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Dominant strategy mechanisms with mutually payoff-relevant private information and with public information
Economics Letters 34 (1990) 109-112 North-Holland
109
Dominant strategy mechanisms with mutually payoff-relevant private information and with public information * Hitoshi
Matsushima
Institute of So&-Economic
Planning
University of Tsukuba, Tsukuba, Ibaraki 305, Japan
Received 19 March 1990 Accepted 13 April 1990
We present a necessary and sufficient condition for truthful revelation in dominant strategies in the collective choice problem with full transferability, where private information is mutually payoff-relevant and there exists verifiable public information. We present a tractable sufficient condition.
1. Introduction We study the n-person collective choice problem originated by Groves (1973) where utilities are fully transferable and agents have private information. It is well known that truth-telling is not a dominant strategy in any efficient mechanism with budget balancing. In this paper, we investigate the general class of environments where there exists verifiable public information about private information. Agents can observe public information after announcing messages in a direct mechanism. The central planner chooses a public decision and a payoff-transfer with budget balancing among the agents, which depend on not only messages but also public information. The pioneering works are Johnson, Pratt and Zeckhauser (1989) and Matsushima (1989). We present a necessary and sufficient condition on a public decision rule, i.e., Condition 1, for the existence of a transfer rule with budget balancing such that truth-telling is a dominant strategy in the associated mechanism. We introduce a tractable informational condition, i.e., Condition 3, which is sufficient for the existence of such a transfer rule. Condition 3 means that for every agent, the conditional distribution of public information differs with respect to her private information. Our results are positive in contrast to the negative results by the previous authors in the class of environments without public information. * This work was supported by grants from the Japan Society for the Promotion of Science. 0165-1765/90/$03.50
0 1990 - Elsevier Science Publishers B.V. (North-Holland)
110
H. Motsushimo
/ Dominant strotegv mechanisms
2. The model N:= {l,..., n} is the finite set of agents. Si is the finite set of possible private ifnormation of agent i. sl={wl,..., w,, } is the finite set of possible public information, where h = #52. Denote S= xiCNSi and s=(sr,..., s,). A state is described by (w, S), and s2 x S is the set of states. Agent i has a probability distribution over 52 X S_i. pi(o, s_~ 1si) is the probability that public information is w and the other agents’ private information is s_~ given that agent i’s private information is si. We denote by q a public decision. Q is the set of feasible q. A decision rule is a function z:QxS-+Q. We assume unrestricted side payment with fuI1 transferability. Agent’s utility is given by ui (q, w, s) + ri, where ri is the transfer to agent i, and ui(q, w, s) is the agent i’s direct return from the public decision q. A detailed discussion about mutually payoff-relevant information can be found in Johnson, Pratt and Zeckhauser (1989). The set of all transfer payments with budget balancing is denoted by v = { r E R” : Ci E Nri = O}. A transfer rule is a function t = ( ti) : L? X S --j V. We introduce a mechanism, where agent i ‘s message space is S, and an outcome function is a decision rule and a transfer rule (z, t). A strategy for agent i is a function c#+: Si + Si - 4 is the set of all strategies for agent i. Let @ = Xi E N Gi and (z, t) to agent i given that agent i’s private 9 = ($1,. . . , &I,). The expected utility of a mechanism information is si when $I is played is
E[q 19,
C { U,(Z(a,(P(s)),WY s) +ti(W, +(s))}Pi(W,
St, Z, t] =
s-i Isi)-
(WA0
A joint strategy I$ is dominant and every si E Si,
+T is the honest strategy
for a mechanism
for agent
(z, t) if for every i E N, every joint
i, which is the identity
mapping,
strategy
(p’ E @
i.e., +F(si) = si for all si E Si.
3. The results It is clear that the honest joint
strategy
(p* is dominant
for a mechanism
(z, t) is and only if for
every (4 +, si), C
{ ti(W,
(%-,) 2
We define
,a:_., {ui(z(wp
w’(s_,,
R. Let A=(Ai)i,N.
+(s)/si)
. I
- ti(W9 cP(s))}Pi(w*
s-i
Isi)
W, S)-Ui(Z(W, +(s)/si)9 WYs)}Pi(“P s-ilsi)e +Cs)J9
G_~) = { sLi E S_i : s_~ = c~_,(s’~)}. Denote Xi : @ X Sr * R + ad We introduce a condition on a decision rule z.
k: 0 X S +
H. Maisushimo
Condition
111
/ Dominant strategy mechanisms
For every X and every k, if for every (i, w, s),
1.
c
c
9 SI,E W’(SL,,$!_,)
(Pttwl
sLils,)
C ‘j(G, S,’E s,
sir ~I’)-CP,(~~ s,
sL,]sl)x,(+j
s,‘, s,)
1
(1)
=k(w, s),
-
s,
C”i(z(w2
G(s)/st’)?
a,
S)P;(W
s-,
Is,)~,(dJ,
s;,
s,‘)
11 (2) 20.
