UNIVERSITY OF ILLINOIS LIBRARY AT URBANA-CHAMPAIGN BOOKSTACKS Digitized by the Internet Archive in 2011 with funding from University of Illinois Urbana-Champaign http://www.archive.org/details/nashsolutionutil364roth ,00 Faculty Working Papers THE NASH SOLUTION AND THE UTILITY OF BARGAINING Alvin E. Roth #364 College of Commerce and Business Administration University of Illinois at Urbana-Champaign FACULTY WORKING PAPERS College of Connerce and Business Adininistration University of Illinois at Urbana-Champaign January 4, 1977 THE NASH SOLUTION AND THE UTILITY OF BARGAINING Alvin E. Roth #364 THE NASH SOLUTION AND THE UTILITY OF BARGAINING by Alvin E. Roth Assistant Professor, Business Administration ABSTRACT It has recently been shown that the utility of playing a game with side payments depends on a parameter called strategic risk posture. The Shapley value is the risk neutral utility function for games with side payments. In this paper, utility functions are derived for bargaining games without side payments, and it is shown that these functions are also determined by the strategic risk posture. The Nash solution is the risk neutral utility function for ba.rgaining games without side payments. THE NASH SOLUTION AND THE UTILITY OF BARGAINING by Alvin E. Roth I. Introduction Recent work has shown that the Shapley value for a game with side payments is a cardinal utility function which reflects the desir- ability of playing different positions in a game, or in different games (cf. Shapley [14], Roth [9]). A player's utility for playing some position in a game is determined in part by his assessment of the payoff he will receive in a class of games with side payments called bargaining games. Given a player's evaluation of these bargaining games, his utility for playing a position in any game with side payments can be determined (cf. Roth [11]). It is desirable to extend these results to games without side payments, since the assumption that side payments can be made is not appropriate in many situations. In this paper we will derive a class of utility functions for playing bargaining games without side payments. Games of this sort are studied by Nash [7], who developed a solution to bargaining games which is an extension of the Shapley value for games with side payments. That is, the Nash solution coincides with the Shapley value for bargaining games with side payments. Somewhat surprisingly, the utility of playing a bargaining game without side payments is determined by the same considerations which determine the utility of playing a game with side payments. Given a player's evaluation of bargaining games with side payments, his utility for bargaining without side payments is determined. II . uti lity Functions for Games with Side Payments This section summarizes the development of utility functions for games with side payments, as presented in [9] and [11]. We begin with some necessary definitions. 2 A game with side payments consists of a set of positions N = {l,...,n} and a superadditive function v from the subsets of N to the real numbers such that v(0) = 0. The function v denotes the amount of wealth which each coalition of players can obtain for itself, and the model assumes that utility is linear in wealth, and that wealth is freely transferable. The set of individually rational outcomes (imputations) is thus the set X = {(x^,...,x )| 5!x^ = v(N), x^ ^ v(i)} Since we shall be interested in comparing different games, we take N to be the common set of positions for all games. Thus n is the largest number of players who can take part in a game (e.g., n could be the population of the world). In order to distinguish which positions have an active role in a given game, define position i to be a dummy in game v if, for all subsets S of N, v(S) = v(SUi) . The positions which are not dummies are called the strategic positions of the game v. If 7t is a permutation of N, denote the image under it of a subset S of N by irS, and define the game nv by Trv(iiS) = v(S). To simplify the exposition, we confine our attention to the class G of non-negative games; i.e., games for which v(S) >_ 0. For any subset R of N, it will be convenient to define a bar- gaining game with side payments, v^^, by ^^^(S) = {^ Qt-^erwise ' ^^ ^^^^ game, all players in R are symmetric, and all players not in R are dummies. Similarly, for