Auction Theory Lecture Note, Daid McAdams, Fall 2008 1 Bilateral Trade ** Reised 10-17-08: An error in the discussion after Theorem 4 has been corrected. We shall use the example of bilateral trade to introduce seeral key ideas of auction theory and mechanism design more generally. Basic setup There is one seller and one buyer, each of whom is risk-neutral. The seller can produce one unit of a perishable good at priate cost c [c, c] while the buyer has priate alue [, ]., c are assumed to be independently distributed, with pdfs, f S (c) and cdfs F B (), F S (c). Assume that these distributions satisfy the monotonicity requirement that 1 F B () and c + F S (c) are increasing in, c. (Interpretations of this assumption will f S (c) emerge later.) The players hae one opportunity to trade. We are interested in whether and when trade will occur. For the moment, we shall abstract from the details of the process that generates such trade, iewing that process as generating a mechanism, and focus on the outcome generated in that mechanism. 1 Namely, conditional on realized alues of (, c), what is the probability x(, c) that the seller produces the good and the buyer consumes it, and what are the expected payments t S (, c) made to the seller and the expected payment t B (, c) made by the buyer? (For simplicity, we assume that the good is neer produced and not consumed.) Payoffs. Since the players are assumed risk-neutral, these probabilities and expectations are sufficient to capture their ex post utility conditional on each realized (, c). Namely: (buyer s ex post utility) x(, c) t B (, c) (seller s ex post utility) t S (, c) cx(, c) Similarly, we may express each player s interim (or interim expected ) utility conditional on just his own priate information as: 1 Mechanisms were defined formally in Econ 302.
Auction Theory Lecture Note, Daid McAdams, Fall 2008 2 (buyer s interim utility) x B () t B () where x B () E[x(, c) ] and t B () E[t B (, c) ] (seller s interim utility) t S (c) cx S (c) where x S (c) E[x(, c) c] and t S (c) E[t S (, c) c] Restrictions. When the players hae limited recourse to an outside source of funds, transfers may be limited to satisfy a budget balance (BB) requirement. The most commonly studied are: (ex post BB) t B (, c) t S (, c) for all (, c) (ex ante BB) E[t B (, c)] E[t S (, c)] A weaker ersion of budget balance is so-called feasibility, which allows for the possibility that the seller may receie less than the buyer pays. If participation is oluntary, payoffs may also be restricted to be positie. This leads to an indiidual rationality (IR) requirement. The most commonly studied are: (ex post IR) x(, c) t B (, c) 0 and t S (, c) cx(, c) 0 for all (, c) (interim IR) x B () t B () 0 and t S (c) cx S (c) 0 for all (, c) Finally, since players are strategic, they must hae an incentie not to seek to change the mechanism outcome. This leads to an incentie compatibility (IC) requirement. The most commonly studied are: (ex post IC) x(, c) t B (, c) x(ṽ, c) t B (ṽ, c) and t S (, c) cx(, c) t S (, c) cx(, c) for all (, c) (interim IC) x B () t B () x B (ṽ) t B (ṽ) and t S (c) cx S (c) t S ( c) cx S ( c) for all (, c) When ex post IC is satisfied, the mechanism is implemented in dominant strategies. When interim IC is satisfied, the mechanism is implemented in Bayesian equilibrium strategies. In the interests of time, we will not discuss dominant strategy implementation, a rich field of its own. Howeer, for the interested student, I think it would be worth your time to find a simple proof of the following result. (Theorem 1 is a special case of Theorem 2, which is proen below.) Theorem 1. No mechanism exists that is efficient and that satisfies ex post IR, interim IC, and ex ante BB.
