Optimal Selling Mechanisms on Incentive Graphs

Transcription

1 Optimal Selling Mechanisms on Incentive Graphs Itai Sher Rakesh Vohra February 5, 2010 Abstract We present a model highlighting one advantage of negotiation over posted prices from the seller s perspective. In the process, we develop a general methodology for studying mechanism design with weaker incentive constraints. To do so, we introduce the notion of an incentive graph, which encodes the relevant constraints. This construction is relevant when agents possess hard evidence, when they partake in a specific form of costly signaling, or when they are boundedly rational. In such settings, we show that the optimal selling mechanism involves price discrimination and study the form which this price discrimination takes. When the incentive graph has a tree structure, we offer a complete solution. Our solution involves an adaptation of the classic notion of a virtual valuation as in Myerson (1981) to a tree. We show that for acyclic graphs, all allocations are incentive compatible, but nevertheless, the revenue maximizing mechanism is in many ways similar to that found in the classic analysis. JEL Classification: D44, D82. Keywords: Incentive graph, incentive compatibility, optimal auctions, price discrimination, negotiation, bounded rationality, virtual valuation, evidence. 1 Introduction What determines a seller s choice between a posted price and a negotiation? We present a model highlighting one advantage of negotiation, and analyze the optimal negotiation from the seller s perspective. In the process, we develop a general methodology for studying mechanism design with weaker incentive constraints. Consider a buyer-seller interaction in which the seller may design the mechanism governing the interaction. Assume for simplicity that the seller s value for the object is known. We are grateful to David Rahman for useful conversations. Economics Department, University of Minnesota. isher@umn.edu. Kellogg School of Management, MEDS Department, Northwestern University. r- vohra@kellogg.northwestern.edu. 1

2 What should the seller do? It seems that the optimum is a posted price. Communication with the buyer ought to be valueless because if the seller were to base the price on anything that the buyer said, the buyer would simply say whatever was necessary to receive the lowest price. If the seller already has some information about the buyer, the seller could base the price on this information, but conditional on this information, nothing that the buyer says should influence the price. However, suppose the buyer tells the seller about a particularly good deal he can get elsewhere. If the buyer is unaware of such a deal, he may not make such a claim. There are many claims that a buyer may or may not be able to produce in a way that is systematically linked to his information. These claims may be about financing constraints, the use to which the buyer plans to put the object or whether he intends to resell. The differential ability to produce claims may result from verifiability, bounded rationality, or scruples. The standard model for analyzing selling mechanisms would impose the constraint of incentive compatibility. A mechanism is incentive compatible if no type has an incentive to mimic any other type. However, in the context of a buyer-seller interaction and also in many other contexts we have just argued that this constraint may be too strong. It supposes that if any buyer can produce an argument that would persuade the seller to lower her price, then all buyers can can produce such an argument. A more realistic assumption is that if a seller decides to lower her price based on a certain argument, then there is a certain class of buyers who would make this argument and thereby get a lower price, but this class does not include all buyers. The model that we propose to study accounts for the kind of situation described above. Rather than imposing all incentive constraints, we impose only a subset of the incentive constraints. One can envisage the possibilities on a directed graph G = (V, E), where the set V of vertices is the set of types. A pair of types (s, t) belongs to the set E of edges if type t can mimic type s. G is an incentive graph. The classic model corresponds to the case in which G is the complete graph: every type can mimic every other type. This paper studies more general possibilities for the incentive graph. Incentive graphs can have any of the interpretations mentioned above; whether a type can mimic another type may have to do with presentation of evidence and other forms of verifiability, costly signaling, scruples, and perhaps most importantly, bounded rationality; manufacturing information which differs from the information that one actually has is a costly activity and should be treated as such. We apply our model to the problem mentioned above, namely to that of finding an optimal selling mechanism for selling a single object to a single buyer. However, our model can also be interpreted as describing the optimal selling mechanism to a population of consumers with unit demand. In the latter case, it is natural to interpret each such consumer as interacting with a salesperson who offers the consumer a price at the end of a negotiation. 2

3 It should be straightforward to extend our methodology and results to optimal auctions, which should really be interpreted as multi-party negotiations involving a single seller and many competing buyers. This is the subject of ongoing research (Sher and Vohra 2010). We find the optimal selling mechanism under two conditions: value monotonicity and value anti-monotonicity. In the former case, a necessary condition for type t being able to mimic type s is that type t has a higher value than type s, and in the latter case, a necessary condition for type t being able to mimic type s is that type t has a lower value than type s. In neither case are these necessary conditions sufficient, which means that types do not generally have a one-dimensional structure. Both cases involve price discrimination, but in each, price discrimination takes a very different form. In general, the exact form of price discrimination depends on the structure of the incentive graph. Under value monotonicity, the optimal mechanism may generally be quite complex. We are able to explicitly derive the optimal mechanism in the special case in which the incentive graph has a tree structure (Proposition 3). In this case we obtain a virtual valuation on a tree analogous to the virtual valuation from the classic study of optimal auctions due to Myerson (1981). When virtual valuations are monotone, then just as in the classic analysis (with a single buyer), the mechanism sells the object to a type only if that type s virtual valuation is nonnegative. However, in contrast to the classic analysis, there is not a single posted price. Rather different types receive different prices because the virtual valuation may turn positive on several different branches of the tree. It is interesting to contrast our analysis with the interpretation given to optimal auctions given by Bulow and Roberts (1989), who interpret each bidder as a market and thereby map the optimal auction problem into the monopolist s problem of third-degree price discrimination. With a single bidder, there is a single corresponding market, and hence no price discrimination. In contrast, in our model, we find something resembling third degree price discrimination with a single buyer or under an alternative interpretation, a single population of buyers. Moreover, the segmentation of different buyer types into different markets with different prices is endogenous, resulting from the seller s optimization. The technical assumption that the incentive graph has a tree structure can be interpreted as saying that each buyer type will ultimately have access to at most one market, although the scope of that market is determined endogenously. In the special case in which the tree is also a linear order so that the model is truly one-dimensional, our solution coincides with the classic solution of a posted price, and there is no price discrimination. 1 At the other extreme, when the incentive graph does not have a tree structure, the optimal mechanism may be more complex. The optimal mechanism may involve random allocation, and may even violate allocation monotonicity. This means that there may be two types s and t such that t has a higher value than s and t can mimic s, but nevertheless, t may receive the object with lower probability than s. 1 This particular conclusion can also be derived from the results of Moore (1984). 3

