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1 '" I " " "" ll '" I'" ' " P"-' I II -«...-.,. ' M»^..l^..l I! Ilium llljlll '^' '^'^ " " '!»"I AD SYMMETRIC SIMPLE GAMES Robert James Weber Cornell University Prepared for: Office of Naval Research February 1973 DISTRIBUTED BY: [m National Technical Information Service U. S. DEPARTMENT OF COMMERCE 5285 Port Royal Road. Springfield Va
2 ' "" pir- ' " "^ '"» ^- P 'I'""!. I i.l.l : > I «i Mi ' ' ' " ' '"......,.,.. -,.;.. * *,.< i-r-vi T^«rf I I ^ 4 1»0 4 f: CO r t» 4 - m Ir t» k Q «fi DEPARTMENT OF OPERATIONS RESEARCH :: ir. r. o o o 0 NATIONAL TECHNICAL INFORMATION SERVICE II S I mi..in.er. I f,,(, Sli.'.gl 1,1 VA 2V1 51 COLLEGE OF ENGINEERING CORNELL UNIVERSITY D D CX, I ITHACA, NEW YORK Urn m n rtmrämiimti' r^.., %
3 wm """ " ' '" ' - mmmmqm jjnaa s.ifii:,d S» 1! Uritx ( I i iti DOCUMENT CONTKOL DATA K & D iniltllv I t.f,.itll.ill'hi <>l llll,-. hn ll I.,' I.il.ui I llul,,11, ,!,. n, ' I, ±,, I r,,1.< I.., I h.,11 I, I,11,, I.,,11,,/) 3^1 OIN A r I N., A (, T i, i 1 *» I - rjmif.if f,,,ift,,,t j Cornell University Department of Operations Research Ithaca, New York J NtHOMT TITLt :.i. HI > o " i s t t ).I i UNCLASSIFIED 2t Cf'OUF' SYMMETRIC SIMPLE GAMES 4. UC SC AlP T I vt NOTtS (Type i>l tvpufl.'ml, in< h,, i \ r Unites) Technical Report 5 *U T MOMi4l ('/ 'ffw njrrv, nm.'ju' tntttut, Ut. f n.,ir- Robert J. Weber g REMOHT D»ir February 1973 I«. COHTMACT OM Cf<Af. T NO N A PHOJEC T NO Ja. TOTAL NO Of f' A G r?b. HO O t fit F Va. O fllclh A T OFi'S» t t'o ^ T ". vjm HI fm5 Technical Report No. 173 this rrpor!) 10. DISTRtOUTION SfATCMENT This document has been approved for publ.'c release and sale; its distribution is unlimited. I) SUCFLI.ML-MAM* N1TL5 W SF' On SO F^ I N L f.*l I J AO^'MACT Operations Research Program Office of Naval Research Arlinpton, Vir^inia A symmetric simple game is an n-person game in which the winninc, coalitions are the coalitions of at least k players. All von Ncumann-Morgenstern stable sets, in which (n-k) players are discriminated, are characterized for these games. DD,"",1473 " '" l S/N aot-obi I ' JL UNCLASSIFIED S**curits t l.i.-.iin.iti'mi All 10-^ mi * m ~wmmmh i mi
4 ,p ~ HIM»pnniiwil'-' nw immippi mi H J)NCLASSIfI D Soc'.r'lv CliisMfu..tion K C Y V O P< U» LINK A HO L t «T L in K II NOLI ** T ^ \u *. C F. O l I rt T game theory cooperative games characteristic function games simple games symmetric games majority games stable sets von Neumann-Morgenstern solutions discriminatory solutions DD. F r.,m73 S/N loj-tl? I BACK) ii^ai IIIII^M mmtmn f Mill J ß UNCLASSIFinD Security Cl.i'.'.if M.it ion
