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In decision theory, the Ellsberg paradox (or Ellsberg's paradox) is a paradox in which people's decisions are inconsistent with subjective expected utility theory. John Maynard Keynes published a version of the paradox in 1921. Daniel Ellsberg popularized the paradox in his 1961 paper, "Risk, Ambiguity, and the Savage Axioms". It is generally taken to be evidence of ambiguity aversion, in which a person tends to prefer choices with quantifiable risks over those with unknown, incalculable risks. Ellsberg's findings indicate that choices with an underlying level of risk are favored in instances where the likelihood of risk is clear, rather than instances in which the likelihood of risk is unknown. A decision-maker will overwhelmingly favor a choice with a transparent likelihood of risk, even in instances where the unknown alternative will likely produce greater utility. When offered choices with varying risk, people prefer choices with calculable risk, even when those choices have less utility. Experimental research Ellsberg's experimental research involved two separate thought experiments: the 2-urn 2-color scenario and the 1-urn 3-color scenario. Two-urns paradox There are two urns each containing 100 balls. It is known that urn A contains 50 red and 50 black, but urn B contains an unknown mix of red and black balls. The following bets are offered to a participant: Bet 1A: get $1 if red is drawn from urn A, $0 otherwise Bet 2A: get $1 if black is drawn from urn A, $0 otherwise Bet 1B: get $1 if red is drawn from urn B, $0 otherwise Bet 2B: get $1 if black is drawn from urn B, $0 otherwise Typically, participants were seen to be indifferent between bet 1A and bet 2A (consistent with expected utility theory) but were seen to strictly prefer Bet 1A to Bet 1B and Bet 2A to 2B. This result is generally interpreted to be a consequence of ambiguity aversion (also known as uncertainty aversion); people intrinsically dislike situations where they cannot attach probabilities to outcomes, in this case favoring the bet in which they know the probability and utility outcome (0.5 and $1 respectively). One-urn paradox There is one urn containing 90 balls: 30 balls are red, while the remaining 60 balls are either black or yellow in unknown proportions. The balls are well mixed so that each ball is as likely to be drawn as any other. The participants then choose a gambling scenario: Additionally, the participant may choose a separate gamble scenario within the same situational parameters: The experimental conditions manufactured by Ellsberg serve to rely upon two economic principles: Knightian uncertainty, the unquantifiable nature of the mix between both yellow and black balls within the single urn, and probability, of which red balls are drawn at 1/3 vs. 2/3. Utility theory interpretation Utility theory models the choice by assuming that in choosing between these gambles, people assume a probability that the non-red balls are yellow versus black, and then compute the expected utility of the two gambles individually. Since the prizes are the same, it follows that the participant will strictly prefer Gamble A to Gamble B if and only if they believe that drawing a red ball is more likely than drawing a black ball (according to expected utility theory). Also, there would be indifference between the choices if the participant thought that a red ball was as likely as a black ball. Similarly, it follows the participant will strictly prefer Gamble C to Gamble D if and only if the participant believes that drawing a red or yellow ball is more likely than drawing a black or yellow ball. It might seem intuitive that if drawing a red ball is more likely than drawing a black ball, drawing a red or yellow ball is also more likely than drawing a black or yellow ball. So, supposing the participant strictly prefers Gamble A to Gamble B, it follows that he/she will also strictly prefer Gamble C to Gamble D, and similarly conversely. However, ambiguity aversion would predict that people would strictly prefer Gamble A to Gamble B, and Gamble D to Gamble C. Ellsberg's findings violate assumptions made within common Expected Utility Theory, with participants strictly preferring Gamble A to Gamble B and Gamble D to Gamble C. Numerical demonstration Mathematically, the estimated probabilities of each color ball can be represented as R, Y, and B. If the participant strictly prefers Gamble A to Gamble B, by utility theory, it is presumed this preference is reflected by the expected utilities of the two gambles. We reach a contradiction in our utility calculations. This contradiction indicates that the participant's preferences are inconsistent with the expected-utility theory. The generality of the paradox The result holds regardless of the utility function. Indeed, the amount of the payoff is likewise irrelevant. Whichever gamble is selected, the prize for winning it is the same, and the cost of losing it is the same (no cost), so ultimately there are only two outcomes: receive a specific amount of money or nothing. Therefore, it is sufficient to assume that the preference is to receive some money to nothing (this assumption is not necessary: in the mathematical treatment above, it was assumed U($100) > U($0), but a contradiction can still be obtained for U($100) < U($0) and for U($100) = U($0)). In addition, the result holds regardless of risk aversion—all gambles involve risk. By choosing Gamble D, the participant has a 1 in 3 chance of receiving nothing, and by choosing Gamble A, a 2 in 3 chance of receiving nothing. If Gamble A was less risky than Gamble B, it would follow that Gamble C was less risky than Gamble D (and vice versa), so the risk is not averted in this way. However, because the exact chances of winning are known for Gambles A and D and not known for Gambles B and C, this can be taken as evidence for some sort of ambiguity aversion, which cannot be accounted for in expected utility theory. It has been demonstrated that this phenomenon occurs only when the choice set permits the comparison of the ambiguous proposition with a less vague proposition (but not when ambiguous propositions are evaluated in isolation). Possible explanations There have been various attempts to provide decision-theoretic explanations of Ellsberg's observation. Since the probabilistic information available to the decision-maker is incomplete, these attempts sometimes focus on quantifying the non-probabilistic ambiguity that the decision-maker faces – see Knightian uncertainty. That is, these alternative approaches sometimes suppose that the agent formulates a subjective (though not necessarily Bayesian) probability for possible outcomes. One such attempt is based on info-gap decision theory. The agent is told precise probabilities of some outcomes, though the practical meaning of the probability numbers is not entirely clear. For instance, in the gambles discussed above, the probability of .... Discover the Yasuhiro Sakai popular books. Find the top 100 most popular Yasuhiro Sakai books.