Allais Paradox

The two pairs of lotteries

Allais set out two binary choices. Each subject picks one option from each pair.

Choice I.

• A: $1,000,000 with certainty.
• B: 10% chance of $5,000,000; 89% chance of $1,000,000; 1% chance of $0.

Choice II.

• A': 11% chance of $1,000,000; 89% chance of $0.
• B': 10% chance of $5,000,000; 90% chance of $0.

Most subjects pick A in Choice I and B' in Choice II. That combination is impossible under expected utility theory — and the inconsistency is the paradox.

Why the EU axioms forbid the switch

Let u(·) be any vNM utility function. The four lotteries decompose into a common 89% block plus an 11% block:

A  = 89% × $1M  + 11% × $1M
B  = 89% × $1M  + 10% × $5M  +  1% × $0
A' = 89% × $0   + 11% × $1M
B' = 89% × $0   + 10% × $5M  +  1% × $0

By the independence axiom, replacing the common 89% block in both alternatives — $1M in pair I, $0 in pair II — cannot reverse the preference. The remaining 11% blocks are identical across pairs:

Pair I  → 11% × $1M   vs   10% × $5M  +  1% × $0
Pair II → 11% × $1M   vs   10% × $5M  +  1% × $0

The independence axiom says these are the same comparison. If A ≽ B, then A' ≽ B'. If B' ≻ A', then B ≻ A. Picking A and B' violates the axiom. Algebraically: A ≽ B requires 0.11·u(1M) ≥ 0.10·u(5M) + 0.01·u(0). B' ≻ A' requires 0.10·u(5M) + 0.01·u(0) > 0.11·u(1M). Both cannot hold.

Yet experimentally, roughly 80% of subjects pick A and B'. The certainty of the $1M in Choice I psychologically swamps the marginal expected-value gain from B (which has E = $1.39M vs A's $1M); but when nothing is certain in Choice II, the higher expected value of B' wins.

A complete payoff comparison

Lottery	10%	1%	89%	Expected value	EU rank (typical u)
A (Choice I)	$1M	$1M	$1M	$1,000,000	Higher if u is concave enough
B (Choice I)	$5M	$0	$1M	$1,390,000	Higher if u is less concave
A' (Choice II)	$1M	$1M	$0	$110,000	Higher if u(1M)/u(5M) > 10/11
B' (Choice II)	$5M	$0	$0	$500,000	Higher otherwise
Independence constraint	A ≽ B ⇔ A' ≽ B'			Cancels the common 89% column
Empirical reversal	A ≻ B & B' ≻ A' (in ~80% of subjects)			Violation

A Paris dinner and a 30-year fight

The paradox debuted at a May 1952 conference on risk in Paris organized by the Centre National de la Recherche Scientifique. Allais — a Paris-based engineer-economist — presented his counterexample to Leonard Savage, the leading EU theorist of the day. Savage, the story goes, picked A and B' himself, then admitted the inconsistency and changed one answer. Allais published the result in Econometrica in 1953 under the title "Le Comportement de l'Homme Rationnel devant le Risque."

The reaction was muted. Allais wrote in French and the American economics establishment dismissed it as a curiosity. Friedman and Savage's expected-utility framework was already dominant, and the Cold War's mathematical economics was confident in the vNM axioms. Allais kept publishing — in 1979 his Expected Utility Hypotheses and the Allais Paradox (with Hagen) was 700 pages of relentless attack — but recognition came slowly.

Kahneman and Tversky's 1979 prospect-theory paper rehabilitated the paradox by using it as exhibit A for descriptive failures of EU. Allais finally won the 1988 Nobel Prize in Economics, partly for the paradox. He was 77, and the Royal Swedish Academy specifically cited his "fundamental contributions to the theory of markets and efficient utilization of resources" — covert acknowledgment that he had been right.

The common consequence effect — Allais generalized

Allais's specific pair is one case of a broader pattern Kahneman and Tversky named the common consequence effect. Consider any preference between L₁ and L₂, both involving a probability p of receiving a common outcome c plus a probability 1−p of a different distribution. The independence axiom says the value of c is irrelevant to the comparison. Empirically, varying c — particularly making it certain (p=1) — flips preferences. Specifically:

• When c is the best outcome (here $1M, the modal payoff in pair I), shifting c to nothing flips you from the safer to the riskier alternative.
• When c is the worst outcome ($0 in pair II), shifting to $1M flips you toward the safer alternative.
• The reversal is sharpest when one alternative is degenerate (certain).

This is the fingerprint of probability weighting — people overweight the discontinuity at p = 1.

Variants and stress tests

• Conlisk (1989) — real money. Ran a scaled-down version with $20 payoffs; reversal rate held around 60% — slightly lower than thought experiments, still large enough to reject EU.
• Slovic and Tversky (1974) — education resistance. Showed subjects the EU argument; most still reversed. The paradox is not a numeracy failure.
• Common ratio effect. Multiplying all probabilities in both lotteries by a common factor (say 0.25) often reverses preferences — another independence violation, distinct from common consequence.
• Huck and Müller (2007) — large stakes. Even with explicit incentive-compatible procedures and several hundred euros at stake, the pattern persisted at 50–70% of subjects.
• Cumulative prospect theory (Tversky-Kahneman 1992). The rank-dependent probability weighting w(F) reproduces the reversal while preserving stochastic dominance.
• Cabanac-Hagen experiments. Replicated the paradox with non-Western subjects to rule out cultural artifacts; reversal remained.

Real-world implications

• Insurance markets. The certainty effect explains why low-deductible insurance — paying high premiums for full coverage — sells robustly even when expected-value analysis says it's expensive. People overweight the certainty of zero-loss outcomes.
• Lottery sales. The complementary effect: people pay over-EV prices for tiny chances of huge payoffs because rare events are overweighted. US lottery sales: $113 billion/year (2024).
• Equity premium puzzle. If investors overweight certain (bond-like) payoffs vs probabilistic (stock-like) ones, they demand a higher premium for stocks than EU predicts — partially closing the Mehra-Prescott gap.
• Plea bargaining. Defendants accept certain shorter sentences in plea deals at rates higher than EU predicts; the certainty effect biases away from risky trial outcomes.
• Drug regulation. Approval criteria favoring known small risks over unknown large benefits reflect a regulator-side certainty bias.
• Climate and policy uncertainty. Choices between certain modest losses (mitigation cost) and probabilistic catastrophic ones (unmitigated warming) systematically violate EU rankings in the Allais direction.

Common pitfalls in interpreting the paradox

• Treating it as a maths error. Even after explanation, most people pick A and B' again. It is a stable preference, not a calculation slip.
• Conflating with risk aversion. Risk aversion (concave u) is consistent with EU; the Allais reversal is a deeper violation of independence, not of monotonicity or concavity.
• Assuming the paradox dissolves at small stakes. Conlisk and others have shown the reversal persists at $20-scale payoffs, though sometimes weaker.
• Mistaking Allais for ambiguity aversion. Allais uses known probabilities throughout; Ellsberg's paradox is the ambiguity-aversion case. Different axioms violated (independence vs sure-thing principle in Savage's framework).
• Forgetting Allais's normative claim. Allais argued the axiom itself was wrong, not that humans are irrational. Whether the "rational" choice is A or B is contested.