A Curious Property of Convex Functions and Mechanism Design

A Curious Property of Convex Functions and Mechanism Design Sergiu Hart August 11, 217 Abstract The solution of a simple problem on convex functions that has nothing to do with mechanism design namely, the largest convex function with given values on the axes makes use of the payoff functions of mechanism design. 1 The S-Transform Let b : R d R be a a real convex function defined on the nonnegative orthant of the d-dimensional Euclidean space; without loss of generality assume that b() =. For each z R d let s(z) := b (z;z) b(z), where b (z;z) := lim δ (b(z δz) b(z))/δ is the directional derivative of b at z in the direction z (see September 212 (first version); October 213 (revised and expanded); November 216 (minor corrections). The author thanks Noam Nisan, Phil Reny, and Benji Weiss for useful discussions. Research partially supported by an ERC Advanced Investigator Grant. The Hebrew University of Jerusalem (Center for the Study of Rationality, Institute of Mathematics, and Department of Economics). E-mail: hart@huji.ac.il Web site: http://www.ma.huji.ac.il/hart 1

Rockafellar 197). The function s satisfies s() = and s(z) = z b(z) b(z) at all points z where b is differentiable (i.e., at almost every z). When d = 1, we have s (z) = zb (z) b(z), and s is a nondecreasing function (for instance, when b is C 2 we have s (z) = zb (z) ); that is however no longer true when d > 1 (cf. Hart and Reny 215). Let d = 2. For every z R 2, let β(t) := b(tz)/t, then β (t) = s(tz)/t 2, and so one can reconstruct the function b from the function 1 s: b(z) := 1 s(tz) t 2 dt. Let Sb denote the function s obtained from b; we will call it the S- transform of b. 2 A Curious Property of the s Functions Let b 1 (x) and b 2 (y) be two nondecreasing convex functions defined on R, with b 1 () = b 2 () =. Assume for simplicity that b 1 and b 2 are continuously differentiable (i.e., in C 1 ; see below for the extension to the nondifferentiable case). Let b (x,y) be the largest convex function b on R 2 such that b(x, ) = b 1 (x) and b(,y) = b 2 (y); we will refer to this as the two-axes condition. The function b is well defined since there is always a function satisfying the two-axes condition (e.g., b(x,y) = b 1 (x) b 2 (y)), and the maximum of the collection of such convex functions is a convex function. 1 Assume for instance that b(z) = for all z in a neighborhood of. 2

Define 2 s 1 (x) := xb 1(x) b 1 (x), s 2 (y) := yb 2(y) b 2 (y), s (x,y) := xb x(x,y) yb y(x,y) b (x,y). Thus s 1 := Sb 1, s 2 := Sb 2, and s = Sb. We write λ for 1 λ. Theorem 1 Assume that 3 sup x s 1 (x) = sup y s 2 (y) >. Then for every x,y > there is < λ < 1 such that ) ( ) b (x,y) = λb 1 λ λb ỹ 2, λ ) ( ) ỹ s (x,y) = s 1 = s 2, λ λ b x(x,y) = b 1, and ( λ) ) ỹ b y(x,y) = b 2. λ Thus, b is the envelope of the b i functions along equi-s lines. What appears curious and intriguing here is that the solution of a simple problem on convex functions that has nothing to do with mechanism design (namely, the largest convex function with given values on the axes) involves the payoff functions from mechanism design (with s the seller payoff function and b the buyer payoff function). Proof. Extend b 1 and b 2 to R 2 : put b 1 (x, ) := b 1 (x) when y = and b 1 (x,y) := otherwise; similarly, put b 2 (,y) := b 2 (y) when x = and 2 The two partial derivatives of b are denoted b x and b y. 3 See below for the extension to the general case. 3

