Private Signals Research Articles

Averaging dynamics, mortal random walkers and information aggregation on graphs

Many models of dynamics on graphs implicitly capture some tendency towards alignment or synchronisation between neighbouring nodes. This mechanism often leads to a tension between independent dynamics at each node preserving local variation, and alignment between neighbours encouraging global smoothing. In this paper, we explore some of the intuition behind this phenomenon by considering a simplified set of dynamics where the states of agents are determined by a combination of private signals and averaging dynamics with their neighbours. We show that outcomes of this mechanism correspond to the behaviour of mortal random walkers on the graph, and that steady state outcomes are captured precisely in an object called the fundamental matrix, which summarises expected visitation between pairs of nodes. The bulk of the paper approximates the elements of the fundamental matrix as a function of the topology of the graph, in the case of undirected and unweighted graphs. In doing so we show intuitively how features such as degree distribution, community structure and clustering impact the trade-off between local variation and global smoothing in the outcomes, and can shed light on more complex instances of dynamics on graph. We consider as an application how the results can be used to predict and better understand the steady state outcomes of an information aggregation process.

Chromosome X-wide Analysis of Positive Selection in Human Populations: Common and Private Signals of Selection and its Impact on Inactivated Genes and Enhancers.

The ability of detecting adaptive (positive) selection in the genome has opened the possibility of understanding the genetic basis of population-specific adaptations genome-wide. Here, we present the analysis of recent selective sweeps, specifically in the X chromosome, in human populations from the third phase of the 1,000 Genomes Project using three different haplotype-based statistics. We describe instances of recent positive selection that fit the criteria of hard or soft sweeps, and detect a higher number of events among sub-Saharan Africans than non-Africans (Europe and East Asia). A global enrichment of neural-related processes is observed and numerous genes related to fertility appear among the top candidates, reflecting the importance of reproduction in human evolution. Commonalities with previously reported genes under positive selection are found, while particularly strong new signals are reported in specific populations or shared across different continental groups. We report an enrichment of signals in genes that escape X chromosome inactivation, which may contribute to the differentiation between sexes. We also provide evidence of a widespread presence of soft-sweep-like signatures across the chromosome and a global enrichment of highly scoring regions that overlap potential regulatory elements. Among these, enhancers-like signatures seem to present putative signals of positive selection which might be in concordance with selection in their target genes. Also, particularly strong signals appear in regulatory regions that show differential activities, which might point to population-specific regulatory adaptations.

A Robust Bayesian Truth Serum for Small Populations

Peer prediction mechanisms allow the truthful elicitation of private signals (e.g., experiences, or opinions) in regard to a true world state when this ground truth is unobservable. The original peer prediction method is incentive compatible for any number of agents n >= 2, but relies on a common prior, shared by all agents and the mechanism. The Bayesian Truth Serum (BTS) relaxes this assumption. While BTS still assumes that agents share a common prior, this prior need not be known to the mechanism. However, BTS is only incentive compatible for a large enough number of agents, and the particular number of agents required is uncertain because it depends on this private prior. In this paper, we present a robust BTS for the elicitation of binary information which is incentive compatible for every n >= 3, taking advantage of a particularity of the quadratic scoring rule. The robust BTS is the first peer prediction mechanism to provide strict incentive compatibility for every n >= 3 without relying on knowledge of the common prior. Moreover, and in contrast to the original BTS, our mechanism is numerically robust and ex post individually rational.

Strategic Trading with Information Acquisition and Long-Memory Stochastic Liquidity

This paper investigates the strategic interaction of information acquisition and noise trading patterns, as well as its significant implications in market equilibrium outcomes. We consider a market where the strategic trader can dynamically acquire costly information about an asset's payoff via private signals instead of endowing him with its full information. Particularly, the noise trading, which provides necessary liquidity for the market, has the feature of long-range dependence. We provide closed-form characterizations of the equilibriums in both opaque and transparent markets depending on whether or not the trader discloses his order flow. We then find that: (1) the endogenous information acquisition improves the price informativeness over time, while the information efficiency is decreasing; (2) the long-memory stochastic liquidity renders excessive price volatility, enlarges the price impact, and slows down information learning; (3) in a transparent market, the trader shall adopt a mixed strategy by adding an additional noise in his trading strategy to prevent rapid information leakage.

