Anyone have any good ideas for poker variants to be played in situations where a deck of cards is either unavailable or too conspicuous? I'm thinking mainly for classes here where you could play undetected on calculators. (Although unfortunately, the best time for me to do this, high school, is several years passed now).

I understand that there are a number of games in The Mathematics of Poker which use random numbers as opposed to cards, but they are explicitly designed to be simple to solve. I'm looking for something more complex. However, it also has to be fun, so something like "seven streets, limit for the first three streets and pot limit for the last four", while complex, isn't really that interesting.

I'm thinking the best variants would make use of numbers in the way regular poker games do not, so hand rankings are less ordinal. Here's one idea:


Game A
Each player generates two random numbers and writes down the first non-zero number for each. There is round of betting (any limit could be played). Players can then "draw" by crossing out one or both numbers and generating two more in the way above. There is another round of betting. If a showdown is reached, the winner is the player whose numbers have the highest product.


Although you'd need to use an honour system for this, I quite like how this fairly simple game is a decent replica of lowball draw games. Two middling numbers make for a good "pat" hand, while a high number and a number card make for a good "draw" hand. If you wanted to, you could always add more digits, more numbers and/or more draws/betting rounds. There's no need to restrict yourself to product either, you could have all sorts of fancy ways of calculating a hand's value.

edit: there are also dice and obviously Yahtzee, but they're conspicuous, too. Although, dice are easier to throw than cards are to shuffle, so more dice games would be interesting, too.

this product thing sounds interesting

Also if [0,1]*[0,1] high low poker sounds too simple to you, I'm eager to hear your insight on that

Hmm no, I'd forgotten about that.

I think that game is best conceptualised as two different high hands splitting the pot. Calling one of them a "low hand" just makes it confusing. There is also no reason it couldn't involve three or more hands, with a split of the pot that's something other than even.

I think I remember proving (though I can't recall or find the proof) that, at least for a simple version of the game, all bluffs should be demibluffs (unless pure bluffs are all you have left). It's like how you should add semibluffs to your range before adding pure bluffs. If [1],[1] is a bet, then so are all values of [x],[1] and [1],[y].

How did you prove it? (that's for the 1-street game I guess?) I remember trying to solve it at the time I posted that thread in the theory forum, but failed. I'd love to see that proof (I hope it's not hopelessly lost)

Jean

Actually, I think it's wrong. I haven't solved the 1 street game, but I think we can learn by looking at the limits (as in extremes).

Firstly, consider the [0,1]*[0,1] game but with unequal division between the high pot and the low pot. There is no reason that the pot has to be split 50/50 between the two. As the division gets more disparate, the more the game resembles a regular high hand game. The optimal strategy for high hand games involve pure bluffs. Clearly, setting aside a billionth of the pot to a low hand isn't likely to make a pure bluff suboptimal. This would occur if you had more hands indifferent to bluffing than you needed to bet with, in which case you may as well bluff the hand with the highest low component.

On the other extreme, consider dividing the pot up into even more pieces, the [1,0]*[1,0}*...[1,0] game, with the pot split amongst a billion hands. Because of law of large numbers, it would be extremely rare for any set of hands to win a significant majority against another set of hands. Hence, you would almost always figure to have close to 50% equity, and thus almost never fold so long as bet sizes were not enormous themselves. In this case I believe this game is very similar to the regular [0,1] no fold game - bet the top half of hands, or bet slightly less than that if raising is an option. Pure bluffs would never have the pot odds to succeed.

Somewhere between the two, then, pure bluffs disappear as an optimal option. I'm not sure whether it's strictly a function of the number of separate pots or other factors like bet sizes, number of streets and number of raises available. I think it's probably the latter.

Hmm ... I don't see this reasoning proving that "there shouldn't be pure bluffs" is wrong?