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Bidding systems refer to auction-style variable pricing in DKP loot distribution systems. As in real life, there is an arbitrary number of methods to conduct an auction. The most popular are:
On loot distribution, everybody writes their bids in the open raid channel. The loot master functions as auctioneer. The highest bidder gets the item and loses the amount of DKP he bid.
When an item is due to be distributed, everyone interested whispers their bid to the loot master. The highest bidder wins the item for the amount bid.
Spend one more
The winner pays not the amount he bid, but just one more than the second highest bidder.
Bidding amounts are in set tiers, e.g. in multiples of 10; players bid at tiers, up to the DKP they possess (or in some cases, the next tier up, resulting in some players have small negative amounts of DKP).
Items may have a static maximum bid value; players may bid no more than this amount, no matter how many DKP they possess.
Items may have a static minimum bid value; the winner MUST spend at least this much, even if they are the only bidder on an item.
Only high priority classes may be allowed to bid on an item, versus any player who can equip an item. This may be used to prevent situations where an individual member of a class values an item more highly (and has more points to bid) than members of a class for whom the item is more traditionally valuable. For example: a Paladin bidding on Drake Fang Talisman and outbidding Rogues who may get more benefit. Class restrictions (or priorities) are traditionally hotly debated.
The subjective value of a drop is not fixed. The same item may be a valued upgrade for one raid member, and a minor sidegrade for another. In an homogenous and fair environment, bidding leads to the best possible item pricing and item distribution. Since everyone is free to bid as much as they can (up to their current DKP score), it's not automatically the person with the highest score who gets the item, if someone with a lesser amount is willing to spend more, he will get it. So items normally go to the raid members who value them most, for whom they are the greatest upgrades. On the other end, even small up- or sidegrades will be distributed and not disenchanted.
Since bidding is a free market, it introduces all problems of a free market. Most problems associated with bidding revolve around finding the right price, and therefore are systematically inherent. The system's balance is always disturbed if an item is sold either too cheap or too expensive.
Playing the System
Bidding systems not only reward contribution to the guilds goals, but also reward "playing" the bidding system well. In order to gain maximum profit it is necessary to "read" the situation, and to correctly assess when an item starts to become cheap (or whether it will remain rare/expensive for quite some time). This may actually develop into a meta-game with a strong luck factor.
The DKP Gap and Collusion
See the extensive discussion in the Loot system article.
A player may get away with an item for a very low price, because every other player on that raid didn't want or need it, even though other players in the system (but not in the raid) would have bid much higher.
Sometimes, one or more players enter a bidding contest with a specific player just to push the price of an item up, and thus drain the point pool of the player in question, ensuring that this player has to stay out of later bidding contests.