<p>The short answer seems to be yes your example violates expected utility... It mostly seems to me like a simple transformation of the first example you gave (but you got rid of the red balls).</p>

<p>As mentioned in other answers expected utility is not equipped to handle uncertainty because it deals with taking expectations and expectations cannot be computed when you don't know probabilities. For that reason, I think it is relevant to my answer to specify the distinction between what risk and uncertainty are.</p>

<ul>
<li>Risk: This deals with facing a set of probabilities over outcomes in which the agent knows/understands the possible outcomes that he is facing</li>
<li>Uncertainty: This deals with an agent facing an unknown set of probabilities over possible outcomes</li>
</ul>

<p>In a previous version of my answer, I wasn't careful in how I thought about and explained these ideas and I have clarified a few ideas in my head since then which hopefully translates to a clarified answer. Now to answer your question:</p>

<p>We have an urn with 60 balls. We are <em>uncertain</em> about the number of black balls and yellow balls in this urn. Imagine we present this urn with the above specified lotteries to an individual (with a monotonic utility function $u$) and ask them to choose between $L_{A'}$ and $L_{B'}$. Without loss of generality, imagine they choose to take $L_{A'}$.</p>

<p>This tells us by revealed preference $L_{A'} \succ L_{B'}$ which then implies</p>

<p>$$u(30) &gt; u(B)$$</p>

<p>where $B$ is the number of black balls that are believed to be in the urn (no probabilities here, this is some number the decision maker arrived at and at this point we don't know how). This implies</p>

<p>$$B &lt; 30$$</p>

<p>Now imagine that you give the agent the choice between $L_{C'}$ and $L_{D'}$ and the agent chooses to face lottery $D'$. Then</p>

<p>$$u(60) &gt; u(30 + (60-B))$$</p>

<p>where $(60-B)$ is the number of yellow balls that the agent must think are in the urn. This implies</p>

<p>$$60 &gt; 30 + (60-B) \Rightarrow B &gt; 30$$</p>

<p>This means that an agent cannot prefer $L_{A'}$ to $L_{B'}$ and prefer $L_{C'}$ to $L_{D'}$ because that would imply $B &gt; 30$ and $B &lt; 30$. It is difficult to talk about where $B$ comes from because it is a hard concept to think about when we are used to working in expected value terms, this agent <em>cannot</em> compute any type of expectation over the sets of values that he thinks $B$ lies in. As an example of this, let me use a max-min approach.</p>

<p>Imagine that when he is presented the urn and lotteries that he is told that the number of black balls is either 15 or 45, but the agent doesn't know with what probability it is 15 and what probability it is 45.</p>

<ul>
<li>Lottery $A'$ will give him $u(30)$ utility units for sure</li>
<li>Lottery $B'$ will give him either $u(15)$ utility units or $u(45)$ utility units.</li>
<li>Lottery $C'$ will give him either $u(75)$ utility units or $u(45)$ utility units.</li>
<li>Lottery $D'$ will give him $u(60)$</li>
</ul>

<p>The agent is worried about uncertainty (an interpretation of the max-min is that he thinks it is a magic urn that tries its best to pay the least amount as possible, so it will choose the value that leaves him worse off if given a choice). Then using this in his decision process when comparing $L_{A'}$ and $L_{B'}$ he thinks these lotteries will pay the following amounts</p>

<ul>
<li>$L_{A'}$ will pay $u(30)$</li>
<li>$L_{B'}$ will pay $\min(u(15), u(45)) = u(15)$</li>
</ul>

<p>thus he chooses lottery $A'$. Now consider $L_{C'}$ and $L_{D'}$. He thinks these lotteries will pay</p>

<ul>
<li>$L_{C'}$ will pay $\min(u(45), u(75)) = u(45)$</li>
<li>$L_{D'}$ will pay $u(60)$</li>
</ul>

<p>thus he would choose lottery $D'$. This is one example of how uncertainty can be treated in a problem like this, but by no means is the only.</p>

<p>Note: Earlier I thought had a piece of my answer that talked about priors over the value of $B$, but this wasn't carefully thought through. You can have multiple values that you might imagine $B$ takes, but as soon as you assign a probability distribution to any of these values then you have left the realm of uncertainty and moved to the realm of risk.</p>

