Solving this question
Think by starting from Pirate 5 only
In order to solve this question. We have to predict what would happened if other pirates vote against. Which means we have to think backwards, becaue the pirate will have to make other priates admit your plan by telling them a better plan than the next one.
This is similar to solving a recursion problem. In order to solve the first case, we have to solve the second case; to solve the second case, we have to solve the third case... Until it reach the base. In this case we start at Pirate 1 and stop at Pirate 5.
Situation of only pirate 5 alives
Let's start up with the situation that everytime most of the pirates vote against. Of course, this is the case that only happens if every pirate are having suicidal thoughts, or without a human brain to think. However that doesn't mean we don't need to consider this case, because all other pirates would make their proposal by infering the next pirate's proposal.
Indeed, the 5th pirate would take all the coins. Even if he is the person who is so kind that would like to giveaway all the coins, he has no one to give.
So in this case he would get all 100 coins.
Situation of pirate 4 and above alive
Pirate 4 would give 100 coins to himself, none for Pirate 5.
Whatever how Pirate 4 proposes, Pirate 5 would vote against. So even if what Pirate 4 proposes is extremely biased, the proposal would still pass because Pirate 4 has the opportunity to vote for himself. Which makes half of the survived people vote "for".
Situation of pirate 3 and above alive
Pirate 3 would give 99 coins to himself, none to Pirate 4, and 1 coin to Pirate 5.
Since Pirate 4 would vote "against" in all situations, Pirate 4 can threaten Pirate 5 that if he also votes "against", he will get nothing from Pirate 4. However if he votes "for", he will get 1 coins which is more than what Pirate 4 will propose.
Situation of pirate 2 and above alive
Pirate 2 would give 99 coins to himself, none to Pirate 3, 1 coin to Pirate 4, and none to Pirate 5.
Pirate 3 would always vote "against", so it is better to give him nothing. Pirate 2 can threaten Pirate 4 that if he vote "against", he will get nothing from pirate 3. Pirate 5 is more expensive than Pirate 4, and we already have two pirates including pirate 2 himself vote "for", which is over 50%.
Situation of all the piates alive.
Pirate 1 would give 98 coins to himself, none to Pirate 2, 1 coin to Pirate 3, none to Pirate 4, and 1 coin to Pirate 5.
This time we will only get 98 coins because we need 3 people's agree in order to pass the proposal.
Pirate 2 would vote "against" at all time. Threaten Pirate 3 to vote or else Pirate 2 won't give him anything. None to Pirate 4 because he is too expensive. 1 to Pirate 5 as he has the same situation as Pirate 3.
Graph
Organize the datas into the table.
| Pirates | Pirate 5 alive | Pirate 4, 5 alive | Pirate 3, 4, 5 alive | Pirate 2, 3, 4, 5 alive | All Pirates alive |
|---|---|---|---|---|---|
| Pirate 5 | 100 | 0 | 1 | 0 | 1 |
| Pirate 4 | - | 100 | 0 | 1 | 0 |
| Pirate 3 | - | - | 99 | 0 | 1 |
| Pirate 2 | - | - | - | 99 | 0 |
| Pirate 1 | - | - | - | - | 98 |
You may notice that for every pirate that will have something to get by the next proposal, the proposal would be give them nothing. On the contrary, if they won't have anything by the next proposal, they will get 1 coin for the current proposal.
Formula of the first person's coin
The total amount of coins, reduce round up half of the total amount of people, plus one more vote from the current pirate who proposals which is always the number one, equals the coins the current pirate can get.
Put it in the Microsoft Excel sheet with the formula of fx=100-ROUNDUP(A2/2,0)+1, and graph it. Here's the result.