Auctions serve the dual purpose of eliciting preferences and allocating resources between competing uses. A less fundamental but more practical reason for studying auctions is that the value of goods exchanged each year by auction is huge. We describe the main kinds of auctions, define strategic and revenue equivalence, analyze optimal bidding behavior, and compare the outcomes from using different types. Read Chapters 19 and 20 of Strategic Play. Lecture 3 on Auctions Violating Revenue Equivalence

When does revenue equivalence fail? The private valuations of bidders might be drawn from probability distributions that are not identical. The theorem does not apply when values are not private, such as when bidders receive signals about the value of the object to them that are correlated with each other. Bidders might be risk averse or risk loving rather than risk neutral. (This is of less importance value maximizing firms.)

Affiliated Valuations The revenue equivalence theorem applies to situations where the valuation of each is bidder is independently distributed. This is not always a valid assumption, because how a bidder values the object on the auction block for his own use might depend on information that another bidder has.

An example: Value of the object not known to bidders Consider a new oil field tract that drillers bid for after conducting seismic their individual explorations. The value of the oil field is the same to each bidder, but unknown. The n th bidder receives a signal s n which is distributed about the common value v, where s n = v +  n and  n  E[v| s n ] – v is independently distributed across bidders. Notice that each drilling company would have more precise estimates of the common valuation from reviewing the geological survey results of their rivals.

The expected value of the item upon winning the auction If the n th bidder wins the auction, he realizes his signal exceeded the signals of everybody else, that is s n ≡ max{s ₁,…,s N } so he should condition the expected value of the item on this new information. His expected value is now the expected value of v n conditional upon observing the maximum signal: E[v n | s n ≡ max{s ₁,…,s N }] This is the value that the bidder should use in the auction, because he should recognize that unless his signal is the maximum he will receive a payoff of zero.

The Winner’s Curse Conditional on the signal, but before the bidding starts, the expectation of the common value is We define the winner’s curse as Although bidders should take the winner's curse into account, there is widespread evidence that novice bidders do not take this extra information into account when placing a bid.

Symmetric valuations We relax independence and consider the class of symmetric valuations, which have two defining features: 1. All bidders have the same utility function. 2. Each bidder only cares about the collection of signals received by the other bidders, not who received them. Thus we may write the valuation of bidder n as:

Revenue Comparisons for Symmetric Auctions We can rank the expected revenue generated in symmetric equilibrium for auctions where valuations are also symmetric. There are two basic results. In a symmetric auction: 1. The expected revenue from a Japanese auction is higher than what an English auction yields. 2. The expected revenue from an English auction exceeds a first price sealed bid auction.

Asymmetric valuations In a private valuation auctions suppose the bidders have different uses for the auctioned object, and this fact is common knowledge to every bidder. Each bidder knows the probability distributions from which the other valuations are drawn, and uses that information when making her bid. In that case the revenue equivalence theorem is not valid, and the auctioneer's prefers some types of auctions over others. What happens if the private valuations of bidders are not drawn from the same probability distribution function?

An example of asymmetry Instead of assuming all bidders appear the same to the seller and to each other, suppose that bidders fall into two recognizably different classes. Suppose there are two cumulative distributions, F ₁ (v) and F ₂ (v) with probability p ₁ and p ₂ respectively. Thus: F(v) = p ₁ F ₁ (v) + p ₂ F ₂ (v) Bidders of type i ∈ {1,2} draw their valuations independently from the distribution F i (v). We assume the supports are continuous and overlap, but that they do not coincide.

An example of asymmetry Instead of assuming all bidders appear the same to the seller and to each other, suppose that bidders fall into two recognizably different classes. Suppose there are two cumulative distributions, F ₁ (v) and F ₂ (v) with probability p ₁ and p ₂ respectively. Thus: F(v) = p ₁ F ₁ (v) + p ₂ F ₂ (v) Bidders of type i ∈ {1,2} draw their valuations independently from the distribution F i (v). We assume the supports are continuous and overlap, but that they do not coincide.

