ECON30019 Chap.14 Allais, Probability Weighting and Ambiguity Aversion
Allais, Probability Weighting and Ambiguity Aversion
The risk lectures name four topics in advance without developing them, so this chapter builds them from the standard published theory on the apparatus the subject does teach.
The independence axiom says that mixing two ranked lotteries with the same third lottery preserves the ranking, and because the representation theorem runs both ways, a choice pattern violating it cannot be represented by any expected-utility function at all. Two classical patterns do exactly that.
The common-consequence effect swaps a shared component out of both options and observes a reversal; the common-ratio effect scales both probabilities by the same factor and observes another. The usual explanation is the certainty effect, that a step onto certainty is worth more than an equal step below it, which motivates replacing probabilities with decision weights.
A separate strand concerns choices where the probabilities themselves are unknown.
What this chapter covers
- 01
Why these topics are listed here, and what they are built from
- 02
The independence axiom recalled, and why the theorem makes a violation fatal
- 03
The common-consequence pattern, and the swap that should have been invisible
- 04
The common-ratio pattern, scaling both probabilities together
- 05
Writing the two choices as inequalities that contradict each other
- 06
The certainty effect as the shared explanation
- 07
Decision weights, and the objective they sit inside
- 08
The inverse-S shape: small probabilities overweighted, large ones underweighted
- 09
The fourfold pattern of risk attitudes, produced by one curve
- 10
Ambiguity aversion, and why it cannot be a belief about the odds
Demonstrate the common-ratio violation
- 2Write the first choice as an inequality: the utility of 3,000 exceeds 0.8 times the utility of 4,000, since the gamble's other branch contributes nothing.
- 1Write the second choice as an inequality: 0.20 times the utility of 4,000 exceeds 0.25 times the utility of 3,000.
- 2Scale the first inequality by 0.25, which preserves its direction: 0.25 times the utility of 3,000 exceeds 0.20 times the utility of 4,000.
- 1Compare: that is exactly the reverse of the second inequality, so the two choices cannot both hold for any utility function, and the usual explanation is the certainty effect.
Key terms
- Common consequence
- A component shared by two options that is swapped for another shared component in both at once. Independence requires the ranking to survive the swap.
- Common ratio
- A pair of comparisons in which every probability is scaled by the same factor, so the ratio between them is unchanged and the ranking must be unchanged too.
- Certainty effect
- The disproportionate weight given to outcomes that are certain relative to outcomes that are merely very likely, which explains both classical reversal patterns.
- Decision weight
- A function of a stated probability used in place of the probability itself, allowing the model to treat a step onto certainty differently from an equal step elsewhere.
- Fourfold pattern
- The combination of risk seeking over small-probability gains, risk aversion over large-probability gains, risk aversion over small-probability losses and risk seeking over large-probability losses.
- Ambiguity aversion
- A preference for a known distribution over an unknown one that cannot be explained by any single set of beliefs about the unknown odds.
Allais, Probability Weighting and Ambiguity Aversion FAQ
Why is a violation of independence worse than an ordinary anomaly?
Because the representation theorem is an equivalence. If a choice pattern violates independence, then no utility function whatsoever can represent it, so the failure cannot be repaired by attributing an unusual risk attitude to the person. That is what makes these patterns a refutation of the model rather than a puzzle about a particular chooser.
How does one weighting function produce four different risk attitudes?
By combining with the value function. Small probabilities are overweighted and large ones underweighted, so a small chance of a large gain is inflated, producing risk seeking, while a large chance of a gain is deflated, producing risk aversion. The same two effects on the loss side reverse the labels, and the result is that buying insurance and buying a lottery ticket are consistent.
Why does the two-urn preference rule out any belief about the odds?
Because it appears whichever colour is named. Preferring the known container when you name the first colour is consistent with thinking that colour rare in the unknown container; preferring it again when you name the second requires thinking the other colour rare too. Since the two must sum to one, no single set of beliefs delivers both.
Exam move
Treat this chapter as an algebra chapter rather than a paradox chapter. The examinable skill is producing two inequalities that contradict each other, so practise the normalisation, the scaling and the comparison until the three lines are automatic. Learn the two directions of the weighting function as a signed statement rather than as a shape, since a reversed direction turns the fourfold pattern inside out.
Finally, get into the habit of classifying a problem as risk or as ambiguity in your first sentence, because that classification decides which model applies.
Working through Allais, Probability Weighting and Ambiguity Aversion in ECON30019? Sia is AskSia’s AI Economics tutor — ask any ECON30019 Allais, Probability Weighting and Ambiguity Aversion question and get a clear, step-by-step explanation grounded in how ECON30019 is taught and assessed. Read this chapter free, then take your hardest questions to Sia.
