$\textbf{Pirates and Gold}$
Five pirates are trying to split up $1000$ gold pieces according to the following rules
- The $1^{st}$ pirate can propose a plan. If his proposal wins the majority's support (including himself), then it is done. Otherwise, he will be instantly killed.
- If the $1^{st}$ pirate is killed, then the $2^{nd}$ pirate will make a proposal. If his proposal wins the majority's support (a tie does not suffice, i.e. he needs at least $3$ votes), it is done. Otherwise, he will be instantly killed.
- The process continues until a proposal is agreed by a majority.
Assuming all these five pirates are intelligent (i.e. always choose the optimal strategy for himself), greedy (i.e. get as much as gold for himself) and ruthless (i.e. the more pirates dead, the better), what will be the final distribution of the gold?
$\textbf{Answer}$
The $1^{st}$ pirate will get $997$ pieces, the $3^{rd}$ will get $1$ piece and either the $4^{th}$ or the $5^{th}$ will get $2$ pieces.
$\textbf{Analysis}$
Let's try to analyze backward. If only the $4^{th}$ and $5^{th}$ pirates are left, then no matter what the $4^{th}$ proposes, the $5^{th}$ will disagree. This makes the $4^{th}$ no way to win the majority. Therefore, the $4^{th}$ will agree to whatever the $3^{rd}$ proposes if the first two pirates are killed. In this case, the $3^{rd}$ will propose to have all the gold pieces to himself.
Hence, the $2^{nd}$ just need to offer the $4^{th}$ and $5^{th}$ one piece each to secure their votes in order to win the majority. In this case, the $3^{rd}$ will have nothing.
Now, the $1^{st}$ can propose $1$ piece to the $3^{rd}$ which the $3^{rd}$ has to agree. (Otherwise, if the $1^{st}$ is killed, he will end up with nothing.) In order to win another vote, the $1^{st}$ can propose to either the $4^{th}$ or the $5^{th}$ two pieces. This will make his proposal more appealing than what the $2^{nd}$ can propose.
$\textbf{Note}$
Backward analysis is effective. Its essence is to simplify the problem. In this case, the result is obvious if there are only two pirates. This in turn makes the result obvious if there are only three pirates, and so on.