Talk:Ellsberg paradox

message: The 'psychological explanation' that has been cut from this page resembles a simple (perhaps too simple) version of the drama-theoretic explanation of ambiguity aversion. I've added a link to this explanation. (I've also corrected the number of red balls from 20 back to 30. 30 is what it should be, without doubt.) 
The drama-theoretic explanation invokes the 'personalization hypothesis'. This says that humans interacting with an entity in a signficant way will consciously or subconsciously personalize that entity -- ie, think of it as another party in an ongoing dramatic interaction. Luck (as in 'Luck be a lady') or death (as in 'Death has come to take me') are almost inevitably personalized in this way. Now what an experimenter presents as an ambiguity will, according to this hypothesis, be experienced as something that's chosen by the (personalized) experimental set-up. An assurance that the set-up will make an impartial, disinterested choice may be distrusted. The resulting 'trust dilemma' is reduced or eliminated by choosing an alternative that requires less trust to be placed in the experimental set-up.Nhoward 19:17, 23 August 2005 (UTC)
thread_title:  Ambiguity aversion and drama theory 

message: In the first section, it says "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", or R > B. However, in the "Mathematical demonstration" section, ">=" is used instead of ">". Is this an inconsitency or am I wrong here? Rbarreira 13:18, 5 February 2007 (UTC)
thread_title:  Inconsistency 

message: In the mathematical demonstration, I don't think it's necessary to require U(100) preferred to U(0). The paradox holds as long as we consider that both U(100) and U(0) are constant throughout. 
Running through the maths you can easily show that 
R[U(100)+U(0)] > B[U(100)+U(0)] 
in the first case and... 
B[U(100)-U(0)] > R[U(100)-U(0)] in the second.
Hence, it's not actually necessary to make a judgement over preferences beyond that A is preferred to B and D to C. Jamesshaw 15:30, 11 May 2007 (UTC)
thread_title:  Mathematical demonstration 

message: NOTE: Revised from first effort.  Thanks for the comment.
If an individual faces a choice between U(x)=5 with certainty and a bet where there's a 0.5 chance that U(x) = 6 and a 0.5 chance that U(x) = 4, his expected utilities are equal.  But if he's making many such bets in sequence, then he doesn't really have to pick one or the other.  He can use a mixed strategy over time as though he were constructing weighted portfolios in each alternative.  And it's possible that this mixed strategy will have higher utility than always picking one or the other.
Let's say a value maximizing investor has the option of allocating between two bets.  Call them A and B.  A delivers $100 with probability 0.5.  B has two cases.  Case 1 has 50% probability, and pays $100 40% of the time.  Case 2 has 50% probability, and pays $100 60% of the time.
Say that the weights on A and B sum to 1.  Call the weight on A = A, so the weight on B = 1-A.  The investor's objective function looks like:
\max\left\{0.5*U(A*\$100) + 0.5*U(A*\$0) + 0.5*0.4*U((1-A)*\$100) + 0.5*0.6*U((1-A)*\$100)\right\}
Which reduces to:
\max\left\{0.5*U(A*\$100) + 0.2*U((1-A)*\$100) + 0.3*U((1-A)*\$100)\right\}
Taking the derivative with respect to A and setting = 0 to maximize, you get:
\$50*U'(A*\$100) - \$20*U'((1-A)*\$100) + \$30*U'((1-A)*\$100) = 0
U'(A*\$100) = 0.4*U'((1-A)*\$100) + 0.6*U'((1-A)*\$100)
The optimal value of A will depend on the choice of U, but I'm pretty sure (though haven't proven) it will almost always be greater than 0.5 for risk averse investors.  For illustration, suppose U(x) = sqrt(x).  Then U'(x) = 1/(2*sqrt(x)).  Inserting this function, multiplying both sides by 2, and taking the reciprocal of both sides gives:
\sqrt{(A*100)} = (5/4)*\sqrt{((1-A)*100)} + (5/6)*\sqrt{((1-A)*\$100)}
Divide by 10 on each side to get:
\sqrt{A} = (5/4)*\sqrt{1-A} + (5/6)*\sqrt{1-A} = (25/12)*\sqrt{1-A}
Square to get:
A = (625/144)*(1-A)
Which gives A = 0.812744 > 0.5.  So A gets a higher weight.  
This doesn't necessarily imply that if you're forced to choose between A or B, you choose A.  However, think of the experiment as being a repeated game and not a 1-off thing.  Rational risk-averse individuals would choose A more often and B less often as if they were putting together a weighted portfolio.
I have to stop now.  Comments/further development welcome. — Preceding unsigned comment added by 128.151.203.76 (talk • contribs) 23:19, 4 July 2008 (UTC)
replies: text: I think your very first equation is wrong. You have that your objective function is (the expected utility of the payoff from bet A) plus (the expected utility of the payoff from bet B). It should be the expected utility of (the payoff from bet A plus the payoff from bet B). So you'll have eight terms (six if you set U($0) = 0, which is fine). For that matter, and for the same reason, I think the comparison between mixed strategy and weighted portfolio is wrong. The expected utility of going one way once and the other way once is not the expected utility of choosing randomly (50:50) twice. --EmbraceParadox (talk) 17:59, 5 July 2008 (UTC)
thread_title:  What if individuals face such bets repeatedly in life and treat them like a portfolio of bets? 

