appear before, 1 f all members of T appear after, and zero otherwse. Clearly, the number of orderngs n whch appears after all the members of T s the same as the number of orderngs n whch appears before all the members of T (thnk of reversng the orderngs), and hence ψ u T = 0. Ths establshes (5), whch mples that n =1 ψ d = T 1 T = 1. It remans to verfy the null player axom.

If s a null player then d(p R ) = d(p R random orderngs, so does R. Therefore, ψ d = 1 d(p R ) = 1 n! n! whch mples that ψ d = 0. R GAMES OF THREATS 9 ) = d(p R ). As R ranges over all the d(p R ) = ψ d, R Appendx A. The value of strategc games A strategc game s a trple G = (N, A, g), where N = {1,..., n} s a fnte set of players, A s the fnte set of player s pure strateges, g = (g ) N, where g : S A R 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: 1 (δg)(s) := max x X S mn y X N\S S g (x, y) S g (x, y). 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 [1] t s proved that γ s the unque map 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 (1) 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) over all k-player coaltons that nclude, then the value of player s the average of δ,k over k = 1, 2,..., n. References [1] Kohlberg, E. and A. Neyman (2017), Cooperatve Strategc Games, Federmann Center for the Study of Ratonalty dscusson paper 706. [2] Shapley, L. (1953), 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. 307 319. [3] Young H.P. (1985), Cost allocaton n far allocaton. In Young, H.O. (ed.), Proceedngs of Symposa n Appled Mathematcs, 33, Amercan Mathematcal Socety, pp. 60 94. 1 Expressons of the form max or mn over the empty set are gnored.