COU 6: Analyzing a Another Model of Learning

In COU 2, we considered a group of persons choosing between two alternatives, A1 and A2, by voting. We supposed that the group uses a voting rule in which A1 is adopted if k or more of the persons vote for A1, and A2 is adopted otherwise, where k is a number between 1 and the number of persons in the group.

We further imagined a member of the group named Amir, who is uncertain about (1) which of the two alternatives is better and (2) how many of the group members other than him will vote for each alternative. Specifically, we imagined that Amir has the following beliefs:

Which alternative is better Probability
A1 \frac{1}{3}
A2 \frac{2}{3}
If A1 is the better alternative:
Outcome Probability
Fewer than k-1 of Amir’s fellow group members think A1 is better \frac{1}{16}
Exactly k-1 of Amir’s fellow group members think A1 is better \frac{3}{16}
More than k-1 or more of Amir’s fellow group members think A1 is better \frac{3}{4}
If A2 is the better alternative:
Outcome Probability
Fewer than k-1 of Amir’s fellow group members think A1 is better \frac{3}{4}
Exactly k-1 of Amir’s fellow group members think A1 is better \frac{1}{16}
More than k-1 or more of Amir’s fellow group members think A1 is better \frac{3}{16}

Imagine that the members of the group cast their votes in sequence, with Amir casting his vote last. Thus when it is Amir’s turn to vote, his uncertainty about how many of his fellow group members will vote for each alternative is resolved. On the basis of how many of his fellow group members vote each way, then, he can update his beliefs about which alternative is better.

This of course is a model of learning. Before the voting starts, Amir has prior beliefs that describe his uncertainty about which alternative is better and how many of his fellow group members will vote for each alternative. Then, Amir sees how many of his fellow group members vote for each alternative. He then forms a posterior beliefs, based on how many of his fellow group members have actually voted for each alternative, about which alternative is better. It looks like this:

Analyze this model of learning by responding to the following prompts.

Prompt 1

Draw a conditional probability diagram that depict Amir’s prior beliefs.

Prompt 2

Suppose that fewer than k-1 of Amir’s fellow group members vote for alternative A1. Use the following table to write Amir’s posterior beliefs about which alternative is better, given that fewer than k-1 of Amir’s fellow group members voted for alternative A1…

Which alternative is better Probability
A1
A2

…and redraw the diagram you drew in response to Prompt 1, marking it up to show how you calculated the values in the table.

Prompt 3

Suppose that exactly k-1 of Amir’s fellow group members vote for alternative A1. Use the following table to write Amir’s posterior beliefs about which alternative is better, given that exactly k-1 of Amir’s fellow group members voted for alternative A1…

Which alternative is better Probability
A1
A2

…and redraw the diagram you drew in response to Prompt 1, marking it up to show how you calculated the values in the table.

Prompt 4

Suppose that more than k-1 of Amir’s fellow group members vote for alternative A1. Use the following table to write Amir’s posterior beliefs about which alternative is better, given that more than k-1 of Amir’s fellow group members voted for alternative A1…

Which alternative is better Probability
A1
A2

…and redraw the diagram you drew in response to Prompt 1, marking it up to show how you calculated the values in the table.

Prompt 5

Suppose Amir decides that he will vote for whichever the alternatives seems most likely to be the better one. So, for instance, if he had to cast is vote purely on the basis of his prior belief…

Which alternative is better Probability
A1 \frac{1}{3}
A2 \frac{2}{3}

…he would vote for A2.

Given your answers to Prompts 2, 3 and 4 above, is there any profile of votes by his fellow group members that could “swing” Amir in favor of voting for A1? Put another way, will Amir believe that A1 is more likely than not to be the better alternative in any of the three posterior beliefs he can have in this model?

Answer “yes” or “no” and explain your answer by referencing the specific numerical values you computed in response to Prompts 2, 3 and 4.

Prompt 6

So far, we’ve imagined that the members of the group cast their votes sequentially and that Amir votes last, so that he gets to see how all his fellow group members vote before casting his own vote. Now imagine instead that everyone is going to cast their votes simultaneously. Specifically, each person will write her vote on a piece of paper, not showing what she writes to anyone else, and then, after everyone has written down her vote, one person will collect the pieces of paper and count the votes.

This new system of simultaneous voting bothers Amir, because he would like to know how everyone else will vote before he casts his own vote. Amir ruminates about this problem for a while, but then comes up with an idea. He thinks:

Hm…If fewer than k-1 of my fellow group members vote for A1, then my vote doesn’t matter. A2 will be selected regardless of how I vote. On the other hand, if more than k-1 of my fellow group members vote for A1, my vote also doesn’t matter. In that event, A1 will be selected regardless of how I vote. The only way my vote matters is if exactly k-1 of my fellow group members vote for A1. In that event, we’ll select A1 if I vote for A1 and we’ll select A2 if I vote for A2. So, even though I can’t know how my fellow group members will vote before I cast my vote, I should cast my vote on the assumption that exactly k-1 of them have voted for A1. Sure, that assumption might be wrong. But if it is, my vote won’t change the outcome anyway!

Using your calculations from the earlier prompts, write the belief Amir should have about which alternative is better on the assumption that exactly k-1 of his fellow group members have voted for A1…

Which alternative is better Probability
A1
A2

Then, notice that Amir’s reasoning will lead him to vote the basis of a belief about which alternative is better that is not grounded in how his fellow group members actually vote. Instead, he will vote on the basis of a posterior belief informed by an assumption that may be false. In about one-half a page of double spaced text, say whether or not you think Amir’s approach is reasonable and defend your position.

Prompt 7

As in Prompt 5, suppose Amir decides to vote for the alternative that he believes is more likely than not to be the best one. Recall that his prior belief is

Which alternative is better Probability
A1 \frac{1}{3}
A2 \frac{2}{3}

So if he votes purely on the basis of his prior belief, he will vote for A2. Now suppose that Amir decides to vote as described in Prompt 6…i.e. on the assumption that exactly k-1 of his fellow group members will vote for A1. Will making this assumption (as opposed to assuming the he has no information about how his fellow group members will vote) cause Amir to “swing” from voting for A2 to voting for A1? Answer “yes” or “no” and explain your answer.

Rubric

For Prompt 1

You get four points on this prompt if the diagram you draw includes all the required information and correctly uses all the relevant graphical conventions.

You get two points if you draw a square with the axes labeled properly and with the space within the square divided up into the appropriate number of boxes, but the relative sizes of any of the boxes are obviously wrong and/or any of the numerical labels for the probabilities are missing or wrong.

You get zero points otherwise.

For Each of Prompts 2, 3 and 4

You get two points if you put the correct response in each cell of the table and you draw a diagram that correctly shows how you computed those values.

You get one point if you draw a diagram that correctly shows how to compute the values in the table, but one or more of the values in the table are missing or incorrect.

you get zero points otherwise.

For Prompt 5

You get two points if you give the correct answer (“yes” or “no”) and you explain your answer by referencing the specific numerical values of the posterior probabilities that are relevant to your answer and employ those values in your answer in the correct way. You get zero points otherwise.

For Prompt 6

You get three points if you write the correct values in the table and if the reasons you give for your position reflect a correct understanding of the situation and Amir’s reasoning.

You get two points if you write the correct values in the table and if the reasons you give for your position reflect an understanding of the situation or Amir’s reasoning that is in any way incorrect.

You get zero points otherwise.

For Prompt 7

You get two points if you give the correct answer and an explanation for your answer that correctly identifies the relevant beliefs and whether each of them holds that it is more likely than not that A1 is the better alternative.

You get zero points otherwise.