Delegation in Veto Bargaining
DDelegation in Veto Bargaining ∗ Navin Kartik † Andreas Kleiner ‡ Richard Van Weelden § June 15, 2020
Abstract
A proposer requires the approval of a veto player to change a status quo. Preferencesare single peaked. Proposer is uncertain about Vetoer’s ideal point. We study Proposer’soptimal mechanism without transfers. Vetoer is given a menu, or a delegation set, tochoose from. The optimal delegation set balances the extent of Proposer’s compromisewith the risk of a veto. Under reasonable conditions, “full delegation” is optimal: Vetoercan choose any action between the status quo and Proposer’s ideal action. This outcomelargely nullifies Proposer’s bargaining power; Vetoer frequently obtains her ideal point,and there is Pareto efficiency despite asymmetric information. More generally, we identifywhen “interval delegation” is optimal. Optimal interval delegation can be a Pareto im-provement over cheap talk. We derive comparative statics. Vetoer receives less discretionwhen preferences are more likely to be aligned, by contrast to expertise-based delegation.Methodologically, our analysis handles stochastic mechanisms. ∗ We thank Nageeb Ali, Wiola Dziuda, Alex Frankel, Sanjeev Goyal, Marina Halac, Elliot Lipnowski, MalleshPai, Mike Ting, and various seminar and conference audiences for helpful comments. Bruno Furtado providedexcellent research assistance. † Department of Economics, Columbia University. Email: [email protected] . ‡ Department of Economics, Arizona State University. Email: [email protected] § Department of Economics, University of Pittsburgh. Email: [email protected] . a r X i v : . [ ec on . T H ] J un . Introduction Motivation.
There are numerous situations in which one agent or group can make propos-als but another must approve them. Legislatures (e.g., U.S. Congress) send bills to executives(e.g., the President), who can veto them. Governmental legislation can be struck down as un-constitutional by the judiciary. Prosecutors choose which charges to bring against defendants,but judges and juries decide whether to convict. A real-estate agent can recommend a houseto his client, but the client must decide to put in an offer; similarly, a search committee canput forward a candidate, but the organization decides whether to hire her.Romer and Rosenthal (1978) present a seminal analysis of such veto bargaining . Theirframework is one of complete information in which a proposer (Congress, government, pros-ecutor, salesperson, search committee) makes a take-it-or-leave-it proposal to a veto player(President, judiciary, judge/jury, customer, organization). Preferences are single peaked. Thatis, rather than “dividing a dollar”, the negotiating parties share some preference alignment.Such preferences are plausible in the contexts mentioned above.Our paper studies veto bargaining with incomplete information. The proposer is uncer-tain about the veto player’s preferences—specifically, which proposals would actually getvetoed. Previous scholars have emphasized this feature’s importance; see Cameron and Mc-Carty’s (2004) survey. But to our knowledge, our paper is the first that takes a general ap-proach to the issue. We do not assume the proposer is restricted to making a single proposal(e.g., Romer and Rosenthal, 1979), nor do we fix any particular negotiating protocol (e.g., oneround of cheap talk in Matthews, 1989; Forges and Renault, 2020). Instead, we consider thepossible outcomes across all possible protocols by taking a mechanism design approach. Ourfocus is on identifying the proposer’s optimum.There are at least two reasons this mechanism design approach is of interest. First, it iden-tifies an upper bound on the proposer’s welfare. Second, as is standard in settings withouttransfers, any (deterministic) mechanism is readily interpreted and implemented as delega-tion . That is, the proposer simply offers a menu of options; the veto player can select anyone or reject them all. Menus or delegation sets are observed in practice in some applicationsof our model. Salespeople show customers subsets of products, and search committees putforward multiple candidates for their organization to choose among. In politics, a bill autho-rizing at most $ x of spending effectively offers an executive who controls the implementingbureaucracy the choice of any spending level in the interval [0 , x ] . Bills can also grant more orless discretion of how to allocate a given level of spending, as encapsulated by former SenatorRuss Feingold in the context of the recent U.S. coronavirus stimulus package: “Congress has1o decide how much discretion it wants to delegate to executive branch officials.” (Washing-ton Post, March 22, 2020.) Main results.
