Part 3 About Legal Tech Investing for Angels: Saint Petersburg Paradox

Math is both beautiful and terrifying, like an army with banners. (@Quentin Solt – this allusion is included for your benefit; not sure how many others will recognize the source).

I had an interesting conversation recently about investing in legal tech, and my interlocutor made a rather sophisticated argument that a rational person would always calculate the expected value of an investment and – if the calculated value was higher than the cost of the investment – the investment should be made. No exceptions.

Sounds perfectly sensible… but there are few rules in math without exceptions.

Enter the Saint Petersburg Paradox.

Imagine a game where you flip a coin. If it is heads, you win, and – here is the interesting part – the payoff doubles on every turn. If it is heads on the first turn, you get $2. On the second turn, you get $4. Third turn - $8… you get the picture.

What is the expected value of this game? In other words, how much should a rational person pay to play it?

Simple enough: take each payoff amount, multiply by the probability of that payoff, and calculate the sum of the resulting values. It is easy to see that the expected value of the game increases on every turn because the expected value of each turn adds another $1 (on the first coin flip, the expected value is $2 x ½ = 1; on the second coin flip, the expected value is $4 x ¼=1, and so on).

This series does not converge – and the expected value of the game goes all the way to infinity.

Rationally, then, you should be prepared to wager any (finite) amount to play it, right?

Wrong. Why?

The distribution of outcomes is non-Gaussian and features a power-law-like heavy tail. For example, if the first heads occurs on the 30th turn, the payoff would be around $1 billion… but the likelihood of it is 1 in a billion, too.

So, here is the practical question buried inside the paradox: not "is the expected value infinite" (it is), but "what is a sane price to pay to play?"

Let’s run the numbers. Paying $1 to play is barely a decision — you are almost certain to come out ahead or flat, and the tiny chance of a large payoff is pure upside. Paying $2 is still comfortable; you are now betting on doing better than the "first flip" outcome, but the odds are still working hard in your favor. Push it to $10, and you've quietly changed the nature of the bet — you now need several heads in a row just to break even, and while that's not unreasonable, it's no longer a no-brainer. It depends on what else you could do with that $10, and how badly you need it back.

Now pay $500. To merely break even, you need nine consecutive heads — roughly a 1-in-500 event. You are no longer investing. You are relying on the tail to bail out the price you already agreed to pay. The math has not changed — expected value is still infinite — but you have changed what you are actually exposed to.

I should note that this has very little to do with whether the underlying business is good.

A legal tech company with real product-market fit, a sharp team, and a genuine wedge into an underserved workflow is a very good coin to be flipping. But the price you pay to enter the game determines whether you are the person wagering $2 or the person wagering $500. At a sane seed valuation, you don't need the company to become the next category-defining outcome to do well — a modest, "first few flips" result already clears your bar. At a valuation that has priced in the eventual billion-dollar outcome, you need the tail — and only the tail — to show up, on schedule, with certainty you don't actually have.

This is where "expected value is infinite, so no exceptions" breaks down as investment advice.

It is worth being precise about what has actually changed here, because it is easy to read this as an argument for lowering your sights and playing it safe. It is not. The $2 flip and the $500 flip have exactly the same access to the billion-dollar outcome on turn thirty — the tail doesn't know or care what you paid to be in the game. What changes is not the ceiling. It is whether you were depending on the tail to bail out a price you had already agreed to. Discipline, in this sense, is not an alternative to seeking real alpha. It is what allows you to keep seeking it without praying for an outcome that is 35 tails away.

None of this is purely a matter of temperament, either. The $2 price and the $500 price are not both sitting on the table for everyone to choose between — they are usually offered to different people, for different reasons, and being shown the $2 round is its own separate skill from being disciplined once you are in the room. And the math itself only works as a guide across many plays, not one. A single well-priced flip is still a single flip; it is the portfolio of flips, played at sane prices, that lets the distribution start working for you instead of against you.

The paradox says the number of heads you need to justify your entry price grows brutally fast, and, at some point, you have silently swapped "reasonable investment" for "lottery ticket dressed as a thesis." Legal tech right now has no shortage of exciting theses. It has, in places, a shortage of investors willing to say the valuation has moved past the point of what the math supports.

Discipline in venture investing is not refusing to play the game, and it is certainly not refusing to play for the biggest possible outcome. It is understanding the distribution, the valuation, and how many flips you need to get ahead. This way, when the tail does show up, you are positioned to benefit from it rather than merely rescued by it. It is also about the right size of the check, but portfolio construction is a separate (equally exciting) topic.