According to economic theory, supported by empirical and laboratory evidence, the equilibrium price of a financial security reflects all of the information regarding the security's value. We investigate the dynamics of the computational process on the path toward equilibrium, where information distributed among traders is revealed stepby -step over time and incorporated into the market price. We develop a simplified model of an information market, along with trading strategies, in order to formalize the computational properties of the process. We show that securities whose payoffs cannot be expressed as a weighted threshold function of distributed input bits are not guaranteed to converge to the proper equilibrium predicted by economic theory. On the other hand, securities whose payoffs are threshold functions are guaranteed to converge, for all prior probability distributions. Moreover, these threshold securities converge in at most n rounds, where n is the number of bits of distributed information. We also prove a lower bound, showing a type of threshold security that requires at least n/2 rounds to converge in the worst case.

This paper studies information aggregation in dynamic markets with a finite number of partially informed strategic traders. It shows that for a broad class of securities, information in such markets always gets aggregated. Trading takes place in a bounded time interval, and in every equilibrium, as time approaches the end of the interval, the market price of a “separable” security converges in probability to its expected value conditional on the traders ’ pooled information. If the security is “non-separable, ” then there exists a common prior over the states of the world and an equilibrium such that information does not get aggregated. The class of separable securities includes, among others, Arrow-Debreu securities, whose value is one in one state of the world and zero in all others, and “additive ” securities, whose value can be interpreted as the sum of traders ’ signals.

Given common priors, no agent can publicly estimate a non-zero sign for the difference between his estimate and another agent’s future estimate. Thus rational agents cannot publicly anticipate the direction in which other agents will disagree with them.

Much evidence supports that financial markets have the ability to aggregate information. When tied to a random variable, a financial market can forecast the value of the random variable. It then becomes a prediction market. We establish a model of prediction markets with aggregate uncertainty, and theoretically characterize some fundamental properties of prediction markets. Specifically, we have shown that a prediction market is guaranteed to converge to an equilibrium, where traders have consensus on the forecast. The best possible prediction a prediction market can make is the direct communication equilibrium. However, prediction markets do not always converge to it. We have proved that a sufficient condition for the convergence to the direct communication equilibrium under our model is that the private information of each trader, conditioned on the state of the world, is identically and independently distributed. Furthermore, if this condition is satisfied, the prediction market converges in at most two rounds. 1

The novel idea of setting up Internet-based virtual markets, information markets, to aggregate dispersed information and predict outcomes of uncertain future events has empirically found its way into many domains. But the theoretical examination of information markets has lagged relative to their implementation and use. This paper proposes a simple theoretical model of information markets to understand their information dynamics. We investigate and provide initial answers to a series of research questions that are important to understanding how information markets work, which are: (1) Does an information market converge to a consensus equilibrium? (2) If yes, how fast is the convergence process? (3) What is the best possible equilibrium of an information market? and (4) Is an information market guaranteed to converge to the best possible equilibrium?

Abstract. Given a finite state space and common priors, common knowledge of the identity of an agent with the minimal (or maximal) expectation of a random variable implies “consensus”, i.e., common knowledge of common expectations. This “extremist ” statistic induces consensus when repeatedly announced, and yet, with n agents, requires at most log 2 n bits to broadcast. Key words: consensus, common knowledge, information pooling, Bayesian learning 1.

Sigatures are on file in the Graduate School. iii In almost all walks of life, predicting uncertain future events plays an essential role in decision-making processes. However, information related to future events frequently exists only as dispersed opinions, insights, and intuitions of individuals. Each individual only knows a little, but aggregating the dispersed information together may make considerable contribution to decision making. This is typical in many domains including business, politics, and entertainment. Therefore, how to aggregate such dispersed information for useful decision support is a crucial task. Markets have shown great potential as one of the most effective mechanisms for gathering distributed information and generating accurate forecasts, often surpassing many existing methods in practice. This research studies information markets, markets that are specially designed for information aggregation and forecasting, from four different perspectives: theoretical examination, experimental evaluation, empirical analysis, and design.

with aggregate uncertainty

We propose a model of prediction markets where participants are biased according to their social relationships. We relax the standard assumption of complete rationality and adopt an arguably more realistic model where agents are disproportionally influenced by their neighbors in a social network. We conduct extensive agent-based simulations of our model. We find that prices in prediction markets remain accurate even when participants are biased and irrational. Moreover, accuracy is robust to changes in many factors, including how individuals are motivated to participate in the market, the way that individuals use public information, individual utility functions, the topology of the social network, and the strength of social influences. Our model can explain the high volume of trade often observed in speculative markets that is hard or impossible to explain under standard market rationality assumptions. Our model can also explain the documented ability of prediction markets to succeed even in the face of biased and irrational participants.

Several recent studies in experimental economics have tried to measure beliefs of subjects engaged in strategic games with other subjects. Using data from one such study (Nyarko-Schotter, 2002) we conduct an experiment where our experienced subjects observe early rounds of strategy choices from that study and are given monetary incentives to report forecasts of choices in later rounds. We elicit beliefs using three different scoring rules: linear, logarithmic, and quadratic. There are differences between the elicited beliefs under quadratic and logarithmic scoring rules in spite of both being proper scoring rules. The (improper) linear scoring rule frequently elicits boundary forecasts as theory predicts, and is poorly calibrated. We compare the forecasts of our trained observers to forecasts of the actual players in the Nyarko-Schotter experiment and identify several differences. There was a significant positive correlation between observer forecasts and the choice behavior in the game under both proper scoring rules, but no significant correlation between the players ’ own forecasts and the actual play. This raises doubts about whether beliefs can be reliably elicited from players who simultaneously have a stake in the target of their forecast, in