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, he lacks first-hand access to the information required to decide how he ought to use his power. By necessity, he spends all his time in government buildings in his nation’s capitol, hundreds of kilometers from the front lines. Thus he has no first-hand access to any of the information he needs to use his power wisely. Information about the condition of his nation’s forces and those of the enemy’s, about what orders are given by his generals and what his nation’s forces do in response to those orders, about how many are killed and wounded and how much ground is taken in each battle – about everything relevant to deciding how the leader should exercise his authority – comes to him second-hand, via reports from military generals, staff in his national intelligence agency and stories in the media.
In this COU, you’ll explore a model in which a person must make a decision on the basis of advice from an expert who has first-hand access to the information the person needs. As the above example suggests, the problems this poses are inherent in the exercise of political authority. Political leaders are by necessity removed from the events they try to control, and thus must choose what to do on the basis of second-hand information from others in-the-know.
Here’s the model: Imagine a leader who must choose between one of two available actions, labeled A and B. Suppose that the effect of the leader’s action on his objectives depends on what we’ll call a “state of the world”. Think of the state of the world as a condition that is beyond the leader’s control and that affects which of his available actions is better from his point of view.
For instance, the leader could be a nation’s commander-in-chief who is deciding whether to order his country’s armed forces to continue a military offensive against an enemy nation, or instead to offer a truce to the enemy and order a halt to offensive actions. In this case, the state of the world could be the strength of the leader’s nation’s armed forces relative to those of the enemy. If his armed forces are strong relative to the enemy’s, the leader is better off ordering his forces to continue fighting. If his armed forces are weak relative to the enemy’s, the better decision is to order his forces to stop fighting and offer a truce.
Whatever the application, the important thing is that the state of the world can have one of two possible values, which we’ll label a and b. If the state of the world is a, action A is the better of the leader’s two available actions. If the state of the world is b, action B is the better of the leader’s two available actions. Specifically, the leader’s utility level, given the action she takes and the state of the world is given by:
\begin{Bmatrix} 1 & \text{if he chooses $A$ and the state of the world is $a$} \\ 0 & \text{if he chooses $B$ and the state of the world is $a$} \\ 0 & \text{if he chooses $A$ and the state of the world is $b$} \\ 1 & \text{if he chooses $B$ and the state of the world is $b$} \end{Bmatrix}
The leader’s problem is that he must choose between his two available actions (A or B) without knowing the state of the world (a or b). Before he chooses, however, he gets a report about the state of the world from another person who we’ll call an advisor. Suppose this advisor, unlike the leader, knows the state of the world. For instance, if we imagine the leader is a commander-in-chief deciding whether to order a military offensive to continue or stop, the advisor might be a military general who by virtue of commanding the armed forces in the field for months knows whether the nation’s forces are strong enough to gain further ground in fighting or so weak that continued combat will lead to their destruction.
Since the state of the world has one of two values (a and b) and the leader knows that, the advisor’s report in effect can claim only one of two things: “the state of the world is a” or “the state of the world is b”. But, unfortunately for the leader, what the advisor claims about the state of the world may or may not correspond to the truth, and the leader has no direct way to verify the advisor’s claim. Moreover, the leader knows that the advisor, even though she knows the state of the world, might lie.
Specifically, the advisor is one of three types: a-biased, b-biased, or honest. If the advisor is a-biased, she will report “the state of the world is a” regardless of whether the state of the world actually is a. If the advisor is b-biased, on the other hand, she will report “the state of the world is b” regardless of whether the state of the world actually is b. Finally, if the advisor is honest, she will report the state truthfully – i.e. she will say the state is a when it is a and b when it is b.
Unfortunately for the leader, all he can know for sure is what the advisor reports – i.e. whether the advisor says “that state of the world is a” or “the state of the world is b”. He cannot directly access either the state of the world or whether the advisor is a-biased, b-biased or honest. Thus, at the moment just before the leader hears the advisor’s report the leader knows that one of six possible things are true:
- The state of the world is a and the advisor is a-biased.
- The state of the world is a and the advisor is b-biased.
- The state of the world is a and the advisor is honest.
- The state of the world is b and the advisor is a-biased.
- The state of the world is b and the advisor is b-biased.
- The state of the world is b and the advisor is honest.
