Epstein: The Theory of Gambling and Statistical Logic (1977)

In his book on games, Richard Epstein opens one of the chapters with the following theorem:

Theorem 1: If a gambler risks a finite capital over a large number of plays in a game with constant single-trial probability of winning, losing, and tying, then any and all betting systems lead ultimately to the same value of mathematical expectation of gain per unit amount wagered. (p. 53)

This is pretty badly written. I could also be interpreted in several different ways, some of which lead to false statements.

Epstein briefly discusses the details of the game he has in mind, but not enough to dispel the most important ambiguities. In a infuriating footnote, he even adds

Proofs of the theorems have been developed in the first edition of this book. (p. 52)

Wonderful. When I find my time machine, I'll read them.

What Does the Theorem Say?

Style aside, here's what seems to be clear:

We are assuming a situation in which a gambler starts with one unit of capital (X₀ = 1), and we then interested in the limit behavior of E[X_n], the expected capital after n games. The claim is that this depends only on the probabilities and odds of the game, not on the bets.

The problem is that it's not quite clear what Epstein takes to be permissible betting strategies. Depending on what you mean by "risking a finite capital over a large number of plays," his theorem can be read at least two very different ways, and only one of them is true.

The Stockpile Reading

The first reading would be one in which the gambler splits a starting capital of 1 into n piles, and then wagers those piles, one at a time, without reinvestment.

This corresponds essentially to playing n identical games at the same time with n small bets. By the linearity of expectations, this will give the same expected gain as a single game with the entire capital (although a higher variance).

So if that's the intended reading, then the theorem is true, but also quite uninteresting. In any realistic setting, a gambler (or investor) has the possibility of reinvesting past gains, and the sum of the bets over time can thus easily add up to more than the size of the starting capital.

The Percentage Reading

Another reading is that the gambler at any specific time can bet any percentage of the capital accumulated at that time. The claim would then be that however the series (b₁, b₂, b₃, …, b_n) of percentages would look, the expected capital after n games will be the same.

That's a false claim. I can choose the constant betting strategy b₁ = b₂ = b₃ = … = b_n = 0 and get an expected loss of 0 regardless of the game. In fact, even if I must bet a positive fraction of my capital in each game, I can still limit my losses to any arbitrarily small number a by betting the fractions

b₁ = (1 – a)/2,   b₂ = (1 – a)/4,   b₃ = (1 – a)/8,   b₄ = (1 – a)/16, …

So even with some quite generous assumptions, the "theorem" is simply false under this reading.

What We Can Say

Here's one thing which is true, though:

Any particular game of this kind will have a doubling rate W which is computable from its odds and its probabilities. After n games, the expected capital of the gambler will lie somewhere between 1 and 2^Wn, regardless of the betting strategy. (This is true both for positive and negative growth rates, although 2^Wn will be larger than 1 in the former case and smaller in the latter.)

If we like, we can thus get a bound on the expectation of the capital growth. But it's an exponential bound, and it can't be improved: For every number within that bound, there is a betting strategy that gives me an expected gain of exactly that amount.

Postscript: A Comment by Peter Griffin

Now, after having written this, I also see that there is a paper by Peter A. Griffin which addresses the ambiguity in Epstein's theorem. He writes that quantity which Epstein claims to be constant is "E(Win)/E(Bet), the average win divided by the average bet," as opposed to "E(Win/Bet), the average of individual win rates" (p. 1541).

This is consistent with my first interpretation of Epstein's claim: A static game gives, of course, a static expected gain per dollar wagered. As Griffin puts it, this rate will quantify "in what direction and how fast the money is flowing," and this flow rate determined solely by the odds and probabilities of the game.

The alternative that one might be interested in is "how, on average, individuals perceive their win rates." This can indeed vary according to betting system, and for the gambler interested in maximizing returns, this quantity is of course the figure to keep your eyes on.