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On the Observational Implications of Knightian Uncertainty.

June 8, 2018

At least since the work of Knight (1921), economists have understood that economic agents may behave differently in risky circumstances, where outcomes are random but governed by known probabilities, as opposed to uncertain circumstances, where risks are unknown. Ellsberg (1961) provides examples that highlight the tendency for some decision makers to be averse to the presence of Knightian uncertainty--or, ambiguity.

In recent years, there has been an explosion of theoretical work developing models that incorporate ambiguity aversion, building off of the seminal contribution of Gilboa and Schmeidler (1989). In the literature to date, Knightian uncertainty has been a factor inserted in a model that could possibly explain puzzling observations. It has served a role analogous to that of dark matter in cosmological models, lurking behind the scenes to explain observed phenomena, never being directly observed. At the same time, a rich literature has evolved exploring the efficiency of betting and prediction markets that price specific events. Following on the early work of Kahneman and Tversky (1979) and Asch, Malkiel and Quandt (1982), the ability of these markets to predict future events has been studied extensively, and a number of empirical anomalies have been identified.

In this paper, we extend the theoretical literature and connect it to the prediction market application. In so doing, we develop more directly observable implications of the presence of Knightian uncertainty than has been achieved previously in the literature, and we provide a method to test for its presence.

While we below will formally derive a model that suggests our test, the intuition of our approach is quite straightforward and can be illustrated using an example from Ellsberg (1961). Suppose that we have two urns. In one urn, we have 50 black balls and 50 red balls. In another urn--the "Knightian urn"--we have 100 balls, but we have no information regarding the proportions. A subject is offered a game. If she pulls a black ball out of the urn, she wins $1. If she pulls a red ball out she wins nothing. The literature has documented a tendency for individuals to prefer the urn with the known probabilities, suggesting that they exhibit the aversion to ambiguity discussed above.

Suppose that an econometrician could observe games played with both of the urns in Ellsberg's game. With a number of repeated trials, the sample proportions from the first urn would fairly rapidly indicate an estimate that the binomial probability of victory is 50 percent. With enough data, one would say that with great confidence. On the other hand, if one observed repeated play with the second, Knightian urn which, after all, has some number of black balls in it, then the sample proportion would also converge to an estimated binomial probability, but that probability would not necessarily be 50 percent.

The observation that motivates this paper stems from this thought experiment. Given a market-derived ex ante probability of a binary event, as one frequently observes in betting markets, there will naturally be circumstances where information is extremely solid, and odds are quite far from 50 percent. There will also be situations where information suggests there is an even match (as with a coin flip), and the contract suggests there is close to a 50 percent chance of either outcome. This often happens, for example, in presidential futures markets in the U.S. after the conventions are over. But it is also possible that there are contracts that suggest that the odds of either outcome are 50 percent because the event is shrouded in ambiguity. If we were to estimate the ex post sample proportions from just these contracts with ex ante 50 percent probabilities, then they could, as in the Ellsberg example above, be anything. If we were to estimate the ex post sample proportions of the high information contracts with probabilities far from 50 percent, the proportions and ex ante probabilities should, if markets are efficient, align. But close to 50 percent, they might not, and if they do not, it is an indication of the presence of Knightian uncertainty. Thus, the pattern by which the relationship between ex post proportions and ex ante probabilities deviates from the 45 degree line becomes informative regarding the presence of Knightian uncertainty. We also discuss the extent to which learning can occur in markets over time. If Knightian uncertainty induces knowledge acquisition, then the relationship between proportions and probabilities will evolve as a market matures, a possibility we explore in the paper.

The next section briefly reviews the literature. In Sections II and III, we draw on the work of Gilboa and Schmeidler (1989) and Dow and Werlang (1992) and develop a model that suggests that the pattern described by our intuitive example would emerge in a market influenced by the present of significant Knightian uncertainty. In Section IV, we provide some high-level evidence that the relationship between ex post proportions and ex ante probabilities is consistent with the predictions of our model. The last section concludes.

I. Literature Review

This paper draws from two different strands in the literature. First, theorists have made remarkable strides in recent years incorporating Knightian uncertainty and ambiguity aversion into models of financial markets.

These models have, according to an exhaustive recent review, "implications for portfolio choice and asset pricing that are very different from those of SEU (subjective expected utility theory) and that help to explain otherwise puzzling features of the data." (1) Ambiguity aversion could help explain the tendency of markets to stop operating during financial crises, for prices to not be completely informative, and even for there to be bank runs. (2)

This branch of the literature has focused on financial markets in general. At the same time, an equally impressive literature has emerged exploring the functioning of prediction markets, which, for the most part, price in the probability of specific binary events. As Thaler and Ziemba (1988) first noted, these prediction markets may be a better laboratory to test cutting edge theories, as they contain contracts with known durations, and observable discrete events that stop the trading. While an equity might live on virtually forever, a presidential election future has a specific end date, and its ability to forecast the outcome can be precisely evaluated.

This second literature has advanced both empirically and theoretically. On the theoretical side, Ali (1977) and Manski (2006) illustrate that the beliefs of bettors may not necessarily yield a market-based probability. More recently, Snowberg and Wolfers (2010) identify the conditions under which prediction market prices coincide with bettors' mean beliefs about probabilities. On the empirical side, prediction markets have been found to be informative regarding the odds of events occurring. Berg et al. (2008), for example, find that the Iowa Electronic Markets outperformed polls in predicting election outcomes. At the same time, markets have been found to exhibit a favorite-longshot bias, with favorites outperforming their odds, and long-shots underperforming (see, e.g., Cain, Law and Peel 2000). A number of possible explanations for this pattern include insider trading (Shin 1992), risk-loving behavior (Weitzman 1965; Quandt 1986), belief dispersion (Gandhi and Serrano-Padial 2015), and imperfect ability to process information (Snowberg and Wolfers 2010).

