Sequential Search Auctions with a Deadline

Transcription

1 Sequential Search Auctions with a Deadline Joosung Lee Daniel Z. Li University of Edinburgh Durham University January, / 48

2 A Motivational Example A puzzling observation in mergers and acquisitions (M&As) is that the dominant selling mechanism is one-on-one negotiation, where no obvious competition among bidders is observed. Betton et al (2008): 95% of the sample deals in the US market ( ) are classi ed as one-on-one negotiation Some Explanations: Boone and Mulherin (BM, 2007): Misleading classi cation; Aktas et al (2010): Negotiations under the threat of an auction 2 / 48

3 Selling Processes in M&As A typical selling processes in M&As Involves several rounds of transactions with a nite deadline; A seller needs to screen and search potential bidders; It can be costly for the seller to contact a bidder; e.g., loss of proprietary information to bidders in due diligence. A classi cation of the M&A selling processes (BM, 2009) One-on-one negotiation: just 1 bidder contacted in round 1 Private controlled sale (auction): a few contacted in round 1 Public full-scale auction: a simultaneous auction 3 / 48

4 Other Examples: Sequential Search with a Deadline Academic recruitments in the UK by the REF2021 deadline Departments aim to build up a strong publication pro le; Conduct several rounds of recruiting campaigns; Search intensity commonly increases when the deadline approaches. Search in marriage markets Sequential matching/dating with a deadline, Search intensity increases overtime. Sequential contest/assignment problems, etc. 4 / 48

5 A More General Question A seller needs to allocate a product among N potential bidders; She needs to complete the transaction within T periods; To contact a bidder, the seller needs to pay a search cost. ) A sequential search problem with a deadline. Q: What would be the optimal (e cient) search mechanism? 5 / 48

6 Literature: One-by-one Sequential Search Weitzman (Ecta, 79): Sequential one-by-one search 6 / 48

7 Literature: One-by-one Sequential Search Weitzman (Ecta, 79): Pandora s problem Pandora faces n closed boxes, each with a random prize V i F i (v); She can open one box in a single period; an opening cost c i ; Search with full recall, i.e., discovered prize always reclaimable; Objective: to maximize the net payo. 7 / 48

8 Literature: One-by-one Sequential Search Pandora s Rule: Allocate a reservation prize vi for each box i, c i = Z v i (x v i ) df i (x). Selection Rule: If a box is to be opened, it should be the unopened box with the highest reservation prize of vi. Stopping Rule: Terminate search whenever the maximum prize discovered exceeds the reservation prizes of all the unopened boxes. Cremer et al (JET, 07): Sequential search auctions 8 / 48

9 Literature: Sequential Search with a Deadline Benhabib&Bull (JPE83); Gal et al (IER81); Morgan et al (RES83) Pandora needs to nish her search within T < n period, One-by-one sequential search may no longer be optimal, She may open several boxes simultaneously in a single period. ) A sequential-&-simultaneous search problem Some properties of the optimal search procedure Search with no recall: cuto prices # + search intensity "; Search with full recall: cuto prices # + search intensity " under some restrictive conditions. 9 / 48

10 This Paper We study a sequential-&-simultaneous search problem; Targets for search are strategic bidders, not non-strategic boxes; We consider both cases of long-lived and short-lived bidders; Many real important problems can be tted into this model. 10 / 48

11 Main Results An optimal sequential search mechanism can be implemented by a sequence of 2nd price auctions. The optimal sequential auction is characterized by decreasing reserve prices and increasing search intensity over time; Robust in both cases of short-lived and long-lived bidders. For long-lived bidders, the optimal reserve prices demonstrate a one-step-ahead property; Generalizing the well-known formula of Weitzman (Ecta, 79) An e cient mechanism is featured by lower reserve prices and higher search intensity than an optimal mechanism. 11 / 48

