Econ 805 Advanced Micro Theory I Dan Quint Fall 2007 Lecture 7 Sept 27 2007 Tuesday: Amit Gandhi on empirical auction stuff p till now, we ve mostly been analyzing auctions under the following assumptions: 1. Bidders (and seller) are risk-neutral 2. Bidders are ex-ante symmetric 3. Bidders types are independent 4. Bidders have private values The next several lectures, we ll be relaxing each of these assumptions. Today, we relax riskneutrality. Next week, we relax symmetry. The following week, we relax both independence and private values. Specifically, today, we compare the first- and second-price auctions when bidders are riskaverse (PATW 4.3.1), and discuss Maskin and Riley, Optimal Auctions with Risk-Averse Bidders. We ve talked some about optimal mechanisms; but in some cases, implementing a complicated direct-revelation mechanism is unrealistic Empirically, most auctions for a single good are either a first- or second-price auction with a reserve price (or something that s strategically equivalent to one of these) These require minimal information on the part of the seller, generally give pretty good performance (We saw with symmetric IPV, with the right reserve price, either is optimal) So a reasonable question when we come to risk-averse bidders: when bidders are risk-averse, which auction performs better, a first- or a second-price auction? Maskin and Riley give a very general formulation of risk-averse bidder preferences with private values; but here s the simplest/most natural one: 1
nder risk-neutrality with private values, we ve been assuming a bidder s payoff is { 0 if lose u = t b if win Instead, have each bidder maximize the expected value of { (0) if lose u = (t b) if win where is some increasing, concave von Neumann-Morgenstern utility function In this setting, we get a nice sharp revenue-ranking result: Theorem 1. Suppose is strictly concave and differentiable. Then In a second-price auction with a reserve price r, bidders bid the same as they would if they were risk-neutral: bidders with valuations below r do not submit serious bids, and bidders with valuations above r bid their value In a first-price auction with a reserve price r, every bidder with types t > r bids higher than in the equilibrium when bidders are risk-neutral With risk-neutral bidders, the two auctions are revenue-equivalent; so with risk-averse bidders, the first-price auction yields strictly higher expected revenue. The proof that b(t i ) = t i is a dominant strategy is exactly the same as before. So risk aversion doesn t change equilibrium bids, and therefore revenue, in a second-price auction. (It does change bidder payoffs, of course.) For the first-price auction, the intuition is this: fixing the opponents bid distribution, my optimal bid in a first-price auction is higher when I m risk-averse. This is because, starting at the bid that maximizes my expected (risk-neutral) gain, raising my bid a little bit more is the same as buying partial insurance that s priced very close to actuarily fair. To put it another way, I m better off when I win than when I lose, which means my marginal utility of wealth is lower when I win; so I m happy to give up more in those cases (by bidding higher) to improve my outcome in some of the cases where I was losing. To see this, let G be the probability distribution of the highest of everyone elses bids. If I m risk-neutral and have private value t, I maximize (t b)g(b) with first-order condition (t b)g(b) G(b) so if b RN is my risk-neutral best-response, g(b RN ) = G(b RN )/(t b RN ) 2
Now suppose I m risk-averse, with some Bernoulli utility function, and have the same type t and am facing the same opponent distribution G. If we normalize (0) = 0, then I maximize with first-order condition Let s plug in b RN and check the sign: (t b)g(b) (t b)g(b) (t b)g(b) (t b RN )g(b RN ) (t b RN )G(b RN ) = (t b RN) t b RN G(b RN ) (t b RN )G(b RN ) By the intermediate-value theorem, (t b) (0) t b = (a) for some a (0, t b), so this is G(b RN ) ( (a) (t b RN ) ) but since a < t b RN and is concave, this is positive. So at the same type and the same opponent bid distribution, I bid higher when I m risk-averse. Of course, that isn t a proof; we need to make sure that this holds up in equilibrium, that is, when everyone else shifts their equilibrium bid functions as well. The proof is from Putting Auction Theory to Work, pages 122-125. Before we get to the proof, a word on notation. Thus far, we ve been using the convention that at a type (or signal) t i, a bidder s value from winning the object is exactly t i Works well with private values; but as we get into common-value and other more general auctions, we ll need bidder s valuation as some function of everyone s types We ll use the notation v i (t) for bidder i s value from winning the object, given a vector of types t = (t 1,..., t N ) We can see IPV as the special case where v i depends only on t i, not other bidders types Once we re assuming that the value of winning the object is v i (t i ), not t i, there s no loss of generality from assuming that signals t i are drawn from a particular distribution, say, the uniform distribution on the interval [0, 1] 3