The connection of these two literatures is promising, as betting markets often exist for events, such as Brexit or elections, for which Knightian uncertainty may well be present. Since they also have finite and determinate life spans, they also allow the econometrician the ability to evaluate their performance ex post. We now turn to illustrating the utility of this approach.

II. A Model of Prediction Market with Ambiguity

A. Setup

EVENTS AND CONTRACTS.

Consider a prediction market for the occurrence of a binary event. There are two all-or-nothing contracts corresponding to the two possible realizations. One contract pays $1 if event A occurs and $0 otherwise, while the other contract pays $1 if the complementary event [A.sup.c] occurs and $0 otherwise. Let [pi] denote the price of contract A. No-arbitrage condition dictates that, in equilibrium, the price of contract [A.sup.c] be 1 - [pi].

TRADERS.

There is a continuum I of competitive traders, each endowed with homogeneous initial wealth w. The net position on contract A held by trader i is denoted by [x.sub.i] [member of] R. (3) Given price [pi], the final wealth wi of trader i is

[mathematical expression not reproducible]

All traders have log utility of their final wealth: u ([w.sub.i]) = ln [w.sub.i].

Beliefs and Ambiguity.

Suppose trader i has a subjective belief that event A occurs with probability q [member of] [0,1]. Then, the subjective expected utility of trader i from holding position [x.sub.i] at price [pi] is given by

U ([pi],[x.sub.i]; q) = qln (w + (1 - [pi]) [x.sub.i]) + (1 - q) ln (w - [pi][x.sub.i]).

However, ambiguity exists, for traders may be uncertain about how likely event A is to occur. We follow Gilboa and Schmeidler (1989) and model ambiguity using the "multiple-prior" framework. Specifically, suppose each trader i considers every probability q [member of] [[q.sub.i] - [??], [q.sub.i] + [??]], where [??] [greater than or equal to] 0, an admissible probability that governs the realization of the binary event. Under this framework, [q.sub.i] represents the "mean" belief of trader i, while [??] is interpreted as a measure of ambiguity. Given price [pi], trader i chooses position [x.sub.i] to maximize the minimum--that is, the worst-case scenario--of all her admissible, subjective expected utilities:

(1) [mathematical expression not reproducible]

Traders are heterogeneous in mean belief. Let the distribution of traders' mean beliefs be characterized by a cumulative distribution function F over interval [[??], 1 - [??]]. That is, for the most pessimistic trader, the worst-case belief that A occurs is probability 0 while, for the most optimistic trader, the best-case belief that A occurs is probability 1.

B. Optimal Demand and Portfolio Inertia

Solving the inner minimization reduces the optimization problem (1) to

[mathematical expression not reproducible]

where sgn (*) is an indicator function that takes the sign of its argument.

The intuition behind the above expression is straightforward. If trader i has a positive position on contract A, then the worst-case scenario would be that event A occurs with probability [q.sub.i] - [??], the lower bound. Similarly, if the position of trader i is negative, then, in the worst-case scenario, event A occurs with the upper-bound probability, [q.sub.i] + [??].

Solving the maximization problem gives the optimal (net) demand for contract A by trader i,

(2) [mathematical expression not reproducible]

as a function of price and mean belief. Therefore, trader i longs contract A when the price is lower than her most pessimistic belief, and shorts contract A when the price is higher than her most optimistic belief. For any price in the intermediate range [[q.sub.i] - [??], [q.sub.i] + [??]], trader i does not participate in the prediction market--the phenomenon of portfolio inertia.

That portfolio inertia arises when investors have maxmin preferences is well known in the finance literature since the work by Dow and Werlang (1992). The setup of this model replicates this phenomenon in the context of prediction markets. In particular, for each trader, the size of price region at which portfolio inertia occurs is given by 2[??]. In other words, the higher the degree of ambiguity, the more inertial the traders' portfolios.

C. Equilibrium

Given price [pi] for contract A and distribution function F of traders' mean beliefs, the aggregate (net) demand for the contract is given by

(3) [mathematical expression not reproducible]

The prediction market is in equilibrium when the aggregate demand for contract A equals zero, that is, [X.sub.F] ([pi]) = 0. The following proposition establishes the equilibrium price. (4)

PROPOSITION 1: Given distribution function F, the equilibrium price [[pi]*.sub.F] is such that

[mathematical expression not reproducible]

When ambiguity is absent (i.e., [??] = 0) the prediction market aggregates the wisdom of crowds:

[[pi]*.sub.F][|.sub.e=0]= EF (q).

That is, the equilibrium price of contract A corresponds to the average of traders' mean beliefs about the occurrence of event A.

In the presence of ambiguity, however, the prediction market does not necessarily aggregate the wisdom of crowds. In particular, it aggregates the wisdom of crowds if and only if the distribution function F is such that [mathematical expression not reproducible]. The next proposition shows that the situations in which such equality happens to hold are topologically rare.

PROPOSITION 2: With ambiguity, the prediction market "rarely" aggregates the wisdom of crowds. Formally, let [DELTA] be the space of probability distributions over [[??], 1 - [??]], endowed with the weak topology. Then, the subset of probability distributions such that the equilibrium price equals the average of traders' mean beliefs is nowhere dense in [DELTA].

We relegate all proofs to the Mathematical Appendix.

Propositions 1 and 2 together suggests that the presence of ambiguity renders the prediction market ineffective in aggregating the beliefs held by heterogeneous traders.

PROPOSITION 3: The equilibrium quantity of trades is strictly decreasing in the degree of ambiguity.

Proposition 3 is a direct consequence of portfolio inertia. As the degree of ambiguity increases, the "inaction range" of each trader i, [[q.sub.i] - [??], [q.sub.i] + [??]], becomes wider. Since each trader is more likely to stay put in a more ambiguous environment, the aggregate trades must be fewer as well. This result is reminiscent of well-known models of ambiguity in financial economics (e.g., Caballero and Krishnamurthy 2008; Guidolin and Rinaldi 2010; Routledge and Zin 2009), which suggest that a significant increase in Knightian uncertainty may contribute to liquidity hoarding and market breakdown.

