Rationality Revisited: The Ellsberg Paradox

Consider the following situation:

In an urn, you have 90 balls of 3 colors: red, blue and yellow. 30 balls are known to be red. All the other balls are either blue or yellow.

There are two lotteries:

  • Lottery A: A random ball is chosen. You win a prize if the ball is red.
  • Lottery B: A random ball is chosen. You win a prize if the ball is blue.

Question: In which lottery would you want to participate?

  • Lottery X: A random ball is chosen. You win a prize if the ball is either red or yellow.
  • Lottery Y: A random ball is chosen. You win a prize if the ball is either blue or yellow.

Question: In which lottery would you want to participate?

If you're like most people you probably chose Lottery A over B and Lottery Y over X. (If you did choose something else, skip ahead to the end section)

Did you know that under the classical models of decision theory, there is no probability distribution under which such a decision would be rational?

Why is that?

Well, to begin with we state the Axiom of Independence in Von Neumann's Model of Rationality which otherwise seem to be obviously correct.

If a person prefers event L to event M, then he always prefers L along with N to M along with N, for any event N.

This seems fairly obvious. If I like Carrots over Soya Sauce, I would definitely prefer Carrots along with Transparent Soap to Soya Sauce along with Transparent Soap.

However, in this case it appears that our decisions suddenly changed our preference for event Red to the event Blue, just as we introduced Yellow into the scene. See below:

Ellsberg Paradox Ellsberg Paradox

Let us formalize this bit with the Axiom of Rationality

Axiom of Rationality: A rational agent aims to maximize his expected utility.

First, we are clear that the probability of getting a red ball is 13\frac{1}{3}.

Let us say that the probability of getting a blue ball is pp and that of getting a yellow ball is 23p\frac{2}{3} - p. Furthermore, for convenience, we assert the utility of the prize to be 11 and not getting it 00.

We calculated the expected utilities of lotteries A and B as follows:

u(A)=1×13+0×p+0×(23p)=13u(A) = 1 \times \frac{1}{3} + 0 \times p + 0 \times (\frac{2}{3} - p) = \frac{1}{3} u(B)=0×13+1×p+0×(23p)=pu(B) = 0 \times \frac{1}{3} + 1 \times p + 0 \times (\frac{2}{3} - p) = p

Thus, if a rational agent did prefer A to B, he believes that u(A)>u(B)    13>pu(A) > u(B) \\ \implies \frac{1}{3} > p

Now, let us set aside this result and turn to lotteries X and Y.

u(X)=1×13+0×p+1×(23p)=1pu(X) = 1 \times \frac{1}{3} + 0 \times p + 1 \times (\frac{2}{3} - p) = 1-p u(Y)=0×13+1×p+1×(23p)=23u(Y) = 0 \times \frac{1}{3} + 1 \times p + 1 \times (\frac{2}{3} - p) = \frac{2}{3}

If a rational agent did prefer Y to X, he believes that u(Y)>u(X)    23>1p    p>13u(Y) > u(X) \\ \implies \frac{2}{3} > 1- p \\ \implies p > \frac{1}{3}

But we just showed that the agent believes that 13>p \frac{1}{3} > p . Thus, choosing A cannot be consistent with choosing Y.

But does it mean we are really irrational? Maybe something was wrong with our axioms of rationality?

What actually drives us to do these choices is actually that we prefer the kind of surity behind those lotteries. Is there a way to formalize this idea?

The answer is yes, fortunately and we boldly propose our new axiom or rationality:

Axiom of Rationality: A rational agent aims to maximize his minimum expected utility.

This, by the way, is called The Maximin Principle of Rationality.

To account for the idea of minimum expected utility, we turn to an idea called Imprecise Probability.

This is an idea to say that we do not just believe that an event occurs with a probability pp but we do believe that events could have a set of possible probabilities (called a credal set) spread over an interval. Just to state an example, we do believe that heads on tails on a coin are equally likely but do we really believe the same thing about, say, a thumb tack?

Let us look at our problem in this light.