4 In the case of value anti-monotonicity, the optimal mechanism looks very different. In this case, it is possible for the seller to achieve perfect price discrimination independently of the structure of the incentive graph. So in this case, we obtain a much stronger result under much weaker assumptions. Aside from the analysis of optimal selling mechanisms, we obtain several results, which, while specialized to buyer-seller interactions in this paper, promise to have broader implications for the more general the study of mechanism design on incentive graphs. A key assumption which is employed in much of our analysis is acyclicity, under which every allocation is incentive compatible. Indeed, acyclicity of an incentive graph can be characterized in terms of the incentive compatibility of all allocations (Proposition 1 and 2), a result closely related to Rochet s (1987) classic characterization of incentive compatibility in terms of cyclic monotonicity. The incentive compatibility of all allocations greatly simplifies the analysis, and represents a polar case. Given that one has already assumed value monotonicity, acyclicity appears to be a weak additional assumption because value monotonicity implies that any cycle in the incentive graph could only pass through types with the same value. 2 The incentive compatibility of all allocations does not mean that anything goes from the seller s point of view. Just because it is possible for the seller to implement any allocation does not mean that it is profitable for the seller to implement any allocation. Under acyclicity, incentives do not impose any constraint on allocations alone, but rather, incentives impose joint constraints payments and allocations. Indeed, as we have explained, there are many similarities between optimal mechanism in the traditional model and under acyclicity, despite the fact that in the latter case all allocations are incentive compatible. This suggests that with respect to revenue maximization, incentive compatibility of allocations may not be as important as one might have thought. We derive a formula expressing the expected revenue of a mechanism as a function of the allocation (Proposition 1) 3. We also introduce a notion of allocation monotonicity mentioned above which generalizes the traditional notion of monotonicity to arbitrary incentive graphs. Under value monotonicity, when the incentive graph has a tree structure, the optimal mechanism satisfies allocation monotonicity, but whereas in the traditional analysis, allocation monotonicity is a consequence of incentive compatibility, in our analysis this is not so, as all allocations are incentive compatible. Allocation monotonicity is rather a consequence of optimization. When we depart from tree structure, allocation monotonicity may fail at the optimum. This paper is related to several lines of research. Our results contribute to the literature 2 A similar comment applies to value anti-monotonicity, although in this case we do not need acyclicity for our result. 3 Because we use a model with a discrete type space, revenue equivalence does not hold. Nevertheless, for any allocation, there is a revenue maximizing mechanism among those implementing that allocation. The revenue formula applies to this revenue maximizing mechanism. 4

5 that compares auctions, posted prices, and negotiations (?,?, Bulow and Klemperer 1996). There is also a small but growing body of research on mechanism design with evidence (Green and Laffont 1986, Singh and Wittman 2001, Forges and Koessler 2005, Bull and Watson 2007, Ben-Porath and Lipman 2008, Deneckere and Severenov 2008, Kartik and Tercieux 2009). The formalism of incentive graphs is related to but distinct the formalism used in these papers. We introduce incentive graphs rather than studying evidence structures, because the former get more directly at the heart of the matter: it is not the case that every type can mimic every other type. We avoid the extraneous structure imposed by models of evidence. On the other hand, the results that we obtain can be translated into results about optimal selling mechanisms with evidence. Severenov and Deneckere (2006) present an analysis which is motivated similarly to our paper but which contains a very different model and results. Like us, Severenov and Deneckere argue that it is not realistic to suppose that every type can mimic every other type, and this may lead a monopolist to converse with an agent, and thereby, price discriminate. However, they present a model in which some agents are strategic and may mimic any other type whereas others are nonstrategic, and the latter must report their information truthfully. This differs sharply from our model in which assumptions about which types can mimic which other types take a very general form. A model which is more formally similar to ours is Celik (2006). Celik studies an adverse selection problem in which higher types can pretend to be lower types but not vice versa, and shows that the weakening of incentive constraints does not alter the optimal mechanism. Celik s model induces precisely the sort of one-dimensional structure in which we find that there is no price discrimination, paralleling his result. This last result on the absence of price discrimination also follows from the analysis of Moore (1984), who studies an optimal auction subject to only that downward incentive constraints. Our main results concerning price discrimination occur on richer incentive graphs and thus differ sharply from the papers of Celik and Moore. A final literature which is related to the current work concerns models of persuasion (Milgrom and Roberts 1986, Shin 1994a, Lipman and Seppi 1995, Glazer and Rubinstein 2004, Glazer and Rubinstein 2006, Sher 2008, Sher 2009). Some papers in that literature notably Milgrom and Roberts (1986) studied buyer-seller relationships in which it is the seller, rather than the buyer, who has private information and can present evidence. Some versions of the model generate unravelling in which all information is revealed, although some do not (Shin 1994b). It is interesting to note that under value anti-monotonicity, our perfect price discrimination result is somewhat analogous to unraveling, 4 whereas with value monotonicity, the seller generally fails to extract the full surplus. The organization of the paper is as follows. Section 2 presents the model. Section 3 4 Note however that one important difference is that whereas in models such as Milgrom and Roberts (1986), the uninformed agent does not commit to a strategy up front, in our model the privately uninformed player does commit. 5