12 Other Related Literature Sequential and dynamic auctions Cremer et al (07) : search auction + full commitment Skreta (15): sequential auction + limited commitment Fershtman & Pavan (17): Sequential matching auctions Negotiations vs auctions in M&As Boone & Mulherin (07; 09); Aktas et al (09); Fidrmuc et al (12). Bulow & Klemperer (09) : auction sequential negotiation Buy-price auctions Reynolds & Wooders (09); Chen et al (16); Zhang (17) Revenue management Board & Skrzypacz (16): full commitment Dilme & Li (16); Mierendor et al (17): limited commitment 12 / 48

13 Other Related Literature Bulow & Klemperer (AER, 09) : Why Do Sellers Prefer Auctions? Bidders need to pay entry costs; Sequential negotiations vs. simultaneous auctions; In sequential negotiations, jump-bid by a bidder deters following entries, which may harm sellers; Therefore, sellers (usually) prefer simultaneous auctions. We propose a sequential search model instead A seller pays search costs A sequential mechanism dominates a simultaneous one Empirical evidences: Atkas et al (2010), etc. 13 / 48

14 Model Setup A seller allocates an indivisible product among N potential bidders. The transaction has to be completed within T periods. To contact bidder i, the seller needs to pay a search cost c i > 0. Bidder values fv i g i2n are n independent draws from F on [0, 1]. The seller is a pro t-maximizer, and all the players are risk-neutral. There is no time discounting. 14 / 48

15 Model Setup F is of IFR, which implies the virtual value ψ (v) = v 1 F (v) f (v) is increasing in v. All the bidders are valuable for the seller, i.e., c i < Z 1 r ψ (x) df (x), for all i 2 N. Bidders are long-lived, e.g., once invited, always participate. 15 / 48

16 A Search Mechanism A seller rst determines a family of disjoint bidder subsets of N, denoted by M T = M 1,, M T ; A sampling rule is a permutation of M T that speci es the order of searching the bidder samples, denoted by M = fm 1,, M T g; A stage mechanism, (Q, P), speci es a pair of an allocation and payment rule for each period t = 1, 2,, T. A search mechanism S = a sampling rule + a stage mechanism. 16 / 48

17 An Optimal Search Mechanism Full commitment assumption: a seller is fully committed to the pre-announced search mechanism of S ) revelation principle. Myerson (1981): When S is incentive feasible, the payment from a truthful bidder i is equal to his virtual value of ψ (v i ); Consider a sample M of bidders, The total search cost is c M = i 2M c i ; X (1) (1) M denotes the M bidders highest value, with distribution F M ; The maximum payment from the M bidders is thus ψ X (1) M ; 17 / 48

18 Optimal Search Mechanism Given the current o er of ψ (v), a seller needs to decides: (1) whether or not to stop searching; (2) if not, which sample of bidders to search in the next period. The seller then solves the following DP problem h J t (v) = nψ (v), EJ t+1 max max M t+1 2N c t n v, X (1) M t+1 oi c Mt+1 o. (1) J t (v): seller s value function of having an o er of ψ (v) in period t; Bidders are long-lived ) fall-back o er is ψ (v). 18 / 48

19 Optimal Search Mechanism The optimal cuto value, ξ t, chosen satis es ψ (ξ t ) = Z 1 0 max fψ (ξ t ), ψ (x)g df (1) M t+1 (x) c Mt+1. (2) For given bidder samples of M T, the optimal search rule is given by: Selection rule: search from the sample with the highest cuto value; Stopping rule: stop whenever a discovered value is greater than the highest cuto value of all the remaining un-inspected bidder samples. Optimal mechanism implementable by a sequence of 2nd price auctions. 19 / 48

20 Sequential Search Auction t = 1: The seller invites M 1 bidders at the cost of c M1, and runs an auction among the N 1 = M 1 bidders, with a reserve price r 1 ; If an e ective bid is submitted, then the transaction ends; If not, the seller continues her search to the next period. t = 2: The seller further invites M 2 N c 1 bidders at the cost of c M 2, and runs an auction among the accumulated N 2 = M 1 [ M 2 bidders, with a new reserve price r 2 ; The seller continues with this process, until the end of period T. 20 / 48

