Learn the theory

Decision analysis is the discipline of choosing between options when the outcomes are uncertain. It turns a fuzzy judgement — "which of these investments is best?" — into explicit numbers you can argue with, change and stress-test.

Expected value

If an outcome is worth V and happens with probability p, its expected contribution is p × V. The analyzer asks you for best-case, most-likely and worst-case scenarios with probabilities that sum to 100%, and computes the probability-weighted annual cash flow for each option. If you enter probabilities that don't sum to 100, they are scaled proportionally and the adjustment is flagged — you are never silently corrected.

NPV and discounting

A pound received in five years is worth less than a pound today, because today's pound could be invested. Net Present Value (NPV) discounts each future year's cash flow back to today at a discount rate (the analyzer discounts at 10% a year over a 10-year horizon by default, and you can set your own rate on the scenarios step) and subtracts the initial cost. The tool also fades your entered growth rate toward a 2% long-run terminal rate across the operating years, so a business cannot compound at heroic rates forever.

Why risk-adjusted value beats percentage ROI

A small pilot with a 300% return on a tiny base creates less value than a large programme returning 30% on a big base. Percentage ROI ignores scale. The analyzer ranks options by risk-adjusted value — the NPV after applying a haircut derived from the risk scores you assign — so absolute value creation, not percentage optics, decides the recommendation.

Sensitivity to probabilities

Every answer here depends on your probability estimates. If moving mostLikely from 50% to 60% flips the recommendation, the decision is genuinely close — worth gathering more evidence before committing. Run the same analysis twice with different beliefs and see how stable the ranking is.

Run a decision analysis yourself

Game theory studies decisions where the best choice depends on what others choose. The building blocks are players (who decides), strategies (what each can do) and payoffs (what each combination of choices is worth to each player).

Dominant strategies

A strategy is strictly dominant if it gives you a strictly better payoff than any other strategy, no matter what the other player does. A weakly dominant strategy is never worse and sometimes better. When you have a dominant strategy, rational play is easy — the analyzer labels these "always better" and "never worse" respectively.

Nash equilibrium

A Nash equilibrium is a pair of strategies where neither player can do better by changing their choice alone — each is already playing a best response to the other. It predicts where stable, self-enforcing outcomes settle, even when neither side likes the result.

The classic illustration is the Prisoner's Dilemma: two suspects can stay silent (cooperate) or confess (defect). Defecting is a dominant strategy for both, so the equilibrium is Defect/Defect — even though both would prefer Silence/Silence. Individual rationality can produce a collectively worse outcome, which is why pricing wars and arms races are so hard to escape.

Some games have no equilibrium in pure strategies — the equilibrium is a mixed strategy, where players randomise with specific probabilities. Matching Pennies is the textbook case: both players mix 50/50. The analyzer computes the mixing probabilities for 2×2 games when no stable pure-strategy pairing exists.

Weak vs strict dominance

Strict dominance gives a unique, robust prediction. Weak dominance can leave several equilibria tied — the analyzer tells you when a stable outcome is tied rather than uniquely preferred, because a tie is a warning that the model alone cannot pick your answer.

Model a payoff matrix yourself

The analyzer is organised so each screen demonstrates one piece of theory:

• Scenario probabilities and outcomes → expected value
• Option comparisons → NPV, discounting and risk-adjusted value
• Payoff matrix → players, strategies and payoffs
• Stable outcomes screen → dominant strategies and Nash equilibrium
• Dynamic game screen → how rankings change when you view strategy, not just money
• Consensus screen → combining the lenses into one defensible recommendation

Open the analyzer and try each screen

• Dixit & Nalebuff, Thinking Strategically — game theory for decision-makers.
• Schelling, The Strategy of Conflict — the origin of focal points and commitment.
• von Neumann & Morgenstern, Theory of Games and Economic Behavior — the founding text.
• OpenStax, Principles of Economics (free online) — accessible chapters on oligopoly and game theory.