The Chip Utility Paradox Resolved
There’s a heated debate in the poker world over the relative value of chips in a poker tournament. In cash game poker the value of a chip is determined and does not fluctuate. Therefore, it is possible to assign a direct monetary value to any gamble. However, as most tournament players know, the value of chips in a poker tournament is constantly changing. Therefore, assigning a value to your chips is crucial in knowing whether you are making a positive expectation value gamble in a tournament.
In this article I will describe the two opposing camps and explain the paradox that seemingly arises from their differing conclusions. If all goes well I may be able to even convince you that there is no paradox at all.
In red corner… The Reverse Chip Value Theorem
The reverse chip value theorem (RCV) is a widely accepted and mathematically grounded model of the value of chips in a poker tournament. This model relies on the concept that although one person will end the tournament with all the tournament chips they will only receive a portion of the overall prize pool. Take a 9 player $100 sit and go tournament with standard payouts of 50%, 30%, 20%. 9 players start with 1500 chips each valued at $100/1500 or 6.66 cents per chip. However, the winner only takes home 50% of the prize pool, so by the end of the tournament the chips are worth only $500/13500 or 3.7 cents per chip. This is a very attractive theory for gamblers because it is based entirely on mathematics. However, this theory has led some well respected poker authors to conclude that because there is less value to the chips that you acquire, than in the chips you already have, a risk adverse strategy is correct.
And in the blue corner… The Chip Utility Argument
Imagine the situation first described by Arnold Snyder in his article “The Implied Discount.”
“It is incorrect to convert chips to dollar values with no consideration for how individual players might use those chips… When the tournament director says, “Shuffle up and deal!” a battle has begun, and chips are nothing more than ammunition… If a chip is a bullet, and I have 500 bullets, and you have 4500 bullets, you can utilize your ammo in many ways that I cannot. So, intrinsically, each of your bullets has a greater value than each of mine purely as a function of its greater utility… The more chips you have, the more each chip is worth.”
Snyder is correct in many ways that when it comes to fast paced tournaments, having extra chips at your disposal provides many advantages, including
a. You can more easily drive opponents out of pots
b. Chips can be used to steal blinds and antes with reduced risk
c. Chips can be used to see more flops
d. Chips can be used to bluff
e. Chips can be used to call down suspected bluffs
f. Chips can be used to bet for value
g. Chips can be used to bet for information
h. Chips can be used to see a players hand at showdown
i. Chips can be used to wait
At appropriate times in a tournament each of these uses could be valuable to a skilled player. Thus, Snyder defines chip utility as a chip value that increases as a skilled player’s chip stack increases
Chip Utility Theorem
1. The more skill you have, the more your chips are worth
2. The more chips you have, the more your skill is worth
Snyder’s assertion is that to a skilled player chips increase in value and for a skilled player correct play dictates that you should act aggressively to acquire more chips whenever possible.
Resolving the Seeming Paradox
So it seems that the two combatants in this battle have drawn their respective lines in the sand and that there is an apparent paradox. Can a bigger chip stack be worth more to a skilled player, while at the same time being worth mathematically less?
Take the situation of a nine player SNG where one player has aggressively eliminated three other players. The chips stacks are 1500 for all players except one who now has 6000 chips. The big stack can bully the table by pushing his stack into the middle without risk of being knocked out of the tournament. By using an equity calculator*, (based on reverse chip values) the big stack’s equity is 33.73% of the prize pool while each other stack represents 13.25% of the prize pool. Although the big stack has 4 times as many chips as his opponents he only has 2.54X the equity. This shows the decreasing value of his additional chips.
What happens if he is involved in a confrontation and loses? His stack drops to 4500 and is worth 28.3% of the prize pool while the 3000 stack jumps to 21.8% equity and all 1500 stacks drop to 12.46% equity. Although he was in a confrontation for a lot of chips the loss only slightly damages his equity in the tournament reducing it by 5%. What about when he wins the confrontation? The eliminated stack loses 1500 chips or 13.25% equity.
The undeniable conclusion is that the short stack is risking significantly more equity with not much payoff upside when they win. The big stack risks 5% equity while the short stack risks 13.25% equity. They are each risking the same amount of chips, but precisely because the big stack’s chips are worth less, they are able to be much more aggressive.
Conclusions
All chips in a stack are actually worth different amounts. The first chip is the most valuable and the last is the least valuable. This is a mathematical fact. By building a dominating chip stack in a poker tournament you are able to exploit the equity inequality of chips because the chips a short stack is forced to wager against you are worth more than the ones you have to wager. Simply put, you are not threatened with elimination and they are.
Essentially there is truth to both camp’s arguments. The proper conclusion is that chips do decrease in value the more you acquire, but it is precisely that decrease that allows the skilled player to take less risk when wielding a big stack and gain utility from their massive pile of chips. After all, the few chips they risk are worth much less than the chips than a short stack’s and so they can use this advantage as a springboard to tournament success.

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