Moreover, the deterrence of trades in a particular way is what causes the failure of the prediction market to aggregate beliefs. Specifically, for any prevailing price [pi], the traders who stay put are those with moderate beliefs such that their inaction ranges cover [pi]. Those who trade have beliefs that are more extreme--either more optimistic or more pessimistic--than the abstainers. Suppose the abstaining traders did trade, the chance of the hypothetical market price, after aggregating the abstaining traders' beliefs, happens to be exactly the same as [pi] is zero.

III. Testable Implications

The previous section has derived the equilibrium results under ambiguity. However, since the degree of ambiguity is not observable, those results cannot be tested directly. In this section, we impose more structures on the model and derive implications that are testable with prediction market data.

A. Preliminaries

Suppose the true probability that event A occurs is given by p [member of] [[??], 1 - [??]]. Traders do not know the true probability, but each holds an interval of (subjective) admissible beliefs--their multiple priors. We assume that, for a mass m [member of] (0,1) of traders, their mean beliefs coincide with p. For the other traders, their mean beliefs are continuously distributed over [[??], 1 - [??]]. Assumption 1 embeds these additional structures into the distribution function F.

ASSUMPTION 1: The distribution function F takes the following form:

[mathematical expression not reproducible]

where F is some continuous distribution function of q over [[??], 1 - [??]].

The functional form of [bar.F] is left unspecified. We let [bar.[PHI]] denote the integral of [bar.F], i.e., [mathematical expression not reproducible].

The interpretation of Assumption 1 is that there are some traders whose beliefs happen to be "correct." One could provide a micro-foundation for this setup by assuming that, prior to trading, the mass m of traders have received imperfect private signals that are partially informative of the true probability, while other traders have received no such signals. With such a micro-foundation, the traders' (multiple) beliefs can be interpreted as their (multiple) posteriors. In this section, we adopt a reduced-form approach and build these details directly into the distribution function F.

One may wonder to what extent the mass of traders with "correct" mean beliefs matters to deriving testable implications, which we address in the remark below:

REMARK 1 (On the "Informed" Traders): As will be clear in Proposition 4, our result does not depend on the value of m. Whether the "informed" mass is extremely small, in which case few traders have the "correct" beliefs, or very large, which means most traders do, the result in unchanged. The essence of the setup is that, as long as at least some traders have the "correct" beliefs, whether they trade or abstain makes a qualitative difference in the equilibrium price.

We will come back to this point after establishing the result in the next subsection.

B. Implications

Since the distribution function F is given and parameterized by the true probability p, applying Proposition 1 allows us to solve for the equilibrium price as a function of p, as shown below.

PROPOSITION 4: Under Assumption 1, the equilibrium price [pi]* (p) is:

1) continuous, with [pi]* ([??]) > [??] and [pi]* (1 - [??]) < 1 - [??];

2) such that [pi]* (p) = [pi] for any p [member of] [[pi]? - [??], [pi] + [??]];

3) strictly increasing for p [??] [[pi] - [??], [pi] + [??]];

where [pi] is identified by [pi] - [bar.[PHI]] ([pi]? + [??]) + [bar.[PHI]] ([pi] - [??]) = 1 - 2[??] - [bar.[PHI]] (1 - [??]).

Figure 1 plots the equilibrium price in a p-[pi] diagram, where the true probability p = [([pi]*).sup.-1] ([pi]) is a correspondence of the equilibrium price [pi]. Specifically, it attains a non-singleton set value when [pi] = [pi], with the size of that set equal to 2[??].

The most important feature of the equilibrium (part 2) is that there exists a range of true probabilities, [[pi]? - [??], [pi] + [??]], within which the market price is not at all responsive to any change in the underlying state of the world. That is, [[pi]*.sup.l] (p) = 0 for any p in that range. Instead of predicting, the prediction market simply "assigns" an uninformative number [pi], the mid-point of the range [[pi] - [??],[pi] + [??]], as the price. The reason for this result is straightforward: Since the traders who have the correct mean beliefs about p are not trading, what exactly those traders think about the true state of the world must not be reflected in the market price.

As noted in the previous remark, the exact value of the mass m does not affect the size of the vertical segment [[pi]? - [??], [pi] + [??]], which equals twice the degree of ambiguity. This vertical segment is the consequence of the traders' portfolio inertia under ambiguity. Therefore, when some traders who have the "correct" beliefs are in their inaction range--no matter how few of them there are--then the beliefs they hold will not be reflected in the market equilibrium. Their abstention, therefore, results in the same price for different underlying true probabilities.

Outside the vertical segment [[pi] - [??], [pi] + [??]], however, the prediction market "works" (part 3). Specifically, if all parameters of the model were known, one would be able to infer the true probability p from the equilibrium market price [pi]* (p). The higher the true probability, the higher the price.

Part 1 of the proposition also shows that, for a true probability that is very high (near 1 - [??]) or very low (near e), the equilibrium price exhibits a favorite-longshot bias commonly observed in the literature (e.g., Cain, Law and Peel 2000): favorite events are under-priced while longshot events are over-priced. The intuition is as follows. For a longshot event where p = [??], for example, if the market price was as low as [??], that would imply all traders' mean beliefs were greater than the prevailing price and, hence, all traders would long the contract, which cannot be an equilibrium. Therefore, the equilibrium price of a longshot must be significantly larger than the longshot's odds.

Note that, although the degree of ambiguity, [??], is not directly observable in reality, Proposition 4 yields implications of the presence of ambiguity that are testable with prediction market data. Suppose an econometrician could conduct a large number of repeated trials for each value of the true probability. Then, with enough data, the ex post sample proportion, denoted by P, would converge to the corresponding true probability, p. It follows that the estimated relationship between P and the market price, [pi], would converge to the graph of the correspondence p = [([pi]*).sup.-1] ([pi]). As in Figure 1, such ideal trials would show a big jump at price level [pi] = [pi], with P being generally below the 45-degree line below but close to [pi], and above it just thereafter. Moreover, since the relationship between P and [pi] fundamentally shifts between the two continuous segments, our result suggests a testable structural change near the jump in at [pi].