CC BY-SA 3.0
2015-05-02T20:15:46.453Z
What if you don't have a belief regarding the exact numbers of balls, but you believe that there is, say a 50% chance that B < 30?  Is that sort of belief not allowed in expected utility theory?  A belief like that isn't really Knightian uncertainty, because it's completely quantifiable.  So what sort of utility theory would accommodate a probability distribution over the different possible numbers of black balls?
Keshav Srinivasan
CC BY-SA 3.0
2015-05-02T20:24:39.160Z
Having a probability distribution over multiple possibilities of black balls is exactly the same. It doesn't matter whether you have expectations of expectations of expectations of the number of black balls. It will eventually spit out a number of balls $B$ that you think are in the urn. If you don't have that then you can't deal with expected utility.
cc7768
CC BY-SA 3.0
2015-05-02T22:31:24.577Z
Well, let's consider my example, and suppose you believe that there's a 50% chance that there are 15 black balls and a 50% chance that there are 45 black balls.  Then are you telling me that it violates expected utility theory to both prefer a guaranteed 30 dollars to a 50% chance of 15 dollars and a 50% chance of 45 dollars, and to prefer a guaranteed 60 dollars to a 50% chance of 45 dollars and a 50% chance of 75 dollars?  I think it's completely consistent with expected-utility theory; it's standard risk-aversion in a vNM utility function.
Keshav Srinivasan
CC BY-SA 3.0
2015-05-03T00:12:58.180Z
Firstly, I have no sense of risk-aversion in my example. My example only treats a risk-neutral version of this problem. Secondly, there isn't very much channel for risk the way that you phrased the problem because each of the lotteries are only a single outcome instead of mixing 2 outcomes, so for my entire argument, I could just add utility functions around both sides of each inequality and it goes through. Thirdly, an important distinction to make is the difference between example and counter-example.
cc7768
CC BY-SA 3.0
2015-05-03T00:15:12.807Z
An example is not sufficient to prove something is true, but a counter-example is sufficient to prove that it is *not* true. What you give in your comment is precisely the one example in which my inequalities turn into equalities and thus it is ok. If you move it outside of that knife-edge case (where the random variables both have expectation 30) then I think you will see that it falls apart. At the very least consider the counter-example of your expectation on the balls is $B = 0$ and $Y=60$
cc7768
CC BY-SA 3.0
2015-05-03T00:29:48.240Z
@cc768 Sorry, I'm confused.  I thought you were making a general claim and I was providing a counterexample.  Weren't you making the claim that regardless of your beliefs about the probability distribution over the different possible numbers of black balls, that doesn't affect the fact that it's inconsistent to prefer both A' to B' and D' to C'?
Keshav Srinivasan
CC BY-SA 3.0
2015-05-03T00:37:08.057Z
Your claim is that you can rationalize these things. My claim is that you can't. It is true that my argument didn't exactly go through because of the possibility of indifference, but you can't strictly prefer $A'$ to $B'$ and $D'$ to $C'$. At most you can be indifferent. In your example, the agent is indifferent between the two. I have changed my answer accordingly by requiring strict preference.
cc7768
CC BY-SA 3.0
2015-05-03T00:41:16.693Z
Well, again, if you believe that there's a 50% chance of 15 black balls and a 50% chance of 45 black balls, why is it inconsistent to both strictly prefer A' to B' and D' to C'?  Isn't that equivalent to strictly preferring both a guaranteed 30 dollars to a 50% chance of 15 dollars and a 50% chance of 45 dollars, and strictly preferring a guaranteed 60 dollars to a 50% chance of 45 dollars and a 50% chance of 75 dollars, which is completely consistent with the vNM axioms?
Keshav Srinivasan
CC BY-SA 3.0
2015-05-03T00:42:46.487Z
Because if you use that example then the expected number of black balls in the urn is 30 meaning you're indifferent between A' and B' and indifferent between C' and D'
cc7768
CC BY-SA 3.0
2015-05-03T00:43:43.390Z
Do you or do you not agree that strictly preferring both a guaranteed 30 dollars to a 50% chance of 15 dollars and a 50% chance of 45 dollars, and strictly preferring a guaranteed 60 dollars to a 50% chance of 45 dollars and a 50% chance of 75 dollars, is consistent with the vNM axioms?
Keshav Srinivasan
CC BY-SA 3.0
2015-05-03T00:46:30.750Z
I'm claiming the way I formulated it that you are indifferent between those bundles and so you don't satisfy my requirement for strict preference.
cc7768
CC BY-SA 3.0
2015-05-03T00:54:19.680Z
I'm not sure what you mean, so let me rephrase my question: assuming that I believe that there's a 50% chance of 15 black balls and a 50% chance of chance of 45 black balls, why does that imply that I'm indifferent between A' and B'?  The mere fact that the expected value of money in A' and B' are equal doesn't imply that the expected utility in A' and B' are equal, does it?
Keshav Srinivasan
CC BY-SA 3.0
2015-05-03T01:02:26.913Z
For instance, suppose the utility function is equal to the natural log of the amount of money.   Then if there's a 50% chance of 15 black balls and a 50% chance of 45 black balls, then the expected utility under A' is equal to ln(30), and the expected utility under B' is equal to (ln(15) + ln(45))/ 2, which is less than ln(30).
Keshav Srinivasan
CC BY-SA 3.0
2015-05-03T02:02:49.767Z
Let us [continue this discussion in chat](http://chat.stackexchange.com/rooms/23388/discussion-between-cc7768-and-keshav-srinivasan).
cc7768
CC BY-SA 3.0
2015-06-13T01:58:26.387Z
By the way, I just posted a question about something we were discussing in our chat:  http://economics.stackexchange.com/q/6099/4447
Keshav Srinivasan
CC BY-SA 3.0
2015-05-02T20:06:59.007Z
2015-05-03T06:17:34.477Z
2015-05-03T06:17:34.477Z
cc7768
<blockquote>
  <p><em>"...one of the major challenges for expected-value theory is the Ellsberg Paradox."</em></p>
</blockquote>

