Hanging On

In a medium buy-in no-limit hold'em tournament a few years ago, I made a decision that seemed like the right one at the time, even though I was sure more experienced players would play differently. With 11 players left in this event, six at my table, and $1,550 in chips to my name, I posted my $600 big blind and $50 ante, leaving me with $900. A well-known tournament player, the chip leader at the time, was on the button and opened for $2,000. It was clear he would have done so with any two cards. The small blind, also short-stacked, folded, leaving me to decide if it was worth defending my blind. If I called and won, I would have $3,600, still not much of a stack. I had K-8 suited, which is a little worse than a 3-2 favorite against a random hand, winning 58.3 percent of the time.

Conventional wisdom holds that because of top-heavy payouts, you should generally be willing to sacrifice a small bump in prize money for a shot at the top few spots. At the time, I thought it would be especially nice to make a final table (it was the largest tournament I'd ever played in). And there was clearly a great opportunity to see two more players bust out before I would be forced all in – based on what I could see, I thought I was about a 3-1 favorite to do so if I could last out the round. So, I folded. As it turns out, only two more hands were played before the tables were combined, so I was right on that count. But was it the right decision? Should I have taken a shot at building an almost workable stack with a 3-2 edge, or was it smart to just grab for that extra bit of prize money? Deep in my heart I felt sure that more experienced players would have called without hesitation. (In fact, a player on the rail even tracked me down after the tournament to berate me for my poor decision.) But I was still not completely sure it had been a bad idea to give up a 3-2 edge to more than double up in favor of a relatively easy $400 increase in prize money – especially given that I didn't see myself having much chance of getting far at the final table either way.

One way of looking at this kind of problem is in terms of your equity in the tournament as a function of your decision. To figure that out, you just need to know your probability of finishing in each position as a function of the decision you make, and the prize distribution. Working out the probabilities, unfortunately, is difficult when you have a short stack. The positions of the blinds at the two tables are critical. The skill of the remaining players and the distribution of chips among them are also very important in assessing your equity. These factors are hard enough to integrate when you're on the spot and have to make a decision, but they're nearly as hard to work out when you have some time later. Although there are some viable approaches to working out tournament equity, it's hard to come up with a systematic way of differentiating between the situations in which skilled vs. unskilled players have the big stacks, or between the relative skill level of the players left to defend their blinds vs. the players who would be attacking them. And, in fact, it's even hard to gather this information on the spot.

What I sometimes find helpful for post-hoc analysis in situations like this is a kind of sensitivity analysis. Before throwing your hands up in frustration, try to get a sense of how sensitive your decision is going to be to inaccurate assumptions. If your decision is clear-cut no matter what your assumptions, you don't need to worry. If your expectation doesn't vary that much no matter what you assume, you can't really make too big a mistake. If, however, your analysis hinges wildly depending on the assumptions you make, it can't be considered reliable.

With a computer and some spreadsheet software, it's easy to experiment with different assumptions to get some idea of how good or bad a decision may have been. Although it's possible we'll end up as confused as when we started, it's also possible we'll discover either that the decision was clear-cut (right or wrong) or that it was inconsequential.

To figure out your equity as a function of some decision, you just need to assign a probability to each position in which you might finish. Multiply each probability by the prize money for that position, and add up those products. That's your equity. The table that follows gives a simple example, using the prize structure from this tournament and percentages that almost seem reasonable for a player in my position after folding.

To really answer the question, we'll need to redo this table multiple times, making multiple different assumptions, and see how they affect the outcome. I won't go into all the details – if you're really interested in the exact numbers, you can try it out yourself – but I will describe some of the basic assumptions I made, and how I varied things. (Note that for calling, it's simplest to assume I win the hand, and then combine 60 percent of that value with 40 percent of $1,300, for the times I call and lose.)

First, I started with the assumption that my probability of winning the tournament should be the same as my proportion of the chips. With roughly $100,000 in chips in play, that would be 0.9 percent if I folded, and 3.6 percent if I called and won. Since part of my intuition was due to my belief that I was outclassed by all of the players at both tables, I also tried fixing that top percentage at a third of its nominal value (0.3 percent and 1.2 percent, respectively), and also setting the top three payouts to zero.

In estimating the probabilities for the remaining spots, there's no way of getting around some subjectivity. I didn't count down all of the other stacks, or check the positions of the players at the other table. I just had my impression that the play there was considerably wilder than at my table. There were also two short stacks at my table who I thought were likely to make all-or-nothing decisions before I would be forced all in. They might not be forced to do so, but I'm sure they were less interested in a $400 bump in prize money than I was. So, they might be inclined to defend their blinds instead of waiting for me to bust out. Given all that, I assumed that there was a 75 percent chance of making the final table if I folded. This may overstate the value of folding, but that's OK.

For calling, I first assumed that if I called and won, my chances of making the final table were increased to 88 percent (being an intimidated wimp makes this guess seem reasonable). And, finally, I assumed that my probability of finishing in the remaining positions was either a simple linear or exponential function. Again, it's clearly an oversimplification, but it's an easy calculation to make to get a general idea of how the relative merits of my two options would vary.

What emerged from this simple exercise in sensitivity analysis was that although the answer is somewhat sensitive to how all of these values are specified, on the whole they favor calling. Although my belief that the top three spots were unachievable would certainly make folding a viable and possibly correct option (by $62 in the best reasonable case I found), most other sets of assumptions favored the call, in the worst case by as much as $700. Although I may never face this exact decision again, I can try to adjust downward the weight I place on survival when the next few steps in prize money aren't very great.

This hand is reasonably illustrative, because most big-little hands, suited or otherwise, have roughly a 60-40 edge over random cards. And small stacks often face this kind of decision. Although these exact results are a little hard to roll into a tidy lesson that could be applied to a similar situation in another tournament, they are still instructive, and helpful in building our intuition about similar situations. It's also reassuring to know that the considerations I thought at the time mandated a fold (mainly the extent to which I was outclassed) actually did skew the outcome in favor of folding. And we have some idea of how big a mistake I did make, although the range is still wide. In cases like this, where you really don't have all the information to figure out the exact numbers, it's still helpful to examine the sensitivity of your analysis. In this case, calling was probably correct, but at worst, a small mistake. Folding may have been a large one. But perhaps, given that I was seriously outclassed by the other players and was willing to sacrifice a few dollars for the enjoyment of making the final table, it wasn't such a terrible decision.

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