Daniel Ellsberg has died, aged 92. I don’t have anything to add to the standard account of his heroic career, except to observe that Edward Snowden (whose cause Ellsberg championed) would probably have done better to take his chances with the US legal system, as Ellsberg did.
In decision theory, the subsection of the economics profession in which I move Ellsberg is known for a contribution made a decade before the release of the Pentagon papers. In his PhD dissertation, Ellsberg offered thought experiments undermining the idea that rational people can assign probabilities to any event relevant to their decisions. This idea has given rise to a large theoretical literature on the idea of ‘ambiguity’. Although my own work has been adjacent to this literature for many decades, it’s only recently that I have actually written on this.
A long explanation is over the fold. But for those not inclined to delve into decision theory, it might be interesting to consider other people who have been prominent in radically different ways. One example is Hedy Lamarr, a film star who also patented a radio guidance system for torpedoes (the significance of which remains in dispute). A less happy example is that of Maurice Allais, a leading figure in decision theory and Economics Nobel winner, who also advocated some fringe theories in physics. I thought a bit about Ronald Reagan, but his entry into politics was really built on his prominence as an actor, rather than being a separate accomplishment.
The simplest of Ellsberg’s experiments is the “two-urn” problem. You are presented with two urns. One contains 50 red balls and 50 black balls. The other contains 100 black or red balls, but you aren’t told how many of each. Now you are offered two even money bet, which pay off if a red ball is drawn from one of the runs. You get to choose which urn to bet on. Intuition suggests choosing the urn with known proportions. Now suppose instead of a bet on red, you are offered the same choice but with a bet on black. Again, it seems that the first urn would be better.
Now, on the information given, the probability of a red ball being drawn from the first urn is 0.5. But what about the second urn. Strictly preferring the first urn for the red ball bet implies that the probability of a red ball being drawn from the second must be less than 0.5. But preferring the first urn for the black ball bet implies that the probability of a red ball being drawn from the second must be more than 0.5. So, there is no probability number that rationalises these decisions.
The title of Ellsberg’s paper was “Risk, Ambiguity and the Savage Axioms”. As a result, the term “ambiguity” has been applied, in contradistinction to risk, to the case when there are no well-defined probabilities. But this was not the way Ellsberg himself used the term. Rather he referred to
the nature of ones information concerning the relative likelihood of events. What is at issue might be called the ambiguity of this information, a quality depending on the amount, type, reliability and unanimity of information, andg iving rise to one’s degree of confidence in an estimate of relative likelihoods.(emphasis added)
I’ve developed this point in a paper whose title Seven Types of Ambiguity is one of numerous homages to William Empson’s classic work of literary criticism. Among these homages, I’d recommend the novel of the same name by Australian writer Elliot Perlman (later a TV series).
The central claim in my paper is that all forms of ambiguity in decision theory may
be traced to bounded and differential awareness. If that sounds interesting, you can read the paper here. If you’re super-interested, I’ll be presenting the paper in a couple of conferences in Europe in July – email me at j.quiggin@uq.edu.au for details.
John Quiggin 
Not my area so naive comments.
Is the issue that in the second urn the selector has no idea of the prospects of getting a particular colour and that they prefer situations where probabilities can be assigned to situations of uncertainty where they cannot be? If that’s true how far does aversion to uncertainty extend? For example suppose in the first urn there were 20 red and 80 Black would they still choose the situation where probabilities are known?
If I had to make the choice say of betting on red my intuition is that context is important. “I’d be suspicious – is the proposer of the task trying to trick me? Is this the bounded awareness you mention? What is the context – nothing is revealed when the problem is set out?
I think early writers on probability used “indifference rules” that assigned equal probabilities to uncertain events. In that case you should be indifferent between choosing between the urns. Selecting the first urn involves no cost.
One of the central points about Ellsberg was his opposition to nuclear war. He knew that in 1958 the US considered a nuclear attack on China over some Taiwan-claimed Islands. This attack would have killed 600 million Chinese. Today a similar war could kill far more. Ellsberg recently argued we are closer to a nuclear conflict than for quite a while – indeed Putin has used his nuclear arsenal as a weapon – a threat – recently.
Ellsberg was a moral man who sought a better world free from the prospect of nuclear annihilation. He zoned in on the key moral issue of our times.
The Urn is a problem for most of us I’d say.
Harry said “Not my area so naive comments” My comments are naive too, yet this is everybody’s area whi makes decisions.
Answers below, I naively suggest. And a simulator linked below.
Harry, you pose a good question; “Is the issue that in the second urn the selector has no idea of the prospects of getting a particular colour and that they prefer situations where probabilities can be assigned to situations of uncertainty where they cannot be?”
It seems there are several (13+?) flavours to solve your question.
“The Urn Problem” is correct… one of the 13! variations listed on Wikipedia Urn problem page –
“Examples of urn problems
1. beta-binomial distribution: as above, except that every time a ball is observed, an additional ball of the same color is added to the urn. Hence, the number of total balls in the urn grows. See Pólya urn model.
