Incentive compatibility without compensation

economics letters ELSEVIER

Economics Letters 47 (1995) 35-39

Incentive compatibility without compensation Pierre de Trenqualye Department of Economics, University of York, Heslington, York Y01 5DD, UK Received 1 March 1994; revised version received 24 June 1994; accepted 6 June 1994

Abstract

Standard public good mechanisms rely on a transferable medium of compensation to overcome incentive compatibility constraints (Groves, Laffont). If no such medium exists, and information is incomplete, the available rules with satisfactory welfare and incentive compatibility properties require a unidimensional set of alternatives. We define a dynamic procedure for two-dimensional economies that is Pareto optimal and individually rational, with incentives as those of rules exploiting compensatory transfers.

Keywords: Incentive compatibility JEL classification: D82

I. Introduction

In a public good economy where the share of the cost financed by each agent is determined exogenously and compensatory transfers are ruled out (e.g. the political-economic model of the European Community investigated recently by Feinstein, 1992), a rule in the class of 'generalized Condorcet winners' described by Moulin (1988), implements Pareto efficient and individually rational allocations with dominant strategies, provided preferences satisfy singlepeakedness. Single-peakedness is seen as restrictive, however, unless the space of alternatives is unidimensional (i.e. there is one public good), which motivates the mechanism provided here for two-dimensional economies. Under complete information, Pareto efficient and individual rational allocations can be implemented with Nash equilibrium (Maskin, 1985). We focus on the incomplete information case and define a rule closely related to the dynamic procedures for public good provision in the literature (e.g. Dr6ze and de la Vall6e Poussin, 1971; Malinvaud, 1972; Laffont and Maskin, 1983; Chander, 1993), which is Pareto optimal, individually rational, convergent, and locally incentive compatible. 0165-1765/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0165-1765(94)00528-A

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P. de Trenqualye / Economics Letters 47 (1995) 35-39

2. The model 2.1. The environment

T h e r e are n agents (n/> 2) choosing an alternative x E R~(k <~2). A g e n t i has preferences r e p r e s e n t e d by the utility function u i : R ~---~R, assumed to be twice continuously differentiable and strictly quasi-concave. Let a 0 E R k be the status quo, or initial solution. We assume that there are ch > 0 , . . . , o~, > 0 , such that the map (x)---~ Zi aiu~(x) has a maximum. 2.2. The one-dimensional case

Suppose k = 1. Consider a rule in the class of generalised C o n d o r c e t winners described by Moulin (1988): each agent is asked to cast a vote for the preferred alternative (or to a n n o u n c e +o0 if utility is increasing, or -o0 if it is decreasing). The rule selects the m e d i a n vote a m o n g the votes by agents and (n - 1) votes (by 'phantom' voters) that coincide with the status quo. T h a t is to say, x * = m e d i a n { v 1 , . . . , v,, z ~ , . . . , z , _ l } , where v,. is the vote of agent i, for i -- 1, n, and zg = a 0, i = 1, n - 1. Because this rule belongs to the class described by Moulin (1988), we know it makes truthful behaviour (i.e. announcing the true m a x i m u m utility point, or the p r e f e r r e d direction) a dominant strategy. It is straightforward to show that the rule is also Pareto efficient and individually rational: if votes are all at one side of the status quo then the rule selects the vote that is closer to the status quo, which is individually rational and Pareto efficient. If there are votes at each side of the status quo, then the status quo is both Pareto efficient and individually rational, and the rule selects it. Theorem 1. The incentive compatible rule defined above is Pareto efficient and individually rational.

3. Two-dimensional economies Let us first give some definitions and an intermediate result. Given the set of vectors (m 1, . . . , m , ) , with mi in R2\{0}, let C ( m 1, . . . , mn) denote the convex cone g e n e r a t e d by all non-negative linear combinations of the vectors m/. Formally, C ( m 1, . . . , m , ) = {x E R 2 Ix = Ei l~imi, ~i ~ 0}. D e n o t e by K ( m l , . . . , ran) the set of 'favourable' directions, that is to say, the set of vectors that m a k e an acute angle with every vector mg, i = 1, n. Formally, K(ml,...,mn)={xER21(x, m i ) > O , for i = l , n } . The set K ( m l , . . . , m , ) is, clearly, a convex cone (see, for example, Rockafellar, 1970). L e m m a 1. Let (m I , . . . , m , ) , with m i ~ R2\{0}, be given. I f there is a vector d ~ R z m a k i n g an acute angle with every vector m i, i = 1, n, [i.e. if K ( m 1, . . . , m , ) is not empty], then there are indices a,b E { 1 , . . . , n} such that C ( m a , . . . , m , ) = C(ma, mo).

