kissingerj payoff table final

Joseph KissingerCase 1 – Payoff Table

Executive Summary

I recommend Tom Roberts.

Background

I have completed my training rotation. I will soon transition into a permanent role and a

new trainee must be chosen to take my place. This trainee will also serve as my junior analyst

during the beginning of my new assignment. I narrowed down the choices to five candidates and

invited each to the office for an interview and aptitude test, which measured the candidates on a

Likert scale of 1 to 7, with 7 being the best, in the four areas in which I rotated during my time in

the training program. Of the four areas, there is a 35% chance I will be permanently placed in

International Investing, a 20% I will be placed in Real Estate, a 35% of being placed in Personal

Finance, and a 10% probability of being assigned to Health Care. The raw data collected from

the aptitude tests can be found in the appendix of this report, in Figure 1: Aptitude Test Results.

The five candidates, all recent MBA graduates, are Samantha Lindle from Wake Forest, Andy

Karoly from Duke, Wendy Lee from UNC, Tom Roberts from Virginia, and Bill Williams from

Clemson. I considered who would be the most qualified candidate for the company, and who

would be the most helpful to me during my first months in my permanent role. This criteria

presented tradeoffs that had to be made to determine who I would recommend to take my place

and be my junior analyst.

A nalysis

The tool used to analyze the data in this report and for this recommendation was a payoff

table. Payoff tables are useful decision making tools when a scenario encompasses a number of

possible outcomes and a variety of possible responses. Payoff tables are also used when looking

at a single decision. To make a decision using payoff tables, we use decision rules to compare

decision alternatives to one another for a given state of nature. The decision rules give us

different looks at the data and a method of ranking each alternative against one another. The

rankings are analyzed and eventually used to make a single decision from a number of

alternatives and possible future states. Below, labeled Table 1: Payoff Table, is the original

payoff table used in this analysis.

Table 1: Payoff Table

35% 20% 35% 10%

International

InvestingReal

EstatePersonal Finance

Health Care B of B B of W E.L. E.V. M/M R

Samantha 20 14 10 21 21 10 16.25 15.4 10Andy 30 9 15 10 30 9 16 18.55 13Wendy 13 19 18 23 23 13 18.25 16.95 17Tom 16 15 18 19 19 15 17 16.8 14Bill 14 11 13 11 14 11 12.25 12.75 16

In the table above, the candidates’ names along the side are the alternatives while the four

business areas along the top of the table are the possible states of nature. The percentages above

the states of nature are the probability of each state occurring, given the chosen alternative. The

payoffs were derived by summing the candidates’ scores on the aptitude tests to get a total score

for each business area. Given the 1-7 scale of the aptitude test, the range of payoffs for each

candidate and business area was 5 to 35. The decision rules can be seen in the last five columns

and from left-to-right read: Best of the Best, Best of the Worst, Equally Likely score, the

Expected Value score, and the MiniMax Regret. Best of the Best shows the highest score for

each candidate, while the Best of the Worst shows each’s lowest score. When we take the

difference between the Best of the Best scores and the Best of the Worst scores we find the range

of payoffs for each candidate. The Equally Likely rule shows the average of each candidates’

scores. We can compare the Equally Likely rule to the midpoint of the range to determine where

the data is clustered. That is, an Equally Likely score greater than the midpoint tells us that most

of the data is on the high end of the range. The Expected Value rule is the weighted average of

each candidates’ payoffs, given the probabilities of each state of nature occurring. When

compared to the Equally Likely score, the Expected Value can tell us the relationship between

the weights and the payoffs. That is, an EV greater than the EL suggests a favorable match of

high payoffs and high probabilities. The MiniMax Regret rule tells us how much, in opportunity

costs, we would be giving up if a certain alternative is chosen. Because the MiniMax Regret and

Best of the Worst rules are conservative measures, they can be compared on a basis of rankings

to see if there is a relationship between the regret and the lowest payoff for each candidate. The

rules are ranked to analyze comparisons and make tradeoffs between payoffs. The rules here

were ranked using green, yellow, and red shading to indicate the winner, second place, and the

loser, respectively. The colors make reading the rules much easier and allow us to begin making

assumptions.