each position i in N, denote by v. the game v,(S) = IX , . .In this game all positions other than i are i^ ^0 otherwise ^ ^ dummies. Denote by v the game v^(S) = for all S, the game in which all positions are dummies. In order to make comparisons between positions in a game and in different games, we shall consider a preference relation defined on the set NxG of positions in a game. Write (i,v)P(j,w) to mean "it is pre- fereable to play position i in game v than to play position j in game w." 3 The letter I will denote indifference, and W will denote weak preference. We will consider preference relations which are also defined on the mixture set M generated by NxG. That is, preferences are also defined over lotteries whose outcomes are positions in a game. Denote by [q(i,v) ; (1-q) (j,w) ] the lottery which with probability q has a player take position i in game v, and with probability (1-q) take posi- tion j in game w. We will henceforth only consider preference relations which have the standard properties of continuity and substitutability on M which insure the existence of an expected utility function unique up to an affine transformation. Denote this function fay 6, and write 6^(v) = 9((i,v)), and e(v) = (G^(v) , . . . ,e^(v) ) . The utility for a position in a game is given by (v) = 8((i,v)) = q^l,((i,v)) - q^^Cr^) ^ ' 'Jab^^l^ - '^ab^^O^ where a, b, r^ , and r are elements of M such that aW(i,v)Wbs and J- u aWr Pr„Wb, and the numbers q v (y) are probabilities defined for any y in M such that aWyWb by yl[q , (y)a; (1-q (y))b] . The elements r^ and r^^ determine the origin and scale, since 6(r^) = 1, and 9(^q) - O- We assume that the preference relation obeys the following three conditions: (2.1) For all i e N, v e G, and for any permutation ti, ir(i,v)I(7Tl,iTv) . (2.2) If i is a dummy in the game v, then (i,v) I(i,v„) . Also (i,v.^)P(i,VQ), and for all (i,v) C NxG, (i,v)W(i,Vj^) . (2.3) For any games v and w, and for any probability q, (i, (qw+(l-q) v) ) I [q (i, w) ; (1-q) (i, v) ] . Condition 2.1 says that the names of the positions don't influ- ence their desirability. Condition 2.2 says that a strategic position in a game is always at least as desirable as a dummy position, and that it is equally undesirable to be a dummy in any game. Condition 2. 3 expresses indifference between playing position i in the game (qw+(l-q)v), and playing position i in game w with probability q or in game v with probability (1-q) . Condition 2.2 also insures that we may choose the natural nor- malization for the utility function 6. In what follows, we will con- sider 9 to be normalized so that 8. (v.) = 1, and Q.iv^) = 0. 11 1 It has been shown [9], [11] that any utility function arising from preferences obeying conditions 2.1, 2.2, and 2.3 has the following properties: Property 1. Symmetry: for any (i,v) e N^G, and for any permutation TT, 9 .(-nv) = 9.(v). ni 1 Property 2. Homogeneity: for any (i,v) c NxQ, and any non-negative number c,- 9.. (cv) = c9 . (v) . Property 3. Additivity: for any v,w e G, e(v + w) = 8(v) + 8 (w) . The utility function fl can now be completely determined by specifying the certain equivalent of playing a bargaining game v , as K one of r strategic players. Let f(r) be a number such that (2.4) (i,v^)I(i,f(r)v^) for i f, R. This expresses indifference between receiving f(r) for certain (as the only strategic player in the game f(r)v,) and being one of r strategic players in the game v . Note that f(l) = 1. Using the terminology of K [9], we say that the preference is neutral to strategic risk if f(r) = 1/r for r = l,...,n. The preference is strategic risk averse if f(r) < 1/r, and strategic risk preferring if f (r) > 1/r The utility 9 for playing an arbitrary game with side payments can now be written in terms of the function f(r). Theorem 1: 6 . (v) = J^ k(t) [v(T) - v(T-i)], ^ TCK n _ _ where k(t) = V (-1)^ ^(^_pf(r). r=t Furthermore, if the prefereiice relation is neutral to strategic risk, then the utility of playing a position in a game is equal to its Shapley value. Corollary 1: If f(r) = 1/r, then 9 (v) - T -^s-x; . v.n b, .^^^^^ _ ^(g.