The value of information

Compare an optimal decision before and after a signal, including its cost.

13 minutes · advanced · Original Poker4Quant lesson

What you will learn

Compute expected value of perfect information in a finite decision model.

Prerequisites: Best responses · Bayes with a frequency table

Information has value through a changed action

Define a prior over states and payoffs for each available action. Without a signal, choose the action with the highest expected payoff. With information, choose a best action separately for each observation and then average the resulting payoffs.

The expected value of perfect information is the difference between the perfect-information optimum and the no-information optimum. If the information is free and can be ignored without changing anything else, its optimal expected value cannot be negative: the original policy remains feasible.

Costs and behavior can change the comparison

Paid information is worthwhile under this risk-neutral objective only when the expected improvement exceeds its cost. A noisy signal usually has less value than perfect state revelation in the same model, but must be evaluated using its actual likelihoods and induced actions.

The free-information comparison assumes the same action set, payoff structure and ability to ignore the signal. In a strategic interaction, public revelation might change opponents’ behavior or constrain commitment, producing a different comparison. Do not silently extend the simple result across those changes.

Worked example

State S has probability 30%. Action A pays +40 in S and −10 otherwise. Safe action B pays 0. What is the value of learning the state perfectly before acting?

Without information, EV(A)=0.30×40−0.70×10=5, while B gives zero. Choose A.
With perfect information, choose A in S and B otherwise. Expected payoff is 0.30×40=12.
Subtract the no-information optimum: 12−5=7.

Perfect information is worth 7 units in this model before any fee. Paying 8 for it would reduce expected value relative to taking the original optimal action.

Hands-on: Construct signal policies and decide whether to acquire

Build joint signal populations, optimize the uninformed baseline and contingent actions, execute both policy channels and distinguish gross information value from net-if-bought value.

Reading signal model…

Common mistakes

Comparing informed behavior with a nonoptimal uninformed action.
Subtracting signal cost only in one state when it is fixed.
Ignoring changes in opponent behavior after public information.

Keep these ideas

Optimize consistently before and after the signal.
Information’s value comes from actions it improves.
Include cost and preserve the comparison’s assumptions.