To analyze the payoff table most effectively and timely, I first checked to make sure there

was nothing on the table that did not need to be analyzed. I noticed that Bill Williams could be

eliminated from the table and my analysis because of domination. An alternative that is

dominated is one that, for each given state of nature, is consistently the loser when compared to

the other alternatives. If an alternative is dominated by at least one other alternative it can be

discarded, as I would never choose a lower payoff given the same probability and state of nature.

For example, in Table 1 we see that Tom has a higher score in all areas than Bill. When this is

the case we would choose Tom over Bill every time. Further, if we were to choose Lindle,

Karoly, or Lee over Tom Roberts, and Tom Roberts is consistently better than Bill Williams,

then any time we prefer a candidate over Tom Roberts we also prefer the same candidate over

Bill Williams. Therefore, Bill can be eliminated from our analysis and consideration. The payoff

table showing Bill to be dominated, with new rankings for the decision rules, is shown below as

Table 2: Domination. While the green, yellow, and red shading again indicate the winner,

second-best, and loser for each decision rule, notice the gray shading indicates Bill is dominated.

Table 2: Domination

35% 20% 35% 10%

International

InvestingReal



Samantha 20 14 10 21 21 10 16.25 15.4 10Andy 30 9 15 10 30 9 16 18.55 13Wendy 13 19 18 23 23 13 18.25 16.95 17Tom 16 15 18 19 19 15 17 16.8 14Bill 14 11 13 11 14 11 12.25 12.75 16

The first alternative to be analyzed is Samantha Lindle from Wake Forest University.

Samantha has the next-to-last Best of the Best (BoB) and Best of the Worst (BoW) scores, at 21

and 10, respectively. This relationship gives us the second largest range, at 11, which could be

concerning. The midpoint of this data range, 15.5, falls just below the equally likely (E.L) score,

at 16.25, indicating that the payoffs are slightly grouped above the midpoint, which is favorable.

The expected value (E.V.) score of 15.4 ranks last and is less than the E.L., indicating an

unfavorable matching of probabilities and payoffs. The regret score of 10 ranks the best on the

table, while our other conservative decision rule, BoW, ranks next-to-last for this alternative. The

disagreement between our conservative rules indicates the need for further inspection of our data

set. The regret table, which can be found in the appendix of this report titled “Table 3: Regret”,

shows why this relationship is in disagreement. With the exception to Andy, each alternative’s

regret score comes from the same state of nature, International Investing. Andy’s score of 30 in

International Investing is significantly greater than any other data point on the table, causing the

regret scores to be skewed by this outlier. The regret score, then, is more of a measure of how

much we would regret recommending someone other than Andy if we ended up in International

Investing, or how much we would regret choosing Andy if we ended up in Health Care. The

major takeaway from the MiniMax Regret score is that it is being adversely affected by an

outlier and thus should not be analyzed at the surface level, but requires further investigating to

fully understand. I will make reference to this throughout my analysis. This alternative causes

concern as the BoB and BoW scores are next-to-last and the range of 11 is the second biggest.

The best payoff for Samantha is 21 and the worst is 10, as shown in the BoB and BoW

decision rules. The other payoffs are 20 and 14, giving us the E.L. score of 16.25. The midpoint

of the range is 15.5, just below the E.L. The E.V. is 15.4 and is lower than the E.L. due to the

highest probability, 35%, of getting the lowest score of 10, and the lowest probability, 10%, of

getting the alternative’s highest payoff of 21. The remaining payoffs of 20 and 14 have

probabilities of 35% and 20%, respectively, thus bringing the E.V. closer to the E.L. and

midpoint. Though the regret score of 10 was the best on the table it is being affected by an

outlier, causing disagreement with our other conservative decision rule, BoW, and therefore is

not accurately measuring regret.

The highest payoff for this alternative is 21 and is a major reward. The lowest payoff is

10 and, conversely, is a major risk. Another risk in Samantha’s lowest payoff is that it is the

second lowest score on the entire table. There is also a major risk in the probabilities, since there

is a 35% chance of receiving Samantha’s lowest payoff. Additionally, there is only a 10% chance

of receiving her highest payoff, a minor reward. The second highest payoff of 20 is certainly a

reward, especially given its 35% chance of occurring, which is also a reward. The remaining

payoff of 14 is considered a very minor risk, as well as the 20% chance of this payoff occurring.

