[SETUP] Gladiatorial Popcorn

[callforjudgement]

7 Town, 2 Mafia. Pregame, the Mafia choose a player to be a Gladiator; that player has to gladiate someone before Day 1 starts (gladiate = make everyone except yourself and a player of your choice unlynchable for the day, this is a public effect). On every subsequent Day, the player who survived the previous Day's gladiation is the new Gladiator, and has to pick someone to gladiate before the thread is unlocked for the next Day, etc.. There's no Night, just a short pause while the Gladiator target is selected. Scum win at 50%.

[popsofctown]

I tend to want to see games like this start with the special mechanic kicking in on day 2. The reason being is I hate to see a game action be based on nothing but meta. And as an auxilliary reason, I really like how a "mountainous" day1 can inform and funnel the rest of the dayplay as it passes through mechanics like this.

A nifty setup as is, although I tend to want to see scum EV above 50%, that's where I am in my life right now.

[callforjudgement]

I'm not sure where the EV is. There's too much WIFOM involved, and that always tends to make EV calculations a nightmare.

I agree that the N0 choice being meta-based is potentially a problem but starting with a "day to get reads" first seems like it might spoil the idea behind the setup (often people get most of their reads from D1 as it is).

[popsofctown]

Can we just make everyone a 1 shot day gladiator and set a 5 day deadline for someone firing theirs (or else scum pick), then a longer deadline for lynching within the pair?

I assumed "Scum win at 50%" was an EV calculation and I thought it was weird but just accepted it. I guess you meant scum win at parity.

[callforjudgement]

Yes, I'm indicating that scum win at parity (I wasn't certain that they could finish off the game otherwise, but in retrospect they probably can.)

Letting anyone do the first gladiation on D1 is interesting, I like that way to start the game.

[mith]

EV is at worst 50% for town, even if Mafia pick the first gladiator - town can choose to always lynch the gladiator, and it reduces to 2:6 Nightless if Mafia choose Town first.

Making it slightly more complicated but still manageable, let's say the Town's only two options are always lynch the gladiator or never lynch the gladiator. Let x be the probability Mafia picks Town as first gladiator, and y be the probability Town always lynches the gladiator. Then:

EV = x*y*(1/2) + (1-x)*y*(3/4) + x*(1-y)*(5/7) + (1-x)*(1-y)*0
= 5/7 x + 3/4 y - 27/28 xy

equilibrium:
y = 20/27
x = 7/9 <- that is, pick a player completely at random

EV = 5/9 (55.5...%)

which is identical to Nightless. 2:6 is the same, EV 1/2. (Mafia can force the Nightless EV just by making random choices for gladiator when needed.)

[mith]

That’s a little oversimplified - Mafia can do a bit better in the always lynch gladiator case by not picking their partner. But the general idea holds (and the EV can be proven by induction) by town making a new (appropriately weighted) decision each time Mafia is lynched as gladiator. This forces Mafia to pick a target at random again.