(This is the normalization/convention used in the Milgrom book) That is, in a general model, bidder i s type is drawn from an arbitrary distribution F i, and leads him to value the object at v i (t i ); we ve been using the normalization that v i (t i ) = t i, Milgrom uses the normalization that F i (t i ) = t i. The mapping between the two notations is very clean: v i in one world is simply F 1 i in the other. That is, start in our world, where v i (t i ) = t i and t i is drawn from the distribution F i. If instead of observing t i, the bidder observes F i (t i ), that is, where his type lies in its distribution, then we re in Milgrom s world. So it s just a notation adjustment. For today, we ll stay in our old notation; once we get into interdependent values, though, we ll move into the other notation. But if you look in PATW for the proof, it s done in the v notation. Lemma 1. If log is concave and differentiable, then the unique symmetric equilibrium β of the first-price auction with reserve price r is the solution to the differential equation with boundary condition β (r) = r. (N 1)f(t) F (t)β (t) The bidder s problem is to maximize = (t β (t)) (t β (t)) Pr(win)(t b) + Pr(lose)(0) Normalize (0) = 0, so the second term vanishes, and take the log, giving the bidder s problem as max log (t b) + log H(b) b where H(b) is the probability of winning, conditional on bidding b. Differentiating with respect to b gives (t b) (t b) + 1 H(b) H (b) If everyone else is bidding according to the equilibrium bid function β, then and so H(b) = Pr(β (t j ) < b j i) = Pr(t j < β 1 H (b) = (N 1) ( F (β 1 (b))) N 2 f(β 1 (b)) ( β 1 (b) j i) = ( F (β 1 (b))) N 1 ) ( (b) = (N 1) F (β 1 (b))) N 2 f(β 1 (b)) 1 We want the first-order condition to hold with equality at b = β (t), or β 1 (b) = t, so plugging this in, the first-order condition becomes (t β (t)) (t β (t)) + 1 F N 1 (t) (N 1)F N 2 1 (t)f(t) β (t) = 0 4 β (β 1 (b))
or (N 1)f(t) F (t)β (t) = (t β (t)) (t β (t)) Bidders with t < r must bid below r, and bidders with t > r must bid at least r (since that gives them a positive probability of winning) but below t, so it s easy to show that β (r) = r. Standard differential-equation stuff says there s a unique solution to the differential equation with the boundary condition β (r) = r, and the equation itself makes it clear any solution must be increasing. Since β (t) < t, the right-hand side is positive, so β is strictly increasing above r. We derived β(t) from the first-order condition, so thinking of the objective function as f(x, t), we showed by construction that f x (β(t), t) = 0. By the chain rule, then, since V = f(x(t), t), V (t) = t f(β(t), t) = f x(β(t), t)β (t) + f t (β(t), t) = 0 + f t (β(t), t) which establishes the envelope theorem. We showed that for risk-neutral bidders, β increasing and satisfying the envelope theorem was sufficient for it to be a symmetric equilibrium; we didn t do the more general proof of sufficiency, but the result is more general, and so β is the unique symmetric equilibrium. Next, we introduce a nice trick for comparing two functions. Lemma 2. (Ranking Lemma.) Consider two continuous, differentiable functions g, h : R R. Suppose g(x ) h(x ), and for x x, Then for all x > x, g(x) > h(x). g(x) = h(x) g (x) > h (x) Proof. Suppose g(x) h(x) for some x > x. Let ˆx inf {s > x : g(s) h(s)}. Since either g(x ) > h(x ) or g(x ) = h(x ) and g (x ) > h (x ), ˆx > x. By continuity, g(ˆx) = h(ˆx), so by assumption, g (ˆx) > h (ˆx), so g(s) < h(s) for s just below ˆx, contradicting the definition of ˆx. This leads us to a proof that equilibrium bids with risk aversion are higher than with risk-neutrality. Lemma 3. Let β be the symmetric equilibrium bidding strategy in a risk-neutral auction, and β the symmetric equilibrium bidding strategy in our risk-averse auction. For t > r, β (t) > β(t). We know that β(r) = β (r) = r. For t > r, we know (N 1)f(t) F (t)β (t) = (t β (t)) (t β (t)) and, since the risk-neutral auction is the same but with (s) = s, (N 1)f(t) F (t)β (t) = 1 t β(t) 5
and so β (t) (N 1)f(t) = = β (t) (t β (t)) t β(t) F (t) (t β (t)) We normalized (0) = 0 and assumed was strictly concave, so for x > 0, so so (x) = x 0 (s)ds > x 0 (x) (x) < 1 x β (t) t β(t) = β (t) (t β (t)) (t β (t)) < (x)ds = x (x) β (t) t β (t) So when β (t) = β(t), β (t) > β (t); and we know that β (r) = β(r) = r, so by the ranking lemma, β (t) > β(t) for all t > t. 6
