Faculty Working Papers THE SHAPLEY VALUE AS A VOU KEUMANN- MORGENSTERN UTILITY Alvin E. Roth #297 College of Commerce and Business Administration University of Illinois at Urbana-Champaign FACULTY WORKING PAPERS College of Comirerce and Business Administratis: University of Illinois at Urb ana- Champaign THE SHAPLEY VALUE AS A VOW KEUMANN- MORGENSTERN UTILITY Alvin E. Roth #297 The Shapley Value as a von Neumann-Morgenstern Utility. by Alvin E. Roth University of Illinois The author wishes to acknowledge some thoughtful comments by David Schmeidle*-, Digitized by the Internet Archive in 2012 with funding from University of Illinois Urbana-Champaign http://www.archive.org/details/shapleyvalueasvo297roth Abstract The Shapley value is shown to be a von Neumann-Morgenstern ut function. The concept of strategic risk is introduced, and it is that the Shapley value of a game equals its utility if and only ±i underlying preferences are neutral to both ordinary and strategic 1 • I ti t r od uc t io n The development of game theory has been closely associated with the axiomatx : treatment of cardinal 5tilities, ever since both were introduced by von Neumann and Morgenstern. In Theory of Games and Eco nomi c Behavior they give a set of axioms which assure that a cardinal utility function can be deduced from ordinal preferences. This cardinal utility is than used to define the characteristic func- tion form of a game. In 1953, in his classic paper 'A Value For n-Person Games', L.S. 5hapley notes that this cardinal utility is defined only for 'simple situations' — prizes, and lotteries over prizes — but not for games them- selves. He says: "At the foundation of the theory of games is the assump- tion that the players of a game can evaluate, in their utility scales, every "prospect" that might arise as a result of a play. In attempting to apply the theory to any field, one iould normally expect to be permitted to include, in the class of "prospects," the prospect ot having to play a v^ame. The possibility of evaluating games is therefore of critical im- portance. So long as the theory is unable to assign values •;o the games typically found in application, only relatively simple situations — where games do not depend on other games — will be susceptible to analysis and solution," He proceeds co give three cardinal conditions which a value for Simes 3hculd satisfy, and to show that there is a unique function which satisfies these conditions. In this paper, we develop a cardinal utility function for games, based on ordinal preferences. Our treatment shall differ from the elementary a Somatization of utiliti s from preference: in that games — the objects over which our preferences are defined—are themselves defined in terms of a cardinal utility. Thus it shall be necessary to insure that the utility function for game? is compatible with the existing utility function which defines the games. We shall show that this requirement leads to a unique utility function for games, and that, when the underlying preference relation is neutral with respect to certain kinds of risk, the utility of a game is equal to the Shapley value. 2* Utility Theory We summarize here an axiomatization of utility presented in [1] . A set K is a mixture set if for any elements a,beM, and for any number p G [0,1], we can associate another element of M denoted by [pa;(l-p)b] called a lottery between a and b. (Henceforth the letters p and q will be reserved for elements of [0,1].) . We assume that lot- teries have the following properties for all a.bsM: [la;0b] - a, [pa;(l-p)b] - [(l-p)b;pa], and (2.1) !