Interpreting prediction market prices as probabilities by Justin Wolfers, Eric Zitzewitz

Interpreting prediction market prices as probabilities by Justin Wolfers, Eric Zitzewitz

1. Introduction 
In a provocative recent paper, Charles Manski (2004) asked “What is the logical 
basis for interpreting the price of an all-or-nothing futures contract as a market 
probability that the event will occur?” Moreover, he notes that “recent papers on 
prediction markets provide no formal analysis showing how such markets aggregate 
information or opinions.” Manski poses an excellent question, and he highlights an 
important topic for research. As prediction markets have become of more widespread 
interest, the prices of contracts tied to events as diverse as the re-election of President 
Bush, the ouster of Saddam Hussein, next month’s non-farm payrolls number, or the 
success of specific products have been interpreted in both academic and popular 
discussion as though interchangeable with “the market’s beliefs”. 
This paper presents an initial response to Manski’s challenge, providing a formal 
model which provides a plausible microfoundation under which one can treat prediction 
market prices as probabilities. Further, we explore deviations from our baseline model, 
and show that for most plausible parameters, prediction market prices at least 
approximate the central tendency of the distribution of beliefs of traders.
 The specific 
model offered by Manski is a special case of our model, and while he emphasizes special 
assumptions that can lead his model to yield counter-intuitive results, we show that 
sensible distributions of preferences and beliefs yield more intuitive results. 
We proceed as follows. The next section sketches two very simple models in 
which prediction market prices coincide exactly with the mean of the belief among 
traders. The following section generalizes the model, showing that prediction market 
prices can deviate from mean beliefs, but that this deviation is typically small. The extent 
of the deviation depends crucially on how widely dispersed beliefs are, and so in the final 
section we present field evidence on this point. To preview, our results suggest that 
while prediction market prices and mean beliefs may diverge, they are typically very 
close. We interpret our results as providing a microfoundation for the claim that 
prediction markets (approximately) efficiently aggregate beliefs. 

2. Two Simple Models 
We consider a simple prediction market in which traders buy and sell an all-or-

nothing contract (a binary option) paying $1 if a specific event occurs, and nothing 
otherwise. There is heterogeneity in beliefs among the trading population, and following 
Manski’s notation, we denote trader j’s belief that the event will occur as qj. These 
beliefs are orthogonal to wealth levels (y), and are drawn from a distribution, F(q). 
Individuals are price-takers and trade so as to maximize their subjectively expected 
utility. Wealth is only affected by the event via the prediction market, so there is no 
hedging motive for trading the contract. 
We first consider the case where traders have log utility, and we endogenously derive 
their trading activity, given the price of the contract is π. Thus, in deciding how many 
contracts, x, to buy, traders solve the following problem: 

Interpreting prediction market prices as probabilities by Justin Wolfers, Eric Zitzewitz