Competition, communication and rating bias

I compare the efficiency of different equilibria reached in a credit rating game that allows for communication between the issuer and two credit rating agencies (CRAs) prior to disclosing ratings. CRAs observe private signals that correlate with asset quality and can learn about each other’s signals by exchanging messages with the issuer. Conflicts of interest lead CRAs to provide biased ratings. When issuer messages are informative about signals, CRAs find it optimal to selectively offer biased ratings based on issuer messages. Messages are informative when the issuer discloses high ratings from both CRAs and profits more from increasing issuance than from selling its worst assets. The equilibrium in which only one CRA provides ratings leads to the highest efficiency when average asset quality is low, agency signals frequently disagree and asset payoffs are skewed.

Information Redundancy Neglect versus Overconfidence: A Social Learning Experiment

We study social learning in a continuous action space experiment. Subjects, acting in sequence, state their beliefs about the value of a good after observing their predecessors’ statements and a private signal. We compare the behavior in the laboratory with the Perfect Bayesian Equilibrium prediction and the predictions of bounded rationality models of decision-making: the redundancy of information neglect model and the overconfidence model. The results of our experiment are in line with the predictions of the overconfidence model and at odds with the others’. (JEL C91, D12, D82, D83)

Persuading Voters in District-based Elections

We focus on the scenario in which an agent can exploit his information advantage to manipulate the outcome of an election. In particular, we study district-based elections with two candidates, in which the winner of the election is the candidate that wins in the majority of the districts. District-based elections are adopted worldwide (e.g., UK and USA) and are a natural extension of widely studied voting mechanisms (e.g., k-voting and plurality voting). We resort to the Bayesian persuasion framework, where the manipulator (sender) strategically discloses information to the voters (receivers) that update their beliefs rationally. We study both private signaling in which the sender can use a private communication channel per receiver and public signaling in which the sender can use a single communication channel for all the receivers. Furthermore, for the first time, we introduce semi-public signaling in which the sender can use a single communication channel per district. We show that there is a sharp distinction between private and (semi-)public signaling. In particular, optimal private signaling schemes can provide an arbitrarily better probability of victory than (semi-)public ones and can be computed efficiently, while optimal (semi-)public signaling schemes cannot be approximated to within any factor in polynomial time unless P=NP. However, we show that reasonable relaxations allow the design of multi-criteria PTASs for optimal (semi-)public signaling schemes. In doing so, we introduce a novel property, namely comparative stability, and we design a bi-criteria PTAS for public signaling in general Bayesian persuasion problems beyond elections when the sender's utility function is state-dependent.

Common-Value Public Goods and Informational Social Dilemmas

We experimentally examine private information and communication in a public goods environment with uncertain returns. We consider a common-value public goods game in which the return to contribution is either high or low. Before contributing, three players observe private signals correlated with the return and send cheap talk messages to one another. There are social gains from truthfulness, but a private incentive to exaggerate. We compare treatments with and without cheap talk, finding that communication is largely truthful and increases efficiency. In further treatments, we increase the incentive to exaggerate and find reduced truthfulness and smaller gains from communication. (JEL C72, D82, D83, H41)

An experiment on network density and sequential learning

We conduct a sequential social-learning experiment where subjects each guess a hidden state based on private signals and the guesses of a subset of their predecessors. A network determines the observable predecessors, and we compare subjects' accuracy on sparse and dense networks. Accuracy gains from social learning are twice as large on sparse networks compared to dense networks. Models of naive inference where agents ignore correlation between observations predict this comparative static in network density, while the finding is difficult to reconcile with rational-learning models.