Instead of the previous precise probability distribution of (13,p,23p) ( \frac{1}{3}, p, \frac{2}{3} - p ) , we are now supposed to be working with the credal probability distribution set {p[0,23]:(13,p,23p)} \left \{ p \in [0,\frac{2}{3}] : ( \frac{1}{3}, p, \frac{2}{3} - p ) \right \}

Similarly, we are supposed to replace the expected utilities with sets of expected utilities.

u(A)={13}u(B)={p[0,23]:p}u(X)={p[0,23]:1p}u(Y)={23} u(A) = \left \{ \frac{1}{3} \right \} \\ u(B) = \left \{ p \in [0,\frac{2}{3}] : p \right \} \\ u(X) = \left \{ p \in [0,\frac{2}{3}] : 1 - p \right \} \\ u(Y) = \left \{ \frac{2}{3} \right \}

Now, we are ready to calculate the minimum expected utilities for the four lotteries.

min(u(A))=13min(u(B))=0 min(u(A)) = \frac{1}{3} \\ min(u(B)) = 0

Clearly, we should choose A over B.

min(u(X))=13min(u(Y))=23min(u(X)) = \frac{1}{3} \\ min(u(Y)) = \frac{2}{3}

And also Y over X!

So much for the idea of imprecise probability and the maximin principle, we have seen that we are now ready to answer some really interesting question. For starters, try explaining why it is rational to buy an insurance even though the expected value is negative,

Note by Agnishom Chattopadhyay
6 years ago

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Does this have any relation to carrots?

Kunal Verma - 6 years ago

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Economists have a hard time studying decision theory because it is hard to be rational for decisions that are not trivial.

In these cases, the experimental design often ends up being much more important, and has significant effects on the day. Some known results are:

  • people are willing to give up expected value for the (perceived) control of destiny. E.g "Hitting this button has a 50% chance of Win / Lose. How much would you pay to hit the button" vs "How much would you pay for me to hit the button"
  • people avoid something when phrased as a loss, compared to when phrased as a gain. E.g. "I give you $10. You will lose $5 if ..." vs "You will get $5 if ..."
  • Other environmental factors / effects.

Calvin Lin Staff - 6 years ago

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Von Neumann is bad and he should feel bad.

Let me explain why:

"If a person prefers event L to event M, then he always prefers L along with N to M along with N, for any event N."

Let's define some events.

L: I have carrots on my sandwich.\color{#EC7300}{L:\ I\ have\ carrots\ on\ my\ sandwich. }
M: I have jelly on my sandwich.\color{#69047E}{M:\ I\ have\ jelly\ on\ my\ sandwich.}
N:I have peanut butter on my sandwich.\color{#D61F06}{N: I\ have\ peanut\ butter\ on\ my\ sandwich.}

If I prefer having carrots on my sandwich\color{#EC7300}{carrots\ on\ my\ sandwich} over having jelly on my sandwich\color{#69047E}{jelly\ on\ my\ sandwich}, that doesn't necessary mean that I prefer having carrots\color{#EC7300}{carrots} and peanut butter\color{#D61F06}{peanut\ butter} on my sandwich over having peanut butter\color{#D61F06}{peanut\ butter} and jelly\color{#69047E}{jelly} on my sandwich. In fact, I hate peanut butter and carrot sandwiches.

Therefore, I denounce the axiom of independence.

Still not convinced? Here's some Python!

Python 3.3:

from random import randint
from time import time
from pickle import load, dump

balls = 90
red = 30
    red_picked, blue_picked, yellow_picked, trials = load(open("balls.p", "rb"))
except FileNotFoundError:
    red_picked = 0
    blue_picked = 0
    yellow_picked = 0
    trials = 0
seconds = float(input("How many seconds to you want to run simulations? "))
end = time() + seconds
while time() < end:
        blue = randint(0, balls-red)
        yellow = balls - (red + blue)
        ball = randint(1, balls)
        if ball <= red:
            red_picked += 1
        elif ball <= red + blue:
            blue_picked += 1
            yellow_picked += 1
        trials += 1
    except KeyboardInterrupt:
        print ("\nSimulations stopped!")
red_chance = red_picked/trials
blue_chance = blue_picked/trials
yellow_chance = blue_picked/trials
print ("Chance of a red ball:", red_chance)
print ("Chance of a blue ball:", blue_chance)
print ("Chance of a yellow ball:", yellow_chance)
print ("Trials observed:", trials)
print ("Lottery A: about {0}% chance of winning.".format(red_chance * 100))
print ("Lottery B: about {0}% chance of winning.".format(blue_chance*100))
print ("Lottery X: about {0}% chance of winning.".format((red_chance+yellow_chance)*100))
print ("Lottery Y: about {0}% chance of winning.".format((blue_chance+yellow_chance)*100))

dump((red_picked, blue_picked, yellow_picked, trials), open("balls.p", "wb"))


