On the benefits of set-asides

Transcription

1 On the benefits of set-asides Laurent Lamy (joint with Philippe Jehiel) Paris School of Economics NUS and HKUST, october 2015

2 Introduction

3 Set-asides: a popular instrument in public procurements: In Japan, set-asides for small and medium enterprises (SMEs) are the only explicit discriminatory instrument to favor SMEs in public procurements. Approximatively two thirds of civil engineering contracts are subject to them. In US, federal procurement contracts between $3,000 and $100,000 are reserved automatically for SMEs, and around $30 billion in federal contracts is awarded annually through some form of explicit set-aside programs. Other example in US: 14% of the timber auction dataset considered by Athey, Coey and Levin (2013) -which includes non-salvage Forest Service sales in California between 1982 and is composed of auctions with set-asides (either first-price or English auctions) while the rest are non-discriminatory auctions.

4 Set-asides: a popular instrument in public procurements: Set-asides (and more generally explicit discrimination between bidders) are banned by European legislation, but are sometimes imposed by the regulators: France antitrust authorities imposed set-asides as a remedy after a merger (the Veolia-Transdev case, public transports): the merged entity commits not to participate in the forthcoming procurements where she will be an incumbent (that discourage the participation from competitors due to its strength). Amaral, Saussier and Yvrande-Billon (2009): when someone would acquire a too large market share if he wins a contract (in the London transportation market). When the contractor was not reliable in the past (in particular if it is the incumbent but also possibly for similar contracts, see the new European directives on procurements, not clear how it will be applied).

5 Set-asides seems detrimental for the revenue? In private value model (i.e. when bidders know their valuations/costs): mechanical in second-price (or equivalently English auctions). not aware of any formal result for first-price auctions with asymmetric bidders, but highly improbable that dropping one bidder increases the revenue (same with the point that mergers or cartels are detrimental). with symmetric bidders, Bulow and Klemperer (1996) provide a bound on how excluding one bidder is detrimental to the seller: they show that adding an optimal reserve price would not compensate the seller from losing a single bidder. In common value models, more participants can be detrimental due to the winner s curse (Bulow and Klemperer, 2002). But we will limit our analysis to private value models!

6 Set-asides: a popular instrument in spectrum auctions Cramton (2013) argues that set-asides were at the core of the success of Canada s spectrum auctions for Advanced Wireless Services (AWS) because they encourage participation from deep-pocketed new entrants such that this pushes the price for both the set-aside and the non-set-aside blocks. In UK 2000 s spectrum auctions, the fact that one licence was blocked for a new entrant is also considered as one of the sources of its success (Jehiel and Moldovanu (2003)). When a single bidder deter all potential entrants from participating as illustrated in the 1994 US spectrum auctions where a license covering all of southern California was offered and where it was publicly known that Pacific Bell, the incumbent company, had a higher valuations that its rivals (see Milgrom (2004)): set asides would have been profitable for sure!

7 Set-asides: an endogenous participation perspective Empirical literature: Evidence that set-asides boost participation from the non-excluded bidders but also increase the total number of bidder (Denes (1997), Nakabayashi (2013) and Athey, Coey and Levin (2013)). Mixed evidence on the expected revenue (costs in a procurement): positive effect in Nakabayashi (2013), negative in Athey, Coey and Levin (2013) From a theoretical perspective, Jehiel and Lamy (2015) shows with a mechanism design approach à la Myerson (1981): if all buyers are potential entrants, no discrimination is optimal, if some buyers are incumbents (i.e. to simplify have no participation costs), it is optimal to discriminate against them. But the optimal form of discrimination is not set-asides and is a slightly unrealistic instrument (but that could be approximated with well-chosen bid subsidies). On the whole, a pretty small literature for an important policy issue. No clear insights about the benefits of set-asides.

8 Some applied policy issues we address as a by-product: Is it good to pre-select a special bidder in a procurement in order to be sure that there will be some participants? Is it good to allow a player in a dominant position to merge with another bidder who previously did not bid/participate in this market in order order to reduce its cost (and become even more dominant)? Is it good to exclude some bidder from a cartel in order to weaken the strength of this cartel? Is it good to have a right-of-first-refusal for a special bidder (in particular the incumbent)? What are the incidence on set-asides policy when we switch from English to first-price auctions? or when entry subsidies (or fees) are introduced? The important novelty is that we address those practical issues in a general setup with endogenous entry.

