## A non-expected glance at markets: financial models and Knightian uncertainty: Marcello Basili and Fulvio Fontini

Recent developments in decision theory have recalled the concept of Knightian uncertainty1 (Knight 1921) as an additional source of financial market failures. Knight's distinction between risk and uncertainty refers to the concept of the vagueness or ambiguity of probability: risk is a situation in which the relative odds of events are known; uncertainty is a condition in which probabilities of events are unknown and no unique assignment of them can be obtained. Consider the famous test of Savage's decision theory as illustrated by Ellsberg (1961). An urn contains 90 balls, 30 of which are red and the other 60 either blue or white. Agents are allowed to extract one ball only. Let Zj = [a if r, f3 if b, X if w] be a bet (or act), such that gives a if a red ball is drawn, f3 if it is blue and X if it is white. There are four possible bets (j = 1,2,3,4), that is Zl = [100 if r, 0 if b, 0 if w]; Z2 = [0 if r, 100 if b, 0 if w]; Z3 = [100 if r, 0 if b, 100 if w]; Z4 = [0 if r, 100 if b, 100 if w]. Agents are asked to choose between two pairs of lotteries, Zl and Z2, then Z3 and Z4. Most (more than 70 percent) have the following strict preferences Zl >- Z2 and Z4 >- Z3. This observed behavior leads to a contradiction since: Zl >- Z2 implies Pr > Ph, while Z4 >- Z3 implies Ph + Pw > Pr + Pw or Pr < Ph, where Pi, i = b, r, w denotes the probability of the event "a ball of color i is drawn."