How much is a dollar worth?

By: Claes Andersson

The value of a $1 chip is of course $1 and the value of a $100 chip is $100. After all, it’s printed on them and that’s what you get when you cash them. However, there are two important ways in which this might not be true. The first has to do with how much you perceive that you need the money and the second has to do with tournaments

The term “utility” which is often used in economics where it underlies consumer behaviour theory is a useful concept in poker not least because the situation is so simple there: the good to be consumed is simply amounts of currency and $1000 buys you exactly 1000 times more stuff than $1; hence $1000 should be exactly 1000 times more worth than $1. Here, in a sense, we have the key to the problem whereas knowing the utility of goods such as a packet of corn flakes or a car can be much more intricate. Who knows, perhaps neoclassical economics even has something to do with reality if applied to poker! Of particular interest here is the related concept of risk behaviour. Utility is however notoriously hard to quantify, but it suffices here to think of it as a numerical value that quantifies how much we like some amount of something. It assumes that in the end we act according to our sense of value, not according to some essential and objective notion of value.

Most of us would agree that if $1000 is all that you own, then losing $1000 appears more than 1000 times more unpleasant than losing $1 out of $1000. As a matter of fact, we may even enjoy losing $1 if we do it in an exciting way such as buying a lottery ticket for a lottery with a high first prize. We may ignore even horrendously short odds because of the dream of the 1st prize and the fact that $1 is worth nothing. In other words, the utility of $1000 is not always 1000 times higher than that of $1 and there are many effects of this on the psychological plane. In some instances losing $1 one thousand times might cause us too little grief and losing $1000 at once might cause us too much grief. That is, we might turn down good odds for a large bet when we shouldn’t and accept bad odds on a small bet when we shouldn’t.

To analyze this, we can use a “utility function”: one that toys with the question “how much do you like $x?” by relating a numeric quantity to the joy you experience when receiving $x. Since poker is about accumulating money over long periods of time, your utility function should be a straight line on such a diagram. It should not be curving off downwards or upwards. If making money is what you want, you will want to use the curvature of others’ utility functions, but your own should be as straight as possible. That is: it should be straight for the values of x in which you bet, since there is hardly a person born whose utility of money is straight everywhere.

Using the utility function in the diagram above (U(amount) = amount + 0.0005 amount*amount) we can make a simple calculation example. Let’s say that this person has $400 and is presented with a double-or-nothing $100 bet. At what probability of winning will this person accept this bet? Common sense tells us that we need to win this bet one time out of two on average to break even from it and thus we should be indifferent to accepting or turning it down at a winning probability of p=0.5. The behaviour of the holder of this utility function, however, works differently because the utility function curves off upwards.

We can investigate the utility function to find out at which probability he or she is indifferent to this lottery. Let’s call the lottery L. We want to check when the utility of the lottery is equal to the utility of the current amount of money (i.e. not accepting the lottery). That is we want to solve the equation U(E(L))=U($400). Now, the lottery can be fleshed out to its expected value dependent on the probability of winning which we call “p”: E(L) = p ($500)+(1-p) $300. E(L) is the average amount of money after such a lottery if the original fortune is $. This says that a proportion p of the times this lottery is performed the person will be left with $400 plus the winnings of $100 amounting to $500 and the rest of the times it will be left with $400 minus the bet which is $300. Taking p* U($500)+(1-p)*U($300)=U($400) and solving the equation for “p” we arrive at p=0.45. This means that this person will accept the bet at a probability lower than even money! Such behaviour is called, not surprisingly, “risk loving” behaviour.

Another typical situation is when the player is the opposite of “risk loving” in which case we say that it is “risk averse” – a risk averse person would require a chance of winning above p=0.5, keeping in mind that p=0.5 is the economic break-even probability here. For the sake of completeness I will also mention that a person who strictly looks at the odds no matter the amounts is said to be “risk neutral”. Most people are however not simply risk lovers, haters or neutral but risk lovers for small amounts and risk haters for large amounts. This can lead us to suspect that somewhere in between we would be reasonably risk neutral and this is exactly the amount-interval in which we should be playing. If I maximize my utility playing this way or the other, isn’t that per definition good? Well, yes, but there is a further problem… any deviation from risk neutrality threatens to cause you to lose money over time and even if you maximize your utility in individual pots you are still not maximizing utility in the famous long run.