Thus, there is a 45% chance of receiving a high reward, which is a reward, and a 55% chance of

receiving a risk, which is a risk. From another point of view, there is an equal chance, 35%, of

receiving Samantha’s lowest payoff and second highest payoff. The 35% chance of receiving her

lowest payoff is a risk. This, and the only 10% chance of receiving Samantha’s highest payoff,

causes me to consider this alternative to be moderately risky.

The next alternative to be analyzed is Andy Karoly from Duke University. This

alternative has the highest BoB score, at 30, and the lowest BoW score, at 9, giving us the widest

possible range of payoffs. This is an early concern. The E.L. score of 16 ranks last on the table,

and is well below the midpoint of 19.5, which is significantly influenced by Andy’s exceptional

BoB score of 30. Because the E.L. is lower than the midpoint, the payoffs should be grouped to

the low end of the range. This alternative has the highest E.V., at 18.55, which looks great at first

but requires further analysis to completely understand. Nonetheless, an E.V. greater than the E.L.

indicates a favorable relationship between high payoffs and high probabilities. Finally, the regret

score of 13 is second best on the table, while the BoW score ranks the worst. This again causes

concern because of the disagreement between conservative rules. However, it is worth noting

that this alternative is the cause for skewed MiniMax Regret scores. While the other alternatives’

regret scores came from International Investing, Andy’s came from Health Care. As mentioned

earlier, this is not accurately measuring regret.

In the data, we see the highest payoff of 30 and the lowest payoff of 9. The other payoffs

are 15 and 10, giving us the worst E.L. score on the table at 16. Though the E.L. is the worst on

the table, the E.V. for this alternative ranks the best because of the low probabilities associated

with the two lowest payoffs, and the high chances of receiving high payoffs. This relationship is

very favorable. Though we would expect a high regret score for the alternative with the worst

BoW score, this alternative’s regret score is second best. This is again because of Andy’s

exceptional score of 30 in International Investing which is causing the regret scores to be

skewed.

This alternative has a reward of a high payoff of 30 and a risk of a low payoff of 9. These

scores are the highest reward and biggest risk on the table. The second highest payoff for Andy is

15 and is considered a very minor reward, while the remaining score of 10 is only marginally

less risky than the low payoff of 9. Looking at the probabilities, there is the greatest chance of

receiving a reward, with each payoff having a probability of 35%. The risks have probabilities of

20% and 10%, giving us a 30% chance of receiving a risk. While the probabilities are in favor

with our payoffs, which is a reward, we must be careful not to focus solely on the data presented

in this way. When looking at the data differently, we see there is a 70% chance of receiving a

reward. This seems great. However, the rewards vary significantly, creating a risk. That is, it is

just as likely to receive a score of 15 as it is to receive a score of 30. While 30 would be the

greatest payoff possible, receiving a payoff of 15, now half of our hopeful payoff of 30, would

hardly seem rewarding. Therefore, I would consider receiving anything other than the payoff of

30 to be a loss. Even if we receive a rewarding payoff of 15, the opportunity cost of losing out on

the other 15 (30-15=15) is too significant for me to overlook. Overall, I would classify this

alternative as a moderately high risk, given a 35% chance of a desirable payoff and a 65%

chance of missing out on the ultra-rewarding payoff of 30.

The third alternative to analyze is Wendy Lee from UNC. This alternative provides us

with both the second-best BoB and BoW scores and this is a favorable agreement. The BoB and

BoW scores give us a range of 10, which is the second best among alternatives. These payoffs

give us a midpoint of 18, which is just below the E.L., at 18.25, which indicates the data is just

slightly grouped above the midpoint. This too is a favorable agreement. However, when we look

at the E.V., though second-best among alternatives, we see it is less than the E.L. This indicates

an unfavorable match of probabilities and payoffs. Finally, this alternative has the worst regret

score due to Wendy’s lowest payoff occurring in International Investing, the area where Andy is

skewing the data.

In the data, we see a high score of 23 and a low score of 13, as shown in the BoB and

BoW rules. The remaining scores are very close to one another, at 18 and 19, which brings the

E.L. score to be 18.25. However, because there is not an equally likely chance of each state of

nature occurring, and because this alternative’s lowest two payoffs are associated with the

highest probabilities, the E.V. of 16.95, though second best on the table, is less than the E.L.