9 Outline Our general model with endogenous entry and then our main insight through an example. Preliminary elements about the Vickrey auction with endogenous entry. (Vickrey auction = second-price auction with the reserve price set at the seller s valuation) Set-asides in the Vickrey auctions Application to mergers and anti-collusion policies Beyond the Vickrey auction: 1/ The Vickrey auction with entry subsidies, 2/ Other bidding mechanisms (in particular procurements followed by a renegociation phase and/or first-price auctions) Further insights on set-asides.

10 Our general model with endogenous entry and our main insight through an example.

11 The auction model with endogenous entry (framed into an auction setup instead of a procurement but equivalent) Two sets of buyers: a finite set of incumbents with valuation distribution Fi I (.), i I (and no participation costs), and some groups of potential entrants with valuation distribution F k (.) and participation costs C k, k = 1,..., K. The seller with reservation value X S posts a second price (or equivalently an English auction) with the reserve price X S (assumption relaxed later). Key market design element: The seller chooses a set-asides policy: which incumbents and groups of entrants to exclude/keep. Free entry: Potential entrants decide to enter simultaneously. Entry is modelled as a Poisson game (the number of participants from each group will be a Poisson distribution) where the equilibrium entry rates are such that expected payoffs are equal to entry costs. Remark: The Poisson model is not crucial, we could have a model with a finite number of potential entrants per group (but large enough to guarantee that they use mixed strategies).

12 A pedestrian example Let X S = 0, a single incumbent with valuation x I > 0, homogenous entrants have all the same valuation x E > 0 and entry cost C < E[x E ]. The equilibrium entry rate µ out > 0 if the incumbent is excluded is e µ out E[xE ] = C. The equilibrium entry rate µ in > 0 if the incumbent is not excluded is e µ in E[max{xE x I, 0}] = C. if a solution exists and µ in = 0. Note that this implies that e µ out µ in = E[x E ] E[max{x E x I, 0}]

13 A pedestrian example The expected revenue is then if the incumbent is excluded: R without Inc = (1 e µ out µ out e µ out ) E[xE ] = E[x E ] (1+µ out) C if the incumbent is not excluded: R with 1 Inc = e µ in 0+µ in e µ in E[min{x E, x I }]+(1 e µ in µ in e µ in ) E[x E ] ( R with 1 Inc = E[x E ] E[x E ] E[max{x E x I, 0}] + µ in ) C From the last equality in previous slide and since e x > 1 + x, we get that R without Inc > R with 1 Inc (this holds for any joint distribution of (x I, x E )). This results holds much beyond the limited case where entrants have all the same valuation (i.e. with either ex-post and ex ante asymmetries between entrants).

14 Our main result Theorem: When there is a single incumbent, then the revenue-optimal set-asides policy in the Vickrey auction consists in excluding the incumbent and keeping all kinds of entrants. The informal intuition: what is the impact on the seller s revenue of raising marginally the valuation of the incumbent in the Vickrey auction? If the incumbent has the highest valuation, no impact since she captures all the surplus she brings. If the incumbent has a valuation below the second highest valuation among the entrant, no impact too. If the incumbent has a valuation between the first and the second highest valuation, the incumbent has a direct (positive) impact on the price, but the increase in the final price corresponds exactly to a reduction of the (winning) entrant payoff. At the end all the rents taken to the entrants (which are fixed ex ante) are paid by the seller through reduced participation. The impact is then globally negative because what matters in this end is only the entry profile of the entrants that would be maximized if they may their decisions as if the incumbent was not there.

15 A pedestrian example (continued) Now consider that there are two (identical) incumbents instead of a single incumbent. If the two incumbents are not excluded, then the equilibrium rate is still µ in but the revenue is now R with 2 Inc = e µ in E[x I ]+µ in e µ in E[x I ]+(1 e µ in µ in e µ in ) E[max{x E, x I }]. and then R with 2 Inc = E[max{x E, x I }] (1 + µ in) C. (1) If the incumbent is excluded (recall): R without Inc = E[x E ] (1 + µ out) C Since µ out > µ in, we get that R with 2 Inc > R without Inc > R with 1 Inc

16 Preliminary elements about the Vickrey auction with endogenous entry. [We introduce notation and extend to a framework with incumbents a result that already appears in Jehiel and Lamy (2015)]

17 Some notation N = (n 1,..., n K ) N K denote a realization of the profile of entrants. Let N +k = (n 1,..., n k 1, n k + 1, n k+1..., n K ).