<p>Hmm, but in what way? Elsberg paradox is <em>not</em> like other related paradoxes surrounding Expected Utility theory, (like Allais' or Machina's).
In the Elsberg paradox <em>something is missing from the framework into which Expected Utility theory confines itself</em>: there are <em>unknown probabilities</em>, or <a href="http://en.wikipedia.org/wiki/Knightian_uncertainty" rel="nofollow">"Knightian Uncertainty"</a> -and Expected Utility theory is not constructed to operate in such an environment. </p>

<p>But then, Expected Utility theory is <em>not</em> refuted by the Ellsberg Paradox. What the paradox does, is bringing to our attention that there exist real-world situations where Expected Utility is <em>inapplicable</em>. I.e. it cannot be a <em>universal</em> theory. But that should not be surprising: what theory in Social Sciences holds universally through space and time?  </p>

<p>The value of the Ellsberg paradox is that it presses us to  modify Expected Utility theory, or come up with some complementary theory to handle cases where there exists "Knightian Uncertainty", and decision makers neither know the actual probabilities nor do they form subjective probabilities -but they decide in ways we have yet to fully understand how to model (although <a href="http://en.wikipedia.org/wiki/Ambiguity_aversion" rel="nofollow">"<strong>ambiguity aversion</strong>"</a>, which is the predominant way to rationalize the Ellsberg Paradox has by now been formalized to a large degree). </p>

<hr>

<p>As regards the decision-making framework you create in the second part of the answer, I note that the <strong>"Ellsberg's Paradox"</strong> is an <em>observed empirical phenomenon</em>. You create another framework and you just say "I think people would prefer this or that" -and you neither have actual observations to back that up, nor even a theoretical and/or mathematical argument as to why you think "people would prefer" what you think they will prefer. So if the framework you created is equivalent to the one characterizing "Ellsberg's Paradox" (and you should write down the mathematics to show that), then I am afraid recorded experience refutes your impression about "what people would prefer" in the framework you created.</p>

