Quantifying the Value of Information

US players are welcome

Sometimes you can put a specific price tag on what information is worth. For instance, say, instead of pocket aces in the big blind, you have pocket kings. Now it’s worth something if your holdem opponent shows before you call, as you can fold if you see aces.

Say your holdem opponent has raised $100 all-in, and you think (correctly) that she would do this with pocket aces through nines, ace-king, and ace-queen. With kings, you’re calling against every poker hand except aces. When she has aces, you’re roughly a 4.5-to-l dog, so calling costs you $64 (assuming, for simplicity, that the $ 1 and $2 blinds get raked away).

$64 = (0.82)(- $100) + (0.18)($100)

So avoiding aces is worth $64. But usually, in Texas Holdem, you won’t see aces; usually you’ll see something else. She can have aces six ways, kings one way, queens through nines six ways each, ace-king eight ways, and ace-queen sixteen ways. The chance you’ll see aces is approximately 11 percent.

0.11 = (6 + 1 + 6 + 6 + 6 + 6 + 8 + 16)

Thus, getting to see if your holdem opponent has aces is worth (0.11)($64) = $7.04. If you had queens instead of kings, the information would be worth twice that, or $14.08, as you would be exactly twice as likely to see a hand you would fold to.

Jacks wouldn’t be worth three times, though, as ace-king and ace-queen could now be made 16 ways each instead of 8 and 16.

Now what if the information weren’t perfectly specific and accurate? Say you have pocket kings again, but instead of showing her cards, your holdem opponent answers the question "Do you have aces?" Half the time that she has aces, she says "yes," and the other half, she says "no." She always says "no" when she doesn’t have aces.

She’ll still have aces 11 percent of the time, but she’ll tell you she has aces only 5.5 percent of the time. So this information is worth (0.055)($64) = $3.52. Even though it’s not perfect, it’s still worth something. The information would be worthless, however, if she tainted her "yes" answers with enough hands you can beat that you would have to make the call regardless of what she says.

Good luck.

Also recommended:

1. Paying for Information in Texas Holdem
2. Imperfect Information and Implied Odds in Holdem
3. Another Opportunity to Gather Information
4. Choosing Starting Hands in Seven-Card Stud
5. Texas Holdem Flops