Combos Cut Long-Shot Bias

Simple betting markets have long suffered from a long-shot bias: long-shots chances are too high, while favorite chances are too low. While many plausible causes of this have been identified, two stand out to me as obvious contributors. First, as favorite bettors have to put up more cash than do long shot bettors, a small fraction of long shot fans can have a disproportionate influence on market odds. Second, as the max possible gain in simple betting markets from betting on favorites is far less than from betting on longshots, favorites bettors are more sensitive to transaction fees, any lack of paying interest on deposits, and the opportunity cost of locking up capital that can’t be used elsewhere for a while. Let me call this second set problems of “leverage”.

Though I’ve known it for decades, I’m not sure if I’ve ever said this before publicly: combinatorial betting markets greatly cut leverage problems.

What are combinatorial markets? You can take N simple markets, each re a discrete variable with D possible outcomes, all subsidized to the same liquidity level via automated market makers, and combine them into a single combinatorial market where one can trade on all possible combinations of these N variables, all at this same liquidity level, and all for no extra financial cost. A simple dumb implementation takes roughly D^N states to store info, and also about D^N steps to compute changes, making this approach feasible today for up to roughly 30 binary variables. More complex implementations allow thousands of variables. And an assumptions interface makes this all understandable to users.

Imagine you have a combinatorial market, where the market chance of A is 99%. You make a $99 favorite bet on A, matching a contrarian who makes a long-shot $1 bet on not A, say at 99-1 odds. Doing this gives you $100 in assets that pay off if A happens, while it gives the contrarian $100 if not A.

You could then use your $100 if A assets to bet on another favorite B. You’ll have to bet on B given A, but as A was at 99% chance, you’ll have nearly the same effect on the B market as if you made a simple unconditional bet on B. After you bet on B given A at 99-1 odds, you’ll then have ~$101 if A and B, which you can continue to use to bet on other topics C.

As you bet in combinatorial markets on new topics using the assets you acquired from prior favorites bets, you’ll want to ask yourself if these events are correlated, and adjust the odds you’ll accept to account for that. But in compensation, you get to search for combinations of events where you most disagree with market odds, for max expected trading gains. And the market liquidity will always be high for all these complex combinations, even if no one else ever has or will traded your particular combination.

If you have many opinions on the topics in this combinatorial market, then risk aversion is likely to be the main limit on your reusing assets from bets on some topics to also bet on other topics. Combine enough bets, even on favorites, and eventually the odds of winning become too low to be tolerable. The same limit applies also to contrarians who bet $1 and acquire $100 if not A, except that their limits come much faster. Combining even two bets at 1-99 odds gives a ~1/10K chance to win bet that is likely too risky for most traders.

In an ordinary betting markets, 99-1 odds on a claim means that one who backs the favorite has to put up ~100x as much capital to support their bet, and at best only gains 1%, while the contrarian might win 100x their investment. This tends to induce a bias favoring long-shots. But in a combinatorial betting market everyone can bet up to the limits of their risk tolerance, either by combining a few long shot bets or by combining far more favorite bets. Both sides can take as much or little risk as they want, and so there is no longer a leverage problem to bias the prices for long-shots.