5 - "" II.,,.. ""W"""""" '- DEPARTMENT OF OPERATIONS RESEARCH COLLEGE OF ENGINEERING CORNELL UNIVERSITY ITHACA, NEW YORK TECHNICAL REPORT NO. 173 February 1973 SYKWETRIC SIMPLE GAMES by Robert Janes Weber D D CV M/ ' This research was supported in part by the Office of Naval Research under Contract Number N A and by the National Science Foundation under Grants GK and GP 32314x. Reproduction in Whole or in Ptrt is Permitted for any Purpose of the United States Government. Jit ~*.* ma - i -
6 "' -~ 11 I K f / A question of interest in the study of n-person games concerns conditions under which a set of players of a game may face discrimination which, in a sense, excludes them from the bargaining process. In [2, 4, 5], discriminatory solutions are given for several classes of games. In this paper, a characterization is given of all discriminatory solutions for n-person games in which all coalitions with at least k players win, and all coalitions with less than k players lose. The symmetric solutions of these games were given in (1J for the case k > j. An n-person game is a function v from the coalitions (subsets) of a set of players N {l,2,...,n} to the reals satisfying v(0) v({i})» 0 for all i t N 0 < v(s) < 1 for all S CN v(n) = 1. It is assumed throughout that n ^ 3. For any real vector x C R, define - S x(s) i x. and define x as the restriction of x to the coordinates in 1 its S. Let X {x R n : x(n) - 1, x. ^ 0 for all i N}. If x,y X, then y dominates x with respect to a non-empty coali'lon S, S S written y dom-x, if y > x and y(s) < v(s). For A c X, define dorn A» {x X: y dom_x for some S c N, y A). A solution, or von Noumann-Morgenstein»table set, is any sot K <- X satisfying K Odom K - X, (1) K fl dorn K - 0. (2)
7 mmmm^mmmi^m^*mmmmmmmmm^wmmmmmimi^*mm^>* "- ' "w*^p»w^wiw»wuwäii^ww»^--...«. -"^npihmmbp^^^j ^ fcii mmmmmmmmw i i 2 Any set K satisfying (1) is said to be externally stable; any set satisfying (2) is internally stable. Motivation for these definitions is given in (3). A symmetric simple game, or (n,k)-game, is an n-person game satisfying v(s) - 0 if S < k v(s) =1 if S >_ k, where S denotes the number of players in the coalition S. Clearly domina- tion in this game can occur only with respect to coalitions of at least k players. A p-discrirripatory solution is a solution D(a....a. ;i,,...,i ) - {x X: x. a. for all 1 1 k <_ p). 1 p 1 p x k k The main result of this paper is a characterization of all m-discriminatory solutions for the (n,n-in)-game. For m < j, let M c N be a set of m players, and let P = N-M. Also let a be a non-nogative m-vector, and write K(a) for D(a;M). Theorem. K(a) is an m-discriminatory solution of the (n,n-m)-game if and only if a(m) {n-m-l)a ; < 1 (3) for all i M. The proof follows a sequence of lemmas. L-.m^a 1. For any a, K(a) is internally stable. Proof. Assume on the contrary that x,y i K(a) and y dom_x. Since y M» a» x M and S > n-m, it follows that S =» P. However, y(m) = x(m) and y(n) «x(n) imply y(p) x(p) and therefore y. < x i for some r ^ P, a contradiction.
