Category: Effects
Type: Preferring a known chance to an unknown one
Origin: Frank H. Knight, Risk, Uncertainty and Profit (1921); Daniel Ellsberg, Quarterly Journal of Economics (1961)
Also known as: Uncertainty aversion; an Ellsberg preference
Type: Preferring a known chance to an unknown one
Origin: Frank H. Knight, Risk, Uncertainty and Profit (1921); Daniel Ellsberg, Quarterly Journal of Economics (1961)
Also known as: Uncertainty aversion; an Ellsberg preference
Quick Answer — Ambiguity Aversion is a preference for a prize whose chance is known over the same prize whose chance is not known. Frank Knight separated measurable risk from unmeasurable uncertainty in 1921, and Daniel Ellsberg set the choice pattern out with urns in 1961. In a weighted American Life Panel table of 3,040 people, 52 percent chose the known box on the first question, 10 percent were indifferent, and 38 percent chose the unknown box. The average score sat 1.8 percentage points away from a matching probability of 50 percent.
What is Ambiguity Aversion?
Ambiguity Aversion is a preference for an option whose probabilities are known over an option with the same prizes whose probabilities are not known. A weather app that says “50 percent” and a forecast that says “it might rain,” for the same afternoon and the same coat, are the everyday picture. The missing number is the whole difference. The unnumbered forecast has not been shown to be wetter. Two jars on a kitchen table do the same work. One card reads “50 red, 50 black.” The other reads “100 red and black, mix unstated.” The prize for a red bean is the same jar to jar.What is at issue might be called the ambiguity of this information, a quality depending on the amount, type, reliability and “unanimity” of information, and giving rise to one’s degree of “confidence” in an estimate of relative likelihoods.That is Daniel Ellsberg, writing in 1961. The chance and the prize are the first two parts of a gamble. Ambiguity is a third part: how solid the information is behind the chance. Prospect Theory describes how people weigh gains and losses once the chances are already numbers. Loss Aversion is the extra weight of a loss inside that weighing. Ambiguity Aversion appears when one of the chances has not been numbered.
Ambiguity Aversion in 3 Depths
- Beginner: Two jars offer the same prize if the bean matches the color you name. One jar is 50 red and 50 black. The other is 100 red and black, mix unstated. Write which jar you take for red, and which jar you take for black.
- Practitioner: Price the known chance and the unknown chance on one page, then price the unknown chance on a page by itself. In Craig R. Fox and Amos Tversky’s 1995 Study 1, the side-by-side means for a hypothetical 24.34 and 17.94 and $18.42.
- Advanced: A first-round tilt toward the known box can sit beside a mean matching probability near one half. In the 2012 panel, 52 percent chose the known box, and the mean of “50 percent minus the matching probability” was 0.018. One standard deviation of the standardized score lines up with 2.0 percentage points of stock-market participation in a weighted regression. The survey did not assign the preference.
Origin
Ambiguity Aversion takes its name from Ellsberg’s 1961 account of choices that refuse a single probability. The split between a known chance and an unknown one is older. Frank H. Knight, Risk, Uncertainty and Profit (Boston: Houghton Mifflin, 1921), is the book Ellsberg cites for that split. Ellsberg summarizes Knight’s “measurable uncertainty,” or risk, as a chance that can be written as a number, and Knight’s “unmeasurable uncertainty” as a chance that cannot. Ellsberg quotes Knight, page 219, on a man who knows an urn holds red and black balls and does not know the numbers. Knight’s own line is that, if a wager is required, the man “would have to act on the supposition that the chances are equal.” Ellsberg says the choices he observed go against that supposition. On page 227 Knight writes that the action which follows an opinion depends as much on confidence in the opinion as on how favorable the opinion is. Ellsberg’s first footnote also prints Kenneth Arrow’s 1951 doubt, from Econometrica, volume 19 (October 1951), pages 417 and 426: Knight’s uncertainties seemed to Arrow to share many properties of ordinary probabilities.I propose to indicate a class of choice-situations in which many otherwise reasonable people neither wish nor tend to conform to the Savage postulates, nor to the other axiom sets that have been devised.Ellsberg, “Risk, Ambiguity, and the Savage Axioms,” Quarterly Journal of Economics, volume 75, number 4 (November 1961), pages 643–669, is the paper. The footnote says the research was done while