>[pa;U-p)b];(i-q)b] - [pqa; (l-pq)b] A preference on K is defined to be a binary relation £ such that for any a.beM either a > b or b -£ a must hold, and if a 2 b and b ^ c then a > c. (We write a> b if a £ b and b ^ a, and a ~ b if a ^ b and b ^ a.) A real valued function u defined on a mixture set M is a utilit y function for the preference %* if it is order preserving (i.e. if ¥a»beM, u(a) > u(h) if and only if a >• b) , and if 2) u([pa;(l-p)b]) « pu(a) + (l-p)u(b). If &* ±3 a preference ordering on a mixture set M, then the follow- ing conditions insure that a utility function exists: (2.3) For any a,b,ceM, the set?, (pi [pa; (l~p)b] ^ c} and {pjc^ ' : ^ai(I"p)1>]} are closed; and j(2.4) If a,a'eM and a ~ a 1 then for any beM, fea;^b] ~ [%a';^b]. The utility function is unique up to an affine transformation. For any clement xeM, the utility of x can be given by ,,/„<> ^ Eabi^l—'UEsbilal U( ' Pab<*i> ~ Pab<*0> where a,b»r-, and r. are elements of M such that a > x ^ b and air r.« ^ t,>, h, and for any ycM such that a£ y >> b, p a b(v) is defined by (2.5) V - l?ab(y)^>' 1 "Pab (y);)b5 ' It can be shown that the numbers P 3b (*) are well defined, and the function i\ independent of the choice of a and b. The fixed elements r and r fv determine the origin and scale of the utility func- tion; not a that u(r-) ■ *-> and u ^ r cP ™ °* 3- The Snap ley Value We summarize here the axiomatization of the value presented in [2], giving first some necessary definitions. 4 Denote by N the universal set of players (or positions) , and de- fine a game to be any function v from the subsets of 8 to the real numbers such that for all sets S.Tc N, v(S) > v(SftT) + v(S-T) and v(0) - 0, where denotes the empty set. The quantity v(S) can be interpreted as the utility obtainable by the coalition S. For convenience we shall assume that N is a finite set, and denote its cardinality by n. (Similarly, the cardinality of sets R, S, and T will be denoted r, s, and t.) A carrier of v is any set TSN such that for all S £ N v(S) - v(T S). The superset of any carrier is itself a carrier. If n is a permutation of N(ie. if tt is a one to one mapping from N to itself) then, for all sets S . By the value of a game v we mean a vector valued function 0(v) - (0 (v), (v)"-0 (v)) which associates a real number t (v) with each position ieK, and which obeys the following conditions. (3.1) For each permutation tt, fl^Cirv) - ± (v). (3.2) For each carrier T of v, £ ± (v) - v(T) . ieT (3.3) For any games v and w, 0(v+w) - 0(v) + 0(v) . Note that (3.3) implies that if (v-w) , v, and w are all games, then 0(v-w) = 0(v) - 0(w). Conditions (3.1) and (3.2) are sufficient to show that for games of the form v defined for any R,S£ N by R V R (S) = £o if R *5 i S s and for any non-negative number c, the value must be /"j n rt / % W r if i e R (3.4) » t (cv, l ) =^ Q ±f , At It can also be shown that any game v can be written as a linei combination of games of the form v p : v = I c t>Cv)v RSN R R where the coefficients c r> (v) are given by c p (v) = Z (-D^SrCT). R TSR Condition (3«3) can now be used to demonstrate the remarkable result that the unique function satisfying conditions (3.1), (3.2). and (3.3) and defined on all games is given by (v) . z ( (3-l)?U-s,)! ) ws)-v(s-l)) . S£N 4. A Utility Function fo r Games For simplicity of presentation we will henceforth confine our attention to the class Z of games which are positive valued > i.e. games 1 for which v(S) > for all S£N. A position i e N is called a dummy for a game v if it is not contained in every carrier. Denote by D <= Z the class of games for which i is a dummy. It will be convenient to identify the games v Q and v ± given for all S S N by fl if i e S v Q (S) - 0; and v ± (S) « £ if iXs> In the game v , all positions are dummies; in v ± all positions but i are dummies . We will be interested in the mixture set H generated by the set Z*"N of strategic po sitions . Thus M consists of all lotteries of the form [p(v,i);(l-p)(w,j)L where (v,i) and (w,j) are elements of Ml. We assume that a preference relation £■ is defined on M which satisfies (2.3) and (2.4). (Read (v,i)^ (w,j) as 'It is preferred to play position i in game v than to play position j in game wV ) The games in the class Z are all defined in terms of some observer's (fixed) utility