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:

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 \(\frac{1}{3}\).

Let us say that the probability of getting a blue ball is \(p\) and that of getting a yellow ball is \(\frac{2}{3} - p\). Furthermore, for convenience, we assert the utility of the prize to be \(1\) and not getting it \(0\).

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

\[u(A) = 1 \times \frac{1}{3} + 0 \times p + 0 \times (\frac{2}{3} - p) = \frac{1}{3}\] \[u(B) = 0 \times \frac{1}{3} + 1 \times p + 0 \times (\frac{2}{3} - p) = p\]

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

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

\[u(X) = 1 \times \frac{1}{3} + 0 \times p + 1 \times (\frac{2}{3} - p) = 1-p\] \[u(Y) = 0 \times \frac{1}{3} + 1 \times p + 1 \times (\frac{2}{3} - p) = \frac{2}{3}\]

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

But we just showed that the agent believes that \( \frac{1}{3} > p \). 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 \(p\) 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 \( ( \frac{1}{3}, p, \frac{2}{3} - p ) \), we are now supposed to be working with the credal probability distribution set \( \left \{ p \in [0,\frac{2}{3}] : ( \frac{1}{3}, p, \frac{2}{3} - p ) \right \} \)

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

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

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

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

Clearly, we should choose A over B.

\[min(u(X)) = \frac{1}{3} \\ min(u(Y)) = \frac{2}{3} \]

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,

## Comments

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TopNewestDoes this have any relation to carrots? – Kunal Verma · 1 year, 3 months ago

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– Agnishom Chattopadhyay · 1 year, 3 months ago

Yes!!Log in to reply

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.

\(\color{orange}{L:\ I\ have\ carrots\ on\ my\ sandwich. }\)

\(\color{purple}{M:\ I\ have\ jelly\ on\ my\ sandwich.}\)

\(\color{red}{N: I\ have\ peanut\ butter\ on\ my\ sandwich.}\)

If I prefer having \(\color{orange}{carrots\ on\ my\ sandwich}\) over having \(\color{purple}{jelly\ on\ my\ sandwich}\), that doesn't necessary mean that I prefer having \(\color{orange}{carrots}\) and \(\color{red}{peanut\ butter}\) on my sandwich over having \(\color{red}{peanut\ butter}\) and \(\color{purple}{jelly}\) on my sandwich. In fact, I

hatepeanut butter and carrot sandwiches.Therefore, I denounce the axiom of independence.

Still not convinced? Here's some Python!

Python 3.3:\(\implies\)

STILL not convinced???

Make your own transparent soap! – Brock Brown · 1 year, 3 months ago

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here

The axiom of independence isn not eaxactly a stupid thing. My apologies for the oversimplified version of the axiom. Check out the formal versionIf 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 – Agnishom Chattopadhyay · 1 year, 3 months ago

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– Vishwathiga Jayasankar · 1 year, 2 months ago

How to add links in replies. Please tell me.Log in to reply

`[link text here](link url here)`

For example...

`[Google](https://www.google.com/)`

Makes this: Google – Brock Brown · 1 year, 2 months ago

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– Brock Brown · 1 year, 3 months ago

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

– Agnishom Chattopadhyay · 1 year, 3 months ago

They have equal chance of being distributed in any probability distribution. :)Log in to reply

Maybe that's dumb though. – Brock Brown · 1 year, 3 months 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:

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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 · 1 year, 3 months ago

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Oh I directly used

Axiom of Independenceto conclude I would take either (A,X) or (B,Y) – Pranjal Jain · 1 year, 3 months agoLog in to reply

– Agnishom Chattopadhyay · 1 year, 3 months ago

But if the lotteries were real, would you choose them, really?Log in to reply

– Pranjal Jain · 1 year, 3 months ago

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

You are not supposed to do something because philosophers have put forward an axiom. The philosophers put forward axioms to model what people do. – Agnishom Chattopadhyay · 1 year, 3 months ago

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That pic... a shot from your balcony? – Krishna Ar · 1 year, 3 months ago

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– Agnishom Chattopadhyay · 1 year, 3 months ago

From our terrace, I couldn't find a relevant image anyhow.Log in to reply

– Krishna Ar · 1 year, 3 months ago

So, I was right :PLog in to reply