# Battaglini, M. (2002), Multiple Referrals and Multidimensional Cheap Talk

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Has article title Multiple Referrals and Multidimensional Cheap Talk
Has author Battaglini, M.
Has year 2002
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## Abstract

In previous work on cheap talk, uncertainty has almost always been modeled using a single-dimensional state variable. In this paper we prove that the dimensionality of the uncertain variable has an important qualitative impact on results and yields interesting insights into the "mechanics" of information transmission. Contrary to the unidimensional case, if there is more than one sender, full revelation of information in all states of nature is generically possible, even when the conflict of interest is arbitrarily large. What really matters in transmission of information is the local behavior of senders' indifference curves at the ideal point of the receiver, not the proximity of players' ideal point.

## Rui's Intro points

• Departure point is the Gilligan/Krehbeil paper or Crawford/Sobel.
• Focused NOT on the open/closed rule angle, but more generally on the conflict (difference in ideal points) between commitee/floor or agent/principle.
• Basic idea: "As someone listening to multiple biased experts, I can ask pointed questions to fully reveal the information that the biased experts have." -- Rui, who says this is a loose, intuitive way of understanding the point.

## Model

• Choice space: $x\in R^{2}$, a pair of reals.
• Outcomes: $y=x+\theta, \theta\in R^{2}$. Note that here, $\theta$ is akin to $\omega$ in Gilligan and Krehbeilh.
• Preferences: Single peaked. Ideal point of receiver is $x^{R}=(0,0$.
• Equilibrium strategy: Perfect Bayesian Equilibrium.
• Equilibrium strategies:
1. Experts reveal truthfully ($s^{i}(\theta)=\theta=(m^{i}_{1},m^{i}_{2})$).
2. R only believes $(m_{1}^{i},m_{2}^{2})$. Meaning: He believes the report of expert 1 on one dimension, and the report of expert 2 on the other dimension.

## Rui's closing comments ;)

• Slight change in assumptions leads to different answers.
• "Degrees of freedom": More dimensions allows more revelation?
• Is this robust? People saying no:
1. Levy and Razin: Study Battaglini model with noise. Think about it this way: Crawford/Sobel show no 1 dimension, 1 sender, no full revelation. Battaglini shows: 2 dims, 2 senders, full revelation. Levy and Razin show: 2 dims, 2 senders, no full revelation.
2. Ambrose and Takahashi: 2 senders, 2 dimensions plus constraints on choice space -- answer is no full revelation. Even if you choose an interval, full revelation breaks down.