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Mechanism design without quasilinearity. (English) Zbl 1466.91074

Summary: This paper studies a model of mechanism design with transfers where agents’ preferences need not be quasilinear. In such a model, (i) we characterize dominant strategy incentive compatible mechanisms using a monotonicity property, (ii) we establish a revenue uniqueness result (for every dominant strategy implementable allocation rule, there is a unique payment rule that can implement it), and (iii) we show that every dominant strategy incentive compatible, individually rational, and revenue-maximizing mechanism must charge zero payment for the worst alternative (outside option). These results are applicable in a wide variety of problems (single object auction, multiple object auction, public good provision, etc.) under suitable richness of type space. In particular, our results are applicable to two important type spaces: (a) type space containing an arbitrarily small perturbation of quasilinear type space and (b) type space containing all positive income effect preferences.

MSC:

91B03 Mechanism design theory
91B26 Auctions, bargaining, bidding and selling, and other market models
91B08 Individual preferences
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[1] Illustration of proof ofTheorem 1for two alternatives withab.
[2] As discussed earlier, TPrichness is a stronger richness requirement than TP+richness. We give a simple example to show thatTheorem 1may not hold with TP+ richness; of course, ifis a complete order, then the two richness notions are equivalent. Example
[3] SupposeA= {a b}andis empty. SupposeD=R+, i.e., the domain is the set of all positive income effect preferences. Fix two numbersκ(a)andκ(b)withκ(a) > κ(b). Define a subset of PIE preferences asD∗:= {R∈D:VR(a (b κ(b))) < κ(b)}. Now consider the following mechanism(f p)onD: for everyR∈D: f (R) p(R)=b κ(b)ifR∈D∗
[4] Indifference diagram of a preferenceRsatisfying PIE withVR(b (a κ(a)))=κ(a). Figure
[5] Illustration of violation of monotonicity offwhenis empty.
[6] Arevenue uniqueness theorem
[7] A domainDsatisfiesrevenue equivalenceif for every pair of DSIC
[8] Failure of revenue equivalence.
[9] A domainDsatisfiesrevenue uniquenessif for every pair of DSIC mechanism(f p)and(fp)ˆonD, we havep= ˆp. Theorem2.Let >0. Every domain that satisfiesTP+richness satisfies revenue uniqueness. Proof. LetDbe a domain that satisfies TP+richness for some >0. Assume, to the contrary, that there exist two DSIC mechanisms(f p)and(f p)such thatp =p. DSIC implies that the posted-price property holds; see the discussion immediately afterDefinition 10. Hence, there exist mapsκandκsuch thatκ(f (R))=p(R)and κ(f (R))=p(R)for allR∈D. Sincep =p, we getκ =κ. We now complete the proof in three steps. Step
[10] In this step, we show thatκandκrespect.8Pick anya b∈Aand supposeb a. PickRsuch thatf (R)=a(by ontoness, this is possible). By incentive compatibility, VR(b (a κ(a)))≤κ(b). SinceRrespects, we haveκ(a) < VR(b (a κ(a))). Combining the two inequalities, we getκ(b) > κ(a)as desired. A similar proof works forκ. Step
[11] In this step, we show that eitherκ > κorκ> κ. Without loss of generality, assume thatκ(a) > κ(a)for somea∈A. We show thatκ(b) > κ(b)for allb∈A. Assume, to the contrary, thatκ(b)≤κ(b)for someb∈A. Letδ >0but sufficiently close to zero. Definevasvx:=κ(x)−δfor allx =bandvb:=κ(b). Sinceδis chosen sufficiently small andκrespects(by Step 1),vrespects. Further,va> κ(a). By OP richness (which is implied by TP+richness), there isR∈Dsuch thatv∈I(R). 8Of course, ifis empty, then there is nothing to prove.
[12] We now complete the proof. Sinceκandκrespectandκ > κ,Lemma 4 · Zbl 1241.83010
[13] Does individual rationality bind?
[14] In this step, we show thatκ(a˜j)≥κ(aj)for allaj∈A. Since(f p)is IR, we know thatκ(a0)≤0. By definition,0= ˜κ(a0)≥κ(a0). Now, we prove the step using induction
[15] We now complete the proof. Pick anyR∈D. Letf (R)=ak. By definition,
[16] The allocations rulesfandf˜in the proof ofTheorem 3can be quite
[17] Atemplate for optimal contract design
[18] Relationship with the quasilinear domain results
[19] Related literature
[20] We show thatv∗respects, but we prove a slightly general assertion, which
[21] Illustration ofv∗for the caseabxy∗c.
[22] We construct the set of preferences R∗:=R∗∈D:VR∗yy∗ κy∗=v∗y∀y =x
[23] As before, using OP richness, chooseR∗∈R∗such thatv∗∈I(R∗). By Step 2, f (R∗)∈ {x y∗}. Further,VR∗(x (y∗ κ(y∗)))=vx∗> VR(x (y∗ κ(y∗)))and monotonicity implies thatf (R∗)=x. Hence, η=infVRˆxy∗ κy∗
[24] We now define two vectorsu v∈R|A|as ux=η+δvx=η−δuy=vy=v∗y∀y =x
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