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Optimal Choice

Whether you're an actuary, investment manager, HR-manager of respected consumer, in real life you often have to take a decision in a situation where you have to pick out "the best" opportunity out of "n" possibilities. Mostly in a situation where you (ex ante) are only limited aware about what you can expect in terms of quality or quantity.

Some examples:

• Making the right investment
  You want to make an investment today. Every day your stockbroker offers you about 20 investment opportunities. Your budget allows you to make only one investment today and it should be the best one. How can you be sure to pick the right one?
  You have only limited time (say about 10 minutes) for each investment to decide; after that, you'll have to wait for the next offer.
   
• Selecting the best candidate for the job
  You’re the HR-manager of a certain company. Your external HRM-advisor promised you to present 10 potential candidates for a certain vacancy. After each candidate you have to decide wetter you accept him/her or wetter you go one for a possible better candidate. You want the best candidate. What can you do?
   
• Applying for the best job
  You solicited for a new job. Six companies have invited you for a visit. After each visit you are obliged to say wetter you take the job or not. You want the best job. What is wise to do?
   
• Buying a new car
  You want to buy a new car. Although the price of that car is fixed, every dealer gives a quick-decision discount.
  You decide to visit 7 dealers. After each visit you have to decide wetter you "buy" or "let go" (the dealer will not accept that you come back later after you came to the conclusion that he was after all the best deal).
  You want the highest discount you can get; How can you manage?
   

Optimal Strategy?

In each of these cases you can ask yourself: what is the optimal strategy? Take the first opportunity or wait until the last? Skipping some opportunities or take the median opportunity?

In decision theory literature these kind of problems are known as "Best Choice Problems" (BCP's). More specific the above decision problems are often described as "The Sultan's Dowry Problem" or "The Secretary Problem".

This type of deceision problem is characterised by the following assumptions:

• You want "the best" choice out of "n" possibilities
• You handle each opportunity after another. After each opportunity you have to decide whether you take the offer or go further (you can’t come back on your decision later).
• You only have limited knowledge about the group of "n" possibilities (in terms of quality or quantity)

---

Solution

The best strategy in these kind of cases is to wait (don’t choose) until the first "m" possibilities of the total number of opportunities "n" have passed. After these "m" possibilities you accept the first offer that is "better" than the one you've had until the moment of decision. The word "better" stands for "better candidate", "better financial offer", etc.

If you’re interested in the mathematical theory behind this kind of problems, click on one of the links below:

• The Secretary Problem
• Mathworld Sultans Dowry Problem
• UCLA (pdf)

---

Best Choice Calculator

Objective The 'Best Choice Calculator' calculates the optimal number of possibilities [m] that you have to let pass, before taking the best one thereafter, to achieve the maximum probability [p] that you indeed will realise the best choice from a given total number of possibilities [n]. Try it out! Explanation calculation output The first column returns "m" The second column returns "p" The input-variable "n" is the total number of possibilities

Total number of possibilities [n] =

[m]       [p] Conclusion

---

Rule of thumb

As you perhaps noticed in using The Best Choice Calculator, there is a close relation between the number of possibilities [n] and the number of opportunities you had to skip [m].

When we have to take decisions in "real life", we do not (yet) have a build-in computer-chip in our head to calculate for each [n] the corresponding value [m]. In this case all you have to do is to memorise the next rule of thumb:

Rule of Thumb

Number of possibilities Number of possibilities to skip  =  3

Or, in plain mathematics:

n m  =  3

Of course the rule of thumb is an approximation.
In the next table you see how [n] and [m] are exactly related for values of n from n=1 to n=

Although it’s nice to have a "rule of thumb", don’t forget to decide on your gutfeeling as well.

Mixing intuition, experience and rules of thumb, guarantees the ultimate best choice.

Original blog by Jos Berkemeijer / July 2002/ Updated : April 2013

Free to copy  

Best Choice Calculator

Whether you're an actuary, investment manager, HR-manager of respected consumer, in real life you often have to take a decision in a situation where you have to pick out "the best" opportunity out of "n" possibilities. Mostly in a situation where you (ex ante) are only limited aware about what you can expect in terms of quality or quantity.

Some examples:

• Making the right investment
  You want to make an investment today. Every day your stockbroker offers you about 20 investment opportunities. Your budget allows you to make only one investment today and it should be the best one. How can you be sure to pick the right one?
  You have only limited time (say about 10 minutes) for each investment to decide; after that, you'll have to wait for the next offer.
   
• Selecting the best candidate for the job
  You’re the HR-manager of a certain company. Your external HRM-advisor promised you to present 10 potential candidates for a certain vacancy. After each candidate you have to decide wetter you accept him/her or wetter you go one for a possible better candidate. You want the best candidate. What can you do?
   
• Applying for the best job
  You solicited for a new job. Six companies have invited you for a visit. After each visit you are obliged to say wetter you take the job or not. You want the best job. What is wise to do?
   
• Buying a new car
  You want to buy a new car. Although the price of that car is fixed, every dealer gives a quick-decision discount.
  You decide to visit 7 dealers. After each visit you have to decide wetter you "buy" or "let go" (the dealer will not accept that you come back later after you came to the conclusion that he was after all the best deal).
  You want the highest discount you can get; How can you manage?
   

Optimal Strategy?

In each of these cases you can ask yourself: what is the optimal strategy? Take the first opportunity or wait until the last? Skipping some opportunities or take the median opportunity?

In decision theory literature these kind of problems are known as "Best Choice Problems" (BCP's). More specific the above decision problems are often described as "The Sultan's Dowry Problem" or "The Secretary Problem".

This type of deceision problem is characterised by the following assumptions:

• You want "the best" choice out of "n" possibilities
• You handle each opportunity after another. After each opportunity you have to decide whether you take the offer or go further (you can’t come back on your decision later).
• You only have limited knowledge about the group of "n" possibilities (in terms of quality or quantity)

---

Solution

The best strategy in these kind of cases is to wait (don’t choose) until the first "m" possibilities of the total number of opportunities "n" have passed. After these "m" possibilities you accept the first offer that is "better" than the one you've had until the moment of decision. The word "better" stands for "better candidate", "better financial offer", etc.

If you’re interested in the mathematical theory behind this kind of problems, click on one of the links below:

• The Secretary Problem
• Mathworld Sultans Dowry Problem
• UCLA (pdf)

---

Best Choice Calculator

Objective The 'Best Choice Calculator' calculates the optimal number of possibilities [m] that you have to let pass, before taking the best one thereafter, to achieve the maximum probability [p] that you indeed will realise the best choice from a given total number of possibilities [n]. Try it out! Explanation calculation output The first column returns "m" The second column returns "p" The input-variable "n" is the total number of possibilities

Total number of possibilities [n] =

[m]       [p] Conclusion

---

Rule of thumb

As you perhaps noticed in using The Best Choice Calculator, there is a close relation between the number of possibilities [n] and the number of opportunities you had to skip [m].

When we have to take decisions in "real life", we do not (yet) have a build-in computer-chip in our head to calculate for each [n] the corresponding value [m]. In this case all you have to do is to memorise the next rule of thumb:

Rule of Thumb

Number of possibilities Number of possibilities to skip  =  3

Or, in plain mathematics:

n m  =  3

Of course the rule of thumb is an approximation.
In the next table you see how [n] and [m] are exactly related for values of n from n=1 to n=

Although it’s nice to have a "rule of thumb", don’t forget to decide on your gutfeeling as well.

Mixing intuition, experience and rules of thumb, guarantees the ultimate best choice.

Original blog by Jos Berkemeijer / July 2002/ Updated : April 2013

Free to copy