message: I'm looking at the "paradox" and it seems obvious that this shouldn't violate any numerical reasoning. My argument: In the first scenario, A is a sured number (30) where B is the 'gamble', but in scenario two, it's the other way around, A is the gamble (30+?), but B is the sured number (60). How could someone interpret this as having a dissonance or not following a simple rule? I think if someone is more risk-taker than not, it's a B-C combo, but if they're not a risk taker, it's an A-D combo. Anyone care to help where a "paradox" comes in? — Preceding unsigned comment added by 69.205.97.220 (talk • contribs) 10:22, 2 March 2009 (UTC)
replies: text: You just proved the paradox.  According to utility theory, if you have a preference for A then you should also have a preference for C.  If you have a preference for B you should also have a preference for D.  Just the opposite of what you suggest.  Of course, the whole point is to show that utility theory does not is not a complete model of decision making.  68.94.91.206 (talk) 20:58, 28 July 2012 (UTC)
replies: text: I must disagree with the statement above (beginning with "You just proved the paradox.") If you have a preference for A (risk averse) than you should have a preference for D. Both choices have a definite expected value. In contrast, choices B and C involve some element of chance and would be better suited to a risk taker.   My intuitive choice was originally A-C and after thinking about it for a while I stuck with my A-C combo despite A-D "technically" being the most risk averse. {I have a risk-averse personality}. I believe the economic theory that ~"The negative effect of a dollar lost is greater than the positive effect of a dollar gained" (Risk aversion/marginal utility) makes the clear logical choice A-D. Although modern psychological theories would suggest that an unexpectedly high number of humans would choose A-C (both "RED" options). Alexbucg (talk) 21:50, 1 March 2013 (UTC)

text: A is to D as B is to C in terms of risk. The argument that those who prefer A should have a preference for C is literally based on the COLOR of the balls, which is irrelevant. The utility theory interpretation incorrectly ties utility to color, when it should be tied to risk. This is not a paradox. Anongork (talk) 08:03, 6 April 2013 (UTC)
thread_title: Why isn't it just comparing risk to knowledge?

message: People are stupid and don't understand math. It's that simple. It's what keeps vegas going.
Or maybe people really do prefer a known bad deal to a deal that probably will be better, but maybe not.
I wonder if the results might vary by culture.  —Preceding unsigned comment added by Paul Murray (talk • contribs) 23:17, 11 March 2009 (UTC)
thread_title:  The simplest explanation. 

message: Chosing A and D is the most rational choice. With the aproach suggested in the article one could be easily fooled into poor choice by using 30 red and 60 black balls. When presented with the first problem, you would logically choose A. Clearly the approach used in the article leads to a poor choice, as chosing C in this situation leads to a lower chance of winning. It's irrational to assume that the choice you made in the first gamble was correct.--Ancient Anomaly (talk) 15:19, 15 December 2010 (UTC)
thread_title:  Paradox?? 

message: The guaranteed probability of getting a red ball is 30/90.
The guaranteed probability of getting a non-red ball is 60/90.
The probability of getting either yellow or black is undefined.
In A the probability of getting $100 is 30/90, in D it is 60/90.
B and C is just trying your luck.  — Preceding unsigned comment added by 94.197.127.135 (talk) 09:01, 23 June 2013 (UTC)
thread_title: Different choices

message: "So, supposing you prefer Gamble A to Gamble B, it follows that you will also prefer Gamble C to Gamble D."
Why does this follow? Based on this false assumption, you will come to the "paradox". 
To understand the "paradox" it is important to understand the assumed conclusion.  — Preceding unsigned comment added by 94.197.127.135 (talk) 09:09, 23 June 2013 (UTC)
thread_title: Unexplained conclusion

message: I would propose that the illustration of the paradox be changed to the one described here: http://ocw.mit.edu/courses/economics/14-123-microeconomic-theory-iii-spring-2010/lecture-notes/MIT14_123S10_notes06.pdf  — Preceding unsigned comment added by Bquast (talk • contribs) 09:03, 3 October 2013 (UTC)
thread_title: Alternative illustration of the paradox