Before conducting the test, the econometrician may not know where the jump would appear, because [pi], given by

[pi] - [bar.[PHI]] ([pi]? + [??]) + [bar.[PHI]] ([pi] - [??]) = 1 - 2[??] - [bar.[PHI]] (1 - [??]),

depends on the distribution of mean beliefs among all prediction market traders. It follows from the above equation that [pi] would be smaller than 0.5 if F is skewed towards the lowest mean belief [??], and larger than 0.5 if F is skewed towards the highest mean belief 1 - [??]. But when [bar.F] is symmetrically distributed over [[??], 1 - [??]], [pi] would be equal to 0.5, which is the following corollary.

COROLLARY 1: Under Assumption 1, if [bar.F] is a symmetric distribution function over [[??], 1 - [??]] (i.e., [bar.F] (1 - x) = 1 - [bar.F] (x) for any x [member of] [[??], 1 - [??]]), then

[pi] = 0.5.

In practice, the empirical chart would precisely follow Figure 1 with the jump at 0.5 in the case of symmetry, but not if asymmetries were present. But even if one might expect skewness to be present for some contracts but not others, the range for the crossover point could be scattered about the neighborhood of 0.5. The aggregation of a large number of contracts, therefore, could push the average [pi] to be in the neighborhood of 0.5. Since each contract would exhibit a similar (if slightly shifted) pattern, the overall pattern should loosely follow Figure 1 if Knightian uncertainty is important in these markets, even though some reflect symmetry whereas others do not.

Accordingly, the theory suggests that the empirical relationship between P and [pi] would contain a testable structural break around the neighborhood of 0.5, where one would expect to see observations scattered below the 45-degree line to the left of the break-point, and above the 45-degree line to the right of it. The presence of a structural break adjacent to 0.5, therefore, would be an indication that Knightian uncertainty is a factor in the market, and would be consistent with the intuition provided in the introduction.

IV. Empirical Evidence

In this section, we provide some high-level evidence that is consistent with the theoretical predictions. (5)

We use the historical data from Intrade, a popular online prediction platform which operated from 2003 to 2013. The platform hosted prediction contracts across wide-ranging categories of events, such as politics (e.g., which candidate would be elected the U.S. president), entertainment (e.g., which movie would win the Academy Award for the Best Picture), current events (e.g., which city would host the Olympic), etc. We collect all those contracts that are on binary events, regardless of their categories, and record how each binary event had turned out.

The aim of the empirical analysis is to estimate the ex post sample proportion, P, of event A's occurrence as a function of the ex ante price, [pi], of contract A. We process the data in the following way. The observations are sorted by price and evenly partitioned into a number of percentile bins. For each percentile bin, we calculate the sample proportion of event A's occurrences whose corresponding prices fall into that bin. Finally, we plot the sample proportions against the mid-points of the corresponding price bins.

If the theory developed in the previous section holds, the following is what one would expect in the empirics. Recall that the value of [??] depends on the distribution of mean beliefs among traders. Since each observation in the dataset is from a certain market with a certain distribution of mean beliefs held by the participating traders, we can interpret each observation as a single draw from the data-generating process associated with a certain version of Figure 1. For a price bin closer to 0, therefore, it is more likely that the observations contained in the bin have been drawn from the left part of Figure 1, i.e., below the break-point. Similarly, for a price bin closer to 1, the observations are more likely to have been drawn from the right part of Figure 1, i.e., above the break-point. More important, when the price bin is near 0.5, the observations are more likely to be from just around the jump, suggesting a structural break.

We start with the empirical evidence from political events, one of the largest categories in the Intrade dataset. These events, like Brexit and U.S. presidential elections, often see a high volume of transactions between bettors. Figure 2, based on a partition into 50 bins (i.e., 2% of observations per bin), plots the sample proportion for all bins against the corresponding price. Since prices evolve in the prediction markets until the random events are realized, the two panels of the figure together capture the effect of timing by showing the estimation for two different dates: (a) the first day market opens to bettors, and (b) the last trading day before the event starts to take place. For example, the market for whether Barack Obama would win the 2008 U.S. presidential election first opened on October 23, 2006. For our purpose, November 3, 2008--the day before the election day--is identified as the last trading day. (6)

We are interested in whether there is a discontinuity in the relationship between the price and the sample proportion, which falls into the well-developed literature about testing structural breaks. For each panel of Figure 2, we conduct three analyses. First, a linear regression assuming no structural breaks is shown as the dashed line in the diagram. Next, we run two types of break-point tests--an F test (7) and a moving sum of residuals (MOSUM) test (8)--against the null hypothesis that there is no structural breaks for the entire sample. Lastly, we re-run the linear regression by estimating the location of one break-point (as suggested by our theory). The estimation returns (i) a linear segment on each side of the estimated structural break, plotted as the solid lines in the diagram, as well as (ii) the location of the structural break, identified by two red dots in the diagram corresponding to, respectively, the last observation of the first segment and the first observation of the second segment. The details of the three analyses are shown in the column "50 bins" of Table 1.

A few remarks on the results follow. First, in both panels, the regression lines without structural breaks fall very close to the 45-degree line, suggesting the overall efficiency of markets in pricing the probabilities of random events. The evidence of market efficiency on the first trading day is remarkable because, for politics, a lot of markets opened a long time--sometimes years--ahead of the resolution of the events. Yet, as the regression table shows, the slopes are statistically significant and very close to 1.