Despite the close grouping of this alternative’s payoffs, Wendy has the worst regret score on the

table. It is interesting that the alternative with the worst regret score has the second-best BoB and

BoW scores. This is because Wendy’s lowest payoff of 13 is in International Investing, where

the outlier lies. Because of the outlier, the regret score was the worst on the table. However,

when looking at the regret table, which again can be seen in Table 3 in the appendix of this

report, we see that this alternative had the highest regret in International Investing, but no regret

in any of the other areas. That is, Wendy has the lowest payoff in International Investing, but is

at least tied for the highest payoff in the other three areas. Because of this, the regret score should

not be considered too heavily.

This alternative has one outstanding reward of 23, its highest payoff, and one very slight

risk of 13, its lowest payoff. The 10% probability of receiving the outstanding payoff is a minor

reward. The remaining scores of 18 and 19, and their 35% and 20% respective probabilities of

occurring are all rewards.. Thus, there is a 65% chance of receiving a reward, which is a reward

in itself. Conversely, there is a 35% chance of receiving a very manageable risk, which is still a

risk but not a major one. While it may be a concern that there is a 70% chance of receiving this

alternative’s worst two payoffs, it is worth noting that Wendy’s next-to-worst payoff of 18, in

Personal Finance, is actually tied for the best score in that area. Actually, while there is a 35%

chance of receiving Wendy’s worst payoff, in International Investing, there is a 65% chance of

receiving the best score possible in the other three areas. This is a major reward. Given the major

rewards and minor risks associated with this alternative, I would classify this alternative as a low

to moderately low risk.

The last alternative to be analyzed is Tom Roberts from the University of Virginia. This

alternative has the lowest BoB score, at 19, but the best BoW score, at 15. This relationship

creates the smallest possible range of payoffs, at 4. With such a small range, we expect the

payoffs to be close to one another. The midpoint for this alternative is 17 and is equal to the E.L.,

suggesting the payoffs may be uniformly distributed around the midpoint. Next, we see that the

E.V., at 16.8, is just below the E.L. and midpoint, indicating a slightly unfavorable match of

probabilities and payoffs. Finally, the regret score, at 14, is next-to-last on the table and is again

caused by the outlier in International Investing.

In the data, we see a high payoff of 19 and a low payoff of 15. This small range suggests

Tom has general knowledge in all of the tested areas, as is reinforced by his remaining scores of

16 and 18. Interestingly, this alternative’s highest and lowest payoffs have the smallest

probabilities of occurring at 10% and 20%, respectively. It is also interesting to see the uniform

distribution of payoffs, as the payoffs lie equally below and above the midpoint. If each state of

nature were equally likely to occur, the E.V. would equal the E.L. However, there is a 55%

chance of the payoff falling below the midpoint, and a 45% chance of receiving a payoff above

the midpoint. This is slightly unfavorable and is the reason the E.V. is just less than the E.L. The

regret score is again being skewed by Andy’s exceptionally high score in International Investing.

This alternative’s highest payoff is a reward, at 19, while the lowest payoff, at 15, is the

risk. The high and low payoffs for this alternative give us the lowest range on the table, which

represents a degree of consistency and can be considered a reward. The 10% probability of

receiving Tom’s highest payoff is a minor reward. Because 15 is the lowest payoff for this

alternative it is considered a risk, though it is not undesirable when compared to other scores in

the Real Estate area. The remaining two scores, 16 and 18, can be considered rewards. The 35%

probabilities associated with the payoffs of 16 and 18 are also rewards. Thus, this alternative

provides us with three rewards and one very manageable risk. When looking at the data

differently, we see that there is an 80% chance of receiving a reward, and a 20% chance of

receiving a risk. I could even argue that the risk of 15 in Real Estate is one worth taking, since it

is the second best score in the Real Estate area. It is also worth noting that this alternative’s

second lowest payoff, 18, is tied for the best payoff in Personal Finance. This is a reward, as is

the 35% chance of this payoff occurring. For all of the reasons listed above and because of the

conservative nature of a low range alternative, I would classify this alternative to be a

conservative, low risk choice.

