Claudia Meo

# Abstract

The possibility to compare information partitions is investigated for economies with asymmetric information. First, we focus on two potentially suitable instruments, the Boylan distance and the entropy, and show that the former does not fit the purpose. Then, we use the entropy associated with the information partition of each trader to construct a partially endogenous rule which regulates the information sharing process among traders. Finally, we apply this rule to some examples and analyze its impact on two cooperative solutions: the core and the coalition structure value.

JEL codes: C49; C71; D51; D82

Corresponding author: Claudia Meo, Dipartimento di Scienze Economiche e Statistiche, Universit` di Napoli Federico II, Naples, Italy, E-mail:
Article note: The author gratefully thanks the Editor and two anonymous referees for their useful comments and suggestions.
Appendix

We collect here some calculations relative to Example 2 provided in Section 2.

Let us consider the set S = {s1, s2, s3} formed by three states of nature s1, s2, and s3 that occur with probabilities q1, q2, q3, respectively, with q1 + q2 + q3 = 1 and q1 > q2 > q3.

The initial information of each trader is displayed below:

P1={s1|s2?s3};P2={s2|s1?s3}?;P3={s3|s1?s2};P4={s1|s2|s3}?.

Let first compute and compare among themselves the Boylan distances D(F1,F4),D(F2,F4), and D(F3,F4).

The algebras generated by these partitions are given by:

F1={X,S,{s1},{s2,s3}}?;F2={X,S,{s2},{s1,s3}}?;F3={X,S,{s3},{s1,s2}}?;F4={X,S,{s1},{s2,s3},{s2},{s1,s3},{s3},{s1,s2}}?.

It holds that:

D(F1,F4)=supF1?F1?infF4?F4?q(F1?F4)+supF4?F4?infF1?F1q(F1?F4)?==0+max{min(q3,1-q3),min(q2,1-q2)}?.

Since q1 > q2 > q3, it cannot be the case that q3>12; analogously, it cannot be that q2>12. Hence:

D(F1,F4)=max(q3,q2)=q2

On the other side, we have:

D(F2,F4)=supF2?F2?infF4?F4q(F2?F4)+supF4?F4?infF2?F2q(F2?F4)?==0+max{min(q3,1-q3),min(q1,1-q1)}=max{q3,min(q1,1-q1)}==min(q1,1-q1)?.

In all cases, it is true that D(F2,F4)>D(F1,F4).

Let us compare now D(F2,F4) and D(F3,F4).

It holds that:

D(F3,F4)=supF3?F3?infF4?F4q(F3?F4)+supF4?F4?infF3?F3q(F3?F4)?==0+max{q2,min(q1,1-q1)}=min(q1,1-q1)?.

Hence, D(F3,F4)=D(F2,F4).

Consider now the entropies associated with partitions P1, P2, and P3.

It holds that:

H(P1)=-q1?log?q1-(q2+q3)?log?(q2+q3)=-q1?log?q1-(1-q1)?log(1-q1)H(P2)=-q2?log?q2-(1-q2)?log?(1-q2)H(P3)=-q3?log?q3-(1-q3)?log?(1-q3)

We want to prove that the following implication is true:

q1>q2>q3?H(P1)>H(P2)>H(P3)

Consider the function H(x)=-x?log?x-(1-x)?log?(1-x) with x?]0,1[. It is strictly increasing whenever x?]0,12[. Therefore, if q1,q2?]0,12[, it follows that H(P1)>H(P2).

For the case q1>12, note that H is symmetric with respect to x=12, that is:

H(12+e)=H(12-e),foralle>0?.

If q1=12+e, then q2=12-e-q3<12-e. Therefore, by the monotonicity:

H(P1)=H(12+e)=H(12-e)>H(P2)

Finally, since q2,q3?]0,12[, the implication q2>q3?H(P2)>H(P3) follows from the monotonicity of the function H.

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