Second, although panel (a) is relatively noisier, panel (b) shows a clear pattern as predicted by our theory: The null hypothesis of no structural breaks is rejected by the MOSUM test in (a) and by both tests in (b), and the break-point estimation shows a significant jump near price level 0.5. The diagram, hence, resembles our prediction shown in Figure 1. For observations in the intermediate price range, one might think that the price is close to 0.5 because traders have solid information suggesting an "even match" between outcome A and [A.sup.c]. It is also possible, however, that the market is shrouded in ambiguity as some traders, albeit partially informed, are reluctant to trade. Just like in the example of a Knightian urn, an intermediate price in this case could mean a wide range of true probabilities. In panel (b), for a price in the break region between 0.57 and 0.69, the sample proportion could be as low as 33%, or as high as 83%. In other words, the degree of ambiguity, [??], in this particular example is about 0.25 (i.e., half of 83% - 33%). Such a magnitude is significant not only statistically--as the rejection of null hypothesis "no structural breaks" implies the rejection of "[??] equals to zero"--but also economically. According to the multiple-prior framework, it would mean that a typical trader would consider all the probabilities within an interval of length 0.5 equally admissible to themself in governing the realization of the binary event. A significant jump near price level 0.5 like the one in panel (b), therefore, is an indication of the presence of Knightian uncertainty. As the linear regression without breaks shows, the specific pattern of observations also causes the regression line to have a slope larger--albeit only slightly--than 1.

Another important difference between the two panels is that, in panel (b), more observations are clustered near price levels 0 and 1. This means, by the last day, more traders hold (posterior) beliefs that some outcome--either A or [A.sup.c]--is very likely to be realized, suggesting a decrease in risks over time. Such a decrease in risks can be a result of information acquisition by the traders, who, until the random events resolve, may have the incentives to learn about the events and update their bets accordingly. Since risks have decreased while ambiguity remains, our empirical evidence also suggests an observational distinction between the concepts of risk and Knightian uncertainty.

Furthermore, the above empirical patterns are to some extent robust against the choice of the number of bins. Figure 3 reproduces the diagram by partitioning the data into 30 bins instead. The column "30 bins" of Table 1 shows the details of the regression and break-point analysis. Overall, the observations drawn earlier still hold. (9)

The observation that ambiguity remains until the last trading day suggests, unlike risks, Knightian uncertainty may be not "learnable" in practice to the traders. The intuition can be illustrated using the Knightian urn where the composition of black and red balls is unknown. Imagine two different scenarios. In the first scenario, a subject observed repeated draws from the same Knightian urn. In this case, the sample proportion over time would reveal the true composition of the two colors because, after all, the composition is fixed over time. In the second scenario, there was an experimenter who replaced the Knightian urn with a new one every time a ball was drawn by the subject. In this case, the sample proportion may not inform the subject of what to expect in the next Knightian urn, simply because the composition of black and red balls in the new urn could be anything of the experimenter's choosing. If the underlying data-generating process--that is, the way the experimenter changed every other Knightian urn--was not learnable to the subject, then the degree of ambiguity would not decrease over time. (10) One might think that, in politics, it is intuitively easy for traders to acquire knowledge--from polls, news reports, political analyses, etc. Yet, our empirical evidence, which is based on a large number of prediction markets about various political events, seems to fit the second scenario, suggesting that the Knightian uncertainty of politics may indeed be not "learnable" through information acquisition.

We now turn to another major category: entertainment events, such as the winners of cinematic awards or the box offices of movies. Figures 4 and 5 reproduce the P-[pi] diagram for 30 bins and 50 bins, respectively, and Table 2 reports the details of the regressions and break-point tests. Although qualitatively similar, the patterns are less pronounced compared to politics. The jump near 0.5 is less clear and, interestingly, the clustering near 0 and 1 is less marked. This evidence suggests less learning in entertainment than in politics, which is understandable since it might be more difficult for bettors to acquire information about the general public's personal tastes of movies and music.

Politics and entertainment together account for over 80% of the Intrade dataset. However, for completeness, we reproduce the empirical evidence with the full sample, as shown in Figures 6 and 7, as well as Table 3. The patterns, essentially by construction, are similar to what we established above.

V. Concluding Remarks

Knightian uncertainty--an important theoretical concept in the literature that is often used to explain observed phenomena--has never been directly evidenced in an empirical setting. In this paper, we have developed a model of a prediction market with ambiguity, where traders have maxmin preferences. We have derived more direct, observational implications of the presence of Knightian uncertainty. Using the historical betting data from Intrade, we have further presented some high-level evidence that is consistent with the prediction of our model. In particular, for price levels close to 0.5, the market-implied, ex ante probability of a random event is not indicative of the ex post sample proportion, suggesting the presence of Knightian uncertainty.

Moreover, our empirical evidence has shown that, although traders seem to have acquired information which leads to a decrease in risks, ambiguity remains until the last trading day, suggesting that the Knightian uncertainty of the world may be not "learnable" to traders. By comparing political events and entertainment events, we have also shown that the empirical patterns we identified are more pronounced in politics than in entertainment.

The evidence we have provided is only preliminary, since the empirics of this paper are based on a single prediction platform that is skewed towards political and entertainment events. In a future, empirical study, we will collect more prediction market data across different platforms and different event types, and we will examine more closely the relationship between the ex post sample proportion and the ex ante price by taking into account the type of events, the time ahead of the resolution of randomness, and other aspects of the betting markets.

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Mathematical Appendix

Proof of Proposition 1

Note that any equilibrium price [pi] has to satisfy (i) [pi] > 2e and (ii) [pi] < 1 - 2e. If (i) does not hold, then [pi]* [less than or equal to] [q.sub.i] + [??] for all i, which means any trader will have either a long position or a zero position--not an equilibrium. Similarly, if (ii) does not hold, no trader will have a long position, which cannot be an equilibrium either.