he was a member of the Society of Fellows at Harvard, in 1957. A version was delivered at the Econometric Society meetings in St. Louis in December 1960. The axioms are those of L. J. Savage, The Foundations of Statistics (New York: Wiley, 1954). Ellsberg names Savage’s Postulate 2 as the sure-thing principle: if yellow pays the same amount in both options, adding yellow should leave the ranking of the other colors as it was. The first comparison uses two urns. Urn I holds 100 red and black balls in a ratio entirely unknown, from 0 red to 100 red. Urn II holds exactly 50 red and 50 black. A bet on a color pays a prize a, and Ellsberg says “say 0,” if it is not. He asks which color you prefer inside each urn, then which urn you prefer for red, and which urn you prefer for black. He calls indifference inside each urn the typical response. Judging from a large number of responses, under absolutely nonexperimental conditions, he says a majority prefers Red II to Red I and Black II to Black I. A small minority prefers the unknown urn for both colors. Either pair refuses a single probability. Preferring the known urn for red would mean red is more likely there. Preferring it for black as well would mean black is more likely there too. The paper prints no sample size and no percentage. Ellsberg adds that a sample of two balls, one red and one black, or of four balls, two and two, does not seem to change the pattern appreciably. Willingness to pay for a draw from Urn I goes down somewhat and, in his words, remains well below willingness to pay for Urn II. He prints no dollar amount for that gap. The second comparison is one urn: 30 red balls, and 60 black and yellow balls in an unknown split. One draw pays as follows. Bet I pays 0 otherwise. Bet II pays 0 otherwise. Bet III pays 100 on black or yellow. Ellsberg calls a very frequent pattern “I preferred to II, and IV preferred to III.” A less frequent pattern runs the other way. Yellow pays $100 in both III and IV, so the sure-thing principle asks the ranking of I against II to survive in III against IV. Both frequent patterns break that ask. Again he prints no percentage. He also says the pattern is not minimax, not a Hurwicz average of the worst and best payoff, and not minimax regret. Each bet in a pair has the same worst payoff and the same best payoff. Ellsberg records that responses differ after people look at the axioms. G. Debreu, R. Schlaifer, and P. Samuelson, in his account, do not violate them, and tend to use the principle of insufficient reason when in doubt. J. Marschak and N. Dalkey violate them cheerfully. H. Raiffa tends to violate them and feels guilty. A number of sophisticated people, Ellsberg writes, decide that they wish to persist. That group included L. J. Savage when Ellsberg last tested him. Ellsberg adds that he has been reluctant to try Savage again. These are named illustrations. They are not a counted sample. On the page numbered 657, Ellsberg defines high ambiguity as very low confidence in one probability distribution, set against other reasonable distributions. He says ambiguity is a subjective variable. Situations likely to produce it, in his list, have scanty, unreliable, or conflicting information, wide disagreement among people, or low stated confidence. Later portfolio theories use several formal models of this preference. Stephen G. Dimmock, Roy Kouwenberg, Olivia S. Mitchell, and Kim Peijnenburg, in the 2016 paper below, name the multiple-prior model of Gilboa and Schmeidler (1989) among them, and they say their own matching probability does not require a choice among those models.
Key Points
Four comparisons keep the prize fixed and move the information. A known chance is not a dislike of variance. A pair of colors can refuse every probability. Adding a color that pays the same on both sides can flip the ranking. A gap that appears when both chances are on the page can close when each chance is priced alone.1
Hold the prize fixed and vary the information
Risk, in Knight’s sense as Ellsberg reports it, is a chance already written as a number. A sure 100 or 15. The known box changed its posted share of purple balls. The unknown box did not. Dimmock, Kouwenberg, Mitchell, and Peijnenburg, Journal of Financial Economics, volume 119, number 3 (March 2016), pages 559–577, call the posted share at which a person switches the matching probability. They define Ambiguity Aversion as 50 percent minus that share. A matching probability of 40 percent means the person treats the unknown box as worth a 40 percent known chance of the same $15. Overconfidence Effect compares a stated chance with later hits. The matching probability is a statement about the unknown box before any ball is drawn.
