Alberto Bisin

Danilo Guaitoli

English

Research Papers in Economics

Information Extraction and Exclusivity
Alberto Bisin
New York University
Danilo Guaitoli
Universitat Autonoma de Barcelona
-
UFAE/IAE Working Papers
n. 546.02
-
UAB 2002
11 Introduction
Much analysis has been recently devoted in the contract literature to economies in
which each agent observes a signal about other agents' private information. This
toghether with correlation among agents' outputs may be used to design information
extraction mechanisms. In some cases it can be shown that the optimal contract
implements full information extraction, and hence the incentive constrained optimum
coincides with the Pareto optimum.
We study the robustness of information extraction mechanism with respect to
economies in which `exclusive' contracts cannot be implemented. By this we mean
situations in which a `principal' or a nancial intermediary cannot observe, monitor or
contract upon all the contractual relationships an agent may enter with other interme-
diaries or agents. This is a very plausible situation if we think of informal or implicit
contracts that are not in general publicly observable. In a previous paper (Bisin and
Guaitoli 1998) we analyzed equilibria with moral hazard and nancial intermediaries
competing in a `non-exclusivity' environment, showing that equilibria are never second
best ecient and very often fail to implement the optimal action. It is interesting to
ask, therefore, whether information extraction in groups of agents may overcome the
serious ineciency generated by non-exclusivity.
The simplest environment to pursue this analysis consists of an hidden action econ-
omy in which: i) agents are organized in pre-specied groups of two individuals (e.g. a
rm); ii) the outputs of agents in the same group are correlated; iii) agents in any group
perfectly observe each other eort. The incentive problem consists in extracting from
any agent information about the eort of the other agent in his group (which only he
observes). This must be done via contracts contingent on the observable realizations
of both agents and their reports.
In such environments we can show that full information extraction is never sus-
tainable at an equilibrium with non-exclusivity, while information is fully extracted at
the incentive constrained optimum, which for these economies coincides with the (rst
best) Pareto optimum.
A contract which fully extracts information in an economy as above described can
be constructed as follows. Suppose the contract designer can ask each agent to send
a message about the eort of the other agent in his group. If contract payos can be
found so that each agent has an incentive to tell the truth (i.e. to report low eort
on the other agent if and only if he actually observed low eort, and viceversa for
high eort), then full information is extracted and contracts can be eectively made
contingent on eort for each agents (indirectly via the other agent's message). Each
agent will then choose the Pareto optimal eort as if his eort were publicly observed.
Note that rewards for the truthful reporting of the low action of the other agent in the
group are never paid by the intermediary in equilibrium since both agents choose the
high eort. Finally, a contract with payos that guarantee truth telling in the agents'
message game can be found provided the outcome of each agent depends enough from
2the other agent's eort as well as from its own.
The contract which implements full extraction just described is not sustainable
though in equilibrium if it cannot be eectively made exclusive. The argument is as
follows. Suppose an intermediary issued this contract. Then there exists a contract (the
`deviating' contract) that acts as a pre-commitment on the part of both agents never to
report the other's low eort. By entering both this contract and the (full information
extraction) optimal contract agents are able to choose the low eort (without being
caught), and hence to enjoy full insurance at a (better than fair) price, while saving on
the eort cost. As a consequence the `deviating' contract makes non-negative prots,
while the (full information extraction) optimal contract makes negative prots.
The `deviating' contract is constructed essentially to insure an agent against the
possibility of being reported (by the other agent) as having chosen low eort, and to
punish an agent for revealing the low eort choice of the other. This type of con-
tracts looks like a social norm which punishes for `informing' on the other members
of the group, independently of their actions. These forms of social norms t quite
well with non-exclusivity environments (since obviously the adoption of a norm is very
dicult to monitor), and are quite common e.g. among extended family members,
among members of various social, religious or intellectual groups, etc.. We argue there-
fore that social norms could be studied as implicit, informal contracts in environment
characterized by non-exclusivity.