b 2 (x,y) := otherwise. Then b 1 and b 2 are convex functions on R 2 ; let b := conv {b 1,b 2 } be their convex hull (see Rockafellar 197, page 37), i.e., the greatest convex function such that b (x,y) b i (x,y) for i = 1, 2. The function b is given by 4 epi b = conv (epi b 1 epi b 2 ), and satisfies (see Theorem 5.6 in Rockafellar 197) { b (x,y) = inf λb 1 (x 1,y 1 ) λb 2 (x 2,y 2 ) : λ(x 1,y 1 ) λ(x } 2,y 2 ) = (x,y). λ 1 For x,y >, the expression λb 1 (x 1,y 1 ) λb 2 (x 2,y 2 ) is finite only when < λ < 1, (x 1,y 1 ) = (x/λ, ), and (x 2,y 2 ) = (,y/ λ), and so { ) ( )} b (x,y) = inf λb 1 <λ<1 λ λb ỹ 2 ; (1) λ for y =, it is finite only when λ = 1, and so b (x, ) = b 1 (x); and for x =, only when λ =, and so b (,y) = b 2 (y). Thus b is in fact the greatest convex function satisfying the axes condition, i.e., b b. Next, the derivative of λb 1 (x/λ) λb 2 (y/ λ) with respect to λ is ) b 1 λb 1 ( λ λ) xλ ) ( ) ( ) ỹ 2 b 2 λb ỹ 2 λ λ ( ) ỹ ) = s 2 s 1. λ λ ( ) y λ 2 This is a nondecreasing function of 5 λ, and it vanishes when ) ( ) ỹ s 1 = s 2, λ λ which yields thus its minimal value. Finally, using the envelope theorem gives b x(x,y) = λb 1(x/λ) (1/λ) = 4 epi f denotes the epigraph of f, i.e., {(z,α) R 2 R : f(z) α}. 5 With s 2 (y/ λ) s 1 (x/λ) nonnegative as λ and nonpositive as λ 1 (recall that s i () = < s i (t) for t large enough). 4

b 1(x/λ) and b y(x,y) = λb 2(y/ λ) (1/ λ) = b y (y/ λ), and thus s (x,y) = λs 1 (x/λ) λs 2 (y/ λ) = λs 1 (x/λ) = λs 2 (y/ λ). Remarks. (a) Geometrically, the graph of the function b is obtained by connecting with straight lines all pairs of points ((x, ),b 1 (x)) and ((,y),b 2 (y)) that satisfy s 1 (x) = s 2 (y). (b) If f 1 and f 2 are the buyer payoff functions in two single-good IC and IR mechanisms thus b 1(x),b 2(y) 1 for all x,y then s 1 and s 2 are the corresponding seller payoff functions. 6 In this case the functions b and s are the payoff functions of the buyer and the seller, respectively, in a two-good IC and IR mechanism. Moreover, along each line connecting (x, ) with (,y) such that s 1 (x) = s 2 (y), the corresponding menu item (q 1,q 1,s) is constant: q 1 (x,y) = b 1(x), q 2 (x,y) = b 2(y), and s(x,y) = s 1 (x) = s 2 (y). In particular, the mechanism corresponding to b is monotonic (i.e., s is a nondecreasing function); moreover, the collection of all allocations q(x, y) = (q 1 (x,y),q 2 (x,y)) is well-ordered, i.e., for any (x,y) and (x,y ), either q(x,y) q(x,y ) or q(x,y) q(x,y ). (c) Alternative characterization: For every c, let h c (x,y) be the largest affine function with h c (, ) = c such that h c (x, ) b 1 (x) and h c (,y) b 2 (y) for all x,y. Then b (x,y) = sup c h c (x,y). (d) If, say, sup x s 1 (x) =: M 1 < M 2 := sup y s 2 (y), then let ȳ be such that s 2 (ȳ) = M 1 ; the characterization of Theorem 1 holds for all (x,y) with y < ȳ, and for y ȳ we have b (x,y) = b 2 (y) (the infimum in (1) is reached as λ ). (e) If the function f i are not C 1, then f i and s i are defined almost every- 6 A function b : R R is a buyer payoff function iff it is convex, its derivatives b (x) lie in the interval [,1], and b() =. Such a function b(x) lies in the convex hull of the functions [x p] for p and the identically function. The corresponding seller payoff function is a nondecreasing function with s() =, which lies in the convex hull of the functions p1 x p for all p and the identically function (without loss of generality we have made s continuous from the right). 5

