## Published and Accepted

**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**

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**78 (2017): 190-205.**

*Journal of Economic Dynamics and Control*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**

*14, n.2 (2019), 437-473*

**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.

**Learning from ambiguous and misspecified models**joint with Massimo Marinacci

*(2019)*

**Journal of Mathematical Economics**,We model inter-temporal ambiguity as the scenario in which a Bayesian learner holds more than one prior distribution over a set of models and provide sufficient conditions for ambiguity to fade away because of learning. Our conditions apply to most learning environments: iid and non-iid model-classes, well-specified and misspecified model-classes/prior support pairs. We show that ambiguity fades away if the empirical evidence supports a set of models with identical predictions, a condition much weaker than learning the truth.

**Swarm intelligence? Stock opinions of the crowd and stock returns,**joint with Matthias Pelster and Bastian Breitmayer

**International Review of Economics and Finance**, (2019)

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.

**The Wisdom of the Crowd in Dynamic Economies,**joint with Pietro Dindo

Accepted @

*Theoretical Economics*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.

## In Revision:

**Price Probabilities:**A class of Bayesian and non-Bayesian prediction rules

Conditionally accepted @

**Economic Theory**

I 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.

**When does ambiguity fade away?**

**,**joint with Jonathan Newton

R&R @

*Economics Letters*Is long-run ambiguity a possible outcome of the multiple-prior learning model? If the prior support is finite, long-run ambiguity is known to be a possible outcome only if the learning problem is misspecified (Marinacci-Massari 2019). Conversely, here we show that, under natural assumptions, ambiguity fades away on most paths if the prior support is naturally rich.

**Working Papers:**

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

Ambiguity-averse decision functionals matched with the multiple-prior learning model are more robust to model misspecification than the standard expected utility with Bayesian learning. However, these criteria fail to deliver robust decisions generically because the multiple-prior learning model inherits the same fragility of Bayesian learning. There are misspecified learning problems in which an ambiguity-averse DM optimally chooses a sequence of ambiguous acts over a sequence of risky acts that would deliver a strictly higher average utility.

**Entropified Berk-Nash Equilibrium**, joint with Jonathan Newton

(Submitted)

Esponda and Pouzo (2016) propose Berk-Nash equilibrium as a solution concept for games that are misspecified in that it is impossible for players to learn the true probability distribution over outcomes. The beliefs that support Berk-Nash equilibrium are, for each player, the learning outcome of Bayes’ rule. However, under misspecification, Bayes’ rule might not converge to the model that leads to actions with the highest objective payoff among the models subjectively admitted by the player. From an evolutionary perspective, this renders the beliefs that support Berk-Nash vulner- able to invasion. Drawing on the machine learning literature, we propose entropified Berk-Nash equilibrium, which is immune to this critique.

**On the survival of traders that underreact to information**

I show that under-reaction is a robust response to model misspecification rewarded by financial markets, rather than an “irrational” attitude that leads to extinction. In a general equilibrium model of a stylized market with a Bayesian and an under-reacting agent with the same information, I show that there are no paths on which the under-reacting agent loses all his wealth against the Bayesian, while there is a large class of misspecified learning settings in which the Bayesian agent loses all his wealth against the under-reacting agent almost surely.

**On the survival of ambiguity averse traders in financial markets**

Model uncertainty is a recognized difficulty every investor must face in financial markets. Different attitudes toward model uncertainty determine different investment strategies and thus different economic performance. While a growing empirical literature uses ambiguity aversion as one explanation for market anomalies, the theoretical counterpart suggests that ambiguity averse trader will not have a long-run impact on prices because of selection arguments. Here I show that the opposite result holds when learning is taken into account. An ambiguous averse Bayesian agent with multiple priors has an evolutionary advantage against a Bayesian agent when given the same information.

## 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.