We formally study a one-dimensional environment. There is status quo policyor action, , that obtains if there is a veto. There are two agents. Proposer’s ideal action is .Vetoer’s ideal action, v , is her private information. We assume Vetoer preferences are repre-sented by a quadratic loss function, but allow Proposer to have any concave utility function. Our first result (Proposition 1) identifies conditions under which it is optimal for Proposerto simply let Vetoer choose her preferred action in the interval [0 , . We call this full delegation ,because Proposer only excludes options that are, from his point of view, dominated by simplyoffering his ideal action no matter Vetoer’s ideal point. Intuitively, full delegation is optimalwhen the specter of a veto looms large; in particular, it is sufficient that the density of Vetoer’sideal point is decreasing on the unit interval. Optimality of full delegation is quite striking:despite Proposer having considerable bargaining and commitment power, it is Vetoer whofrequently gets her first best. This is a telling manifestation of private information’s conse-quences. Unlike in most other settings that confer information rents, however, full delegationimplies ex-post Pareto efficiency.The opposite of full delegation is no compromise : Proposer only offers his ideal action, .Of course, Vetoer can veto and choose the status quo . Proposition 2 gives conditions foroptimality of no compromise. It is sufficient, for example, that Proposer has a linear lossfunction—so, in the relevant region of actions, [0 , , he only cares about the mean action—and the density of Vetoer’s ideal point is increasing in this region. This case juxtaposes nicelyagainst the aforementioned decreasing-density condition for full delegation.Both full delegation and no compromise are boundary cases of interval delegation : Proposeroffers a menu of the form [ c, . Interval delegation is interesting for multiple reasons. Amongthem are that such delegation sets are simple to interpret and implement, and they turn out tobe tractable for comparative statics. Proposition 3 provides conditions under which intervaldelegation is optimal. These are met, in particular, when Proposer has a linear or quadraticloss function and Vetoer’s ideal point distribution is logconcave (Corollary 3).We show that, under reasonable conditions, optimal interval delegation yields a Paretoimprovement over singleton proposals, even when cheap talk is allowed and full delegation Proposer’s risk attitude is important because he faces uncertainty about the final action. Among determin-istic mechanisms, Vetoer’s risk attitude is irrelevant, so quadratic loss entails no restriction beyond symmetryaround the ideal point.
2s not optimal (Proposition 5). We trace the intuition to Proposer being more willing to com-promise when he can offer Vetoer an interval of options rather than only singletons.We develop two comparative statics, restricting attention to interval delegation — eitherjustified by optimality or otherwise. First, what happens when Proposer becomes more riskaverse? Proposition 4(i) establishes that Vetoer is given more discretion: the optimal thresholdin the interval delegation set (i.e., that denoted c above) decreases. Intuitively, Proposer offersa larger set of options to mitigate the risk of a veto. Second, what about when Vetoer becomesmore ex-ante aligned with Proposer? Formally, we consider right shifts in Vetoer’s ideal pointdistribution, in the sense of likelihood ratio dominance. Proposition 4(ii) establishes that dis-cretion decreases: the optimal interval delegation threshold increases. Intuitively, this is be-cause Proposer is less concerned that a veto will occur. Although these comparative staticsappear natural, it is the structure of interval delegation that allows us to establish them. Contrast with expertise-based delegation.
The second comparative static mentioned abovecontrasts with a key theme of the expertise-based delegation literature following Holmstr ¨om(1977, 1984). In that literature, an agent is given discretion over actions because her private in-formation is valuable to the principal; the principal limits the degree of discretion because ofpreference misalignment. One version of the so-called Ally Principle says that a more alignedagent receives more discretion. Holmstr ¨om (1984) establishes its validity under reasonablygeneral conditions, so long as delegation sets take the form of intervals. In our setting, thereason Proposer gives Vetoer discretion is fundamentally different from that in Holmstr ¨om:it is not to benefit from Vetoer’s expertise; rather, Proposer trades off the risk of a veto withthe extent of compromise. (In jargon, our delegator has state-independent preferences, bycontrast to the state-dependent preferences in most of the literature following Holmstr ¨om.)Hence we find less discretion emerging when there is, in a suitable sense, more ex-ante pref-erence alignment.
