This section derives the equilibrium bidding strategy under standard preferences. Stan-dard preferences are outcome-based, i.e., they consider only realized payoff outcomes.
Furthermore, bidders only consider their own payoff and are consistent with EUT, imply-ing Bayesian rationality. In addition, under standard preferences, the bidder has a global utility function (DellaVigna 2009). Hence, standard preferences are represented by the following monotone utility function
uSP(x) = u(x) (4.2.1)
for every x ∈ R. A bidder’s risk attitude is characterized by the curvature of his utility function. A bidder is averse if and only if his utility function is concave; he is
risk-6Note that B and P have the same elements where b1= pn, b2= pn−1, . . . , bn= p1.
116 4. PREFERENCES AND DECISION SUPPORT IN BIDDING
seeking if and only if his utility function is convex (Gollier 2001). The specific bidding function depends on the functional form of expression (4.2.1). We focus on standard preferences represented by a power-utility function of the form u(x) = xβ in deriving the bidding function.
First-Price Sealed-Bid Auction
In the FPSBA, bidder i decides on his bid bi ∈ P facing the following trade-off. On the one hand, a higher bid makes winning more likely as it increases the chance to exceed the other bidder’s bid bj. On the other hand, the payoff in case of winning is smaller the higher the bid. The optimal bid bi,∗ that balances this trade-off and maximizes the bidder’s expected utility is given by
bi,∗= arg max
bi∈P
uSP(vi− bi) Pr{bj < bi} + uSP(vi− bi) 12 Pr{bj = bi}. (4.2.2) The first term on the right-hand side corresponds to the utility in case that bidder i wins. The second term on the right-hand side is the expected utility in case of a tie, which is broken with equal probability. In case that bidder i loses the auction, he does not receive the item and pays nothing. In Proposition 2, FP denotes the first-price sealed-bid auction.
Proposition 1 [Equilibrium FPSBA – SP] There exists a sequence {zSPk }k∈{1,...,n} such that
where bk+1 = bk+ δ and b1 = 0, constitutes an equilibrium bidding strategy.
The proof is relegated to Appendix 4.A.1. The outline of the proof is as follows.
Following Chwe (1989) and Cai, Wurman, and Gong (2010), we first construct the se-quence {zSPk }k∈N ⊂ [0, 1] that partitions the type space into intervals. We then use this sequence to apply the bidding strategy (4.2.6). The sequence {zkSP}k∈N is derived by as-suming that the bidder bids bk in equilibrium, i.e., no other bid should be a better choice for the bidder. Since the winning probability and the utility function are both monotonic in b, it suffices to compare bk−1 and bk+1 with bk. This gives us the inequalities needed to recursively compute the sequence {zkSP}k∈N. With these, the bidding strategy from Proposition 2 constitutes an equilibrium bidding strategy.
Dutch Auction
For the dynamic course of the DA, we adopt the modeling approach of Bose and Daripa (2009). In the DA, the seller starts the auction with the highest ask p1. She then
4.2. THEORY 117
approaches each bidder sequentially asking whether or not the bidder accepts that ask.
Which bidder is asked first is randomly determined at the beginning of each offer. Each bidder has the same chance to be asked first. In case that the bidder who is asked first rejects the offer, the seller offers the same ask to the other bidder.
Facing the current ask, bidder i has the following trade-off. On the one hand, he can accept the offer and stop the auction. In this case, he receives the item with certainty.
On the other hand, he can reject the offer hoping for a better one. In this case, he could make a greater payoff but also faces the risk that the other bidder stops the auction before he is asked again. At each ask pk, bidder i decides whether to accept or to wait for the next offer.
We begin by comparing the utility the bidder earns if he accepts now in period k, i.e.,
uSP(vi− pk), (4.2.4)
to the expected utility if he waits for the next price, that is,
E[uSP(vi− pk+1)] = Hki · uSP(vi− pk+1), (4.2.5) where Hki is the probability given distribution F (v) that bidder i receives the item at price pk+1 given that he refuses the price pk. The probability Hki consists of two parts:
the probability φik under F that i obtains the item at the next price pk+1 given that it is still available at that price, i.e., that it has not been sold at price pk; and the probability ρik under F that the item is actually available at price pk+1 given that bidder i refused price pk. Consequently, we have Hki = φik· ρik.
Proceeding with this comparison for all prices, we obtain the inequalities needed to construct the same sequence {zkSP}k∈N ⊂ [0, 1] as in the FPSBA that determines the following equilibrium bidding strategy.
Proposition 2 [Equilibrium DA– SP] There exists a sequence {zkSP}k∈{1,...,n} such that
βSP,DA(v) =
where pk+1 = pk− δ and pn= 0, constitutes an equilibrium bidding strategy.
The detailed proof is relegated to Appendix 4.A.2. While the bidding strategies of the FPSBA and the DA look a bit different, they yield the same equilibrium bids for all valuations under standard preferences. Hence, the two formats are strategically equivalent (Vickrey 1961). This implies that both formats yield the same realized revenue.7
7Note that the revenue equivalence theorem only yields the same expected revenue; and this is true only under very restrictive assumptions. Hence, strategic equivalence is stronger because not only expected but actually realized revenues are the same.
118 4. PREFERENCES AND DECISION SUPPORT IN BIDDING