CC BY-SA 3.0
2015-04-26T00:50:34.770Z
Your answer doesn't answer my question at all.  I was just guessing what I think most people would do in my scenario, but that's not the question I was asking.  My actual question was just, is preferring A' to B' and D' to C' inconsistent with expected utility theory or not?  And if it is consistent with expected utility theory, why does it not imply that the results of Ellsberg's paradox are also consistent with expected utility theory?
Keshav Srinivasan
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2015-04-26T00:55:21.700Z
This is the second time that you are changing a question of yours by posting a "clarification comment" on an answer.
Alecos Papadopoulos
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2015-04-26T00:58:21.020Z
I'm not changing my question.  I'm saying the same thing I said above: "So my question is, does it violate expected utility theory to prefer both A' to B' and D' to C'? I think it's consistent with expected utility theory. Assuming I'm right, couldn't you just change the currency from "dollars" to "lottery tickets" everywhere in my example, where a lottery ticket entitles you to a 1/90 chance of getting a 100 dollars?  That would transform my example into the example in Ellsburg's paradox. So where's the flaw in my reasoning?"
Keshav Srinivasan
CC BY-SA 3.0
2015-04-26T04:26:55.287Z
Alecos Papadopoulos did answer your question:
"there exist real-world situations where Expected Utility is inapplicable"
Without beliefs about the distribution you cannot have N-M utility functions.
Given any beliefs a 'rational' consumer could not prefer both A' to B' and D' to C'.
Real-life people are not very good at dealing with probability (and use heuristic algorithms), so it is quite possible that real-life A' to B', D' to C' preferences exist.
Giskard
CC BY-SA 3.0
2015-04-26T05:18:27.327Z
@denesp My question isn't asking how applicable expected-utility theory is to the real world, it's about A', B', C', and D'.  Now if you can prove that someone who is vNM-rational cannot prefer both A' to B' and D' to C', I suggest you post an answer of your own.
Keshav Srinivasan
CC BY-SA 3.0
2015-06-13T01:59:54.707Z
By the way, I just posted another question related to vNM utility functions: http://economics.stackexchange.com/q/6099/4447
Keshav Srinivasan
CC BY-SA 3.0
2015-04-26T00:00:38.647Z
2015-04-26T00:00:38.647Z
Alecos Papadopoulos
<p>Formally Ellsberg paradox is defined in a different choice environment that expected utility. Ellsberg paradox is a problem for Savage subjective utility where there is ambiguity. Superficially they look the same because the have the same affine form, but in Savage the probabilities over states are not objective (are not a primitive) rather they are part of the representation. Also Expected Utility is defined over lotteries, while Savage Subjective Utility is defined over "acts".
A nice explanation of Ellsberg paradox and Savage Expected Subjective Utility is given in
<a href="http://lesswrong.com/lw/9e4/the_savage_theorem_and_the_ellsberg_paradox/" rel="nofollow">http://lesswrong.com/lw/9e4/the_savage_theorem_and_the_ellsberg_paradox/</a></p>

CC BY-SA 3.0
2015-04-29T10:44:31.497Z
OK, but this still doesn't answer my question.  Even if we use Savage's axioms rather than the vNM axioms, the question remains, is preferring both A' to B' and D' to C' inconsistent with Savage's axioms?
Keshav Srinivasan
CC BY-SA 3.0
2015-04-28T11:51:38.523Z
2015-04-28T11:51:38.523Z
user157623
<p>The <a href="http://en.wikipedia.org/wiki/Von_Neumann%E2%80%93Morgenstern_utility_theorem" rel="nofollow">von Neumann-Morgenstern</a> theorem states that, assuming a person's preferences under risk satisfy certain rationality axioms, then there exists a utility function u, the von Neumann utility function, such that the person will tend to maximize the expected value of u.  For this reason, the hypothesis that people satisfy the von Neumann-Morgenstern rationality axioms is known as expected utility theory.  Now one of the major challenges for expected utility theory is the Ellsberg Paradox.  It goes as follows.</p>

<p>Suppose you have an urn which has a total of 90 balls, 30 of which are red and each of the other 60 balls is either black or yellow.  And suppose a ball is drawn at random from the urn.  Then would you rather have lottery A, where you get 100 dollars if a red ball is drawn, or lottery B, where you get 100 dollars if a black ball is drawn?  Most people would prefer lottery A.  And would you rather have lottery C, where you get 100 dollars if you draw a red or yellow ball, or lottery D, where you get 100 dollars if you draw a black or yellow ball?  Most people would prefer lottery D.  But the thing is, preferring both lottery A to lottery B and lottery D to lottery C is inconsistent with expected utility theory; see <a href="http://en.m.wikipedia.org/wiki/Ellsberg_paradox" rel="nofollow">this Wikipedia article</a> for the proof.</p>