By using Theorem 1 in Fan (1956) in the same way as d’Aspremont and Gerard-Varet Legros and Matsushima (1990), we can check that
(1982) and
For every decision rule z, there exists a transfer rule t such that o * is dominant for (z, t) Theorem I. if and only if z satisfies Condition 1. The drawback is that Condition 1 involves not only the probability structure { p, }; E N but also the utility functions { 24,}, E N. We introduce an informational condition independent of z and { u, }; E N. Condition 2. For every h, if h, (+, si, s,‘) > 0 for some i E N, some s, E S, and some s,’ f s,, then there exists no k such that the equality (1) holds for all (i, w, s). It is clear that Condition 2 is sufficient for Condition 1. It is also straightforward from Theorem 1 in Fan (1956) that proposition 2. Condition 2 ho& if and on& if there exists a transfer rule t such that for every i E N, every s, E Si, every s f s, and every Q E @, c { t;(% (m.s-,)
+WsJ
-t&J?
We show below that Condition
2 is tractable.
s-i Is;) ‘0.
We will write pi(w Is,) = C,_,pi(w,
(3)
s_, Is,), and
Isi?-..? Pi(whls~))e
P,]s,l=(Pi(wI Condition
+(s)/s,‘)}P;(c4
3.
For every i E N, for every s, E S, and every s,’ # s,,
Pi[Sil #P,[SIl* Theorem 3.
Condition 2 is equivalent
to Condition
3.
Proof. We show that Condition 3 is necessary for Condition 2. Suppose that Condition 3 does not hold. There exist i E N, s,* and s,* * # si* such that p;[s,* ] = p,[s,* * 1. For every j E N, choose ij
112
H. Matswhima
/ Dominant strategy mechanisms
arbitrarily, and denote by 4 the strategy for agent j such that Gj(sj) = ij for all sj E S,. Choose X so that A;(& s,*, s**)=X;(+, s,**, SF) > 0, and A;($, sit s,!) = 0 otherwise, and for every j E IV/{ i} and every ($J, sj, sj), A,(+, sj, sJ> = 0. Notice that the left-hand side of the equality (1) is equal to zero for all (i, w, s). This means that there exists k which satisfies the equality (1) for all (i, w, s), where k(o, s) = 0 for all (w, s). This is a contradiction of Condition 2. Next, we show that Condition 3 is sufficient for Condition 2. Suppose that Condition 3 holds. For every i E N, we define a function pi : s2 X S, -+ R so that for every (w, si),
cliCwY ‘I)=
-{1-Pi(wIs,)}2- C P,(w’Isi)2. O’#W
Notice from Condition 3 and the literature of the scoring-rule problem that for every i E N, every si E S, and every s,’ f si,
See Matsushima (1989) for the detailed proof. We define t = (ti) in the way that for every (i, w, s),
where IX, E Nti( w, s) = 0 for all (w, s). Such a t satisfies the inequality (3). From Proposition 2, we know that Condition 2 holds. •I
References d’Aspremont, C. and L.-A. Gerard-Varet, 1982. Bayesian incentive compatible beliefs, Journal of Mathematical Economics 10, 83-103. and related systems Fan, K., 1956, On systems of inequalities, in: H.W. Kuhn and A.W. Tucker, eds., Linear inequalities (Princeton University Press, Princeton, NJ) 99-156. Groves, T., 1973, Incentive in teams, Econometrica 41, 617-663. Johnson, S., J. Pratt and R. Zeckhauser, 1989, Efficiency despite mutually payoff-relevant private information: The finite case, Mimeo. Legros, P., and H. Matsushima, 1990, Efficiency of partnerships with stochastic output: A characterization result, Mimeo. Matsushima, H., 1989, Dominant strategy mechanisms with budget balancing revisited: Public information and scoring rules, Mimeo. Matsushima, H. 1990, Unique bayesian implementation with budget balancing, Mimeo. Myerson, R.B., 1979, Incentive compatibility and the bargaining problem, Econometrica 47, 61-73.
