The Undenial Allotment Proposition!

What were rules of allotment process?

The eldest should allot ingot like 9, 0, 1, 0, 2 among 5 bankers.

Let's name bankers as Banker 5, Banker 4 ...... Banker 1 according to decreasing order of their ages.

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CASE 1 : 

Suppose there are only 2 bankers left then the youngest will always deny whatever elder offers so that he can take away all 12 ingots on his turn.

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CASE 2 : 

If case is reduced to 3 bankers then the eldest knows that the youngest is not going to agree with him in any case. With that, the eldest will be dismissed and case reduced to CASE 1 where youngest can take away all. 

So, the eldest here proposes allotment as 11, 1, 0. The Banker 2 has no option than to accept this proposal otherwise he won't get anything if case is reduced to 2 bankers as in CASE 1 above.

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CASE 3 : 

With 4 bankers, the eldest would propose allotment 9, 0, 2, 1.

In the case, Banker 3 will never accept any proposal as after banker 4 is dismissed he would be getting 11 ingots as in CASE 2 above.

The Banker 2 will happily agree with the eldest as he would be getting 1 more ingot than the CASE 2. 

And Banker 1 knows he will be getting nothing if the case is reduced to 3 bankers as in CASE 2. So, he too will agree with the eldest in the case.

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CASE 4 : 

With 5 bankers, the eldest i.e. Banker 5 should propose allotment 9,0,1,0,2.

Obviously, any how Banker 4 is going to deny any proposal as he wants the distribution among 4 bankers where he will be getting 9 ingots as in CASE 3 above. 

And if Banker 5 is eliminated and the case is reduced to CASE 3 where 4 bankers are left then Banker 3 knows he won't be getting anything. So, better he should be happily agree this proposition where he is getting at least 1 ingot.

Finally, offering Banker 5 one more extra ingot than the case where 4 bankers will be left, makes him in favor of this proposition.

Notice that the Banker 5 has to give 3 ingots at least to banker 2 to get his vote as he will be getting 2 ingots in case of 4 bankers as in CASE 3. Whereas, in the same case Banker 3 is not getting anything & would be happily agree if getting 1 ingot at least in this case. 


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