PoE

Process of Elimination, also known as PoE, is a tactic by which a player's role can be deduced by eliminating all other players from possibly having that same role. This is often possible due to the existence of reports on other players, or counterclaims.

There are two different types of PoE: "universal" PoE, which refers to when a player's role is confirmed for all players, or "point of view" PoE in which another player's role is only confirmed in the perspective of one player. See FYPOV and FMPOV for more insight into that.

The following examples will cover some very basic applications and analysis of PoE being utilized.

Basic PoE: Reports

There are 4 live players - Players A, B, C and D. Between them, there are 2 Millers, 1 Cop, 1 Godfather.

PlayerA, the uncc'd cop, has guilty reports on both PlayerC and PlayerD.
As only millers will show up guilty of the living players, and the godfather would not, it can be deduced via PoE that PlayerB is the godfather.

This deduction is universal for all players. PlayerB is the confirmed godfather from the perspective of every player in the game.

Basic PoE: CC's

There are 5 live players - Players A, B, C, D and E. 1 Doctor. 1 Cop. 1 Bomb. 2 Nilla Mafia.
PlayerA is uncc'd cop, but has no live reports. PlayerB is uncc'd doc. Players C and D both claim Bomb. Player E does not claim.

As it is known that only ONE of PlayerC or PlayerD is mafia (one is real bomb, one is mafia cc'ing), but there are 2 mafia left in the game, it can be deduced due to the cc's (and the lack of cc's in the case of cop and doc) that PlayerE is the second mafia, the partner of either PlayerC or PlayerD.

Again, this is a "universal" use of PoE which holds true from the perspective of every living player in the game.

Perspective PoE

There are 5 live players - Players A, B, C, D and E. 1 Doctor. 1 Cop. 1 Blue. 2 Nilla Mafia.
Player A is uncc'd cop, with no live reports. PlayerB and PlayerC claim doc. PlayerD claims blue. Player E does not claim doctor or cop.
Lets look at this scenario from the perspective of PlayerD, the real blue. PlayerD does not know which of PlayerB or PlayerC is the real doctor, however he knows that one of them is the real doctor. This eliminates PlayerE from being doctor. The fact that Cop is uncc'd means PlayerE is also eliminated from being cop. PlayerD knows that he is the last living blue, and therefore this eliminates PlayerE from being blue too. Therefore, from PlayerD's perspective, PlayerE is mafia, simply for not claiming cop or doctor, even if he hasn't directly claimed blue yet. It is logical that PlayerE does go on to claim blue, and hence this confirms PlayerD's suspicions.
Whilst from PlayerD's perspective, PlayerE is confirmed mafia, this PoE is not universal, since every other villager in the game cannot outright eliminate either PlayerD or PlayerE from being the last blue, hence they have to scumhunt rather than use PoE. This is the most basic example of perspective PoE, and in a situation as simple as this, it's just a way of looking at how direct cc's can come about without claims.

As a tactic

When discussing the effectiveness of PoE as a tactic, it's important to differentiate between universal PoE and perspective PoE. Universal PoE is, obviously, one of the most powerful tactics in the game for both factions.

Town Strategy
The above examples are all small scale, but through singling out players as having/not having certain roles, mafia can be confirmed well into the early stages of the game. This is why it is important to keep track of all claimed roles (the role tracker is good), as well as keep any reports that do not pertain to confirmed mafia alive.

In the first example above, it can be seen that the Godfather is caught through keeping guilty reports alive, even though guilties may have been possible mafia in earlier stages had the setup been Basic Variant, a popular setup where PoE is critical.
PoE is also a good way to speed up games, or increase the odds of a lynch being successful. Players should, every day, analyze all past reports next to surviving roles and attempt to narrow down lynches. You may be in an autowin situation and not even know it!
But also remember that PoE is just as useful for narrowing down lynches as it is for confirming Mafia. This can instill confidence in the town when scumhunting, as well speed up any scumhunting since it only has to be done between a reduced number of players. Odds are also improved.
This said, always remember the limitations of PoE. In multisetups, no reveals or janitor setups, you can't single out roles to every living player nearly as effectively, since you do not know what roles are, in fact, alive. Only eliminate if it's truly impossible!
Perspective PoE is not nearly as powerful in determining games, and it's not recommended that you scream about a player being confirmed mafia when they aren't from the perspective of any other player, and do little else. What you should do however is use this knowledge to pinpoint scumtells on the other player which you can utilize to have the remaining members of the town lynch your confirmed mafia. You can also seek out interaction between this player and others, and attempt to deduce which player is most likely to be their mafia partner.

Mafia Strategy
PoE can hugely benefit mafia too - remember that the tactic can be used to single out players as any role, not just as mafia. It's critical that the mafia too mark out the roles of all claimed/confirmed players in the game, so come night time, they can utilize the process of elimination to determine, or narrow down, players who have PR roles or make for desirable kills. If you're hunting for the cop, and all players save for you , your mafia partner and player X have claimed blue, player X must be the cop.
However caution must be utilized when using this tactic - remember that the town is uniformed as to ALL roles except for those claimed (they obviously don't know who the mafia are), so as far the town is concerned - the cop can be one of three players. Don't, therefore, say that "Player X is confirmed cop" in the event that you haven't killed him yet. The exclusion of oneself/another (a potential mafia partner) in any PoE is a common way that mafia tend to slip up!
That said, it's obviously a good idea to avoid giving town the chance to utilize PoE at all, and hence find themselves with better odds of lynching you. Try not to CC unnecessary roles where your decision can confirm a partner, or clear a townie due to them not being involved in a cc. Nonetheless, mafia should also reflect on PoE every day, and think to themselves "if I WAS [town role], who would be confirmed/likely mafia by PoE from my perspective?", and set out to "prove" these players as mafia. Failing to do so isn't going to make you look town by any means.
- Moxie