26 juillet 2012 La psychologie du poker

I’ve touched on this before but I have found that this concept is one of the most overlooked phenomenons in the game of poker. With the WSOP in full swing I will focus on the importance of this in tournament poker but many of the proofs and examples apply just as equally to cash game situations. You will see that chips lost are worth more both mathematically and psychologically then chips gained.

The Math

In a tournament everyone starts with the same amount of chips. Let’s say 10,000 chips. The easiest way to show that chips lost are worth more than chips gained is math: if you lose 5,000 chips you have lost 50% of your starting chips (100% of your current stack), while if you gain 5,000 chips and have 15,000 chips you have only increased your stack by 33%. It is a very simple concept, yet most people look at each hand as if the positive and negative equity are equal.

If I am 50% to win a 5,000 chip pot on the first hand, most would say that the coin flip’s expected value is even; I propose that it is not. Here is where the math gets a little complicated. If you win you have 1.5x your starting stack, and if you lose you have .5x your starting stack. If you make this play 10 times and win 50% of the time, you will end up with 1.5^5 x .5^5 or 7.59375 x .03125 = .2373. Put more simply, if you risk half your stack in ten hands, win 50% of those hands, and lose 50% of those hands, your 10,000 chip stack will be down to 2,373 chips.

The actual percentage to come out ahead isn’t merely 51% to win. Rather, it is closer to 63.5%. In that case you would win 5,000 chips, 1.5^6.35 = 13.127 and lose 5,000 chips .5^3.65 = .0797. Over ten hands risking half your stack each time, mathematically, you should expect results of 13.127 x .0797 or 10,462 chips. As an 80% favorite over ten hands for half your stack, you will end up at 64,072 chips. I know that “half your stack” is not a typical hand, but I use it to illustrate the misnomer of coin flips in tournaments. Putting your money in as a slight leader helps you much less when you win, compared to how much it hurts when you lose.

The variable that affects EV (Expected Value) differently than expected is the common practice of “table stakes.” In tournaments, everyone has a limited number of chips and once those are lost they are out of the game. Even in cash games you can limit your losses to what is on the table and each hand is independent of your total bankroll because you decide the maximum at risk.

This table stakes concept also changes the game in that, when playing against players with smaller stacks, you are always limited to winning the lower stacks’ chip stack. Even if I have accumulated 100,000 chips in a tournament, if I am playing against someone who has 10,000 chips, 10,000 is now my maximum bet. To cite our example above, even if I win those 5,000 chips on the first hand, I haven’t increased my ability to gain chips on future hands because the most chips my opponent can lose at the table is 10,000 (and now one individual is down to only 5,000, after we win that first 5,000 half-stack pot). However if I lose the first hand I have limited myself to playing for only 5,000 chips even though there is 95,000 more in chips on the table.

I know this is simplifying situations that are never so black and white but the same principles exist regardless of the number of opponents in a hand or the level of the blinds. That said, as the blinds and antes increase in a tournament you are generally getting better odds because you are not risking 5,000 chips to win 10,000 chips; you are more likely risking something along the lines of 5,000 chips to win 16,000 chips with the dead money in the pot bringing you to 2.1^5 x .5^5 for an expected chip count of 12,762 over ten attempts rather than 1.5^5 or the dreaded 2,373.

The other factor to realize is that the more races for all your chips the lower your expected value. Even as an 80% favorite if you are all in four hands in a row your odds of winning all four hands and still remaining in the tournament is only 41%. If you up that to ten hands your chance of survival drops to 11%. The goal is to get all the chips in play, but what players neglect to realize is that in a tournament there are many opportunities to win chips without getting all in, even the heaviest favorite can lose, but when your opponents fold you always win.

Of course, there is more than just the math to think about when looking at the effects of risking your chips early. Having a big chip stack in a tournament is a big psychological edge, with the ability to bully smaller stacks and to be safe from elimination when placing big bets. However, early in a tournament, with the blinds so low, the intimidation of a big stack is not what it is later in the tournament with every hand being a possible elimination. Losing half your stack early though is a long, uphill climb. You can get lost in the process of just trying to get back to even, taking more unnecessary risks to compensate for the ones you have already taken.

With the maximum value of every hand based on the lowest stack, if you win significant chips early, you can’t even use them. If your opponent has 10,000 chips, then they are not concerned whether you have 15,000 or 50,000; they are concerned that they are playing for 10,000. On the other hand, if you drop below the average, every one of your chips increases in value and none are safe from risk.

When dealing in limited opportunities, every hand has more value and by playing more small hands and less big ones you are taking the same financial risk without putting the psychological weight on one or two circumstances. While doubling up is satisfying, it doesn’t give you the same rhythm and momentum that winning several small pots does. Losing a big pot drains you psychologically while losing a few smaller pots are easier to brush off.

Whether you look at the psychology of a poker player or the math of the game, chips won are not worth nearly as much as chips lost. The truth is that the joy of winning a single hand is minimal next to the pain of being knocked out of a tournament. Doubling up, while satisfying, has won you nothing until you make it to the later stages of a tournament. In the same way, accumulating chips is worthless unless ultimately you can take those chips with you to the end.

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