21 The Model This sequential mechanism is characterized by (r, M): A sequence of reserve prices r = fr t g 1tT = fr 1, r 2,, r T g ; A sequence of bidder samples M = fm t g 1tT = fm 1, M 2,, M T g. 21 / 48

22 The Model: Assumptions Full commitment assumption: seller is fully committed to the pre-announced mechanism of (r, M) Auctions in the format of 2nd price auction Bidding true value is a weakly dominant strategy for a bidder. Symmetric cuto equilibrium for bidders, ξ = fξ t g 1tT A bidder in period t bids if his value v ξ t, and waits otherwise. 22 / 48

23 An Example of T=1: Simultaneous Search Suppose c i = c, then the seller s problem Z 1 max π (r, m) = ψ (x) df m (1) (x) mc, r,m r ψ (x) " ) Pro t π (r, m) is concave in m; Optimal sample size is denoted by m ; Optimal reserve is r, such that ψ (r ) = 0. For bidders, ξ = r constitutes a symmetric cuto equilibrium. 23 / 48

24 An Example of T=2 Solve by Backward Induction: In the last period of t = 2 If bidder i bids, his expected payo of bidding is U b 2 (v) = F (1) N 2 nfig (ξ 2) (v r 2 ) + I fv ξ2 g R v ξ 2 (v x) df (1) N 2 nfig (x). If he waits, his payo of waiting is Ū 3 (v) = 0. He bids i his value v r 2 ) ξ 2 = r 2 is a cuto equilibrium. 24 / 48

25 An Example of T=2 t = 1: If a bidder bids, his expected payo of bidding is U b 1 (v). If he waits, his payo of waiting is Ū 2 (v) = max U b 2 (v), Ū 3 (v). At the cuto value ξ 1 of the bidder, we have U b 1 (ξ 1 ) = Ū 2 (ξ 1 ) = U b 2 (ξ 1 ), which implies the following cuto condition F (1) N 1 nfig (ξ 1) (ξ 1 r 1 ) = R ξ 1 r 2 F (1) N 2 (x) dx. nfig 25 / 48

26 An Example of T=2 The cuto value conditions for bidders de ne an onto relation (r, M), (ξ, M). The seller maximizes the expected pro t, s.t. the cuto conditions. The solutions are Optimal cuto values: ξ1 > r 1 > ξ 2 = r, where for ξ 1, Z 1 m 2 c = [ψ (x) ψ (ξ1)] df m 2 (x). ξ 1 Optimal sample size m 1 < m / 48

27 An Example: T=2, N=3, Uniform Distribution Let c i = c = 1/16. If the seller chooses (m 1, m 2 ) = (2, 1), then π (2, 1) = If she chooses (m 1, m 2 ) = (1, 2), the expected pro t is π (1, 2) = , and the sampling strategy of (1, 2) is optimal. Let c i = c = 1/32. If the seller chooses (m 1, m 2 ) = (2, 1), then π (2, 1) = If she chooses (m 1, m 2 ) = (1, 2), the expected pro t is π (1, 2) = Then the sampling strategy of (1, 2) is still optimal. 27 / 48

28 T-period Model: IC for Bidders In period t, the value function of bidder i s expected payo n o Ū t (v) = max Ut b (v), Ū t+1 (v), where the expected payo of bidding is U b t (v) = F (1) N t nfig (ξ t) (v r t ) + I fv ξt g Z v ξ t (v x) df (1) N t nfig (x). In period t + 1, the bidder may either bid or wait; Note that Ū T +1 (v) = 0, then ξ T = r T. 28 / 48