Methodology.
We hope some readers will find our analysis interesting on a methodologicallevel. While it is convenient and economically insightful to describe our substantive resultsin terms of optimal delegation sets, the formal problem we study is one of mechanism de-sign without transfers. Our analytical methodology builds on the infinite-dimensional Lan-grangian approach advanced by Amador, Werning, and Angeletos (2006) and Amador andBagwell (2013). Unlike these authors and many others, including the important contributions The comparative static may fail absent interval delegation (e.g., Alonso and Matouschek, 2008).
3y Melumad and Shibano (1991) and Alonso and Matouschek (2008), we also cover stochasticmechanisms. That is, we allow for mechanisms in which Vetoer may choose among lotteriesover actions. We view such mechanisms as not only theoretically important, but also relevantin applications. For instance, in the hiring application mentioned earlier, the search committee(Proposer) can offer the organization (Vetoer) an option of hiring “the best available candidatewith at least five years’ work experience in that country”. Both parties would view this optionas a lottery over some subset of candidates.Stochastic mechanisms can sometimes be optimal in our framework. Nevertheless, weestablish that our sufficient conditions for full delegation, no compromise, and interval dele-gation (Propositions 1, 2, and 3) ensure optimality of these (deterministic) mechanisms evenamong stochastic mechanisms. Furthermore, by permitting stochastic mechanisms, our suffi-cient conditions are shown to also be necessary for a class of Proposer’s utility functions thatinclude linear and quadratic loss. Our approach to handling stochastic mechanisms shouldbe useful in other delegation problems.Recently, Kolotilin and Zapechelnyuk (2019) have introduced balanced delegation problems ,which are delegation problems in which certain extreme actions or outside options must beincluded. Our setting fits into their general framework, as one can assume the status quomust be part of the delegation set. Kolotilin and Zapechelnyuk derive a general equivalencebetween such problems and monotone Bayesian persuasion problems. More concretely, theyshow how some results from the latter literature (e.g., Kolotilin, 2018; Dworczak and Martini,2019) can be brought to bear on “linear” balanced delegation problems. Our approach ofdirectly studying the delegation problem is complementary and has some advantages. First,it permits insights absent said linearity: this is most evident in our full delegation result.Second, we believe it provides some more transparent economic intuitions. Third, unlikeKolotilin and Zapechelnyuk (2019), we can address stochastic mechanisms and necessity ofour sufficient conditions. At a broader level, note that by contrast to us, Kolotilin and Za-pechelnyuk (2019) highlight applications concerning expertise-based delegation (i.e., withstate-dependent delegator preferences). Zapechelnyuk (2019) applies their methodology toa quality certification problem that, he shows, maps into a delegation problem in which the A qualification is appropriate: both Amador et al. (2006) and Amador and Bagwell (2013) allow for moneyburning, identifying conditions under which optimal mechanisms do not employ that instrument; see Amador,Bagwell, and Frankel (2018) as well. Stochastic mechanisms are equivalent to money burning for certain prefer-ence specifications, but in general they are not equivalent. Ambrus and Egorov (2017) discuss settings in whichmoney burning can be optimal. This linearity requires that the utilities of Proposer and (all types of) Vetoer, viewed as a function of theaction, have the same curvature.
Outline.
The rest of the paper proceeds as follows. Section 2 presents our model. Section 3contains our main results on the conditions for optimality of full delegation, no compromise,and, more broadly, interval delegation. Section 4 develops comparative statics and makescomparisons with other mechanisms. Section 5 discusses some applications. Section 6 con-cludes. All proofs are in the appendices.
2. Model
We consider a classic bargaining problem between two players, a proposer (he) and a vetoplayer (she), who jointly determine a policy outcome or action a ∈ R . In a manner elaboratedbelow, Proposer makes a proposal that Vetoer can either accept or reject. If Vetoer rejects, astatus-quo action is preserved; we normalize the status quo to .We assume both players have single-peaked utilities. Proposer’s utility is u ( a ) that is con-cave, maximized uniquely at a = 1 (essentially a normalization), and twice differentiable atall a (cid:54) = 1 . Unless indicated explicitly, we use ‘concave’, ‘increasing’, etc., to mean ‘weaklyconcave’, ‘weakly increasing’, etc. We will sometimes invoke a restriction to the followingsubclass of Proposer preferences, which stipulates a convex combination of the widely-usedlinear and quadratic loss functions.