function u for 'sit le situations'. The preference relation over the set M is assumed to be] o the same observer. In order to make this notion consistent, tpose the following restrictions on the preferences. (4.1) For all i £ N, v £ Z and for any permutation 7T, (v,i) ~ (irv,iri) (4.2) If v £ D. then (v,i) ~ (v Q ,i), and for every v £ Z and i £ N, (v,i)<5 (v Q ,i) and (v ± ,i) > (v Q ,i). (4.3) "For any number c > 1, and for every v £ Z s i £ N, (v,i) - [|(cv,i);(l-^)(v ,i)l. "We that this differs slightly from the usual definition of a dummy 7 Condition (4.1) merely says that the. names of the positions do not determine CheiK desirability in a game. Condition (4.2) say:< elicit: playing any position in any game (in the class Z) is at least as desir- able as being a dummy in any game, and chat there is some strategic position, namely (v. ,i) s which is strictly preferable to being a dummy. Condition (A. 3) takes note of the fact that games are defined in terms of a utility function. It says that if two strategic positions are identical except for the fact that the utility obtainable in one is a positive multiple of the utility obtainable in the other, then t first is indifferent to the appropriate gamble between the second, and the prospect of receiving zero. This is virtually the definition of utility, as given by equation (2.2). We are now in a position to define a utility function for strateg position, which we shall call strate gic utility to distinguish it from the utility function u used to define the games. Such a function ex^st;:, since the preference ;b satisfies (2.3) and (2,4) by assumption. Tne strategic utility of a game v is the vector 0(v) - 0,(v), 6L(v),...0 (v v j), where i l n e ± (v) 5 0<(v,D) - Pao(^Q) Pab< r l) - Pab< r 0> for probabilities Pab(') as * n (2.5) and for a,b s r 1 , r e M such that a^ (v, i)-^b, and a ^ r_ > l. '%- b. Fixing r- = (v. »i) and r_ ■ (v^,i) i U X 1 U ) - 0, Condition (4.2) insures that we always take b ~ r^, so that P a b(r«) = for all a £ M. We can also prove the following lemma Lemma JL: For any permutation TT, and for every i e N, . (rrv) - 8 Proof: Immediate from the order preserving properties of utility functions, and from condit .) . / Lemm a _2: For r c '2 0, and for eny i £ K, e.(cv) - c9. (v). Pr oof : Without loss of generality, take c > 1. Case I; (cv,i) X r, = (v.,i). • — "^ ± i Take a - (cv,i) and b = r « (v-.i). Then G,(cv) = „ P ab ((cv,i>) m 3 Pab( r i> Pab< r ]/ (v,i) ~ [- (cv,i);(l--) r ], so p ab ((v,i)) «- . Consequently e (v) . Pab((v,D) . I e (cv) . i ;; c c i Case II: Let. r, - (v. ,i) X (cv,i) . Take a^r., , b=r r> . Then p a ,(r, ) = 1, and so B ± (cv) - p ab (cv,i). But (v,i)~[— (cv,i) ; (1- — )r Q ]~[-[p ab ((cv,i))a; (i-p^ v ((cv,i))b] ;(1- — )b] by definition of p , («)• But by condition ar> c ab (2.1), this, is equal to [--p ( (cv,i))a; (1- ^p ((cv,i))b]. So c ao c ao e i (v) = Pab ((v - 1); ' = W (CV >- 1)) -£e t < CT >- Our procedure so far has been, in effect, to imbed the set of simple prospects in the mixture space of games. Any simple prospect whose utilitj u is equal to some value c > can be identified with the game cv. , since (4.4) 0.(cv.) - c. 1 1 This follows from Lemma 2 and the fact, that G.(v J > = 1 and 0. (v_) = 0. I i 1 By Lemma 1, any position i yields the same result. Condition (4.4) says that the strategic utility is simply an extension of the utility function a which defines the games. The utility function is unique (for a fixed u) , since condition (4.4) sets the origin and scale. In order to evaluate for other elements of the set M (i.e. for games not of the form cv ) we must investigate the risk posture of the preference relation ^ . 