6 shows that under acyclicity all allocations are incentive compatible and presents a formal for revenue in terms of the allocation. Section 4 presents the optimal mechanism under tree structure. Section 5 shows how allocation monotonicity and deterministic allocation may fail when the tree structure assumption is violated. Section 6 states the perfect price discrimination result under value anti-monotonicity. Proofs of the main theorems and supporting lemmas are presented in the Appendix. 2 Model Let T be the set of types. For our purposes, it is technically convenient to assume that the set T of types is finite. We expect that analogous results could be derived for a continuum of types. In general, a type represents an agent s information. In this application, the agents are bidders and the information that bidders have concerns their value for the object on auction. Let v t be the value of type t and π t be the probability of type t. We refer to the map v = (v t : t T ) as a valuation and π = (π t : t t) is a probability distribution. Next consider a directed graph G = (V, E), where V is a set of vertices and E is a set of directed edges. We set V = T so that the vertices are types and E encodes a mimicking relation. For any types s and t, (s, t) E means that type t can mimic type s. The mechanism must provide incentives for t to refrain from mimicking s only if (s, t) E. G is an incentive graph. It is technically convenient to assume that the mimicking relation is irreflexive so that for all t T, (t, t) E. However, we define the corresponding reflexive relation E := E {(t, t) : t T } so that for all t T, (t, t) E. A path (from t 0 to t n ) is a sequence of types τ = (t 0, t 1,..., t n ) containing at least two distinct types with (t i 1, t i ) E for i = 1,..., n. If t 0 = t n, then τ is a cycle. A polytree is a directed graph containing at most one path between every pair of vertices. It is convenient to define the related graph Ĝ = (T, Ê) where: (s, t) Ê [(s, t) E and u T \ {s, t}, (s, u) E (u, t) E] (1) We have found the following properties useful. Acyclicity G does not contain any cycles. Value Monotonicity (s, t) E v s v t. Tree Structure G is acyclic and Ĝ is a polytree. Tree structure is stronger than acyclicity, and many of our results employ the latter only. The justification for the analysis of acyclicity and tree structure is technical, in terms of the properties of optimal mechanisms that they induce. In particular, Propositions 1 and 2 6

7 show the significance of acyclicity, and Proposition 3 concerns optimal mechanisms under tree structure. However, tree structure also has a nice interpretation as discussed in Section 4. In contrast, the assumption of value monotonicity has a more substantive interpretation. Value monotonicity means that it is a necessary condition for type t to be able to mimic type s that type t has a (weakly) higher value than type s. It is important to note that if type t has a higher value than type s, this is not a sufficient condition for t to be able to mimic s. The fact that we impose only a necessary and not a sufficient condition allows for much richer and more interesting possibilities. The opposite assumption, in which lower value types can mimic (some) higher value types, but not vice versa, is handled in Section 6. It is interesting to observe how value monotonicity interacts with acyclicity. Under value monotonicity, if there were a cycle, then all types on the cycle must have the same value. This suggests that acyclicity may be a weak assumption in the presence of value monotonicity. A mechanism is a tuple (q, p) = (q t, p t : t T ), where q t [0, 1] is the probability that type t will get the object, and p t R + is the price paid by type t. A mechanism is optimal if it solves the linear program: max p tπ t s, t T, with (s, t) E, q t v t p t q s v t p s Incentive Compatibility t T, q t v t p t 0 Individual Rationality t T, 0 q t 1 Probability Constraints (2) The main difference between the traditional analysis and our analysis is that whereas in the traditional analysis, incentive compatibility constraints would apply to every pair of types, in our analysis, incentive compatibility constraints only apply to pairs of types s, t such that (s, t) E. Thus, we recover the traditional analysis when G is the complete graph, or in other words, the graph in which every ordered pair (s, t) of types belongs to E. In our model, a mechanism (q, p) is said to be incentive compatible if it satisfies the constraints of the linear program (2), so that we use this term to refer to the weaker set of constraints encoded by the incentive graph. An allocation q is incentive compatible if there exists a payment scheme p such that (q, p) is incentive compatible. It is convenient to introduce a zero type 0 T such that v 0 = 0 and for all other types t, (0, t) E. This is without loss of generality and allows us to treat the individual rationality constraint as a special case of the incentive compatibility constraints (provided that we also assume p 0 0). In what follows we also always make the harmless assumption that q 0 = 0. In the traditional model when G is the complete graph the form of the program (2) is justified by the revelation principle. We can argue for a version of the revelation principle 7