Auction Theory Lecture Note, Daid McAdams, Fall 2008 3 Bayesian implementation What outcomes can arise when we restrict attention to mechanisms in which players adopt Bayesian equilibrium strategies (interim IC) and neer get negatie interim payoffs (interim IR), and in which the buyer s payment must on aerage equal the seller s price (ex ante BB)? In many contexts, one may want to restrict attention further, say to mechanisms satisfying ex post IR and ex post BB. Clearly, the set of outcomes that can arise in such mechanisms is smaller than that which can arise under interim IR and ex ante BB. Efficiency s Budget-Balance Theorem 2. Suppose that < c. No mechanism exists that implements the efficient outcome and that satisfies interim IR, interim IC, and ex ante BB. Proof. For simplicity, assume that c m and c m. Suppose for the sake of contradiction that there exists an efficient mechanism satisfying interim IR, interim IC, and ex ante BB. Efficiency requires that trade occurs iff > c. Thus, x B () Pr c ( > c) F S () and x S (c) Pr ( > c) 1 F B (c). By interim IC for the buyer, arg maxṽ x B (ṽ) t B (ṽ). The firstorder condition of this maximization requires that dtb () dxb () f S (). Integrating, we may therefore express the buyer s expected transfer solely in terms of the transfer that he receies upon haing the lowest type: t B () t B () + ṽ dxb (ṽ) dṽ (1) Now, by efficiency (and using the simplifying assumption that c), buyertype has interim payoff t B () since he is neer awarded the object. Interim IR therefore requires that t B () 0 so that t B () ṽ dxb (ṽ) dṽ for all. Recalling that dxb () f S (), we may re-write this as t B () cf S (c)dc (2) Similarly, for the seller, c arg max c t S ( c) cx S ( c) requires that dts (c) dc
Auction Theory Lecture Note, Daid McAdams, Fall 2008 4 c dxs (c) dc cf B (c). Integrating, we get that t S (c) t S (c) + c c c dxs ( c) d c (3) dc Again by efficiency, seller-type c neer sells the object and so gets interim payoff t S (c) which must be non-negatie by interim IR. We conclude that t S (c) c d c for all c, which may be re-written as c dxs ( c) c dc Putting this together, t S (c) c c (4) E[t B (, c)] E[t S (, c)] m m m m >c >c t B () ( m m m t S (c)f S (c)dc ) cf S (c)dc cf S (c)dc >c (c )f S (c)dc < 0 m m ( m f S (c)dc c ) f S (c)dc contradicting the assumption of ex ante budget balance (as well as the weaker assumption of ex ante feasibility). By Theorem 2, efficient trade must be subsidized in enironments with a strategic buyer and seller haing priate information. This is the famous result of Myerson and Satterthwaite (1983). For example, one way to implement an efficient outcome is through a Vickrey auction. 2 The buyer and seller each submit a sealed bid, call them ṽ and c. If ṽ c, then the seller keeps the object and no payments are made. Otherwise, if ṽ c, the buyer receies the object, the buyer pays c, and the seller receies ṽ. 2 Such mechanisms are also known as Groes mechanism, piot mechanism, or Vickrey- Clarke-Groes mechanism (VCG) honoring seminal contributions in Vickrey (1961), Clarke (1971), and Groes (1973).
Auction Theory Lecture Note, Daid McAdams, Fall 2008 5 Theorem 3. The Vickrey auction implements the efficient outcome and satisfies ex post IR and ex post IC. It is not budget balanced, but its ex ante expected budget imbalance is the smallest possible in any efficient mechanism satisfying interim IR and interim IC. Characterizing interim IR and interim IC mechanisms Theorem 4. A mechanism with outcome {x(, c), t B (, c), t S (, c)} can be implemented in Bayesian equilibrium strategies, subject to interim IR and interim IC, iff three sets of conditions are satisfied: (monotone) x B () is increasing in and x S (c) is decreasing in c (local IC) 3 dt B () dxb () and dts (c) dc c dxs (c) dc (IR at the bottom ) x B () t B () 0 and t S (c) cx S (c) 0 Proof. Necessity. Monotonicity. Consider any two buyer-types 1 > 2. Interim IC requires that both: 1 x B ( 1 ) t B ( 1 ) 1 x B ( 2 ) t B ( 2 ) (5) 2 x B ( 2 ) t B ( 2 ) 2 x B ( 1 ) t B ( 1 ) (6) Summing these conditions, the payment terms cancel (since they do not depend on buyer types) and we are left with the requirement that ( 1 2 )(x B ( 1 ) x B ( 2 )) > 0. We conclude that x B ( 1 ) x B ( 2 ) > 0. The argument is similar for the seller. Local IC and IR at the bottom. Shown in the proof of Theorem 2. Sufficiency. Interim IC. We need to show that buyer-type does not strictly prefer to mimic type ṽ for all ṽ. Type s incremental expected
