A theoretical exercise, and a bit of a puzzle

Suppose we invent a strange new (and, I'm afraid, quite boring) variation of Texas hold'em. In this game, there is no real betting, no decisions to be made--just an ante from every player to make the pot. Everybody gets their two cards, then the five community cards are dealt, and the best hand takes the pot.

Obviously, nobody is going to show a profit in this game in the long run; players will overall just trade chips back and forth with a neutral expected value, and a net loss if there is a house rake. But here's the question: In the ranking of hold'em starting hands, which ones will be long-term winners?

I'm not sure why, but I started wondering about this question today. Actually, what I started off asking myself was slightly different: What is the worst starting hand that will, on average, be the best one at the table? (Mammas, don't let your babies grow up to be poker players. It warps their brains so that they start thinking about questions like this while eating an otherwise normal lunch.) I decided that I could really only answer that if I defined "best" as being equivalent to "the most likely to win over repeated trials if all five community cards are seen every time without additional rounds of betting." That's why I invented my new poker game.

I used Poker Stove (http://www.pokerstove.com/), with some trial and error, to figure this out. I assumed a ten-handed game. The results surprised me.

It turns out that any of the following hands has the highest equity (i.e., just over 10%) against nine opponents' random hands, and would therefore be correct to bet on in my warped little game (if you were given the option to bet or not after seeing your cards--which you're not):

• Any pocket pair.
• Any suited ace.
• Any unsuited ace down to A-8. [Plus A-5; see comments.]
• Any suited king.
• Any unsuited king down to K-9.
• Any suited queen.
• Any unsuited queen down to Q-9.
• Any suited jack.
• Any unsuited jack down to J-9.
• 10-9 offsuit. (This is the lowest unpaired, unsuited hand that is profitable.)
• Any suited ten down to 10-5. [Should be down to 10-4; see comments.]
• Any suited nine down to 9-5.
• Any suited eight down to 8-5.
• Any suited seven down to 7-4.
• Any suited six down to 6-4. [Should be 6-3; see comments.]
• Suited 5-4. [And 5-3; see comments.]
• Suited 4-3.

Yes, the lowly 4-3 of clubs (or any other suit) will, over several thousand trials, show a small profit played against nine random hands in this game.

I realize that this information is pretty functionally useless as hold'em is actually played. Sure, 10-5 suited in my invented game is profitable, but try playing it in a real hold'em game whenever it is dealt to you, and you will almost surely show a long-term loss. There are two main reasons for this, I think. First is that you're never up against nine random hands. As soon as we give players the option to put in money or not after seeing their hole cards, there is a selection process by which the worst starting hands are weeded out. This reduces the amount of -EV money going in, and improves the average range of hands against which your cards have to hold up.

Second is that hands like a suited 10-5 are likely to put you in situations in which you have to make very difficult decisions on later streets, and difficult decisions mean lots of room for error, which errors will cost you money.

Still, I found it very surprising that many sets of cards generally reckoned to be so pathetic as to be instant folders in a real game would actually be profitable in the artificial construct in which they merely have to hold up against nine random hands slightly more often than one time in ten.

It gives new life to that most common of all donkey refrains, "But they were suited!"


Caveat: I have some degree of doubt about my list, because it leads to a counterintuitive conclusion. Let me explain. There are 1326 possible hold'em starting hands, all equally likely to occur. The list of supposedly profitable hands above includes a total of 538 starting hands, if I've done the math right. For example, there are 78 pairs (6 different possible pairs for each of the 13 ranks of cards). There are 48 possible suited aces (from each of four suits, the ace can go with any of the 12 remaining ranks). There are 72 ways to make an unsuited ace down to A-8 (12 ways to make an unsuited A-K, 12 ways to make an unsuited A-Q, etc.). Add up all of those on my list and you get 518. (In my list, something like "unsuited jack" means that the higher card is a jack, so I'm not counting, say, all the J-K combinations there, because they were previously counted in the kings categories.) But that's about 39% of the possible starting hands. It strikes me as doubtful that 39% of the possible starting hands could all show a long-term profit when matched against 9 opponents. Logically, I would expect the profitable list for a ten-handed game to include only a hair under 10% of the possible starting hands.

To check this, I ran the same sort of calculation for a heads-up version of the same game. I won't bore you with the list of what Poker Stove says are positive equity hands, but it comes out to 658 possible starting hands, which is just a tad under 50% of the 1326 possible starting hands--exactly what I would have guessed (given some small margin of error for the equity calculations).

So now I'm not sure whether Poker Stove is spitting out incorrect equity numbers in the 10-handed situation (I'm using the Monte Carlo simulation because the calculation against 9 random hands otherwise takes way too long for me to wait), or there's something wrong with my intuition that 39% of starting hands can't be profitable in that situation. Maybe the solution lies in ties--that is, that most of the equity in that 39% consists of hands that end up tying with at least one other hand, and if Poker Stove instead spat out its numbers based on winning the whole pot outright, the range of hands on my list would indeed be just under 10% of the possible starting hands. That's my best guess for now, but I really don't know. I'm going to have to mull it over some more. Comments from other brains will be appreciated. Mine hurts too much right now to give it any more thought today.