^j j SCN ^' III. Bargaining Games wit hout S ide Paym ents, and Nash's S olution An n-person bargaining game without side payments is defined by a compact convex subset A of n-dimens:ional Euclidean space, and a point s contained in A. Any point x - (x , ...,x ) contained in A represents the von Neuioann-Korgenste? n utility available to each player as the result of some feasible agreeiiient, and the set A represents the set of all feasible utility payoffs. The point s = (s^,...,s ) i"epresents the utility of the "status quo" — that is, s gives the utility level achieved by each player in the absence of any agreement. For simplicity, v*e will assume that the set A contains only individually rational agreements: i.e., if x e A, then x ^ s. We will also assume that the origin of the utility function for each player (position) is equal to the status quo payoff; i.e., we assume s. = for all i e N. Denote the class of all such bargaining games by H. An element of H will be denoted by the feasible set A, with the status quo being understood to be the origin. As in the previous section, take N = {!,..., n} to be the common set of positions for all bargalnir-g games. The- set R of strategic positions in a bargaining game A is the set R={icNl SxcA such that X. 7^ 0}. A position which is not in the set R is a dummy for the game A. A Nash solution to the bargaining problem is a function F, defined on bargaining games, which associates with each bargaining game A a single feasible outcome F(A) f A, and which obeys the following four conditions: (3.1) Linearity: For any bargaining garae A and positive real numbers a, , . . . ,a , if B = { (a, x, , . . . ,a :;c ) | (x., , . . . ,x ) e A] then 1 n ixnnJ. n F.(B) = a.F.(A) for i - l,...,fi. (3.2) Independence of irrelevant alternatives: If A and B are bargain- ing games and B contains A, and if F(B) e A, then F(B) = F(A) . (3.3) Symmetry: Let R be the set of strategic positions in a game A, and suppose that for every permutation it of N such that irR = R, X € A implies that iix f A. Then F. (A) = F , (A) . r TTi h for all strategic positions i e R, then F(A) "^ x. (3.4) Pareto Optimality: If x and y are elements of A, and y. > x. Nash [ 7 ] proved the following theorem. Theorem 2: There is a unique function F which satisfies conditions 3.1-3.4. For a bargaining game A, F(A) is the unique element x e A such that it x > IT y. for every y ^ x iCR ieR ^ in A, where R is the set of strategic positions of the game A. Thus the Nash solution picks the point x in S which maximizes the geometric average of the payoffs x. for ieR- Note that F(A) = if and only if R is empty. It has recently been shown that this con- dition can replace Pareto optimality in the characterization of the 9 Nash solution (Roth [10]). That is, we have the following theorem. Theorem 3: The Nash solution is the unique function F which satisfies conditions 3.1-3.3, and the condition that F(A) = only when R is empty. 8 IV. The Utility of Bargal7;ting In this sectior we will consider the utility of bargaining, by considering a preference relation ? defined on the set of positions in bargaining games without side payx^ents. Specifically, take P to be a preference relation defined on N^U, and on the mixture set M' generated by NxH, It will be convenient to define, for each set R contained in N, the set A -■ {x| 1 X. ± 1, X. >_ if i y. then (i,A )P{i,A ), and if R is a 1 ■ 1 X y non-empty subset of N such that R =/ (i), then (i,A. )P(i,A ) . (4.3) If B = {(a^x^ , . . . ,a x )! x c A; for a. > for i = l,...,n, and 11 *nn j=" -■ ».» if a^ > 1, then (i,A)I[ (I/a. ) (i, B) ; (l~l/a^)(i,A^^) ] . (4.4) If A c B c C, and (i,A)I(i,C) then (i,A)I(i,B). Condition 4.4 expresses indifference to irrelevant alternatives. It says that if ^ player is indifferent between playing in a game A, or in a game C with a larger set of Eeasibie alternatives, then he is also indifferent between playing A or any game B which contains A and is con- tained in C. Condition 4.3 simply says that, if the payoffs available in a game are multiplied by positive constants, then a player is indif- ferent between playing one game, or participating in the appropriate lottery involving the new game. Conditions 4.1 and 4.2 are similar in form and content to conditions 2.1 and 2.2. If e is a utility function reflecting preferences which obey the above conditions, then it has the follccjing properties. Lemma 1: If i is a dummy in A, then 6 . (A) = 0, and if x. = y., then Proof: This follows immediately from condition 4.2. Lemma 2: If B = { (a^ x, , . . . ,a x )i x € A} where all a. > 0, then i 1 n n ' 2 6 . (B) = a . 