Utility Function for Chips

First off i don't want this to come off as disrespect in any way. I think 2+2 has contributed more to poker theory than all other published materials combined.

That said, when sklansky wrote tournament poker he made alot of value statements based off the decreasing marginal value of chips. (Your first chip is worth the most and every chip after is worth less than the one before it.) The value statements made were those in the nature of "avoid unnecessary gambles".
I believe those value statements should have been made off the utility function for chips in the tournament instead of the value of those chips. Individuals have their own utility functions, and thus some players should gamble more than others in some situations. If you play a certain sized stack poorly gambling to get a bigger (or smaller) stack may increase your ability to prefrom in the tournament and thus increase your utility even though the additional chips are worth less or lost chips are worth more.

[ianlippert]

I've been thinking about this, im not sure if thats the right way to look at it. I think traditionally its thought that if you have 50% of the chips you have a 50% chance of winning. But it seems like the way top tourney players play, its more important to collect as many chips as possible because its more like 50% of the chips will give you a 60% chance of winning. Because the tourneys are hevily weighted to the final spots, its seems like you should go big or go home. I dunno mabey some of the better tourney players can comment on this.

The idea for tournaments is not to place in the money; it is to finish in the top three, where the real money is sitting.

[AaronBrown]

I think you are making two different points.

First is that the individual preferences of a player might overwhelm calculations based the mathematics of the tournament. While that could be true in principle, I don't think it's very important in practice. Whether you love or hate risk, in precisely what flavor, will determine whether you enter the tournament, but once you're in, it almost always makes sense to try to win as much money as possible. One guy might care only about the dollars won, another might be willing to take a lower expected value for a better chance to win first place, another might take a lower expected value to reduce the chance of being eliminated early. But only in extreme cases would these considerations make the marginal utility of chips increasing.

There are two strong reasons that the marginal utility of tournament chips decreases. First is that some of the prize money is given to people who do not finish first. That means some of the value of your second chip is wasted, and more of each subsequent chip, because it decreases your chance of winning some of the lesser prizes by increasing the chance of winning first prize.

The second relative advantage of a shortstack is that you may win a pot when a hand that could beat you folds after you are all-in.

I think your second point is that different playing styles favor different stacks. One guy might play at his best when shortstacked, another might be best at exploiting the advantage of a large stack. Again, that is true in principle, but I can't believe it is important enough to change the overall declining marginal utility of chips.

[Xhad]

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The second relative advantage of a shortstack is that you may win a pot when a hand that could beat you folds after you are all-in.

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More importantly, being all-in allows you to see a showdown with a hand you would "normally" be forced to fold on an early street.

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I think your second point is that different playing styles favor different stacks. One guy might play at his best when shortstacked, another might be best at exploiting the advantage of a large stack. Again, that is true in principle, but I can't believe it is important enough to change the overall declining marginal utility of chips.

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One important point that is not discussed in TPFAP is the fact that the guy with more chips can take more +EV gambles precisely because his risk of ruin is lower. For example, if a 55% edge is for my entire stack on the bubble I can't take it, but if it's for 1/10th or even 1/5th of my stack I probably can.

[NMcNasty]

Its true that if one player plays well with a large stack he should be slightly more inclined to take gambles than someone who doesn't. This does not however in any way change the fact that the strategic value of chips decreases. It doesn't matter how useful a large stack is to you, the difference in value between 2k in chips and 3k chips will always be more significant strategically than the difference between 10k in chips and 11k in chips. Also a "utility function" simply measures how happy something makes you. Happiness does not necessarily mean a strategic advantage, there are plenty of players who get way too happy with their big stacks and spew away all their chips thinking they can bully the table.

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Also a "utility function" simply measures how happy something makes you. Happiness does not necessarily mean a strategic advantage, there are plenty of players who get way too happy with their big stacks and spew away all their chips thinking they can bully the table.

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I was intending to use utility function of chips in a tournament setting not in a true economic sense of happiness but rathar in a poker sense of the utility of chips for a player to maximize the cash payout in the tournament.

I believe that a larger stack also brings more advantage. Players make LESS moves against a big stack because they can afford to take all +EV gambles and my utility increases when i have other players more strongly avoiding gambling with me.

Example that is true in many situations, not intended for extreme situational analysis:
I am the same stack as player A and have what player A correctly believes is 88, he has AK and for some reason i would have a good enough read on him to make my decision accordingly. If we are near the bubble and i make the first raise and he raises allin, i should fold (assuming i am not cuffing myself to the bet), so my stack determines HIS play. If i have 3x his stack and make the same play, i should not fold and so he should not raise. Now i win the blinds with no risk.

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If you play a certain sized stack poorly gambling to get a bigger (or smaller) stack may increase your ability to prefrom in the tournament and thus increase your utility even though the additional chips are worth less or lost chips are worth more.

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Let's say a player plays a short stack well. Does this mean that he will perform better with this short stack than with a larger stack. I argue not he will not. Even though he may play his stack better relative to how a different player would play this same stack he is always better off with more chips. In order to win the tournament one must win all the chips. Since this is the goal of tournament players then each step closer they come to the final goal must make them better off. The law of diminishing marginal returns must apply since although each chip makes one better off, each chip gained subsequently has less impact upon the player's stack.

The statement that a large stack may help one perform in a tournament is undenaibly true. Large stacks win you more chips and allow you to absorb losses more readily. This does not mean that chips' value AT THE MARGAIN is increasing, rather the TOTAL UTILITY of the large stack is greater than the TOTAL UTILITY enjoyed by a player on a short stack.

[AaronBrown]

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In order to win the tournament one must win all the chips. Since this is the goal of tournament players then each step closer they come to the final goal must make them better off.

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If the goal is to win the tournament, and we ignore the short-stack advantage, then the marginal utility of chips is constant. If the goal is to maximize money winnings, then the marginal value of chips decreases.

Aaron_C and Xhad have posted advantages of large stacks, both of which are based on players not maximizing their expected value of money winnings. While both of these can be valid in principle, I stick to my intuition that they are secondary factors compared to the big two.