COU 5: Advice

Imagine the political leader of a nation at war. The leader has the ultimate authority to direct whether and how the war continues. But, as the nation’s political leader, she lacks first-hand access to the information required to decide how she ought to use her power. By necessity, she spends all her time in government buildings in her nation’s capitol, hundreds of kilometers from the front lines. Thus she has no first-hand access to any of the information she needs to use her power wisely. Information about the condition of her nation’s forces and those of the enemy’s, about what orders are given by her generals and what her nation’s forces do in response to those orders, about how many are killed and wounded and how much ground is taken or lost in each battle – i.e., about everything relevant to deciding how the leader should exercise her authority – comes to her second-hand, via reports from military generals, staff in her national intelligence agency and stories in the media.

This leader’s reliance on second-hand reports is ubiquitous in politics. In all political institutions, some persons have the information and expertise needed for decisions, while others have the authority to make those decisions. Thus if information and expertise are to inform the exercise of authority, there must be some way for the well-informed to credibly communicate what they know to those in power.

But, for better or worse, the credibility of any such communication is always questionable. After all, if some piece of information determines whether a decisionmaker prefers one action to another, then the ability to change that decisionmaker’s beliefs about that information amounts to the power to control her decision. Thus credibility is itself a form of power, and thus everyone with an interest in a decisionmaker’s actions wants credibility, regardless of their actual expertise or inclination to honesty. How, then, does communication work when the credibility if experts is questionable? In this COU, you’ll explore a parameterized model in which a leader makes a decision on the basis of advice from an expert who has only partial credibility for accuracy and honesty.

The model depicts the interaction of two persons: a leader who must choose which of two policies to implement – labelled policy A and policy B –, and an advisor who gives the leader advice before the leader chooses her action. Events in the model proceed as follows:

The leader has prior beliefs about which of the two policies – again, policy A or policy B – is in fact the best policy.
The advisor gives advice – specifically, he either says ‘A is best’ or says ‘B is best’.
The leader forms posterior beliefs, in light of the advisor’s statement, about which policy is in fact the best policy.
The leader chooses whether to implement policy A or instead implement policy B.

Notice that, like the model in Lesson 6, COU 4, this is a model of learning and decisionmaking, depicting the leader first updating her beliefs, and then making decision.

Part 1

We’ll begin with a non-parameterized version of the model. Suppose the leader’s prior beliefs about which of the two policies is in fact best, and about the likelihood of each of the advisor’s statements given which policy is best are described by the following Marginal-Conditional Diagram:

This diagram depicts a situation in which the leader believes that the advisor is only partially credible. More precisely, the leader believes the advisor is biased in favor of policy B in a way that distorts the truth. To see this, first consider what occurs when policy B is in fact the best policy (depicted in the right-hand portion of the diagram). In this case, the advisor will tell the truth, saying ‘B is best’ with certainty (i.e. with probability 1). On the other hand, when policy A is in fact best (left-hand portion of the diagram), there is some chance the advisor will falsley say ‘B is best’. Specifically, conditional on the event that A is in fact best, the advisor will (truthfully) say ‘A is best’ with probability \frac{1}{3} but (falsely) say ‘B is best’ with probability \frac{2}{3}.

All this means more specifically that the statement ‘B is best’ by the advisor is not entirely credible. When she hears it, the leader will believe that there is some chance that B really is in fact the best policy and the advisor is telling the truth, but there is also some chance that A is the best policy and the advisor is either misinformed, deluded or lying.

Recall that once the advisor makes his statement, the leader has two actions available: She can implement policy A or implement policy B. Suppose that the leader’s utility level depends on which policy she implements and which policy is in fact best as follows: \begin{cases} 1 & \text{if the policy the leader implements is the policy that is in fact best.}\\ 0 & \text{if the policy the leader implements is the policy that is not in fact best.} \end{cases} For instance, if the leader implements policy A and policy A is in fact the best policy, the leader’s utility level is 1. On the other hand, if the leader implements policy A but B is in fact the best policy, the leader’s utility level is 0.

Under this assumption, the leader gets her highest possible expected utility by implementing whichever policy she believes is more likely to be the best policy. With this fact in mind, re-examine the diagram describing the leader’s beliefs…

…and notice that the leader’s prior belief about which policy is best puts probability \frac{2}{3} on the event that A is in fact the best policy and probability \frac{1}{3} on the event that B is in fact the best policy. Thus if the leader had to choose which policy to implement without first hearing from the advisor, she would implement policy A. Further, notice that if the advisor says ‘A is best’, the leader will be absolutely certain that A is in fact the best policy, since the only circumstance in which the biased-towards-B advisor says ‘A is best’ is when A is in fact best.