Suppose the leaders believes – again, just prior to hearing the advisor’s report – that the state of the world is a with probability \frac{1}{2} and b with probability \frac{1}{2}. Further, the advisor is honest with probability \frac{1}{2}, a-biased with probability \frac{1}{2}\alpha and b-biased with probability \frac{1}{2}(1-\alpha), where \alpha (the Greek letter pronounced “alf-ah”) is a number between 0 and 1. Therefore, the leader believes just before he hears the advisor’s report that the relative likelihoods of each of the six possibilities above are:
| Possibility | Probability |
|---|---|
| The state of the world is a and the advisor is a-biased. | \frac{1}{4}\alpha |
| The state of the world is a and the advisor is b-biased. | \frac{1}{4}(1-\alpha) |
| The state of the world is a and the advisor is honest. | \frac{1}{4} |
| The state of the world is b and the advisor is a-biased. | \frac{1}{4}\alpha |
| The state of the world is b and the advisor is b-biased. | \frac{1}{4}(1-\alpha) |
| The state of the world is b and the advisor is honest. | \frac{1}{4} |
Recall that an a-biased advisor reports “the state of the world is a” regardless of the actual state of the world, a b-biased advisor reports “the state of the world is b” regardless of the actual state, and an honest advisor reports the state of the world just as it is. Since the leader does not know the state of the world or the advisor’s type, then, the advisor’s report is also an uncertain event for the leader at the moment just before the leader hears her report. Moreover, the advisor’s report is distributed conditional on the state of the world and the advisor’s type. For instance, conditional on the event that the state of the world is a and the advisor is a biased, the advisor’s report is distributed as follows:
| Report | Probability |
|---|---|
| “the state is a” | 1 |
| “the state is b” | 0 |
On the other hand, conditional on the event that the state of the world is a and the advisor is b-baised, the advisor’s report is distributed as follows:
| Report | Probability |
|---|---|
| “the state is a” | 0 |
| “the state is b” | 1 |
Since there are two possible states of the world (a or b) and the advisor is one of three possible types (a-biased, b-biased or honest), there are six possible conditional distributions of the advisor’s report. Rather than listing all six tables describing these conditional distributions, we’ll depict the entire conditional probability distribution over the state of the world, the advisor’s type and the advisor’s report in a conditional probability diagram:
There is an aspect of this conditional distribution that you have not seen in any of conditional distributions in the lessons or COUs so far, so make sure you understand how to interpret it in the diagram: Conditional on the state of the world and the advisor’s type, one of the two possible reports occurs with probability 1, and the other occurs with probability 0. Therefore, each section of the diagram corresponding to a state of the world and a type for the advisor is “split” into only one portion along the diagram’s vertical axis. For instance, consider the the portion of the diagram depicting the distribution of the advisor’s report conditional on the event that the state of the world is a and the advisor is honest:
If the state of the world is a and the advisor is honest, the advisor reports “the state of the world is a” with probability 1, and “the state of the world is b” with probability 0. Thus, the portion of the vertical axis taken up by the event that the report is “the state of the world is a” is 1, and none of the vertical axis is taken up by the event that the report is the “state of the world is b”.
Part 1
Given the leader’s prior belief diagrammed above about the distribution of the state of the world, the advisor’s type and the advisor’s report, the leader can learn from the advisor’s report to form posterior beliefs about the the state of the world. In response to Prompts 1A and 2A below, you’ll calculate these posterior beliefs.
Prompt 1A
Fill in the following table with the expressions describing the leader’s posterior beliefs about the state of the world in the event that the advisor reports “the state of the world is a”…
| State of the World | Probability |
|---|---|
| a | |
| b |
…and mark up this diagram…
…to show how you computed the expressions for the blank cells in the table.
Prompt 1B
Fill in the following table with the expressions describing the leader’s posterior beliefs about the state of the world in the event that the advisor reports “the state of the world is b”…
| State of the World | Probability |
|---|---|
| a | |
| b |
…and mark up this diagram…
…to show how you computed the expressions for the blank cells in the table.
Prompt 1C
Draw a graph that shows the value of the parameter \alpha on the horizontal axis and the values of two expressions on the vertical axis at each value of \alpha:
- P(\text{state of the world is $a$} | \text{advisor reports "the state of the world is $a$"}) – i.e. the leader’s posterior belief about the probability that the state of the world is a given that the advisor reports “the state of the world is a”.