2
Ask both colors before you name a probability
In Ellsberg’s two-urn comparison, the majority pattern he describes prefers Urn II for red and Urn II for black. One probability for Urn I cannot sit below one half for red and below one half for black. The practical test is the same pair of questions. A supplier with a published on-time rate, and a new supplier with no rate, are not yet a probability judgment if you would also reject the new supplier when the question is about the other outcome. Ellsberg prints no share for the majority. The 0 are the stakes he offers as an example, in the words “say 0.”
3
Add a column that pays the same on both sides
In the 30-and-60 urn, yellow pays $100 in bet III and in bet IV. The sure-thing principle says the person who preferred red to black should still prefer red-or-yellow to black-or-yellow. The frequent pattern Ellsberg reports prefers red, the known 30 out of 90, and then prefers black-or-yellow, the known 60 out of 90. The known total is attractive on both questions, and the ranking flips. A committee that adds the same bonus to both vendors, and then reverses which vendor it likes, has reproduced the shape of that flip. The paper gives no count of how often the flip occurred.
4
Price the unknown chance once beside the known one, and once alone
Fox and Tversky, “Ambiguity Aversion and Comparative Ignorance,” Quarterly Journal of Economics, volume 110, number 3 (August 1995), pages 585–603, argue that the gap is produced by a comparison. Study 1 gave 141 Stanford undergraduates a hypothetical problem, for class credit. Bag A held 50 red poker chips and 50 black. Bag B held 100 red and black chips in an unknown proportion. The person guessed a color, then drew a chip, and a match paid a hypothetical 24.34 (standard error 2.21), vague bet 9.51 more for the clear bet, t(66) = 6.00, p < .001. The printed means differ by 17.94 (standard error 2.50, N = 35), vague bet $18.42 (standard error 2.87, N = 39), t(72) = −0.12. The interaction is z = 2.42, p < .01. The 67, the 35, and the 39 sum to 141. These are stated prices for a hypothetical ticket. They are not cash paid for a chip.
Applications
Use the known chance as a line you can read. Keep each dollar and each share inside the question that produced it.Write both colors before you reject the unknown mix
For a course with a published grade distribution and a new course with none, or for a supplier with an on-time rate and a supplier without one, write the choice for one outcome and the choice for the other. Ellsberg’s majority pattern rejects the unknown urn for red and for black. That pair is the signal that no single probability fits. He prints no percentage for the pair.
Price the unfamiliar option on a second page
Fox and Tversky’s Study 1 states a 100 ticket, after a guess of color.
Separate a missing rate from a dislike of variance
A coin at known 50 percent, set against a sure amount, is a risk question. The panel’s matching probability sets a known share against an unknown mix and holds the $15 prize fixed. In the 2016 regressions, foreign shares and employer shares move in opposite directions as the ambiguity score rises. A single dislike of ups and downs does not predict both signs. The shares are associations, spelled out in the case below.
Read a side-by-side premium as side-by-side
Study 4 asked 189 pedestrians at Berkeley, paid with a California lottery ticket for a five-minute survey, what they would pay for hypothetical 22.74 in San Francisco against 17.79 against 15.84 higher for San Francisco, t(89) = 5.05, p < .001. Alone, at least 60 degrees was 21.07, t(87) = 0.17, and under 60 degrees was 17.29, t(87) = 0.13. The paper says the sum gap is $1.52, t(87) = 0.19. People pay more to bet on the familiar city when the unfamiliar city is on the same page.
