Lets consider a 'smaller' variant of the pirate game (https://en.wikipedia.org/wiki/Pirate_game):
There are three pirates with a chest of 100 gold coins. They now have to decide how to split the coins.
The pirate world's rules of distribution say that the most senior pirate first proposes a plan of distribution. The pirates, including the proposer, then vote on whether to accept this distribution. If the majority (including the proposer) accepts the plan, the coins are dispersed and the game ends. In case of a tie vote, the proposer has the casting vote. If the majority rejects the plan, the proposer is thrown overboard from the pirate ship and dies, and the process repeats with next most senior pirate making a new proposal to begin the distribution again.
$\textbf{Q:}$ Is there more than one (even two) bckward induction outcome for this game? As of now, I only have the outcome: $(99,0,1)$ from backward induction. Some help and insight will be great!
$\textbf{Edit/Addition:}$
Are the outcomes $(99,1,0)$ and $(0,100,0)$ possible as well by backward induction? If someone can illustrate this via a game tree that wil be great, as I am really clueless on this.
Here's another outcome: $X=(100,0,0)$.
Let A, B, C denote the three pirates with descending seniority.
Here, C does not have a strict incentive to vote for $X$, but he doesn't have a strict incentive to vote against it either. He gets $0$ by voting either way. So voting for $X$ is also rational. Therefore $X$ is also an outcome of backward induction.