Recommendation

Comparing the first two alternatives analyzed, Samantha and Andy, the highest payoff for

each is 21 and 30, respectively, and Andy wins. When looking at the lowest payoffs, we see

Samantha has a low of 10 while Andy has a low of 9, making Samantha the winner of the BoW

scores. Considering this, I prefer Andy over Samantha as I would risk losing one point to

possibly receive a nine point higher payoff. When we look at the probability of the highest

payoff for each alternative, we see Samantha’s high score of 21 has only a 10% chance of

occurring, while Andy’s high score of 30 has a 35% chance of occurring. The preference here

again favors Andy, as both his payoff and probability are higher than Samantha’s. Looking at the

chance of the lowest payoff, we see Samantha’s lowest payoff of 10 has a 35% probability of

occurring. Meanwhile, Andy has only a 20% chance of receiving his lowest payoff, at 9. This

comparison again favors Andy as there is a greater chance of receiving Samantha’s low payoff.

Though Samantha’s lowest payoff is higher than Andy’s lowest payoff, Andy’s probabilities

give us a better chance of receiving a payoff other than his lowest. The second best payoffs for

Samantha and Andy are 20 and 15, respectively. Both alternatives’ second highest payoffs have

35% probabilities associated with them, favoring Samantha as we would have the same chance

of receiving an additional 5 points with Samantha. Though this favors Samantha, it is not enough

to change my overall preference for Andy because, if I were to choose Samantha and receive her

payoff of 20, in International Investing, I would actually be missing out on an additional 10 in

payoffs if we had chosen Andy and been placed in International Investing. While there is a 45%

chance of receiving Samantha's highest two payoffs, there is a 70% chance of doing so with

Andy. Thus, there is a 55% chance of receiving one of Samantha’s lowest two payoffs, and a

30% chance of receiving one of Andy’s lowest two scores. This favors Andy. Also, with Andy,

we have a good chance to receive an outstanding reward, while we maintain the same chance at

receiving significantly lower payoffs with Samantha. We also see that Andy gives us a 70%

chance of receiving a payoff of 15 or higher. With Samantha, there is a 55% chance we receive a

payoff of 14 or less. Further, there is a 45% chance of receiving a payoff of 20 or higher with

Samantha, but I would be willing to increase my probability to 70% to receive a payoff of 15 or

higher, with potential for the highest payoff on the table. Overall, my preference is for Andy due

to the reasons listed above, and his better match of strong probabilities and high payoffs when

compared to Samantha.

Next, I will compare Andy to Wendy. The highest payoffs for Andy and Wendy are 30

and 23, respectively. Andy again wins this comparison, as he will continue to do because of his

exceptional score in International Investing. The lowest payoffs for Andy and Wendy are 9 and

13, respectively, and this favors Wendy. Though Wendy’s lowest payoff has the best chance of

occurring, her score of 13 is only two points lower than Andy’s second best score of 15.

Therefore, Wendy’s biggest risk is only marginally worse than Andy’s second best reward. This

heavily favors Wendy. The ranges for these alternatives are 21 for Andy and 10 for Wendy.

Because of the better low-end payoff and the smaller range, I prefer Wendy over Andy. When

we consider the probabilities, there is a 35% chance of receiving Andy’s highest payoff and a

10% chance of receiving Wendy’s highest payoff. This favors Andy greatly as his highest payoff

is significantly greater than Wendy’s, with a much better chance of occurring. Also in favor of

Andy is the probability of receiving each alternative’s lowest payoff, as Andy has a 20% chance

and Wendy has a 35% chance. This would be enough to change my preference to Andy if we did

not realize that Wendy’s worst payoff is not that risky when compared to Andy’s payoffs, and if

we did not look further into the probabilities. Though the probabilities are in Andy’s favor when

we look at the highest and lowest payoffs, the overall probabilities favor Wendy. While there is a

70% chance of receiving a reward from Andy, and a 65% chance of receiving a reward from

Wendy, there is also a 65% chance of receiving a better payoff from Wendy. Actually, Wendy’s

payoffs are the best scores in Real Estate, Personal Finance, and Health Care, while Andy has the

highest score in only International Investing. Because of this, I would take my chances with

Wendy as I would rather have the best score in three areas than a significantly better score in one

area. This is another reason why I prefer Wendy over Andy, as the training program suits

generalists better than it would specialists as trainees are rotated through different business areas.

