1. Imagine two countries, A
and B, that have a disputed piece of territory between them. Tension is mounting.
The ministers of country A meet in emergency session to plot their course of
action. They conclude that there are really only two options open to them. One
is to offer to negotiate the dispute with country B, and the other is to attack
swiftly in an attempt to take the territory by force. The main uncertainty is
over what country B will do. Will country B negotiate in good faith, or will it
try to attack first?
The ministers of A agree that
the best thing would be to attack while B is preparing to negotiate. The worst
possibility would be to have B attack as they are trying to negotiate. In that
case, B would get most of the territory. However, if both countries attack
simultaneously, being fairly equally matched, they will probably each end up with
about half the territory, after fighting a short war. If B would negotiate,
they could probably end up agreeing to split the territory without a war.
A. Set up a matrix for the decision problem from the standpoint of country A. Rank the outcomes using 1st for the best and 4th for the worst.
A’s Matrix would look like this:
|
|
B Negotiates |
B Attacks |
|
Negotiate |
2nd |
4th |
|
Attack |
1st |
3rd |
B. Which RCT decision strategy applies (e.g., maximizing expected utility, play it safe, gamble, best option, satisfactory option)? What action does this strategy indicate? (If more than one strategy applies, discuss all of them.)
Given the rankings in the matrix, A would apply the best option strategy, because regardless of whether B negotiates or attacks, A would get a better result by attacking. More precisely, A would get its 1st outcome over its 2nd if B negotiates and its 3rd over its 4th if B attacks. Thus, A would attack.
C. Country B is in exactly the same position as country A. If it applies the same decision strategy, what will the outcome be?
Since B would also decide to attack by applying the best option strategy, A and B would be faced with their 3rd-ranked outcome (both countries attacking).
D. Return to question B above. Suppose country A believed that the probability that country B would decide to negotiate was 2/3. Assuming that country A is still making its decision according to the simple dictates of RCT, how would having this information affect its decision making strategy?
If A wanted to summarize all the available information, its matrix would now look like this:
|
|
B Negotiates |
B Attacks |
|
Negotiate |
P = 2/3 2nd |
P = 1/3 4th |
|
Attack |
P = 2/3 1st |
P = 1/3 3rd |
However, s ince we’re working with a situation in which there is a best (or dominating) option, the probabilities we now have would not effect A’s decision. Attacking would still lead to the best result whether B negotiates or B attacks. Another way of thinking about why the probabilities can be ignored is to choose some scale to rank your utilities and plug in values for rankings. 2/3 of your highest ranked outcome will be higher than 2/3 of the 2nd-ranked outcome and 1/3 of your 3rd-ranked outcome will be higher than 1/3 of your 4th-ranked outcome. Thus, the sum of 2/3 of the 1st-ranked and 1/3 of the 3rd-ranked will always be higher than the sum of 2/3 of the 2nd-ranked and 1/3 of the 4th-ranked.
2. Use as the basis for the
following questions the matrix for the Prisoner's Dilemma presented in Lecture
14, slides 3 and 4. (Or you can use the one in Plous on p. 247 if you change
the box where Plous writes "5 years" to one saying "4
years" for each prisoner.)
A. Let's now explore the "Feel Your Pain" resolution. (As you will surmise from the last question in this section, our results will be somewhat inconclusive.) Suppose the first prisoner (Able) is so empathetic that he feels the second prisoner's pain in an amount equal to his own. (So if Baker gets a 10 year sentence, that would be just as bad as if Able himself got 10 years.)
i) Set up a new decision matrix for the perfectly empathetic Able and insert Able's new utilities. Now apply RCT. Which decision strategy now applies - or can you tell? Discuss.
We need to add Able’s “costs” together with Baker’s, so Able’s revised matrix would look like this:
|
|
Baker Confesses |
Baker Doesn’t Confess |
|
Confess |
-4 + -4 = - 8 years |
0 + -10 = -10 years |
|
Don’t Confess |
-10 + 0 = -10 years |
-1 + -1 = -2 years |
Now there is no best (or dominating) option for Able. Using the gambler’s strategy Able would choose not to confess, because he’d give himself the chance of feeling the pain of two years in prison. The playing-it-safe strategy does not apply to this situation, because there’s no way to avoid the worst outcome. That is, in both cases (confess and don’t confess) he’d have the chance of getting the worst outcome (i.e., feeling 10 years in prison).