Chance of a red ball: 0.3332976169108931
Chance of a blue ball: 0.3333817695820553
Chance of a yellow ball: 0.3333817695820553
Trials observed: 225946482
Lottery A: about 33.32976169108931% chance of winning.
Lottery B: about 33.33817695820553% chance of winning.
Lottery X: about 66.66793864929484% chance of winning.
Lottery Y: about 66.67635391641106% chance of winning.

STILL not convinced???

Make your own transparent soap!

Brock Brown - 6 years ago

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The axiom of independence isn not eaxactly a stupid thing. My apologies for the oversimplified version of the axiom. Check out the formal version here

If the balls are distributed across red, yellow and blue, there is nothing wrong about being indifferent among the options. While I really appreciate simulations to answer probability questions, this is not exactly the case here. The question is about what you should be assuming when you do not know what the probability distribution is.

I came to know about this idea from this course: Introduction to Mathematical Philosophy

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How to add links in replies. Please tell me.

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@Vishwathiga Jayasankar [link text here](link url here)

For example...
Makes this: Google

Brock Brown - 6 years ago

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Oh, I misunderstood the question then. So the remaining balls do not have an "equal chance" of being either yellow or blue?

Brock Brown - 6 years ago

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@Brock Brown They have equal chance of being distributed in any probability distribution. :)

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@Agnishom Chattopadhyay If they have an equal chance of being distributed in any probability distribution, could we potentially imply that the chance of a blue ball is equal to the chance of having a yellow ball? For each of the infinite probability distributions that exist, we can swap the conditions of the chance of a yellow ball with the chance of a blue ball. It's like how if I have an infinite number of hotel rooms on the east side of the lobby and an infinite amount of hotel rooms on the west side of the lobby. On the east side we could have all probability distributions (rooms) where the chance of blue is more than or equal to that of a yellow ball, and on the west side we could have all of the distributions where the chance of a yellow ball is greater than or equal to the chance of a blue ball.

Maybe that's dumb though.

Brock Brown - 6 years ago

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That pic... a shot from your balcony?

Krishna Ar - 6 years ago

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From our terrace, I couldn't find a relevant image anyhow.

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So, I was right :P

Krishna Ar - 6 years ago

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Oh I directly used Axiom of Independence to conclude I would take either (A,X) or (B,Y)

Pranjal Jain - 6 years ago

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But if the lotteries were real, would you choose them, really?

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Yes, I would. Because I didn't knew there's something like this Axiom. It was an intuition.

Pranjal Jain - 6 years ago

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@Pranjal Jain Okay.

You are not supposed to do something because philosophers have put forward an axiom. The philosophers put forward axioms to model what people do.

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I think people tend to choose insurances with negative expected values because poor people tend to value a dollar more than rich people. Many people's utility function follow logarithmic or related patterns. Because of the concave nature of the logarithm function, the expected value of a person's satisfaction by "playing it safe" can exceed that of taking a risk whose expected value is slightly above the safe option.

For example, suppose you have million dollars, and must choose among 2 options: you can either lose $1, or toss a coin, in which you win a million dollars if you toss heads, and go broke if you toss tails. Any rational person would choose the former option, despite its lower expected value.

Here's a page about Utility functions: http://www.econ.ucsb.edu/~tedb/Courses/Ec100C/VarianExpectedUtility.pdf

Alex Li - 6 years ago

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A nice explanation. Sounds more like an aggregation paradox (Simpson's, Arrow's) rather than an aversion anomaly. The independence axiom is intuitively obvious in some contexts and not-so-justifiable in others.

Anil Menon - 2 months, 3 weeks ago

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