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Numbers, Part 2

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This edition of Numbers will look at various town strategies in the NewbieGame (2 Mafia, 1 Cop, 1 Doctor, 3 Citizen).

Introduction

Note: This article only applies to the original Open Setup Newbie Game. The modified C9 setup and other variations may not have a worthwhile no lynch strategy, and I do not recommend it even in the original setup. This is a purely theoretical discussion.

It has been suggested that the town's best strategy is a cop claim/no lynch one. The cop comes out the first day, the town doesn't lynch until either a mafia is found, the doctor dies, or it's lynch-or-lose, and then the cop makes investigations protected by the doctor while the mafia frantically try to kill the doctor so they can get at the cop.

Abbreviations

Roles:

  • M - Mafia
  • C - Cop
  • D - Doctor
  • T - Townie/Town
  • I - checked Innocent
  • X - Dead Townie, k - kill

Actions:

  • i - investigates
  • ly - lynches
  • nl - no lynch

The Numbers

Winning probabilities are from the Mafia perspective.

Overall Probability: Mafia wins 141/720 + 114/720 = 255/720 = 17/48, around 35%. Town wins 31/48, around 65%.

  1. M k D (1/4) - 40/180 + 26/180 + 75/180 = 141/180
    1. C i M (2/6) - T ly M - M k C - T nl - M k T - 2/3
    2. C i D (1/6) - 4/15 + 9/15 = 13/15
      1. T ly M (2/5) - M k C - T nl - M k T - 2/3
      2. T ly T (3/5) - 1
    3. C i T (3/6) - 2/6 + 3/6 = 5/6
      1. T ly M (2/4) - M k C - 2/3
      2. T ly T (2/4) - 1
  2. M k T (3/4) - 50/1080 + 45/1080 + 58/1080 + 75/1080 = 228/1080 = 19/90
    1. C i M (2/6) - T ly M - 5/36 + 0 = 5/36
      1. M k D (1/3) - 0 + 2/12 + 3/12 = 5/12
        1. C i M (1/4) - 0
        2. C i D (1/4) - 2/3
        3. C i T (2/4) - 1/2
      2. M k T (2/3) - 0 + 0 + 0 + 0 = 0
        1. C i M (1/4) - 0
        2. C i D (1/4) - 0
        3. C i X (1/4) - 0 !
        4. C i T (1/4) - 0
    2. C i D (1/6) - 8/36 + 1/36 = 9/36 = 1/4
      1. M k D (1/3) - 1/3 + 1/3 = 2/3
        1. C i M (2/4) - T ly M - M k C - 2/3
        2. C i T (2/4) - 2/3
      2. M k T (2/3) - 0 + 1/24 + 0 = 1/24
        1. C i M (2/4) - 0
        2. C i X (1/4) - 1/6 !!
        3. C i T (1/4) - 0
    3. C i X (1/6) - 15/90 + 14/90 = 29/90
      1. M k D (1/3) - 8/30 + 5/30 + 8/30 = 21/30 = 7/10
        1. C i M (2/5) - T ly M - M k C - 2/3
        2. C i D (1/5) - 2/6 + 3/6 = 5/6
          1. T ly M (2/4) - 2/3
          2. T ly T (2/4) - 1
        3. C i T (2/5) - 2/3
      2. M k T (2/3) - 0 + 1/30 + 3/30 + 1/30 = 5/30
        1. C i M (2/5) - T ly M - 0 !
        2. C i D (1/5) - 1/6 !!
        3. C i X (1/5) - 1/2 !!!
        4. C i T (1/5) - 1/6 !!
    4. C i T (2/6) - 10/72 + 4/72 + 1/72 = 15/72 = 5/24
      1. M k D (1/3) - 3/12 + 2/12 + 0 = 5/12
        1. C i M (2/4) - T ly M - M k C - 1/2
        2. C i D (1/4) - 2/3
        3. C i T (1/4) - 0
      2. M k I (1/3) - 2/24 + 1/24 + 1/24 = 4/24 = 1/6
        1. C i M (2/4) - T ly M - 1/6 !
        2. C i D (1/4) - 1/6 !!
        3. C i T (1/4) - 1/6 !!
      3. M k T (1/3) - 0 + 0 + 1/24 = 1/24
        1. C i M (2/4) - 0
        2. C i D (1/4) - 0
        3. C i X (1/4) - 1/6 !!

! - Before town lynches Mafia, everyone claims. Either there will be 2 Doctor claims or 2 Townie claims. Cop checks one of those claims, and knows the last Mafia the next day.

!! - Situation is 2 Mafia, 1 Cop, 1 Doctor, 1 Townie, but the Cop knows either the Doctor or the Townie is innocent. Best play for the town is to pick one of the three unchecked and force a claim; the Mafia must both claim the same as the other unchecked.

There is a 2/3 chance of the Mafia being picked to claim, in which case there is a 1/2 chance of making the right claim. If the Mafia makes the wrong claim, the town wins (as the checked innocent reveals and gets that Mafia lynched, then the Cop checks between the two unconfirmed players). Since the innocent will always make the "right" claim if picked, the town's best strategy is to lynch one of the two that weren't picked (partial probability says that it's more likely a right claim is innocent, since the Mafia have a chance to make a wrong claim). So, there's a 1/2 chance that the innocent will be lynched and the Mafia wins, for a 1/6 total. The Mafia always loses if the innocent is picked, as one of the Mafia will get lynched and then the Cop checks among the two remaining unconfirmed players.

!!! - The tricky bit here is that there is a Doctor still alive, so claims can help the town. If both Mafia claim Townie, then there's a 1/3 chance of them winning (the town lynches the innocent townie claim). If both claim Doctor, there's also a 1/3 chance (same logic). However, if one claims Townie and one claims Doctor, there's a 1/2 chance for the Mafia now (pick one of the pairs and flip a coin). So that's the Mafia's best strategy.

Without claims, the Mafia wins 5/6 (town has to lynch correctly twice in a row, at 1/2 and 1/3), so this is better for the Town.

Other strategies will be added later.