Week 5: Position and the value of information

A clean decision-theory result — the value of perfect information — used to put a number on why acting last is worth something

22 March 2027

“Position is valuable” is a claim, not a fact#

Every poker text says acting last is an advantage. This week formalizes why, using a general decision-theory result before connecting it back to the table.

The value of information, generally#

Suppose a decision has two possible actions, and the world is in one of two states with known probabilities p and 1 − p. Each (action, state) pair has a known payoff. Two ways to play it:

• Blind: commit to one action before the state is revealed. Best case: the single action with the highest expected payoff.
• Informed: see the true state first, then act. Best case: the best action for that state, every time.

Value of information = E[best action per state] − E[best single blind action]

This difference can never be negative — seeing the true state and still being free to take the blind-optimal action is always an option, so the informed side can only match or beat the blind side.

A toy number#

Villain’s river hand is Value 60% of the time, a Bluff 40% of the time. Hero can Call (win $100 vs a bluff, lose $50 vs value) or Fold (always $0):

Blind (commit to one action for both states): EV(call) = 0.6×(−50) + 0.4×(100) = +$10; EV(fold) = $0. Best blind action is call, worth $10.

Informed (know villain’s exact holding before acting): fold vs value (0, beats −50), call vs bluff (100, beats 0). Expected value = 0.6×0 + 0.4×100 = $40.

Value of information ≈ $40 − $10 = $30 per hand, in this toy model.

Connecting it to position#

Nobody gets to see villain’s actual hole cards. But acting last means seeing villain’s action — a bet, a check, a sizing — before committing, on every street of the hand, not just once. That’s not the clean perfect-information case above; it’s a repeated, partial-information version of it. The player who never has to act first never has to reveal a decision before extracting whatever signal the other player’s action carries.

A general structural point sits underneath the numbers too: in any sequential decision, the player choosing last can always fall back on whatever the blind player would have done, and is never forced to do worse. That’s the same asymmetry the toy example measures in dollars.

Before next week#

The toy example assumed a perfect signal — hero simply knows villain’s state. Real betting actions are a partial signal: a big river bet is more likely, not certain, to mean a strong hand. Week 6 formalizes how to turn a partial signal like that into an updated probability, using Bayes’ rule.