Condition LQ.
For some γ ∈ [0 , , u ( a ) = − (1 − γ ) | − a | − γ (1 − a ) . Vetoer’s utility is represented by − l ( | v − a | ) , where l ( · ) is strictly increasing. So her utilityis symmetric around the unique ideal point v . For tractability, we assume l ( | v − a | ) = ( v − a ) .A subset of our results will rely only on Vetoer’s ordinal preferences, for which the choiceof quadratic loss entails no loss of generality given that Vetoer’s utility is symmetric around Permitting a point of nondifferentiability allows the linear loss function u ( a ) = −| − a | . When we write u (cid:48) (1) subsequently, it refers to the left-derivative when u is not differentiable at 1. v is Vetoer’s private information. We accordinglyrefer to v as Vetoer’s type . It is drawn from a cumulative distribution F whose support is aninterval [ v, v ] , where we permit v = −∞ and/or v = ∞ . We assume F admits a continuouslydifferentiable density f , and that f ( · ) > on [0 , . All aspects of the environment except thetype v are common knowledge. If v were common knowledge, this model would reduce tothat of Romer and Rosenthal (1978).Naturally, it is in Proposer’s interests to elicit information from Vetoer about v . For exam-ple, they might engage in cheap talk communication (Matthews, 1989), possibly over multiplerounds, or Proposer might make sequential proposals, and so on. To circumvent issues aboutexactly how the bargaining ensues, we take a mechanism design approach. Following the rev-elation principle, we consider direct revelation mechanisms, hereafter simply mechanisms.A deterministic mechanism is described by a real-valued function α ( v ) , which specifies theaction when Vetoer’s type is v , and must satisfy the usual incentive compatibility (IC) andindividual rationality (IR) conditions. IC requires that each type v prefers α ( v ) to α ( v (cid:48) ) for any v (cid:48) (cid:54) = v ; IR requires that each type v prefers α ( v ) to the status quo . Notice that any determin-istic mechanism is equivalent to the Proposer offering a (closed) menu or delegation set A ⊆ R ,and Vetoer choosing an action from A ∪ { } . We will also consider the more general classof stochastic mechanisms , which specify probability distributions over actions for each Vetoertype, with analogous IC and IR constraints to those aforementioned. Stochastic mechanismsare theoretically important because the revelation principle does not justify focussing only ondeterministic mechanisms. As noted in the introduction, they may also be relevant for appli-cations. A notable contribution of this paper is to establish conditions under which, despitethe absence of transfers, stochastic mechanisms cannot improve upon deterministic ones. We highlight that our model is one of private values: Vetoer’s type does not directly affectProposer’s preferences. This is by way of contrast with the delegation literature initiated byHolmstr ¨om (1984), in which a principal gives discretion to an agent because of the agent’sexpertise, i.e., because they have interdependent preferences. We could extend our modeland analysis to incorporate this expertise-based delegation or discretion aspect, but one ofour main themes is that discretion will emerge even when that is absent. Remark 2 below explains why stochastic mechanisms can be optimal; Example E.1 in Appendix E elaborates.Alonso and Matouschek (2008, p. 281) provide a related example in their framework without a veto option; seealso Kov´aˇc and Mylovanov (2009, Section 4).
6s mentioned in the introduction, the mechanism design approach we take can be viewedas identifying an upper bound on Proposer’s welfare. That said, as also mentioned there, wefind the implementation via delegation sets quite realistic in various contexts.
