Axiomatic analysis of auction rules

An auction is an example of a resource allocation rule. The auction procedure operates by each player being required to submit a sealed bid for the object. We assume, for convenience, that all bids are positive and that no two bids are equal. The problem of determining the buyer to whom the object has to be assigned on the basis of the bidding profile, leads to an allocation rule. Our initial result concerns axiomatically characterising all allocation rules, which assigns the object to the bidder whose bid has a pre-specified rank among all the bids that the seller receives. We call such allocation rules, rank based allocation rules. We use two properties to characterise such allocation rules: the allocation rule should be order preserving; it should be anonymous. Once we introduce a pricing rule along with an allocation rule, we have a well-defined auction rule. We axiomatically characterise auction rules, which allocates the object to the highest bidder. We invoke the assumptions of competitiveness and non-responsiveness to characterise the Vickrey auction rule and its generalisation respectively.