8 under the following assumption: Transitivity [(s, t) E and (t, u) E] (s, u) E Transitivity is consistent with all of the other assumptions mentioned above. (In particular, transitivity is consistent with tree structure, because tree structure assumes that Ĝ given by (1) is a polytree; it is not assumed that G itself is a polytree). However, while transitivity simplifies some proofs, most of our results do not require transitivity, and therefore, to gain a deeper insight into the mathematical structure of the problem, we usually omit this assumption. 3 Revenue Generated by an Allocation Because the type space is discrete, our model will generally not satisfy revenue equivalence. However, for any fixed incentive compatible allocation, there exists a revenue maximizing payment scheme among those implementing q. For any type t, let P (t) be the set of paths in G from 0 to t. If τ = (t 0, t 1,..., t n ) is an element of P (t), then we use the notation η(τ) = n and τ i = t i. In this section, we construct this revenue maximizing payment profile, deriving a revenue formula which is analogous to the standard revenue formulas that come out of smooth continuous models. Proposition 1 Assume acyclicity. Then every allocation q is incentive compatible. Moreover the revenue maximizing payment scheme among those implementing q is such that each type t s payment is given by: Proof. In Appendix. η(τ) p t = v t q t max{ (v τi v τi 1 )q τi 1 : τ P (t)} (3) i=1 This proposition shows that in contrast to the traditional analysis, under acyclicity, incentive compatibility does not constrain allocations. However, this does not mean that anything goes from the seller s point of view. Just because it is possible for the seller to implement any allocation does not mean that it is profitable for the seller to implement any allocation. Incentives do not impose any constraint on allocations alone, but rather, incentives impose joint constraints payments and allocations. (3) gives us a payment formula which is somewhat analogous to standard mechanism design. Proposition 1 is closely related to the characterization of incentive compatibility in terms of cyclic monotonicity due to due to Rochet (1987). In the model studied by Rochet every type can mimic every other type. In the current setting, where the possibilities for mimicking are limited, the analog to Rochet s no negative cycles condition is always vacuously satisfied under acyclicity because this case, there are no cycles. 8

9 We also derive a converse of the first statement in Proposition 1: Proposition 2 Consider an incentive graph G that contains a cycle. Then there exists a valuation v such that in (G, v), not every allocation is incentive compatible. Proof. In Appendix. Acyclicity is exactly the assumption that decouples incentive compatibility of allocations from profit maximization. In contrast to more complex incentive graphs failing acyclicity, acyclicity allows us to construct the revenue maximizing payments implementing any allocation recursively, starting at the bottom of the incentive graph via (3). It is a virtue of our model that it helps us to uncover such a simple recursive structure. Next we note a useful corollary of Proposition 1 which allows us to reformulate the problem of finding an optimal mechanism in such a way that payments are expressed in terms of allocations. Corollary 1 Assume acyclicity. (q, p ) is an optimal mechanism if and only if (q, p ) solves: s.t. t T, max (q,p) p t π t (4) η(τ) p t = v t q t max{ (v τi v τi 1 )q τi 1 : τ P (t)} (5) i=1 t T, 0 q t 1 (6) 4 Tree Structure This section derives the optimal mechanism under the assumptions of tree structure and value monotonicity. In combination with the existence of the zero type, tree structure implies that for all t T \ 0, there exists exactly one type s T such (s, t) Ê.5 Define ϕ(t) be equal to this unique type s, and define ϕ(0) := 0. Define a set E of edges by: (s, t) E G contains a path from s to t. We also define a corresponding reflexive relation E := E {(t, t) : t T }. Lemma 1 Assume value monotonicity, transitivity, and tree structure. (q, p ) is an opti- 5 Assume for contradiction that there were two distinct types r and s such that (r, t), (s, t) Ê. Since (0, s) T and G is acyclic and finite, there is a maximal length path from 0 to s in G. This path is also a path from 0 to s in Ĝ. Likewise, there is a path from 0 to r in Ĝ. Because (r, t), (s, t) Ê, this implies that there are two distinct paths from 0 to t in Ĝ, contradicting tree structure. 9

10 mal mechanism if and only if (q, p ) solves: Proof. In Appendix. s.t. t T, max (q,p) p t π t (7) p t = v t q t s T :(s,t) E (v s v ϕ(s) )q ϕ(s) (8) s, t T with (s, t) E, q s q t (9) t T, 0 q t 1 (10) (9) is a monotonicity constraint, which plays an interesting role in the lemma. Recalling that tree structure implies acyclicity, Proposition 1 implies that every allocation is incentive compatible, so the monotonicity constraint is not a consequence of incentive compatibility; indeed, the monotonicity constraint rules out feasible allocations. However, the ruled out allocations cannot be optimal. The monotonicity constraint is included because for all allocations satisfying the monotonicity constraint, (5) from Corollary 1 simplifies to (8) under tree structure, so that we may then remove the maximization from (5). The following property and corollary allow us to relate our model to the traditional model. Allocation monotonicity [v s v t and (s, t) E] q s q t. Recall the traditional model may be viewed as the special case of our model in which for all types, s, t, (s, t) E. Corollary 2 If every type can mimic every other type, as in the traditional model, the optimal mechanism satisfies allocation monotonicity. Likewise, value monotonicity, transitivity, and tree structure together imply allocation monotonicity at the optimum. However notice a disanalogy. In the traditional model, allocation monotonicity is a consequence of incentive compatibility; indeed not only the optimal mechanism but all incentive compatible mechanisms are allocation monotone. In contrast, in our model, allocation monotonicity is not a consequence of incentive compatibility not all incentive compatible mechanisms are allocation monotone but rather, allocation monotonicity is a consequence of optimization. In Corollary 2, one can substitute the assumption of monotone virtual valuations (defined below) for that of transitivity (see Proposition 4 below). Next we characterize the optimal mechanism under tree structure and value monotonicity. Our characterization is closely analogous to the classic characterization of optimal auctions due to Myerson (1981). For each t T, define: Π t = s T :(t,s) E π s 10