Substitute (2) into (3) and rewrite the aggregate demand as

[mathematical expression not reproducible]

Hence, [X.sub.F] ([pi]) = 0 if and only if

[mathematical expression not reproducible]

where the last step follows from integration by parts. Simplifying and rearranging terms yields the stated expression in the proposition. Q.E.D.

Proof of Proposition 2

Let G be the space of distribution functions over [[??], 1 - [??]], endowed with the Levy metric l, where

l([G.sub.1], [G.sub.2]) [equivalent to] inf {[epsilon] > 0 | [G.sub.1] (q - [epsilon]) - [epsilon] [less than or equal to] [G.sub.2] [less than or equal to] [G.sub.1] (q + [epsilon]) + [epsilon] for all q [member of] [[??], 1 - [??]]}

for any [G.sub.1],[G.sub.2] [member of] G. Let F be the subset of G that satisfies [[pi]*.sub.F] = [E.sub.F] (q) for any F [member of] F. Since the Levy metric metrizes the weak topology (11) the proposition is equivalent to the claim that F is nowhere dense in (G,[member of]).

Note that F is closed. Since a set is nowhere dense if and only if the complement of its closure is dense, (12), it remains to be shown GF is dense, that is, for any point in G, there is a sequence from G \ F converging to that point. It is thus enough to show, for any F [member of] F and any [delta] > 0, there exists some G [member of] G \ F such that l (F, G) < [delta].

F is non-decreasing since it is a distribution function. It follows that

[mathematical expression not reproducible]

We show prove the results by examining two cases.

Case 1: [mathematical expression not reproducible].

Given [delta] > 0, we construct a distribution function G from F as

[mathematical expression not reproducible]

where [[delta].sub.1], [[delta].sub.2] > 0 are such that function g [equivalent to] G - F satisfies conditions

[mathematical expression not reproducible]

and

max {g ([E.sub.F] (q) - [??] - [[delta].sub.1]), - g ([E.sub.F] (q) + [??] + [delta]2)} = [[delta]/2].

It is easily verified that G is a mean-preserving spread of F, with two new atoms created at points [E.sub.F] (q) - e - [[delta].sub.1] and [E.sub.F] (q) + e + [[delta].sub.2]. By construction, this implies that

[mathematical expression not reproducible]

where the last equality holds because F [member of] F, and the inequality holds because g ([E.sub.F] (q) + [??]) < 0, which implies [mathematical expression not reproducible]. Since [mathematical expression not reproducible] G(q)dq < [??], G [member of] G\F. Finally, let [rho] be the uniform metric, that is,

[rho] ([G.sub.1], [G.sub.2]) = sup {|[G.sub.1] (q) - [G.sub.2] (q)|| q [member of] [[??],1 - [??]]}

for any [G.sub.1], [G.sub.2] [member of] G. By construction, [rho](F, G) = [[delta]/2]. Since the Levy metric is bounded by the uniform metric from above, that is, l([G.sub.1],[G.sub.2]) [less than or equal to] [rho]([G.sub.1],[G.sub.2]) for any G1, G2 [member of] G, we have l (F, G) [less than or equal to] [[delta]/2] < [delta].

Case 2: [mathematical expression not reproducible].

Given [delta] > 0, we construct a distribution function H from F as

[mathematical expression not reproducible]

where [[delta].sub.3],[[delta].sub.4] > 0 are such that function h [equivalent to] H - F satisfies conditions

[mathematical expression not reproducible]

and

max {[[delta].sub.3], -h ([E.sub.F] (q) + [??] + [[delta].sub.4])} = [[delta]/2].

It is easily verified that H is a mean-preserving spread of F, with two new atoms created at points [E.sub.F] (q) - [??] and [E.sub.F] (q) + [??] + [[delta].sub.4]. By construction, this implies that

[mathematical expression not reproducible]

where the last but second equality holds because F [member of] F, and the last equality follows from the construction of H. Since [mathematical expression not reproducible]. Finally, similar toCase 1, we have [rho](F, H) = [[delta]/2] and, hence, l(F, H) < [delta]. Q.E.D.

Proof of Proposition 3

Decompose [X.sub.F] ([pi]) into the aggregate supply (shorts) SF ([pi]) and the aggregate demand (longs) [D.sub.F] ([pi]), where

[mathematical expression not reproducible]

and [S.sub.F] ([[pi]*.sub.F]) = [D.sub.F] ([[pi]*.sub.F]) in equilibrium. We show that an increase in [??] shifts the supply curve inwards. That is,

[mathematical expression not reproducible]

Similarly, an increase in e shifts the demand curve inwards (i.e., [mathematical expression not reproducible]). It follows that the equilibrium quantity of trade--[S.sub.F]([[pi].sub.F]), or [D.sub.F]([[pi]*.sub.F])--has to be smaller as the degree of ambiguity increases. Q.E.D.

Proof of Proposition 4

Let [PHI] denote the integral of F, i.e., [mathematical expression not reproducible]. It follows from the definition of F that

[mathematical expression not reproducible]

where [bar.[PHI]] is the integral of [bar.F]. The equilibrium condition becomes

[pi] = EF (q) + [PHI] ([pi] + [??]) - [PHI] ([pi] - [??]) - [??] = 1 - 2[??] - [PHI] (1 - [??]) + [PHI] ([pi] + [??]) - [PHI] ([pi] - [??]),

where the second equality follows from integration by parts. Since [PHI] (q) has a kink at point p, the equilibrium price depends on the position of p relative to [pi] + [??] and [pi] - [??].

Case 1: [pi] - [??] [less than or equal to] p [less than or equal to] [pi] + [??].

The equilibrium condition is rewritten as

[pi] = 1 - 2[??] - (1 - m) [bar.[PHI]] (1 - [??]) - m (1 - [??] - p) + (1 - m) [bar.[PHI]] ([pi] + [??]) + m ([pi] + [??] - p) - (1 - m) [bar.[PHI]] ([pi] - [??]).