18 Some notation N = (n 1,..., n K ) N K denote a realization of the profile of entrants. Let N +k = (n 1,..., n k 1, n k + 1, n k+1..., n K ). I I a set of incumbents and for i I, we let I i = I \ {i}

19 Some notation N = (n 1,..., n K ) N K denote a realization of the profile of entrants. Let N +k = (n 1,..., n k 1, n k + 1, n k+1..., n K ). I I a set of incumbents and for i I, we let I i = I \ {i} F (1:N I ) (x) denote the CDF of the first order statistic among the set of entrants N and the set I I of incumbents.

20 Some notation N = (n 1,..., n K ) N K denote a realization of the profile of entrants. Let N +k = (n 1,..., n k 1, n k + 1, n k+1..., n K ). I I a set of incumbents and for i I, we let I i = I \ {i} F (1:N I ) (x) denote the CDF of the first order statistic among the set of entrants N and the set I I of incumbents. P(N µ) = K k=1 eµ k [µ k ] n k n k! denote the probability of the realization N when the entry rate vector is µ

21 Some notation N = (n 1,..., n K ) N K denote a realization of the profile of entrants. Let N +k = (n 1,..., n k 1, n k + 1, n k+1..., n K ). I I a set of incumbents and for i I, we let I i = I \ {i} F (1:N I ) (x) denote the CDF of the first order statistic among the set of entrants N and the set I I of incumbents. P(N µ) = e K k=1 µk K [µ k ] n k k=1 n k! denote the probability of the realization N when the entry rate vector is µ (µ, I ) = ent N NK P(N µ) Vk (N +k, I ) C k denote the expected (ex ante) payoff of a group k buyer Π ent k

22 Some notation N = (n 1,..., n K ) N K denote a realization of the profile of entrants. Let N +k = (n 1,..., n k 1, n k + 1, n k+1..., n K ). I I a set of incumbents and for i I, we let I i = I \ {i} F (1:N I ) (x) denote the CDF of the first order statistic among the set of entrants N and the set I I of incumbents. P(N µ) = e K k=1 µk K [µ k ] n k k=1 n k! denote the probability of the realization N when the entry rate vector is µ (µ, I ) = ent N NK P(N µ) Vk (N +k, I ) C k denote the expected (ex ante) payoff of a group k buyer Π ent k (µ, I ) = inc N NK P(N µ) Vi (N, I ) denote the expected (ex ante) payoff of the incumbent i Π inc i

23 Some notation N = (n 1,..., n K ) N K denote a realization of the profile of entrants. Let N +k = (n 1,..., n k 1, n k + 1, n k+1..., n K ). I I a set of incumbents and for i I, we let I i = I \ {i} F (1:N I ) (x) denote the CDF of the first order statistic among the set of entrants N and the set I I of incumbents. P(N µ) = e K k=1 µk K [µ k ] n k k=1 n k! denote the probability of the realization N when the entry rate vector is µ (µ, I ) = ent N NK P(N µ) Vk (N +k, I ) C k denote the expected (ex ante) payoff of a group k buyer Π ent k (µ, I ) = inc N NK P(N µ) Vi (N, I ) denote the expected (ex ante) payoff of the incumbent i Π inc i W (N, I ) := 0 max{x, X S }df (1:N I ) (x) as the expected (interim) gross welfare

