Optimum type auction for seller's profit maximisation I am trying to formulate an auction, in which sellers will create a cartel and ask for the highest possible price that buyers pay. Practically, I would have a multi-part game in which in each part I will increase the price until the buyer abstains. Do you know what is the optimum kind of auction for this scenario?
Thanks
 A: The type of auction you are describing is a Dutch Auction. You could alternatively use a Vickrey Auction, as honest bidding is a weak Nash Equilibrium. The way it works is that the highest bidder wins the item, but pays the second highest price. Vickrey actually proved this is strategically equivalent to a descending price oral auction. 
Vickrey's Theorem: In a Vickrey auction, bidding one's true valuation for the object being sold is a weakly dominant strategy.
Proof: Let $v_{i}$ be bidder $i$'s valuation of the object, and let $x$ be the highest bid of the opponents. If $x > v_{i}$, then player $i$ should not increase his bid above $x$, as he would lose more than by not bidding. Similarly if $x < v_{i}$, then bidder $i$ must bid $x + \epsilon$ for some $\epsilon \in (0, v_{i} - x]$.  As another player $j$ may have $v_{j} = v_{i}$, player $j$ can iterate on the $\epsilon$ argument and bid higher than $x + \epsilon$. So taking the $sup$ of this sequence yields $v_{i}$. That is, if $x = v_{i}$, then bidding $v_{i}$ will result in a tie and a chance at winning the object. Bidding less will surely lose player $i$ the auction, and bidding more will result in a net loss.