109
Dominant strategy mechanisms with mutually payoff-relevant private information and with public information * Hitoshi
Matsushima
Institute of So&-Economic
Planning
University of Tsukuba, Tsukuba, Ibaraki 305, Japan
Received 19 March 1990 Accepted 13 April 1990
We present a necessary and sufficient condition for truthful revelation in dominant strategies in the collective choice problem with full transferability, where private information is mutually payoff-relevant and there exists verifiable public information. We present a tractable sufficient condition.
1. Introduction We study the n-person collective choice problem originated by Groves (1973) where utilities are fully transferable and agents have private information. It is well known that truth-telling is not a dominant strategy in any efficient mechanism with budget balancing. In this paper, we investigate the general class of environments where there exists verifiable public information about private information. Agents can observe public information after announcing messages in a direct mechanism. The central planner chooses a public decision and a payoff-transfer with budget balancing among the agents, which depend on not only messages but also public information. The pioneering works are Johnson, Pratt and Zeckhauser (1989) and Matsushima (1989). We present a necessary and sufficient condition on a public decision rule, i.e., Condition 1, for the existence of a transfer rule with budget balancing such that truth-telling is a dominant strategy in the associated mechanism. We introduce a tractable informational condition, i.e., Condition 3, which is sufficient for the existence of such a transfer rule. Condition 3 means that for every agent, the conditional distribution of public information differs with respect to her private information. Our results are positive in contrast to the negative results by the previous authors in the class of environments without public information. * This work was supported by grants from the Japan Society for the Promotion of Science. 0165-1765/90/$03.50
0 1990 - Elsevier Science Publishers B.V. (North-Holland)
110
H. Motsushimo
/ Dominant strotegv mechanisms
2. The model N:= {l,..., n} is the finite set of agents. Si is the finite set of possible private ifnormation of agent i. sl={wl,..., w,, } is the finite set of possible public information, where h = #52. Denote S= xiCNSi and s=(sr,..., s,). A state is described by (w, S), and s2 x S is the set of states. Agent i has a probability distribution over 52 X S_i. pi(o, s_~ 1si) is the probability that public information is w and the other agents’ private information is s_~ given that agent i’s private information is si. We denote by q a public decision. Q is the set of feasible q. A decision rule is a function z:QxS-+Q. We assume unrestricted side payment with fuI1 transferability. Agent’s utility is given by ui (q, w, s) + ri, where ri is the transfer to agent i, and ui(q, w, s) is the agent i’s direct return from the public decision q. A detailed discussion about mutually payoff-relevant information can be found in Johnson, Pratt and Zeckhauser (1989). The set of all transfer payments with budget balancing is denoted by v = { r E R” : Ci E Nri = O}. A transfer rule is a function t = ( ti) : L? X S --j V. We introduce a mechanism, where agent i ‘s message space is S, and an outcome function is a decision rule and a transfer rule (z, t). A strategy for agent i is a function c#+: Si + Si - 4 is the set of all strategies for agent i. Let @ = Xi E N Gi and (z, t) to agent i given that agent i’s private 9 = ($1,. . . , &I,). The expected utility of a mechanism information is si when $I is played is
E[q 19,
C { U,(Z(a,(P(s)),WY s) +ti(W, +(s))}Pi(W,
St, Z, t] =
s-i Isi)-
(WA0
A joint strategy I$ is dominant and every si E Si,
+T is the honest strategy
for a mechanism
for agent
(z, t) if for every i E N, every joint
i, which is the identity
mapping,
strategy
(p’ E @
i.e., +F(si) = si for all si E Si.
3. The results It is clear that the honest joint
strategy
(p* is dominant
for a mechanism
(z, t) is and only if for
every (4 +, si), C
{ ti(W,
(%-,) 2
We define
,a:_., {ui(z(wp
w’(s_,,
R. Let A=(Ai)i,N.
+(s)/si)
. I
- ti(W9 cP(s))}Pi(w*
s-i
Isi)
W, S)-Ui(Z(W, +(s)/si)9 WYs)}Pi(“P s-ilsi)e +Cs)J9
G_~) = { sLi E S_i : s_~ = c~_,(s’~)}. Denote Xi : @ X Sr * R + ad We introduce a condition on a decision rule z.
k: 0 X S +
H. Maisushimo
Condition
111
/ Dominant strategy mechanisms
For every X and every k, if for every (i, w, s),
1.
c
c
9 SI,E W’(SL,,$!_,)
(Pttwl
sLils,)
C ‘j(G, S,’E s,
sir ~I’)-CP,(~~ s,
sL,]sl)x,(+j
s,‘, s,)
1
(1)
=k(w, s),
-
s,
C”i(z(w2
G(s)/st’)?
a,
S)P;(W
s-,
Is,)~,(dJ,
s;,
s,‘)
11 (2) 20.