- P(\text{state of the world is $a$} | \text{advisor reports "the state of the world is $b$"}) – i.e. the leader’s posterior belief about the probability that the state of the world is a given that the advisor reports “the state of the world is b”.
As described in the video in the Lesson 5 about analyzing parameterized models of learning, there are three things your graph must correctly depict about each of these expressions: (1) the value of the expression when \alpha is at its minimum value of 0, (2) the value of the expression when \alpha is at its maximum value of 1, (3) whether the expression increases, decreases, increases then decreases or decreases then increases as \alpha increases from its minimum value to its maximum value.
HINT: Consider the expression
\frac{\frac{1}{4}x + \frac{1}{4}}{\frac{1}{4}x + \frac{1}{4} + \frac{1}{4}x } where x is some number between 0 and 1. It’s difficult to tell whether this expression, as written, increases or decreases as x increases, because x appears in both the numerator and the denominator. We can re-arrange the expression so that x appears only in the denominator as follows:
First, add and subtract \frac{1}{4} to the denominator like so: \frac{\frac{1}{4}x + \frac{1}{4}}{\frac{1}{4}x + \frac{1}{4} + \frac{1}{4}x + \frac{1}{4} - \frac{1}{4}} Notice that this expression has exactly the same value as the original expression, since we both added and subtracted \frac{1}{4} to the denominator. Now notice that the expression \frac{1}{4}x + \frac{1}{4} appears once in the numerator and twice in the denominator. More specifically, we can re-write the whole express as: \frac{\frac{1}{4}x + \frac{1}{4}}{2 \left(\frac{1}{4}x + \frac{1}{4}\right) - \frac{1}{4}} Now divide the both the numerator and denominator by \frac{1}{4}x + \frac{1}{4} like this: \frac {\frac{\frac{1}{4}x + \frac{1}{4}}{\frac{1}{4}x + \frac{1}{4}}} {\frac{2 \left(\frac{1}{4}x + \frac{1}{4}\right) - \frac{1}{4}}{\frac{1}{4}x + \frac{1}{4}}} Notice that this expression is exactly the same as the original expression, since all we have done is divided both the numerator and denominator by the same number. Further, simplifying this new version of the expression it becomes: \frac {1} {2 - \frac{\frac{1}{4}}{\frac{1}{4}x + \frac{1}{4}}} Now notice that as x increases \frac{1}{4}x + \frac{1}{4} increases. Thus as x increases the expression \frac{\frac{1}{4}}{\frac{1}{4}x + \frac{1}{4}} decreases. Thus as x increases the expression 2 - \frac{\frac{1}{4}}{\frac{1}{4}x + \frac{1}{4}} increases. Thus as x increases the expression \frac {1} {2 - \frac{\frac{1}{4}}{\frac{1}{4}x + \frac{1}{4}}} decreases. This last expression is equivalent to our original expression and thus we have our answer!
Prompt 1D
In about half a page of well-written, double-spaced text, describe how a change in the value of \alpha changes the effect of the advisor’s report on the leader’s posterior belief about the state of the world.
Rubric for Part 1
For Each of Prompts 1A and 1B
You get…
- 2 points if you write the correct values for both blank cells in the table and if you provide a diagram that correctly shows how to compute those values.
- 1 point if you write a diagram that shows that you used the correct procedure to compute the values in the table, but one or more of those values is missing or incorrect.
- 0 points otherwise.
For Prompt 1C
You get…
…four points if you draw a graph with the axes properly labeled and that shows the two posterior probabilities, each one clearly labeled, and with the level of each posterior probability correct at the minimum and maximum values of \alpha, and with the direction of change in each one as \alpha changes correct.
…three points if you meet all the criteria for four points, except the direction of change in one or the both posterior probabilities as \alpha changes is incorrect.
…two points if you meet all the criteria for three points except the level of exactly one of the two posterior probabilities is incorrect at either the minimum or maximum value of \alpha is incorrect, but not both.
…zero points otherwise.
For Prompt 1D
You get…
…four points on Prompt 1D only if you got four points on Prompt 1C and if what you write is accurate, coherent and well-written.
…three points on Prompt 5 only if you got four points on Prompt 4 and if what you write is accurate and coherent and but has errors in grammar, spelling or usage that make it anything less than perfectly easy to understand.
…zero points otherwise.