Within the related literature, we refer in particular to Ma (1988), where he suggests
a mechanism to implement the rst best in a similar economy. Our rst proposition
replicates his conclusion but through a dierent mechanism, providing therefore a dif-
ferent proof of the result. For the non-exclusive case, Holmstrom and Milgrom (1990)
discuss the consequences of unregulated trade among agents but in a context where
agents collude and choose jointly their eort as a syndicate. What we have in our
model is not collusion, but non-cooperative Nash equilibrium choices of both eort
and `side-contracts'.
2 The economy
More specically the economy we study is as follows. It is a hidden action economy
which lasts two periods, t 2 f0;1g; agents' preferences value consumption in period 1
only: u1(c1)   v(e); and eort is private information. Also, eort can take two values,
e 2 fa;bg, with v(a) > v(b). Agent i's endowment wi
1 takes values wH, wL, with
wH > wL. Agents are grouped in couples (i;X(i)) (X is a one-to-one and onto map1).
The probability distribution of agent i's endowment is aected by both i's and X(i)'s
eort: ei;eX(i)(si), for si 2 fH;Lg. We denote with e;e0 = e;e0(H) the probability
that agent i has the high endowment H; given that he chose eort e and X(i) choose
eort e0 (note that e;e0 refers to agent i's probability, even though the index i does
1Agent i and X(i) are symmetric. For simplicity we report the notation only for agent i.
3not appear in the notation). Agent i receives a private information signal which fully
reveals eX(i) (and symmetrically for agent X(i)). Agent i then sends a message mi
with value in fa;bg about eX(i) (and symmetrically for agent X(i)). Finally, contract
j's payo to agent i is denoted dj = fd
j
H;mi;mX(i);d
j
L;mi;mX(i)g. Note that contracts only
pay o in period 1 (since agents only consume in period 1) and depend on agents'
messages regarding their partner's eort.2
3 Information extraction
We are now ready to characterize the incentive constrained optimum for the information
extraction economy when exclusive contracts can be implemented.
Proposition 1 In the information extraction economy with exclusivity, assume
(1   aa)
aa
<
(1   ba)
ba
<
(1   ab)
ab
<
(1   bb)
bb
: (1)
Then the incentive constrained optimal allocation is unique and achieves the Pareto
optimum:
cH = cL = EwH + (1   E)wL
where E 2 fa;bg is the eort choice at the Pareto optimum.
Proof. See Appendix.3
The contracts fdsEgs2S; E2fa;bg2 which implement the Pareto optimum must extract
information on ei from X(i); for any i 2 I; hence the dependence of the contract payo
on the messages is non trivial: ds;E 6= ds;E0, 8s 2 fH;Lg; E; E0 2 fa;bg2.
The proof of Proposition 1 constructs a contract which implements full information
extraction (the only non-trivial case is when at the Pareto optimum both agent choose
the high eort, E = (a;a)). This contract has the property that it rewards agent i
for truthfully reporting agent X(i)'s low eort (and viceversa). In equilibrium then
full information is extracted and, as a consequence, agents choose the high eort and
are provided with full insurance contracts at fair price, (1   aa)=aa. Condition (1)
guarantees that a contract which pays agent i a positive amount in expected value
2In general payos should be allowed to depend also on the partner's realized outcome, as in
`relative performance evaluation' models (cf. Hart-Holmstrom (1987)). Here we omit it for simplicity,
since (as we prove) it is not needed for the optimal contract to reach the rst best (Proposition 1),
while it would not change qualitatively the main consequence of non-exclusivity, i.e. the impossibility
of full information extraction (Proposition 2).
3Ma (1988) shows that full information extraction is the unique Perfect Nash Equilibrium of the
optimal contract problem, allowing for general implementation mechanisms (e.g. sequential mech-
anisms). Proposition 1 is a stronger result because it proves Subgame Perfect Implementation via
simultaneous mechanisms (a subset of the mechanisms allowed in Ma (1988)).
4when X(i) has chosen the low eort, and a negative expected amount when X(i) has
chosen the high eort, does in fact exists. Moreover the reward for truthful reporting
the low action of the other agent in the group is never paid by the intermediary in
equilibrium since both agents choose the high eort.
