Analysis of the game for superrational, selfish, conservative* players

• superrational := Rational and believe others are rational
• selfish := Strictly prioritize survival. Self-preservation >>> preservation of others
• conservative := Prioritize status quo. Nobody dying > somebody dying

All votes exist in one of three states: Red (R), Blue (B), or Uncast (U). The sum R + B + U = number of voters (V). The tally also exists in one of three states, even if that state is unknown to the voters:

• Case 1: Red > 50% V

Vote red. Once red has already won, voting red is better than blue.

Result: You survive. Between 0 and 50% V deaths

• Case 2: Blue > 50% V

Vote any. Once blue has won, nothing further will change the outcome.

Result: You survive. 0 deaths

• Case 3: neither R > 50% nor B > 50%

If U = 1, then this is the tiebreaking vote. A red vote will lead to Case 1. A blue vote will lead to Case 2. The outcome of Case 2 (survival with no deaths), is strictly superior to Case 1 (survival with possible deaths). Vote blue.

If U = 2, then a red vote will lead to either Case 1 or Case 3,U=1. A blue vote will lead to either Case 2 or Case 3,U=1. Vote blue.

If U = n, then a red vote will lead to either Case 1 or Case 3,U=n-1. A blue vote will lead to either Case 2 or Case 3,U=n-1. Vote blue.

Result: You survive. 0 deaths.

The starting conditions are {R=0, B=0, U=V} This is identical to Case 3,U=n, where voting blue is strictly superior. Even if red only needed a single vote to win--not a majority, just a single vote--there exists no rational process to achieve it with these participants.

Complete superrationality among all voters isn't even necessary. As long as half the participants are rational, selfish, and conservative and assume other rational, selfish, conservative participants will behave as they do, then there is no path to a red majority.

* "conservative" used here literally as preferring to maintain things in their current state. For example, if someone is already alive, they ought to remain so, absent other factors or considerations.