The influence of communication structure on opinion dynamics in social networks with multiple true states

In order to study the influence of communication structure on opinion dynamics in social networks with multiple true states, a social learning model is proposed in which the social network is composed of strongly connected communities and some uninformed individuals outside the communities. Multiple communities correspond to multiple underlying true states. At each time step, the individual in a community receives his private signal and updates his opinion to be a weighted linear combination of his own Bayesian posterior opinion and the opinions of his neighbors, while the uninformed individual outside the communities has no private signal and only takes the neighbors’ weighted average opinion as his own opinion. Simulation results show that in the social network environment with multiple true states, for the uninformed individual, the opinion will depend on the weighted average opinion of his neighbors, and for the individual in the strongly connected community, the opinion dynamics will depend on the communication types between communities. If communities are not connected, he will achieve asymptotic learning. If communities are unidirectionally connected, he will learn the true state of the root community. If communities are bidirectionally connected, his opinion will appear chaos phenomenon and oscillate within a certain range. So, in a society with multiple true states, the absence of communication between communities is conducive to asymptotic learning and bidirectional communications between communities will hinder asymptotic learning.

Non-monotone social learning

We revisit the canonical binary-state model of social learning to investigate the possibility of non-monotone learning: ceteris paribus, with some predecessor(s) switching to actions that reveal greater confidence in one state of the world, agents nevertheless become more confident in the other. A necessary and sufficient condition for non-monotone learning is provided in an illustrative setting where agents are either uninformed or partially informed by binary private signals. In a general setting with continuous private signals, we obtain a sufficient condition for non-monotone learning that does not rely on the shape of the underlying information structures apart from a couple of simple boundary requirements. As a result, a social learning process that completes eventually can be well approximated in a way such that non-monotone learning and anti-imitation are bound to happen.

Informationally Efficient Prices are not Allocationally Efficient

We consider a dynamic market with two firms that sell competing common-value products. The firms offer both products to an infinite set of rational consumers. Each consumer observes a conditionally independent and identically distributed private signal about the product qualities. Consumers enter the market sequentially, observe the market history, and, given the current prices, decide which product to buy. We assume that, with a constant frequency, prices are set to be informationally efficient (i.e., adjusted to reflect the accumulated market information) and establish that a higher frequency of price adjustments implies strictly lower expected welfare.

Hidden Experts in the Crowd: Using Meta-Predictions to Leverage Expertise in Single-Question Prediction Problems

Modern forecasting algorithms use the wisdom of crowds to produce forecasts better than those of the best identifiable expert. However, these algorithms may be inaccurate when crowds are systematically biased or when expertise varies substantially across forecasters. Recent work has shown that meta-predictions—a forecast of the average forecasts of others—can be used to correct for biases even when no external information, such as forecasters’ past performance, is available. We explore whether meta-predictions can also be used to improve forecasts by identifying and leveraging the expertise of forecasters. We develop a confidence-based version of the Surprisingly Popular algorithm proposed by Prelec, Seung, and McCoy. As with the original algorithm, our new algorithm is robust to bias. However, unlike the original algorithm, our version is predicted to always weight forecasters with more informative private signals more than forecasters with less informative ones. In a series of experiments, we find that the modified algorithm does a better job in weighting informed forecasters than the original algorithm and show that individuals who are correct more often on similar decision problems contribute more to the final decision than other forecasters. Empirically, the modified algorithm outperforms the original algorithm for a set of 500 decision problems.This paper was accepted by Yan Chen, decision analysis.

Liquidity Formation and Preopening Periods in Financial Markets

This paper studies the role of preopening periods in liquidity formation and welfare in financial markets. Because no transaction occurs during these preopening periods, their economic significance could be questioned. We model a market where costly participation and asymmetric information prevent latent liquidity from being expressed. At equilibrium, risk‐averse insiders use preopening periods to better coordinate supply and demand of liquidity by communicating liquidity needs, thus improving welfare. Partial or full communication of private signals by the insider with the asset at preopening periods does not always enhance liquidity formation, but improves welfare through reducing adverse selection risk faced by the outsider, and increasing the likelihood of her entry. Our findings have implications for portfolio management and the design of financial markets.

Persuading Large Investors

A regulator who designs a public stress test to avert default of a distressed bank via private investment must account for large investors' private information on the bank's state. We provide conditions for crowding-in (crowding-out) so that the regulator offers more (less) information to better-informed investors. Crowding-in obtains if investors' private information is not too discriminating of the state. We show that the region of the common prior is consequential: if crowding-in occurs for ex-ante optimistic investors then crowding-out follows if they were instead pessimistic. Investors' value from more precise private signals may come from their effect on the public test's precision.