8 ^^ ^^^^^^^^^^^^»..., i.,-,.,^..,,,...,,,.,...,11,i iii i iipmii I», ii,,,...,. i..., ".-^ i'.-.i wnm - MMMMMiaMMMi^.^MMBa^ Lemma 2. Suppose x i K(a). Then x dorn K(a) if and only if a(m) x(s 0 p) > 1 or x Sf> * / a 5^ (4) for all S CN with Isl» n-m. Proof. If y t K(a) and y donu.x, then T >^ n-m. Take any S ^T with S - n-m. Then y don 0 c x, and x 5^ < y 5^ - a 5^. Since 2m<n, S 0? t 0 S and therefore a(m) x(s 0 P) < y(m) y(s H p) * i. This establishes the sufficiency of (4). To establish necessity, assume that (4) fails for some S. Let y. a. i M yj x i (i-a(m)-x(s n p))/ s n p i c s n P y. = 0 i P - S Then y K(a), y donux and therefore x dom K(a). Lemma 3. Let T(X) - (i M x i ^ ou). If T(X) - M, then x K(a) U dorn K(a) If T(X)?M, then x K(a) U dom K(a) if and only if a(m) x(s n p) > 1 (5) for all S fn with S n-m and T(X) Hs» 0. Proof. Assume T(X) M, and let» x(m) - o(m). If B 0 then x K(a) If 6 > 0. let
9 UJLii.iiiHiininijiiiiilvmqn^Wltn^lWiiiWIiiP i III i mil IUIIP mu 11,1 11,..II...U..».., ' " «' ' ' ' P n " '. >». ^ - * mm y i '^ i M y. = x. /(n-in) i C P Then y C K(a) and y donux. Fhe remainder of the lemma is simply a restate- ment of Lemma 2. Lemma 4. There exists x K(a) Ü dom K(a) with T(X) T if and only if a(m) S H p (l-a(t))/(n-m) >_ 1 (6) for all S c N such that s > n-m and = 0. Proof. For any x X, let Xi -a i i C T y. 0 i M-T y. - x i (x(m) - a(t))/ P i P If x satisfies (5), then y clearly also satisfies (5). Therefore by Lemma 3, there exists x K(a) U dom K{a) with T(X)» f if and only if there exists some y such that y(p) - l-a(t) a(m) y(s n P) > 1 (7) for all S CN with s n-m and S Hi 0. By the symmetry of (7). such a y exists if and only if (7) is satisfied when
10 !.,...-..,,,.,,.., "-" ' mm» ~ * ^.. - fciaa! * m n y. - (l-afj))/\p\ i C P. This establishes the lemma. Proof of theorem. Observe that (6) is satisfied if and only if it is satisfied when S H p is minimized, that is S fl P n - 2m T. Therefore, in view of the preceeding lemmas, K(a) is externally stable if and only if a(m) (n-2m*t)(l-a(t))/(n-m) < 1 for all T»T M, whore T t. Replacing T with M-T, this condition is equivalent to t a(m) (n-m-t)a(t) < t (8) for all T c M with T t > 0. For t 1, this is exactly the condition (3) of the theorem. For t > 1, (3) implies t a(m) (n-m-t)a(t) < t a(m) t (n-m-l)ä < t, where a max (a.). Thus (8) is equivalent to (3), completing the proof of X i6m the theorem. Comments. 1. With slight modifications to the proof, the theorem may be shown to hold for all 0 m n The neoren characterizes all m-discriminatory solutions to the (n.n-no-game. It is t-jily verified that the game has no k-discrimioatory solutions f->r k ;* m.
11 P"' '" '' ' " " " "~ : Hl»I- I I I.I I I»!! 1. III.LXII.I.HI1TI,...,,,.11,.Uli.,.,,,.,,..! REFERENCES [1] R. Bott, "Symmetric Solutions to Majority Games". Annals of Mathematics Study No. 28 (Princeton, 1953) [2] M. H. Hebert, "Doubly Discriminatory Solutions of Four-Perscn Constant-sum Games", Annals of Mathematics Study No. 52 (Princeton, 1964), [3] J. von Neumann and 0. Morgenstern, Theory of Games and Economic Behavior, Princeton University Press, Princeton, N.J., [4] G. Owen, "n-person Games with only 1, n-1, and n-person Coalitions", Proc. Amer. Math. Soc. 19 (1968), [5] R. J. Weber, "Discriminatory Solutions for (n^^l-games", Technical Report No Cornell University "
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