We now formally define Proposer’s problem. Let M ( R ) denote the set of Borel probabil-ity distributions on R , and M ( R ) be the subset of distributions with finite expectation andfinite variance. Denote by δ a the degenerate distribution that puts probability 1 on action a . Astochastic mechanism—or simply a mechanism without qualification—is a measurable func-tion m : [ v, v ] → M ( R ) , with m ( v ) being the probability distribution over actions for type v . To reduce notation, for any deterministic mechanism α : [ v, v ] → R , we also denote themechanism v (cid:55)→ δ α ( v ) by α . For any integrable function g : A → R , let E m ( v ) [ g ( a )] denote theexpectation of g ( a ) when a has distribution m ( v ) . We only consider mechanisms m for which v (cid:55)→ E m ( v ) [ a ] is integrable. Define the subset of mechanisms S := (cid:8) m : [ v, v ] → M ( R ) | m (0) = δ and ∀ v < v (cid:48) : E m ( v ) [ a ] ≤ E m ( v (cid:48) ) [ a ] (cid:9) . That is, S consists of mechanisms in which type gets the status quo and a higher type re-ceives a higher expected action. The first requirement is implied by IR, since Vetoer can al-ways choose the status quo. The second is implied by IC, since Vetoer’s utility − ( v − a ) isequivalently represented by av − a / ; singlecrossing difference in ( a, v ) yields monotonicityof E m ( v ) [ a ] in v from standard arguments (elaborated in fn. 9 below).Proposer’s problem is: max m ∈S (cid:90) E m ( v ) [ u ( a )]d F ( v ) (P)s.t. E m ( v ) (cid:2) av − a / (cid:3) = (cid:90) v E m ( s ) [ a ]d s ∀ v ∈ [ v, v ] . (IC-env)As noted above, it is without loss to restrict attention to mechanisms in S . The constraint(IC-env) captures the additional content of IC, beyond monotonicity, via an analog of thestandard envelope formula. Note that since IC requires that no type prefer type ’s lottery We endow M ( R ) with the topology of weak convergence and the corresponding Borel σ -algebra. There is no loss in restricting attention to M ( R ) instead of M ( R ) because no type would choose a lotterywith infinite mean or variance, given that the status quo is available. Formally, using quadratic utility, IC requires ∀ v, v (cid:48) , E m ( v ) [ av − a / ≥ E m ( v (cid:48) ) [ av − a / , and IR requires ∀ v , E m ( v ) [ av − a / ≥ . An IC mechanism m thus satisfies IR if and only if m (0) = δ . It follows that m satisfies IC ’s IR constraint requires that it receive action (captured in S ), everytype’s IR constraint is implied by type ’s. An optimal mechanism is a solution to problem (P).If we restricted attention to deterministic mechanisms, the analogous problem for Pro-poser would be: max α ∈A (cid:90) u ( α ( v )) d F ( v ) (D)s.t. vα ( v ) − α ( v ) / (cid:90) v α ( s )d s, where A := { α : [ v, v ] → R | α (0) = 0 and α is increasing } . Any deterministic mechanism α that is IC has a corresponding (closed) delegation set A α := (cid:83) v α ( v ) . Conversely, any delegation set A has a corresponding deterministic mecha-nism α A where α A ( v ) is the action in A ∪ { } that type v prefers the most (with ties brokenin favor of Proposer). Note that α A satisfies IC and IR. While our formal analysis works withmechanisms, it is easier and more economically intuitive to describe our main results, whichconcern certain deterministic mechanisms, using delegation sets.We emphasize some terminology: an optimal deterministic mechanism (or an optimal dele-gation set ) is a solution to problem (D). But when we say that a deterministic mechanism (ordelegation set) is optimal , we mean that it solves problem (P), i.e., no stochastic mechanism canstrictly improve on it. Remark . We capture veto power via a standard interim IR constraint. An alternative, as inCompte and Jehiel (2009), would be to allow Vetoer to exercise her veto even after the mech-anism determines an action. This is stronger than just ex-post IR because it also strengthensthe IC constraint: when type v mimics type v (cid:48) , v may veto a different set of allocations than v (cid:48) would, and so the action distribution that v evaluates the deviation with is not m ( v (cid:48) ) . Whichform of veto power is conceptually appropriate depends on the application. But any IC andIR deterministic mechanism also satisfies the ex-post veto constraint. Hence, the sufficientconditions we provide below for optimality of delegation sets would remain sufficient. (cid:5) and IR if and only if m ∈ S and the envelope formula (IC-env) holds. To confirm this, let V ( v ) := E m ( v ) [ av − a / .Mechanism m is IC if and only if V ( v ) = max v (cid:48) E m ( v (cid:48) ) [ av − a / , which holds if and only if V is convex and V ( v ) = V (0) + (cid:82) v E m ( s ) [ a ]d s (Milgrom and Segal, 2002). Consequently, m is IC and IR if and only if E m ( v ) [ a ] isincreasing in v , (IC-env) holds, and m (0) = δ . .2. Preliminary Observations Consider delegation sets. Notice first that there is no loss for Proposer in including hisideal action in the delegation set: for any Vetoer type v , either it does not affect the chosenaction, or it results in a preferable action. Next, there is no loss for Proposer in excludingactions outside [0 , : shrinking a delegation set A that contains to A ∩ [0 , only results ineach type choosing an action closer to . We have: Lemma 1.