Maskin and Riley, Optimal Auctions with Risk-Averse Bidders So Maskin and Riley is a pretty long paper, and the math is pretty hard, so we re not going to go into all the details. As a general point, they mention that the introduction of risk-averse bidders changes the seller s problem in two ways: Since the seller is risk-neutral and bidders are risk-averse, the seller can profit by selling the bidders insurance. That is, relative to a standard, say, second-price auction, the seller can offer a deal to transfer some surplus from the bidder i wins case to the bidder i loses case, at less than fair value, and the bidder will still accept the deal In addition, risk-aversion gives the seller another way to punish high types who bid low, allowing the seller to extract more of their surplus (or making them easier to screen) The first would lead toward the seller removing all risk from the bidders, but the second makes it optimal to leave some. Maskin and Riley give a very general formulation of risk-averse preferences, then make a bunch of assumptions, and then give several examples of more narrowly-defined formulations that satisfy all their assumptions. Their general framework is that bidders are symmetric; have independent types θ i ; and have two utility functions, one, u, for when they win the object, which is a function of wealth and θ; and one, w, for when they don t win the object, which is a function only of wealth. They normalize starting wealth to 0, so bidders maximize E {H i (s)u( β i (s), θ i ) + (1 H i (s))w( α i (s))} where θ i is bidder i s true type s is a vector of all the bidders reported types H i is the probability that bidder i gets the object given reports s β i is what he pays if he gets it α i is what he pays if he doesn t get it and the expectation is taken over everyone elses types, given their equilibrium strategies Except that also, β and α are allowed to be stochastic, so the expectation is taken over their realizations as well. 7
They make two sets of assumptions on the utility functions u and w. Assumption A, are very standard: The first set, u and w three times differentiable u and w increasing in wealth w(0) = 0 (normalization) u and w concave (risk aversion) u increasing in θ (higher types want the object more) The second set ( Assumption B ) are harder to interpret, but they give some examples where they hold: u xθ < 0 u θθ < 0 u x ( t 1, θ) < w x ( t 2 ) u( t 1, θ) > w( t 2 ) u xθθ 0 u xxθ 0 The give four cases where assumptions A and B will hold, to make the case that the assumptions aren t too crazy. Case 1 Certain Quality, Equivalent Monetary Value This is the case we looked at already u( t, θ) = (θ t) and w( t) = ( t), where is concave and increasing. Case 2 Certain Quality, Additive tility, No Equivalent Monetary Value This is a generalization of case 1, where u( t, θ) = (θ + Ψ( t)), w( t) = (Ψ( t)), with and Ψ concave and increasing. In both these cases, assumptions A and B are satisfied if is increasing, concave, and 0. Case 3 same as Case 1, but with uncertain quality This is a generalization of case 1 where the actual value of the object is stochastic, but higher types value it more stochastically. u( t, θ) = E v θ (v t) and w( t) = ( t), where the distribution of v is increasing in θ, that is, for θ > θ, the distribution of v given θ first-order stochastically dominates the distribution of v given θ. In this case, A and B hold under some additional assumptions. 8
Case 4 Intensification This is when higher θ also leads to higher marginal utility of income u( t, θ) = (θ + 1)(θ t), w( t) = ( t). In this case, A and B hold if the coefficient of absolute risk aversion ( / ) is nonincreasing and greater than 2 everywhere. Some Preliminary Results Theorems 2-4 establish that at a given reserve price, the first-price auction outperforms the second-price auction this is what we already proved for Case 1, they show it generally under Assumptions A and one more technical condition. Theorem 5 shows that if the seller is also risk-averse, he still prefers the first-price auction. (This basically combines two results we knew: one, risk-averse seller with risk-neutral buyers prefers first-price auction; two, risk-neutral seller with risk-averse buyers prefers first-price; so it makes sense that risk-averse seller with risk-averse buyers would also prefer first-price.) Theorem 6 is that the seller does not gain by fully insuring the buyers. In general, when you have a risk-neutral principal with a risk-averse agent, there s profit to be made by the principal from effectively selling insurance to the agent in this case, insuring a bidder of each type against the uncertainty created by