Information transmission in nested sender–receiver games
<Para ID="Par1">We introduce a “nestedness” relation for a general class of sender–receiver games and compare equilibrium properties, in particular the amount of information transmitted, across games that are nested. Roughly, game <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$B$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>B</mi> </math> </EquationSource> </InlineEquation> is nested in game <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$$A$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>A</mi> </math> </EquationSource> </InlineEquation> if the players’ optimal actions are closer in game <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$$B$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>B</mi> </math> </EquationSource> </InlineEquation>. We show that under some conditions, more information is transmitted in the nested game in the sense that the receiver’s expected equilibrium payoff is higher. The results generalize the comparative statics and welfare comparisons with respect to preferences in the seminal paper of Crawford and Sobel (Econometrica 50(6):1431–1452, <CitationRef CitationID="CR6">1982</CitationRef>). We also derive new results with respect to changes in priors in addition to changes in preferences. We illustrate the usefulness of the results in three applications: (i) delegation to an intermediary with a different prior, (ii) the choice between centralization and delegation, and (iii) two-way communication with an informed principal. Copyright Springer-Verlag Berlin Heidelberg 2015
