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Online Brokers

stock market

Five online brokers, Fidelity, Scottrade, Charles Schwab, TD Waterhouse, and Ameritrade must distribute amongst themselves 100 shares of   the world’s most expensive stock listed on any market in the world : Berkshire Hathaway Inc. (NYSE:BRK.A)

The online brokers have committed to a strict order of seniority:
Ameritrade is superior to Charles Schwab, who is superior to Fidelity, who is superior to TD Waterhouse, who is superior to Scottrade.

The online brokers also follow strict rules of share distribution, which are : the most senior online broker should propose a distribution of shares.
The online brokers, including the senior broker, then vote on whether to accept this distribution.
If the proposed allocation is accepted by a majority vote, it happens.
If not, the proposer must resign, and the next most-senior online broker makes a new proposal to begin the system again.

In the event of a tie vote, the most senior online broker has the casting vote.

Online brokers base their decisions on three factors, in order of priority:
First of all, each online broker does not want to resign.
Second, each online broker wants to maximize the number of shares he receives.
Third, all things being equal, a broker would prefer to force the most-senior broker to resign.

Question : What is the number of shares each online broker receives?

Answer :
Ameritrade: 98 shares
Charles Schwab: 0 shares
Fidelity: 1 share
TD Waterhouse: 0 shares
Scottrade: 1 share

Perhaps Ameritrade will have to allocate little if any to himself  for fear of being forced to resign thus allowing feewer brokers to share between themselves. However, this is impossible.

If you work backwards, here is the scenario.
If all brokers except TD Waterhouse and Scottrade have resigned, TD Watrehouse suggest 100 sahres for himself and 0 sahres for Scottrade.
TD Waterhouse  has the casting vote, and so this is the allocation.

If there are three left (Fidelity, TD and Scottrade) Fidelity realizes that TD will offer Scottrade 0 shares in the next round; therefore, Fidelity must offer Scottrade 1 share in this round to make Scottrade vote with him, and get his allocation through. Therefore, when only three are left the allocation is Fidelity:99, TD:0, Scottrade:1.

If Charles Schab, Fidelity, TD Waterhouse and Scottrade remain, Charels Schwab is aware of this when he makes his decision.
To avoid forced resignation, Charles Schwab can simply offer 1share  to TD.
The support only by TD is sufficient because he has the casting vote.
Thus he proposes Charles Schwab:99, Fidelity:0, TD:1, Scottrade:0.
One consideration could be  Charles Schwab:99, Fidelity:0, TD:0, Scottrade :1, as Scottrade is sure he won't get more, if any, if he forces Charles Schwar to resign.
But, as each online broker is eager to force the oithers into resignation, Scottrade would prefer to push Charles Schwab to resign, to obtain the same amount of shares from Fidelity.

Ameritrade  knows all these things, he can count on Fidelity and Scottrade's support for the following allocation, which is the final solution.
And the online brokers live happily ever after.

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