If you think about it, a lot of businesses thrive on this exact psychological mechanism: you are much more likely to accept to pay a small monthly fee than a large fee up front even if the larger fee is considerably smaller than the small fees summed up. In the same way, you do not bother to look at the odds for lotteries and gambling for tiny amounts. In other words, people are profiting from your curved utility function so you should do your best at keeping it straight. Now, none of us can or should have a straight utility function in reality: not upwards for risk of fluctuating into economical dire straits and not downwards for the risk of leading very boring lives. For example it appears rational that if someone offered you 1000000 times the money at 1:10000 odds but you can only bet everything you own you would not take these excellent odds simply because unless a miracle happens you will be dead broke (of course, unless you can get someone wealthy enough to be able to weather such losses to stake you). What you need to do is to select betting options in an interval where your own personal utility function happens to be straight. Only then can you make the decisions that will end you up winning in the long run. This is hardly any news – anyone knows that you should not play too low or too high, this article simply goes into more details about exactly why.

Now, when we have discussed utility functions we can move to the case where the utility function of chip amounts is in fact not supposed to be straight: tournaments. The value of a tournament is simply the sum of the probabilities of finishing in all possible places times the payout of these places. For example, say that you have probabilities 0.3, 0.2 and 0.10 of finishing 3’rd, 2’nd and 1’st respectively and probability 0.5 of finishing out of the money. Furthermore, let’s say that the payouts are $100, $200 and $350. This gives us an expected value of the tournament of 0.3*$100+0.2*$200+0.1*$350=$30+$40+$35=$105. Now, these probabilities can never be known with any good accuracy but the line of reasoning is still important: every time we make an action we will change these probabilities and even if we don’t know them exactly we might be able to guess roughly how our actions affect them.

There are a number of ways to conceptualize a tournament and one of them is to state that the objective is to maximize the expected number of players that you out-last. Thus, the stack size is coupled to the expected value of the tournament because it affects how many more blinds you can pay. Every time a blind is paid a round has been played and the chance is some players have been knocked out. If you live to pay the last blind, you have won the whole thing. Now, why do we wait to pay blinds? The answer is: in order to wait until we again come in a position to increase the number hands we can expect to last. We can, however, not wait forever because of the blinds which keep coming and keep increasing. Related is hence the question: what hand quality can I expect to get before I am blinded out or my stack is reduced enough that I can not attack the blinds? With a large stack you can last more hands and thus the expected best hand you will see before you have to move is better than if you are short stacked. This logic is particularly visible when you are extremely short stacked: if I’m considering mucking my hand from UTG and the blind will cripple me next hand, I must ask myself whether I can really expect to get anything better when the blind comes around. In fact, if I have just enough to put in a small raise now, I must even have a substantially better hand on the blind since I won’t be able to pick up the blinds without a showdown then. If you have 83o then you better wait for the blind because there is a good chance that this hand will be sufficiently better. If on the other hand you have KTo you will want to move right there. This logic can be applied throughout. One can ask oneself the question: do I have to play this hand?

There is furthermore an inherent advantage of playing few hands in a tournament. Every time you play a hand there is a chance that you will be trapped and bust out or be crippled or reduced enough that you will start playing worse. Let’s compare two scenarios: first that you play 50 hands with a probability of 0.05 of going bust every hand or that you play 30 hands more restrictively with only a probability of 0.04 of busting out. In the first case, your chance of surviving is a little more than 7.5% and in the second case it is almost 30%! Of course, if you do get a large stack, your chance of busting out of becoming crippled goes down, but this example still illustrates the principle. The difference might not be very visible during the game, the second player is a little bit tighter and folds a little bit more often, but the effect it has on your probability of reaching top positions is tremendous!