where. For every t let x t := inf(x : s 1 (x) t} and y t := inf{y : s 2 (y) t} (could be ), then the graph of b (x,y) is obtained by connecting with straight lines all pairs of points ((x t, ),b 1 (x t )) and ((,y t ),b 2 (y t )). This includes the case of (d) above where sup x s 1 (x) and sup y s 2 (y) may be different (if, say, x t = and y t is finite, then the line becomes {((x,y t ),b 2 (y t )) : x }, i.e., in (x,y) space it is parallel to the x-axis). Moreover, we have s 1 (x) = s 2 (y) = s (x,y) = t 1 x x t =1 dt = t 1 y y t =1 dt = t 1 x x t y y t =1 dt = 1 x x t 1 dt, 1 y y t 1 dt, 1 x x t y y t 1 dt, and so, for random variables X and Y, E [s 1 (X)] = E [s 2 (Y )] = E [s (X,Y )] = [ X P [ Y P P ] 1 dt, x t ] 1 dt, y [ t X Y ] 1 x t y t dt. (f) In the symmetric case where b 1 (t) = b 2 (t), it is easy to see that b(x,y) = b 1 (x y) = b 2 (x y). This is used in Theorem 28 in Hart and Nisan (212). (g) When b 1(x) [, 1] for all x then b 1 lies in the closed convex hull generated by the functions [x p] for p. Similarly for b 2. However, this does not imply that b lies in the closed convex hull of the functions 6

[a 1 x a 2 y p] with a 1,a 2 [, 1] and p. An example: b 1 (x) = max {, 12 } x 1,x 3 b 2 (y) = max {, 25 } y 1,y 3 b(x, y) = max {, 12 x 25 } y 1,x y 3. Then b(x,y) = 1 [ 3 3 2 x 6 ] 5 y 3 2 [ 3 3 4 x 9 = 1 [x 45 ] 2 y 2 3 5 ] 1 y 3 ] [ 5 6 x y 1 3 (in the first decomposition, where the weights 1/3 2/3 = 1, we have the linear coefficients 3/2, 6/5 > 1; in the second decomposition, where all linear coefficients are 1, we have 1/2 3/5 > 1). Note that b 1 (x) = 1 2 [x 2] 1 2 [x 4] b 2 (y) = 2 [ y 5 ] 3 [ y 1 ] 5 2 5 3 3 A Two-Good Revenue Maximization Problem Theorem 2 Let F be a two-dimensional cumulative distribution function with density function f. Assume that there is a = (a 1,a 2 ) such that f(x,y) = when x < a 1 or y < a 2, and for (x,y) a the function f(x,y) is differentiable and satisfies xf x (x,y) α 1 f(x,y) and yf y (x,y) α 2 f(x,y) (2). 7

for some α 1,α 2 with α 1 α 2 = 3. Then, to maximize revenue, it suffices to consider functions b as obtained from Theorem 1. Proof. Let b correspond to a two-dimensional IC and IR mechanism, then R(b, F) = sup (xb x (x,y) yb y (x,y) b(x,y))f(x,y) dx dy. M>a 1,a 2 a 2 a 1 For each y we integrate by parts the xb x (x,y)f(x,y) term: a 1 b x (x,y)xf(x,y) dx = [b(x,y)xf(x,y)] M a 1 = b(a 1,y)a 1 f(a 1,y) b(m,y)mf(m,y) a 1 a 1 b(x,y) (f(x,y) xf x (x,y)) dx. b(x,y) (f(x,y) xf x (x,y)) dx Similarly for the yb y (x,y)f(x,y) term; altogether (do not forget the b(x,y)f(x,y) term) we get r M (b) = a 1 M a 2 b(a 1,y)f(a 1,y) dy a 2 a 2 a 2 a 1 a 1 a 2 b(m,y)f(m,y) dy M a 1 a 1 b(x,a 2 )f(x,a 2,y) dx b(x,m)f(x,m) dx b(x,y) ( α 1 f(x,y) xf x (x,y)) dx dy b(x,y) ( α 2 f(x,y) yf y (x,y)) dy dx (we split the 3b(x,y)f(x,y) term into two parts, α 1 b(x,y)f(x,y)α 2 b(x,y)f(x,y), which appear in the last two integrals). Fixing the functions b(,a 2 ) and b(a 1, ), and thus the first two integrals above, in order to maximize r M (b) we should take b(x,y) as large as possible (all the coefficients of b in the other integrals are nonnegative), and thus b is the maximal convex function with the given values on the axes. Finally, let 8

