Bayesian Nash Equilibrium in First-Price Auctions
Consider the first-price sealed-bid auction when item values are private and independently and identically distributed. Assume there are N >= 2 bidders competing to buy a single unit of an auctioned item. Assume that the seller's reservation price is zero. Let the value of the auctioned item to bidder j, where j = 1, 2, ..., N, be vj. If bidder i submits the highest bid b* then she buys the auctioned item, pays a price equal to b*, and realizes an economic gain equal to vj - b*.
How much should any bidder j, with item value vj, bid in the auction? Let's imagine how a bidder might think about deciding how much to bid. First consider a bid equal to 0. A winning bid equal to 0 would maximize the surplus vi - bi. But a bid of 0 has a probability equal to 0 of being the highest bid. Thus, bidding 0 is sure to give the bidder a 0 surplus from bidding in the auction. Bidding an amount equal to item value vi will have positive probability of yielding a surplus equal to 0 because bi = vi implies vi - bi = 0. Furthermore, bids greater than vi can yield negative surplus. And negative bids are inadmissible. So a bidder will want to bid a positive amount that is less than the item value vi. So far, so good. But the bidder must choose a precise amount to bid.
Let's think about increasing the amount to bid from a starting conjectural bid of 0. As the conjectured bid increases, the probability of winning with the bid would increase but the surplus from winning would decrease. So the bidder must seek to balance the desirable increase in probability of winning with the undesirable decrease in surplus. But how can she find the specific bid that provides the optimal balance? The answer must depend on the number N - 1 of other bidders competing in the auction and the bidding strategies they use. Our representative bidder needs to discover a bidding strategy that is a best reply to the strategies used by her competitors in the auction. Since their bids depend on their item values, information that is unknown to our representative bidder, our representative bidder needs to discover a Bayesian Nash Equilibrium (BNE) bidding strategy.
Vickrey (1961) first developed a BNE bid function for the first-price sealed-bid auction. He showed that is (1) all bidders are risk neutral and (2) private item values are independently drawn from a uniform distribution on [0,v(bar)] then the BNE bid function is
bi = (N-1)/N * vi
Thus, given assumptions (1) and (2) above, if a bidder knows that all of her rivals will bid amounts equal to (N-1)/N times their values, then her best reply is to use the same bidding strategy.
But what if bidders may not be risk neutral? Cox, Roberson, and Smith (1982) and Cox, Smith, and Walker (1982) generalized Vickrey's model to the case where bidders can have constant relative risk averse utility functions other than the risk neutral case modeled by Vickrey. Thus, assumption (2) above is maintained but assumption (1) is replaced by: (1') each bidder has power function utility of surplus (vi - bi)ri and the ri are independently drawn from some distribution on (0, rBAR] where rBAR >= 1. In that case, Vickrey's bid function is generalized to
bi = (N-1)/(N-1+ri) * vi
for vi <= (N-1+ri) vBAR / (N-1+rBAR) and it is a strictly increasing concave function for higher values when ri < rBAR.
The above constant relative risk averse model (CRRAM) was further generalized by Cox, Smith, and Walker (1998) and Cox and Oaxaca (1996), as follows. Let an individual bidder's utility for surplus in the auction depend on an (m-1) dimensional characteristic vector thetai that is independently drawn from a probability distribution. Bidders' preferences are assumed to be everywhere log-concave; thus, if a bidder's preferences are locally risk-preferring then they must be "less convex" than the exponential function. Therefore, this model is named as the log concave model (LCM) because ln u(vi-bi, thetai) is assumed to be strictly concave in auction surplus vi - bi. Thus, different bidders can be risk averse, risk neutral, or risk-preferring. Furthermore, an individual bidder can be risk neutral in some parts of the payoff space, risk averse in some other parts, and/or risk-preferring in other parts. The BNE bid function for the LCM does not have a closed form that can be exhibited, as can (most of) the bid function for CRRAM. But the LCM bid function has known testable properties. Cox, Smith, and Walker (1988) demonstrate that more risk averse bidders will bid more than less risk averse bidders with the same auction value. Cox and Oaxaca (1996) demonstrate that the slope of the bid function is everywhere greater than 0 and less than 1.