Managementplaza: The Best Choice

') frame2.document.write(comm); frame2.document.close(); frame1.document.write(' No output !') frame1.document.close(); } } function calculate() { maxnum = document.socform.num.value * 1; var ia,ma,ta,p,q; frame1.document.write(' ') var s= Array() , p=Array() s[0]=0; p[0]=0; maxm=1; for (k=1; k < maxnum; k++) { s[k]=s[k-1]+1/(maxnum-k); p[k]=s[k]*(maxnum-k)/maxnum; if (p[k]p[k-2] && k>1) { maxp=p[k-1]; maxm=1+maxnum-k; } } } maxp=p[maxnum-maxm]; p[maxnum]=1/maxnum; for (m=0; m < s.length; m++) { stro="" var rgbcl; rgbcl=p[maxnum-m]*255/maxp; rgbcla=255; if (m==maxm) {rgbcla=170}; if (m==maxm-1) {rgbcla=210}; if (m==maxm+1) {rgbcla=210}; if (m==maxm-2) {rgbcla=230}; if (m==maxm+2) {rgbcla=230}; stro= stro+""; stro=stro+printround(m,4,0)+" "+printround(100*p[maxnum-m],9,1)+"%"+ " "; if (m==maxm) {stro = stro + "[Max.Value]"}; stro= stro+""; stro= stro+"
" frame1.document.write(stro); } maxmd=maxm; posd="ies" if (maxm==1) {maxmd=""; posd="y";}; comm='Conclusion
'+ ''+"Skip the first "+maxmd+ " possibilit"+posd+" and take the best one thereafter, to reach the highest possible probability of P="+printround(100*maxp,5,1)+" % to get the "best" solution in the total of "+ maxnum +" possibilities you are offered.
"+ "So there is always a change of "+printround((100-100*maxp),5,1)+ " % that you don't succeed in selecting "the best one" and that the best solution was part of the first "+maxmd+ " opportunit"+posd+ "." frame2.document.write(' ') frame2.document.write(comm); frame1.document.close(); frame2.document.close(); } function checkandcalculatet(){ maxnum = 1+document.socform.numt.value * 1; tota=''; maxk=maxnum; for (kk=1; kk < maxk; kk++) { maxnum=kk; var ia,ma,ta,p,q; var s= Array() , p=Array() s[0]=0; p[0]=0; maxm=1; for (k=1; k < maxnum; k++) { s[k]=s[k-1]+1/(maxnum-k); p[k]=s[k]*(maxnum-k)/maxnum; if (p[k]p[k-2] && k>1) { maxp=p[k-1]; maxm=1+maxnum-k; }}} maxp=p[maxnum-maxm]; if (maxnum==1) {maxp=1} tota=tota+"" } tota=tota+"

"+printround(maxnum,8,0)+"

"+printround(maxm,8,0)+"

"+printround(maxp*100,9,1)+"

" frame3.document.write(' ') frame3.document.write(tota); frame3.document.close(); }

Making Decisions

The best choice out of 'n' unknown opportunities

In real life you often have to take a decision in a situation were you have to pick out "the best" opportunity out of "n" possibilities in a situation were you (ex ante) do not know much about what you can expect in terms of quality or quantity.

For example:

Making the right investment
You want to make an investment today. From your stockbroker you get each day about 20 opportunities to invest. But you want to make only one investment this day 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.
Choosing the best candidate for the job
You�re the personnel-manager of a certain company. You make a deal with your external HRM-advisor that he will deliver you this month 10 potential candidates for the vacancy you have. After each candidate you have to decide wetter you accept him 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 given you an invitation for a visit. After each visit you are obliged to say wetter you take the job or don�t. 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 wont accept that you come back later after you came to the conclusion that he was after all the cheapest).
You want the highest discount; How can you manage?

In each of these cases you can ask yourself: what is the optimal strategy? Take the first opportunity or wait until the last? Skipping the first 2 opportunities and than take the next one that is better?

In literature (management science) these kind of problems are known as "Best Choice Problems" (BCP's). BCP's are packaged in descriptions like "The Sultan's Dowry Problem" or "The Secretary Problem".

BCP's are 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 wetter you take the offer or go further (you can�t go back and take an earlier opportunity after all).
You don�t have enough knowledge about the group of "n" possibilities self (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:

Best Choice Calculator

The Best Choice Calculator calculates the optimal number of possibilities that you have to let pass [m] 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!

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]
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Rule of thumb

As you perhaps noticed in using The Best Choice Calculator, there is a close relation between the number op 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]. But don�t worry, 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=

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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.

J.N. Berkemeijer / July 2002
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