How to evaluate arbitrage opportunities in the 2024 US Presidential Election Markets

In the aftermath of Joe Biden's debate debacle, there's been several days of furious activity in Polymarket's election markets. Most notably, the price of contracts in a Biden win have dropped from $0.34 to a current (as of June 30th) new low of $0.18, while the price of a Donald Trump win have gone from $0.56 to $0.65. In addition, the markets for Biden withdrawing from the election, which had been as low as $0.13 on April 30 and had never previously gone above $0.30 are nearing 50/50 ($0.47 as I write this). With a variety of breaking news stories that could cause rapid price movements in these markets, and a lot of strong opinions that could lead to market participants letting emotion guide their trades, these markets are ripe for arbitrage opportunities. I'll try to write about more of these in the near future, but for now I'll focus on the broadest one. Are the election winner markets and the market for Biden withdrawing from the election telling us the same thing? Since this is subject to change at any time, rather than simply telling you the current answer to that question, I'm going to provide a simple formula for how to answer this question. By multiplying the implied chance of Biden winning the election by the implied chance of Biden NOT dropping out, we can calculate the 'chance that Biden wins if remains in the election'. This should be very close to the inverse of the chance that Trump will win the election (maybe slightly lower, since there's also some chance that Trump would drop out or theoretically that RFK somehow pulls a huge upset and wins a three person race). Here's the current math... Trump to win: 65% Biden to drop out: 47% Biden NOT to drop out: 53% (100% - 47%) Biden to win: 18% Biden to win if he doesn't drop out: 34% (18%/53%) Since 34% is very slightly less than the inverse of Trump's 65%, we can see that these markets are consistent with each other, and don't point to any arbitrage opportunities. Now imagine a scenario where everything else stays the same, but the 'Biden to win' market briefly rebounds to 30%. The revised math would look like this: Trump to win: 65% Biden to drop out: 47% Biden NOT to drop out: 53% (100% - 47%) Biden to win: 25% Biden to win if he doesn't drop out: 47% (25%/53%) Since 47% if much higher than the inverse of Trump's 65%, we'd have a situation where one of the markets must be very, very wrong. One could potentially set up profitable arbitrage trades by taking the 'no' side of Biden to win and either the yes side of Trump to win or the no side of Biden to drop out.