Isn’t it worth it to take a shot at getting a really big stack? Well, common wisdom has it that if only very few places are paid then this is truer than in standard tournaments with a more flat pay-out. This has to do with the fact that unless you have a very large stack your chance of winning or finishing in the top few places is of course pretty small, and with much probability mass on positions paying zero this follows as an effect. However, to assess in the general case how much some more chips will help you, let’s look at another way of conceptualizing the situation. Say that “a shot” costs some amount of chips. A “shot” is here to be viewed as something like: first putting in a raise, and then to go if you hit the flop or bluff if the situation arises. Now, how many shots do you have left and what is your probability of winning a pot when you take a shot? What is the difference between having one and two shots? Between 4 and 5 shots? Say that you go into situations where your probability of winning is 70%, for example you raise your hand so there is a good chance of a fold and if you get called you have 40% to win because only really good hands will call you. Now, the chance of winning in one shot is 70%, and with two shots the chance of missing both shots is 0.3*0.3 so the chance of winning at least once is 91%. This is a substantial gain. However, if we compare the difference between 4 and 5 shots we see that 4 shots gives us 99.19 percent of winning at least once and 5 shots gives us 99.757% -- a miniscule difference. Now, of course, the blinds are rising so the number of shots keep getting reduced so the benefit of a big stack is in reality a little bit better than pictured here. Still the trend is clear: if you have 10000 chips, another 1000 chips is a lot less worth to you than 1000 chips if you only have 1000 chips to begin with! It is not the amount of chips, it is more like the percent change in stack size that matter. This seems simple but there is another twist to it. If you have 10000 chips and your opponent has 4000 and you bet 1000, the call is larger than the bet was – your opponent forfeits more probability of staying in than you do by paying 1000 chips. The result: it is always worse to lose a chip than what it is good to win one. This is one way of looking at what the “big stack advantage” really is. This, of course, is quite different from ring games where, as discussed, winning and losing a chip are (should be) the exact opposite things.

This seems to suggest that one should hardly ever play a hand in a tournament: the small stack because it is expensive and the large stack because more chips are not worth much. This is not so and it has to do with the previously mentioned expectations that we can have on future hand strength. Sure, we forfeit more probability when moving with a small stack, but then the quality of the hands we can expect to see in the future is also considerably smaller. Say that the probability of catching a good hand to move with is 5% per hand, then a player who can expect to last only three rounds will have an 78.5% probability of seeing at least one such hand during this time. Someone who can last 15 rounds, on the other hand, looks at a probability of 99.95% chance of getting a hand that good. Clearly, to maximize pay-off the smaller stack must be less picky than the large hand. You should not look for, say, the top 5% hands, you should look for the best hand you have a fair chance of seeing before you’re out.

Exhausting the consequences of these factors is impossible but there are a number of additional phenomena that should be considered. The first is that it often becomes rational for a large stack to make calls and raises with weak hands nevertheless. This is for two reasons: first, the large stack’s owner knows that the small stack’s owner is not very picky with hands and second, late in a tournament an important aspect of outlasting players is to knock them out your self. For example, if you call 10% of your stack for a 25% percent to bust a player out the probability of staying in that you lose by doing so is likely not by far large enough to outweigh the benefit of knocking this player out of the game. I don’t know how many times I’ve been yelled at after busting premium hands with rags – how could I not see I was behind? The answer was that I did understand very well I was sorely behind, but the benefit of that player busting out together with the still not tiny chance of actually beating a premium hand all in with rags outweighed it. Add to this the benefit of showing off that attacking your blinds is not a safe thing to do – you will watch the blinds trickle in a lot more often if you display this behaviour (don’t overdo it of course).

Another consequence has to do with re-buys and add-ons. You pay a fixed amount to receive more chips and when you are buying from zero or from a small stack size, these chips buy you a great deal of expected value from the tournament. The add-on, however, is only worthwhile doing if your stack is pretty small. If your stack is big, you are likely just donating money: you increase smaller stacks’ expected payout more than your own. Sure, you gain a little in probability of getting far but as was illustrated before, this probability increase multiplied with the payouts likely does not compensate the money you pay.