Puzzle time - shrinking board edition

[jon-nyc]

The board of directors for the Acme Acute Angles Company has grown too large — 50 members, now — and its members have agreed to the following reduction protocol. The board will vote on whether to (further) reduce its size. A majority of "ayes" results in the immediate ejection of the newest board member, then another vote is taken, and so on. If at any point half or more of the surviving members vote "nay," the session is terminated and the board remains as it currently stands.

Suppose that each member places the highest priority on personally remaining on the board, but, given that, agrees that the smaller the board, the better.

To what size will this protocol reduce the board?

[Klaus]

Sounds like a prisoner's dilemma-like situation.

I'd say the solution is 2 or 50. I don't think it is anything between.

Should the solution take into account that everyone acts completely rational and knows that everyone else acts completely rational, too? Furthermore, I assume you rule out "deals" among subsets?

[jon-nyc]

You do allow for rational people that assume others are rational. The answer is neither 2 nor 50.

[Klaus]

||
Here's the situation for smaller groups, starting with group size 2.

In the following, the number n denotes the n-th oldest board member. "gs" stands for group size. The notation "1,2 vs 3" means "1,2 vote ayes, 3 votes no".

gs = 2 -> termination with 1 and 2 remaining
gs= 3 -> 1,2 vs 3 -> 3 kicked out
gs = 4 -> 1,2 vs 3,4 -> termination with 1-4 remaining
gs = 5 -> 1,2,3,4 vs 5 -> 5 kicked out
gs = 6 -> 1,2,3,4 vs 5,6 -> 6 kicked out
gs = 7 -> 1,2,3,4 vs 5,6,7 -> 7 kicked out
gs = 8 -> 1,2,3,4 vs 5,6,7,8 -> termination with 1-8 remaining

So it looks like the powers of 2 are the termination points.

Based on that reasoning, I'd say that 32 members remain.
||

[jon-nyc]

Yep

[Klaus]

So, what do I win?

[jon-nyc]

Bragging rights for beating Horace and Ax to an answer.

[Klaus]

I wondered what happens if a stale mate results in kicking out a member, not termination.

Turns out that in this case the powers of two minus one (1,3,7,15,31 etc.) are the termination points.

[Klaus]

What I meant with my question about "deals" is something like this.

Let's assume group size 4. In the strategy described above it would be a termination point.

But 3 could make a deal with 1 and 2: I'll vote for kicking out 4 if you promise to vote for termination in the next round.

[jon-nyc]

Yeah but they could break their promise and be further ahead on their goals

[jon-nyc]

It’s fun to think about though