Auction

An auction allocates goods using submitted messages and a specified payment rule. Private-value auctions model each bidder's valuation as its own private information. Equilibrium bids depend on the allocation and payment rules, not only on valuations.

In a standard all-pay auction, every player pays its bid or effort cost, and the highest bid receives the prize. With a value $v_i$, unit effort cost, and winning probability $x_i(b)$, player $i$ has quasilinear utility $v_i{x}_i(b)-b_i$. Equal highest bids require an explicit tie rule.

For effective prizes $A_1\ge A_2>0$ and unit effort costs, equilibrium effort CDFs on $[0,A_2]$ are $G_1(b)=b/A_2$ and $G_2(b)=1-A_2/A_1+b/A_1$. The weaker player has an atom at zero. Incremental utilities are $A_1-A_2$ and zero, and winning probabilities are $1-A_2/(2A_1)$ and $A_2/(2A_1)$. Direct payoff indifference and exclusion of larger bids verify the Nash equilibrium. A further player with effective prize at most $A_2$ cannot profit by entering against these distributions.

Each bidder submits one private bid; the highest bidder wins and pays its own bid, while losers pay nothing. A multi-unit version can award one item to each of the highest bidders, each paying its own bid. In a monotone symmetric equilibrium, a bidder can imitate another type's bid when checking incentive constraints.

For a risk-neutral single-parameter bidder with incentive-compatible type reports, interim utility satisfies $u^{\prime }(\theta )=G(\theta )$ wherever the winning probability is continuous. Consequently expected payment is determined by allocation probabilities and the utility of the lowest type. The common normalization $u(0)=0$ must be justified, not obtained from allocation alone.

Two auctions with the same interim allocation probabilities and the same lowest-type utilities have the same interim expected payments under the usual risk-neutral single-parameter incentive conditions. Equal realized payments are not required. This conclusion follows directly from the interim payment identity.

A private-value auction gives each bidder its own value for receiving an item; the value is determined by its own type rather than by another bidder's information. An independent private values model also assumes independence between types.

A bidder with unit demand values receiving one item but obtains no additional value from extra identical units. Selecting two winners means awarding one item to each of two bidders, not two units to one bidder.

This model assigns independent private valuation types to the bidders. Symmetry adds identical type distributions and bidder roles. These assumptions determine interim winning probabilities from the valuation distribution in a monotone symmetric equilibrium.