24 Some notation N = (n 1,..., n K ) N K denote a realization of the profile of entrants. Let N +k = (n 1,..., n k 1, n k + 1, n k+1..., n K ). I I a set of incumbents and for i I, we let I i = I \ {i} F (1:N I ) (x) denote the CDF of the first order statistic among the set of entrants N and the set I I of incumbents. P(N µ) = e K k=1 µk K [µ k ] n k k=1 n k! denote the probability of the realization N when the entry rate vector is µ (µ, I ) = ent N NK P(N µ) Vk (N +k, I ) C k denote the expected (ex ante) payoff of a group k buyer Π ent k (µ, I ) = inc N NK P(N µ) Vi (N, I ) denote the expected (ex ante) payoff of the incumbent i Π inc i W (N, I ) := 0 max{x, X S }df (1:N I ) (x) as the expected (interim) gross welfare The expected total welfare (net of the entry costs): TW (µ, I ) := N N K P(N µ) W (N, I ) K k=1 µ k C k

25 Equilibrium conditions and the revenue Equilibrium free entry conditions: for any k, we have Π ent k (µ, I ) = 0 if µ k > 0. (resp. ) (resp. =) We let µ (I, E) the equilibrium entry profile. By plugging the entry equilibrium conditions, we get the following expression for the seller s revenue: R(µ (I, E), I ) = TW (µ (I, E), I ) i I Π inc i (µ (I, E), I ). revenue = gross welfare minus the rents of the incumbents minus the entry costs of the potential entrants.

26 The fundamental property of the Vickrey auction The payoff of a bidder corresponds exactly to his contribution to the welfare (this is true ex post, i.e. for any realization of the valuations) For the potential entrants from group k, this implies from an ex-ante perspective: TW (µ, I ) µ k = P(N µ) Vk ent N N K (N +k, I ) C k = Π ent k (µ, I ). For the incumbent i I, this implies from an ex-ante perspective: Π inc i (µ, I ) = TW (µ, I ) TW (µ, I i ).

27 Some fundamental preliminaries (mild extension of Jehiel and Lamy, 2015) Lemma µ (I, E) Arg max µ R K + TW (µ, I ) for any I I. [To establish the lemma we show in particular that µ TW (µ, I ) is globally concave]. Proposition The welfare-optimal set-asides policy involves no exclusion. Corollary 1 Without incumbents, the revenue-optimal set-asides policy involves no exclusion. [Jehiel and Lamy (2015) adopt a mechanism design approach à la Myerson (1981) and show the much stronger result that the Vickrey auction (without exclusion) is the optimal mechanism.] Corollary 2 With full competition among incumbents, the revenue-optimal set-asides policy involves no exclusion.

28 Set-asides in the Vickrey auction [The main building-block of the paper]

29 The single incumbent case In equilibrium, the expression of the revenue reduces to: R(µ (I, E), I ) = TW (µ (I, E), ) The objective of the seller is thus to maximize the total welfare as if the incumbent was not there. From the previous fundamental lemma, this is done if the incumbent is excluded. (formally this is because: R(µ (, E), ) = TW (µ (, E), ) = max µ R K + TW (µ, ))

30 Some comments A possible application: procurements where contracts are renewed periodically and the previous winner is always a bidder. (But we have to be a bit cautious...) It is a bad idea to pre-select only one bidder in order to avoid that the contract remains unsold. Technically, the Poisson assumption avoids equilibrium miscoordination but it is not a so crucial assumption (finite number of potential entrants would work too) Our fundamental lemma and our main result as a corollary extend to other multi-object environments (see Jehiel and Lamy, 2015)

31 The multiple incumbents case In equilibrium, the expression of the revenue can be also expressed as: R(µ (I, E), I) = TW(µ (I, E), I i ) Π inc j j I i (µ (I, E), I). to be compared with R(µ (I i, E), I i ) = TW (µ (I i, E), I i ) j I i Π inc j (µ (I i, E), I i ). We have a new channel: excluding bidder i would modify the rents of the other incumbent in I i. Some informal intuitions: consider that some strong incumbent j has for sure a higher valuation that all the other incumbents It is detrimental to exclude any weak incumbent if the strong incumbent is not-excluded (because the equilibrium entry rates do not change when such inefficient incumbents are dropped while they can reduce the rent of the strong incumbent). It is not clear-cut for the strong incumbent: because we have the usual gain of exclusion that the entry rates will be optimal (first term), on the other hand, the weak incumbents will have rents (second term). If µ (I, E) is close to µ (I i, E), then the first effect will be negligible (especially because TW (µ (I i,e),i i ) = 0). If there is strong enough competition between µ k the incumbents, it is still detrimental to exclude the incumbent.