Case Study
In mid-March 2012 the RAND American Life Panel opened a module that closed in mid-April. The panel is an internet sample of U.S. households. Households without internet at recruitment were given a laptop and wireless service. Dimmock, Kouwenberg, Mitchell, and Peijnenburg use the panel’s survey weights for the summary statistics and analyses they report. The module asks people to choose between Box K, purple and orange balls in a known mix, and Box U, the same two colors in an unknown mix. The sequence moves the known share until the person is close to indifference. One choice was selected at random and played for a chance to win 23,850 in total to 1,590 of the 3,258 people who received the module. The question is whether that preference lines up with stocks the same people hold. Of the 3,258, 188 were dropped for finishing in under two minutes, and 30 more did not complete the ambiguity questions. Table 3, N = 3,040, is the weighted first-round choice between a known 50 percent and the unknown box: 52 percent ambiguity-averse, 10 percent indifferent, 38 percent ambiguity-seeking. The continuous score, 50 percent minus the matching probability, has mean 0.018, standard deviation 0.213, minimum −0.440, median 0.030, and maximum 0.470. Under their definition the mean matching probability is 48.2 percent, and the median is 47 percent. The first-round 52 percent and the mean of 0.018 answer different questions. One says which box won the first question. The other says how far the matching share sits from one half. Portfolio questions came from other panel modules fielded between 2008 and 2014, on overlapping subsets. The participation regression keeps people with demographic information: N = 2,943, from 2,683 households, after the 188, the 30, and 97 people with missing socio-demographic information. The model is a weighted probit with household-clustered standard errors. Controls include age and age squared, gender, white, Hispanic, married, children, health, education, employment, family income, wealth, defined-contribution and defined-benefit coverage, financial literacy, trust, a risk-aversion score, question order, check-question errors, and missing-data indicators. Nonbinary variables are standardized. The marginal effect on the standardized ambiguity score is −0.020 (standard error 0.01). The text beside the table says one standard deviation is associated with a 2.0 percentage point lower probability of owning stocks or stock mutual funds, 8.7 percent relative to a baseline of 23 percentage points. The introduction prints 8.6 percent relative to a baseline rate of 23 percent. A first-round choice of the known box, entered as a yes-or-no indicator, has a marginal effect of −0.039. The authors compare the 2.0 point move with 0.41 standard deviations of wealth, which they put at $238,000. Among people who already own stock, a Tobit on the fraction of financial assets in equities, N = 731, has a marginal effect of −0.040. The authors read that as 4.0 percentage points, 7.8 percent relative to a conditional average allocation of 51.4 percentage points. The full-sample Tobit, N = 2,943, prints −0.079. That cell is a different row from the 4.0 point sentence. Foreign shares and employer shares are holdings outside a 401(k). A one-standard-deviation higher score is associated with a 2.6 percentage point lower probability of foreign stock in the column that includes nonowners, N = 779, against a baseline of 13.3 percentage points, and with an 8.0 point lower probability among owners, N = 258, against a baseline of 27.0. Employer stock, limited to people employed by someone else, moves the other way: plus 1.4 percentage points in the broader column, N = 664, against a baseline of 5.0, significant at the 10 percent level, and plus 11.7 points among owners, N = 155. The paper prints no baseline beside the 11.7. The authors cite French and Poterba (1991) for the home-bias puzzle. This page does not carry country equity shares from that paper. The lesson is a measured preference with a small average and a real scatter, lined up with portfolio differences in a cross-section. It is a preference relative to risk, because both boxes pay the same prize. The boundary is the design. Nobody was assigned an ambiguity score. The stock questions and the urn questions do not share one date. The 2.0 point figure is the weighted association in the 2,943-person regression. The standard deviation of 0.213 belongs to the N = 3,040 table, and the regression standardizes its own sample.Boundaries and Failure Modes
Ambiguity Aversion is the wrong name when the two chances are never shown together, when a first-round share is treated as a large average discount, and when a cross-section is retold as a lever someone pulled. The first boundary is the page layout. In Fox and Tversky’s Study 1 the alone prices do not show the side-by-side gap. Study 2 put real money on a smaller version: a bag with one red and one green Ping-Pong ball, against a bag with two balls that could each be red or green. People guessed a color. A correct guess paid 19.50 down to 1.21, t(51) = 2.70, p < .01. The noncomparative comparison is t(50) = −0.61. The interaction is z = 1.90, p < .05. Study 4’s temperature bets, above, shrink from a stated 1.52 when each city is priced alone. The authors say this study’s gap comes from a lower price on the less familiar city. Studies 1 and 2, they say, raised the clear bet more than they cut the vague one. Status Quo Bias is a preference for the option already in place. A known chance can be the new option on the page. The second boundary is the average. Table 3’s 52 percent chose the known box first. The mean score is 0.018, on a scale whose minimum in that table is −0.440 and whose maximum is 0.470. Check