There is an optimal delegation set A satisfying ∈ A ⊆ [0 , . It would also be without loss to assume that a delegation set contains the status quo, . Wedon’t do so, however, because it is convenient to sometimes describe optimal delegation setswithout including .For any a ∈ (0 , , the delegation set A = [ a, strictly dominates the singleton { a } because A results in preferable actions for Proposer when v > a . This simple observation highlightsthe significance of giving Vetoer discretion, despite our model shutting down the expertise-based rationale that the literature initiated by Holmstr ¨om (1984) has focused on.While Proposer always wants to include action in the delegation set, he faces a tradeoffwhen including any action a ∈ (0 , . Allowing Vetoer to choose such an action a reduces theprobability of a veto (or any action less than a ) but also reduces the probability that Vetoerchooses an action even higher than a , which Proposer would prefer to a .
3. Delegation and Optimal Mechanisms
In light of Lemma 1, we refer to the delegation set [0 , as full delegation . Note that full del-egation does impose some constraints on Vetoer. But the constraints are minimal: only actionsoutside the convex hull of the status quo and Proposer’s ideal point are excluded. Given theveto-bargaining institution, an outcome of full delegation starkly captures how Vetoer’s pri-vate information can corrode Proposer’s bargaining or agenda-setting power. Indeed, if thetype distribution’s support is [0 , , then Vetoer gets her first best. Regardless of the support,however, full delegation results in ex-post Pareto efficiency, unlike in most other settings thatconfer information rents. Full delegation thus contrasts sharply with the outcome under com-plete information (Romer and Rosenthal, 1978), in which case Proposer would make Vetoerwith ideal point v ∈ (0 , / indifferent with exercising the veto while getting his own ideal9ction from types v ∈ (1 / , . It also contrasts with the outcome under incomplete informa-tion were Proposer restricted to making a singleton proposal. In that case the proposal wouldlead to a veto by some subinterval of types v ∈ [0 , , hence to ex-post Pareto inefficiency, andall Vetoer types would be weakly worse off, many strictly. It is thus of interest to know when full delegation is optimal. The following quantityconcerning the concavity of Proposer’s utility will play a key role in our analysis: κ := inf a ∈ [0 , − u (cid:48)(cid:48) ( a ) . Proposition 1 (Full delegation) . Full delegation is optimal if κF ( v ) − u (cid:48) ( v ) f ( v ) is increasing on [0 , . (1) Conversely, under Condition LQ, full delegation is optimal only if (1) holds.
Consider sufficiency in Proposition 1. If we had restricted attention to deterministic mech-anisms and assumed that Proposer’s utility is a quadratic loss function, then the result wouldfollow from a result in Alonso and Matouschek (2008), even though their model does nothave a veto constraint and, as such, highlights expertise-based delegation. To make the con-nection, we observe that when u ( a ) = − (1 − a ) , condition (1) is equivalent to Alonso andMatouschek’s “backward bias” being convex on [0 , . Their Proposition 2 then implies thatif { , } is contained in the optimal delegation set, then the interval [0 , is contained in theoptimal delegation set. But recall from Lemma 1 that in choosing among delegation sets, Pro-poser need not offer any action outside [0 , and can offer his ideal point ; moreover, he mayas well offer the status quo . It follows that full delegation is an optimal delegation set.We therefore emphasize that Proposition 1 establishes optimality among more generalProposer preferences and stochastic mechanisms. The proposition directly implies:
Corollary 1.
