Journal of Economic Theory 87, 429433 (1999) Article ID jeth.1999.2545, available online at http:www.idealibrary.com on
A Note on Matsushima's Regularity Condition Kim-Sau Chung* Department of Economics, University of Wisconsin, 1180 Observatory Drive, Madison, Wisconsin 53706 kchungssc.wisc.edu Received June 30, 1998; revised March 5, 1999
This note first uses a simple example to show that H. Matsushima's (1991, J. Econ. Theory 54, 198203) regularity condition, to the contrary of his claim, actually does not imply C. d'Aspremont and L.-A. Gerard-Varet's (1979, J. Public Econ. 11, 2545) compatibility condition. It then proves a stronger version of Matsushima's proposition, namely that efficient public decision rules can be truthfully implemented with budget-balancing mechanisms under the weak regularity condition. Journal of Economic Literature Classification Numbers: C72, D82. 1999 Academic Press
1. INTRODUCTION Matsushima [3] investigated the Bayesian collective choice problem with full transferability. He introduced the regularity condition under which efficient public decision rules (PDRs) can be truthfully implemented with budget-balancing mechanisms. He also claimed that the regularity condition implies d'Aspremont and Gerard-Varet's [2] compatibility condition (the same claim also appeared in Matsushima [4]). This note first uses a simple example to show that Matsushima's claim is false, and it then proves a stronger version of Matsushima's proposition, namely that efficient PDRs can be truthfully implemented with budget-balancing mechanisms under the weak regularity condition. As the general setup of the problem is well understood in the literature, we shall introduce the notation and definitions without much comment. The reader is encouraged to consult Matsushima's original paper for further details.
* I thank an associate editor and two referees for comments.
429 0022-053199 30.00 Copyright 1999 by Academic Press All rights of reproduction in any form reserved.
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N is a finite set of agents with |N| 3. Q is a finite set of feasible public decisions. For any i # N, S i is a finite set of possible types of agent i. S= _i # N Si is the set of states. For any i # N, q # Q, s i # S i , and transfer r i , agent i 's utility is u i (q, s i )+r i . A PDR is a function g: S Ä Q. An efficient PDR is a PDR g* such that \s # S, g*(s) # argmax q # Q i # N u i (q, s i ). A transfer rule is a function t: S Ä R |N| , with the i th component, t i (s), to be interpreted as a transfer to agent i at state s # S. A direct mechanism is a pair (g, t). The Groves mechanism is the pair (g*, t~ ), where g* is an efficient PDR, and \i # N, \s # S, t~ i (s)= k{i u k ( g*(s), s k ). For any i # N, a pure strategy is a function _ i : S i Ä S i . The truth-telling strategy profile is the vector _*=(_ *) i i # N such that \i # N, \s i # S i , _ i* (s i )=s i . For any i # N and s i # S i , p( } | s i ) is agent i 's belief on S &i conditional on s i , and p^ i ( } | s i , j) is agent i 's belief on S &i& j conditional on s i (notice that p^ i ( } | s i , j) can also be regarded as an |S &i& j |-dimensional vector). For any agent i # N, state s i # S i , PDR g, transfer rule t, and strategy profile _, v i (_, s i | g, t)= s &i # S&i [u i (g(_(s)), s i )+t i (_(s))] p(s &i | s i ) is agent i 's conditional expected utility. Definition 1. The regularity condition is satisfied if there exist i{ j # N such that the vectors p^ i ( } | s i , j), s i # S i , are linearly independent. Definition 2. Denote k: S Ä R, * i : S i _S i Ä R + , and *=(* i ) i # N . The compatibility condition is satisfied if for any k and *, if \i # N, \s # S, s$i # Si * i (s i , s$i )[ p i (s &i | s i )& p i (s &i | s$i )]=k(s), then k( } )#0.
2. AN EXAMPLE This example shows that the regularity condition, to the contrary of Matsushima's claim, actually does not imply d'Aspremont and GerardVaret's compatibility condition. Let N=[1, 2, 3]. For any i # N, Si =[a i , b i ]. The agents share the following common prior on S: p(a 1 , a 2 , a 3 )=0.15,
p(a 1 , a 2 , b 3 )=0.125,
p(a 1 , b 2 , a 3 )=0.1,
p(a 1 , b 2 , b 3 )=0.125,
p(b 1 , a 2 , a 3 )=0.1,
p(b 1 , a 2 , b 3 )=0.125,
p(b 1 , b 2 , a 3 )=0.15,
p(b 1 , b 2 , b 3 )=0.125.
MATSUSHIMA'S REGULARITY CONDITION
431
TABLE I k(a 1 , a 2 , a 3 )=
0.1=
0.1_* 1 (a 1 , b 1 )=
k(a 1 , a 2 , b 3 )=
0=
0_* 1 (a 1 , b 1 )=
0.1_* 2 (a 2 , b 2 )=
0.05_* 3 (a 3 , b 3 )
0_* 2 (a 2 , b 2 )= &0.05_* 3 (b 3 , a 3 )
k(a 1 , b 2 , a 3 )= &0.1= &0.1_* 1 (a 1 , b 1 )= &0.1_* 2 (b 2 , a 2 )= &0.05_* 3 (a 3 , b 3 ) k(a 1 , b 2 , b 3 )=
0=
0_* 1 (a 1 , b 1 )=
0_* 2 (b 2 , a 2 )=
0.05_* 3 (b 3 , a 3 )
k(b 1 , a 2 , a 3 )= &0.1= &0.1_* 1 (b 1 , a 1 )= &0.1_* 2 (a 2 , b 2 )= &0.05_* 3 (a 3 , b 3 ) k(b 1 , a 2 , b 3 )=
0=
0_* 1 (b 1 , a 1 )=
0_* 2 (a 2 , b 2 )=
0.05_* 3 (b 3 , a 3 )
k(b 1 , b 2 , a 3 )=
0.1=
0.1_* 1 (b 1 , a 1 )=
0.1_* 2 (b 2 , a 2 )=
0.05_* 3 (a 3 , b 3 )
k(b 1 , b 2 , b 3 )=
0=
0_* 1 (b 1 , a 1 )=
0_* 2 (b 2 , a 2 )= &0.05_* 3 (b 3 , a 3 )
The regularity condition is satisfied because p^ 1 (a 2 | a 1 , 3)=0.55,
p^ 1 (a 2 | b 1 , 3)=0.45,
p^ 1 (b 2 | a 1 , 3)=0.45,
p^ 1 (b 2 | b 1 , 3)=0.55.
