I agree to TheCaptain that if no ones stops the loop the game is a draw with respect to rule 104.4b. I disagree related to the statement that this loop cannot be stopped.
From my understanding of resolving abilities, timing and priority the following is happening:

1. I tap SD and put the ability on the stack to deal 1 damage to itself.
1.1 all player pass
1.2 topmost object of stack (see 1) will resolve.
2. One damage is dealt to SD, this has two effects according to rules 118.3d and 118.3e: 1 damage is marked on SD and I get 1 life
2.1 Trigger condition for SDs ability is met: Whenever Stuffy Doll is dealt damage, it deals that much damage to the chosen player. This triggered ability is put on the stack (see rule 116.2a, 603.1, 603.2, 603.3)
2.2 all player pass
2.3 topmost object of stack (see 2.1) will resolve.
3. One damage would be dealt to the player which is replaced by Pariahs effect to redirect it to the SD.
3.1 Step 2 is repeated.
3.2 Step 2.1 is repeated. SD damage on stack
3.3 Step 2.2 is repeated. all player pass
3.4 Step 2.3 is repeated. topmost stack object resolves
...
Since all players receive priority in between all players are able to cast something with instant speed. Demistify or swords to plowshares or seal of cleansing should break this loop.

Did I misunderstand anything related to stack and resolving spells and abilities?

104.4b If a game that’s not using the limited range of influence option (including a two-player game) somehow enters a “loop” of mandatory actions, repeating a sequence of events with no way to stop, the game is a draw. Loops that contain an optional action don’t result in a draw.

116.2a Triggered abilities can trigger at any time, including while a spell is being cast, an ability is being activated, or a spell or ability is resolving. (See rule 603, “Handling Triggered Abilities.”) However, nothing actually happens at the time an ability triggers. Each time a player would receive priority, each ability that has triggered but hasn’t yet been put on the stack is put on the stack. See rule 116.5.

603.1. Triggered abilities have a trigger condition and an effect. They are written as “[Trigger condition], [effect],” and begin with the word “when,” “whenever,” or “at.” They can also be expressed as “[When/Whenever/At] [trigger event], [effect].”

603.2f An ability triggers only if its trigger event actually occurs. An event that’s prevented or replaced won’t trigger anything.
Example: An ability that triggers on damage being dealt won’t trigger if all the damage is prevented.

603.3. Once an ability has triggered, its controller puts it on the stack as an object that’s not a card the next time a player would receive priority. See rule 116, “Timing and Priority.” The ability becomes the topmost object on the stack. It has the text of the ability that created it, and no other characteristics. It remains on the stack until it’s countered, it resolves, a rule causes it to be removed from the stack, or an effect moves it elsewhere.


[Edited by knarf_the_dwarf on 4/Dec/12 at 8:15AM]