Auction Theory Lecture Note, Daid McAdams, Fall 2008 6 payoff from message relatie to ṽ is: ( x B () t B () ) ( x B (ṽ) t B (ṽ) ) ( x B () x B (ṽ) ) ( t B () t B (ṽ) ) ṽ ṽ ṽ dx B ( ) dx B ( ) ṽ ṽ dt B ( ) (7) dxb ( ) (8) ( ) dxb ( ) 0 (9) (8) follows from (7) by local IC, while (9) holds by monotonicity since dx B ( ) 0. Thus, type prefers not to deiate. The proof of interim IC for the seller is similar. [ d Interim IR. By local IC, x B () t B () ] x B () + dxb () x B () 0. So, x B () t B () x B () t B () 0 for all, where the second inequality follows is by IR at the bottom. dtb () It is worth pausing to reflect upon the power of Theorem 4 for applied work examining mechanisms. First, if an outcome function satisfies the three conditions of the theorem, then that outcome can be implemented in Bayesian equilibrium strategies in the corresponding direct reelation mechanism. Second, and perhaps een more powerful, Theorem 4 proides a shortcut that allows one to guess and erify equilibrium strategies in auctions. Suppose one has a guess about the allocation that will result in equilibrium. This determines the allocation probability functions which, in turn by local IC, determine the expected payments. Knowledge of expected payments can often then be used to back out player strategies. If these inferred strategies do indeed lead to the conjectured allocation, then Theorem 4 tells us that no bidder has any profitable deiation among the range of submitted bids. 4 Indeed, these are the only strategies that might possibly implement the conjectured allocation in Bayesian equilibrium! 4 ALERT: In the first ersion of this note, I incorrectly stated that Theorem 4 tells us that the inferred strategies constitute a Bayesian equilibrium. This is not correct. It remains possible that a bidder could hae a profitable deiation at some price outside of the range of bids made in these strategies. Often it is easy to rule out such bids. For example, imagine that bids in a first-price auction are submitted in the range [0, 1]. Since a bid of 1 always wins, any higher bid is strictly dominated.
Auction Theory Lecture Note, Daid McAdams, Fall 2008 7 Virtual aluations and optimal mechanisms So-called irtual aluations play an important role in auction theory and mechanism design more generally. (irtual alue) The buyer s irtual alue ψ B () 1 F B () (irtual cost) The seller s irtual cost ψ S (c) c + F S (c) f S (c). Theorem 5. Suppose that a mechanism with outcome {x B (, c), x S (, c), t B (, c), t S (, c)} can be implemented in Bayesian equilibrium strategies, subject to interim IR and interim IC. Then the net expected payments made by the players takes the form: E [ t B (, c) ] E [ t S (, c) ] E [ x(, c) (( 1 F ) ( B () c + F ))] S (c) f S (c) ( x B () t B () ) ( t S (c) cx S (c) ) (10) where the last two terms of (10) are zero when IR at the bottom is binding. Proof. The ex ante expected welfare created from trade is E[x(, c)( c)], and the last two terms of (10) are the expected profit (or rent or information rent ) of the lowest buyer type and highest seller type. Note that [ex ante expected buyer rent] + [ex ante seller rent] [total surplus] - [ex ante expected payment by the buyer] + [ex ante expected payment to the seller], and that [total surplus] E[x(, c)( c)]. Thus, after re-arranging, [ (10) ( is equialent to)] showing that the sum of the player s rent equals E x(, c) 1 F B () + F S (c). More than that, we will show that f S (c) [ ] [ex ante expected buyer rent] E x(, c) 1 F B () and [ex ante expected [ ] buyer rent] E x(, c) F S (c). f S (c) ( d Consider the buyer. Recall that, by local IC, x B () t B () ) x B (). Thus, a buyer s rent takes the form xb ( ) + [rent of -type]. So, it suffices for us to show that E[ xb ( ) ] E[x B () 1 F B () ], and
