Roberts Theorem with Neutrality. A Social Welfare Ordering Approach

: A Social Welfare Ordering Approach Indian Statistical Institute joint work with Arunava Sen, ISI

Outline Objectives of this research Characterize (dominant strategy) implementable social choice functions in quasi-linear environments when agents have multidimensional types. Use social welfare ordering approach, as in aggregation problems with utility information ( richer versions of the Arrovian aggregation problem). Investigate if this approach works for any restricted domain. Our results A simple proof of Roberts theorem with neutrality. Roberts theorem with neutrality holds for a particular restricted domain.

Our Approach Step 1: Every implementable social choice function satisfies a condition called positive association of differences (PAD) - Roberts, 1979. Step 2: If the domain is unrestricted or the set of all non-negative types then: Step 2a: If a social choice function satisfies PAD and is neutral, then it induces an ordering on the domain. Step 2b: This ordering satisfies three axioms: weak Pareto, invariance, and continuity. Step 2c: Every ordering satisfying these axioms can be characterized as weighted welfare maximizers. We can use classical results in aggregation theory such as Blackwell and Grishick (1954) and Milnor (1954).

The Model Finite set of m alternatives: A = {a, b, c,...}. Assume m 3. Set of agents N = {1, 2,..., n}. Type of agent i: t i = (t a i, tb i,..., ) - a vector in Rm. Type profile of agents: t (n m matrix) - n vectors in R m. Possible types of agent i is T and possible type profiles of agents is T. t a is the column vector corresponding to alternative a. Possible column vector of any type profile matrix is C T.

Dominant Strategy Implementation A social choice function (SCF) is a mapping f : T A. A payment function is a mapping p : T R n, where p i (t) denotes the payment of agent i when the type profile is t T. A social choice function f is implementable if there exists a payment function p such that for all i N, for all t i T i we have t f (t) i p i (t) t f (s i,t i ) i p i (s i, t i ) s i, t i T. What social choice functions are implementable?

A Property of Implementable SCFs A social choice function f satisfies positive association of differences (PAD) if for every s, t T such that f (t) = a with s a t a s b t b for all b a, we have f (s) = a. Lemma (Roberts, 1979) Every implementable social choice function satisfies PAD.

Roberts Theorem A social choice function f satisfies non-imposition if for every a A, there exists t T such that f (t) = a. Theorem (Roberts, 1979) Suppose C T = R n. If an implementable social choice function satisfies non-imposition, then there exists weights λ R n + \ {0} and a deterministic real-valued function κ : A R such that for all t T, [ f (t) arg max λ i ti a + κ(a) ] a A i N

A Choice Set Assumption: C T = R n (unrestricted domain) or R n +. The choice set of an SCF f at every type profile t is C f (t) = {a A : ε 0, f (t a + ε, t a ) = a}. Lemma If f is implementable, then for all type profiles t, f (t) C f (t).

Social Welfare Ordering A social welfare ordering (SWO) R f induced by a social choice function f is a binary relation on C T defined as follows. The symmetric component of R f is denoted by I f and the antisymmetric component of R f is denoted by P f. Pick x, y C T. We say xp f y if and only if there exists a profile t with t a = x and t b = y for some a, b A such that a C f (t) but b / C f (t). We say xi f y if and only if there exists a profile t with t a = x and t b = y for some a, b A such that a, b C f (t).

Why R f is an Ordering Lemma Let f be an implementable social choice function. Consider two type profiles t = (t a, t b, t ab ) and s = (s a = t a, s b = t b, s ab ). a) Suppose a, b C f (t) and a C f (s). Then b C f (s). b) Suppose a C f (t) but b / C f (t). Then b / C f (s).

Why R f is an Ordering - Neutrality A social choice function f is neutral if for every type profile t T and for all permutations ϱ on A such that t s, where s is the type profile due to permutation ϱ, we have ϱ(f (t)) = f (s). Proposition Suppose f is an implementable and neutral social choice function. The relation R f induced by f on C T is an ordering.

Three Axioms for an Ordering An ordering R on C T satisfies weak Pareto (WP) if for all x, y C T with x y we have xpy. An ordering R on C T satisfies invariance (INV) if for all x, y C T and all z R n such that (x + z), (y + z) C T we have xpy implies (x + z)p(y + z) and xiy implies (x + z)i (y + z). An ordering R on C T satisfies continuity (C) if for all x C T, the sets U x = {y C T : yrx} and L x = {y C T : xry} are closed in R n.

An SWO Satisfies these Axioms Proposition Suppose f is an implementable and neutral social choice function. Then the social welfare ordering R f induced by f on C T satisfies weak Pareto, invariance, and continuity.

Characterizing the Ordering Suppose an ordering on C T satisfies WP, INV, and C. Can we say anything about the ordering? See d Aspremont and Gevers (2002). Proposition (Blackwell and Girshick (1954)) Suppose an ordering R on R n satisfies weak Pareto, invariance, and continuity. Then there exists weights λ R n + \ {0} and for all x, y R n xry i N λ i x i i N λ i y i. We show that this result holds even if R is an ordering on R n +.

Theorem Suppose f is an implementable and neutral social choice function. Then there exists weights λ R n + \ {0} such that for all t T, f (t) arg max a A λ i ti a. i N

Final Comments and Future Directions If we impose anonymity (permuting rows), then λs become equal (i.e., f is efficient) in Roberts theorem. An easy proof using our approach and an elegant result of Milnor (1954) exists for this case. Can we drop neutrality and prove the general version of Roberts theorem using our approach? Note: Our results are slightly more general than Roberts theorem with neutrality since our result holds for R n + also. Can we extend this approach to private goods settings (where every SCF satisfies PAD)?

Anonymity A social choice function f is anonymous if for every t T and every permutation σ on the row vectors (agents) of t we have f (σ(t)) = f (t). An ordering R on C T satisfies anonymity if for every x, y C T and every permutation σ on agents we have xiy if x = σ(y). Lemma Suppose f is implementable and anonymous. Then, R f satisfies anonymity.

A Characterization of Efficiency Theorem Suppose f is implementable, neutral, and anonymous. Then f is the efficient social choice function.

Sketch of Proof The proof is an adaptation of an elegant proof of Milnor (1954). Suppose we have an ordering R which satisfies WP, INV, and Anonymity. Consider n = 3 and take x = (5, 3, 4) and y = (6, 1, 5). We show that xiy. Consider x = (3, 4, 5) and y = (1, 5, 6). By Anonymity, xix and yiy. So x and y ranked same way as x and y. Consider x = (2, 0, 0) and y = (0, 1, 1). By INV, x and y ranked same way as x and y. Repeating this, we will finally get (0, 0, 0) and (0, 0, 0) to conclude xiy. If x = (5, 3, 7) and y = (6, 1, 5), we construct x = (5, 3, 7) and y = (7, 2, 6). By WP y Py, and like before we show xiy to conclude xpy.