message: Are you told both bets at the same time?  If not, do you complete the first draw before being informed of the second bet?  If you are told sequentially, you might choose A out of risk aversion, and then make your choice on the second bet influence by framing effects created by the first decision (Kahneman's book discusses this).  
Furthermore, if you are told the bets sequentially, there remains the hypothesis that the person giving you the wagers is behaving sharply (i.e. trying to psych you out).  Since you don't know the second bet is coming, when the first bet is offered, if you are defending yourself against sharp behaviour, you'll assume there are no black balls and pick A.  The sharp operator will, of course, have placed 60 black balls in the urn, expecting you to pick A.  Then when the second offer is posed, you can at most win $100 total.  Defending yourself against sharp thinking you'll pick D (by now realizing that you were tricked into not taken the best choice on the first bet).  In general, you'll always suspect that the person making the offer is, at that step in the game, one step ahead of the chooser, and you'll make the picks viewed as most immune from this disadvantage.  
Additionally, for mathematical rigour, in the case where the second offer is made after a choice is taken on the first offer, whether the second offer posed can be influenced by either the choice/selected ball from the the first offer, otherwise explanations invoking an analysis of the psychology of the person posing the offers is insufficiently posed.  
Finally, for completeness is really ought to be stated whether the chooser is left wondering whether there might possibly be a third offer on the same urn.  
In the case where you are told that there are two bets only, and you get to make both your selections in tandem, a rational analysis remains subject to your utility function for the different pay-offs, which can reasonably be linear, concave, or convex depending on your personal circumstance (consider The Gift of the Magi by O. Henry).  This does not necessarily have anything to do with risk aversion. — MaxEnt 19:17, 6 March 2015 (UTC)
thread_title:  The statement of the problem is far from sufficiently clear 

message: Hey, I'm currently working - together with a fellow student - on the German version of the Article for a seminar. (Our current Version can be found here) For this I created some graphics which could be used to vizualize the 2 Versions of the Ellsberg Paradox. If I have time at the end of the semester I could also add some of the stuff we have written for the German Wiki here.
Another thing: The section "Possible explanations" is pretty much taken from Lima Filho, Roberto IRL (July 2, 2009). "Rationality Intertwined: Classical vs Institutional View" Available at SSRN 2389751: pp. 5–6. doi:10.2139/ssrn.2389751. Thats why I added a citation at the End of the section, but basically it covers the whole section (and also I'm not sure if the Ben-Haim, Yakov (2006) citation there is appropriate, the original paper doesn't have it).
best --Daimpi (talk) 19:29, 28 June 2015 (UTC)
thread_title:  Citations and Illustration 

message: The experimentally observed choices have a rather mundane explanation — provided the utility function is concave, and the utility function is applied to expectations.  It is easy to see in a simplified version of the game: 
* There are two urns with 6 balls each, one with RRBBBY balls, the other with RRBYYY.  
* You choose one urn (at random: they are randomized before you choose).
* After this you are given 2 choices: $300 for R, or $300 for B.
In the first case the utility of the expected gain is U($100), in the second it is either U($50), or U($150); here U is the utility function.  With concave U (as usually assumed) the first choice is better.
(This is OR, but as far as I can see, this is more or less identical to formula `(1)` in this result of quick googling. --Ilya-zz (talk) 08:11, 3 April 2021 (UTC)
thread_title:  Utility function explanations seem to be completely wrong 

message: I am a university student completing a Wikipedia editing assignment (Wikipedia Course Link for reference: https://outreachdashboard.wmflabs.org/courses/UQ/ECON3430_2021_(Semester_One_2021)) and I have made the most recent edit to this Wiki Page (26/4/21: 3:56pm). I have made some subtle changes to the wording of the earlier portion of the page and extended slightly upon the 'Decisions under uncertainty aversion section' as well as added an image and Academic Paper section towards the bottom.
The markers viewing the work will review the changes I have made and grade me, I am asking everyone if they could please withhold from making further edits until the 10/5/21 to allow ample time for the tutors to see my version of the work and grade me. 
Regards.  — Preceding unsigned comment added by WHn457 (talk • contribs) 06:13, 26 April 2021 (UTC)
replies: text: That's a terrible suggestion. An encyclopedia shouldn't care about your course. Also, your tutors can just look at the version that you edited, no need to stop changing it afterwards. — Preceding unsigned comment added by 141.70.3.46 (talk) 06:07, 2 July 2021 (UTC)
thread_title:  Withhold from editing briefly 

message: This article was flagged for tone in December 2010 with this edit, though the OP has given no indication what the problem is. Anyway, it looks OK to me, so I have removed it. If anyone feels the tone is still an issue, they should replace the template, and say what (in their opinion) is wrong with it. I trust this is OK with everyone. Moonraker12 (talk) 15:32, 27 July 2022 (UTC)
thread_title:  Templates 

message: The final para contains two self-contradictory statements.  "The work was made public in 2001, some 40 years after being published" and "The book is considered a highly-influential paper".  To be made public means the same as to be published.  A book can't normally be considered a paper.  I don't know what the writer is trying to say here.  Could someone who does know, suggest a more sensible way of formulating it?  Andy Denis 12:43, 17 June 2023 (UTC)
thread_title:  Possible corrections to final para