4 Non-exclusive contracts
We now consider the information extraction economy when exclusivity cannot be im-
posed on contracts. This will be the case whenever the principal is unable to monitor
agents' trades with other intermediaries or among themselves, or even when informa-
tion about such trades is not veriable.
Proposition 2 shows though that there exists a contract (which we construct in the
proof) that acts as a pre-commitment on the part of both i and X(i) to never report
the other's low eort. By entering both this contract and the incentive constrained
optimal contract agents are able to choose the low eort (without being caught), and
hence to enjoy full insurance at the (better than fair) price (1 aa)=aa, while saving
on the eort cost.
Referring to the contract which implements the Pareto optimum (cf. proof of
Proposition 1) as contract d, we have the following result.
Proposition 2 In the information extraction economy with non-exclusivity, assume
condition (1) and E = (a;a) at the Pareto optimum. There exists then a contract d0
such that if the set of contracts fd;d0g is issued:
 all agents buy both contracts
 all agents choose the low eort, i.e. (ei;eX(i)) = (b;b)
 each agent i reports the high eort for agent X(i) and viceversa, i.e. (mi;mX(i)) =
(a;a).
Proof. See Appendix.
Whenever contracts d and d0 are simultaneously issued, then, the intermediary
issuing the incentive constrained optimal contract d will make negative prots, while the
intermediary issuing contract d0 will make zero prots (d0 can obviously be perturbed
to generate small positive prots).
Contract d0 essentially insures agent i against the possibility that agent X(i) reveals
that he has chosen low eort, and punishes agent i for revealing the low eort choice
of X(i) (and viceversa).
It is also clear from the proof that contracts of the form of d0 can be constructed
to upset any base contract designed to extract information by agent i on agent X(i)'s
eort.
5References
Bisin, A., and D. Guaitoli (1998): \Moral Hazard and Non-Exclusive Contracts", UPF
Working Paper n.345.
Hart, O., and B. Holmstrom (1987): \The Theory of Contracts", in T. Bewley (ed.),
Advances in Economic Theory, Cambridge University Press, pp. 71-156.
Holmstrom, B., and P. Milgrom (1990): \Regulating Trade Among Agents", Journal
of Institutional and Theoretical Economics, 146, pp. 85-105.
Ma, C. T. (1988): \Unique Implementation of Incentive Contracts with Many Agents",
Review of Economic Studies, pp. 555-571.
6Appendix
Proof of Proposition 1. The proof is by construction of an optimal contract d (we
concentrate on the only non-trivial case in which, at the Pareto optimum, E = (a;a)).
Take ds;a;a = dFI
s , 8s 2 S, such that
wH + d
FI
H = wL + d
FI
L and a;ad
FI
H + (1   a;a)d
FI
L = 0
i.e. dFI
s are the payos associated with the (full insurance) Pareto optimal contract.
Also take ds;a;b = dFI
s   P, ds;b;a = dFI
s + ^ ds, ds;b;b = dFI
s + ^ ds   P, where ^ d = (^ dH; ^ dL)
satises:
a;au
i(wH + d
FI
H + ^ dH) + (1   a;a)u
i(wL + d
FI
L + ^ dL) = U
FI   a;a
a;bu
i(wH + d
FI
H + ^ dH) + (1   a;b)u
i(wL + d
FI
L + ^ dL) = U
FI + a;b
b;au
i(wH + d
FI
H + ^ dH) + (1   b;a)u
i(wL + d
FI
L + ^ dL) = U
FI   b;a
b;bu
i(wH + d
FI
H + ^ dH) + (1   b;b)u
i(wL + d
FI
L + ^ dL) = U
FI + b;b
with hk > 0, h;k 2 fa;bg; and UFI = ui(wH + dFI
H ) = ui(wL + dFI
L ). A sucient
condition for ^ d to exist (i.e. for the equations above to be satised) is
(1   a;a)
a;a
<
(1   b;a)
b;a
<
(1   a;b)
a;b
<
(1   b;b)
b;b
(construct ^ d to have negative expected value if (e;e) = (a;a) or if (e;e) = (b;a), and
positive expected value if (e;e) = (a;b) or (e;e) = (b;b)). For ^ d small enough there
exist then !hk > 0, h;k 2 fa;bg, such that with messages (b;b)
a;au
i(wH + d
FI
H + ^ dH   P) + (1   a;a)u
i(wL + d
FI
L + ^ dL   P) = U
P   !a;a
a;bu
i(wH + d
FI
H + ^ dH   P) + (1   a;b)u
i(wL + d
FI
L + ^ dL   P) = U
P + !a;b
b;au
i(wH + d
FI
H + ^ dH   P) + (1   b;a)u
i(wL + d
FI
L + ^ dL   P) = U
P   !b;a
b;bu
i(wH + d
FI
H + ^ dH   P) + (1   b;b)u
i(wL + d
FI
L + ^ dL   P) = U
P + !b;b
with UP = ui(wH + dFI
H   P) = ui(wL + dFI
L   P).