The question, then, is what happens when the advisor says ‘B is best’? More specifically, is the advisor in this model sufficiently credible that the statement ‘B is best’ will cause the leader to choose B when otherwise (either if she did not hear from the advisor or if the advisor said ‘A is best’) she would choose A? Develop your answer by working through the following prompts.

Prompt 1A

Use a table like the following to write the leader’s posterior beliefs about which policy is in fact best conditional on the event that the advisor says ‘B is best’…

Event	Probability
A is in fact best
B is in fact best

…and mark up a copy of the Marginal-Conditional Diagram above to show how you computed you answers. Remember, you are not required to simplify any fractions you compute!

Scoring for Prompt 1A

For each of the two cells of the table:

You get 3 points if you write a correct expression (regardless of whether or how it is simplified) and a diagram demonstrating that you used the correct procedure to compute that expression.
You get 1 point if you write a correct expression (regardless of whether or how it is simplified) but provide no diagram or a diagram that fails to demonstrate the use of a correct process to compute that expression.
You get 0 points if you do not write a correct expression.

Prompt 1B

Using the above assumptions about the leader’s utility function and the results you calculated in response to Prompt 1A, fill out the following table to describe the probability distribution over the outcomes that can occur and leader’s utility level when the advisor says ‘B is best’ AND the leader then implements policy A…

Outcome	Probability	Leader's Utility
The policy implemented is the policy that is in fact best.
The policy implemented is not the policy that is in fact best.

…THEN write the expression for the leader’s expected utility from implementing policy A when the advisor says ‘B is best’, remembering that you do not have to simplify any fractions in this expression.

Scoring for Prompt 1B

You get 1 point for each of the four table cells in which you provide a correct expression (regardless of how or whether it is simplified), plus 1 additional point for providing a correct expression (regardless of how or whether it is simplified) for the expected utility. Note that you do not get that additional point if the expression for the expected utility you write is incorrect, even if it is produced by correctly applying the expected utility formula to values from the table that are incorrect.

Prompt 1C

Continuing to apply the same utility function for the leader, use the results you calculated in response to Prompt 1A, to fill out the following table to describe the probability distribution over the outcomes that can occur and leader’s utility level when the advisor says ‘B is best’ AND the leader then implements policy B…

Outcome	Probability	Leader's Utility
The policy implemented is the policy that is in fact best.
The policy implemented is not the policy that is in fact best.

…THEN write the expression for the leader’s expected utility from implementing policy B when the advisor says ‘B is best’, remember that you do not have to simplify this expression.

Scoring for Prompt 1C

Prompt 1D

When the advisor says ‘B is best’, which action yields the leader the highest possible expected utility?
Explain how you know the answer with reference to your answers to prompts 1B and 1C.
Say whether or not the advisor suffiently credible in this model to change the leader’s decision from what it would be otherwise by saying ‘B is best’.

Scoring for Prompt 1D

You get one point for correctly answering each of prompts (i), (ii) and (iii). Note that a correct explanation in response to (ii) requires correct expressions for the relevant expected utilities.

Part 2

In this part, you’ll use a parameterized model to depict variation in the advisor’s credibility to the leader, and the effects of that variation on how the leader responds to the advisor.

Let \alpha (the Greek letter pronounced “al-fah”) be a number between \frac{1}{2} and 1, let \beta (the Greek letter pronounced “bay-tah”) be a number between 0 and 1, and suppose that the leader’s prior beliefs about which policy is in fact best and the distribution of the advisor’s statement conditional on which policy is in fact best is given as follows:

There are a two things to make sure you understand about this conditional probability distribution:

First, consider the parameter \alpha. This is the prior probability that the leader assigns to the event that A is in fact the best policy. Recall that we assume that \alpha is between \frac{1}{2} and 1. Thus before the leader hears what the advisor has to say, the leader believes that A is more likely than B to be the best policy.
Second, consider the parameter \beta. This is the leader’s belief about the extent of the advisor’s “bias” in favor of policy B. More specifically, it is the probability that the advisor will lie and say ‘B is best’ when in fact A is best. Recall that \beta ranges from 0 to 1. At the the extreme of \beta = 0, the leader believes the advisor always tells the truth – i.e. the advisor is perfectly credible (or, if you prefer, “unbiased”). At the other extreme when \beta = 1, the leader believes there is no relationship between the truth and what the advisor says. Specifically, the leader believes the advisor will always say ‘B is best’ regardless of which policy is actually best.
Third, a increase \beta depicts a decrease in the advisor’s credibility. More precisely, as \beta increases, the likelihood that the advisor is incorrect or lying when he says ‘B is best’ decreases.

