Since the answer is too long, I am posting it in the form of a thread.
The standard answer to this problem is: Pirate No. 1 gives 1 gem to No. 3, 2 gems to either No. 4 or No. 5,
and keeps 97 gems for himself, that is, the distribution plan is (97, 0, 1, 2, 0) or (97, 0, 1, 0, 2). Now let's look at my analysis:
First, start with Pirate No. 5, because he is the safest, with no risk of being thrown into the sea, so his strategy is also the simplest: it is best if all the previous pirates are dead, then he can have all 100 gems for himself.
Next, look at No. 4. His chance of survival completely depends on whether there are people left in front of him, because if Pirates 1 to 3 are all fed to the sharks, then with only No. 4 and No. 5 left, no matter what distribution plan No. 4 proposes, No. 5 will definitely vote against it to let No. 4 be fed to the sharks so he can take all the gems. Even if No. 4 tries to survive by appeasing No. 5 and proposes a plan like (0, 100) giving No. 5 all the gems, No. 5 might still consider that keeping No. 4 is dangerous and vote against it to feed him to the sharks. Therefore, a rational No. 4 should not take such a risk or rely on No. 5's random choice; he can only support No. 3 to absolutely ensure his own life.
Now, consider No. 3. Following the above logic, he will propose a distribution like (100, 0, 0) because he knows that No. 4, even with nothing, will still unconditionally support him and vote in favor. Together with his own vote, he can securely get all 100 gems.
However, No. 2 also reasons through No. 3's distribution plan and will propose a plan of (98, 0, 1, 1). Compared to No. 3's plan, in this plan No. 4 and No. 5 can at least get 1 gem each. Rational No. 4 and No. 5 will naturally think this plan is more beneficial to them and will support No. 2, not wanting No. 2 to be out and No. 3 to take over the distribution. Hence, No. 2 can happily take 98 gems.
Unfortunately, Pirate No. 1 is not someone to underestimate. After reasoning through, he also sees No. 2's distribution plan. His strategy is to abandon No. 2, give 1 gem to No. 3, and 2 gems to either No. 4 or No. 5, that is, propose a distribution plan of (97, 0, 1, 2, 0) or (97, 0, 1, 0, 2).Since the allocation plan of No. 1 can bring more benefits to No. 3, No. 4, or No. 5 compared to the plan of No. 2, they will vote to support No. 1. With No. 1's own vote, 97 gems can easily fall into No. 1's pocket.