Part 2
Suppose that after the leader has heard the advisor’s report and formed his posterior belief about the state of the world, he must choose between his two available actions, A and B. Recall that the leader’s preferences over his available actions depend on the state of the world. Specifically, his utility level given his action and the state of the world is given by:
\begin{Bmatrix} 1 & \text{if he chooses $A$ and the state of the world is $a$} \\ 0 & \text{if he chooses $B$ and the state of the world is $a$} \\ 0 & \text{if he chooses $A$ and the state of the world is $b$} \\ 1 & \text{if he chooses $B$ and the state of the world is $b$} \end{Bmatrix}
Notice that regardless of the state of the world and the action taken by the leader, there are only two utility levels that can occur for the leader: 1 and 0. Thus, each of the leader’s available actions results in a probability distribution over her utility level like this:
| Utility Level | Probability |
|---|---|
| 1 | |
| 0 |
Prompt 2A
Suppose the advisor’s report is “the state of the world is a”. Using the leader’s posterior beliefs you computed in response to Prompt 1A and the utility function above, fill in the following tables to describe the distribution over the leader’s utility level resulting from each of her available actions…
| Leader’s Utility Level | Probability |
|---|---|
| 1 | |
| 0 |
| Leader’s Utility Level | Probability |
|---|---|
| 1 | |
| 0 |
…and explain, step-by-step, how you computed the cells in the tables.
Prompt 2B
Continue to suppose that the advisor reports “the state of the world is a”. Use your answer to Prompt 2A to compute the expressions for leader’s expected utility from each of her available actions…
| Action | Expected Utility |
|---|---|
| A | |
| B |
…and explain, step-by-step, how you computed your answers.
Prompt 2C
Suppose the advisor’s report is “the state of the world is b”. Using the leader’s posterior beliefs you computing in response to Prompt 1B and the utility function above, fill in the following tables to describe the distribution over the leader’s utility level resulting from each of her available actions…
| Leader’s Utility Level | Probability |
|---|---|
| 1 | |
| 0 |
| Leader’s Utility Level | Probability |
|---|---|
| 1 | |
| 0 |
…and explain, step-by-step, how you computed the cells in the tables.
Prompt 2D
Continue to suppose that the advisor reports “the state of the world is b”. Use your answer to Prompt 2C to compute the expressions for leader’s expected utility from each of her available actions…
| Action | Expected Utility |
|---|---|
| A | |
| B |
…and explain, step-by-step, how you computed your answers.
Prompt 2E
Do the leader’s preferences over her available actions (A and B) depend on whether the advisor reports “the state of the world is a” or “the state of the world is b”? For instance, does he get a higher expected utility from action A than from action B when the advisor reports “the state of the world is a”, but a higher expected utility from action B than from action A when the advisor reports “the state of the world is b”? State your answer, specifying how the expected utilities of the available actions are ordered relative to one another for each of the two possible reports of the advisor, and explain how you know it is correct with specific reference to the results you calculated in response to Prompts 2B and 2D.
Rubric for Part 2
For each of Prompts 2A through 2E…
- You get two points if all the answers you give are correct and you give a correct explanation for how you computed the results.
- You get one point if some or all of your answers are missing or incorrect, but you give an explanation that shows that you correctly understand the process for computing the result.
- You get zero points otherwise.
Part 3
If you correctly answered the prompts above, you know that the leader’s preferences over her available actions depend on the report she receives from the advisor, but do not depend on the value of the parameter \alpha. This is because the model as specified so far is symmetric in one important respect. Recall the leader’s utility function: \begin{Bmatrix} 1 & \text{if he chooses $A$ and the state of the world is $a$} \\ 0 & \text{if he chooses $B$ and the state of the world is $a$} \\ 0 & \text{if he chooses $A$ and the state of the world is $b$} \\ 1 & \text{if he chooses $B$ and the state of the world is $b$} \end{Bmatrix} Think about the leader choosing between her two available actions, A and B. Suppose the leader asks himself, “what if the state of the world is a but I choose action B? How bad would that be?” Examining the utility function above, you can see that if the state of the world is a but the leader chooses B instead of A, his utility will be lower by 1. Similarly, if the state of the world is b, but the leader chooses action A instead of B, his utility level will be lower by 1. In this sense, the stakes of the leader’s choice between his two available actions are symmetric. Choosing B incorrectly (i.e. when the state is a) is neither better nor worse than choosing A incorrectly (i.e. when then state is b).