Rearranging terms and dividing both sides by 1 - m yields

[pi] - [bar.[PHI]] ([pi] + [??]) + [bar.[PHI]] ([pi] - [??]) = 1 - 2[??] - [bar.[PHI]] (1 - [??]).

Case 2: p > [pi] + [??].

The equilibrium condition is rewritten as

[pi] = 1 - 2[??] - (1 - m) [bar.[PHI]] (1 - [??]) - m (1 - [??] - p) + (1 - m) [bar.[PHI]] ([pi] + [??]) - (1 - m) [bar.[PHI]] ([pi] - [??]).

Rearranging terms yields

(A1) [mathematical expression not reproducible].

Note that the left-hand side of equation (A1) is strictly increasing in [pi]. Thus, the solution [pi]* to the equation is a continuous and strictly increasing function of p. Furthermore, as p [right arrow] [pi] + [??], where [pi] is the equilibrium price in Case 1, the right-hand side of equation (A1) converges to 1 - 2[??] - [bar.[PHI]] (1 - [??]) + [[pi]m/[1-m]], and the solution to the equation converges to [pi]. In other words, the equilibrium price is continuous at point p = [pi] + [??].

Next, we show [pi]* (1 - [??]) < 1 - 2[??], which implies [pi]* (1 - [??]) < 1 - [??] in part 1 of the proposition. Let LHS ([pi]) and RHS (p) denote the left- and right-hand sides of equation (A1), as functions of [pi] and p, respectively. Note that

[mathematical expression not reproducible]

Since LHS is strictly increasing in [pi], the solution to the equation when p = 1 - [??] must be smaller than 1 - 2[??].

Case 3: p < [pi] - [??].

The equilibrium condition is rewritten as

[pi] = 1 - 2[??] - (1 - m) [bar.[PHI]] (1 - [??]) - m (1 - [??] - p) + (1 - m) [bar.[PHI]] ([pi] + [??]) + m ([pi] + [??] - p) - (1 - m) [bar.[PHI]] ([pi] - [??]) - m ([pi] - [??] - p).

Rearranging terms yields

(A2) [mathematical expression not reproducible].

Similar to Case 2, the solution [pi]* to equation (A2) is continuous and strictly increasing in p, and it converges to [pi] as p [right arrow] [pi] - [??]. Hence, the equilibrium price is continuous at point p = [pi] - [??] as well.

Next, we show [pi]* ([??]) > 2[??], which implies [pi]* ([??]) > [??] in part 1 of the proposition. Again, let LHS ([pi]) and RHS (p) denote the left- and right-hand sides of equation (A2). Note that

[mathematical expression not reproducible]

where the last inequality holds because [bar.[PHI]] is the integral of distribution function [bar.F] over [[??], 1 - [??]]. Since LHS is strictly increasing in [pi], the solution to the equation when p = [??] must be larger than 2[??]. Q.E.D.

Proof of Corollary 1

Recall that [pi] is identified by equation

[pi] - [bar.[PHI]] ([pi] + [??]) + [bar.[PHI]] ([pi] - [??]) = 1 - 2[??] - [bar.[PHI]] (1 - [??]).

The symmetry of F implies - [bar.[PHI]](1 - x) = (x - [??]) - [bar.[PHI]] (x) for any x [member of] [[??], 1 - [??]]. Thus, the equilibrium condition becomes

[pi] - [[pi] + [bar.[PHI]] (1 - [pi] - [??])] + [bar.[PHI]] ([pi]? - [??]) = 1 - 2[??] - [1 - 2[??] + [bar.[PHI]] ([??])] [??] [bar.[PHI]] (1 - [pi]? - [??]) - [bar.[PHI]] ([pi]? - [??]) = [bar.[PHI]] ([??]) = 0,

to which [pi] = 0.5 is the only solution. Q.E.D.

Data Appendix

The historical data of Intrade was archived by Ipeirotis (2013) and is available on GitHub. Table 4 lists all the categories of events and the number of markets within each category. We complete the dataset by creating an outcome variable and recording how each random event had turned out. The outcome equals 1 if an event occurs, and it equals 0 if its complement event occurs.

Some markets have correlated outcomes, because they are about the same, uncertain circumstances. For example, concerning the 2012 U.S. Republican Party presidential nominee, there are 53 separate markets corresponding to 53 possible winners, including Mitt Romney, Rick Santorum, Ron Paul, Newt Gingrich, and "any other individual" not specified by the prediction platform. To avoid such correlation in the observations, for each group of these correlated markets, we randomly select one market into the aggregate sample and disregard the rest.

The total number of selected markets included in the final analysis also shown in Table 5. The table lists the number of observations--the total as well as the number of observations per percentile bin--for political events, entertainment events, and the full sample. The dataset is skewed towards political and entertainment events, as the two categories together accounts for 82% of the full sample.

By KEVIN A. HASSETT AND WEIFENG ZHONG (*)

(*) Hassett: Chairman, Council of Economic Advisers. Zhong (corresponding author): American Enterprise Institute, 1789 Massachusetts Ave NW, Washington, DC 20036, weifeng@weifengzhong.com. We appreciate helpful comments from Julian Chan, Jon Hartley, and Joe Sullivan, and we thank Cody Kallen for excellent research assistance. The research of this paper was completed while Mr. Hassett was at the American Enterprise Institute. This paper does not represent the findings or views of the Council of Economic Advisers or the United States. All errors are our own.

(1) See Epstein and Schneider (2010), p. 315.

(2) See Caballero and Krishnamurthy (2008), Caballero and Simsek (2013), Guidolin and Rinaldi (2010), Routledge and Zin (2009), and Ulrich (2013).

(3) In practice, trader i can long and/or short contract A and/or contract [A.sup.c], but some strategies are mathematically equivalent. For example, holding [m.sub.i] > 0 units of contract A and [n.sub.i] > [m.sub.i] > 0 units of contract [A.sup.c] would be equivalent to holding mi units of cash, 0 unit of contract A, and [n.sub.i] - [m.sub.i] > 0 units of contract [A.sup.c]. Therefore, without loss of generality, we let a single decision variable [x.sub.i] = [m.sub.i] - [n.sub.i] (which could be positive, zero, or negative) represent the net position held by trader i.