In class discussion it emerged that it was not intuitively obvious to some that the Confess/Confess box should be - 8 instead of simply - 4. Maybe it would help to think of the dilemma as dollars in fines instead of years in prison.
ii) Suppose Baker's decision matrix remains as before (Baker is NOT empathetic) while Able is perfectly empathetic. What is likely to be the outcome after both Able and Baker make their choices (still assuming they are completely isolated from one another)? Does it matter whether or not Baker knows that Able is perfectly empathetic?
As stated in the original prisoner’s dilemma, Baker would choose to confess. Assuming Able will gamble and not confess, Baker would get 0 years in prison and Able now gets slammed with feeling a 10 year prison sentence. In this case, Baker does not need to know that Able is perfectly empathetic, because using RCT he would choose to confess without that information.
iii) What if Able and Baker are both perfectly empathetic and know that about each other?
If Baker is now perfectly empathetic, his matrix will look like Able’s revised matrix in part (i). Using the gambler’s strategy, Baker would choose not to confess. Since Baker and Able now know about the perfect empathy of their partners in crime, they should stick with not confessing, because they would then get their 1nd-ranked choice (feeling the pain of two years in prison).
iv) What if Able and Baker are each perfectly empathetic but mistakenly believe of the other that they are NOT at all empathetic?
Let’s take Able’s case first. Being perfectly empathetic, he would choose not to confess. Thinking that Baker was not at all empathetic, Able would reason that Baker will confess (as dictated in the original prisoner’s dilemma situation), thus leaving Able feeling 10 years in the slammer. Able would then revise his original decision and confess, because doing so would lead to the better outcome (i.e., feeling 8 years in prison). Baker is going through the same process of reasoning, so in the end, we’d have both Able and Baker confessing, and again, they’d be left with their 2nd-ranked outcome (getting 8 years in prison).
v) Return to question (i) above. What does RCT dictate if Able believes that there is only a 1/3 chance that Baker would confess? What if Able believes that there is a 9/10 chance that Baker would confess?
Plugging in the 1/3 probability, A’s matrix would now look like this:
|
|
Baker Confesses |
Baker Doesn’t Confess |
|
Confess |
P = 1/3 -8 years |
P = 2/3 -10 years |
|
Don’t Confess |
P = 1/3 -10 years |
P = 2/3 -2 year |
Since there is no dominating or best option in this situation, we can calculate the expected utilities:
EUConfess = (1/3)(-8) + (2/3)(-10) = -2.66 + (-6.66) = -9.33
EUDon’t Confess = (1/3)(-10) + (2/3)(-2) = -3.33 + (-1.33) = -4.66
Using the maximize expected utility strategy, Able would choose to not confess.
If Able believes there is a 9/10 chance that Baker would confess, we get the following the expected utilities:
EUConfess = (9/10)(-8) + (1/10)(-10) = -7.2 + (-1) = -8.2
EUDon’t Confess = (9/10)(-10) + (1/10)(-2) = -9 + (-.2) = -9.2
Applying the maximize expected utility strategy, Able would revise his decision and confess.
vi) Just to think about (no need to write out an answer): Is just empathy enough to guarantee a good outcome to PD situations?
B. Let us now explore the Moral Imperative remedy for PD situations. Suppose Able has been brought up in a tightly-knit mob family where children have been told from Day 1 to NEVER tell the police anything. (Perhaps a more realistic case would be American soldiers taken as prisoners of war who are honor bound never to tell the enemy anything except their name, rank and serial number.)
i) If both Able and Baker follow the mobsters’ code, what will the outcome be?
Neither Able nor Baker would confess if each follows the mobsters’ code, so they’d be in prison a year each.
ii) Suppose only Able is bound by the mobster code, what will the outcome be?
If only Able is bound to the code, he’d choose not to confess. Baker would confess by applying RCT, so Baker wouldn’t go to prison at all and Able would have to suffer through 10 years in the Big House.
iii) Suppose Able is bound by the mobsters’ code and Baker is not. Suppose Able is also perfectly altruistic and believes that the probability that Baker will confess is 9/10. What conflict might Able now face? Would it matter whether Able had also been brought up his mobster family to believe that he had a duty to always do what is best for himself and his partners in crime no matter what?