However, by setting * 1 (a 1 , b 1 )=1,
* 1 (b 1 , a 1 )=1,
* 2 (a 2 , b 2 )=1,
* 2 (b 2 , a 2 )=1,
* 3 (a 3 , b 3 )=2,
* 3 (b 3 , a 3 )=0,
we have k( } )0 (see Table I). Therefore our example violates the compatibility condition. Note that our example also satisfies Aoyagi's [1] Assumption 1. While Aoyagi was not sure about ``the exact relationship between the compatibility condition and Assumption 1,'' we now know that neither the compatibility condition nor Aoyagi's Assumption 1 implies the other.
3. THE WEAK REGULARITY CONDITION Matsushima proved that efficient PDRs can be truthfully implemented with budget-balancing mechanisms under the regularity condition. 1 Using the same trick as in Aoyagi [1], one can prove that Matsushima's proposition holds even under a weaker version of the regularity condition. 1 It should be noted that Matsushima's original proof contains a typo. On page 201, lines 79, the correct sentence should be: ``For every s$1 # S 1 , we define a function ;[s$1 ]: S 1 Ä R such that for every s 1 # S 1 , ;[s$1 ] (s 1 )= s &1 # S &1 i # N t~ i (s &1 , s$1 ) p 1 (s &1 | s 1 ).''
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Definition 3. The weak regularity condition is satisfied if there exist i{ j # N such that for any s i {s$i # S i , p^ i ( } | s i , j){ p^ i ( } | s$i , j). The weak regularity condition is the same as the first half of Aoyagi's Assumption 1. If the second half of Aoyagi's Assumption 1 is also satisfied, then any PDR is truthfully implementable even if agents' types are mutually payoff-relevant and the PDR is not efficient. As Aoyagi pointed out, the weak regularity condition is significantly simpler than the regularity condition, as the latter involves checking the linear independence of conditional probability vectors. Suppose that the weak regularity condition holds, and let i, j be as defined in the condition. Aoyagi proved that there exists a function y: S & j Ä R such that \s i {s$i # S i , y(s &i& j , s i ) p^ i (s &i& j | s i , j)
: s &i&j # S &i&j
>
y(s &i& j , s$i ) p^ i (s &i& j | s i , j).
: s &i&j # S &i&j
Define ,= min si {s$i # Si
#= max si {s$i # Si
= min si {s$i # Si
[ y(s &i& j , s i )& y(s &i& j , s$i )] p^ i (s &i& j | s i , j),
: s &i&j # S &i&j
:
: [t~ k (s &i , s i )&t~ k (s &i , s$i )] p i (s &i | s i ),
s &i # S &i k{i
:
[u i ( g*(s &i , s i ), s i )&u i (g*(s &i , s$i ), s i )] p i (s &i | s i ).
s &i # S &i
Pick a positive number : such that :,&#+>0. Such an : exists because ,>0. Define \s # S, t^ i (s)=:y(s &i& j , s i )& : t~ k (s), k{i
t^ j (s)=t~ j (s)&:y(s &i& j , s i ), and \k{i, j, t^ k (s)=t~ k (s).
MATSUSHIMA'S REGULARITY CONDITION
433
It is clear that t^ is budget balancing. For any _ i {_ i* and any s i # S i such that _ i (s i ){_ i* (s i )=s i , v i (_*, s i | g*, t^ )&v i (_*_ i , s i | g*, t^ ) =:
[ y(s &i& j , s i )& y(s &i& j , _ i (s i ))] p^ i (s &i& j | s i , j)
: s &i&j # S &i&j
&
: [t~ k (s &i , s i )&t~ k (s &i , _ i (s i ))] p i (s &i | s i )
:
s &i # S &i k{i
+
:
[u i (g*(s &i , s i ), s i )&u i (g*(s &i , _ i (s i )), s i )] p i (s &i | s i )
s &i # S &i
:,&#+ >0. The rest of the proof that ( g*, t^ ) is incentive compatible now follows from the incentive compatibility of the Groves mechanism (g*, t~ ).
REFERENCES 1. M. Aoyagi, Correlated types and Bayesian incentive compatible mechanisms with budget balance, J. Econ. Theory 79 (1998), 142151. 2. C. d'Aspremont and L.-A. Gerard-Varet, Incentives and incomplete information, J. Public Econ. 11 (1979), 2545. 3. H. Matsushima, Incentive compatible mechanisms with full transferability, J. Econ. Theory 54 (1991), 198203. 4. H. Matsushima, Bayesian monotonicity with side payments, J. Econ. Theory 59 (1993), 107121.
