I'd be interested to see your math on those. I did the numbers on a nerfed version of JK/Doctor (basically Lover IC can't be killed while JK lives, but JK has no ability otherwise; JK also won't be countered by Mafia but the first scum on the chopping block will force him out with a claim) and ended up with 48:52, so I don't doubt it's balanced, but I'm curious how you dealt with all the possible claim shenanigans and mind games.
Spoiler: maths
D1: IC chooses lynch to avoid giving info about lover.
2/8 - Mafia, claim JK, JK counter, lynch Mafia (assumes JK will never be countered by Mafia), kill JK
D2:
1/6 - Town win
5/6 - Lynch VT, kill Lovers, 1:4 = EV 7/15
EV 5/9
1/8 - JK, claim
2/7 - Mafia, kill JK, EV 5/9
5/7 - Town, kill JK
2/6 - Mafia, kill Lovers, 1:4 = EV 7/15
4/6 - Town, kill lovers, 2:3 = EV 2/15
EV 11/45
EV 1/3
5/8 - Town
N1:
1/6 - JK, EV 11/45
1/6 - Lover, 2:1:4, should be EV 11/45?
4/6 - Town
D2:
2/6 - Mafia claim JK counter lynch Mafia kill JK EV 1/2
1/6 - JK, claim
2/5 - Mafia, kill JK, EV 1/2
3/5 - Town, kill JK, EV 1/6
EV 3/10
3/6 - Town
N2:
1/4 - JK, EV 1/6
1/4 - Lover, EV 1/6
2/4 - Town, 2:2:1:1 massclaim EV 1/4?
EV 5/24
EV 77/240
EV 319/1080
EV 514/1080
FTF can be hard on scum, but I certainly don't have a problem putting a theoretically balanced setup in just because of that. I do like it better with the JK - JK is stronger than Doctor in a normal situation, but the Lover IC makes it really tough for him to be unpredictable. (AP may well be right though, that the JK should just try to block the kill rather than protecting the IC.)
D1: IC chooses lynch to avoid giving info about lover.
2/8 - Mafia, claim JK, JK counter, lynch Mafia (assumes JK will never be countered by Mafia), kill JK
D2:
1/6 - Town win
5/6 - Lynch VT, kill Lovers, 1:4 = EV 7/15
EV 5/9
1/8 - JK, claim
2/7 - Mafia, kill JK, EV 5/9
5/7 - Town, kill JK
2/6 - Mafia, kill Lovers, 1:4 = EV 7/15
4/6 - Town, kill lovers, 2:3 = EV 2/15
EV 11/45
EV 1/3
Same.
5/8 - Town
N1:
1/6 - JK, EV 11/45
1/6 - Lover, 2:1:4, should be EV 11/45?
4/6 - Town
D2:
2/6 - Mafia claim JK counter lynch Mafia kill JK EV 1/2
1/6 - JK, claim
2/5 - Mafia, kill JK, EV 1/2
3/5 - Town, kill JK, EV 1/6
EV 3/10
3/6 - Town
N2:
1/4 - JK, EV 1/6
1/4 - Lover, EV 1/6
2/4 - Town, 2:2:1:1 massclaim EV 1/4?
EV 5/24
EV 77/240
EV 319/1080
EV 514/1080
If JK is able to stop the kill n1, then we have 2 Mafia, 2 Lovers, 1 JK, 4 Town. (2:7)
D2: IC decides who to lynch again,
2/7 - Mafia, claim JK, JK counter, lynch Mafia (assumes JK will never be countered by Mafia), kill JK
1/6 - Town win
5/6 - Lynch VT, kill Lovers, 1:4 = EV 7/15
EV 5/9
The issue is that if we assume the JK has the ability to stop the kill, we have to also assume that the Mafia has the ability to kill the IC, and find the equilibrium probabilities for each strategy. (Plus take into account that if the JK stops the kill, there's a pretty good chance whoever he targeted is scum and so will probably go ahead and claim that. This is probably a bigger factor which will weigh things a bit more toward the town, though it's hard to say how much without getting into the strategy equilibrium.)
[edit]Anyway, that's an issue for another thread. I think we all agree that it is theoretically balanced.[/edit]
Ah, good deal. At some point I'd like to get back to applying Nash Equilibria to simple setups with complex claiming strategies (e.g. 2 Mafia, 1 Cop, X Townies is just a complete mess).
Good point, and actually I found an error right at the end (the last case EV didn't get multiplied by 5/8) so it should have been 37:63. Now I'm getting 41:59 with the JK trying to block the kill if there is only one scum left (but otherwise always protecting the IC).
Giving the JK the ability to block also greatly affects the four-dead-townies case. Working that out now.
For 2:1:4,
D2:
2/7 - Mafia, claims JK, JK counters, Mafia lynched, JK shot, results in 1:4 EV 7/15
1/7 - JK, claims JK
====2/6 - Mafia, JK shot, results in 1:4 EV 7/15
====4/6 - Town, JK shot, results in 2:3 EV 2/15
====EV 11/45
4/7 - Town
====N2: JK making a save does nothing to improve town's EV
====1/4 - JK, results in 2:3 EV 2/15
====3/4 - Town, results in 2:1:2
========D3: JK claims (makes no difference in EV whether JK claims or not since LyLo and Mafia will fakeclaim JK)
========2/4 - Mafia
============1/3 - JK saves, town win
============2/3 - JK shot, resulting in 1:2 EV 1/3
============EV 5/9
========EV 5/18
====EV 29/120
EV 193/630
So, my equilibrium calculation can't be correct, since one of the perfect strategy outcomes (JK always protects the IC, scum never shoots IC) should be equal to the EV calculated above, but it's not.
I probably made an error in my calculations somewhere.