2. binomial distribution: the distribution of the number of successful draws (trials), i.e. extraction of white balls, given n draws with replacement in an urn with black and white balls.[3]
3. Hoppe urn: a Pólya urn with an additional ball called the mutator. When the mutator is drawn it is replaced along with an additional ball of an entirely new colour.
4. hypergeometric distribution: the balls are not returned to the urn once extracted. Hence, the number of total marbles in the urn decreases. This is referred to as “drawing without replacement”, by opposition to “drawing with replacement”.
5. multivariate hypergeometric distribution: the balls are not returned to the urn once extracted, but with balls of more than two colors.[3]
6. geometric distribution: number of draws before the first successful (correctly colored) draw.[3]
7. Mixed replacement/non-replacement: the urn contains black and white balls. While black balls are set aside after a draw (non-replacement), white balls are returned to the urn after a draw (replacement). What is the distribution of the number of black balls drawn after m draws?
8. multinomial distribution: there are balls of more than two colors. Each time a ball is extracted, it is returned before drawing another ball.[3] This is also known as ‘Balls into bins’.
9. negative binomial distribution: number of draws before a certain number of failures (incorrectly colored draws) occurs.
10. Occupancy problem: the distribution of the number of occupied urns after the random assignment of k balls into n urns, related to the coupon collector’s problemand birthday problem.
11. Pólya urn: each time a ball of a particular colour is drawn, it is replaced along with an additional ball of the same colour.
12. Statistical physics: derivation of energy and velocity distributions.
13. The Ellsberg paradox.
wikipedia /wiki/Urn_problem
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“Seven types of ambiguity”
John Quiggin
February 26, 2023
“Abstract [repeated sentences]
“Reducing ambiguity to a purely technical property of preferences misses much of the insight in Ellsberg’s (1961) paper, as well as in more recent developments in the study of differential awareness. In this paper, syntac- tic approaches to ambiguity are used to illustrate the point that Reducing ambiguity to a purely technical property of preferences misses much of the insight in Ellsberg’s (1961) paper, as well as in more recent developments in the study of differential awareness.”
…
” 7 Concluding comments
“In this paper, it has been argued that the semantic interpretation of ‘ambiguity’ to mean the absence of well-defined subjective probabilities is restrictive and unhelpful. Whereas the usual interpretation implies full awareness of the state space, a correct understanding of ambiguity must be linked to bounded awareness. The use of syntactic as well as semantic approaches helps to illustrate the central point of the paper: that ‘ambiguity’is an ambiguous concept, but nonetheless a powerful one.”
Click to access 662.pdf
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“Ambiguity and awareness: a coherent multiple priors model.”
Simon Grant
Ani Guerdjikova
Jihn Quiggin
February 20, 2019
Abstract
“Ambiguity in the ordinary language sense is that available information is open to multiple interpretations. We model this by assuming that individuals are unaware of some possiblities relevant to the outcome of their decisions and that multiple probabilities may arise over an individual’s subjective state space depending on which of these possibilities are realized.
“We formalize a notion of coherent multiple priors and derive a representation result that with full awareness corresponds to the usual unique (Bayesian) prior but with less than full awareness generates multiple priors.
“We show when information is received with no change in awareness, each element of the set of priors is updated in the standard Bayesian fashion (that is, full Bayesian updating).
“An increase in awarenss, however, leads to an expansion of the individual’s subjective state and (in general) a contraction in the set of priors under consideration.”
Click to access quiggin.pdf
[Link in JQ’s page below would not load for me]
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“Urn probability simulator
“This calculator simulates the urn (or box with colored balls) often used for probability problems, and can calculate probabilities of different events.”
https://planetcalc.com/7679/
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After skimming, I wanted to read some research with both “bounded awareness and ambiguity”.
Guess who came #1 in results?
JQ’s papers with “ambiguity” in the title:
Grant, Simon, Guerdjikova, Ani and Quiggin, John(2021).
Ambiguity and awareness: a coherent multiple priors model.
Grant, Simon, Kline, J. Jude and Quiggin, John(2018).
Contracting under uncertainty: a principal-agent model with ambiguity averse parties.
Grant, Simon, Kline, J. Jude and Quiggin, John(2014).
A matter of interpretation: ambiguous contracts and liquidated damages.
Grant, Simon, Kline, J.Jude and Quiggin, John(2012).
Differential awareness, ambiguity, and incomplete contracts: A model of contractual disputes.
Quiggin, John (2007).
Ambiguity and the value of information: An almost-objective events analysis.
All linked at:
https://economics.uq.edu.au/profile/10210/john-quiggin
Hedy Lamarr’s bright idea for radio control of torpedoes faces the big problem that radio waves propagate very badly in water, especially if it’s salty. The US Navy tested a radio-controlled torpedo (Hammond) in the 1930s, but did not adopt it. The standard method of steering a torpedo since the 1960s has been through a thin spooled signal wire. Curiously, this is an Australian invention (Louis Brennan, 1877 – see Wikipedia), though again his design, in which the two wires also propelled the torpedo by being pulled from a spool in the torpedo like a child’s toy, was not practical for wide use.
