Ellsberg paradox
Encyclopedia
The Ellsberg paradox is a paradox
in decision theory
and experimental economics
in which people's choices violate the expected utility hypothesis
.
An alternate viewpoint is that expected utility theory does not properly describe actual human choices. Likewise, utility theory clashes with the Saint Petersburg Paradox which better explains how humans make their choices.
It is generally taken to be evidence for ambiguity aversion
. The paradox was popularized by Daniel Ellsberg
, although a version of it was noted considerably earlier by John Maynard Keynes
.
Ellsberg raised two problems: 1 urn problem and 2 urn problem. Here, 1 urn problem is described, which is the better known one.
containing 30 red balls and 60 other balls that are either black or yellow. You don't know how many black or how many yellow balls there are, but that the total number of black balls plus the total number of yellow equals 60. The balls are well mixed so that each individual ball is as likely to be drawn as any other. You are now given a choice between two gambles:
Also you are given the choice between these two gambles (about a different draw from the same urn):
This situation poses both Knightian uncertainty
– whether the non-red balls are all yellow or all black, which is not quantified – and probability – whether the ball is red or non-red, which is ⅓ vs. ⅔.
Since the prizes are exactly the same, it follows that you will prefer Gamble A to Gamble B if and only if
you believe that drawing a red ball is more likely than drawing a black ball (according to expected utility theory). Also, there would be no clear preference between the choices if you thought that a red ball was as likely as a black ball. Similarly it follows that you will prefer Gamble C to Gamble D if, and only if, you believe 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, then drawing a red or yellow ball is also more likely than drawing a black or yellow ball. So, supposing you prefer Gamble A to Gamble B, it follows that you will also prefer Gamble C to Gamble D. And, supposing instead that you prefer Gamble B to Gamble A, it follows that you will also prefer Gamble D to Gamble C.
When surveyed, however, most people strictly prefer Gamble A to Gamble B and Gamble D to Gamble C. Therefore, some assumptions of the expected utility theory are violated.
where
is your utility function. If 
(you strictly prefer $100 to nothing), this simplifies to:
If you also strictly prefer Gamble D to Gamble C, the following inequality is similarly obtained:
This simplifies to:
This contradiction indicates that your preferences are inconsistent with expected-utility theory.
In addition, the result holds regardless of your risk aversion
. All the gambles involve risk. By choosing Gamble D, you have a 1 in 3 chance of receiving nothing, and by choosing Gamble A, you have 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, 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 comparison of the ambiguous proposition with a less vague proposition (but not when ambiguous propositions are evaluated in isolation).
. 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 a red ball is 30/90, which is a precise number. Nonetheless, the agent may not distinguish, intuitively, between this and, say, 30/91. No probability information whatsoever is provided regarding other outcomes, so the agent has very unclear subjective impressions of these probabilities.
In light of the ambiguity in the probabilities of the outcomes, the agent is unable to evaluate a precise expected utility. Consequently, a choice based on maximizing the expected utility is also impossible. The info-gap approach supposes that the agent implicitly formulates info-gap models for the subjectively uncertain probabilities. The agent then tries to satisfice
the expected utility and to maximize the robustness against uncertainty in the imprecise probabilities. This robust-satisficing approach can be developed explicitly to show that the choices of decision-makers should display precisely the preference reversal which Ellsberg observed.
Another possible explanation is that this type of game triggers a deceit aversion mechanism. Many humans naturally assume in real-world situations that if they are not told the probability of a certain event, it is to deceive them. People make the same decisions in the experiment
that they would about related but not identical real-life problems where the experimenter would be likely to be a deceiver acting against the subject's interests. When faced with the choice between a red ball and a black ball, the probability of 30/90 is compared to the lower part of the 0/90-60/90 range (the probability of getting a black ball). The average person expects there to be fewer black balls than yellow balls because in most real-world situations, it would be to the advantage of the experimenter to put fewer black balls in the urn when offering such a gamble. On the other hand, when offered a choice between red and yellow balls and black and yellow balls, people assume that there must be fewer than 30 yellow balls as would be necessary to deceive them. When making the decision, it is quite possible that people simply forget to consider that the experimenter does not have a chance to modify the contents of the urn in between the draws. In real-life situations, even if the urn is not to be modified, people would be afraid of being deceived on that front as well.
A modification of utility theory to incorporate uncertainty as distinct from risk is Choquet expected utility, which also proposes a solution to the paradox.