11 Π t is analogous to the complementary cumulative distribution function on one-dimensional domains. It is interesting to note that if we also define the analog of the cumulative distribution Π t = s T :(s,t) E π s, then in contrast to the case of a one-dimensional domain, when E does not induce a linear order on types, in general Π t 1 Π t. Assume tree structure, and for each t T, define: ψ t := v t s ϕ 1 (t) (v s v t ) Π s (11) π t ψ t is a virtual valuation on a tree, analogous to the standard notion of a virtual valuation as in Myerson (1981), but adapted to the tree structure of the incentive graph. Unlike Lemma 1 which is used in the proof of Proposition 3 Proposition 3 does not require transitivity. Proposition 3 Assume value monotonicity and tree structure. Assume moreover: Monotone Virtual Valuation (s, t) E ψ s ψ t. 6 Then the optimal mechanism (qt, p t : t T ) is such that: { { qt 1, if ψ t 0; = p min{v s : (s, t) E, ψ s 0}, if ψ t 0; t = 0, otherwise. 0, otherwise. (12) Proof. In Appendix. This proposition characterizes the optimal mechanism under tree structure and value monotonicity. It is analogous to the characterization of Myerson (1981), which when specialized to the case of a single buyer says that the object should be sold whenever the buyer s virtual valuation a notion analogous to marginal revenue (see Bulow and Roberts (1989)) is nonnegative, and moreover should be sold at a price equal to the lowest value at which the virtual valuation is nonnegative. In the traditional model with a single buyer, the optimal mechanism is therefore a posted price and the seller s problem is analogous to the monopoly problem. However, it is important to notice the differences between our solution and the solution of Myerson (1981). In particular, Proposition 3 does not imply that there is a single posted price. As in Myerson (1981), the object is sold once the virtual valuation turns nonnegative, but in contrast, the virtual valuation may turn nonnegative in several different branches 6 The following weaker assumption is also sufficient: Single-Crossing Virtual Valuations (s, t) E (ψ s 0 ψ t 0) The proposition is formulated in terms of the Monotone Virtual Valuations assumption because (i) it is more familiar from the study of optimal auctions, and relatedly, (ii) if one were to prove an analogous result with multiple buyers, one would need to use Monotone Virtual Valuations, as Single-Crossing Valuations would not be sufficient. 11

12 on the tree, and the valuations at which it turns nonnegative may differ, leading to price discrimination. Consider the following diagram: Figure 1: An Incentive Graph t can mimic s whenever there is a directed path from s to t. (Formally, this means that there is an edge in the corresponding incentive graph not only for every arrow in the above figure, but also for every sequence of arrows; for example, there is an edge from 1 to 4. We have not drawn additional corresponding arrows to avoid cluttering the graph. Because of the omitted arrows, the above figure literally shows the graph Ĝ defined by (1) rather than G. It is Ĝ which is required to be a tree for the tree structure assumption to be satisfied.) In this example the number assigned to a type corresponds to that type s value. An optimal mechanism might charge a price of 3 to types 3, 4, and 7, a price of 2 to types 2, 5, and 6, and not sell the object to types 0 and 1. This would be optimal if the virtual valuation turned positive at vertices 2 and 3. Alternatively, if the virtual valuation was still negative at vertices 2 and 3, the optimal mechanism would impose perfect price discrimination on types 4, 5, 6, and 7, and not serve the other types. A third possibility would be that the virtual valuation would turn positive at vertex 1, in which case there would be a single posted price of 1, which would apply to every type; only type zero would not purchase the object. It is interesting to contrast our analysis with the interpretation given to optimal auctions given by Bulow and Roberts (1989), who interpret each bidder as a market and thereby map the optimal auction problem into the monopolist s problem of third-degree price discrimination. With a single bidder, there is a single corresponding market, and hence no price discrimination. In contrast, in our model, we find something resembling third degree price discrimination with a single buyer or under an alternative interpretation, a single population of buyers. Moreover, the segmentation of different buyer types into different markets with different prices is endogenous, resulting from the seller s optimization. The technical assumption that the incentive graph has a tree structure can be interpreted as saying that each buyer type will ultimately have access to at most one market, although the scope of that market is determined endogenously. 12

13 5 Failure of Allocation Monotonicity and Determinism In this section, we present an example showing that properties of the optimal mechanism may be very different when the incentive graph fails to satisfy tree structure. Let T = {1, 2, 3, 4}. Assume transitivity and antisymmetry. Consider the following diagram, illustrating the mimicking relation: Figure 2: An Incentive Graph Failing to Satisfy Tree Structure Suppose that type s can mimic type t if there is a directed path from s to t. So in particular, type 4 can mimic all other types, type 3 can mimic all types except 4, and types 1 and 2 can only mimic type 0, and type 0 cannot mimic any other type. The above example does not satisfy tree structure because in the graph Ĝ = (T, Ê), there are two paths leading from 0 to 3, one passing through 1, and the other passing through 2. Suppose, moreover that the numbers of the types also represent their values for the object so that for t = 0, 1,..., 4, v t = t. Finally the probabilities of the types satisfy the following relations: π 2 π 4 π 1 π 3 > π 0 = 0 where means is much greater than. One can show that the unique optimal mechanism is given by the following table: t q t p t /3 2/ /3 2/ A mechanism (q, p) is deterministic if q t {0, 1} for all types t, meaning that each type either does or does not receive the object with probability 1. This example violates both determinism and allocation monotonicity. Determinism is violated because both types 1 13

