6 research outputs found
Informational Substitutes
We propose definitions of substitutes and complements for pieces of
information ("signals") in the context of a decision or optimization problem,
with game-theoretic and algorithmic applications. In a game-theoretic context,
substitutes capture diminishing marginal value of information to a rational
decision maker. We use the definitions to address the question of how and when
information is aggregated in prediction markets. Substitutes characterize
"best-possible" equilibria with immediate information aggregation, while
complements characterize "worst-possible", delayed aggregation. Game-theoretic
applications also include settings such as crowdsourcing contests and Q\&A
forums. In an algorithmic context, where substitutes capture diminishing
marginal improvement of information to an optimization problem, substitutes
imply efficient approximation algorithms for a very general class of (adaptive)
information acquisition problems.
In tandem with these broad applications, we examine the structure and design
of informational substitutes and complements. They have equivalent, intuitive
definitions from disparate perspectives: submodularity, geometry, and
information theory. We also consider the design of scoring rules or
optimization problems so as to encourage substitutability or complementarity,
with positive and negative results. Taken as a whole, the results give some
evidence that, in parallel with substitutable items, informational substitutes
play a natural conceptual and formal role in game theory and algorithms.Comment: Full version of FOCS 2016 paper. Single-column, 61 pages (48 main
text, 13 references and appendix
Optimal Scoring Rule Design
This paper introduces an optimization problem for proper scoring rule design.
Consider a principal who wants to collect an agent's prediction about an
unknown state. The agent can either report his prior prediction or access a
costly signal and report the posterior prediction. Given a collection of
possible distributions containing the agent's posterior prediction
distribution, the principal's objective is to design a bounded scoring rule to
maximize the agent's worst-case payoff increment between reporting his
posterior prediction and reporting his prior prediction.
We study two settings of such optimization for proper scoring rules: static
and asymptotic settings. In the static setting, where the agent can access one
signal, we propose an efficient algorithm to compute an optimal scoring rule
when the collection of distributions is finite. The agent can adaptively and
indefinitely refine his prediction in the asymptotic setting. We first consider
a sequence of collections of posterior distributions with vanishing covariance,
which emulates general estimators with large samples, and show the optimality
of the quadratic scoring rule. Then, when the agent's posterior distribution is
a Beta-Bernoulli process, we find that the log scoring rule is optimal. We also
prove the optimality of the log scoring rule over a smaller set of functions
for categorical distributions with Dirichlet priors
The Possibilities and Limitations of Private Prediction Markets
We consider the design of private prediction markets, financial markets designed to elicit predictions about uncertain events without revealing too much information about market participants' actions or beliefs. Our goal is to design market mechanisms in which participants' trades or wagers influence the market's behavior in a way that leads to accurate predictions, yet no single participant has too much influence over what others are able to observe. We study the possibilities and limitations of such mechanisms using tools from differential privacy. We begin by designing a private one-shot wagering mechanism in which bettors specify a belief about the likelihood of a future event and a corresponding monetary wager. Wagers are redistributed among bettors in a way that more highly rewards those with accurate predictions. We provide a class of wagering mechanisms that are guaranteed to satisfy truthfulness, budget balance on expectation, and other desirable properties while additionally guaranteeing epsilon-joint differential privacy in the bettors' reported beliefs, and analyze the trade-off between the achievable level of privacy and the sensitivity of a bettor's payment to her own report. We then ask whether it is possible to obtain privacy in dynamic prediction markets, focusing our attention on the popular cost-function framework in which securities with payments linked to future events are bought and sold by an automated market maker. We show that under general conditions, it is impossible for such a market maker to simultaneously achieve bounded worst-case loss and epsilon-differential privacy without allowing the privacy guarantee to degrade extremely quickly as the number of trades grows, making such markets impractical in settings in which privacy is valued. We conclude by suggesting several avenues for potentially circumventing this lower bound