'" I " " "" ll '" I'" ' " P"-' I II -«...-.,. ' M»^..l^..l I! Ilium llljlll '^' '^'^ " " '!»"I AD-757 354 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. 22151
' "" 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 14850 -Urn m n rtmrämiimti' r^.., %
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,..-..1.,!,. 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 14850 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 N00014-67-A-0077-0014 6. 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 22217 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 01 01 -aot-obi I ' JL UNCLASSIFIED S**curits t l.i.-.iin.iti'mi All 10-^ mi * m ~wmmmh i mi
,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 0101 -loj-tl? I BACK) ii^ai IIIII^M mmtmn f Mill J ß UNCLASSIFinD Security Cl.i'.'.if M.it ion
- "" 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/4 26 1973' This research was supported in part by the Office of Naval Research under Contract Number N00014-67-A-0077-0014 and by the National Science Foundation under Grants GK 29838 and GP 32314x. Reproduction in Whole or in Ptrt is Permitted for any Purpose of the United States Government. Jit ~*.* ma - i -
- --. - - - - ---"' -~ 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)
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.
^^ ^^^^^^^^^^^^»..., 1... -..i.,-,.,^..,,,...,,,.,...,11,i iii i iipmii I», ii,,,...,. i...,...1.. ".-^ 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
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 5 0 1 = 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
1. -...!.,...-..,,,.,,.., "-" ' 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-2. 2. 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.
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). 319-323. [2] M. H. Hebert, "Doubly Discriminatory Solutions of Four-Perscn Constant-sum Games", Annals of Mathematics Study No. 52 (Princeton, 1964), 345-375. [3] J. von Neumann and 0. Morgenstern, Theory of Games and Economic Behavior, Princeton University Press, Princeton, N.J., 1944. [4] G. Owen, "n-person Games with only 1, n-1, and n-person Coalitions", Proc. Amer. Math. Soc. 19 (1968), 1258-1261. [5] R. J. Weber, "Discriminatory Solutions for (n^^l-games", Technical Report No. 175. Cornell University. 1973. - - "