Being bound to the mobsters’ code, Able would feel a duty not to confess. Yet if he is perfectly altruistic, Able would be looking the matrix described in A(i) above. Applying the 9/10 probability to his situation, the maximize expected utility strategy would dictate that Able cooperate (as explained in A(v) above). Hence, Able has a conflict between the dictates of his mobster code and those of RCT.
If Able also felt a duty to do what is best for himself and his partners in crime no matter what, there still appears to be no clear resolution of the conflict, because what’s best for himself conflicts with what’s best for his partner in crime.
iv) Just to think about (no need to write out an answer): What are some limitations on solving the PD by invoking moral imperatives?
3. You have not been given any problems like the following one before. However, the web provides a table of "coefficient of relatedness" values (which are used in "Hamilton's Rule" for the prediction of altruistic behavior). With this information in hand you should approach this problem like an SAT question.
A. Suppose a retired biologist named Ham decides to make out her will using as a recipe for distribution the coefficients of relatedness found in Evolutionary Theory (ET). Ham has 4 surviving children and 5 cousins. If Ham's total estate is worth $550,000, how much should she leave to each of her 9 descendants?
Here’s a table derived from the web reading:
Relationship |
r |
|
Parent-Offspring |
0.5 |
|
Grandparent-Grandchild |
0.25 |
|
Cousin-Cousin |
0.125 |
What’s important to notice is that the value of r for Parent-Offspring (.5) is 4 times greater than the value of r for Cousin-Cousin (.125). This tells us that Ham should leave four times more of her estate to each child than to each cousin. In other words, if a cousin gets X amount of the estate, a child should get 4X of the estate.
We now need to take into account the total number of recipients: 4 children and 5 cousins.
X = amount left to one cousin
4X = amount left to one child
Based on this information, we can set up the following equation:
(4 children)(4X) + (5 cousins)(X) = $550,000
16X + 5X = $550,000
21X = $550,000
X = $26,190
The amount left to one cousin is $26,190. Therefore, the amount left to one child is 4($26,190) = $104,760.
B. Ham decides to think in more detail about her will. Of
her 4 children she reckons that 2 of them will never amount to anything from a
reproductive point of view because she is certain they will never have any off-spring. Of the
remaining children, she believes one is 75% likely to have a nice brood of 6
kids. She reckons that the odds are two to one that her fourth child will have
3 kids. She doesn't know her cousins well enough to say but people in their
general socio-economic class have on average 0.9 children each. Now how should
Ham divide her estate if her goal is to use her money to support the
contributions of her own genes to future generations?
With this new information, Ham will only have to consider leaving money to 2 children. Let’s call them A and B. A has a 2/3 chance of having 3 children, and B has a .75 chance of having 6 children. Each of these grandchildren has a .25 coefficient of relatedness to Ham. For the 5 cousins, each has a 100% chance of having .9 children. The value of r associated with each of these children will be .125 * .5 = .0625. Again the amount given to each child will be 4 times more than that given to each cousin, so we can set up our equation as follows:
(Amt. To A) + (Amt. To B) + (Amount to all Cousins) = $550,000
(2/3)(3)(4X) + (.75)(6)(4X) + (1)(.9)(5X) = $550,000
8X + 18X + 4.5X = $550,000
30.5X = $550,000
X = $18,032
Note that in this problem X stands for the amount that a cousin would get BEFORE taking into account the fact that they only have 0.9 child on average. So each cousin will actually get (.9) of $18,032 or $16, 228. A will get 8X = $144,256 and B will get 18X = $324,576.
C. Suppose Ham were going to give advice to Able re how
empathetic he should be to Baker in the PD situation discussed in the above
problem. Suppose Baker is Able's grandchild. Draw up a decision matrix for the
PD situation described in Problem #2 above if one assumes that Able's degree of empathy is based on the strength of
"blood ties".