In this and the next parts of this COU, you’ll explore the implications of a version of the model in which the stakes for the leader’s choice are asymmetric.
Suppose that the leader’s prior beliefs about the state of the world, the advisor’s type and the advisor’s report are exactly as in the original model, described by this diagram:
However, the effect of the state of the world and the leader’s action on her utility level is given by a slightly different utility function. Given the state of the world and the leader’s action, the leader’s utility level is given by:
\begin{Bmatrix} \gamma & \text{if he chooses $A$ and the state of the world is $a$} \\ 0 & \text{if he chooses $B$ and the state of the world is $a$} \\ 0 & \text{if he chooses $A$ and the state of the world is $b$} \\ 1 & \text{if he chooses $B$ and the state of the world is $b$} \end{Bmatrix} where \gamma (the Greek letter pronounced “gamm-ah”) is a number greater than 1.
The key thing to understand about this utility function is that it depicts a situation in which it is worse for the leader to choose B incorrectly than for the leader to choose A incorrectly. To see this, notice that if that state is a but the leader chooses action B instead of action A, he gets a utility level of 0 instead of \gamma. On the other hand if the state is b but he chooses A instead of B, he gets utility level 0 instead of 1. Since \gamma is a number bigger than one, the consequences of the former mistake are larger than the consequences of the latter mistake.
The only difference between this version of the model and the one you analyzed in Parts 1 and 2 of this COU is the utility function. Thus, all of the correct answers to Prompts 1A, 1B, 1C and 1D are correct applied to this version of model. Specifically, the leader’s posterior beliefs about the state of the world after he observes the advisor’s report in this version of the model are exactly the same as those in the version you worked with in Parts 1 and 2.
So, to analyze this model, take the answers you formulated to Prompts 1A and 1B and apply them to the following.
Prompt 3A
Here again is the leader’s utility function for this version of the model:
\begin{Bmatrix} \gamma & \text{if he chooses $A$ and the state of the world is $a$} \\ 0 & \text{if he chooses $B$ and the state of the world is $a$} \\ 0 & \text{if he chooses $A$ and the state of the world is $b$} \\ 1 & \text{if he chooses $B$ and the state of the world is $b$} \end{Bmatrix}
Notice that regardless of the state of the world and the action taken by the leader, there are three utility levels that can occur for the leader: 1, 0 and \gamma. Thus, each of the leader’s available actions results in a probability distribution over his utility level like this:
| Utility Level | Probability |
|---|---|
| \gamma | |
| 1 | |
| 0 |
Suppose the advisor’s report is “the state of the world is a”. Using the leader’s posterior beliefs you computed in response to Prompt 1A and the utility function for this version of the model, fill in the following tables to describe the distribution over the leader’s utility level resulting from each of her available actions…
| Utility Level | Probability |
|---|---|
| \gamma | |
| 1 | |
| 0 |
| Utility Level | Probability |
|---|---|
| \gamma | |
| 1 | |
| 0 |
…and explain, step-by-step, how you computed the cells in the tables.
Prompt 3B
Continue to suppose that the advisor reports “the state of the world is a”. Use your answer to Prompt 3A to compute the expressions for leader’s expected utility from each of her available actions…
| Action | Expected Utility |
|---|---|
| A | |
| B |
…and explain, step-by-step, how you computed your answers.
Prompt 3C
Suppose the advisor’s report is “the state of the world is b”. Using the leader’s posterior beliefs you computing in response to Prompt 1B and the utility function for this version of the model, fill in the following tables to describe the distribution over the leader’s utility level resulting from each of her available actions…
| Utility Level | Probability |
|---|---|
| \gamma | |
| 1 | |
| 0 |
| Utility Level | Probability |
|---|---|
| \gamma | |
| 1 | |
| 0 |
…and explain, step-by-step, how you computed the cells in the tables.
Prompt 3D
Continue to suppose that the advisor reports “the state of the world is b”. Use your answer to Prompt 3C to compute the expressions for leader’s expected utility from each of her available actions…
| Action | Expected Utility |
|---|---|
| A | |
| B |
…and explain, step-by-step, how you computed your answers.
Rubric for Part 3
For each of Prompts 3A through 3D…
- You get two points if all the answers you give are correct and you give a correct explanation for how you computed the results.