14 and 3 receive the object with probability 2/3. Moreover allocation monotonicity is violated because (2, 3) E but q 2 = 1 > 2/3 = q 3. We summarize the example s significance: Proposition 4 In the traditional model (with a single buyer), if every type can mimic every other type, then the optimal mechanism satisfies allocation monotonicity and determinism. Likewise, under value monotonicity, monotone virtual valuations, and tree structure, the optimal mechanism always satisfies allocation monotonicity and determinism. When these assumptions fail, optimal mechanisms may violate allocation monotonicity and determinism. 6 Value Anti-Monotonicity This section considers the case which is the opposite of value monotonicity. Value Anti-Monotonicity (s, t) E v s v t. In other words, it is a necessary condition for t to mimic s that t have a lower value than s. Proposition 5 Assume value anti-monotonicity. Then the optimal mechanism induces perfect price discrimination; each type is allocated the object and pays a price equal to that type s value. 7 Appendix Proposition 1 Fix an allocation q = (q t : t T ), and let p = (p t : t T ) be the revenue maximizing payment scheme implementing q (assuming that q is implemented by some payment scheme, which we will verify below). Individual rationality implies that p 0 0 (recall that we assume q 0 = 0). On the other hand, if p 0 > 0, since the zero type cannot mimic any other types, it would be possible to reduce p 0, thereby increasing expected revenue without violating the incentive compatibility or individual rationality constraints. Define l(t) := {s T : (s, t) E }. (E is defined in Section 4). Next we argue that for all t T \ 0, p t must satisfy: v t q t p t = max{v t q s p s : (s, t) E} (13) Incentive compatibility implies that v t q t p t max{v t q s p s : (s, t) E}. Assume for contradiction that v t q t p t > max{v t q s p s : (s, t) E}. Then it would be possible to increase p t, raising expected revenue without violating any of t s incentive constraints or individual rationality constraint (because (0, t) E and p 0 = 0). Incentive constraints for 14

15 other types would only become easier to satisfy. This contradicts the assumption that p is the revenue maximizing payment scheme implementing q, establishing (13). Above we have assumed that there exists a payment scheme (and indeed, a revenue maximizing payment scheme) implementing q. We now verify this fact. It follows from acyclicity, (s, t) E l(s) < l(t). This implies that in the right hand side of (13), all terms of the form p s are payments for types s with l(s) < l(t). This along with p 0 = 0, implies that there exists a unique solution to the set of equations (13) (where we have one such equation for each t T ). Moreover (13) and the fact that for all t T \ 0, (0, t) E immediately implies that at this solution all incentive constraints and individual rationality constraints are satisfied. Finally, we prove that p t satisfies (3) by induction on l(t). If l(t) = 0, then t = 0. We have already established that p 0 = 0, which is equivalent to (3) for t = 0 because v 0 = 0. Next suppose that (3) holds for all t with l(t) k for k > 0. We prove that (3) holds when l(t) = k. v t q t p t = max{v t q s p s : (s, t) E} = max{(v t v s )q s + v s q s p s : (s, t) E} η(τ) = max{(v t v s )q s + max{ (v τi v τi 1 ) q τi 1 : τ P (s)} : (s, t) E} i=1 η(τ) = max{ (v τi v τi 1 ) q τi 1 : τ P (t)} i=1 where the first equality follows from (13), and the third inequality follows from the inductive hypothesis, and the fact that by acyclicity, (s, t) E l(s) < l(t). Proposition 2 Consider a cycle τ = (t 0, t 1,..., t n ) in G with t 0 = t n. Because E is irreflexive, t 0 t 1. We may also choose the cycle so that for all i {2,..., n 1}, t 0 t i. Choose a valuation so that v t0 = 1 and for all i {1,..., n 1}, v ti = 2. Choose an allocation q so that q t0 = 1 and for all i {1,..., n 1}, q ti = 0. Assume for contradiction that q is incentive compatible. Then there exists a payment profile p such that for all i {2,..., n 1} v ti q ti p ti v ti 1 q ti 1 p ti 1 Since q ti = 0 for i {1,..., n 1}, this is equivalent to: p tn 1 p tn 2 p t1, 15

16 which, in turn, implies that p t1 p tn 1 (14) Incentive compatibility implies that: p t1 = v t1 q t1 p t1 v t1 q t0 p t0 = 2 p t0 1 p t0 = v t0 q t0 p t0 v t0 q tn 1 p tn 1 = p tn 1 Together these two inequalities imply p t1 2 p t0 > 1 p t0 p tn 1. However, this contradicts (14). Lemma 1 We begin by arguing that: [(s, t) E and qt < qs] v t qt p t > v t qs p s (15) Assume for contradiction that (15) is violated for some pair (s, t). Incentive compatibility then implies that p t < p s. Now assume for contradiction that v t qt p t = v t qs p s. Then redefine qt := qs and p t := p s. This will increase expected revenue without violating any incentive or individual rationality constraints; in particular, transitivity implies that each type has (weakly) fewer deviations (in terms of price-probability pairs) than before the redefinition. However this contradicts the optimality of the mechanism that we started off with, establishing (15). Next, we argue that the allocation in any optimal mechanism (q, p ) satisfies allocation monotonicity. Acyclicity (a consequence of tree structure) and value monotonicity imply that in order to establish allocation monotonicity, it is sufficient show that: (s, t) Ê q s qt (16) Assume for contradiction that (16) is violated for some pair (s, t). Consider any r T with (r, t) E. Acyclicity implies that there is a path from r to t in Ĝ.7 Tree structure implies that any path in Ĝ from r to t must pass through s. Transitivity then implies that (r, s) E. Maintaining the assumption that (s, t) violates (16), assume for contradiction that: v t qt p t = v t qr p r. (17) 7 In particular, acyclicity implies that there is a maximal length path from r to t in G. Any such maximal length path must also be a path in Ĝ. 16