6 . (A) . Proof: Suppose a, >^ 1. Then by condition 4.3, 9 . (A) = e^[(l/a_j^)(i,B);(l-l/ap(i,AQ)] = (l/a^)e^(B) + (1-1/a^) 9^(Aq) 10 = (l/a.)6.(B). Suppose a. < 1. Vixen let b. = 1/a. for 1 1 X 3 J i = 1, . . . ,n. ITien A = { (,b, y, , . . . ,b y ) ! y C B}, and b. > 1. J ' ' linn' 1. So e.(B) = (l/b.)6.(A) = a.e,(A). Lemma 3: For any x, 6 . (A ) = x. . IX 1 Proof: Let y be the vector such that y. = x. and y. = for i ?^ i. Then lemma 2 implies that 6 . (A ) = 9.(x.A.) = x.9.(A.) ■^ lyxxxiix = X., and lemma 1 implies that 6 . (A ) = 9 . (A ). X X X X y As is the case for games with side payments, the function G will be completely determined by the posture towards strategic risk. For bargaining games with side payments, condition 2.4 stated (i,v_)I(i, f (r)v.) for i e R. The equivalent condition for bargaining games without side paymen ts is (4.5) (i,A„)J(i,f(r)A.) for i f R.' K X This expresses indifference between playing a strategic position in the game A^ (as one of r strategic players) or receiving the utility f(r) for certain (as the only strategic player in the game A.)- By condition 4.2 we know that f(l) = 1, and < f(r) < 1 for r > 1. As in the case of games with side payments, we say the preference relation is neutral to strategic risk if f(r) = 1/r, averse to strategic risk if f(r) < 1/r, and strategic risk preferring if f(r) > 1/r. We will show that the Nash solution is the utility function reflecting risk neutrality. An immediate consequence of condition 4.5 is that e.(A^) = f(r). More generally, we have the following result. Lemma 4: If B„ = {y > 01 )' b.y. < 1, y. = for i i! R} where b. > xeR 11 for each i e R then 6 . (B„) = f(r)/b. for i € R. 1 R 1 Proof: B^ - {(a,x_,...,a x )j x c K.^ where a. = 1/b , for R 11 n n ' K" ], 1 1 e R and a . = 1 for j ^ R. So leinma 2 implies that 8 . (B ) J 1 K a.6.(V = nr)/b.. We can nov? specify the function 6 for an arbitrary bargaining game A. Theorem 4: If A is a bargaining game with R the set of strategic posi- tions, then for k e R, G, (A) = x, , where x is the unique element of A such that II x. > n y. for all y c A such icR leR that y ^ x, where q = (q^,...,q ) is any non-negative vector such that q, = f ( r) and Y q. = 1. xcR The element x named in the theorem maximizes the geometric average with weights q , ...,q over the set A. (Tlie weighted geometric average is concave, so it has a unique maximum of A.) The statement of the theorem implies that 6, (A) = x, depends only on q. . Explicitly, the following technical proposition follows as a corollary of the theorem. ^i Pi Proposition: If x maximizes n x. and y maximizes JT y. over the ICR ^ i€R ^ set A, where q and p are non-negative vectors such that I q i = I Pi = 1' then :<^ = y if q = p . ieR "- ieR ^ R K ic K Proof of theorem: Let A be a bargaining game without side payments, and let R cr N be the set of strategic positions of A, and let k e R. Let q = (q , ,..,q ) be a non-negative 12 vector such that T q. = J q. = 1, and q, = f(r) > 0. Let X be the element of A which maximizes n x. . That i(^R .^ q, q^ is, II X." = c > n y. for all y e A nuch that y t' x. iCR ^ ieR ^ Let H = {yj II y. >^ c} = {y| 1 q. log y. >^ log c}. iCR ^ IfR ^ ^ Tlien H and A are convex sets whose intersection is the point x, and so there is a plane which separates H and A. This plane is the tangent to H at x, i.e., the set T = {zl z.n = x-n} where n = (q /x , ,..,q /x ) . So T = {z] (q,/x,)z, + ... + (q /x )z = T q = 1}. ' ^1 1 i n n n .'-.^ i Let B = {z\ (q /x )z + ... + (q /x )z £ D • Then J_Ji„JL iLltLl A cr B, since T separates A from H. Lemma 4 implies that e^(B) = (Xj^/qj^)f(r) = (x^^/f(r))f(r) = x^. So (k,B)l(k,A^) , since 9 (A ) = 6, (B) = x, . Thus we have A c: A c: B, and (k,A )I(k,B). By condition 4.4, this implies that (k,A )I(k,A), and so 9 (A) = x . This completes the proof. X K. K. Corollary 2: When f (r) = 1/r, Q is equal to Nash's solution. Proof: If A is a bargaining game with strategic positions R, then for k e R, 9, (A) = x where x maximizes 11 x. "^ ^ ieR ^ on A, for q, = f(r) = 1/r and ^ q, = 1. In particular, ■^ ICR "- X maximizes II x. , and, since r > 0, x maximizes n x. . iC R le R Thus we have shown that the utility of playing a bargaining game without side payments is determined by the posture towards strategic 13 risk. Since the Shapley value and the Nash solution agree on bargaining games with side payments, it is natural to observe that they result from the same risk posture. The treatment presented here permits us to observe not only the similarities between utility functions for games with and without side pajrments, but the differences as well. The most significant difference seems to be that, for bargaining games without side payments, there is no parallel to condition 2.3 for games with side payments. Tliat is, if A, B, and C are bargaining games without side payments, such that 11 111 C=yA + -T-B = {(2X + ^y)j xCA, ye B}, then the utility of a lottery between A and B is not in general equal to the utility of C. That is 6.