A conceptual experiment Suppose each bidder sees his valuation, but does not immediately learn whether other bidders comes from the high or low probability distribution. At that point the bidding strategy cannot depend on which probability distribution the other valuations come from. She forms her bid b(v). Then each bidder is told what type of person the other bidders are, type i ∈ {1,2}, that is whether their valuations were drawn from F ₁ (v) and F ₂ (v). How should she revise her bid?

Intuition Suppose F ₁ (v) and F 2 (v) have the same support, but F 1 (v) first order stochastically dominates F 2 (v), that is F ₁ (v) 6  F 2 (v). When a bidder learns that his valuation is drawn from F 2 (v) (respectively F ₁ (v)), he deduces the other one is more likely to draw a higher (lower) valuation than himself, realizes the probability of winning falls (rises), so adjusts his bid upwards (downwards). The intuition is in first price auctions to bid aggressively from weakness and vice versa.

Bidding with differential information Another type of asymmetry occurs when one bidder knows more about the value of the object being auctioned than the others. What happens if they are asymmetrically informed about a common value? An extreme form of dependent signals occurs when one bidder know the signal and the others do not. How should an informed player bid? What about an uninformed player?

Second price sealed bid auctions The arguments we have given in previous lectures imply the informed player optimally bids the true value. The uninformed player bids any pure or mixed distribution. If he wins the auction he pays the common value, if he loses he pays nothing, and therefore makes neither gains or losses on any bid. This implies the revenue from the auction is indeterminate.

Perspective of the less informed bidder in a first price auction Suppose the uninformed bidder always makes the same positive bid, denoted b. This is an example of a pure strategy. Is this pure strategy part of a Nash equilibrium? The best response of the informed bidder is to bid a little more than b when the value of the object v is worth more than b, and less than b otherwise. Therefore the uninformed bidder makes an expected loss by playing a pure strategy in this auction. A better strategy would be to bid nothing.

A theorem on first price sealed bid auctions The argument in the previous slide shows that the uninformed bidder plays a mixed strategy in this game. One can show that when the auctioned item is worth v the informed bidder bids:  (v) = E[V|V  v] in equilibrium, and that the uninformed bidder chooses a bid at random from the interval [0, E[V]] according to the probability distribution H defined by H(b) = Prob[  (v)  b]

Return to the uninformed bidder If the uninformed player bids more than E[V], then his expected return is negative, since he would win the auction every time v E[V]. We now show that if his bid b < E[V], his expected return is zero, and therefore any bid b < E[V] is a best response to the informed player’s bid. If the uniformed bids less than E[V] and loses the auction, his return is zero. If he bids less than E[V] and wins the auction, his return is E[V|  (V) < b] – b = E[V| V <  -1 (b) ] – b =  (  -1 (b) ) – b = 0

Return to the informed bidder Since the uninformed player bids less than E[v] with unit probability, so does the informed player. Noting that  (w) varies from v to E[v], we prove it is better to bid  (v) rather than  (w). Given a valuation of v, the expected net benefit from bidding  (w) is: H(  (w))[v -  (w)] = Pr{V  w}[v -  (w)] = F(w)[v -  (w)] Differentiating with respect to w, using derivations found on the next slide, yields F’(w)[v - w] which is positive for all v > w and negative for all v < w, and zero at v = w. Therefore bidding  (v) is optimal for the informed bidder with valuation v.

The derivative Noting it follows from the fundamental theorem of calculus that and so the derivative of F(w)[v -  (w)] with respect to w is :

Lecture Summary We described several types of auction mechanisms. We showed conditions under which auctions are strategically equivalent, revenue equivalent, and compared revenue from different auctions when these conditions do not hold. We showed how to derive the optimal bidding rule for several private value auctions, and explained the nature of the winner’s curse in common value auctions.