We just outline the proof. Suppose the vector d makes an acute angle with all given vectors. The line through the origin with normal having the same direction as d, contains the whole

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cone C ( m l , . . . , mn) on one side, and shares only the origin with the cone. Clearly, any vector in { m l , . . . , m,} is a non-negative combination of two vectors in the boundary of this cone. The result follows.

3.1. Mechanism and result Agents are required to announce, at each time t, the normalised gradient of their utility function at the current state x(t), or the vector zero if the existing state is their ideal point (i.e. the point at which utility is maximised). That is, agent i announces

mi(x, t) -

grad ui(x, t)

Ilgrad ui(x, t)][

'

if grad(ui(x))

(0, 0),

and mi(x, t) = 0, otherwise.

The state at time t, x(t), is revised as represented by the following differential equations: ~(ma(x, t) +mb(X, t)), A(t) = [.0, where

(a,b)

are

ifK(ml(x, t ) , . . . , m,(x, t)) is not empty, if K(m 1, . . . , mn) is empty,

the indices of two agents,

C ( m a ( X , t), m b ( X , t)) = C ( m l ( x , t ) , . . .

a~b,

a, b E { 1 , . . . . , n }

such that

, m n ( X , t)). W e say in this case that agents a and b a r e

'boundary agents' at time t. (The lemma guarantees that the mechanism is well defined.) Notice that in two-agent economies, this mechanism coincides with the one defined in de Trenqualye (1992).

Definition. A dynamic mechanism is 'Pareto optimal' if and only if its stationary points are Pareto optimal, 'convergent' if any solution of the system of differential equations defined by the mechanism converges to a stationary point, 'individually rational' if the utility of each individual increases along the trajectory, and 'strongly locally individually incentive compatible' if revelation of the true normalised gradient (or the vector zero) is a local dominant strategy (Fujigaki and Sato, 1981; Champsaur and Rochet, 1983; Laffont and Maskin, 1983).

Theorem 2. In a two-dimensional economy as previously defined the dynamic mechanism above is Pareto efficient, individually rational, convergent, and strongly locally individually incentive compatible. Notice that our mechanism relies on incentives that are weaker than those characterising mechanisms based on dominant strategies. But this cannot generally be avoided in economies with two (or more) dimensions (Gibbard, 1973; Satterthwaite, 1975; Satterthwaite and Sonnenschein, 1981).

Proof. We show (1) Pareto efficiency; (2) convergence; (3) individual rationality; (4) incentive compatibility. (1) Consider first Pareto efficiency. From the strict concavity assumption on utility functions it follows that x ~ R 2 is Pareto efficient if and only if for any vector d E R 2 there is an index

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i E {1, . . . , n} such that (d, grad ui(x ) ) ~< 0. Suppose x is a stationary point. T h e n , either o n e or m o r e a n n o u n c e m e n t s are zero (i.e. the stationary point x is the ideal point of one or m o r e agents), or there is no direction that makes an acute angle with all gradients. This proves Pareto efficiency. (2) We now show convergence. We can restrict attention to the case where 2(t) ~ 0, for all t > 0. T a k e an arbitrary index i ~ { 1 , . . . , n}. By definition of the mechanism: ui(t) =

(grad(ui(x)) ' grad(u.)

grad(uv) Hgrad(ua)l [ + ][grad(Ub)]l),

where the two vectors, grad(u,)/llgrad(u,)H and grad(Ub)/llgrad(Ub)]l are such that grad(ua) g r a d ( u i ( x ) ) = Aa

grad(Ub)

