FOR COMPACT CONVEX SETS

Transcription

1 PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 49, Number 2, June 1975 ON A GAME THEORETIC NOTION OF COMPLEXITY FOR COMPACT CONVEX SETS EHUD KALAI AND MEIR SMORODINSKY ABSTRACT. The notion of complexity for compact convex sets introduced by Billera and Bixby is considered. It is shown that for zz > 3 there are sets in R" of complexity zz. Also for zz = 3 the maximal complexity is Introduction. The following definitions occur in the mathematical model of market games. Let u., «,,, zz be zz concave continuous real valued functions defined on the z?z-dimensional Euclidean unit cube lm. An allocation (of the commodities) is a matrix (x) = (x!.), i = 1, 2,, zzz; / = 1, 2,, n, satisfying the conditions: (1) x>>0 for every z and /, and (2) V =lx\= 1 for i= 1,2,-.., m. We define the function zz from the set of allocations to R" by u((x[)) = (uy(x\, x\,...,x)),.,«(*». * ",*;)). The image of zz is a compact subset of R" (see [l]). A subset VCR" is attainable if for some zz, V = ip R": p < zz(x) for some allocation ix)\. (For p = ip y,, p ) and q = iq y,, qn), we say that p < q if p < <7 for i = 1, 2,, zz.) Another way to define V is V = Image (zz) - R". (R" = ix = (xj,, xn)\x. > 0, i = 1,, zz!.) Intuitively we think of n traders ("players") in a market of zzz commodities. u.(xy, x,, x ) represents the utility that the ;'th player has if he holds the bundle of commodities (x., x,, x ), where x. represents his portion of the total amount available of the z'th commodity. In a recent paper by Billera and Bixby it was shown that for any compact convex set C C R", the set V = C - R*l is attainable for some iu,, u~,, u) T I n on Im, where m can be chosen so that m < nin - l). They defined the coz7z- Received by the editors November 13, 1973 and, in revised form, May 13, 1974 and May 24, AMS (MOS) subject classifications (1970). Primary 90A15, 52A20; Secondary 90D12. Key words and phrases. Compact convex sets, concave utility functions, complexity, market games. Copyright American Mathematical Society 416

2 A GAME THEORETIC NOTION OF COMPLEXITY 417 plexity of V (or C) to be the minimal ttz for which this is true, and the complexity of n (com(zz)) to be the maximal complexity of an attainable set in R". It follows from the work of Billera and Bixby that com (tí) < 72(72 - l). It is easy to see that com (2) = 1, and in their paper they gave an example of an attainable set in R3 with complexity 2, i.e. 2 < com (3) < 6. One of them conjectured that com (zz) = n - 1 for all 72. In 2 we prove that for every n, n > 3, there is an attainable set of complexity n and, therefore, com (72) > 72 for n > 3. 3 we show that com(3) = 3. We are grateful to Nimrod Megiddo who made valuable comments about the problem. following 2. Complexity in R". Lemma 1. Suppose V* is an attainable set in R", n > 3, satisfying the properties: (a) the 72+1 points p* =(a, a,, a), p* = (1, 0,, 0), P* = (0, 1, 0, -.., 0),... P* = i0,, 0, 1) belong to V* with 0 < a< 1; (b) for each p = iy y,, y ) V*. if for some j, 1 < ;' < 72, y. > a, then y, < a for every k 4 j- Then com V* > n. n Proof. Assume that u = iuy,, u ) are the utility functions on lm that generate V*. Let (x) = (x;.) be the allocation for which zz(x) = p*, i.e. u {x',, x' ) = a for /: = 1, 2,, 72. Consider the bundle of the 72th player in this allocation, (x",, x"). Let / be the set of indices i for which I zzz J X.» 1. If / 4 0, assume without loss of generality that ] = \k + 1,, m\. We claim that there is an index j,, 1 < i < k, such that x\y = 0 for every /', /' 4 1» n. If not, put 8. = 2.,} nxi > 0 for 1 < i < k and 8 = min1< < S.. Consider, for the 72th player, the set of bundles B = \ix* + f., x" + r,,, x" + r ): \r.\ < 8 for I = 1, 2,, m\ D lm; ra zzz zzz ' z1 ' ' B is an open set in the relative topology of I"1. Since all the bundles of B can be allocated to the 72th player without changing the bundle of the

