Ellsberg’s Urns, explained.
Ellsberg's paradox explores ambiguity aversion: people may prefer bets with known probabilities over bets whose probabilities are unspecified.
Why it happens
Risk with known odds and uncertainty about the odds are different information states. Assigning an unknown probability its midpoint is a modeling choice, not knowledge that the midpoint is correct. Preferences can depend on that distinction.
Ellsberg used urn choices to study ambiguity: a decision maker may treat unknown probabilities differently from known ones.
Read the result
Compare the known urn with the full range of outcomes consistent with the ambiguous urn. Treat the displayed hidden composition as one illustrative possibility, not information available when you made the initial choice.
A worked example
Two urns, different information
One urn is known to contain equal numbers of red and black balls. Another has the same total but an undisclosed color split.
A red-ball bet has a known one-half chance in the first urn; its chance in the second is not established by the total alone.
Preferring the first can reflect a response to missing probability information, rather than a difference in the stated reward.
OPTIONAL DEEPER DETAILGo deeper: inside the model
Inside this model
The known urn is 50% red. For the unknown urn, a red share is drawn once per session uniformly from [50−width/2, 50+width/2] percent; its value stays hidden until you play. Each urn pays 10 on red, 0 otherwise. The chart shows payoff under every possible unknown composition, not a claim about your preferences.
Where this idea is useful
A practical use
Compare a contract with known failure odds with one whose probability cannot be estimated confidently from available data.
A common misconception
“An unknown chance is automatically fifty-fifty.”
Equal plausibility of verbal descriptions does not identify the urn's composition. A midpoint requires an additional assumption.
What this explanation leaves out
- The random composition is a teaching device. Ellsberg's point concerns unknown beliefs and preference patterns, not a known uniform distribution over urn compositions.
How is ambiguity different from ordinary risk?
Ordinary risk often describes uncertainty about an outcome with a specified probability model. Ambiguity concerns uncertainty about which probabilities or model to use.
Are you uncertain about the outcome, the probability, or both?
Associated thinkers
Further reading
Explore the original research or the teaching reference behind this experiment.