A short tutorial on identity politics and the spoils system

One of the enduring mysteries of identity politics and the spoils system is "Why do minor players willingly accept such small slices of the pie?"

This simple exercise in logic might explain that conundrum.  (Hat tip Mrs ERJ)

The Three Pirate Problem
Suppose there are three pirates.  Suppose they looted a ship and had 100 pieces of gold to distribute.

They agree that might-makes-right, so the largest pirate will be the first to propose a way to distribute the loot.  If half, or more, of the pirates agree to the plan then that is what they go with.  If less than half agree to the plan then the largest pirate is thrown overboard and the largest of the two remaining pirates gets to propose his plan.

Assuming that all pirates are greedy, rational and want to live, what plan will be implemented?

Approaching problems from the other direction
This is a problem best solved in reverse.

Suppose that the largest pirate's plan was rejected by the other two.  He would be pitched out of the boat.

Then the second largest pirate would propose that he get all 100 pieces of gold and that the smallest pirate get none.  Since half of the pirates agree with the plan (the second largest pirate is all for it) that is what would go into effect.

The second largest pirate knows this and will not agree to any plan put forth by the largest pirate unless the plan is for the second largest pirate get all 100 pieces of gold.  All he has to do is wait.

The smallest pirate knows that if he does not vote for the largest pirate's plan that he will get nothing.  Therefore, he will vote for the plan even if the spoils are small.

Thinking this through, the largest pirate knows that any amount he offers the second largest pirate will have no material bearing on the outcome.  Therefore he offers the second largest pirate NOTHING.

He also knows that offering the smallest pirate even the most miserly of offers will be superior to what the smallest pirate will get if the second largest pirate is allowed to make a proposal.  So the largest pirate'[s plan is 99 pieces of gold for himself, 0 pieces of gold for the second largest pirate and 1 piece of gold for the smallest pirate.

There!  Do the outcomes seen in the spoils system make any more sense?  Politicians are pirates!