"Can you put a price on information?"
The club room fell silent at Yuki's question.
Aoi answered with interest. "You can. There's a field called information value theory."
"How do you measure it?"
"By how much the expected utility of decision-making changes before and after getting information."
Riku was confused. "Sounds difficult."
"Let's think with a concrete example," Aoi wrote on the whiteboard.
"Tomorrow, I'm going out. If it rains, I need an umbrella. If sunny, I don't. But I don't know the weather."
"Check the weather forecast?" Yuki said.
"Yes. We can calculate how much value that weather forecast has."
Aoi drew a diagram.
"With umbrella: If rain, comfortable (utility 10), if sunny, heavy (utility 5) Without umbrella: If rain, get wet (utility 0), if sunny, comfortable (utility 10)"
"If rain probability is 50%, what's the best action without information?"
Riku calculated. "With umbrella expected utility 7.5, without 5. So take it."
"Correct. Now, with perfect weather forecast?"
"If rain is known, take umbrella (utility 10), if sunny, don't (utility 10). Expected utility is 10."
"Difference is 2.5. That's the value of perfect information."
Yuki was surprised. "Information itself has value."
"Yes. By reducing uncertainty, better decisions can be made."
Mira wrote in her notebook. "Value of Information = E[utility with info] - E[utility without info]"
"Precise definition," Aoi acknowledged.
Riku thought. "What about imperfect information?"
"Good question. If weather forecast accuracy is 70%, information value also decreases."
Aoi continued calculating. "Using Bayesian updating, calculate posterior probability. Then find expected utility."
Yuki wrote in the notebook. "Information value depends on accuracy."
"Exactly. Perfect information has highest value, completely random information has zero value."
Mira supplemented. "Information value in decision-making context"
"Important point," Aoi nodded. "Information value depends on context. Same information has different value depending on situation and decision-maker."
Riku gave an example. "Stock price information has value for investors, but meaningless to students."
"Exactly. Information value is subjective and context-dependent."
Yuki thought deeply. "So conversations in this club room also have value?"
"Of course," Aoi smiled. "For Yuki, knowledge of information theory improves future decision-making. That increase in expected utility is this conversation's value."
"But can't be measured in money."
"True. Monetary value and information value are different. But they're related."
Aoi continued. "Companies pay for market research. Because that information improves product development decisions."
"They calculate cost-effectiveness," Riku understood.
"Yes. If information gathering cost exceeds information value, it's wasteful."
Mira said quietly. "Sometimes ignorance is optimal"
The four fell silent. It was a deep insight.
"Ignorance is optimal?" Yuki asked back.
"When information acquisition cost is too high, not knowing can be better," Aoi explained.
"Seeking perfect information too much prevents decision-making."
Riku nodded. "Analysis paralysis."
"Yes. Information theory also teaches when to stop gathering information."
Yuki summarized. "Information value is measured by uncertainty reduction and decision improvement. But cost should also be considered."
"Perfect summary," Aoi said.
"Measuring information value. That might have been today's meeting's value."
Mira smiled. It was rare.
Outside the club room, dusk was deepening. An after-school meeting measuring information value. It itself had high information value.