Ilgrad(ua)ll "-[- /~b ]lgrad(Ub)H ,

with ha~>0, hb>~0 , (ha, h 6 ) ~ ( 0 , 0 ) ) . T h e n /is(t)>0, for i = 1, n. Define the function V(t) = E i aiui(t ). Clearly, V(t) is increasing. By assumption, it is b o u n d e d . Since the mapping: x--* Zi ague(x) is strictly quasi-concave and (by assumption) has a m a x i m u m , any set of the f o r m {x[ Zi a~ui(x) ~> c, c E R} is compact. T h e n the trajectory x(t), t > 0, stays in a c o m p a c t set. T o prove that x(t) converges we show that it has only one limit point. A s s u m e to the contrary that p and q are two different limit points of x(t). Since rig(t)>0, we have u,(p) > u,(q) and u~(q) > ua(p), a contradiction. (3) Individual rationality follows from ui(t)> 0 w h e n e v e r 2 ( 0 ~ 0. (4) Finally, let us prove that the rule is strongly locally individually incentive compatible (SLIIC). Without loss of generality we focus on agent 1 and assume that the current state x is not the ideal point (i.e. the m a x i m u m utility point) of agent 1. Let gl d e n o t e agent l's n o r malised gradient. First, assume K(g 1, m 2 , . . . , m,) is empty. T h e n a truthful a n n o u n c e m e n t gives zero payoff to agent 1. Besides, the direction of any revision makes an acute angle with all a n n o u n c e m e n t s , and therefore with the vectors ( m 2 , . . . , m n ) . Consequently, it c a n n o t m a k e an acute angle with gl (as K is empty). A truthful a n n o u n c e m e n t maximises the payoff. Secondly, suppose K(g~, m 2 , . . . , m,) is not empty. T h e n agent l's payoff is higher u n d e r m~ = g~ than u n d e r m I = 0 (as m I = 0 stops revisions). Let us n o w check that agent 1 does not find it beneficial to deviate from truthful behaviour in other ways. Since K ( g l , m 2 , . . . ,m,,) is not empty, then K(m2,... ,m,) is not empty. Suppose first that K ( m 2 , . . . , m,) is one-dimensional. That is to say, suppose that there exists e E R 2 such that K ( m 2 , . . . , m , ) = {x E R2]x = / z e , / z I>0}. T h e n the allocation is revised according to: 2 ( 0 = rn 1 + e/I]e]]. T h e n i l l ( t ) = (grad(ul(x)), m~ + e/Hell ). F r o m the C a u c h y - S c h w a r z inequality: (grad(Ul(X)), m 1) ~< Hgrad(Ul(X))l] Ilml]]. T h e m a x i m u m rate of change in utility is therefore achieved by announcing m 1 = g~. Suppose now that K(m 2, . . . , m~) is not one-dimensional. Let a and b d e n o t e the indices of two agents such that C(m,, m b ) = C ( m 2 . . . . , m,). Without loss of generality we represent a message by the angle it makes with a given axis. Let 0, = 0 represent agent a's message, 0b agent b's message, 01 agent l's truthful a n n o u n c e m e n t , and 0 the actual message sent by agent 1. We distinguish two cases: (i) 0 ~< 01 ~< 0b; (ii) 0b < 01 < ~. [K(g~, m 2 , . . . , m,) is otherwise empty.]

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Case (i). In this case gl E C(ma, mb). T h e n the payoff u n d e r truthful behaviour, H * , is lI* = til(t ) = Ilgrad(ul(x))ll{cos(01)+ cos(0b - 0 1 ) } . Distinguish four conditions: (a) 0 ~< 0 ~< Oh; (b) Ob (Ob -- 01), H * > H b. If (c), t h e n K ( m l , . . . ,ran) is empty, leading to zero payoff. If (d), then agents 1 and b are ' b o u n d a r y agents' and the payoff is H a = t i l ( / ) = Ilgrad(ul(x))ll{cos(01 - oh) + cos(0 - 0 l ) ) . Since cos(0 - 01) < c o s ( 0 1 ) , / / * >/70Case (ii): ob < 01 < ~r. That is to say, suppose that the normalised gradient of agent 1 is not in C(ma, mb). If m 1 = gl, then agents 1 and a are b o u n d a r y agents. A g e n t l's payoff u n d e r truthful behaviour, H * , is then H*=til(t)=llgrad(ul(x))ll(l+cos(O~) ). As before, we distinguish four conditions: (a) 0 ~< 0 < Oh; (b) 0 b ~ 0 < '~T; (C) 71"~ 0 ~ "ff + Ob; (d) 7r + 0b < 0 < 27r. If (a), then agents a and b are b o u n d a r y agents. T h e n til(t ) = [[grad(ul(x))l[(cos(ol)+ cos(01 - Ob)}, and the payoff is lower than u n d e r truthful behaviour. If (b), then 1 and a are b o u n d a r y agents and the m a x i m u m is achieved with m I = g~. If (c), then K ( m l , . . . , ran) is e m p t y and agent l's payoff is zero. Finally, if (d), then 1 and b are ' b o u n d a r y agents'. T h e payoff is til(t ) = Ilgrad(Ul(X))l] (cos(0~ - 0b) + cos(01 - 0)}. Since cos(01 - - Ob) < 1 and cos(0~ 0 ) < c o s ( 0 1 ) , H o < ~ I I *. Q . E . D .

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