Question

2. [Uncertainty and risk] A DM is presented with two jars. Jar 1 has 50 red and 50 blue balls. Jar 2 consists of 100 total balls each of which is either red or blue but the colors are in an unknown proportion. An experiment consists of drawing a single ball from each jar. The DM faces the following two choices. Choice 1 is between option 1a which pays $100 if the Jar 1 ball is red, and option 1b which pays $100 if the Jar 2 ball is red (and $0 otherwise). Choice 2 is between option 2a which pays $100 if the Jar 1 ball is blue, and option 2b which pays $100 if the Jar 2 ball is blue (and $0 otherwise). Suppose the DM chooses 1a over 1b and 2a over 2b (and has a strict preference in each case).

Are these choices consistent with subjective EU? In other words, does there exist a probability distribution over the contents of Jar 2 (that is, a belief that the proportion of Red balls is p and of Blue balls is 1-p) such that, given these beliefs, the choices of the DM can be rationalized by expected utility? If so, provide the subjective probabilities that rationalize the choices. If not, argue that there are no such probabilities. [Note: Because the outcomes are only $0 and $100, risk preferences play no role here. That is, all utility functions for which u($100) > u($0) are observationally equivalent on these choices. Notice that in order for the curvature of the utility function to be relevant, one would need to consider at least three wealth levels. This is the reason that I am asking only about probabilities and not also about the utility function in this question.]

Answer 1

JAR 1:

• 50 red balls & 50 blue balls.
• Thus probability of getting a red ball is   and probability of getting a blue ball is also .

JAR 2:

• Total 100 balls with unknown probablity.
• We assume the probability of getting a red ball is "p" and probability of getting a blue ball is "(1-p)".

There exist a probability distribution over the contents of Jar 2.

Let X denotes drawing a red ball from Jar 2, then X follows binomial distribution i.e, Bin(100,p)

Now, the DM faces the following two choices.

Choice 1

• option 1a which pays $100 if the Jar 1 ball is red, and
• option 1b which pays $100 if the Jar 2 ball is red (and $0 otherwise).

Choice 2

• option 2a which pays $100 if the Jar 1 ball is blue, and
• option 2b which pays $100 if the Jar 2 ball is blue (and $0 otherwise).

Suppose the DM chooses 1a over 1b and 2a over 2b.

The above options consider the two possible cases

• p>1/2
• p<1/2

When p>1/2 the probability of getting a red ball from jar2 increases and consequently probability of getting a blue ball from jar2 decreases. This implies choice1 is more consistent than choice 2 i.e, winning 100$ over option 1b increases than over option 2b.

Whereas, when p<1/2 the probability of getting a blue ball from jar2 increases and consequently probability of getting a red ball from jar2 decreases. This implies choice2 is more consistent than choice 1 i.e, winning 100$ over option 2b increases than over option 1b.

Note: Option 1a and option 2a are both equally likely since the probability of getting a red ball from jar1 is equal to the probability of getting a blue ball from jar1.