3 418 EHUD KALAI AND MEIR SMORODINSKY first player, it follows from property (b) that zz is bounded by a. on B. So zz has a local maximum in B and by concavity it must be a global maximum on Va. But this contradicts the requirement that p* is attained and establishes the truth of the claim. Observe that x. > 0, for if xjy = 0 then x?, = 1 and z'j > k, which is not the case. Next, repeat the above argument with the 2nd and 72th players instead of the 1st and 72th players to conclude that there is an index z', 1 < i- < k, such that x\2 = 0 tot every j, j 4 2,n, and x2 > q. Notice this implies z, 4 i2- Continuing inductively, we find z,,, i _, all distinct and all between 1 and k such that x\. = 0 for every / 4 I, n. This implies immediately that m > k > n - 1. Observe that for each player j 4 n, there are 72-2 commodities from which he gets zero quantity. Suppose that m = n - 1 so that J = 0. If this is the case, assign the roll of player 72 to player 1 to conclude that there is an index z such that x" > 0 and x\ = 0 for every j, j 4 1, w. This implies that z = z,. But since the choice of player 1 is arbitrary, we can also conclude that i = i.. Since n > 3, this is a contradiction, and the proof of the Lemma is completed. Remark. It follows from the proof of Lemma 1 that in the case where 77Z = n > 3, the allocation for the point p* is the unit matrix up to a renaming of the commodities. Theorem 1. The attainable set V in R", n > 3, generated by the convex hull of the 72+1 points Pl = ivi,v2,,la), p* = ii,o,o,.-.,o), P2* = (o, 1,0,...,o), -, p* = io,---,o,d has complexity n. Proof. To show that the complexity of V is at least 72, it is enough to show that V satisfies condition (b) of Lemma 1. Let p be a point in the convex hull of \p^,, p*\; then p = "=0 X p*, where X > 0 and 2^"=0^/ = 1. Denoting p = iyy,, yn), we then have for z 4 ] > which proves property (b). y. J1 + y. J j = A 0 + A. 1 + A. j < 1, ' To show that com Vn < n, we exhibit 72 utility functions from /" to R that generate V. Define

4 A GAME THEORETIC NOTION OF COMPLEXITY 419 zz.(x,,, x )'=: \x. + min x. I z l ' zz -, I z ; I 2 L IS/i«J for i = 1, Denote by V the attainable set of zz = (zzj,, zzn). pg is attained at the allocation that assigns to each player / the total amount of the ;'th commodity. For every player ;', p* is attained at the allocation that assigns to player ; the total amount of all the commodities. Therefore V 3 V. To show the converse inclusion, we start with an arbitrary allocation x = (xjj and show that the image of x under zz is a point p = iyy,, y ) which is less than a point in the convex hull of ip*,, p*\. Without loss of generality assume that for every player /, x1. = ß. if i 4 ji x\ = a- and a> ß.. Consider the point p which is the image of the allocation that assigns to every player different from 1 and 2 the same amounts as in x, assigns to player 1 (ctj - ß Jt 0,, 0) and assigns to player 2 iß2 + ßy, ß y, ß2 + ßy,, ß2+ ßy). Also consider the point p obtained in the same way as p, by interchanging the roles of players 1 and 2. pisa convex linear combination of p and p. Now starting from p and p and making the changes in the bundles of the 2nd player (or 1st in the case of p ) and the 3rd player, and so on, we finally arrive at allocations where all the off diagonal entries are zero except for one row. We infer that p is a convex linear combination of the images of such allocations of the following form: xl = y and xí = 0 ií i 4 j fot every j'4 k; x* = 1 and xk = 1 - y for i 4 k (where k is the row with the off diagonal nonzero entries). If p is the image of such an allocation, then p = iy y,, yn), where y. = Y2y ii j 4 k and yk=v2 +ÍV2- My) = 1 - Viy- So p = yp* + (l - y)p*, and we conclude that p is a convex linear combination of ip*,, p*). 3. Complexity in R3. In this section we prove that any attainable set V in R5 can be attained by three continuous, concave, monotone in each variable, utility functions on /^. Theorem 2. com(3) = 3- Proof. In view of Theorem 1 we have to show only that com (3) < 3- Let C be a compact convex set in R. Assume without loss of generality (see [l, p. 262]) that maxíyj: iy y, y 2, y A) C tot some y2> y^\ = 1 and that miniy,: (y,, y 2, y A C tot some y,, ya = 0. Also assume the same conditions for the 2nd and 3rd players. Let V = C R\. Consider the auxiliary function on /,

5 420 EHUD KALAI AND MEIR SMORODINSKY gix y, x2) = max\s R: ix y + 8, 1 - X. +8, 1 - x2) e V\. It is easy to verify that g is continuous and concave. Now we shall define utility functions on I* and prove that these utility functions generate V. UyiXy, X2, Xj) = Xy+ g^, X^, uax, x, xa - x. + g(l - x,, xa, zz3(xr x2, x3) = min x2, x^. Let V be the attainable set of zz = iuy, zz,, zz,). To see that V ^ V, consider any point y = iy., y 2, y A C. Choose 8Q such that y y + z5q + y, + (5n = 1. It follows that 0 < y. + 8Q < I, i = 1, 2. Consider the allocation.i... * >r,.. i - *1 = >1 + S0' *2 = 2 -y3' x3 = 0> x, = y + (5, x, = 0, x, = 1 - y,, 1 ' 2 0' 2 3 '3 x^ = o, *2 = y3' *)->* A straightforward calculation of zz((x'.)) shows that zz((xp) = y. To prove that V C V, let (x7.) = x be any allocation. It is easy to see that Uy, u2, u, ate monotonically increasing in each variable, so without loss of generality it can be assumed that x* = 0, x2 = 0, x^ = 0, and x^ = x3. If z5 = g(xj, 1 - xp, then there is a point (y1? y2, y A V such that yj=xj+z5,y2 = 1-Xj + S and y, = x3. But these y 's are the values of the zz.'s for the given allocation x. REFERENCE 1. Louis J. Billera and Robert E. Bixby, A characterization of Pareto surfaces, Proc Amer. Math. Soc. 41 (1973), DEPARTMENT OF STATISTICS, TEL AVIV UNIVERSITY, TEL AVIV, ISRAEL