32 The multiple incumbents case Difficult to derive general insights. But in a first step we derive simple conditions (independence, symmetry among entrants, technical restrictions on the shape of the CDFs) guaranteeing that Π inc j (µ (I, E), I ) < Π inc j (µ (I i, E), I i ) namely excluding an incumbent increases the rent of the other incumbents. The entry effect does not counterbalance the direct effect of facing one incumbent less.

33 The set of assumptions 1. Valuations are drawn independently on the same support [x, x], 2. There is a single group of entrants characterized by the distribution F and the entry cost C (namely K = 1), 3. The function x 1 j I i, 4. The function x log[f I I 1 Fj (x) Fj I (x) 1 F (x) i (x)] 1 Fi I (x) is decreasing on (x, x) for any 1 F I i (x) 1 F (x) is decreasing on (x, x). Proposition Under the above set of assumptions: the rents of the non-excluded incumbents in I i increases when bidder i is excluded. This works e.g. when the CDFs F I j are weighted sum of F k, k = 1,...K.

34 Excluding small incumbents is detrimental Proposition: Let us parameterize the distribution of incumbent i by the parameter λ such that Fi I (x) = [G(x)]λ and assume that the previous set of assumption holds. Then if λ is small enough then the revenue raises when incumbent i is excluded. TW (µ (I, E), I i ) TW (µ (I i, E), I i ) is of order λ 2 since TW (µ (I i,e),i i ) µ = 0. Second order effect.

35 No clear-cut insights with multiple incumbents We can build examples such that: Excluding each incumbent is marginally detrimental, but excluding all incumbents is profitable. With multiple incumbents, we can build a general class of examples where excluding a small incumbent is good (but where the other incumbent are also small!)

36 Application to mergers and anti-collusion policies. [This discussion applies (or adapts) our main result about the benefit of excluding a single incumbent]

37 Mergers Two very different kinds of mergers: Internal merger: two bidders that previously compete in the market merge (horizontal mergers). Standard model for such mergers: if two bidders with valuations v 1 and v 2 merge then the valuation of the merged entity is max{v 1, v 2 }. Usually considered as detrimental for the seller (in models with exogenous entry). Nevertheless Li and Zhang (2015) show that it may be different in models with endogenous entry (and for first-price auctions). External merger: one bidder merge with an external firm to have lower cost (example, a vertical merger, can be viewed as an investment to upgrade his valuation). Typically not discussed in the literature. With exogenous entry, it is beneficial for the seller in the second price auction (and very improbable that it differs in the first price auction).

38 What our analysis brings to such debates. Proposition for external mergers In the model with symmetric entrants (K = 1) and with a single incumbent, the revenue increases when the incumbent gets weaker (according to first-order stochastic dominance). As a corollary: If there is a single incumbent who propose to merge with some firms, then it is better for revenue to prevent the merge. This result strongly contrasts with what we would obtain with exogenous entry Proposition for internal mergers If all the incumbents propose to merge, then it is better either to reject the merge or to exclude all the incumbents. This justifies why we should propose as a remedy to prevent the merged entity to participate in markets where she is an incumbent.

39 Cartels. Efficient cartels can be interpreted as a merger... and we can thus reinterpret the previous results.

40 Beyond the Vickrey auction: 1. The Vickrey auction with entry subsidies. 2. Other bidding mechanisms.

41 Extension to other auction formats? Our results extend straightforwardly to any mechanism which is payoff-equivalent from an ex ante perspective and for any possible entry profile µ and any vector of incumbents I such that the expected profit of the incumbent i (resp. a group k entrant) is still equal to Π inc i (µ, I ) (resp. Π ent k (µ, I )). From the well-known payoff equivalence Theorem, if valuations are independent, any mechanism which assigns the good efficiently and leaves no rents to buyers with null valuation is payoff-equivalent to the Vickrey auction. An example is the case of the first-price auction (with the reserve price X S ) when all bidders are symmetric, where the underlying valuation distribution F is continuously differentiable, and where the set of entrants is publicly observed at the bidding stage. [we stress that it no longer works if the set of entrants is not observed since it would create an asymmetry...]

