Each bid function maps the player's value (in the case of a buyer) or cost (in the case of a seller) to a bid price.
The payoff of each player under a combination of strategies is the expected utility (or expected profit) of that player under that combination of strategies.
Usually, but not always, a private values model assumes that the values are independent across bidders, whereas a common value model usually assumes that the values are independent up to the common parameters of the probability distribution.
Game-theoretic models of auctions and strategic bidding generally fall into either of the following two categories.
In a private value model, each participant (bidder) assumes that each of the competing bidders obtains a random private value from a probability distribution.
Standard auctions require that the winner of the auction is the participant with the highest bid.
A nonstandard auction does not require this (e.g., a lottery).
Both the private value and common value models can be perceived as extensions of the general affiliated values model.
When it is necessary to make explicit assumptions about bidders' value distributions, most of the published research assumes symmetric bidders.
An important example (which does not assume independence) is Milgrom and Weber's "general symmetric model" (1982).
The first formal analysis of auctions was by William Vickrey (1961).
This optimal auction format is defined such that the item will be offered to the bidder with the highest valuation at a price equal to their valuation, but the seller will refuse to sell the item if they expect that all of the bidders' valuations of the item are less than their own.