(5) See the Data Appendix for details.

(6) On election days, November 4 in this example, exit polls are informative even before the result is determined. We thus treat it as part of the resolution of uncertainty and, hence, exclude it from our data.

(7) The F test is an extension of the Chow test (Chow 1960), against the alternative hypothesis of an unknown break-point. See, e.g., Andrews (2003) and Andrews and Ploberger (1994) for details.

(8) The MOSUM test analyzes the moving sum of residuals and detects whether a strong shift of the fluctuation process exists. See, e.g., Chu, Hornik and Kuan (1995b, a) for details.

(9) We have checked other variations between 30 and 50, which yield similar results (omitted to limit space). Obviously, the number of bins should be neither too small (which would leave too few points in the diagram), nor too large (which would leave too few observations per bin).

(10) See Epstein and Schneider (2007) for a theoretical treatment of learning under ambiguity.

(11) See, e.g., Huber and Ronchetti (2009), p. 28.

(12) See, e.g., Sutherland (1975), p. 64.
Table 1--Estimation and Test of Structural Breaks: Politics

Dependent variable: ex post sample proportion Independent variable: ex
ante price

                                50 bins       30 bins

(a) first trading day
 slope (no breaks)               0.953           0.948
                                (0.046)         (0.047)
 tests of "no breaks"
  F test                      not rejected    not rejected
                                (4.232)         (6.818)
  MOSUM test                  rejected        rejected
                                (1.417)         (1.331)
 structural break estimation
  slope (segment 1)              0.680           0.641
                                (0.151)         (0.142)
  break region                  [0.45, 0.47]    [0.43, 0.47]
  slope (segment 2)              0.893           0.839
                                (0.117)         (0.121)
(b) last trading day
 slope (no breaks)               1.03            1.04
                                (0.022)         (0.026)
 tests of "no breaks"
  F test                      rejected        rejected
                              (223.660)        (90.312)
  MOSUM test                  rejected        rejected
                                (1.804)         (1.586)
 structural break estimation
  slope (segment 1)              0.614           0.749
                                (0.033)         (0.041)
  break region                  [0.57, 0.69]    [0.60, 0.75]
  slope (segment 2)              0.464           0.318
                                (0.077)         (0.124)

For regression coefficients, standard errors are in parentheses
For F test and MOSUM test, test statistics are in parentheses.

Table 2--Estimation and Test of Structural Breaks: Entertainment

Dependent variable: ex post sample proportion Independent variable: ex
ante price

                              50 bins        30 bins

(a) first trading day
 slope (no breaks)              0.933          0.932
                               (0.045)        (0.047)
 tests of "no breaks"
  F test                      not rejected   not rejected
                               (4.832)        (3.886)
  MOSUM test                  not rejected   not rejected
                               (0.870)        (0.782)
 structural break estimation
  slope (segment 1)             0.864          0.773
                               (0.099)        (0.106)
  break region                 [0.50, 0.50]   [0.48, 0.50]
  slope (segment 2)             0.876          0.847
                               (0.130)        (0.128)
(b) last trading day
 slope (no breaks)              1.04           1.03
                               (0.031)        (0.029)
 tests of "no breaks"
  F test                      rejected       rejected
                              (18.348)       (25.542)
  MOSUM test                  rejected       not rejected
                               (1.728)        (0.956)
 structural break estimation
  slope (segment 1)             0.680          0.679
                               (0.089)        (0.075)
  break region                 [0.46, 0.49]   [0.44, 0.49]
  slope (segment 2)             0.991          1.00
                               (0.083)        (0.068)

For regression coefficients, standard errors are in parentheses
For F test and MOSUM test, test statistics are in parentheses.

Table 3--Estimation and Test of Structural Breaks: Full Sample

Dependent variable: ex post sample proportion Independent variable: ex
ante price

                              50 bins        30 bins

(a) first trading day
 slope (no breaks)              0.918          0.927
                               (0.028)        (0.029)
 tests of "no breaks"
  F test                      not rejected   rejected
                               (8.104)       (11.156)
  MOSUM test                  not rejected   not rejected
                               (1.049)        (0.970)
 structural break estimation
  slope (segment 1)             0.730          0.725
                               (0.077)        (0.070)
  break region                 [0.45, 0.47]   [0.44, 0.48]
  slope (segment 2)             0.871          0.882
                               (0.071)        (0.067)
(b) last trading day
 slope (no breaks)              1.03           1.03
                               (0.019)        (0.021)
 tests of "no breaks"
  F test                      rejected       rejected
                              (42.064)       (18.113)
  MOSUM test                  rejected       not rejected
                               (1.362)        (1.141)
 structural break estimation
  slope (segment 1)             0.858          0.915
                               (0.036)        (0.040)
  break region                 [0.62, 0.68]   [0.65, 0.75]
  slope (segment 2)             0.680          0.604
                               (0.094)        (0.152)

For regression coefficients, standard errors are in parentheses
For F test and MOSUM test, test statistics are in parentheses.

Table 4--Intrade Data: Event Categories and Number of Markets.

Event category              Number of markets

Art                                60
Business                           43
Chess                              52
Climate & Weather                 861
Construction & Engineering          9
Current Events                   1540
Education                           1
Entertainment                    8715
Fine Wine                           5
Foreign Affairs                    87
Legal                             310
Media                              10
Politics                         5460
Real Estate                         2
Science                            20
Social & Civil                     30
Technologies                       65
Transportation                     11

Table 5--Intrade Data: Number of Observations in Final Analysis.

Event category     Total      Observations per bin
                observations  (50 bins)  (30 bins)

Politics            897          18         30
Entertainment      1157          23         39
Full sample        2509          50         84
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