We introduce a framework for studying the equilibrium effects of machine learning. Agents process information using a Chow and Liu (1968) tree, a widely-used machine learning procedure that admits a closed-form solution. We apply the model to an asset market with dispersed information based on Hellwig (1980). The price mechanism fails to aggregate the information extracted by the algorithm, even approximately. While there are partial equilibrium benefits from access to algorithms, the equilibrium price aggregates less information than the rational equilibrium. Equilibrium typically features diverse world-models, demands, and utilities, even with ex ante identical agents.

Individuals fail to recognize that two weak positive signals of a rare event together constitute strong evidence for the event. We demonstrate the dramatic effect of replacing Bayesian aggregation of signals with commonly used aggregation procedures in a simple model of opinion formation and a model of strategic voting.

A BSTRACT : A decision maker observes multiple signals about an event. He has information about the frequency of the event given each individual signal and wishes to update his beliefs about the event. We examine the problem experimentally and identify some of the commonly used procedures for signal aggregation. These procedures are for the most part inconsistent with Bayesian updating. We apply some of these procedures to a well-known panel game that has previously been studied under the standard Bayesian assumptions and find that the properties of the equilibria differ significantly.

We study the effects of diverse beliefs on equilibrium securitization under risk neutrality. We provide a simple characterization of the optimal securities. Pooling and tranching of assets emerges in equilibrium as a consequence of the traders' diverse beliefs about asset returns. The issuer of securities tranches the asset pool, and traders sort among the tranches according to their beliefs. We show how the traders' disagreement about the correlation of asset returns is a key factor in determining which assets are pooled.

We present a decision-theoretic analysis of an agent’s understanding of the interdependencies in her choices. We provide the foundations for a simple and flexible model that allows the misperception of correlated risks. We introduce a framework in which the decision maker chooses a portfolio of assets among which she may misperceive the joint returns, and present simple axioms equivalent to a representation in which she attaches a probability to each possible joint distribution over returns and then maximizes subjective expected utility using her (possibly misspecified) beliefs.

We study an agent who chooses a profile of actions between which she may misperceive the correlation. Our agent cannot be modeled by reducing every action profile to an act, as implied by the usual monotonicity axiom. We introduce a novel framework that explicitly considers action profiles and axiomatically characterize a model that relaxes monotonicity but retains the rest of the expected utility axioms. Our agent acts as if she attaches a probability to each possible correlation structure and then maximizes expected utility using her (possibly misspecified) beliefs. This representation nests several models used in the behavioral game theory literature to consider imperfect inference, including correlation neglect.

This paper discusses community enforcement in infinitely repeated, two-action games with local interaction and uncertain monitoring. Each player interacts with and observes only a fixed set of opponents, of whom he is privately informed. The main result shows that when beliefs about the monitoring structure have full support, efficiency can be sustained with sequential equilibria that are independent of the players' beliefs. Stronger results are obtained when only acyclic monitoring structures are allowed or players have unit discount rates. These equilibria satisfy numerous robustness properties.

Under sequential voting, voting late enables conditioning on which candidates are viable, while voting early can influence the field of candidates. But the latter effect can be harmful: shrinking the field increases not only the likelihood that future voters vote for one's favorite candidate, but also that they vote for an opponent. Specifically, if one's favorite candidate is significantly better than all others, then early voting is disadvantageous and all equilibria are equivalent to simultaneous voting. Conversely, when some other candidate is almost as good, then any Markov, symmetric, anonymous equilibrium involves sequential voting (and differs from simultaneous voting).

We analyze a simple model of an asset market, in which a large rational trader interacts with "noise speculators" who seek short-run speculative gains, and become active following a prolonged episode of mispricing relative to the asset's fundamental value. The model gives rise to price patterns such as bubble dynamics, positive short-run correlation and vanishing long-run correlation of price deviations from the fundamental value. We argue that this example model sheds light on the question as to whether rational speculators abet or curb price fluctuations.

