Pricing the Forecast

Working the umbrella with a perfect forecast prices the forecast exactly — and shows the price collapsing when the odds go extreme.

With a reliable forecast, the agent decides inside each world: told rain — three mornings in ten — carry, 80; told dry, leave, 100. Expected utility with the forecast: 0.3 × 80 plus 0.7 × 100 — 94. Without it, the best was Carry's 87. The forecast is worth 94 minus 87 — 7, and paying up to 7 for it is rational. Now push rain to 0.9: with the forecast, 0.9 × 80 plus 0.1 × 100 — 82; without it, Carry already scores 81. The same perfect forecast is now worth 1 — because near-certain rain means the forecast almost never changes the act.