Auction Theory Lecture Note, Daid McAdams, Fall 2008 8 similarly for the seller. But this follows from integration by parts: E[ x B ( ) ] x B ( )f B ( ) x B ( ) ( 1 F B ( ) ) E[x B () 1 F B () ] The argument for the seller is similar, and omitted. Intuition for Theorem 5. Consider the problem of a social planner that absorbs any budget imbalance. This social planner would like to maximize the realized surplus from trade, while maximizing the amount of money that it is able to extract from the mechanism (or minimizing the amount of money that it must pump into the mechanism). If the realized alues of the players are (, c) with > c, the planner would like to induce trade. Howeer, doing so makes mimicking type more profitable for all buyer types with alues greater than, and makes mimicking type c more profitable for all seller types with alues less than c. So, there is a trade-off. On one hand, inducing this trade creates surplus c, surplus that the social planner might be able to extract from the buyer and seller. The expected pie grows due to this effect by ( c)f S (c). On the other hand, all higher buyer-types must hae their rent increase, leading to more of the pie being gien to the buyer as rent. So, the buyer s ex ante expected rent grows by (1 F B ())f S (c), since this is the probability that a higher buyer type would play against seller-type c. Similarly, the seller s ex ante expected rent grows by F S (c). Oerall, the ex ante expected reenue to the social planner attributable to this trade is proportional to (diiding by f S (c)) ( c) 1 F B () F S (c) f S (c) ψb () ψ S (c) (11) Virtual surplus rule. Let us refer to ψ B () ψ S (c) as the irtual surplus that is created when trade occurs gien buyer alue and seller cost c. The ex ante expected reenue to the social planner is therefore just the
Auction Theory Lecture Note, Daid McAdams, Fall 2008 9 ex ante expected irtual surplus from trade, E[x(, c)(ψ B () ψ S (c))]. In particular, the ex ante budget balance requirement is one that the social planner has zero ex ante expected surplus. Theorem 6. Consider the special case, c U[0, 1]. Subject to the requirements of interim IR, interim IC, and ex ante BB, the mechanism that maximizes social welfare is one in which trade occurs iff > c + x, where x is defined implicitly by: ( ψ B () ψ S (c) ) dc 0 (12) (,c):>c+x In the uniform case of Theorem 6, note that ψ B () ψ S (c) 2( c) 1 while d Pr( c x)/dx x for all x [0, 1]. (In class, we shall discuss a graphical interpretation.) After a change of ariables, x can therefore be defined implicitly as the non-zero solution of 1 x (2x 1) x (x ) ( 2 2 3 x 2) 1 0, i.e. x 3/4. We conclude that the socially efficient mechanism induces trade iff c > 1. 4 Implementing the (constrained) socially efficient mechanism. An issue of practical interest is whether an outcome of interest can be implemented in equilibrium in a particular game. In the case of bilateral trade with uniform distributions, the (constrained) socially efficient mechanism is indeed implemented within a ery simple and appealing game. Uniform example continued. Consider a game in which the buyer and seller each announce prices, ṽ and c, trade occurs iff ṽ > c at a transaction price equal to the aerage of the announced prices, p ṽ+ c. It is an equilibrium of this game for the buyer to follow bidding strategy ṽ 2 + 1 and 3 12 2 c 2c+ 1. An important feature of these strategies is that ṽ c c+ 1, 3 4 4 so trade occurs exactly when specified in Theorem 6... this is a socially efficient mechanism subject to the constraints of ex ante, interim IR, and interim IC! (Of course, since the payment receied by the seller is always equal to the payment made by the buyer, this outcome induced in this equilibrium in fact satisfies the stronger conditions of ex post BB and ex post IR.) Proof sketch: Consider seller s incenties to slightly shade down his bid by one marginal unit. He will lower his probability of winning by 3/2 and,
Auction Theory Lecture Note, Daid McAdams, Fall 2008 10 since he would hae tied on this margin, he will lose ṽ on these marginal units. Since ṽ 1 1/4 ( 1/4), this amounts to losing. On the other 3 2 hand, he will lower his payment when he wins by 1/2, and this happens with probability 1/4. Readings. Myerson and Satterthwaite (1983), Efficient Mechanisms for Bilateral Trading.