Playing multiple hands in a poker game.

Question

I was thinking about how great an advantage a player would have if they were allowed to play multiple hands simultaneously in a single game of poker. In other words, all hands get dealt normally except some set of the hands may share information and must bet from a single (albeit larger than the rest of the players') pool of money.

My initial reaction was that this is a huge advantage. Being able to know what some of the other hands are? I wish! But after I thought more about it I changed my mind. It's still undoubtedly an advantage but not as great as I first though.

Let's say you are playing 5 out of 9 hands on the game. You have a 5/9 chance of winning any given round. But you are also paying 5 times the blind and any bets. So at face value, before any information sharing, this is still a completely fair game.

When you take into account the ability to share information you will have the opportunity to fold some of your weaker hands early (ie. hands which will likely, maybe definitely, not beat some of your other hands). This is where I believe the primary, non-psychological, advantage lies. You are able to increase your reward/risk ratio this way.

So now I'm wondering how one would come to an optimal strategy for playing poker in this manner? Are we able to quantify the advantage achieved by playing like this? What other ways you can use the extra information to gain an advantage?

Answer 1

One advantage is the one you identify-that you can fold the weaker of your hands. Let's take an extreme-you do the blind bets for all your hands, but then when you get the whole cards you are told somehow which of your five hands will (after all the cards are turned) be the best. That hand (following the Monty Hall argument) has $\frac 59$ chance of winning the pot. You win from all the other players, but only pay losses once. Of course, you don't know for certain which of your hands is best, so the advantage isn't that large, but you are still way ahead.

The other advantage is knowing eight other cards. If one of your hands is a pair of kings, you are worried about losing to a pair of aces. If you see one or two aces among the eight other cards, the chance that some other hand will have a pair of aces is decreased substantially. The same goes for completing straights and flushes. You can adapt your strategy to how likely your opponent is to have certain hands.