29 Envelope Theorem: Cuto Equilibrium for Bidders A symmetric cuto equilibrium ξ always exists, no matter the sequence of cuto s is monotonic or not. Lemma (cuto equilibrium) For given (r, M), in each period t T, there exists a unique ξ t such that each bidder i 2 N t bids i his true value v ξ t. Furthermore, (Ūt+1 (v) if v < ξ t, Ū t (v) = Ū t+1 (ξ t ) + R v F (1) ξ t N t nfig (x) dx if v ξ t. (3) Equivalent representations of a mechanism: (r, M) () (ξ, M). 29 / 48

30 Cuto Equilibrium for Bidders In a symmetric equilibrium with declining cuto s, Ut b (ξ t ) = Ū t+1 (ξ t ) = Ut+1 b (ξ t ). The equilibrium cuto ξ t is uniquely determined by F (1) N t nfig (ξ t r t ) = F (1) N t+1 nfig (ξ t+1) (ξ t+1 for t < T ; and ξ T = r T. r t+1 ) + R ξ t ξ t+1 F (1) N t+1 nfig (x)dx (4) Lemma (declining cuto s) For given (ξ, M) with declining equilibrium cuto values, the sequence of reserves fr t g 1tT is determined by (4), and r t is also declining in t. 30 / 48

31 Expected Auction Pro t For declining equilibrium cuto s, the expected net auction revenue π (ξ, M) = T F (1) N t 1 (ξ t 1 ) [R t (N t ) c Mt ], t=1 R t (N t ) auction revenue in period t, conditional on it happens. The seller s problem is to choose a mechanism (ξ, M) to maximize her expected pro t π, subject to the IC constraints of the bidders. 31 / 48

32 Lemma (expected pro t) For given (ξ, M) with declining equilibrium cuto values, the expected auction pro t is π (ξ, M) = T R ξt 1 ξ ψ t (x) df (1) N t (x) t=1 + T t=1 h F (1) R i 1 N t 1 (ξ t 1 ) ψ ξ (x) df (1) t 1 M t (x) c Mt. (5) π (ξ, M) is quasi-concave in ξ. In period t, N t 1 weak bidders compete with M t strong bidders. 32 / 48

33 Optimal Cuto Values Proposition (optimal cuto s) For given (ξ, M) with declining equilibrium cuto s, the optimal sequence of fξ t g 1tT is the unique solution to c Mt+1 = Z 1 ξ t and ξ T = r, for t = T. [ψ (x) ψ (ξ t )] df (1) M t+1 (x), for 1 t < T ; (6) Some economic intuitions A generalization of the well-known formula of Weitzman (1979) c i = Z v i (x v i ) df i (x). 33 / 48

34 Optimal Cuto Values: Some Properties Denote ξ (M) the optimal cuto value for searching a set M of bidders, c M = Z 1 ξ (M ) h 1 F (1) M (x) i dψ (x). Lemma (property of optimal cuto ) 1) For any M, M 0 N with the cardinality jmj = jm 0 j, c M < c M 0 =) ξ (M) > ξ M 0. 2) For any M, M 0 N with c M = c M 0, jmj < M 0 =) ξ (M) < ξ M / 48

35 Optimal Sample Sizes Substituting ξ (M t ) into π (ξ, M), we get π (M) = where ξ (M T +1 ) r. T Z ξ (M t ) h i 1 F (1) t=1 ξ N (M t+1 ) t (x) dψ (x). (7) It incorporates the result in stationary and in nite-horizon (SIH) search problems as a special case, where the optimal pro t π SIH (ξ ) = ψ (ξ ). 35 / 48

36 Optimal Sample Sizes To x the idea of search intensity, let c i = c ) all the bidders are ex ante homogeneous; Denote m t the sample size of M t. Search intensity in period t is then measured by sample size m t. 36 / 48

37 Optimal Sample Sizes If c i = c, then ξ (m) is decreasing in m. Proposition (optimal sampling) Suppose c i = c for all i 2 N. In an optimal search mechanism of (ξ, M), the sequence of optimal sampling sizes is increasing in t, that is, mt mt+1, for t = 1, 2,, T / 48