<p>I'd like to understand this logic a little better.  Consider a new urn that just has 60 balls, and each ball is either black or yellow.  Then would you rather have lottery A', where you get 30 dollars guaranteed, or lottery B', where you get a dollar for every black ball in the urn?  I think most people would prefer lottery A'.  And would you rather have lottery C', where you get 30 dollars plus a dollar for every yellow ball in the urn, or lottery D', where you get 60 dollars guaranteed?  I think most people would prefer lottery D'.  </p>

<p>So my question is, does it violate expected utility theory to prefer both A' to B' and D' to C'?  I think it's consistent with expected utility theory.  Assuming I'm right, couldn't you just change the currency from "dollars" to "lottery tickets" everywhere in my example, where a lottery ticket entitles you to a 1/90 chance of getting a 100 dollars?  That would transform my example into the example in Ellsburg's paradox.  So where's the flaw in my reasoning?</p>

<microeconomics><utility><expected-utility><risk-aversion><social-choice>
Question about the Ellsberg Paradox in Expected Utility Theory
Keshav Srinivasan
# Question about the Ellsberg Paradox in Expected Utility Theory

(Asked by: username_0 on 2015-04-25)

The [von Neumann-Morgenstern](http://en.wikipedia.org/wiki/Von_Neumann%E2%80%93Morgenstern_utility_theorem) theorem states that, assuming a person's preferences under risk satisfy certain rationality axioms, then there exists a utility function u, the von Neumann utility function, such that the person will tend to maximize the expected value of u. For this reason, the hypothesis that people satisfy the von Neumann-Morgenstern rationality axioms is known as expected utility theory. Now one of the major challenges for expected utility theory is the Ellsberg Paradox. It goes as follows.

Suppose you have an urn which has a total of 90 balls, 30 of which are red and each of the other 60 balls is either black or yellow. And suppose a ball is drawn at random from the urn. Then would you rather have lottery A, where you get 100 dollars if a red ball is drawn, or lottery B, where you get 100 dollars if a black ball is drawn? Most people would prefer lottery A. And would you rather have lottery C, where you get 100 dollars if you draw a red or yellow ball, or lottery D, where you get 100 dollars if you draw a black or yellow ball? Most people would prefer lottery D. But the thing is, preferring both lottery A to lottery B and lottery D to lottery C is inconsistent with expected utility theory; see [this Wikipedia article](http://en.m.wikipedia.org/wiki/Ellsberg_paradox) for the proof.

I'd like to understand this logic a little better. Consider a new urn that just has 60 balls, and each ball is either black or yellow. Then would you rather have lottery A', where you get 30 dollars guaranteed, or lottery B', where you get a dollar for every black ball in the urn? I think most people would prefer lottery A'. And would you rather have lottery C', where you get 30 dollars plus a dollar for every yellow ball in the urn, or lottery D', where you get 60 dollars guaranteed? I think most people would prefer lottery D'.

So my question is, does it violate expected utility theory to prefer both A' to B' and D' to C'? I think it's consistent with expected utility theory. Assuming I'm right, couldn't you just change the currency from "dollars" to "lottery tickets" everywhere in my example, where a lottery ticket entitles you to a 1/90 chance of getting a 100 dollars? That would transform my example into the example in Ellsburg's paradox. So where's the flaw in my reasoning?

**Tags:** microeconomics, utility, expected-utility, risk-aversion, social-choice


---

## 3 answers

### Answer by username_1 on 2015-05-02. Score: 4

The short answer seems to be yes your example violates expected utility... It mostly seems to me like a simple transformation of the first example you gave (but you got rid of the red balls).

As mentioned in other answers expected utility is not equipped to handle uncertainty because it deals with taking expectations and expectations cannot be computed when you don't know probabilities. For that reason, I think it is relevant to my answer to specify the distinction between what risk and uncertainty are.