Able will now have to consider how many years to add onto the value of each outcome based on the value of r for Grandparent-Granchild (.25). For example, instead of adding another –4 years to the confess-confess case, Able would add on (-4)(.25). The matrix would now look like this:
|
|
Baker Confesses |
Baker Doesn’t Confess |
|
Confess |
-4 + -4(.25) = - 5 years |
0 + -10(.25) = -2.5 years |
|
Don’t Confess |
-10 + 0(.25) = -10 years |
-1 + -1(.25) = -1.25 years |
Just as in the original case, there is no best (or dominating) option. Now, however, Able can choose to either play-it-safe or gamble. Using play-it-safe, Able would stay away from the possibility of getting 10 years in prison and choose to confess. Using gambler’s strategy, Able would not confess so that he would have the possibility of only getting 1.25 years in prison.
4. (The presentation of the
following problem is seriously over simplified but the point it makes has some
validity.) A psychologist in a mental hospital is trying to decide what
clinical diagnosis to attach to a patient. There are two obvious choices: one
is to classify the patient as having a borderline case of Delusional Disorder;
the other is to select the category of Paranoid Schizophrenic. The clinician
thinks that there's a good probability (say 70%) that the less pathological
diagnosis will turn out to be the correct one. However, being a good student of
RCT she also has to consider the costs and benefits to her reputation in the
clinic is she picks the wrong diagnosis.
If she concludes that the patient is sicker than what might first appear, then the clinic will be proud of any future progress the patient makes and not blame her for any lack of progress. If, however, she pronounces the patient less sick than might at first appear, the rest of the staff will expect progress and a failure (particularly a dramatic one such as a suicide or violent attack) could be a disaster to her career.
A. Draw up a simple two-by-two decision matrix for the psychologist and rank order the outcomes. What choice is dictated by RCT?
|
|
DD Diagnosis turns out to be Correct |
PS Diagnosis turns out to be Correct |
|
Say it's a Borderline case of Delusional Disorder (DD) |
3rd |
4th |
|
Say it's a case of Paranoid Schizophrenia (PS) |
1st |
2nd |
Here is how I arrived at the above matrix. First, I assumed that the doctor was only considering two possible diagnoses. Here's how I arrived at the rankings. The phrase "disaster to her career" plus suicide/violent attack seemed to clearly signal the worst outcome (4). "Proud of progress" signaled to me that overestimating how serious the patient's condition was would be of high value (1), because it would then appear to the clinis that the patient was progressing! Now, as is often the case, figuring out to assign the 2 and 3 rankings is more difficult. I used the first sentence to come up with the 2 ranking. But note that in this problem the assignment of 2 and 3 doesn't affect the question of whether there is a dominant strategy.
Applying the best option strategy, RCT dictates that the doctor diagnose the patient with Paranoid Schizophrenia, despite the fact that the doctor is 70% confident that the correct diagnosis is a borderline case of delusional disorder.
B. The patient is also interested in how the diagnosis turns out and has a choice as to whether to try to appear as normal as possible during the interview thereby hoping to receive the milder label of borderline delusional or whether to dramatize his erratic thought patterns, thus hoping to receive a diagnosis that reflects a more disturbed state.
The patient has a great fear (paranoia?) of being locked up in an institution even though there's nothing really much wrong with him. On the other hand, he would take a certain glee in fooling all the shrinks into thinking he was pretty much OK if he were indeed stark raving mad. He doesn't particular like the idea of being treated as mad even if he were mad, but on the other hand it would be quite boring to both be more or less and normal and also treated as being more or less normal.
B. Draw up a decision matrix for the patient. What choice is dictated by RCT?
|
Receive DD Diagnosis, which is actually correct |
Receive DD Diagnosis, which is in fact incorrect |
Receive PS Diagnosis, which is actually correct |
Receive PS Diagnosis, which is in fact incorrect |
Act normal like a Borderline case of |
Hi prob 3 |
Hi prob 1 |
Lo prob |
Lo prob |
Act erratic like a case of Paranoid Schizophrenia (PS) |
Lo prob |
Lo prob |
Hi prob 2 |
Hi prob 4 |
In this case we have to consider four outcomes as indicated above. The probability assignments are based on the assumption that if he acts normal he is more apt to get a DD diagnosis, while if he acts erratic he's more apt to get the PS diagnosis. The rankings above were extracted from the description but they are incomplete! However, the most important thing to notice is that in the patient's matrix, the probabilities of the outcomes are affected by which action he selects. This is quite different from what happens in the classic PD or clash-of-wills scenario. The choice dictated by RCT will depend on the actual values of the probabilities. There can be no best/dominating option in such a case.