38 E cient Mechanism In an e cient mechanism of (ξ, M), again, it is necessary that the sequence of cuto s, fξ t g 1tT, is declining over time. The e cient search procedure can be implemented by ) a sequence of 2nd price auctions with properly set reserve prices. 38 / 48

39 E cient Mechanism For given (ξ, M), a symmetric cuto equilibrium always exists; If the equilibrium cuto is declining over time, The equilibrium reserve r t is uniquely determined, by (4); The expected social welfare is W (ξ, M) = T R ξt 1 t=1 + T t=1 ξ t xdf (1) N t (x) F (1) h R i 1 (1) N (ξ t 1 t 1 ) xdf ξt 1 M t (x) c Mt. 39 / 48

40 E cient Cuto s Proposition (e cient cuto s) For given (ξ, M) with declining equilibrium cuto values, the e cient sequence of fξt g 1tT is the unique solution to c Mt+1 = Z 1 ξ t and ξ T = 0 for t = T. (x ξ t ) df (1) M t+1 (x), for 1 t < T ; (8) 40 / 48

41 E cient Cuto s De ne ξ (M) as the e cient cuto for searching M bidders, c M = Z 1 ξ (M ) [x ξ (M)] df (1) M (x) = Z 1 ξ (M ) Optimal cuto is higher than the e cient one. Proposition h 1 F (1) M (x) i dx. For given M, optimal cuto values are higher than e cient ones, ξ (M t ) > ξ (M t ), for 1 t T. 41 / 48

42 E cient Sample Sizes Substituting ξ (M t ) into W (ξ, M), we get W (M) = T Z ξ (M t ) h i 1 F (1) t=1 ξ N (M t+1 ) t (x) dx. Increasing search intensity in an e cient mechanism. Proposition Suppose c i = c for all i 2 N. If a sampling strategy M is optimal, then the e cient sample size mt is increasing in t, that is, m t m t+1, for t = 1, 2,, T / 48

43 E cient Sample Sizes Optimal sample size is greater than the e cient one. Corollary For given cuto ξ, the optimal sample size is greater than the e cient one, m (ξ t ) m (ξ t ), for 1 t T. 43 / 48

44 Short-lived Bidders: No Recall Short-lived bidder: A bidder, if invited, will participate in the transaction only once, sequential search with no recall; Equilibrium cuto strategy: ξ t = r t, for t = 1, 2,, T. The seller s problem, when bidders are short-lived, is h n oi Ĵ t (v) = max nψ (v), EĴ t+1 max r, X (1) M t+1 2Nt c M t+1 c Mt+1 o, (9) where the fall-back revenue of continuing search is 0 = ψ (r ). 44 / 48

45 Optimal Search Mechanism Proposition In an optimal search mechanism with homogeneous short-lived bidders, the optimal cuto value ˆξ t is decreasing, and the optimal sample size ˆm t is increasing over time. That is, for all t = 0, 1,, T 1, ˆξ t > ˆξ t+1, ˆm t ˆm t+1. Intuition: the continuation value of search, which determines the optimal cuto value, is decreasing overtime. 45 / 48

46 Optimal Search Mechanism Proposition For given sampling strategy M, the optimal cuto value for short-lived bidders is lower than that for long-lived bidders. Speci cally, for 0 t < T 1, ˆξ t < ξ t, and ˆξ T = ξ T = r. Intuition: the seller s bargaining power (fallback revenue) is smaller in the case of short-lived bidders. 46 / 48

47 Conclusion Optimal search mechanism can be implemented by a sequence of second-price auctions; The optimal sequential auction is characterized by # reserve prices and " search intensity over time; It is robust in both cases of short-lived and long-lived bidders. For long-lived bidders, the optimal reserve prices demonstrate a one-step-ahead property It generalizes the well-known formula of Weitzman (Ecta, 79) The e cient mechanism is featured by lower reserve prices and higher search intensity than the optimal mechanism. 47 / 48

48 Thank you! 48 / 48