- You get one point if some or all of your answers are missing or incorrect, but you give an explanation that shows that you correctly understand the process for computing the result.
- You get zero points otherwise.
Part 4
In both the ‘asymmetric’ version of the model studied in Part 3 of this COU and the ‘symmetric’ version of the model studied in Parts 1 and 2, the leader believes that the state is more likely to be a than it is to be b when the advisor reports “the state of the world is a”. Thus, since in the asymmetric version of the model it is worse for the leader to incorrectly choose B when the state a than it is to incorrectly choose A when the state is b, the leader will prefer action A over action B when the advisor reports “the state of the world is a” regardless of the values of the parameters \alpha and \gamma.
But the situation is different (again, in the asymmetric version of the model), when the advisor reports “the state of the world is b”. Since the consequences for the leader of incorrectly choosing B are worse than the consequences of incorrectly choosing A, the leader will only be willing to choose B instead of A when he thinks that the likelihood that the state of the world is b is much higher than the likelihood that the state of the world is a. How much higher depends on the values of the parameters \alpha and \gamma.
Specifically, one can apply the correct answers to the prompts above to show that if the advisor reports “the state of the world is b”, then the leader gets a higher expected utility from choosing B than from choosing A if and only if the following inequality holds:
\frac{\frac{1}{4}(1-\alpha) + \frac{1}{4}}{\frac{1}{4}(1-\alpha) + \frac{1}{4} + \frac{1}{4}(1-\alpha)} \geq \gamma \frac{\frac{1}{4}(1-\alpha)}{\frac{1}{4}(1-\alpha) + \frac{1}{4} + \frac{1}{4}(1-\alpha)}
Prompt 4A
Re-arrange this inequality so that \gamma is alone (without any other numbers or parameters) on one side of the inequality, and an expression of the parameter \alpha is on the other side.
Prompt 4B
If you correctly re-arranged the inequality presented in Prompt 4A, you’ve shown that a threshold implication applies to this model. Specifically, there is a threshold on the value of the parameter \gamma, given by an expression of \alpha. If the parameter \gamma is less than this threshold, the leader responds to the advisor’s report. Specifically, the leader prefers action A to action B when the advisor reports “the state of the world is a”, and the leader prefers action B to action A when the advisor reports “the state of the world is b”. On the other hand, if the parameter \gamma is greater than this threshold, the leader ignores the advisor’s report. Specifically, the leader prefers action A to action B regardless of what the advisor reports.
Using your answer to Prompt 4A, add the expression of \alpha to the following diagram that depicts this threshold implication.
Prompt 4C
The threshold on the value of \gamma that determines whether or not the leader responds to the advisor’s report is an expression of \alpha. Draw a graph in which \alpha is on the horizontal axis and the value of the expression for the threshold is plotted on the vertical axis. As always, your graph must (1) correctly show the value of the threshold when \alpha is at its minimum value of 0, (2) correctly show the value of the threshold when \alpha is at its maximum value of 1, and (3) correctly show the direction in which the threshold changes as \alpha increases from 0 to 1.
Prompt 4D
The threshold on \gamma that determines whether the leader is responsive to the advisor’s report increases as the parameter \alpha increases. Recall that \alpha captures the likelihood that the advisor is a-biased instead of b-biased. Thus as \alpha increases, the probability that a report that “the state of the world of b” is as lie decreases. Put another way, as \alpha gets large, it becomes very unlikely that the advisor is b-biased. Thus the leader believes that a report that the state of the world is b is likely to occur only when the state of the world actually is b, since “the state of the world is b” is unlikely to be a lie.
In between half a page and one page of double spaced, well-written text, explain why the threshold increases as \alpha increases. Use no mathematical expressions in what you write. Also, do not refer to \alpha and \gamma. Instead, refer to the concepts they represent.
Rubric for Part 4
You can earn up to 2 points for the combination of all of what you write in response to Prompts 4A, 4B, 4C and 4D. You earn 2 points if your answers to 4A, 4B and 4C are all correct and what you write in response to Prompt 4D is accurate and meets all of the requirements stated in the prompt. You earn 1 point if what your answer Prompt 4D is accurate and meets all the requirements stated in the prompt, but the answers you give to any of Prompts 4A, 4B and 4C are missing or incorrect. You earn 0 points otherwise.