There are many examples of multi-outcome prediction markets, for example election markets with more than two candidates, or sports championship markets with dozens of teams.

What is the best way to implement a multi-outcome prediction market?

The simplest way is to effectively ignore the fact that there are multiple outcomes, breaking up the market into a bunch of separate binary markets, one for each outcome. Each outcome-market is an independent instrument with its own order flow and processing.

This seems to be the most common approach, taken by for example intrade, IEM, racetracks, and most financial exchanges. IMHO, it’s the wrong way, for three reasons.

1. Splitting up a market can hurt liquidity. In a split market, there are effectively two ways to do everything (e.g., buy outcome 1 equals sell outcomes 2 through N), so traders may not see the best price for what they want to do, and orders may not fill at the best price available. There may even be orders that together constitute an agreeable trade, yet are stuck waiting in separate queues.
2. A split market may also slow information propagation. Price changes in one outcome do not directly affect prices of other outcomes; it’s left to arbitrageurs to propagate logical implications.
3. Finally, a naïve implementation of a split market may limit traders’ leverage, forcing them set aside more money than necessary to complete a set of trades. For example, on IEM, short selling one share at $0.99 requires that you have $1 in your account, even though the most you could possibly lose in this transaction is $0.01. The reason is that to short sell on IEM you must first buy the bundle of all outcomes for $1, then sell off the outcome that you don’t want.

IEM has possibly the worst implementation, suffering from all three problems.

Intrade’s implementation is slightly better: they at least handle leverage correctly.

Newsfutures is smarter still.¹ They generate phantom bids to reflect the redundant ways to place bets. For example, if there are bids for outcomes 2 through N that add up to $0.80, they place a phantom ask on outcome 1 for $0.20. A trader who accepts the ask, buying outcome 1 for $0.20, actually sells outcomes 2 through N behind the scenes, an entirely equivalent transaction. Chris Hibbert has a more elaborate methodology for eking out as much liquidity as possibly using phantom bids, an approach he has implemented plans to implement in his Zocalo platform.

Yet phantom bids are a band-aid that cannot entirely heal a fractured market. Still missing is the ability to trade bundles of outcomes in a single transaction.

For example, consider the US National Basketball Association championship market, with 30 teams. A split market (possibly with phantom bids) works great for betting on individual teams one at a time, but is terribly cumbersome for betting on groups of teams. For example, betting that a Western conference team will win requires 15 separate transactions. A common fix is to open yet another market in each popular bundle, however this limits choice and exacerbates all three problems above.

Bundling is especially useful with interval bets. For example, consider this bet on the peak price of gasoline through September 2008, broken up into intervals $3-$3.25, $3.25-$3.40, etc. In order to bet that gas prices will peak between, say, $3.40 and $4.30, you must buy all six outcomes spanning the interval, one at a time. (Moreover, you must sum the six outcome prices manually to compute a price quote.)

Fortunately, there is a trading engine that solves all three problems above and also allows bundle bets…

It’s linear programming!

Bossaerts et al. call it combined value trading. Baron & Lange, Lange & Economides and Peters et al. call it a parimutuel call market. Fortnow et al. and Chen et al. describe it in the context of combinatorial call markets.

Whatever you call it, the underlying principle is relatively straightforward, and it seems inherently the right way to implement a multi-outcome market. Yet I’ve rarely seen it done. The only example I know of is the now defunct economic derivatives markets run by Longitude, Goldman Sachs, and Deutsche Bank.

The set up of the linear program is as follows. Each order is associated with a decision variable x that ranges between 0 and 1, encoding the fraction of the order that the auctioneer can accept.² There is one constraint per outcome that ensures that the auctioneer never loses money across all outcomes. The choice of objective function depends on the auctioneer’s goals, but something like maximizing the fill fraction makes sense.

Once the program is set up, the auctioneer solves for the x variables to determine which orders to accept in full (x=1), which to accept partially (0<x<1), and which to reject (x=0). The program can be solved either in batch mode, after waiting to collect a number of orders, or in continuous mode immediately as new orders arrive. Batch mode corresponds to a call market. Continuous mode corresponds to a continuous auction, a generalization of the continuous double auction mechanism of the stock market.

Each order consists of a price, a quantity, and an outcome bundle. Traders can just as easily bet on single outcomes, negations of outcomes, or sets of outcomes (e.g., all Western Conference NBA teams). Every order goes into the same pool of liquidity no matter how it is phrased.

Price quotes are queries to the linear program of the form “at what price p will this order be accepted in full?” (I believe that bounds on the dual variables of the LP can be interpreted as bid and ask price quotes.)

Lange & Economides and Peters et al. devise clever ways to make prices unique rather than bid-ask ranges, by injected a small subsidy to seed the market at the onset.

Note that Hanson’s market scoring rules market maker also elegantly solves all the same problems as the LP formulation, including handling bundle bets. However, the market maker requires a patron to subsidize the market, while the LP auctioneer formulation is budget balanced — that is, can never lose money.

Also note that I am not talking about a combinatorial-outcome market here. In this post, I am imagining that the number of outcomes is tractable — small enough so that we can explicitly list, store, and compute across all of the outcomes. A true combinatorial-outcome market, on the other hand, has an exponentially large number of outcomes making it impossible to even list them all explicitly, and forcing all calculations to operate on an implicit representation of outcomes, for example Boolean combinations of base events.

	1Apparently worked out in conjunction with Brian Galebach, a mathematician and Newsfutures fan extraordinaire who runs the prediction contest probabilitysports.com.
	2Alternatively, the variables can range between 0 and q, where q is the quantity of shares ordered.