[y A;y B] = -rO . (A) + :r6 . (B) / 6.(C). A discussion of this phenomenon in the context of the Nash solution is given by Harsanyi [2, pp. 330- 332]. Nash originally interpreted his solution as applying to players of equal bargaining ability, but subsequently modified this interpretation [7,8], Our results support Nash's original interpretation. The attitude of neutrality to strategic risk, which gives rise to the Nash solution as a utility function, simply expresses a player's belief that he will receive the average reward in a bargaining situation. As we have seen, any other risk posture gives rise to a utility function different from the Nash solution. FOOTNOTES 1. For related results, see Roth [12,13]. 2. We speak of "positions" rather than the more customary "players" since we are interested here only in the structural properties of the game. We shall be concerned with the problem of evaluating the different positions from the point of view of a player who must choose among different positions. 3. So alb means neither aPb or bPa, and aWb means aPb or alb. 4. A mixture set has the properties that for all a,b e M [la;Ob] = a, [qa;(l-q)b] = [(l-q)b;qa], and [q[pa; (l-p)b] ; (l-q)bi = [pqa; (l-pq)b]. (Cf Herstein and Milnor [4].) 5. Cf Herstein and Milnor. 6. A utility function has the property that u(a) > u(b) if and only if aPb. An expected utility function on a mixture space has the property that u( [qa; (l-q)b]) = qu(£) + (l-q)u(b). That is, the utility of a lottery is its expeci:ed utility. 7. Cf Herstein and Milnor. 8. The cardinality of sets R, S, T is denoted r, s, t. 9. This statement of the theorem makes use of the fact that we have already assumed individual rationality. 0. Harsanyi and Selten [3, lemma 10. 1] and Kalai [5 J both show that weighted geometric averages of this sort obey all of Nash's conditions except symmetry. 15 11. For different approaches to the bargaining problem see Brito, et. al. [1] or Kalai and Smorodinsky [6]. REFERENCES 1. Brito, D.L.; Buoncristiani, A.M., and Intriligator, M.D., "A New Approach to the Nash Bargaining Problem," Econometrica , (to appear). 2. Harsanyi, J.C. "Games with Incomplete Information Played by 'Bayesian' Players Part II: Bayesian Equilibrium Points," Manage- ment Science , vol. 14, no. 5, January 1968, pp. 320-334. 3. Harsanyi, J.C, and S el ten, R., "A Generalized Nash Solution for Two Person Bargaining Games with Incomplete Information," Management Science , vol. 18, no. 5, January Part 2, 1972, pp. 80-106. 4. Herstein, I.W. , and Milnor, J., "An Axiomatic Approach to Measurable Utility," Econometrica, vol. 21, 1953, pp. 291-297. 5. Kalai, E., "Nonsyrcmetric Nash Solutions and Replications of 2- Person Bargaining," International Journal of Game Theory (to appear). 6. Kalai, E., and Smorodinsky, M. , "Other Solutions to Nash's Bargaining Problem," Econometrica , vol. 43, 1975, pp. 513-518. 7. Nash, J.F., "The Bargaining Problem," Econometrica , vol. 18, 1950, pp. 155-162. 8. Nash, J.F., "Two Person Cooperative Games," Econometrica , vol. 21, 1953, pp. 128-140. 9. Roth, A.E., "The Shapley Value as a von Neumann Morgenstern Utility," Econometrica (to appear) . 10. Roth, A.E., "Individual Rationality and Nash's Solution to the Bar- gaining Problem," Mathematics of Operations Research (to appear). 17 11. Roth, A.E., "Bargaining Ability, the Utility of Playing a Game, and Models of Coalition Formation," mimeograph, 1976. 12. Roth, A.E., "Utility Functions for Simple Games," mimeograph, 1976. 13. Roth, A.E,, "A Note on Values and Multilinear Extensions," Naval Research Logistics Quarterly (to appear). 14. Shapley, L.S., "A Value for n-Person Games," Annals of Mathematics Study , vol. 28, 1953, pp. 307-317.