## Published and forthcoming

**Comment on If you're so smart, why aren't you rich? Belief selection in complete and incomplete markets**

Econometrica, 81, 849-851, 2013.

Market Selection studies the evolution of capital shares among traders with heterogeneous beliefs operating in a market. For the case of complete markets, allowing for heterogeneous discount factors and learning, Theorem 8 in Blume and Easley (2006), henceforward B-E, provides a sufficient condition for the wealth share of a trader to converge to 0 (vanish). I show by means of a counterexample that Theorem 8 is incorrect.

**Markets with heterogeneous beliefs: A necessary and sufficient condition for a trader to vanish**

*Journal of Economic Dynamics and Control*78 (2017): 190-205.

Formerly:

**Do not follow a weak leader**

What does it take to survive in the market? Previous literature has proposed sufficient con- ditions for a trader to vanish, which depend on pairwise comparisons of traders’ discounted beliefs. We propose a novel condition which focuses on the ratio of traders’ discounted beliefs and (approximate) equilibrium prices. Unlike existing conditions, ours is both necessary and sufficient for a trader to vanish and delivers the exact rate at which vanishing traders lose their consumption-shares. As an application, we analyze the performance of two intuitive behavioral strategies: the “Follow the Leader Strategy” that prescribes mimicking the beliefs of the most successful trader, and the “Follow the Market Strategy” that pre- scribes to use beliefs which coincide with the state price density. Further, we show that the relative performance of vanishing traders cannot be studied in isolation. Our analysis high- lights an intuitive point obscured by the existing conditions: trading in financial markets is qualitatively different from bilateral trading.

**Market Selection in Large Economies: A Matter of Luck**

forthcoming @

*Theoretical Economics*

In a general equilibrium model with a continuum of traders and bounded aggregate endowment, we investigate the Market Selection Hypothesis that markets favor traders with accurate beliefs. Contrary to known results for economies with (only) finitely many traders, we find that risk attitudes affect traders’ survival and that markets can favor “lucky” traders with incorrect beliefs over “skilled” traders with accuratebeliefs. Our model allows for a clear distinction between luck and skills and it shows that market selection forces induce efficient prices even when accurate traders do not survive in the long-run.

## Working Papers:

**The Wisdom of the Crowd in Dynamic Economies,**joint with Pietro Dindo (Submitted Apr.19)

Formerly:

**Market Selection and the Wisdom of the Crowd Revisited**

The Wisdom of the Crowd applied to financial markets asserts that prices, an average of agents' beliefs, are more accurate than individual beliefs. However, a market selection argument implies that prices eventually reflect only the beliefs of the most accurate agent. In this paper, we show how to reconcile these alternative points of view. In markets in which agents naively learn from equilibrium prices, a dynamic Wisdom of the Crowd holds. Market participation increases agents' accuracy, and equilibrium prices are more accurate than the most accurate agent. If we replace naive learning with Bayes' rule, this positive result disappears.

**Learning from ambiguous and misspecified models**joint with Massimo Marinacci (Submitted Apr.19)

We model inter-temporal ambiguity as the scenario in which a Bayesian learner holds more than one prior distribution over a set of parameters and provide necessary and sufficient condition for ambiguity to fade away because of learning. Our condition applies to most learning environments: iid and non-idd model-classes, well-specified and misspecified model-classes/prior support pairs. It shows that a Bayesian agent does not suffer from long-run ambiguity if and only if the data support a unique model.

**Price Probabilities:**A class of Bayesian and non-Bayesian prediction rules (Submitted, Dec 2018)

We use the standard machinery of dynamic general equilibrium models to generate a rich class of probabilities and to discuss their properties. This class includes probabilities consistent with Bayes’ rule and known non-Bayesian rules. If the prior support is correctly specified, we prove that all members of this class are as good as Bayes’ rule in terms of likelihood. If it is misspecified, we demonstrate that those rules that underreact to new information can significantly outperform Bayes’. Because under- reaction is never worse and sometimes better than Bayes’, we question the common opinion that Bayes’ rule is the only rational way to learn and propose a valid alternative.

**Swarm intelligence? Stock opinions of the crowd and stock returns,**joint with Matthias Pelster and Bastian Breitmayer (R-R @ International Review of Economics and Finance, Submitted, Nov 2018)

We find that crowds’ analyses of stocks, disclosed on a social trading platform, provide explanatory power for stock returns. Exploiting a novel dataset that contains more than 14.9 million individual stock assessments for 10,452 stocks over the period from August 1, 2007, to July 15, 2015, our study shows that social trading platforms add valuable information for explaining future and abnormal stock returns. This research contributes to the literature on social media and stock markets by (i) analyzing precise stock assessments disclosed on a social trading platform and (ii) providing new evidence for the wisdom of the crowds in financial markets. We provide a theoretical rationale and comprehensive robustness tests of our results.

**Ambiguity, Learning, and Raiffa’s critique**(Submitted, Feb 2019)

We present evidence of non-trivial interactions between learning and ambiguity. We find that ambiguous averse decision functional do not produce robust out-of-sample predictions when matched with learning models. There are situations in which the illusion of learning induces an ambiguous averse DM to optimally choose a sequence of ambiguous acts over a sequence of risky acts which would deliver a higher average utility.

On the survival of traders that underreact to information (with Pietro Dindo)

An accredited hypothesis about financial markets is that evolutionary forces favor “ratio- nal” Bayesian agents over “irrational” non-Bayesian agents. We challenge this hypothesis demonstrating that, ceteris paribus, agents that underreact to information have a weak evolutionary advantage against Bayesian. If Bayesian and non-Bayesian agents are given the same prior in- formation and observe the same data, the non-Bayesian survive on every path while there are paths on which the Bayesian agents vanish.

## Older Projects:

**A proof of Tanaka’s formula for Random Walk that does not rely on the discrete time version of Ito’s lemma.**

I present a simple proof of Tanaka’s formula for Random Walk that does not rely on the discrete time version of Ito’s lemma.

**Price Dynamics and Market Selection in Small and Large Economies**(JMP)

This paper analyzes and characterizes the dynamics of wealth-share and equilibrium price in a stochastic general equilibrium model with heterogeneous consumers. The characterization enables a comparison between probabilistic learning and price evolution, revealing that prices incorporate “sparse” information efficiently. Results on wealth-share are obtained by comparing traders’ optimal investment-consumption plans against their prices. This novel approach extends recent results in the literature by providing a sufficient and necessary condition for a trader to vanish. The results are applied to survival in iid, survival in large economies, and survival of traders that follow strategies commonly observed in real markets.

**Bayesian updating and bounded memory**

I use the market selection setting to illustrate some surprising features of Bayesian Learning that arise when the posterior does not converge. In particular I show that, given the same prior, Bayesian procedures (namely, Bayesian Mixture and Bayesian Factor models) that for the general case are asymptotically equivalent are not equivalent. Moreover I prove that in this setting it is optimal to forget some (all) the past observations. This result is new and contradicts the generally accepted idea that the forecasts obtained by a Bayesian procedure improve as the number of past observations increases.