Paradox
Similar to Circular reasoning, A paradox is a seemingly true statement or group of statements that lead to a contradiction or a situation which seems to defy logic or intuition...
in decision theory
Decision theory
Decision theory in economics, psychology, philosophy, mathematics, and statistics is concerned with identifying the values, uncertainties and other issues relevant in a given decision, its rationality, and the resulting optimal decision...
and experimental economics
Experimental economics
Experimental economics is the application of experimental methods to study economic questions. Data collected in experiments are used to estimate effect size, test the validity of economic theories, and illuminate market mechanisms. Economic experiments usually use cash to motivate subjects, in...
in which people's choices violate the expected utility hypothesis
Expected utility hypothesis
In economics, game theory, and decision theory the expected utility hypothesis is a theory of utility in which "betting preferences" of people with regard to uncertain outcomes are represented by a function of the payouts , the probabilities of occurrence, risk aversion, and the different utility...
.
An alternate viewpoint is that expected utility theory does not properly describe actual human choices. Likewise, utility theory clashes with the Saint Petersburg Paradox which better explains how humans make their choices.
It is generally taken to be evidence for ambiguity aversion
Ambiguity aversion
In decision theory and economics, ambiguity aversion describes an attitude of preference for known risks over unknown risks. People would rather choose an option with fewer unknown elements than with many unknown elements. It is demonstrated in the Ellsberg paradox In decision theory and...
. The paradox was popularized by Daniel Ellsberg
Daniel Ellsberg
Daniel Ellsberg, PhD, is a former United States military analyst who, while employed by the RAND Corporation, precipitated a national political controversy in 1971 when he released the Pentagon Papers, a top-secret Pentagon study of U.S. government decision-making in relation to the Vietnam War,...
, although a version of it was noted considerably earlier by John Maynard Keynes
John Maynard Keynes
John Maynard Keynes, Baron Keynes of Tilton, CB FBA , was a British economist whose ideas have profoundly affected the theory and practice of modern macroeconomics, as well as the economic policies of governments...
.
Ellsberg raised two problems: 1 urn problem and 2 urn problem. Here, 1 urn problem is described, which is the better known one.
The 1 urn paradox
Suppose you have an urnUrn problem
In probability and statistics, an urn problem is an idealized mental exercise in which some objects of real interest are represented as colored balls in an urn or other container....
containing 30 red balls and 60 other balls that are either black or yellow. You don't know how many black or how many yellow balls there are, but that the total number of black balls plus the total number of yellow equals 60. The balls are well mixed so that each individual ball is as likely to be drawn as any other. You are now given a choice between two gambles:
| Gamble A | Gamble B |
|---|---|
| You receive $100 if you draw a red ball | You receive $100 if you draw a black ball |
Also you are given the choice between these two gambles (about a different draw from the same urn):
| Gamble C | Gamble D |
|---|---|
| You receive $100 if you draw a red or yellow ball | You receive $100 if you draw a black or yellow ball |
This situation poses both Knightian uncertainty
Knightian uncertainty
In economics, Knightian uncertainty is risk that is immeasurable, not possible to calculate.Knightian uncertainty is named after University of Chicago economist Frank Knight , who distinguished risk and uncertainty in his work Risk, Uncertainty, and Profit:- Common-cause and special-cause :The...
– whether the non-red balls are all yellow or all black, which is not quantified – and probability – whether the ball is red or non-red, which is ⅓ vs. ⅔.
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.Since the prizes are exactly the same, it follows that you will prefer Gamble A to Gamble B if and only if
If and only if
In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
you believe that drawing a red ball is more likely than drawing a black ball (according to expected utility theory). Also, there would be no clear preference between the choices if you thought that a red ball was as likely as a black ball. Similarly it follows that you will prefer Gamble C to Gamble D if, and only if, you believe 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, then drawing a red or yellow ball is also more likely than drawing a black or yellow ball. So, supposing you prefer Gamble A to Gamble B, it follows that you will also prefer Gamble C to Gamble D. And, supposing instead that you prefer Gamble B to Gamble A, it follows that you will also prefer Gamble D to Gamble C.
When surveyed, however, most people strictly prefer Gamble A to Gamble B and Gamble D to Gamble C. Therefore, some assumptions of the expected utility theory are violated.
Mathematical demonstration
Mathematically, your estimated probabilities of each color ball can be represented as: R, Y, and B. If you strictly prefer Gamble A to Gamble B, by utility theory, it is presumed this preference is reflected by the expected utilities of the two gambles: specifically, it must be the case that
where
is your utility function. If (you strictly prefer $100 to nothing), this simplifies to:
If you also strictly prefer Gamble D to Gamble C, the following inequality is similarly obtained:
This simplifies to:
This contradiction indicates that your preferences are inconsistent with expected-utility theory.
Generality of the paradox
Note that the result holds regardless of your utility function. Indeed, the amount of the payoff is likewise irrelevant. Whichever gamble you choose, 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: you receive a specific amount of money, or you receive nothing. Therefore it is sufficient to assume that you prefer receiving some money to receiving nothing (and in fact, 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 your risk aversion
Risk aversion
Risk aversion is a concept in psychology, economics, and finance, based on the behavior of humans while exposed to uncertainty....