* Risk: This deals with facing a set of probabilities over outcomes in which the agent knows/understands the possible outcomes that he is facing
* Uncertainty: This deals with an agent facing an unknown set of probabilities over possible outcomes

In a previous version of my answer, I wasn't careful in how I thought about and explained these ideas and I have clarified a few ideas in my head since then which hopefully translates to a clarified answer. Now to answer your question:

We have an urn with 60 balls. We are *uncertain* about the number of black balls and yellow balls in this urn. Imagine we present this urn with the above specified lotteries to an individual (with a monotonic utility function $u$) and ask them to choose between $L\_{A'}$ and $L\_{B'}$. Without loss of generality, imagine they choose to take $L\_{A'}$.

This tells us by revealed preference $L\_{A'} \succ L\_{B'}$ which then implies

$$u(30) > u(B)$$

where $B$ is the number of black balls that are believed to be in the urn (no probabilities here, this is some number the decision maker arrived at and at this point we don't know how). This implies

$$B < 30$$

Now imagine that you give the agent the choice between $L\_{C'}$ and $L\_{D'}$ and the agent chooses to face lottery $D'$. Then

$$u(60) > u(30 + (60-B))$$

where $(60-B)$ is the number of yellow balls that the agent must think are in the urn. This implies

$$60 > 30 + (60-B) \Rightarrow B > 30$$

This means that an agent cannot prefer $L\_{A'}$ to $L\_{B'}$ and prefer $L\_{C'}$ to $L\_{D'}$ because that would imply $B > 30$ and $B < 30$. It is difficult to talk about where $B$ comes from because it is a hard concept to think about when we are used to working in expected value terms, this agent *cannot* compute any type of expectation over the sets of values that he thinks $B$ lies in. As an example of this, let me use a max-min approach.

Imagine that when he is presented the urn and lotteries that he is told that the number of black balls is either 15 or 45, but the agent doesn't know with what probability it is 15 and what probability it is 45.

* Lottery $A'$ will give him $u(30)$ utility units for sure
* Lottery $B'$ will give him either $u(15)$ utility units or $u(45)$ utility units.
* Lottery $C'$ will give him either $u(75)$ utility units or $u(45)$ utility units.
* Lottery $D'$ will give him $u(60)$

The agent is worried about uncertainty (an interpretation of the max-min is that he thinks it is a magic urn that tries its best to pay the least amount as possible, so it will choose the value that leaves him worse off if given a choice). Then using this in his decision process when comparing $L\_{A'}$ and $L\_{B'}$ he thinks these lotteries will pay the following amounts

* $L\_{A'}$ will pay $u(30)$
* $L\_{B'}$ will pay $\min(u(15), u(45)) = u(15)$

thus he chooses lottery $A'$. Now consider $L\_{C'}$ and $L\_{D'}$. He thinks these lotteries will pay

* $L\_{C'}$ will pay $\min(u(45), u(75)) = u(45)$
* $L\_{D'}$ will pay $u(60)$

thus he would choose lottery $D'$. This is one example of how uncertainty can be treated in a problem like this, but by no means is the only.

Note: Earlier I thought had a piece of my answer that talked about priors over the value of $B$, but this wasn't carefully thought through. You can have multiple values that you might imagine $B$ takes, but as soon as you assign a probability distribution to any of these values then you have left the realm of uncertainty and moved to the realm of risk.

