# On Information Aggregation and Interim Efficiency in Networks

## Abstract

**:**

## 1. Introduction

## 2. Model

#### 2.1. Preferences and Information Structure

#### 2.2. Network Structure

**Example**

**1.**

#### 2.3. Optimal Actions

## 3. Equilibrium and Social Welfare

**Proposition**

**1**(Welfare Loss Function)

**.**

## 4. Informative Content of Signal Profiles and Efficiency

**Lemma**

**1.**

**Proposition**

**2.**

**Example**

**2.**

**Proposition**

**3.**

## 5. Conclusions

## Acknowledgments

## Conflicts of Interest

## Appendix

**Proof**

**of Lemma 1.**

**Proof**

**of Proposition 2.**

**Proof**

**of Proposition 3.**

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^{1.}For example, suppose that the profitability of some investment activity depends on an uncertain exogenous state of the world and on the aggregate investment. Here, investors would like to pick investment strategies that match both the exogenous variable and the other investors’ strategies as well.^{2.}The “beauty contest” terminology comes originally from a well-known parable by ([1], Chapter 12). Following the seminal contribution of [2], “beauty contest” games have been extensively used to explore a wide range of phenomena in a number of settings, including investment games ([3,4]), financial markets ([5]), monopolistic competition ([6]), or models of political leadership ([7]), among others.^{3.}For example, under a terrorist attack threat, the analyst wishes to assess which is the most likely location of the attack but also wants to come up with locations not very distant from those predicted by other analysts. In this way, counterterrorism measures could be more effective to prevent the attack.^{4.}Ref. [4] have investigated in a very comprehensive way the social value of information in an ex-ante efficiency benchmark without restrictions in the form of local interactions and where the agents have both private and public sources of information. For that environment, they have shown that whether more informative content increases or decreases welfare depends on whether equilibrium is efficient under both complete and incomplete information or only under incomplete information. Their contribution highlights that understanding the social value of information depends crucially on the notion of efficiency used. Without a well-specified efficiency benchmark, assessing the social value of information follows the folk theorem that “everything goes” in a second-best world. Assessing the social value of information with complementarities in the presence of networks remains a question far from understood.^{5.}Of course, signals cannot be unconditionally independent because all of them depend on the state of the world.^{6.}For instance, suppose for some agent i, the profile $s(i)=\mathit{s}\in {\mathcal{S}}_{g}$ is the subset of signals $\left\{{s}_{1},{s}_{3},{s}_{7},{s}_{100}\right\}$. Then, we will consider $s(i)=\mathit{s}=({s}_{1},{s}_{3},{s}_{7},{s}_{100})$ for technical tractability.^{7.}Formally, in the current content, the network is minimally (directedly) connected if for each neighborhood $g(i)\in g([0,1])$ there exists another neighborhood $g(h)\in g([0,1])$ such that $g(i)\cap g(h)\ne \varnothing $. I thank an anonymous referee for pointing out an error in an earlier version of this definition. This notion of minimally connectedness is equivalent to having, for any two different agents, at least a directed path in the network that connects them.^{8.}Since this is an average over all subsets of signals, $\overline{E}\left[\theta \right]$ equivalently indicates the average posterior expectation on θ over all agents.^{9.}For applications where one considers instead a relatively small number of subsets of signals, the law of large numbers cannot be reasonably invoked to compute averages of expectations on the state. In these cases, keeping track of the higher-order beliefs that are required to characterize equilibria follows a completely different approach. In particular, under certain conditions, one can make use of the iterated application of a knowledge index matrix. The idea of using a knowledge index matrix to track individual arbitrarily higher-order beliefs in a network was originally proposed by [9]. An application of the knowledge index matrix to information acquisition problems in small populations has been recently provided by [8].

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**MDPI and ACS Style**

Jimenez-Martinez, A. On Information Aggregation and Interim Efficiency in Networks. *Games* **2017**, *8*, 15.
https://doi.org/10.3390/g8010015

**AMA Style**

Jimenez-Martinez A. On Information Aggregation and Interim Efficiency in Networks. *Games*. 2017; 8(1):15.
https://doi.org/10.3390/g8010015

**Chicago/Turabian Style**

Jimenez-Martinez, Antonio. 2017. "On Information Aggregation and Interim Efficiency in Networks" *Games* 8, no. 1: 15.
https://doi.org/10.3390/g8010015