Just as in Part 1, assume the leader’s utility level depending on which policy she implements and which policy is in fact best is given by: \begin{cases} 1 & \text{if the policy the leader implements is the policy that is in fact best} \\ 0 & \text{if the policy the leader implements is the policy that is not in fact best} \\ \end{cases} Under this assumption the leader gets her highest possible expected utility by implementing whichever policy she believes is more likely to be the best policy. With this in mind, recall that the parameter \alpha is the prior probability the leader puts on the event that A is in fact the best policy, and that we assume that \alpha is between \frac{1}{2} and 1. This means that if the leader were forced to choose which policy to implement without first hearing from the advisor, she would implement policy A. Further, if the advisor says ‘A is best’, the leader will believe that A is in fact best with certainty, because the advisor only says ‘A is best’ when A is in fact best. Therefore, the operative question in this model is whether the advisor is sufficiently credible to persuade the leader to implement policy B by saying ‘B is best’ when otherwise (if the leader were to act without first hearing from the advisor or if the advisor says ‘A is best’) the leader would implement policy A. Identify the conditions under which this is the case by answering the following prompts.

Prompt 2A

Assuming that the leader’s beliefs are given by the conditional probability diagram above, Use the following table to write the expressions for the posterior probabilities the leader will assign to the event that A is in fact best and to the event that B is in fact best conditional on the event that the advisor says ‘B is best’…

Event	Probability
A is in fact best
B is in fact best

…and mark up a copy of the Marginal-Conditional Diagram above to show how you computed you answers. Remember, you are not required to simplify any fractions you compute!

Scoring for Prompt 2A

For each of the two cells of the table:

You get 3 points if you write a correct expression (regardless of whether or how it is simplified) and a diagram demonstrating that you used the correct procedure to compute that expression.
You get 1 point if you write a correct expression (regardless of whether or how it is simplified) but provide no diagram or a diagram that fails to demonstrate the use of a correct process to compute that expression.
You get 0 points if you do not write a correct expression.

Prompt 2B

Using your answer to prompt 2A and a graph like this…

…draw a graph showing for each possible value of \beta the posterior probability the leader assigns to the event that B is in fact best conditional on the event that the advisor says ‘B is best’.

Your graph need not correctly display the exact shape of the posterior probability as a function of \beta. However, it must correctly show:

The level of the posterior probability at \beta = 0 (when the advisor is completely credible).
The level of the posterior probability at \beta = 1 (when the advisor is completely non-credible), with a label on the vertical axis giving the correct expression for this level.
Using a label on the horizontal axis and a dashed vertical line from this level, the expression for the value of \beta at which the posterior probability that B is in fact best is exactly equal to \frac{1}{2}.

Scoring for Prompt 2B

To get full credit on this prompt, you must draw a graph that does all of the following:

Has a vertical axis that ranges from 0 to 1, with the level \frac{1}{2} clearly labeled and marked with a dashed horizontal line extending across the graph.
Has a horizontal axis that ranges from 0 to 1.
Has the path of a function drawn in which:
1. The level of the function at 0 on the horizontal axis is clearly marked on the vertical axis by a correct expression (regardless of how or whether it is simplified) for that level.
2. The level of the function at 1 on the horizontal axis is clearly marked on the vertical axis by a correct expression (regardless of how or whether it is simplified) for that level.
3. The point on the horizontal axis at which the function intersects the level \frac{1}{2} on the vertical axis is clearly marked with a correct expression (regarldess of how or whether it is simplified) for the value of \beta at which that intersection occurs.
4. The sign of the functions slope (either positive or negative) is correctly drawn over the entire domain of the function.

If your graph does not satisfy either (a) or (b), it gets zero points. If your graph satisfies (a) and (b), you get 1 point for each of (c)(i), (c)(ii), (c)(iii) and (c)(iv) that your graph satisfies.

Prompt 2C

Using the graph you drew in response to Prompt 2B, describe in complete and coherent sentences how an increase in the parameter \beta changes the effect of the statement ‘B is best’ by the advisor on the leader’s belief about whether A or B is in fact the best policy.