By using Theorem 1 in Fan (1956) in the same way as d’Aspremont and Gerard-Varet Legros and Matsushima (1990), we can check that
(1982) and
For every decision rule z, there exists a transfer rule t such that o * is dominant for (z, t) Theorem I. if and only if z satisfies Condition 1. The drawback is that Condition 1 involves not only the probability structure { p, }; E N but also the utility functions { 24,}, E N. We introduce an informational condition independent of z and { u, }; E N. Condition 2. For every h, if h, (+, si, s,‘) > 0 for some i E N, some s, E S, and some s,’ f s,, then there exists no k such that the equality (1) holds for all (i, w, s). It is clear that Condition 2 is sufficient for Condition 1. It is also straightforward from Theorem 1 in Fan (1956) that proposition 2. Condition 2 ho& if and on& if there exists a transfer rule t such that for every i E N, every s, E Si, every s f s, and every Q E @, c { t;(% (m.s-,)
+WsJ
-t&J?
We show below that Condition
2 is tractable.
s-i Is;) ‘0.
We will write pi(w Is,) = C,_,pi(w,
(3)
s_, Is,), and
Isi?-..? Pi(whls~))e
P,]s,l=(Pi(wI Condition
+(s)/s,‘)}P;(c4
3.
For every i E N, for every s, E S, and every s,’ # s,,
Pi[Sil #P,[SIl* Theorem 3.
Condition 2 is equivalent
to Condition
3.
Proof. We show that Condition 3 is necessary for Condition 2. Suppose that Condition 3 does not hold. There exist i E N, s,* and s,* * # si* such that p;[s,* ] = p,[s,* * 1. For every j E N, choose ij
112
H. Matswhima
/ Dominant strategy mechanisms
arbitrarily, and denote by 4 the strategy for agent j such that Gj(sj) = ij for all sj E S,. Choose X so that A;(& s,*, s**)=X;(+, s,**, SF) > 0, and A;($, sit s,!) = 0 otherwise, and for every j E IV/{ i} and every ($J, sj, sj), A,(+, sj, sJ> = 0. Notice that the left-hand side of the equality (1) is equal to zero for all (i, w, s). This means that there exists k which satisfies the equality (1) for all (i, w, s), where k(o, s) = 0 for all (w, s). This is a contradiction of Condition 2. Next, we show that Condition 3 is sufficient for Condition 2. Suppose that Condition 3 holds. For every i E N, we define a function pi : s2 X S, -+ R so that for every (w, si),
cliCwY ‘I)=
-{1-Pi(wIs,)}2- C P,(w’Isi)2. O’#W
Notice from Condition 3 and the literature of the scoring-rule problem that for every i E N, every si E S, and every s,’ f si,
See Matsushima (1989) for the detailed proof. We define t = (ti) in the way that for every (i, w, s),
where IX, E Nti( w, s) = 0 for all (w, s). Such a t satisfies the inequality (3). From Proposition 2, we know that Condition 2 holds. •I
References d’Aspremont, C. and L.-A. Gerard-Varet, 1982. Bayesian incentive compatible beliefs, Journal of Mathematical Economics 10, 83-103. and related systems Fan, K., 1956, On systems of inequalities, in: H.W. Kuhn and A.W. Tucker, eds., Linear inequalities (Princeton University Press, Princeton, NJ) 99-156. Groves, T., 1973, Incentive in teams, Econometrica 41, 617-663. Johnson, S., J. Pratt and R. Zeckhauser, 1989, Efficiency despite mutually payoff-relevant private information: The finite case, Mimeo. Legros, P., and H. Matsushima, 1990, Efficiency of partnerships with stochastic output: A characterization result, Mimeo. Matsushima, H., 1989, Dominant strategy mechanisms with budget balancing revisited: Public information and scoring rules, Mimeo. Matsushima, H. 1990, Unique bayesian implementation with budget balancing, Mimeo. Myerson, R.B., 1979, Incentive compatibility and the bargaining problem, Econometrica 47, 61-73.