Zerosum DKP
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Zerosum DKP means that DKP is distributed based on the value of items that drop. All members in a zerosum DKP start with a score of 0. When an item is given to a raid member, they have to pay for it in DKP (going negative if necessary). These DKP spent are immediately distributed among all the raid members. Thus the amount of DKP, i.e. the sum of all members in the system, is always zero. The idea here is that inflation is impossible, since no points enter the system from the outside.
Although "zerosum" describes only a method of earning DKP, and it would in theory be perfectly possible to combine this income method with bidding for finding the item prices, nearly no raids actually do that. Instead, about every reallive zero sum system uses fixed prices, thus the terms "zero sum" and "zero sum, fixed price" have nearly become synonyms.
Criticisms
In its pure form, zerosum has the disadvantage that learning new content is not rewarded. There can be no bonuses for anything  punctuality, killing bosses etc.
A much bigger problem with zerosum systems is that they don't actually do their job properly. Zerosum was invented to fight Inflation, but the underlying problem is not solved by Zerosum. The deeper problem with Inflation is that it's impossible for new raid members to ever compete for a topoftheline drop. In nonzero sum systems the effect is very obvious, if an average run yields 40 DKP, and the top 10 members each have 1000 DKP or more, it's a rather simple matter for a newbie to figure out when he will get his first highend item (maybe after 1 year).
In zerosum systems, the very same deeper problem exists too. The fact that no points enter the system from the outside doesn't prevent veterans from building a large point lead over new members. The system maintains its advertised properties as long as the group is homogenous. If the whole raid starts out with similar levels of equipment, and everybody does the same number of runs, players will stay close together. But if there are differences (and there always are), if for example some run members have filled some equipment slots with PvP items or drops from heroics, these members will earn more DKP than they spend. If newbies with subpar equipment enter the system, they will feed DKP to the wellequipped raid members, see The DKP Gap.
In summary, the obvious brick wall which every new member can clearly see in the standard systems is replaced by a glass wall, which in the long term has the very same effect, but more subtle and not so obvious.
The frequently raised concern that new players enter the system at 0 points, right in the middle of the veteran players, and thus can take drops away from veterans, is usually a nonissue. After a few runs, newbies will regularly find themselfes at the very end of the ladder, because they will get quite a lot of drops which the veteran run members just don't need any more. Most raids actually formalize this dynamic by introducing a rule that newcomers may only get uncontested drops on their first few runs.
Point Bank
One method to remedy the problem that no bonuses are possible is to introduce the bank as a member in the system. If the bank takes part in every raid (as 11., 21., 26. or 41. member), a pool of DKP will be available which can be handed out as rewards. This same pool can also be used to "buy" all BOE loot from the raid. If the income of the bank is too low, it's possible to give the bank more than one share.
Taxes
A good solution to the DKP gap are taxes, in zerosum as in any other loot system. Of course nobody likes taxes, but in zerosum they have the nice effect to be sometimes negative (i.e. increase players point score).
Relational DKP
Relational DKP systems are a more radical alternative. They allow more flexibility on gaining points, while still being fair to newbies and veterans.
Rewarding raid time
As criticised above, Zerosum systems normally reward only presence at the moment of loot distribution. Beyond the bank solution, there is a more sophisticated system to reward the total time spent raiding, and even to weigh instances differently. To do this, the DKP spent are not immediately awarded to the raid members, but instead held back until the end of the raiding week. At the end of the week, all spent DKP are distributed to the raid members based on the weighted amount of time they spent. An example:
Assume a raiding week is spent as follows:
Wednesday: SSC: 4 hours, 25 ppl. Hydross and Lurker down. 4 drops, 80DKP spent Thursday: SSC: 4 hours, 25 ppl. No bosskill, 1 random drop for 10DKP Sunday: Gruul & Magtheridon, 3 hours, 25 ppl, 8 drops, 1 disenchant for 140 DKP Monday: TK: 4 hours, 25 ppl. Voidreaver down, 3 drops, 60 DKP Tuesday: SSC: 3 hours, 25 ppl, no drops
Each evening, the run is fully added to the database, all items are assigned to players and spent DKP are subtracted immediately as well. Only DKP earned will remain at 0 until the end of the week. After the week is over (on Tuesday night), first the total DKP spent is calculated:
Total DKP: 80 + 10 + 140 + 60 + 0 = 290 DKP
Assume that the Thursday, Monday and Tuesday runs were labeled as progress runs, so their hours get a factor of 1.5 as bonus. Then the weighted number of hours spent is calculated:
(4*1)+(4*1.5)+(3*1)+(4*1.5)+(3*1.5)=23.5 weighted hours
So each hour of raiding is worth the following amount per raid member:
290/23.5 = 12.34
Thus the individual run members will receive (standard zerosum reward in brackets):
SSC(Wednesday ): 12.34 / 25 * 4 = 1.97 (3.2) SSC(Thursday): 12.34 / 25 * 4 * 1.5 = 2.96 (0.4) Gruul/Magth(Sunday): 12.34 / 25 * 3 = 1.48 (5.6) TK(Monday): 12.34 /25 * 4 * 1.5 = 2.96 (2.4) SSC(Tuesday): 12.34 / 25 * 3 * 1.5 = 2.22 (0)
For control, multiply the DKP handed out per player with the number of players in the runs and compare the sum to total DKP:
1.97*25+2.96*25 +1.48*25+2.96*25+2.22*25 = 289.75, which is close enough to the total amount of DKP spent (290), thus correct
Comparing the points earned in this system with the rewards from zero sum, it can be clearly seen that progress (wipe) runs have become much more interesting, while farm runs were demoted. Even if no weight was used, farm runs just have the same value per hour as wipe runs. With the introduction of the weight, guild masters and run leaders have a very finely scalable instrument to make runs more or less rewarding.
There are two known drawbacks to this system. The first is that points are not handed out immediately. If two players competing for the same drop are very close to each other, so that one run could make a difference on who gets the item, a system which is updated daily will produce different results from this one. Whether this effect is really a problem can be discussed, because it affects everybody equally.
The second drawback is that when rewards is solely based on attendance time, a run with a lot of interruptions and afk time will be rewarded just like a raid with many corpse runs. This can be taken into account by a weight  if the run leader feels that there was a lot of idle time, a factor smaller than one could be used as weight for that run.