Econ 805 Advanced Micro Theory I Dan Quint Fall 2008 Lecture 14 October 23 2008 Pretty much everything we ve done this semester has implicitly assumed an asymmetry between seller and buyers Buyers have private information and act strategically Sellers have no private information they may act strategically in choosing how to run an auction but are not players in the game once the auction starts In many settings this makes sense the seller either has no private information or has an incentive to reveal it or has no reason to act strategically However In particular if we want to think of auctions as analogies for markets more generally as in the information aggregation literature it may make sense to treat buyers and sellers more symmetrically Today we ll consider double auctions auctions where both buyer and seller have private information and act strategically 1 The k Double Auction We begin with the k double auction which is considered in the Satterthwaite and Williams paper Bilateral Trade with the Sealed Bid k Double Auction Existence and Efficiency This can be thought of as a two sided extension of the sealed bid private value auction The model is this One risk neutral buyer with private value vb drawn from a distribution Fb on 0 1 with density fb which is everywhere positive One risk neutral seller with private value vs drawn from Fs on 0 1 density fs which is everywhere positive The buyer submits a sealed bid b at the same time the seller submits a sealed bid s If b s trade occurs at a price of kb 1 k s for some fixed k 0 1 hence the name k double auction When k 0 the buyer faces a second price auction with one opponent whose bid is s As usual the buyer has a dominant strategy of bidding his value b vb Given that each type of seller generally has a unique best response so there s a unique nontrivial equilibrium In fact it turns out to be equivalent to the seller making his optimal take it or leave it offer and the buyer simply deciding whether or not to trade at the price I said a unique nontrivial equilibrium In two sided models like these a no trade equilibrium always exists for example the buyer always bidding 0 and the seller always bidding 1 Nontrivial just refers to an equilibrium with a positive probability of trade Similarly when k 1 the same holds in reverse The seller has a dominant strategy of bidding his value s vs since he doesn t affect the price only whether trade occurs the buyer has a unique best response and so there s a unique nontrivial equilibrium which is the same as if the buyer made a take it or leave it offer and the seller simply accepted or declined 2 However it turns out that when k 0 1 there are lots of equilibria That is lots of nontrivial equilibria First a couple of regularity assumptions fb and fs are nonzero everywhere on 0 1 b vb vb 1 F fb vb strictly increasing continuous differentiable This should look familiar it s our old marginal revenue term vs vs Ffss v strictly increasing continuous differentiable This is the seller s analog of s the buyer s marginal revenue A pair of distributions Fb Fs is called admissible if these conditions are satisfied Since under these conditions you can recover the distribution Fb from the function 1 Fb vb Fs vs fb vb and Fs from fs vs Satterthwaite and Williams sort of treat these as the primitives not the distributions it doesn t make a difference Satterthwaite and Williams focus on a class of well behaved equilibria of the k double auction Letting B and S denote the buyer s and seller s strategies respectively S and W restrict attention to equilibria where S 0 1 0 1 and B 0 1 0 1 are both continuous and strictly increasing vs S vs 1 for all vs 0 B vb vb for all vb S is differentiable for vs 0 B 1 S vs vs for vs B 1 B is differentiable for vb S 0 1 B vb vb for vb S 0 So S and B are assumed to be increasing continuous and differentiable on the range where they matter that is at types who could profitably trade given the opponent s strategy For types that cannot profitably trade that is when vb is below the range of S or vs above the range of B any bid which never wins is a best response S and W normalize bidders in that range to just bid their value Strategies satisfying these conditions are called regular 3 S and W prove the following Theorem 1 Let Fs Fb be admissible Regular strategies S B are an equilibrium of the k double auction if and only if for all vb S 0 and vs B 1 they satisfy the differential equations Fs vs B 1 S vs S vs kS 0 vs fs vs and S 1 B vb B vb 1 k B 0 vb 1 Fb vb fb vb To see this let G be the distribution of seller s bids S vs and consider a buyer s optimization problem choosing b to maximize Z b vb kb 1 k s g s ds s Differentiating with respect to b gives vb b g b kG b 0 Now note that G b Pr S vs b Pr vs S 1 b Fs S 1 b and therefore g b fs S 1 b 1 S 0 S 1 b Plugging these into the FOC gives vb b fs S 1 b 1 S 0 S 1 b or vb b kS 0 S 1 b kFs S 1 b Fs S 1 b fs S 1 b Since we got here from the buyer s problem this holds at b B vb or vb B 1 b B 1 b b kS 0 S 1 b Fs S 1 b fs S 1 b Since this holds for every b in the range of S let b S vs for some vs and we re left with B 1 S vs S vs kS 0 vs Fs vs fs vs the first of our two differential equations Doing the same thing with the seller s optimization problem given the distribution of B vb gives the second one So these two differential equations follow from the two players first order conditions it s clear then that if S B is to be an equilibrium with differentiable strategies they re 4 necessary That they re sufficient is a little tougher but the proof is basically the same as all the proofs we ve done where the envelope theorem is sufficient for equilibrium in a one sided auction S and W prove it in the appendix Now here s where it gets funky Take b and treat it like a parameter and let vb b and vs b S 1 b Differentiating with respect to b v s dvs …