God gave you guts: don't let him down

To dig deeper into game theory, consider the game of guts, sometimes played at poker tables, although it's not poker. You ante and are dealt a five-card poker hand. You look at it, then take one chip of the agreed denomination under the table, and come up with a fist, either with the chip in it or not. When everyone has one fist on the table, the fists are opened. If no one came up with a chip, everyone takes his ante back. If one player came up with a chip, he gets all the antes. If more than one player came up with a chip, all those chips are added to the antes, and the player with the best poker hand among the bettors wins the pot.

I've simplified slightly. In real guts if no one comes up with a chip, everyone antes again and another hand is played, and if only one player comes up with a chip, his hand has to beat a new hand dealt from the deck to collect the pot; otherwise, his chip and the antes are left in for another round. To further simplify, I'm going to consider only two players and assume that the betting chip is the same denomination as the ante chips (usually it would be larger).

If you are dealt a royal flush, you will obviously bet. You cannot lose, and you might win. Game theory tells us to assume your opponent will do the same thing. Now consider a king high straight flush. If you bet and the other player has a royal flush, you are going to lose one chip, the chip you had in your fist. I'm not counting the ante you will also lose. A common way to make a mistake in computing poker strategies is to mix up the accounting. You can measure profit and loss from before or after the ante, or any other point, but you have to be consistent. I prefer to set the zero point after you've put in the ante but before any other bets are made. That's the best way to think about it-that any money already put in the pot is no longer yours. It's not a loss if you lose it; it is a profit if you get it back.

If you bet your king high straight flush, and the other player has any worse hand, you are going to gain at least one extra chip by betting. If the other player bets, you'll win three chips (the antes plus his bet) by betting, when you would have had zero by folding. If the other player doesn't bet, you'll win two chips (the antes) by betting, when you would have won one chip (your ante back) by folding. Since there are four royal flushes that beat you and 2,598,952 hands you can beat, and if you win you get at least as much as the amount you lose when you lose, you should clearly bet.

To be more precise, we should eliminate the hands that are impossible, given your holding. That means comparing three possible royal flushes against 1,533,933 possible hands you can beat, but it's still an easy choice. So we will play king high straight flushes and assume the other player will as well.

We can work our way down, hand by hand, using this logic until we get to a hand that can be beaten by the same number of hands it can beat. There's no hand for which this is exactly true; ace/king/queen/ jack/two is as close as we can get. There are 1,304,580 hands this good or better, and 1,294,380 worse hands. Pretend that exactly 50 percent of the hands the other player might have beat it, and exactly 50 percent lose to it. Obviously, we should play this hand, since 50 percent of the time it will cost us one chip, and 50 percent of the time it will win us either one or three chips. You, and by assumption the other player, should bet on any hand that is ace/king/queen/jack/two or better. This represents 50.2 percent of the hands you could be dealt. It's over 50 percent because you bet when you hold the median hand.

Now let's start at the other end, with the worst possible poker hand: Seven/five/four/three/two, not all of the same suit. Most lowball games allow you to count ace as low and ignore straights and flushes, in which case five/four/three/two/ace is the worst (best) hand. But guts is a high-card game, so you'd certainly use the ace as high and insist on your straights and flushes.

Seven/five/four/three/two doesn't beat anything. If you bet, you will lose one chip if the other player bets and gain one if he doesn't. But from the preceding analysis, we know he will bet more than half the time. So we fold this hand. The next-worst hand is seven/six/ four/three/two. This could only beat seven/five/four/three/two, and we know that will be folded, so we get only one chip. There is no way for us to win three chips, so we're in the same position as seven/five/ four/three/two, and we fold. We can use this logic all the way up to ace/king/queen/ten/nine. So the game theory solution is to bet any pair or better, or ace/king/queen/jack anything, but fold on ace/king/ queen/ten/nine or worse.

Дата добавления: 2015-10-26