> **Comments:**
> **username_0**: What if you don't have a belief regarding the exact numbers of balls, but you believe that there is, say a 50% chance that B < 30?  Is that sort of belief not allowed in expected utility theory?  A belief like that isn't really Knightian uncertainty, because it's completely quantifiable.  So what sort of utility theory would accommodate a probability distribution over the different possible numbers of black balls?  
> **username_1**: Having a probability distribution over multiple possibilities of black balls is exactly the same. It doesn't matter whether you have expectations of expectations of expectations of the number of black balls. It will eventually spit out a number of balls $B$ that you think are in the urn. If you don't have that then you can't deal with expected utility.  
> **username_0**: Well, let's consider my example, and suppose you believe that there's a 50% chance that there are 15 black balls and a 50% chance that there are 45 black balls.  Then are you telling me that it violates expected utility theory to both prefer a guaranteed 30 dollars to a 50% chance of 15 dollars and a 50% chance of 45 dollars, and to prefer a guaranteed 60 dollars to a 50% chance of 45 dollars and a 50% chance of 75 dollars?  I think it's completely consistent with expected-utility theory; it's standard risk-aversion in a vNM utility function.  
> **username_1**: Firstly, I have no sense of risk-aversion in my example. My example only treats a risk-neutral version of this problem. Secondly, there isn't very much channel for risk the way that you phrased the problem because each of the lotteries are only a single outcome instead of mixing 2 outcomes, so for my entire argument, I could just add utility functions around both sides of each inequality and it goes through. Thirdly, an important distinction to make is the difference between example and counter-example.  
> **username_1**: An example is not sufficient to prove something is true, but a counter-example is sufficient to prove that it is *not* true. What you give in your comment is precisely the one example in which my inequalities turn into equalities and thus it is ok. If you move it outside of that knife-edge case (where the random variables both have expectation 30) then I think you will see that it falls apart. At the very least consider the counter-example of your expectation on the balls is $B = 0$ and $Y=60$  
> **username_0**: @cc768 Sorry, I'm confused.  I thought you were making a general claim and I was providing a counterexample.  Weren't you making the claim that regardless of your beliefs about the probability distribution over the different possible numbers of black balls, that doesn't affect the fact that it's inconsistent to prefer both A' to B' and D' to C'?  
> **username_1**: Your claim is that you can rationalize these things. My claim is that you can't. It is true that my argument didn't exactly go through because of the possibility of indifference, but you can't strictly prefer $A'$ to $B'$ and $D'$ to $C'$. At most you can be indifferent. In your example, the agent is indifferent between the two. I have changed my answer accordingly by requiring strict preference.  
> **username_0**: Well, again, if you believe that there's a 50% chance of 15 black balls and a 50% chance of 45 black balls, why is it inconsistent to both strictly prefer A' to B' and D' to C'?  Isn't that equivalent to strictly preferring both a guaranteed 30 dollars to a 50% chance of 15 dollars and a 50% chance of 45 dollars, and strictly preferring a guaranteed 60 dollars to a 50% chance of 45 dollars and a 50% chance of 75 dollars, which is completely consistent with the vNM axioms?  
> **username_1**: Because if you use that example then the expected number of black balls in the urn is 30 meaning you're indifferent between A' and B' and indifferent between C' and D'  
> **username_0**: Do you or do you not agree that strictly preferring both a guaranteed 30 dollars to a 50% chance of 15 dollars and a 50% chance of 45 dollars, and strictly preferring a guaranteed 60 dollars to a 50% chance of 45 dollars and a 50% chance of 75 dollars, is consistent with the vNM axioms?  
> **username_1**: I'm claiming the way I formulated it that you are indifferent between those bundles and so you don't satisfy my requirement for strict preference.  
> **username_0**: I'm not sure what you mean, so let me rephrase my question: assuming that I believe that there's a 50% chance of 15 black balls and a 50% chance of chance of 45 black balls, why does that imply that I'm indifferent between A' and B'?  The mere fact that the expected value of money in A' and B' are equal doesn't imply that the expected utility in A' and B' are equal, does it?  
> **username_0**: For instance, suppose the utility function is equal to the natural log of the amount of money.   Then if there's a 50% chance of 15 black balls and a 50% chance of 45 black balls, then the expected utility under A' is equal to ln(30), and the expected utility under B' is equal to (ln(15) + ln(45))/ 2, which is less than ln(30).  
> **username_1**: Let us [continue this discussion in chat](http://chat.stackexchange.com/rooms/23388/discussion-between-username_1-and-username_0-username_0).  
> **username_0**: By the way, I just posted a question about something we were discussing in our chat:  http://economics.stackexchange.com/q/6099/4447  

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### Answer by username_2 on 2015-04-26. Score: 3

> *"...one of the major challenges for expected-value theory is the Ellsberg Paradox."*

Hmm, but in what way? Elsberg paradox is *not* like other related paradoxes surrounding Expected Utility theory, (like Allais' or Machina's).
In the Elsberg paradox *something is missing from the framework into which Expected Utility theory confines itself*: there are *unknown probabilities*, or ["Knightian Uncertainty"](http://en.wikipedia.org/wiki/Knightian_uncertainty) -and Expected Utility theory is not constructed to operate in such an environment.