17 (15) implies that qr qt < qs. We have: v t (qs qr) v s (qs qr) p s p r, (18) where the first inequality follows from value monotonicity, and the second inequality follows from incentive compatibility and the fact that (r, s) E. Incentive compatibility, (17) and (18) then imply v t qt p t = v t qs p s, contradicting (16). This implies that (17) cannot hold, which means that for all r T with (r, t) E, v t qt p t > v t qr p r. But this implies that it would be possible to increase p t, increasing expected revenue, without violating any of t s incentive constraints or t s individual rationality constraint (because we can set r = 0), and this would only make the incentive constraints of all other types easier to satisfy. However, this contradicts the optimality of (q, p ), establishing (16) and hence allocation monotonicity. Next we argue that η(τ) (v s v ϕ(s) )q ϕ(s) = max{ (v τi v τi 1 )q τi 1 : τ P (t)} (19) s T :(s,t) E i=1 Observe that transitivity implies that E = E. Recursively, define ϕ 0 (t) := t and ϕ n (t) = ϕ(ϕ n 1 (t)). Tree structure implies that for all s T with (s, t) E, there exists some number k such that s = ϕ k (t). 89 In particular, 0 = ϕ n (t) for some number n. For the path τ = (ϕ n (t), ϕ n 1 (t),..., ϕ 0 (t)) P (t), the left-hand side of (19) is equal to η( τ) i=1 (v τi v τi 1 )q τi 1 It follows that the left-hand side of (19) is less than or equal to the right-hand side. We now prove the other inequality in (19) by induction on l(t) := {s T : (s, t) E }. When l(t) = 0 (or equivalently, t = 0), this inequality holds vacuously. Next consider t with l(t) = h for some h > 0. Choose s with (s, t) E. As explained above, s = ϕ k (t) for 8 If s = t, then s = ϕ 0 (t). Otherwise, noting that by transitivity, (s, t) E. Find a maximal length path from s to t in G. This path is also a path in Ĝ. That s = ϕk (t) follows from the fact that for every r T, (ϕ(r), r) is the unique edge entering into r in Ĝ. 9 k must be chosen so that k n for n such that ϕ n (t) = 0 since otherwise ϕ k (t) is not defined. 17

18 some number k > 0. Then observe that: η(τ) (v t v s )q s + max{ (v τi v τi 1 )q τi 1 : τ P (r)} (v t v s )q s + = = i=1 k (v ϕ i 1 (t) v ϕ i (t))q s + i=1 r T :(r,s) E (v r v ϕ(r) )q ϕ(r) k (v ϕ i 1 (t) v ϕ i (t))q ϕ i (t) + i=1 s T :(s,t) E (v s v ϕ(s) )q ϕ(s) r T :(r,s) E (v r v ϕ(r) )q ϕ(r) r T :(r,s) E (v r v ϕ(r) )q ϕ(r) The first inequality follows from the inductive hypothesis, the second inequality follows allocation monotonicity, and the last equality follows from acyclicity the fact, discussed above that (r, t) E r = ϕ j (t) for some number j. Since the right-hand side of (19) is equal to the first term in the above derivation form some choice s T with (s, t) E, it follows that the left-hand side of (19) is greater than or equal to the right-hand side. This establishes (19). Lemma 1 now follows from Corollary 1. Proposition 3 Let Program M be the program (2). Then Program M is derived from Program M by replacing the edge set E by the edge set E = Ê and program M is the program derived from Program M be replacing the edge set E by the edge set E = E. We have G = (T, E), and define G = (T, E ) and G = (T, E ). Observe that E E E. This implies that the feasible set for Program M is contained in the feasible set for Program M, which is in turn contained in the feasible set for Program M. All three programs have the same objective. It follows that if the mechanism given by (12) is optimal in both M and M, then it is also optimal in M. Define Ê and Ê via (1) relative to E and E respectively. It is straightforward to show that Ê = Ê = Ê. 10 This implies that because G satisfies tree structure, G and G also satisfy tree structure. Moreover, it also implies that if ϕ (t) is the unique element s such 10 The proof is as follows. It is immediate from (1) that Ê Ê. On the other hand, if (s, t) Ê, then for all u T \ {s, t}, if (s, u) Ê E, then (t, u) E, which implies that (t, u) Ê. This implies that (s, t) Ê. So Ê = Ê. Next choose (s, t) Ê. Then (1) and acyclicity of E implies that (s, t) is the unique path from s to t in G. This implies that (s, t) E and for all u T \ {s, u}, (s, u) E (u, t) E. So (s, t) Ê, and Ê Ê. Finally choose (s, t) Ê. This implies that there is a path from s to t in G, but for all u if there is a path from s to u in G, then there is no path from u to t in G. This implies that in fact (s, t) E, and for all u T \ {s, t}, (s, u) E (u, t) E. So (s, t) Ê. This implies that Ê = Ê. 18

19 that (s, t) Ê, and ϕ (t) is the unique element s such that (s, t) Ê, then ϕ = ϕ = ϕ. Next observe that because E = Ê is a polytree, there is a unique path in G from s to t. Note that Proposition 1 does not directly apply to the optimal mechanism problem with incentive graph G because Proposition 1 made use of the assumption that (0, t) E for all t T \ 0. However, it is straightforward to modify the proof so that the conclusion of Proposition 1 applies to G. 11 Given this modification, Corollary 1 also applies to G. However, in G, the unique path from 0 to t is (ϕ n (t), ϕ n 1 (t),..., ϕ 0 (t)) for n such that ϕ n (t) = 0. (The notation ϕ k (t) was defined in the proof of Lemma 1). It now follows from Proposition 1 that Lemma 2 (q, p ) is the optimal mechanism in program M if and only if (q, p ) maximizes p tπ t subject to 0 q t 1 and for all t T. p t = v t q t s T :(s,t) E (v s v ϕ(s) )q ϕ(s), (20) The equality in (20) uses the fact that because G satisfies tree structure (s, t) E s = ϕ k (t) for some number k n. Let (q, p ) = (qt, p t : t T ) be an optimal mechanism either in Program M or in Program M. Observe that: p t π t = qt v t π t (qt v t p t )π t (21) = qt v t π t π t (v s v ϕ(s) )qϕ(s), s T :(s,t) E where in M, the second equality follows from (20) and the fact that E = E and in M, the second equality follows from Lemma 1 applied to G, using the fact the graph G is transitive and that E = E. 11 The modification is as follows. First note that for every t T \ 0, there is a path from 0 to t in G. To see this, note that (0, t) E, and any maximal length path in G from 0 to t is also a path in G. This implies that it is still the case in G that 0 is the unique type t such that l(t) = 0. Next, in arguing that any optimal mechanism satisfied (13), we used the fact that if for some type t, the equality in (13) were not satisfied, then we could increase p t without violating the individual rationality constraint using the fact that (0, t) E. We would have to replace this argument by a straightforward argument by induction on l(t) (valuated within G ) using value monotonicity. Moreover in arguing that the unique payment scheme satisfying all equations of the form (13) is individually rational, a similar inductive argument could be used. 19

