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Logarithmic Market Scoring Rules for Modular Combinatorial Information Aggregation
 Journal of Prediction Markets
, 2002
"... In practice, scoring rules elicit good probability estimates from individuals, while betting markets elicit good consensus estimates from groups. Market scoring rules combine these features, eliciting estimates from individuals or groups, with groups costing no more than individuals. ..."
Abstract

Cited by 111 (6 self)
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In practice, scoring rules elicit good probability estimates from individuals, while betting markets elicit good consensus estimates from groups. Market scoring rules combine these features, eliciting estimates from individuals or groups, with groups costing no more than individuals.
Eliciting Properties of Probability Distributions
 In Proceedings of the ninth ACM conference on electronic commerce
, 2008
"... We investigate the problem of incentivizing an expert to truthfully reveal probabilistic information about a random event. Probabilistic information consists of one or more properties, which are any realvalued functions of the distribution, such as the mean and variance. Not all properties can be e ..."
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Cited by 37 (6 self)
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We investigate the problem of incentivizing an expert to truthfully reveal probabilistic information about a random event. Probabilistic information consists of one or more properties, which are any realvalued functions of the distribution, such as the mean and variance. Not all properties can be elicited truthfully. We provide a simple characterization of elicitable properties, and describe the general form of the associated payment functions that induce truthful revelation. We then consider sets of properties, and observe that all properties can be inferred from sets of elicitable properties. This suggests the concept of elicitation complexity for a property, the size of the smallest set implying the property.
Information elicitation for decision making
, 2011
"... Proper scoring rules, particularly when used as the basis for a prediction market, are powerful tools for eliciting and aggregating beliefs about events such as the likely outcome of an election or sporting event. Such scoring rules incentivize a single agent to reveal her true beliefs about the eve ..."
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Cited by 10 (5 self)
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Proper scoring rules, particularly when used as the basis for a prediction market, are powerful tools for eliciting and aggregating beliefs about events such as the likely outcome of an election or sporting event. Such scoring rules incentivize a single agent to reveal her true beliefs about the event. Othman and Sandholm [16] introduced the idea of a decision rule to examine these problems in contexts where the information being elicited is conditional on some decision alternatives. For example, “What is the probability having ten million viewers if we choose to air new television show X? What if we choose Y? ” Since only one show can actually air in a slot, only the results under the chosen alternative can ever be observed. Othman and Sandholm developed proper scoring rules (and thus decision markets) for a single, deterministic decision rule: always select the the action with the greatest probability of success. In this work we significantly generalize their results, developing scoring rules for other deterministic decision rules, randomized decision rules, and situations where there may be more than two outcomes (e.g. less than a million viewers, more than one but less than ten, or more than ten million).
An initial implementation of the turing tournament to learning in two person games
 Games and Economic Behavior
, 2006
"... and of the National Science Foundation (Grant #SES0079301) for help in funding the experiments. We ..."
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Cited by 7 (0 self)
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and of the National Science Foundation (Grant #SES0079301) for help in funding the experiments. We
Eliciting Objective Probabilities via Lottery Insurance Games
 Computational Mathematics Laboratory, Rice University
, 1993
"... Since utilities and probabilities jointly determine choices, eventdependent utilities complicate the elicitation of subjective event probabilities. However, for the usual purpose of obtaining the information embodied in agent beliefs, it is su#cient to elicit objective probabilities, i.e., proba ..."
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Cited by 1 (1 self)
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Since utilities and probabilities jointly determine choices, eventdependent utilities complicate the elicitation of subjective event probabilities. However, for the usual purpose of obtaining the information embodied in agent beliefs, it is su#cient to elicit objective probabilities, i.e., probabilities obtained by updating a known common prior with that agent's further information. Bayesians who play a Nash equilibrium of a certain insurance game before they obtain relevant information will afterward act regarding lottery ticket payments as if they had eventindependent riskneutral utility and a known common prior. Proper scoring rules paid in lottery tickets can then elicit objective probabilities.
[Extended Abstract]
"... We investigate the problem of truthfully eliciting an expert’s assessment of a property of a probability distribution, where a property is any realvalued function of the distribution such as mean or variance. We show that not all properties are elicitable; for example, the mean is elicitable and th ..."
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We investigate the problem of truthfully eliciting an expert’s assessment of a property of a probability distribution, where a property is any realvalued function of the distribution such as mean or variance. We show that not all properties are elicitable; for example, the mean is elicitable and the variance is not. For those that are elicitable, we provide a representation theorem characterizing all payment (or “score”) functions that induce truthful revelation. We also consider the elicitation of sets of properties. We then observe that properties can always be inferred from sets of elicitable properties. This naturally suggests the concept of elicitation complexity; the elicitation complexity of property is the minimal size of such a set implying the property. Finally we discuss applications to prediction markets.
The Query Complexity of Scoring Rules
, 2012
"... Proper scoring rules are crucial tools to elicit truthful information from experts. A scoring rule maps X, an expertprovided distribution over the set of all possible states of the world, and ω, a realized state of the world, to a real number representing the expert’s reward for his provided inform ..."
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Proper scoring rules are crucial tools to elicit truthful information from experts. A scoring rule maps X, an expertprovided distribution over the set of all possible states of the world, and ω, a realized state of the world, to a real number representing the expert’s reward for his provided information. To compute this reward, a scoring rule queries the distribution X at various states. The number of these queries is thus a natural measure of the complexity of the scoring rule. We prove that any bounded and strictly proper scoring rule that is deterministic must make a number of queries to its input distribution that is a quarter of the number of states of the world. When the state space is very large, this makes the computation of such scoring rules impractical. We also show a new stochastic scoring rule that is bounded, strictly proper, and which only makes two queries to its input distribution. Thus, using randomness allows us to have significant savings when computing scoring rules. 1