The Ellsberg paradox is a puzzle concerning human rationality. It postulates a scenario involving an urn with 90 balls inside - 30 red ones, while the remaining 60 can be yellow or black. A player gets to draw a ball from the urn, and receives a payoff dependent on the color. The player is then given options for various bets. In practice, humans are shown to display inconsistent preferences as to which bets are worth taking - this is completely irrational behavior under any reasonable theory of probability. In this post I'll ponder this question, and postulate a model of bounded rationality characterized by "problem boundaries" which may shed some light on this puzzle.

The Ellsberg paradox consists of the following situation. An urn contains 90 balls - 30 red, and the remainder are yellow and black. Let us denote by $n_y$ and $n_b$ the number of yellow and black balls.

The player is offered two potential gambles:

• A1 Draw a ball. If the ball is red, the player receives $100. • A2 Draw a ball. If the ball is black, the player receives$100.

The player is then offered two alternative gambles:

• B1 Draw a ball. If the ball is red or yellow, the player receives $100. • B2 Draw a ball. If the ball is black or yellow, the player receives$100.

Lets work this out mathematically. The expected payout of these bets is:

• A1 The payoff is $100 \cdot 30 / 90 = 33$
• A2 The payoff is $100 \cdot n_b / 90$.
• B1 The payoff is $100 \cdot (30+n_y)/90$
• B2 The payoff is $100 \cdot (n_b + n_y)/90 = 100 \cdot 60/90 = 66$

From these formulae, we can draw the following conclusions. Suppose the player considers A1 better than A2. This means that $n_b < 30$, and hence $n_y = 60 - n_b > 30$. From this, we can conclude that if the player considers A1 better than A2, they must believe that $30 + n_y > 30+30$, and thus B1 must be better than B2.

In real life, people tend to prefer option A1 over A2 and option B2 over B1, contradicting the laws of arithmetic.

# Bounded Rationality

The theory of Bounded Rationality supposes that the ability of humans to reason rationally is bounded - for whatever reason, humans are incapable of thinking through every possibility.

I propose one possible mechanism for this. It's based on the concept of variable scope in computer science.

Specifically, I suppose that each subproblem is solved in it's own subroutine. The routine for solving A (i.e., determining whether A1 is better or worse than A2) looks like this:

def solveA = {
nb_guess <- chooseConservatively
payoff_a1 = 100.0 * 30 / 90.0
payoff_a2 = 100.0 * nb_guess / 90.0
if (payoff_a1 > payoff_a2) {
return "choose A1"
} else {
return "choose A2"
}
}


The function chooseConservatively is an oracle function which chooses nb_guess in some sense "conservatively" to avoid the risk of losing too much for the current problem only.

The routine for solving problem B looks like this:

def solveB = {
ny_guess <- chooseConservatively
payoff_b1 = 100.0 * (30+ny_guess) / 90
payoff_b2 = 100.0 * 60 / 90
if (payoff_b1 > payoff_b2) {
return "choose B1"
} else {
return "choose B2"
}
}


The key part of the theory is at the end of the lexical scope (i.e., the last }), the intermediate variables vanish. So during the execution of solveA, the value of nb_guess is available. Once the solution is computed, and the return ... statement is reached, the intermediate variables (namely nb_guess, payoff_a1 and payoff_a2) exit the local scope and are lost.

Because of this, there can be no relation between nb_guess and ny_guess. This allows nb_guess to pessimistically estimate the payoff probability in problem A, while ny_guess pessimistically estimates the payoff probability in problem B.

# Loss of scope can amplify other biases

If this "scoping" theory of reasoning is true, this would suggest a significant enhancement of other cognitive biases. Specifically, consider a motivated reasoning process leading to a desirable conclusion C, using an intermediate step B. However, the intermediate step B implies an undesirable conclusion D. The scoping model of bounded rationality predicts that a person will be less likely to conclude B => D, therefore not B.

A concrete example, based on an actual argument I've heard many times.

Two consenting adults should be able to do whatever they like in the privacy of their own bedroom. Therefore, anti-sodomy laws should be repealed.

The logical argument being made here is the following.

1. Sodomy is a private act between adults.
2. Private acts between consenting adults should not be regulated.

Yet strangely, the opponent of anti-sodomy laws is rarely persuaded by his own argument when the private act is one adult paying another adult $7.24/hour ($0.01 below the US minimum wage, for those outside the US) to clean his bedroom, or one adult providing another adult with drugs not approved by the FDA. I propose that having the intermediate principle vanish from the lexical scope is one possible explanation for this inability to reason.

Another example of motivated reasoning, where the illogical results are a bit more mathematical, comes from Austrian economics. The argument (and I'm paraphrasing an in-person conversation) goes something like this:

Private-sector actors are reasonably efficient at allocating capital to it's most productive uses - specifically uses with the highest rates of return. Loose money policies by the Federal Reserve caused a flood of money, thereby inducing capitalists to malinvest in housing - the result was investments in housing which had a negative rate of return.

This argument takes a similar form. The logical conclusion of the first clause is that a flood of money should not cause malinvestment - after all, why would those private-sector actors to suddenly lose their ability to funnel money to sectors with high rates of return, or at least keep the money in cash form rather than putting it into sectors with negative rates of return?