In my limited understanding, once you shift from a scenario where some estimation of probability is simply not possible, you could take the Bayesian position and integrate over a non informative (Jeffties prior), but that is in my opinion sidestepp the fundamental issue, I.e. you simply cannot know the extent of relative risk your ultimate decision entails. In other words, how should you approach such an ill-defined problem, one that in its worst incarnation involves two implacable foes, locked in a war danse macabre? Is the rational actor the one who never bends, or is it one who concedes ground?
The existential risk of total annihilation, versus generations of intolerable occupation, how can these alternatives be rationally quantified? Do the citizens get a say in this, or does some leader assume the role of totaliser and decision maker?
My personal feeling is that such existential evaluation and decision is somewhat beyond anything that a mathematical analysis is capable of determining. At its ultimate, such problems are like playing a single hand of poker, with no way to determine the rules of the game or indeed the cards that are actually in the deck; at that point it’s all about what you would be willing to literally gamble, in the absence of any informative knowledge, beyond what you personally are willing to lose. I truly think that games of meta-rationality were perhaps the first realisation (and then denial) that the ultimate war would be a pure contest of a game of chicken. I think Daniel understood this, almost from the start.
I find it hard to engage with a problem that makes trolley puzzles look like police procedurals.
“You are presented with two urns. One contains 50 red balls and 50 black balls. The other contains 100 black or red balls, but you aren’t told how many of each.” In what universe of real problems is it possible (a) to know for sure that Urn 2 contains 100 red or black balls (b) to have no information whatever about the proportions? Outside constructed games and puppet philosophical demons, information that always comes bundled with information how. The other thing is that for practical purposes, perfect decision rules are usually unobtainable. But since humans are constructed to make bad decisions all the time (Kahneman), there is massive room in real life for improvement.
I was just listening to an interview with Daniel Ellsburg on the BBC’s Hardtalk, recorded last year. Ellsberg said that he did not expect that whistle blowers such as Edward Snowden would have avoided prison, quoting the examples of Chelsea Manning and Julian Assange. Times have changed.
Also, was not one of the reasons that Ellsberg went free was the burglary of his psychiatrist’s office looking for Ellsberg’s records?
I agree that Ellsberg was lucky, both because the government screwed up in multiple ways and because the political and legal atmosphere was more open. Snowden would have faced much longer odds of avoiding jail. OTOH, it’s notable that Ellsberg himself, at 90, sought a prosecution that would have enabled him to challenge the constitutionality of the Espionage Act, as applied to suppress domestic dissent.
That is an interesting puzzle. I am glad I don’t have to take that test.
James, that battle film was quite beautifully filmed (from last week). I think I liked the filming better than the computer stuff in the LOTR films. I’ll have to look up that battle. (I wasn’t sure who I am supposed to root for.)
I wonder when we humans are going to start getting smarter? Knowing about the errors made in the 60s didn’t stop us from messing up again in Afghanistan, f.e. I guess we had better information – but we still made what seem to me to have been poor decisions. It would be great if one of these wars could be the last one. It just seems like a stupid way to decide things.
Last Friday I listened to a well attended presentation by Tony Aspromourgos at the University of Sydney on Adam Smith (300th anniversary). A few days later I read JQ’s paper on 7 types of ambiguity.
The ‘division of labour’ was of course mentioned under the heading Adam Smith’s contribution to economics, in particular developmental economics.
So I now offer a reason for ‘restricted awareness’: It is a consequence of the division of labour. As for developmental economics, the ‘state space’ relevant to human survival (ie state of nature) is shrinking every year according to the labour segment, called natural science, because the annual usage of natural resources exceed the reproduction capacity of nature. (This is accumulation of real debt.)
Does this mean we, ca 8.something billion minus the natural science specialists, are becoming dumber in the sense of increasing restricted awareness as development progresses? This question goes beyond the abstract discrete topology on a hypothetical state space and the undefined notion of ‘outcomes’. The latter surely would be interpreted differently by the labour segment ‘natural science’ and the labour segment ‘finance.’
As to the JQ’s paper, there some editorial points, which no doubt will be picked up without my input. Otherwise I assume it is meant to be an interpretative survey paper.
I feel dumber every day. I feel a large proportion of my fellow citizens are even dumber every day than I am and that, I must say, takes some doing. But how else am I to interpret the unwillingness to address climate change seriously and the willingness to spread a dangerous new disease without any infection controls other than a seriously leaky vaccine?
Ernestine’s division of labor example neatly divides into those who measure real stuff in real scientific dimensions and those who measure notional stuff in the fictitious dimension of the numéraire. The control of our society and economy by capital and finance (meaning by the holders of those quantities and levers) is at the root of our problems.
This is a system which has to, and will, collapse in stunning fashion in order to destroy peoples’ faith in it. This will happen inevitably as the gap widens between where the real systems are heading and where the formal and notional systems claim we are adding.
New theories of governance and economics need to be prepared, or revived and updated as the case may be, for the day when they are required and people are prepared to countenance something new. At some serious stage of the collapse, which is coming inevitably, people will clamor for something new.
If intentions and good will of the filler or bringer of the second urn are not known, the first urn feels preferable.