the other bidders types. However, in an auction, this insurance interferes with the seller s ability to extract greater surplus from higher types; Theorem 6 says that with case 1 preferences, a perfect insurance auction generates the same expected revenue as a second-price auction; and we already know this is lower than a first-price auction. Back to the Optimal Auction Problem As in the other papers we ve been looking at recently, Maskin and Riley use direct revelation mechanisms. They limit themselves to symmetric auctions. (No loss of generality, since if an asymmetric auction was optimal, they could randomly permute the players ahead of time and end up with a symmetric auction with the same revenue.) First, they consider auctions where bidder i s payment does not depend on θ j, only on θ i and whether or not he is awarded the object. So let G(θ i ) be probability of winning (expectation taken over other types, assuming truthful revelation) b(θ i ) be payment you make conditional on winning a(θ i ) be payment you make conditional on not winning The seller s expected revenue, then, is N [G(θ)b(θ) + (1 G(θ)a(θ)]dF (θ) 9
So to calculate the optimal deterministic auction, this is what the seller maximizes, subject to the usual constraints: individual rationality (everyone is willing to play the game) and incentive compatibility (truthful revelation is an equilibrium). Theorem 7 is basically that for a probability-of-winning function G to be feasible, it must be that the probability of winning at type θ is less than or equal to the probability that you have the highest type; and that if G is nondecreasing, this is sufficient. Theorems 8 and 9 are where they solve for the optimal auction. Make Assumptions A and B and one more technical condition. Theorems 8 and 9 state that if the solution to maximizing [G(θ)b(θ) + (1 G(θ))a(θ)]dF (θ) over the choice variables G, a, and b, subject to the envelope equation (equivalent to incentive compatibility, as we ve seen before), individual rationality, and G increasing and bounded above (their feasibility condition)... If this solution satisfies a technical condition, then it s the optimal auction, not just among deterministic auctions, but among all feasible auctions. (In the case of risk-neutrality, the technical condition they require collapses to regularity; with risk-aversion, it s hard to interpret exactly, but it s basically a limit on how fast F can decline, or a limit on how concave the type distribution F can be.) nder these same conditions, then, the rest of the paper gives a partial characterization of the optimal auction: the optimal auction is deterministic bidder i s payment depends only on his type and whether he gets the object, not the other bidders types; and the seller doesn t use unnecessary randomness to punish low types in order to screen high types but on the other hand, marginal utility is lower when winning then when losing at all but the highest type so except for the highest possible type, bidders are not fully insured the highest type, however, is perfectly insured bidders of all types strictly prefer when they win to when they lose for types with a positive probability of winning, the probability of winning, and the payment when you win, are strictly increasing in type as for a(θ), the payment you make when you don t get the object... in a neighborhood of the lowest type that every wins, losers make a payment that is positive and increasing; but under an additional condition, in a neighborhood near the highest possible type, losers are subsidized (get paid by the seller) For the special case of Case 1 preferences with decreasing absolute risk aversion, they give a further characterization: Bidders pay more when they win than when they lose 10
Bidders always pay something when they win Expected revenue from a given bidder is increasing in his type There are types who never win, so the object is not always sold They also give an interesting interpretation of the case with only 1 buyer, so it s just a buyer-seller game. They point out that with a risk-averse buyer, the type space is divided into three intervals: low types, who don t get the object; medium types, who get the object with positive probability less than 1; and high types, who get the object for sure. They offer another interpretation, which is that G(θ) < 1 corresponds to selling an object of lower quality, since G(θ) could correspond to getting something for sure, but that object falling apart with positive probability. So they argue that with risk-averse buyers, it s optimal for a monopolist to sell less-than-the-highest-quality goods to some types, even if quality costs nothing to improve, since this improves the ability to extract more surplus from the higher types. 11