5. Risk Posture We shall distinguish between two kinds of risk. Ordinary risk in- volves the uncertainty which arises from the chance mechanism involved in lotteries, while strategic risk involves the uncertainty which arises from the interaction in a game of the strategic players (i.e. those who are not dummies) . The preference ^ is averse to strategic risk if for every R ? N and all i e R, (v.,i) X (rv_,i), where v_ is defined as in section 3. This 1 K K means that it is preferable to receive a utility of i for certain (in a game with no other strategic players) than, to negotiate how to distri- bute a utility of r among r players. If the preference is reversed, we say it is risk preferring to strategic risk. The. preference relation tl is neutral to strategic risk if for all R ? N, and every i e R, (5.1) (v.,i) - (rv R ,i). The preference ^ is averse to ordinary risk if for all i £ N, and v,w e Z, ((pv-r(l-p)v) ,i) > [p(w,i) ;(l-p) (v,i)} i.e. if it is preferable 10 to play the game (pw + (i-p)v) than to have a lottery which results in the game w with probability p and the game v with probability (1-p) . Similarly, the preference is neutral to ordinary risk if for all games v, w £ 2, and for every i £ ft (5.2) ((pw+(l-p)v),i) ~ [p(w,i);(l-p)(v,i)}. We can now prove the following lemmas. Lemma 3 : If the preference relation >> is neutral to strategic risk, then \~ for i e R 1 K (0 for i\R. Proof : If i\R, then ± (v R ) = by (4,2) and the fact that v R £ D ± . If i £ R, then 0, (v_) - ~ by (5.1) and Lemma 2. L K r Lemma 4 : If the preference relation £ is neutral to ordinary risk, then 0(v+w) - 0(v) + 0(w). Proof: For each i £ N, 0. (v4w) « b, ( 2 Gflr+J»r) ) - 2 (V+W x i. •»- by Lemma 2. But ©.(V+V) - *2 \M + ^ 6 ± (w) by (5.2) and (2.2), so 0.(v+w) = 0.(v) + (w). Thus for a preference relation^ which has the properties (2.3), (2.4), (4.1), (4.3), we can prove the. following: Theorem : The strategic utility is equal to the Shapley value if and only if the preference relation £ is neutral with re- spect to both ordinary and strategic risk. Proof: Bv (3.5) and Lemmas 2 and 4, 9(v) - Z c (v)0(v ) , where the num- rcn bers c R (v) are given by (3.6). But 0.(v R ) - 0.(v R ) by Lemma 3 and (3.4). So 0.(v) - S c n (v) 0,(0 - 0. (v). RCN R X R * 11 6. Discussion We have shown that the Shapley value is a risk neutral utility function. This fact sheds some lighc on the axioms which define the value, particularly axioms (3.2) and (3.3). Lemmas 3 and A make it clear that these two axioms are intimately related to the two kinds of risk neutrality. Perhaps it will also serve to illuminate some of the properties of the value. For instance, the relationship between the value and the competitive allocations of a market game might be better under- stood if conditions could be determined under which preferences tend towards risk neutrality as the competitiveness of the market increases. It may also prove fruitful to investigate strategic utility func- tions arising from preferences which are net risk neutral. In partic- ular, the concept of strate gic risk seems likely to yield interesting results in this regard. A simple case involves a linear posture to strategic risk, given by (6.1 (kv ,i) ~ (rv R ,i), where k is a non-negative number. Risk neutrality is just the special case k - 1; the preference is risk, averse if k < 1, and risk preferring if k > 1. If is the strategic utility which reflects condition (6.1) , then t v f - for i e R 1 K / for i e R and consequently, for an arbitrary game v, e ± (v) - k0 ± (v). 12 More formally, for preferences which obey conditions (2,3), (2.4), (4.1), (4.2), and (4.3), we have the following generalization of the main theorei . Proposition : If the preference is neutral to ordinary risk, and linear (with coefficient k) to strategic risk, then e.(v) - k0.(v). i i 13 Bibliography [1] Herstein, I.N. and Milnor, J.: "An Axiomatic Approach to Measur- able Utility," Econometrica, 21 (1953), 291-297. [2] Shapley, L.S.: 7 A Value ior n-Person Games', Annals of Math Studies 28 (1953) Kuhn and Tucker, eds., 307-317. [3] Von Neumann, J. and Mor gens tern, 0. ; Theory of Games and Sconondc Behavior Princeton. (Princeton University Press). 1944, 1947, 1953,