But then, Expected Utility theory is *not* refuted by the Ellsberg Paradox. What the paradox does, is bringing to our attention that there exist real-world situations where Expected Utility is *inapplicable*. I.e. it cannot be a *universal* theory. But that should not be surprising: what theory in Social Sciences holds universally through space and time?

The value of the Ellsberg paradox is that it presses us to modify Expected Utility theory, or come up with some complementary theory to handle cases where there exists "Knightian Uncertainty", and decision makers neither know the actual probabilities nor do they form subjective probabilities -but they decide in ways we have yet to fully understand how to model (although ["**ambiguity aversion**"](http://en.wikipedia.org/wiki/Ambiguity_aversion), which is the predominant way to rationalize the Ellsberg Paradox has by now been formalized to a large degree).

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As regards the decision-making framework you create in the second part of the answer, I note that the **"Ellsberg's Paradox"** is an *observed empirical phenomenon*. You create another framework and you just say "I think people would prefer this or that" -and you neither have actual observations to back that up, nor even a theoretical and/or mathematical argument as to why you think "people would prefer" what you think they will prefer. So if the framework you created is equivalent to the one characterizing "Ellsberg's Paradox" (and you should write down the mathematics to show that), then I am afraid recorded experience refutes your impression about "what people would prefer" in the framework you created.

> **Comments:**
> **username_0**: Your answer doesn't answer my question at all.  I was just guessing what I think most people would do in my scenario, but that's not the question I was asking.  My actual question was just, is preferring A' to B' and D' to C' inconsistent with expected utility theory or not?  And if it is consistent with expected utility theory, why does it not imply that the results of Ellsberg's paradox are also consistent with expected utility theory?  
> **username_2**: This is the second time that you are changing a question of yours by posting a "clarification comment" on an answer.  
> **username_0**: I'm not changing my question.  I'm saying the same thing I said above: "So my question is, does it violate expected utility theory to prefer both A' to B' and D' to C'? I think it's consistent with expected utility theory. Assuming I'm right, couldn't you just change the currency from "dollars" to "lottery tickets" everywhere in my example, where a lottery ticket entitles you to a 1/90 chance of getting a 100 dollars?  That would transform my example into the example in Ellsburg's paradox. So where's the flaw in my reasoning?"  
> **username_3**: username_2 username_2 did answer your question: "there exist real-world situations where Expected Utility is inapplicable" Without beliefs about the distribution you cannot have N-M utility functions. Given any beliefs a 'rational' consumer could not prefer both A' to B' and D' to C'. Real-life people are not very good at dealing with probability (and use heuristic algorithms), so it is quite possible that real-life A' to B', D' to C' preferences exist.  
> **username_0**: @denesp My question isn't asking how applicable expected-utility theory is to the real world, it's about A', B', C', and D'.  Now if you can prove that someone who is vNM-rational cannot prefer both A' to B' and D' to C', I suggest you post an answer of your own.  
> **username_0**: By the way, I just posted another question related to vNM utility functions: http://economics.stackexchange.com/q/6099/4447  

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### Answer by username_4 on 2015-04-28. Score: 2

Formally Ellsberg paradox is defined in a different choice environment that expected utility. Ellsberg paradox is a problem for Savage subjective utility where there is ambiguity. Superficially they look the same because the have the same affine form, but in Savage the probabilities over states are not objective (are not a primitive) rather they are part of the representation. Also Expected Utility is defined over lotteries, while Savage Subjective Utility is defined over "acts".
A nice explanation of Ellsberg paradox and Savage Expected Subjective Utility is given in
<http://lesswrong.com/lw/9e4/the_savage_theorem_and_the_ellsberg_paradox/>

> **Comments:**
> **username_0**: OK, but this still doesn't answer my question.  Even if we use Savage's axioms rather than the vNM axioms, the question remains, is preferring both A' to B' and D' to C' inconsistent with Savage's axioms?  

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