Andy may be exceptional in International Investing, but his other scores suggest he may be more

of a specialist and could be disengaged in areas other than where his strengths lie. Wendy brings

more stability to the table, which is probably best since I do not know which area I will be placed

in, and because I would much rather make a conservative first hire decision, rather than a high-

risk, high-reward choice that may not work out. Also, it is important to realize that though there

is a 70% chance of receiving one of Wendy’s two lowest payoffs, her second lowest payoff of 18

is the best in Personal Finance. For these reasons, my overall preference is with Wendy. Next,

we will compare Wendy to Tom to make a final recommendation.

The highest payoffs for Wendy and Tom are 23 and 19, respectively, and this favors

Wendy. The lowest payoffs for Wendy and Tom are 13 and 15, respectively, and this favors

Tom. While these comparisons do not tell us a lot just yet, I prefer Tom over Wendy because his

small range of only 4 is much better than Wendy’s range of 10 and I can therefore count on Tom

to be a more stable generalist than Wendy. When we consider the probabilities, we see that

Wendy has a 35% probability of receiving her lowest payoff, while Tom has a 20% chance of his

lowest payoff. Therefore, not only does Tom have a higher low score than Wendy, but we are

also less likely to receive Tom’s lowest payoff than we are Wendy’s lowest payoff. We can also

see that there is an 80% chance of receiving a reward from Tom, and a 65% chance of receiving

a reward from Wendy. While these are both favorable probabilities, Tom’s chances are better.

We also see that, for International Investing and Personal Finance, Tom has an equal chance as

Wendy at receiving better payoffs. Wendy’s better payoffs lie in areas with the lowest

probabilities. This places favor on Tom as his scores are not much worse than Wendy’s in Real

Estate and Health Care and there is the lowest chance of being placed in these areas. That is,

Wendy’s best rewards lie in areas we are less likely to be placed in, creating opportunity costs if

we chose Wendy and were not placed in one of her best areas, which has a 70% chance of

happening. Because of Tom’s small range and distribution of payoffs, the 80% chance of

receiving a reward, the equal chance of receiving Tom’s better payoffs in International Investing

and Personal Finance, the 70% chance I am placed in one of those two areas, and because he has

the highest BoW score, a conservative measure, my overall preference is for Tom as he presents

himself as a capable generalist, scoring fairly well in all four areas despite the difficulty of the

aptitude tests. With this in mind, I believe it would be best to choose Tom as he is capable of

making an impact in any of the four areas in which I may be placed.

Appendix

Figure 1: Aptitude Test Results

International Investing Real Estate

Personal Finance

Health Care

Aptitude 7 3 1 5Analytical Training 2 3 3 2Experience 1 2 1 3Creativity 4 5 4 6Interest 6 1 1 5


Personal Finance

Health Care

Aptitude 6 1 5 2

Analytical Training 7 2 4 3Experience 5 2 4 1Creativity 5 3 1 3Interest 7 1 1 1


Personal Finance

Health Care



Personal Finance

Health Care



Personal Finance

Health Care


Samantha Lindle from Wake Forest:

Andy Karoly from Duke

Wendy Lee from UNC

Tom Roberts from Virginia

Bill Williams from Clemson

Table 1: Payoff Table

35% 20% 35% 10%

International

InvestingReal



Samantha 20 14 10 21 21 10 16.25 15.4 10Andy 30 9 15 10 30 9 16 18.55 13

Wendy 13 19 18 23 23 13 18.25 16.95 17Tom 16 15 18 19 19 15 17 16.8 14Bill 14 11 13 11 14 11 12.25 12.75 16

Table 2: Domination

35% 20% 35% 10%

International

InvestingReal



Samantha 20 14 10 21 21 10 16.25 15.4 10Andy 30 9 15 10 30 9 16 18.55 13

Wendy 13 19 18 23 23 13 18.25 16.95 17Tom 16 15 18 19 19 15 17 16.8 14Bill 14 11 13 11 14 11 12.25 12.75 16

Table 3: Regret

International Investing Real Estate Personal Finance Health Care

Samantha 10 5 8 2

Andy 0 10 3 13Wendy 17 0 0 0Tom 14 4 0 4Bill 16 8 5 12

kissingerj payoff table final

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