If JK is on the chopping block d1 and claims and there is no cc, then IC should reveal lover and tell the JK to go for the block.
1/8 - JK, claim
====2/7 - Mafia (same as above) EV 17/27
====5/7 - Town
========N1: (Scum will be shooting the JK, since killing JK gives town EV 94/315 and killing the Lovers give town 2:1:4 EV 193/630)
========1/6 - JK blocks, lynches Mafia
============N2: 1:1:6
============1/5 - JK blocks, Town win
============4/5 - JK dies, 1:6 EV 7/15
============EV 43/75
========5/6 - JK dies, 2:6 EV 8/35
============2/6 - Mafia, kill Lovers, 1:4 EV 7/15
============4/6 - Town, kill Lovers, 2:3 EV 2/15
============EV 11/45
========EV 202/675
====EV 124/315
2/8 - Mafia, claim JK, JK counter, lynch Mafia (assumes JK will never be countered by Mafia), lover claim
====1/6 - JK blocks kill, Town win
====5/6 - JK killed
========D2:
========1/6 - Town win
========5/6 - Lynch VT, kill Lovers, 1:4 = EV 7/15
========EV 5/9
====EV 17/27
1/8 - JK, claim
====2/7 - Mafia (same as above) EV 17/27
====5/7 - Town
========N1: (Scum will be shooting the JK, since killing JK gives town EV 94/315 and killing the Lovers give town 2:1:4 EV 193/630)
========1/6 - JK blocks, lynches Mafia
============N2: 1:1:6
============1/5 - JK blocks, Town win
============4/5 - JK dies, 1:6 EV 7/15
============EV 43/75
========5/6 - JK dies, 2:6 EV 8/35
============2/6 - Mafia, kill Lovers, 1:4 EV 7/15
============4/6 - Town, kill Lovers, 2:3 EV 2/15
============EV 11/45
========EV 202/675
====EV 124/315
5/8 - Town
====N1:
====1/6 - JK, EV 11/45
====1/6 - Lover, 2:1:4, EV 193/630
====4/6 - Town
========D2:
========2/6 - Mafia claim JK counter lynch Mafia
============1/4 - JK block kill, Town win
============3/4 - kill JK EV 1/2
============EV 5/8
========1/6 - JK, claim, IC reveals Lover.
============2/5 - Mafia
================1/4 - JK block kill, Town win
================3/4 - kill JK, EV 1/2
================EV 5/8
============3/5 - Town
================1/4 JK blocks, Lynches Mafia
====================1/3 JK blocks again, Town win.
====================2/3 JK dies, EV 1/3
====================EV 5/9
================3/4 JK dies, EV 1/6
================EV 19/72
============EV 49/120
========3/6 - Town
============N2:
============1/4 - JK, EV 1/6
============1/4 - Lover, EV 1/6
============2/4 - Town, 2:2:1:1 massclaim (Mafia JK/Town), lynch townie claim
================1/2 - Mafia, Town Win
================1/2 - Town, EV 1/4 (JK block kill, then pick correct scum from JK claims)
================EV 5/8
============EV 19/48
========EV 683/1440
====EV 0.4080026455
EV ~ 46.16154%
If JK protects randomly among the non-IC, and scum never shoots IC:
2/8 - Mafia (same as above) EV 17/27
1/8 - JK (same as above) EV 124/315
5/8 - Town
====N1:
====1/6 - JK
=========1/7 JK blocks
=============D2: 2:1:6
=========6/7 JK dies, 2:6 with Lovers remaining, EV 11/45
====1/6 - Lover
========2/7 - JK blocks kill
============D2: 2:1:6
============2/7 - Mafia, claims JK, JK counters, Mafia lynched, lover claim
================1/5 - JK saves, Town wins
================4/5 - JK dies
====================1/5 - Lynch Mafia, Town win
====================4/5 - Town, Lovers shot, results in 1:3 EV 1/3
====================EV 7/15
================EV 43/75
============1/7 - JK, claims JK, IC reveals Lover, Mafia will shoot JK
================2/6 - Mafia
====================1/5 - JK saves, Town win
====================4/5 - JK dies; EV 7/15
================4/6 - Town
====================1/5 - JK saves, lynches Mafia
========================N3: Mafia shoots JK
========================1/4 - JK blocks, Town wins
========================3/4 - JK dies, results in 1:5, EV 1/2
========================EV 5/8
====================4/5 - JK dies, results in 2:5 with Lovers remaining, EV 2/15
====================EV 139/600
================EV 31/100
============4/7 - Town
================N2: 2:1:5 with Lovers remaining
================1/5 - JK shot
====================1/6 JK blocks
========================D3: 2:1:5 with Lovers remaining
========================2/6 - Mafia, EV 5/8
========================1/6 - JK, claims
============================2/5 - Mafia, EV 5/8
============================3/5 - Town............etc, etc
It looks like it's going to make a difference of 1.5% ish, but taking this into account, Mafia will also employ a mixed strategy that involves shooting the IC sometimes (very rarely).
Which probably means there's some optimal equilibrium that involves both JK and the mafia using mixed strategy that puts the final EV something like 47.2%?
One last thing for completeness, in case we do ever come back to it. In the cases where JK will try to block the kill, Mafia would typically have the option of submitting a no kill, which has a good chance of getting a townie lynched for free (there is no downside to doing this if the JK is trying to block the kill 100% of the time; worst case is JK targetted the scum and he would have been lynched either way). So there's another equilibrium to find there, unless the scum are compelled to shoot someone.
[aside]I'm wondering if the more interesting solution to the whole No lynch/No kill dilemma in general isn't to have a "scum can't no kill twice in a row" rule. Allowing scum to no kill some of the time makes for a much more interesting interaction with protective/blocking roles.[/aside]