We do not need to impose market clearing conditions on ^ d since in equilibrium,
we will show that, given d, E = (ei;eX(i)) = (a;a). The game played by agents is
represented in Figure 1.
< Figure 1 >
The reader can check that, proceeding by backward induction, the unique Nash
Equilibria of the four simultaneous last-stage games (i.e. the message games) are
respectively
(a;a); (b;a); (a;b); (b;b):
We can then construct the rst stage simultaneous game's payo matrix:
7< Figure 2 >
whose Nash Equilibrium is (ei;eX(i)) = (a;a), for P large enough. }
Proof of Proposition 2. The proof is by construction. Take a contract d0 as follows:
d0
s;e;e0 = ~ ds + d00
s;e;e0, with ~ ds such that ^ ds + ~ ds = 0; 8s. Also d00
s;e;e0 satises: d00
s;a;a =
d00
s;a;b = 0, and d00
s;b;a = d00
s;b;b = d00
s such that
a;au
i(wH + d
FI
H + d
00
H) + (1   a;a)u
i(wL + d
FI
L + d
00
L) = U
FI + a;a
a;bu
i(wH + d
FI
H + d
00
H) + (1   a;b)u
i(wL + d
FI
L + d
00
L) = U
FI + a;b
b;au
i(wH + d
FI
H + d
00
H) + (1   b;a)u
i(wL + d
FI
L + d
00
L) = U
FI + b;a
b;bu
i(wH + d
FI
H + d
00
H) + (1   b;b)u
i(wL + d
FI
L + d
00
L) = U
FI   b;b:
Again, such a contract d00
s exists if
(1   a;a)
a;a
<
(1   b;a)
b;a
<
(1   a;b)
a;b
<
(1   b;b)
b;b
:
Again, for d00
s small enough there exist 
hk > 0, h;k 2 fa;bg, such that
a;au
i(wH + d
FI
H + d
00
H   P) + (1   a;a)u
i(wL + d
FI
L + d
00
L   P) = U
P + 
a;a
a;bu
i(wH + d
FI
H + d
00
H   P) + (1   a;b)u
i(wL + d
FI
L + d
00
L   P) = U
P + 
a;b
b;au
i(wH + d
FI
H + d
00
H   P) + (1   b;a)u
i(wL + d
FI
L + d
00
L   P) = U
P + 
b;a
b;bu
i(wH + d
FI
H + d
00
H   P) + (1   b;b)u
i(wL + d
FI
L + d
00
L   P) = U
P   
b;b:
We can then follow the steps of the proof of Proposition 1 to show the following. If
both agents (i;X(i)) buy both contracts (d;d0) (see the game in Figure 3, then in
equilibrium (ei;eX(i)) = (b;b).
< Figure 3 >
Similarly, it can be shown that if only agent i buys both contracts (d;d0), while
agent X(i) buys d, then in equilibrium (ei;eX(i)) = (a;b). If only agent X(i) buys both
contracts (d;d0), while agent i buys d, then in equilibrium (ei;eX(i)) = (b;a).
The portfolio choice is determined by the Nash equilibrium of the following game:
< Figure 4 >
Hence both agents (i;X(i)) buy both contracts (d;d0) and play (ei;eX(i)) = (b;b) (note
that in equilibrium the intermediary issuing contract d0 has zero prots). }
8

Information Extraction and Exclusivity

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