We model a dynamic, competitive market, where in every period, risk-neutral traders trade a one-period bond against an infinitely lived asset, with limited short-selling of the long-term asset. Traders lack structural knowledge and use different "incomplete theories," all of which give statistically correct beliefs about next period's market price of the long-term asset. The more theories there are in the market, the higher is the equilibrium price of the long-term asset. Investors with more complete theories do not necessarily earn higher returns than those with less complete ones, who can earn above the risk-free rate. We provide two necessary conditions for a trader to earn above the risk-free rate.

This article studies market competition when firms can influence consumers’ ability to compare market alternatives through their choice of price “formats.” In our model, the ability of a consumer to make a comparison depends on the firms’ format choices. Our main results concern the interaction between firms’ equilibrium price and format decisions and its implications for industry profits and consumer switching rates. In particular, market forces drive down the firms’ profits to a “constrained competitive” benchmark if and only if the comparability structure satisfies a property that we interpret as a form of “frame neutrality.” The same property is necessary for equilibrium behavior to display statistical independence between price and format decisions. We also show that narrow regulatory interventions that aim to facilitate comparisons may have an anticompetitive effect. JEL Codes: C79, D03, D43.

This article studies market competition when firms can influence consumers' ability to compare market alternatives through their choice of price "formats." In our model, the ability of a consumer to make a comparison depends on the firms' format choices. Our main results concern the interaction between firms' equilibrium price and format decisions and its implications for industry profits and consumer switching rates. In particular, market forces drive down the firms' profits to a "constrained competitive" benchmark if and only if the comparability structure satisfies a property that we interpret as a form of "frame neutrality." The same property is necessary for equilibrium behavior to display statistical independence between price and format decisions. We also show that narrow regulatory interventions that aim to facilitate comparisons may have an anticompetitive effect.

The paper discusses community enforcement in infinitely repeated two-action games with local monitoring. Each player interacts with and observes only a fixed set of partners, of whom he is privately informed. The main result shows that for generic beliefs efficiency can be sustained in a sequential equilibrium in which strategies are independent of theplayers' beliefs about the monitoring structure. Stronger results are obtained when players are arbitrarily patient and payoffs are evaluated according to Banach-Mazur limits, and when players are impatient and only acyclic monitoring structures are allowed.

We analyze the structure of a society driven by power relations. Our model has an exogenous power relation over the set of coalitions of agents. Agents determine the social order by forming coalitions. The power relations determine the ranking of agents in society for any social order. We study a cooperative game in partition function form and introduce a solution concept, the stable social order, which exists and includes the core. We investigate a refinement, the strongly stable social order, which incorporates a notion of robustness to variable power relations. We provide a complete characterization of strongly stable social orders.

We analyze a model of market competition in which two identical firms choose prices as well as how to present, or “frame”, their products. A consumer is randomly assigned to one firm, and whether he makes a price comparison with the other firm is a probabilistic function of the firms’ framing strategies. We analyze Nash equilibria in this model. In particular, we show how the answers to the following questions are linked: (1) Are firms’ choices of prices and frames correlated? (2) Can firms earn payoffs in excess of the max-min level? (3) Does greater consumer rationality (in the sense of better ability to make price comparisons) imply lower equilibrium prices? We also argue that our model provides a novel account of the phenomenon of product differentiation.

We study a model in which "useless" luxury goods are a mechanism for redistribution.

In the jungle, power and coercion govern the exchange of resources. We study a simple, stylised model of the jungle that mirrors an exchange economy. We define the notion of jungle equilibrium and demonstrate that a number of standard results of competitive markets hold in the jungle.

In this paper, we construct a universal type space for a class of possibility models by imposing topological restrictions on the players’ beliefs. Along the lines of Mertens and Zamir [International Journal of Game Theory, 14 (1985) 1] or Brandenburger and Dekel [Journal of Economic Theory 59 (1993) 189], we show that the space of all hierarchies of compact beliefs that satisfy common knowledge of coherency (types) is canonically homeomorphic to the space of compact beliefs over the state of nature and the types of the other players. The resulting type space is universal, in the sense that any compact and continuous possibility structure can be uniquely represented within it. We show how to extend our construction to conditional systems of compact beliefs.

We study a model in which being more powerful does not necessarily imply being wealthier.