M. If F is a product distribution (i.e., the goods values are independent: F = F 1 F 2, with densities f 1 and f 2, respectively), then condition (2) becomes xf 1(x) α 1 f(x) and yf 2(y) α 2 f(y) for α 1 α 2 = 3; cf. Wang and Tang (217). If moreover F 1 = F 2 (i.e., i.i.d. goods), then b is symmetric and 7 b(x,y) = b(a,x y a) = b(x y a,a), and so b corresponds to bundling; cf. Theorem 28 in Hart and Nisan (212). Remark. The resulting function s is monotonic; cf. Hart and Reny (215). 4 Higher Dimensions 4.1 One-Dimensional Axes Conditions Let b i : R R be convex functions with b i () =, and let b : R d R be the maximal convex function such that b(xe i ) = b i (x) for every x and 1 i d, where e i is the i th unit vector in R d. Theorem 1 and its proof easily generalize to any dimension d. Theorem 3 Assume that sup x s i (x) = S > for all i. Then for every x R d there are < λ i < 1 with d i=1 λ i = 1 such that b (x) = d ( ) xi λ i b i, i=1 λ i ( ) s xi (x) = s i, and λ ( i ) b i(x) = b xi i. 7 Make the change of variables x = x a and y = y a. λ i 9

4.2 Higher-Dimensional Axes Conditions In general, given boundary conditions b i, the maximal convex function b is given by (cf. (1)) { b (x) = inf λ i b i (y i ) : i i λ i y i = 1, i λ i = 1, λ i The parallel of Theorems 1 and 3 is slightly more complicated. }. (3) We illustrate it with the special case where d = 3 and the given boundary conditions are b 1 (,x 2,x 3 ),b 2 (x 1,,x 3 ) and b 3 (x 1,x 2, ). If the infimum in (3) is attained at a point where all λ i >, then we have y i j R for i j with x = 3 i=1 λ iy i and (µ 1,µ 2,µ 3,ν) such that s i (y i ) = ν = s (x) b i (y i ) = µ x j = b (x) j x j b (x) = 3 ( λ i b ) i y i. (If some λ i = then the corresponding first-order conditions become inequalities.) i=1 If in addition we are in the symmetric case where b i β for all i, then b is also symmetric, and it is given by b (x) = { β ( 1 2 (x 1 x 2 x 3 ), 1 2 (x 1 x 2 x 3 ) ), if x i 1 2 (x 1 x 2 x 3 ) for all i, β(x j x k,x i ), if x i 1 2 (x 1 x 2 x 3 ) for some i. Note that x 1 (x 1 x 2 x 2 )/2 is equivalent to x 1 x 2 x 3 (and in this case we get λ 1 = in (3)). 1

References Hart, S. and N. Nisan (212), Approximate Revenue Maximization with Multiple Items, The Hebrew University of Jerusalem, Center for Rationality DP-66; arxiv 124.1846; revised (217). Hart, S. and P. J. Reny (215), Maximal Revenue with Multiple Goods: Nonmonotonicity and Other Observations, Theoretical Economics 1, 893 922. Rockafellar, T. R. (197), Convex Analysis, Princeton University Press. Tang, P. and Z. Wang (217), Optimal Mechanisms with Simple Menus, Journal of Mathematical Economics 69, 54 7. 11