Scoring for Prompt 2C

A full-credit answer amounts to clear and coherent sentences that together accurately describe how an increase in the parameter \beta changes the effect of the statement ‘B is best’ on the leader’s beliefs. Note that the effect to be described is on the leader’s beliefs, not on the leader’s action! Further, what you must describe is an effect – i.e. the change or difference in beliefs that result from the advisor saying ‘B is best’ instead of saying ‘A is best’.

You get 4 points for fully meeting these expectations, 2 points for mostly but not entirely meeting them, 0 points otherwise.

Prompt 2D

Using your answer to Prompt 2A…

Write the expression for the leader’s expected utility from implementing policy B when the advisor says ‘B is best’.
Write the expression for the leader’s expected utility from implementing policy A when the advisor says ‘B is best’.
Write the inequality that describes the conditions under which the leader’s expected utility from implementing policy B is greater than or equal to her expected utility from implementing policy A when the advisor says ‘B is best’.
Re-arrange that inequality so that the paramter \beta is by itself on one side of the inequality.

Scoring for Prompt 2D

You get one point for each of items (i) through (iv) for which you give a correct answer.

Prompt 2E

If you correctly answered Prompt 2D, then you know that this model has a threshold implication. Specifically, there is a threshold on the parameter \beta such that when \beta is on one side of this threshold, the advisor’s statement has no effect on which policy the leader prefers to implement, and when \beta is on the other side of this threshold, the leader prefers to implement policy A when the advisor says ‘A is best’ and prefers to implement policy B when the advisor says ‘B is best’.

To show this, draw a diagram using a horizontal line that shows the possible values of \beta like this:

On your horizontal line, draw a tic mark with a label for the expression for the value of \beta describing the value of \beta at which the leader’s expected utilities from her available actions are equal when the advisor says ‘B is best’. Finally, label each side of the tic mark with whiskers and textual labels describing how the leader’s policy action depends on the statement by the advisor for values of \beta on each side of the tic-mark.

Scoring for Prompt 2E

A full-credit diagram meets all of the following criteria:

It is a line depicting the possible values of \beta, ranging from 0 on the left to 1 on the right.
There is a tic-mark somewhere in the middle of the line labelled with the correct expression (whether or however simplified) for the value of \beta at which the leader’s expected utilities from each of her available actions are equal when the advisor says ‘B is best’.
There are whiskers noting the two ranges of values of \beta on each side of the tic-mark, along with textual labels for each whisker. Each labels correctly and fully describe how the leader’s policy action depends on the statement by the advisor for values of \beta in the respective range.

You get 4 points if you draw a diagram that meets all three criteria. You get 2 points if your diagram meets all of the criteria, except that the expression on the tic-mark is incorrect or missing. You get 0 points otherwise.

Prompt 2F

If you correctly answered Prompt 2E, then you know that the threshold on the parameter \beta that determines whether the leader will implement policy B when the advisor says ‘B is best’ is a function of the parameter \alpha. Recall that \alpha is the prior probability the leader puts on the event that A is in fact best, and that it is a number between \frac{1}{2} to 1. Thus this model depicts a relationship between the leader’s prior beliefs and the level of credibility required for the advisor’s statement to affect the leader’s action. Specifically, notice that at one extreme when \alpha = \frac{1}{2}, the leader is maximally uncertain about which of the two policies is best, believing that they are equally likely to be the best policy. At the other extreme, where \alpha = 1, the leader is completely certain before hearing anything from the advisor that A is the best policy. Thus as \alpha ranges from \frac{1}{2} to 1, the leader becomes increasingly certain, prior to hearing from the advisor, that A is the best policy.

How does the leader’s degree of certainty affect the amount of credibility the advisor has to have to in order to persuade the leader to implement policy B by saying ‘B is best’? Using your answer to 2E, write complete and coherent sentences that describe how an increase in the value of the parameter \alpha changes the level of credibility required for the statement ‘B is best’ by the advisor to persuade the leader to implement policy B.

Scoring for Prompt 2F

A full-credit answer amounts to clear and coherent sentences that together accurately describe how a change in the parameter \alpha changes the threshold depicted in a correct answer to Prompt 2E. Note that the effect you must describe is on the threshold on the parameter \beta (depicting the credibility of the advisor) at which the advisor’s statement affects the leader’s action.

You get 4 points for fully meeting these expectations, 2 points for mostly but not entirely meeting them, 0 points otherwise.