Numbers, Part 3

When considering balance while designing a game, I began to think about the odds of a doctor protecting the mafia target. I began with a basic setup of 1 mafia kill, 1 doctor who cannot self protect, and 0 role blockers. Here's the expression that resulted:

(T-1) / (T*(P-1))

The number of town minus one (players that could be targeted by both the doctor's protection and the mafia's kill), divided by the number of town (mafia targets) multiplied by the number of players minus one (doctor targets).

Thus a basic 9 player setup of 2 mafia goons, 1 doctor, and the rest town (cops don't factor into this probability) with a town lynch going into the first night . . . the odds that the doctor will protect the player who is targeted by the mafia is:

(T-1) / (T*(P-1))
(6-1) / (6*(8-1))
5 / (6*7)
5 / 42
11.9%

Moving on to the next basic setup: 1 Godfather (role blocker), 1 mafia kill, and 1 doctor who cannot self protect. The resulting expression is only slightly expanded:

((T-1)*(T-2)) / (T*(T-1)*(P-1))

The number of town minus one (players that could be targeted by both the doctor's protection and the mafia's kill) multiplied by the number of town minus two (a factor of remaining two non-doctor players so that the doctor is neither killed nor role blocked), divided by the number of town (mafia night kill targets) multiplied by the number of town minus one (mafia role blocking targets) multiplied by the number of players minus one (doctor targets).

This can, of course be reduced to the following expression:

(T-2) / (T*(P-1))

If night actions are applied in the order of role blocking, protection, then kills and assuming a town lynch going into the first night with the basic 9 player setup . . . the odds that the doctor will protect the player who is targeted by the mafia and he is not role blocked is:

(T-2) / (T*(P-1))
(6-2) / (6*(8-1))
4 / (6*7)
4 / 42
9.5%

This is a lower percentage as compared to our no role blocker setup. If it is a closed or semi-open setup, that decrease is justified by it leading to the knowledge of a role blocking player. That is what constitutes the balance of this F11 Newbie setup.