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An Entropy-Based Information Sharing Rule for Asymmetric Information Economies

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.

References

Allen, B. 1983. "Neighboring Information and Distributions of Agents' Characteristics under Uncertainty." Journal of Mathematical Economics 12: 63-101, https://doi.org/10.1016/0304-4068(83)90051-4.Search in Google Scholar

Allen, B. 1991. "Market Games with Asymmetric Information and Nontransferable Utility: Representations Results and the Core." Center for Analytic Research in Economics and the Social Sciences Working Paper #91-09. University of Pennsylvania.Search in Google Scholar

Allen, B. 2006. "Market Games with Asymmetric Information: the Core." Economic Theory 29: 465-87, https://doi.org/10.1007/s00199-005-0070-6.Search in Google Scholar

Aumann, R. 1987. "Correlated Equilibrium as an Expression of Bayesian Rationality." Econometrica 55: 1-18, https://doi.org/10.2307/1911154.Search in Google Scholar

Boylan, E. 1971. "Equiconvergence of Martingales." The Annals of Mathematical Statistics 42: 552-9, https://doi.org/10.1214/aoms/1177693405.Search in Google Scholar

Cabrales, A., O. Gossner, and R. Serrano. 2013. "Entropy and the Value of Information." The American Economic Review 103: 360-77, https://doi.org/10.1257/aer.103.1.360.Search in Google Scholar

Correia-da-Silva, J., and C. Hervis-Beloso. 2007. "Private Information: Similarity as Compatibility." Economic Theory 30: 395-407, https://doi.org/10.1007/s00199-005-0066-2.Search in Google Scholar

Einy, E., O. Haimanko, D. Moreno, and B. Shitovitz. 2005. "On the Continuity of Equilibrium and Core Correspondences in Economies with Differential Information." Economic Theory 26: 793-812, https://doi.org/10.1007/s00199-004-0577-2.Search in Google Scholar

Glycopantis, D., and N. C. Yannelis. 2005. "Equilibrium Concepts in Differential Information Economies." In Differential Information Economies. Studies in Economic Theory, Vol. 19, edited by D. Glycopantis and N. C. Yannelis, 1-53. Berlin, Heidelberg, New York: Springer-Verlag.Search in Google Scholar

Hubert, L., and P. Arabie. 1985. "Comparing Partitions." Journal of Classification 2: 193-218, https://doi.org/10.1007/bf01908075.Search in Google Scholar

Koutsougeras, L. C., and N. C. Yannelis. 1993. "Incentive Compatibility and Information Superiority of the Core of an Economy with Differential Information." Economic Theory 3: 195-216, https://doi.org/10.1007/bf01212914.Search in Google Scholar

Krasa, S., A. Temimi, and N. C. Yannelis. 2003. "Coalition Structure Values in Differential Information Economies: Is Unity a Strength?." Journal of Mathematical Economics 39: 51-62, https://doi.org/10.1016/s0304-4068(02)00083-6.Search in Google Scholar

Owen, G. 1977. "Values of Games with a Priori Unions." In Essays in Mathematical Economics and Game Theory, Berlin: Springer-Verlag.Search in Google Scholar

Radner, R. 1968. "Competitive Equilibrium Under Uncertainty." Econometrica 36: 31-58, https://doi.org/10.2307/1909602.Search in Google Scholar

Rand, W. M. 1971. "Objective Criteria for the Evaluation of Clustering Methods." Journal of the American Statistical Association 66: 846-50, https://doi.org/10.1080/01621459.1971.10482356.Search in Google Scholar

Regnier, S. 1965. "Sur quelques aspects mathematiques des problemes de classification automatique. Mathematiques et Sciences humaines." 82, 1983, 13-29, reprint of I.C.C. Bulletin 4: 175-91.Search in Google Scholar

Schwalbe, U. 1999. The Core of Economies with Asymmetric Information. Lecture Notes in Economics and Mathematical Systems. Berlin, Heidelberg, New York: Springer-Verlag.Search in Google Scholar

Shannon, C.E. 1948. "A Mathematical Theory of Communication." The Bell System Technical Journal 27: 379-423, https://doi.org/10.1002/j.1538-7305.1948.tb01338.x.Search in Google Scholar

Wilson, R. 1978. "Information, Efficiency and the Core of an Economy." Econometrica 46: 807-16, https://doi.org/10.2307/1909750.Search in Google Scholar

Yannelis, N.C. 1991. "The Core of an Economy with Differential Information." Economic Theory 1: 183-98, https://doi.org/10.1007/bf01211533.Search in Google Scholar

Received: 2020-02-04
Accepted: 2020-06-18
Published Online: 2020-10-14

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