Non-Bayesian updating in a social learning experiment

In our laboratory experiment, subjects, in sequence, have to predict the value of a good. The second subject in the sequence makes his prediction twice: first (“first belief”), after he observes his predecessor's prediction; second (“posterior belief”), after he observes his private signal. We find that the second subjects weigh their signal as a Bayesian agent would do when the signal confirms their first belief; they overweight the signal when it contradicts their first belief. This way of updating, incompatible with Bayesianism, can be explained by the Likelihood Ratio Test Updating (LRTU) model, a generalization of the Maximum Likelihood Updating rule. It is at odds with another family of updating, the Full Bayesian Updating. In another experiment, we directly test the LRTU model and find support for it.

Learning from like-minded people

We propose a social learning model where agents choose who to communicate with based on the trade-off between collecting information and influencing others' actions. Like-minded people reveal more truthful information, but talking to someone different makes an agent more likely to influence another person's action, because like-minded people will take similar actions anyway. In our model agents start with heterogeneous priors about an unknown state of the world. First agents form coalitions of communication partners. Then everyone observes a private i.i.d. signal and sends a message to her partner. Finally everyone takes an action based on her prior, her private signal, and her partner's message. Our main finding is that when signals and actions are binary, agents form assortative coalitions, even though assortative coalitions are inefficient for social welfare.

Feddersen and Pesendorfer meet Ellsberg

The Condorcet Jury Theorem formalises the “wisdom of crowds”: binary decisions made by majority vote are asymptotically correct as the number of voters tends to infinity. This classical result assumes like-minded, expected utility maximising voters who all share a common prior belief about the right decision. Ellis (Theor Econo 11(3): 865–895, 2016) shows that when voters have ambiguous prior beliefs—a (closed, convex) set of priors—and follow maxmin expected utility (MEU), such wisdom requires that voters’ beliefs satisfy a “disjoint posteriors” condition: different private signals lead to posterior sets with disjoint interiors. Both the original theorem and Ellis’s generalisation assume symmetric penalties for wrong decisions. If, as in the jury context, errors attract asymmetric penalties then it is natural to consider voting rules that raise the hurdle for the decision carrying the heavier penalty for error (such as conviction in jury trials). In a classical model, Feddersen and Pesendorfer (Am Politi Sci Rev 92(1):23–35, 1998) have shown that, paradoxically, raising this hurdle may actually increase the likelihood of the more serious error. In particular, crowds are not wise under the unanimity rule: the probability of the more serious error does not vanish as the crowd size tends to infinity. We show that this “Jury Paradox” persists in the presence of ambiguity, whether or not juror beliefs satisfy Ellis’s “disjoint posteriors” condition. We also characterise the strictly mixed equilibria of this model and study their properties. Such equilibria cannot exist in the absence of ambiguity but may exist for arbitrarily large jury size when ambiguity is present. In addition to uninformative strictly mixed equilibria, analogous to those exhibited by Ellis (Theor Econo 11(3): 865–895, 2016), there may also exist strictly mixed equilibria which are informative about voter signals.

All-pay auctions with private signals about opponents’ values

We study all-pay auctions where each player observes her private value as well as a noisy private signal about the opponent’s value, following Fang and Morris’s (J Econ Theory 126(1):1–30, 2006) analysis of winner-pay auctions with multidimensional private signals. A unique symmetric monotonic equilibrium exists if the signal is not informative enough. When the signal is sufficiently informative, there exists a symmetric non-monotonic equilibrium in which all types of players randomize in overlapping supports. The revenue is lower than that in the standard independent private value setting. Fixing the signal’s informativeness, the all-pay auction raises lower revenue than the second-price auction, whereas the revenue ranking between the all-pay and the first-price auction is ambiguous.

Simple bets to elicit private signals

This paper introduces two simple betting mechanisms—top‐flop and threshold betting—to elicit unverifiable information from crowds. Agents are offered bets on the rating of an item about which they received a private signal versus that of a random item. We characterize conditions for the chosen bet to reveal the agents' private signal even if the underlying ratings are biased. We further provide microeconomic foundations of the ratings, which are endogenously determined by the actions of other agents in a game setting. Our mechanisms relax standard assumptions of the literature, such as common prior, and homogeneous and risk neutral agents.