Everyone hates Math!

I mean real math. People don't mind doing calculations, or using formulas. But when I try to get someone to verify some logic, I stun people. Just asking, "does this make sense?" seems to offend people.

My math senior project is to review, explain, and extend an article from a math journal. I'm doing The Last Round of Betting in Poker by Jack Cassidy from The American Mathematical Monthly. It seems like even Cassidy doesn't want to do any of what I've come to know as real math. He doesn't give explicit definitions. He doesn't clearly state his assumptions. He doesn't support many claims.

One problem I have with it is that it uses the [0,1] game, which is beautiful and elegant when representing a uniform distribution of hands (0 being the best hand, 1 being the worst). That way, the probability of a random hand y in [0,1] being greater than or equal to a fixed hand x can be expressed as

P(x≤y) = y.

But, in Cassidy's article, he states he is using the [0,1] game, but the only thing he seems to use it for is the notation of ≤ to mean better than or equal to. And then, instead of referring to the [0,1] game he claims to be using, in examples he translates directly from probability (which he writes as F(y)) to a hand in the game for which the probability for having a hand equal to or better than that is equal to F(y).

So, I don't really have time to do it right now, but I would really love to examine if his results really are optimal solutions (neither opponent can adjust to improve their overall expectation), see if this pair of optimal solutions is unique - in other words, see if more than one pair of optimal solutions exists (maybe they will all satisfy the equations given in his article).

Anyways, the strategy seems brittle. He gives results that when translated to the common [0,1] game (not exactly what he uses) with bet size equal to pot size, and uniform distributions, we get the strategies (1/6, 1/2, 1/12) for P1 and (1/3, 1/2, 1/6) for P2. This means P1 should bet the best 1/6 of hands, and bluff the best 1/12 of hands, and call with hands better than or equal to 1/2 (1/3 of starting hands if opponent bets). Similarly for P2. But I wonder, can either player improve their expectation? If so, they are not an optimal pair. I know that if P1 or P2 decided not to bluff at all, it then becomes wrong for P2 to call so much. As it is, it does not matter where the call values are (1/2 seems almost arbitrary). But if 1/2 is the right value, then value betting more or less hands should reduce the expectation, and bluffing more or less hands should do the same.

I have to finish my senior project now though, so I won't worry about these details now.

Let me know if you know anything about this.