Intro to Prediction Markets, Part 0: What are Prediction Markets?

A prediction market is a venue where people buy and sell contracts whose payoff is determined by the outcome of a specified future event. The event in question is usually a discrete, objective outcome, monitored by that market's resolution mechanism, often called an oracle. Based on that outcome, a contract resolves to $0 or $1. Prediction markets are similar to sportsbooks, in that they are event contracts, and financial derivatives, in that the trade claims on future outcomes. By contrast, the price in a sportsbook is fixed by the house (i.e. not tradable), but derivatives price continuously and are less event driven. Comparisons will be made to them due to greater familiarity by the public, but the object of this blog series is to study prediction markets.

Contract Mechanics

• One can buy YES or NO contracts, with each taking one side or the other of that binary resolution. You may sell contracts you already have, but the big venues (Kalshi, Polymarket) do not allow naked shorting, unlike traditional markets.
• The contracts trade at a price $p \in (0,1)$. This is the (market-)implied probability of the event. This differs from the real probability $\theta$.
• How prices are set warrants a blog post in itself, and do not impact our discussion today.
• How the listed prices and the price you pay (bid-ask spread), also do not impact today's discussion, and so is also deferred.
• The return per dollar $$R = \begin{cases} (1-p)/p & \text{with probability } \theta, \\ -1 & \text{with probability } 1-\theta \end{cases}.$$ The expected return per dollar risked is $$\mathbb E[R] = \frac{\theta - p}{p}.$$ This formula is foundational to the evaluation of strategies, which will be discussed in later posts, but fundamentally, your returns depend on the relative error of the market from reality.

Market Taxonomy

Prediction markets like to ask objectively answerable questions, but most of those questions do not have a singular YES/NO answer. As such, compositions of these markets allows for a greater approximation of reality. Each kind of market introduces their own pricing structure, which we introduce with an eye towards arbitrage.

Binary Markets

This structure in the section above is called a binary market and is most well studied because it is the simplest to consider. It is the building block for the more complicated prediction markets, and it affects the analysis we perform. There is no arbitrage unique to a binary market, but one can consider cross-venue arbitrage where two binary contracts on the same underlying are mispriced enough that buying YES/NO across venues ensures a profit.

Examples: Will X market go up or down? Who will win this head-to-head contest?

Categorical / Winner-Take-All

This is a set of $n$ mutually exclusive contracts, only which one can be YES. Binary markets are the special case of $n=2$. If we price each contract as $p_k$, then we cannot have that $$\sum_{k=1}^n p_k < 1,$$ otherwise we could perform arbitrage by buying one of each contract for less than a dollar, then winning a dollar for our trouble. Consequently, $\sum_k p_k \geq 1,$ where sum exceeding one happens often. This difference is called the overround, vigorish or vig (from the Russian выигрыш "gain" or "winnings" via the Yiddish vyigrysh).

Examples: Who will win a tournament of several competitors (e.g. elections)? Who will be fired next?

PDF / Buckets

This is when the outcome space is partitioned into $n$ non-overlapping intervals, each with its own binary contract. In aggregate, these contracts approximate an implied probability mass/density function (PDF). As before, no-arbitrage implies $\sum_{k=1}^n p_k \geq 1.$ Strategies here can bet on the shape of a distribution.

Examples: What will today's high/low temperature be? Will asset X fall in this range? How many people will vote?

CDF / Survival Function / Digital Calls

This is a set of contracts arranged by several strikes $K_1 < K_2 < \cdots$ and ask if the underlying $X$ will exceed each strike. In turn, the price estimates the survival function $S(K) = P(X>K)$. These are also known as digital calls in options markets. The cumulative mass/density function (CDF) can be computed by $F(K) = 1 - S(K) = P(X<K).$ For simplicity, I will refer to both as the CDF.

We can transform these into a PDF market by estimating the density $$f(K) = -\frac{\partial S}{\partial K} = \frac{\partial F}{\partial K}.$$ This estimation also works in reverse, allowing CDF and PDF markets to be spoken about equivalently. For example, If the prices are non-monotonic, i.e. $$S(K_1) < S(K_2) \quad \text{for } K_1 < K_2,$$ then we could buy YES on the cheaper, higher strike, and buy NO on the pricier, lower one, one would always have a positive payout, which is called model-free dominance arbitrage. Mathematically, non-monotonicity of the CDF implies a negative PDF, but a PDF must naturally be positive, and we are pocketing the difference left between the negative PDF value and 0.

Examples: Will the national debt exceed X? Will team A beat team B by more than X points? (the spread in a traditional sportsbooks is supposed to capture the median value).

Compound / Parlay

Some contracts depend on multiple events happening at once. A key edge in these markets is understanding the distinction between the marginal probabilities and the joint distribution. If the events are independent, one should expect $P_j = P(\bigcap_k A_k)= \prod_k p_k$. If a market incorrectly assumes independence for correlated events, this creates a potential opportunity for a good trade, or vice versa. And in any case, if events are dependent, the Frechet bounds still apply to produce $$\max\left\{0,\sum_k p_k - (n-1)\right\} \leq P_j \leq \min_k p_k,$$ with breaks resulting in arbitrage.

Examples: Will political party A win both the House AND the Senate? Traditional parlays, like one sees in sportsbooks.

The Role of Taxonomy

Mathematically, one can view this taxonomy of prediction markets as pricing different indicator functions.

The different kinds of markets will be a persistent feature throughout future blog posts, especially when it comes to strategic development. We've already seen the different flavors arbitrage can have depending on the structure, and the structure will also inform which Option-Greek analogs are sensible. The evaluation of strategies will also radically depend on market structure, and we'll devote a few early blogs to breaking down the failures of classical metrics before rebuilding a framework for the different markets. Finally, if prediction markets are used as part of a broader strategy, the structure of the contract will constrain how we can use it.