Rational Decisions
Binmore on when Bayesian decision theory applies and when it does not
Summary
Binmore, a game theorist, examines the foundations of rational decision theory and argues that its standard apparatus is being applied far outside the conditions under which it is valid. His organising distinction, borrowed from Savage, is between a small world, where all states, consequences and probabilities can be specified in advance, and a large world, where they cannot.
His argument is that Bayesian decision theory is correct and unassailable in small worlds and has no justification in large ones. Savage himself said applying his theory to a large world would be utterly ridiculous, and Binmore takes this seriously. He works through the standard justifications for Bayesian priors, including the principle of insufficient reason and various axiomatic derivations, and argues each fails when the state space itself is not known. He also addresses the Ellsberg paradox, ambiguity aversion, and the question of what a probability means when there is no frequency to ground it.
He is critical of both the Bayesian orthodoxy in economics, for extending the theory beyond its warrant, and of some behavioural economics, for treating deviations from a theory that does not apply as evidence of irrationality.
The book is technical and requires mathematics. It offers a diagnosis more than a replacement, since Binmore's positive proposal for large worlds is tentative. For investors the value is precise: it tells you when probability-based reasoning is well founded and when assigning a number is a gesture rather than an analysis, which is the question that most quantitative risk work avoids.
Key takeaways 7
- Bayesian decision theory is valid in small worlds where all states and consequences can be specified in advance, and has no justification outside them.
- Savage, who constructed the theory, regarded applying it to large worlds as indefensible, and economics has ignored that restriction.
- Assigning a subjective probability to an event whose state space is not known is a gesture rather than an analysis.
- Ambiguity aversion is a reasonable response to situations where probabilities are not available, not a departure from rationality.
- Behavioural economics sometimes treats deviations from a theory that does not apply as evidence of irrationality, which misdiagnoses the problem.
- The principle of insufficient reason cannot ground priors, because the answer depends on how the possibilities are described rather than on anything about the world.
- Rationality in a large world consists in reasoning carefully about a situation you cannot fully specify rather than in maximising an expected utility you cannot compute.
Who should read it
Analytically minded investors and risk practitioners who need to know when probability-based decision analysis is well founded and when it is not.
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