. All the gambles involve risk. By choosing Gamble D, you have a 1 in 3 chance of receiving nothing, and by choosing Gamble A, you have 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, 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
Ambiguity aversion
In decision theory and economics, ambiguity aversion describes an attitude of preference for known risks over unknown risks. People would rather choose an option with fewer unknown elements than with many unknown elements. It is demonstrated in the Ellsberg paradox In decision theory and...
which cannot be accounted for in expected utility theory. It has been demonstrated that this phenomenon occurs only when the choice set permits 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 which the decision-maker faces – see Knightian uncertaintyKnightian uncertainty
In economics, Knightian uncertainty is risk that is immeasurable, not possible to calculate.Knightian uncertainty is named after University of Chicago economist Frank Knight , who distinguished risk and uncertainty in his work Risk, Uncertainty, and Profit:- Common-cause and special-cause :The...
. That is, these alternative approaches sometimes suppose that the agent formulates a subjective (though not necessarily Bayesian
Bayesian probability
Bayesian probability is one of the different interpretations of the concept of probability and belongs to the category of evidential probabilities. The Bayesian interpretation of probability can be seen as an extension of logic that enables reasoning with propositions, whose truth or falsity is...
) probability for possible outcomes.
One such attempt is based on info-gap decision theory
Info-gap decision theory
Info-gap decision theory is a non-probabilistic decision theory that seeks to optimize robustness to failure – or opportuneness for windfall – under severe uncertainty, in particular applying sensitivity analysis of the stability radius type to perturbations in the value of a given estimate of the...
. 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 a red ball is 30/90, which is a precise number. Nonetheless, the agent may not distinguish, intuitively, between this and, say, 30/91. No probability information whatsoever is provided regarding other outcomes, so the agent has very unclear subjective impressions of these probabilities.
In light of the ambiguity in the probabilities of the outcomes, the agent is unable to evaluate a precise expected utility. Consequently, a choice based on maximizing the expected utility is also impossible. The info-gap approach supposes that the agent implicitly formulates info-gap models for the subjectively uncertain probabilities. The agent then tries to satisfice
Satisficing
Satisficing, a portmanteau "combining satisfy with suffice", is a decision-making strategy that attempts to meet criteria for adequacy, rather than to identify an optimal solution...
the expected utility and to maximize the robustness against uncertainty in the imprecise probabilities. This robust-satisficing approach can be developed explicitly to show that the choices of decision-makers should display precisely the preference reversal which Ellsberg observed.
Another possible explanation is that this type of game triggers a deceit aversion mechanism. Many humans naturally assume in real-world situations that if they are not told the probability of a certain event, it is to deceive them. People make the same decisions in the experiment
Experiment
An experiment is a methodical procedure carried out with the goal of verifying, falsifying, or establishing the validity of a hypothesis. Experiments vary greatly in their goal and scale, but always rely on repeatable procedure and logical analysis of the results...
that they would about related but not identical real-life problems where the experimenter would be likely to be a deceiver acting against the subject's interests. When faced with the choice between a red ball and a black ball, the probability of 30/90 is compared to the lower part of the 0/90-60/90 range (the probability of getting a black ball). The average person expects there to be fewer black balls than yellow balls because in most real-world situations, it would be to the advantage of the experimenter to put fewer black balls in the urn when offering such a gamble. On the other hand, when offered a choice between red and yellow balls and black and yellow balls, people assume that there must be fewer than 30 yellow balls as would be necessary to deceive them. When making the decision, it is quite possible that people simply forget to consider that the experimenter does not have a chance to modify the contents of the urn in between the draws. In real-life situations, even if the urn is not to be modified, people would be afraid of being deceived on that front as well.
A modification of utility theory to incorporate uncertainty as distinct from risk is Choquet expected utility, which also proposes a solution to the paradox.
See also
- Allais paradoxAllais paradoxThe Allais paradox is a choice problem designed by Maurice Allais to show an inconsistency of actual observed choices with the predictions of expected utility theory.-Statement of the Problem:...
- Experimental economicsExperimental economicsExperimental economics is the application of experimental methods to study economic questions. Data collected in experiments are used to estimate effect size, test the validity of economic theories, and illuminate market mechanisms. Economic experiments usually use cash to motivate subjects, in...
- Subjective expected utilitySubjective expected utilitySubjective expected utility is a method in decision theory in the presence of risk, promoted by L. J. Savage in 1954 following previous work by Ramsey and von Neumann...
- Utility theory
- Ambiguity aversionAmbiguity aversionIn decision theory and economics, ambiguity aversion describes an attitude of preference for known risks over unknown risks. People would rather choose an option with fewer unknown elements than with many unknown elements. It is demonstrated in the Ellsberg paradox In decision theory and...