42 The Vickrey auction with entry subsidies Now the equilibrium entry profile depend also of the fees (denoted f R K ): µ (I ; f ). The previous expression for the revenue is (almost) unchanged: R(µ (I ; f ), I ) = TW (µ (I ; f ), I ) i I Π inc i (µ (I ; f ), I ) = TW (µ (I ; f ), I i ) Π inc i (µ (I ; f ), I ). i I i With a single incumbent, we still get R(µ (I ; f ), I ) = TW (µ (I ; f ), ) max TW (µ, ) = R(µ (I ; f ), ) µ R + K Proposition: The revenue-optimal set-asides and fee policy consists in excluding the incumbent (and keeping all kinds of entrants and having no fees. However, we also have max µ R K + R(µ, I i ) max µ R K + R(µ, I ) and thus Proposition: Excluding one incumbent is always (weakly) dominated by the policy that consists in imposing optimal fees and keeping the incumbent.

43 The Vickrey auction with entry subsidies On the whole, our results with fees say that our general insight that set-asides are unambiguously good in presence of a single incumbent and should be considered with great cautiousness with multiple incumbents carries over when fees are allowed. Remark: with symmetric entrants (i.e. K = 1), optimal fees always take the form of a partial reimbursement of the entry cost (an intrument that is observed in practice, in the Veolia-Transdev merge, the merged entity finance a reimursement fund!)

44 A specific general class of allocation rules: Definition: For any strictly increasing continuously differentiable function φ : [X S, ) N [X S, ) with φ(x S ) X S, the φ-allocation rule consists in assigning the good in the following way: When all bidders have a valuation below X S, the seller keeps the good, When the incumbent has a valuation below X S and there is at least one entrant with a valuation above X S, the good is assigned to the entrant with the highest valuation, When the incumbent has a valuation above X S and there is no entrants with a valuation above X S, the good is assigned to the incumbent, When the valuation of the incumbent x I and the highest valuation among the n entrants x E are both above X S, the good is assigned to the incumbent if and only if φ(x I, n) x E. An auction mechanism leading to the φ-allocation rule and where only the winner pays is called a φ-auction. We say then that a φ auction advantages (resp. disadvantages) the incumbent if φ(x) x (resp. φ(x) x).

45 The function φ: Highest valuation among the entrants areas with an inefficient allocation the entrant gets the good function φ the seller keeps the the incumbent gets the good

46 In the Vickrey auction, we had: The fundamental property of the Vickrey auction: Π inc i (µ, I ) = TW (µ, I ) TW (µ, I i ) which further implies R(µ (I, E), I) = TW(µ (I, E), ) In a φ-allocation rule, we have The key property of φ-allocation rule (after some calculus): Π inc i (µ, I ; φ) = TW (µ, I ; φ) TW (µ, ; φ)+ (φ(x) x)(1 F I (x))d[g µ(φ(x))] X S + X S φ(x S ) (x X S )d[g µ(x)] where G µ is the CDF of the highest valuation among the entrants. which further implies R(µ (I, E), I) TW(µ (I, E), ) if the φ auction disadvantages the incumbent which further implies R(µ (I, E), I) TW(µ (I, E), ) if the φ auction advantages the incumbent

47 Mechanism that are not ex post efficient Proposition: If the allocation rule favors the incumbent (φ(x) x) then the revenue-optimal set-asides policy consists in excluding the incumbent and keeping all kinds of entrants Comment: if the allocation rule disadvantages the incumbent, we have two opposite effects. The optimal auction has been derived in Jehiel and Lamy (2015): it is a φ allocation rule that disadvantage the incumbents (formally we have φ(x) = x 1 F I (x) ). From Jehiel and Lamy s (2015) f I (x) optimal result, we know thus that it would be detrimental to exclude the incumbent.

48 Applications to ex post failure/renegociation Bajari, Hioughton and Tadelis (2014): in procurements, there is a huge gap between the bids and the final true payment (due to adaptation costs). Remark: on ebay, some buyers may not buy the good at the end or may try to renegociate the price. This is a case that would allow a seller to exclude her from the future auction she organizes.