20 Next, observe that: π t (v s v ϕ(s) )qϕ(s) = (v s v ϕ(s) )qϕ(s) π t s T :(s,t) E (s,t) T T :(s,t) E = (v s v ϕ(s) )qϕ(s) π t (22) s T { :(s,t) E } = s v ϕ(s) )qϕ(s) s T(v = s T(v s v ϕ(s) )q ϕ(s) Π s :(s,t) E π t (21) and (22) imply: p t π t = qt v t π t s v ϕ(s) )q ϕ(s) Π s s T(v = qt v t π t (v s v t )qt Π s = (v t = ψ t q t π t s ϕ 1 (t) s ϕ 1 (t) (v s v t ) Π s )qt π t π t Lemma 2 now implies that the optimal allocation q in Program M is the one given in (12). Monotone virtual valuations imply that the allocation in (12) satisfies allocation monotonicity, and then Lemma 1 and the fact that G is transitive imply that this allocation is also optimal in Program M. Next I argue that the payment scheme defined by (8) (or equivalently (20)) is the same as the one defined in (12). If ψ t < 0, then monotone virtual valuations immediately implies that type t s payment is zero according to both formulas. If ψ t > 0, then let r T be a minimal element of the set {s T : (s, t) E, ψ s 0}, where minimality is assessed with respect to the partial order whose graph is given by E. Tree structure implies that there is a unique such minimal element and value monotonicity implies that v r = min{v s : (s, t) E, ψ s 0}, and for all s T with (s, r) E, qs = 0. Monotone virtual valuations imply that for all s T, with (r, s) E, qs = 1. Moreover, tree structure implies that the set {s T : (s, t) E } is linearly ordered by E (considered as an ordering relation), and that the E -minimal element s of {s T : (r, s) E, (s, t) E } is such 20

21 that ϕ(s ) = r. It follows that: v t qt (v s v ϕ(s) )qϕ(s) = v t s T :(s,t) E = v r, {s T :(r,s) E,(s,t) E } (v s v ϕ(s) )q ϕ(s) implying that (8) (or equivalently (20)) and (12) lead to the same value for p t, and hence the mechanism in (12) is optimal in both Programs M and M. As explained above, this implies that this mechanism is optimal in Program M as well. References Ben-Porath, E., and B. Lipman (2008): Unpublished Manuscript. Implementation and Partial Provability, Bull, J., and J. Watson (2007): Hard Evidence and Mechanism Design, Games and Economic Behavior, 58, Bulow, J., and P. Klemperer (1996): Auctions vs. Negotiations, American Economic Review, 86, Bulow, J., and J. Roberts (1989): The Simple Economics of Optimal Auctions, Journal of Political Economy, 97, Celik, G. (2006): Mechanism Design with Weaker Incentive Compatibility Constraints, Games and Economic Behavior, 56, Deneckere, R., and S. Severenov (2008): Mechanism Design with Partial State Verifiability, Games and Economic Behavior, 64, Forges, F., and F. Koessler (2005): Communication Equilibria with Partially Verifiable Types, Journal of Mathematical Economics, 41, Glazer, J., and A. Rubinstein (2004): On the Optimal Rules of Persuasion, Econometrica, 72, (2006): A Study in the Pragmatics of Persuasion: A Game Theoretical Approach, Theoretical Economics, 1, Green, J., and J. Laffont (1986): Partially Verifiable Information and Mechanism Design, Review of Economic Studies, 53,

22 Kartik, N., and O. Tercieux (2009): Implementation with Evidence: Complete Information, Unpublished Manuscript. Lipman, B., and D. Seppi (1995): Robust Inference in Communication Games with Partial Provability, Journal of Economic Theory, 66, Milgrom, P., and J. Roberts (1986): Relying on Information of Interested Parties, Rand Journal of Economics, 17, Moore, J. (1984): Global Incentive Constraints in Auction Design, Econometrica, 52, Myerson, R. (1981): Optimal Auction Design, Mathematics of Operations Research, 6, Rochet, J.-C. (1987): A Necessary and Sufficient Condition for Rationalizability in a Quasi-Linear Context, Journal of Mathematical Economics, 16, Severenov, S., and R. Deneckere (2006): Screening when Some Agents are Nonstrategic: Does a Monopoly Need to Exclude?, RAND Journal of Economics, 37, Sher, I. (2008): Credibility and Determinism in a Game of Persuasion, Unpublished Manuscript. (2009): Persuasion and Limited Communication, Unpublished Manuscript. Sher, I., and R. Vohra (2010): Optimal Auctions on Incentive Graphs, Unpublished Manuscript. Shin, H. (1994a): The Burden of Proof in a Game of Persuasion, Journal of Economic Theory, 64, (1994b): News Management and the Value of Firms, Rand Journal of Economics, 25, Singh, N., and D. Wittman (2001): Implementation with Partial Verification, Review of Economic Design, 6,