Challenge Problem: Mutual Induction

In this challenge problem, you will "solve" a simple game using induction.

Learning Objectives

Topic: Induction. Remember that completing a challenge problem on a topic does not count toward proficiency for that topic.

Pair Project:$$Yes $($You may optionally work with ONE partner$)$

Instructions

Let's consider a game involving the distribution of points among competing players. In our game $($whose name I cannot remember!$)$ one player draws some number of loot cards and proposes a distribution of them among the various players. The players then vote Yes or No on whether to accept this distribution of the loot. If at least half vote Yes, the round ends and each player gets the assigned loot. Otherwise, the player who proposed the distribution is out for the round and the next player starts over with the same cards but can now make a new proposed distribution. $($The player who was eliminated neither gets loot nor votes this round.$)$ We will consider a simplified version where the "first player" simply gets a random number of points to distribute rather than cards with special abilities.

Your Challenge:

Notice that unlike in other induction examples we have players that can make choices. If we want to consider all possible paths of choices, there are exponentially many to keep track of$.$ However, we can assume that all players would like to guarantee themselves as many points as possible for the round. This allows us to reason about what other players will do$.$

Using induction on the number of players N$,$ show what players must do in order to maximize their loot for one round. I am interested in both a formula to calculate the final points distribution and an inductive proof that this formula is correct.