questions, asked after the match was estimated, were inconsistent for 30.4 percent of people on the first check and 14.0 percent on the second. The participation regression controls for those errors. The authors say results are stronger for people whose checks agreed, and they do not print a second participation effect beside that sentence. In August 2013 a separate module of 500 people let half of them pick the winning color. Mean matching probabilities were 0.479 and 0.459, p = 0.31. That module is not Table 3. Competence has the same shape of a limit. Heath and Tversky (1991) are the source Dimmock and coauthors cite for the idea that a missing chance bites harder where a person feels less competent. In their Table 8, people who called their stock-market knowledge “very low” — more than 30 percent of respondents, the cutoff they chose because the share exceeded 30 percent — show a participation association of −4.6 percentage points per standard deviation. The higher-knowledge association is −1.2 points and is not significant. For literacy, the printed marginal effects are −0.033 when at least one of three questions is missed, and −0.009 otherwise. The second of those is not significant. The authors say the difference between the high and low groups is not statistically significant, so the table does not establish that the groups differ. In their account, people with high financial literacy own about 90 percent of financial wealth and make up about half the sample. The participation association in the literate group is the insignificant −0.009. The third boundary is what the associations are. The selling window in their crisis item is 1 October 2008 through 11 May 2009. People who both bought and sold are counted only when they sold more than they bought. Conditional on owning equities before the crisis, one standard deviation of the score, measured in 2012, is associated with a 4.5 percentage point higher probability of that net sale, 67.2 percent relative to a baseline of 6.7 percentage points. The urn date is after the sale window. Their appendix calibration of another model puts the equity-premium difference at 0.3 percentage points with heterogeneous agents and 0.5 with one ambiguity-averse agent, and at 1.2 when the sample is limited to people who passed both checks. Those are calibration outputs. In the same participation table, the risk-aversion score has a marginal effect of +0.019. The authors call that relation odd and say a few extremely risk-seeking answers carry it. Removing those answers leaves an insignificant negative relation. Base Rate Neglect is a known rate pushed aside by a vivid case. The score on this page is what people do when the rate is missing.Common Misconceptions
Three retellings travel with the urn. The known chance is treated as a dislike of risk. Ellsberg’s prose is treated as a percentage. The panel’s 52 percent is treated as a large discount, and as the cause of a country’s home bias.Ambiguity Aversion is risk aversion under another name
Ambiguity Aversion is risk aversion under another name
A choice between a sure amount and a coin at a known 50 percent is a choice between known chances. Ellsberg’s pair, and the panel’s two boxes, hold the prize fixed and change whether the mix is posted. The matching probability is defined so that a person’s dislike of variance is differenced out of the comparison. In the 2016 tables, a higher ambiguity score lines up with less foreign stock and more employer stock. Those two signs are the authors’ evidence that the score is doing different work from risk aversion.
Ellsberg measured a percentage, and the axioms failed for everyone he asked
Ellsberg measured a percentage, and the axioms failed for everyone he asked
The 1961 paper says “a large number of responses,” “the majority,” and “a very frequent pattern,” under nonexperimental conditions. It prints no sample size and no percentage. The $100 stakes are introduced with the word “say.” Debreu, Schlaifer, and Samuelson are his examples of people who did not violate the axioms. The minority pattern prefers the unknown urn for both colors. Savage appears as one person who, when last tested, wished to persist in the violating choices.
The panel shows that Americans give up a large chance, and that this causes home bias
The panel shows that Americans give up a large chance, and that this causes home bias
Table 3’s 52 percent is the weighted first-round choice in N = 3,040. The mean of 50 percent minus the matching probability is 0.018. The 2.0 point participation figure is the weighted association in N = 2,943. The introduction’s 8.6 percent and the table text’s 8.7 percent are two printings of that 2.0 against a 23 point baseline. Foreign-stock baselines on this page are 13.3 and 27.0 percentage points inside the 2016 columns. They are holdings outside a 401(k) in those samples.
Related Concepts
Ambiguity Aversion sits beside theories that already have a number for the chance, beside the extra weight of a loss, beside staying with the current option, beside a known rate that a story crowds out, and beside confidence checked against a record.Prospect Theory
Gains and losses, once the chances are numbers. Ambiguity Aversion begins where one chance has not been numbered.
Loss Aversion
A loss weighs more than a gain of the same size. The urn comparison holds the prize fixed and changes the information.
Status Quo Bias
A preference for the option already in place. A posted chance can be the new option on the page.
Base Rate Neglect
A known rate is underused once a vivid case arrives. Ambiguity Aversion is the response when the rate is missing.
Overconfidence Effect
A stated chance checked against later hits. Ellsberg’s confidence is confidence in the information behind the chance.