49 Example 1: a model with ex post failure suppose that each bidder (with valuation x) presents a risk of failure 1 p. In case of failure: the seller keeps the good and there is no monetary transfers Reinterpretation: the true valuation (with respect to the welfare) is indeed p x + (1 p)x S. On the contrary, it is still a dominant strategy to bid x in the auction. The second price auction with r = X S is no longer the pivotal mechanism but belongs to our general class of φ-allocation rule. Let p I (p E ) the reliability rate of the incumbent, we have then φ(x) = X S + p E p I (x X S ). φ(x) x if and only if p E p I, i.e. if the incumbent is less reliable. Proposition: If the incumbent is less reliable than the entrants, then it is profitable to exclude him in the second price auction with the reserve price X S.

50 Example 2: ex post renegociation suppose that the incumbent (resp. an entrant) if the final price is p will indeed pay β 1 I (p) (resp. β 1 E (p)) where β I and β E are two increasing functions with β I (X S ) = β E (X S ) = X S. then it is a dominant strategy to bid β I (x) (resp. β E (x)) for an incumbent (resp. an entrant) with valuation x. Reinterpretation: The second price auction with r = X S is no longer the pivotal mechanism but rather a second-price auction with bid subsidies We have then φ(x) = β 1 E (β I (x)). φ(x) x if and only if β I (x) β E (x), i.e. if the incumbent is a better negociator ex post. Proposition: If the incumbent is a better renegociator than the entrants, then it is profitable to exclude him.

51 Applications to standard auction formats The first price auction when the set of entrants is not disclosed disadvantage the incumbent: no clear-cut result. The first price auction with the right of first refusal advantages the incumbent: excluding the incumbent is thus profitable The first price auction with symmetric entrants, when the number of entrants is disclosed and when the incumbent has a weaker distribution (reverse hazard rate dominance) advantages the incumbent: excluding the incumbent is thus profitable The first price auction with symmetric entrants, when the number of entrants is disclosed and when the incumbent has a stronger distribution (reverse hazard rate dominance) disadvantages the incumbent: no clear-cut result second price auction with bid subsidies: if the incumbent is advantaged, excluding him is profitable

52 Further insights on set-asides.

53 On the benefits of excluding entrants For sure, it is suboptimal to exclude all kinds of potential entrants together in the Vickrey auction. However, if there are some incumbents, then it could be profitable to exclude some entrants. The rationale for excluding a group of entrants is that those could dissuade other entrants to participate. (nevertheless without incumbent such a rational never occur as shown before) The intuition is that we should keep those entrants that have a greater ability to reduce the rents of the incumbent (see next example).

54 Excluding entrants : a first example One incumbent with (deterministic) valuation x I. with probability q (0, 1), all entrants have the valuation x E > x I. With probability (1 q), entrants from group 1 (resp. 2) have the valuation x E 1 < x I (resp. x E 2 < x I ). We assume that C 1 < C 2 (such that group 1 bidders contribution more to the welfare than group 2 bidders) We assume that x E 1 < x E 2 (such that group 2 bidders reduce more the rents of the incumbent than group 1 bidders) If the parameters are calibrated such that x E x I > C 2 and C 2 C 1 small enough, then it is profitable to exclude bidders from group 1.

55 Excluding entrants : a surprising example We are able to build an example where it is profitable to exclude a group of potential entrants that have higher valuation distribution and lower entry costs... The intuition in words: a strong group with perfect correlation between their valuations (it creates some cannibalization that reduces the entry rates among this group) a weak group with independent valuations (less cannibalization and we obtain at the end more entrants who enter and reduces the rents of the incumbent)

56 Conclusion Many of the most important practical issues in auction design concern the interaction of the design and entry decisions. [...] Models with entry and asymmetric bidders have received much less attention than symmetric models, despite the great influence of asymmetries among bidders on entry. [Paul Milgrom (2004).] This is especially important since 1/ the answers are radically different in environments with exogenous versus endogenous entry. 2/ the empirical literature stresses the importance of endogenous entry.

57 Conclusion Jehiel and Lamy (2015) adopts an optimal mechanism design approach à la Myerson (1981) in a framework with both potential entrants and incumbents. We have analyzed here a specific (commonly used) instrument: set-asides. [Note that some of our answers do not depend on the valuation distribution] For further research, the analysis of other instruments.