1
Test a preference for order
State completeness and transitivity, find the pair that breaks transitivity on a relation grid, and compute what a cycle costs a chooser per lap.
Course · Philosophy
Decision and Rationality
Lay a choice out as acts against states and see what each rule says. Money pumps, dominance, maximin and regret, expected utility and the value of information, then five puzzles in which the rule seems to give the wrong answer, each computed to the figure at which it turns.
A ≻ B, B ≻ C, C ≻ A a cycle: pumpable
maximin: best of the row minimums
regret = column best − your payoff
EU(a) = Σ P(s)·u(a, s) take the largest
VPI = expected best − best expected
Pascal wins when M > c / p
What this course is about
- Subject
- Philosophy
- Lessons
- 10
- Format
- Interactive
Overview
What you will be able to do
By the end you can lay a decision out as a table, say what each rule recommends and why they differ, and compute the point at which a famous paradox turns.
2
Choose without probabilities
Apply dominance, maximin, maximax and minimax regret to one table, and find an instance in which the rules disagree.
3
Choose with probabilities
Compute expected utility, the probability at which a choice flips, the effect of a concave utility, and the value of perfect information.
4
Pin down a paradox
Show that the Allais choices are inconsistent with any utilities, that the Ellsberg choices fit no single proportion, and where Pascal's wager and Newcomb's problem turn.
5
Price an infinite expectation
Show that the St Petersburg game's expected value diverges, and compute the finite value once the bank's bankroll is capped.
Lessons
10 lessons
Preference comes first, then the rules that need no probabilities, then the rules that do, and the paradoxes come last because each one presses on a different premise of what came before.
Preference, Transitivity and the Money PumpA cycle of preferences can be priced, and the price is paid every lap.Not marked
The Decision Matrix and DominanceLay acts against states, and drop any act that another beats everywhere.Not marked
Maximin, Maximax and Minimax RegretThree ways to choose when you cannot say how likely any state is.Not marked
Expected Value and Expected UtilityWeight each payoff by the probability of its state, add, and take the largest.Not marked
Risk Aversion and the Value of InformationA concave utility makes a sure thing beat a fair gamble, and prices what a forecast is worth.Not marked
The Allais Paradox and the Sure-Thing PrincipleTwo common choices test the same expression for opposite signs, so no set of utilities can make both.Not marked
Ambiguity and the Ellsberg UrnPreferring the known bet in both of two choices fits no single guess about the unknown proportion.Not marked
Pascal's WagerA cost, a probability and a reward fix the threshold at which wagering wins, and an infinite reward removes the threshold.Not marked
Newcomb's ProblemTwo valid arguments recommend opposite acts, and the predictor's accuracy fixes where the evidential one turns.Not marked
The St Petersburg GameA game with an infinite expected value is worth only a modest price once the bank's bankroll is capped.Not marked
Recommended background
Knowledge and Evidence, for credence as a degree of belief and for the rule that probabilities add to 1.
Practice suggestions
Work forward. “Expected Value and Expected Utility” assumes you can read a decision table, which “The Decision Matrix and Dominance” teaches, and the paradoxes assume expected utility.
Change the numbers. Every lab here is a decision table you can edit, and most lessons turn on a figure at which a choice flips. Move a probability or a payoff across that figure and watch the recommendation change; it is faster than reading the argument for it.
Hold the paradoxes loosely. Each lesson states the argument for the puzzling choice at its strongest and shows where it conflicts with the rule. Whether to give up the choice or the rule is left to you, on purpose.
What it does not cover
The representation theorems. This course states what a consistent chooser's utilities are and uses them; it does not prove that preferences satisfying a list of axioms must be representable by expected utility.
Causal decision theory in full. Newcomb's problem is computed under the evidential rule and under dominance, which is as far as the table goes; the machinery for the causal alternative is not built.
Continuous probability and unbounded utilities. Every table is finite, and every probability a fraction. Pascal's mugging, which needs an unbounded utility, is named in the last course and not worked.
Behavioural economics as a science. The Allais and Ellsberg patterns are treated as logical puzzles about consistency; how often people show them, and why, is a question for experiment and not for a table.