Full delegation is optimal if the type density is decreasing on [0 , . In particular, it is sufficient that the type density is unimodal with a negative mode. Toobtain intuition for the corollary, consider removing any interval ( a, a ) from a delegation set Note that vetoes can have positive probability even under full delegation; this is the case if and only if
Prob( v < > . In a model without an outside option, Kov´aˇc and Mylovanov (2009) provide sufficient conditions for cer-tain delegation sets to be optimal when stochastic mechanisms are allowed and Proposer has a quadratic lossfunction. that contains [ a, a ] . This change induces Vetoer with type v ∈ ( a, a ) to choose between a and a . Due to her symmetric utility function, Vetoer will choose a when v ∈ (cid:0) a, a + a (cid:1) , which harmsProposer, while Vetoer will choose a when v ∈ (cid:0) a + a , a (cid:1) , which benefits Proposer. When thetype density is decreasing on [ a, a ] , the former possibility is more likely; in fact, the pruneddelegation set induces an action distribution that is second-order stochastically dominated. Since Proposer’s utility is concave, he prefers the original delegation set A . As any (closed)delegation set can be obtained by successively removing open intervals, full delegation is anoptimal delegation set. While this explanation applies among delegation sets, Proposition 1implies that Corollary 1 holds even allowing for stochastic mechanisms.Removing an interval increases the expected action when the type density is increasing,but it also increases the probability of a lower action. Thus, when Proposer is risk averse,it can be optimal to not remove an interval even if the density is increasing on that interval.This explains why condition (1) is weaker than f decreasing on [0 , . In general, removing aninterval is optimal only if the density is increasing quickly relative to Proposer’s risk aversion.This suggests that full delegation is optimal whenever Proposer is sufficiently risk averse.Proposition 1 allows us to formalize the point using the Arrow-Prat (Arrow, 1965; Pratt, 1964)coefficient of absolute risk aversion. Corollary 2.
Full delegation is optimal if Proposer is sufficiently risk averse, i.e., if inf a ∈ [0 , − u (cid:48)(cid:48) ( a ) /u (cid:48) ( a ) is sufficiently large. Consider now necessity in Proposition 1. If Proposer has a linear loss utility, then our pre-ceding discussion explains why a delegation set A containing [ a, a ] ⊆ [0 , should be prunedto A \ ( a, a ) if the type density is increasing on this interval: the expected action increases.Hence, the converse of Corollary 1 holds for linear loss utility. For quadratic loss utility (andwith additional smoothness assumptions), Alonso and Matouschek’s (2008) Proposition 2 im-plies that that a weaker condition, F ( v ) − (1 − v ) f ( v ) increasing on [0 , , is necessary for full Let G X denote the cumulative distribution of the action induced by A , G Y denote that induced by A \ ( a, a ) ,and let a mid = ( a + a ) / . Since F is the distribution of v , it holds that G X ( a ) = G Y ( a ) for a / ∈ ( a, a ] , G X ( a ) = F ( a ) on [ a, a ] , and G Y ( a ) = F ( a mid ) for a ∈ [ a, a ) . Consequently, for any a ∈ [ a, a mid ] , G Y ( a ) ≥ G X ( a ) and (cid:82) a [ G Y ( t ) − G X ( t )] d t ≥ . Furthermore, for a ∈ ( a mid , a ] , (cid:90) a [ G Y ( t ) − G X ( t )] d t = (cid:90) aa [ F ( a mid ) − F ( t )] d t ≥ (cid:90) aa [ F ( a mid ) − F ( t )] d t ≥ , where the last inequality follows from Jensen’s inequality because F is concave on [ a, a ] . We conclude that G X second-order stochastically dominates G Y .If the type density is not decreasing on [ a, a ] , then second-order stochastic dominance does not hold, butProposer is hurt by pruning ( a, a ) if the expression in condition (1) is increasing on [ a, a ] . We note that were Vetoer’s lowest type v ∈ (0 , , contrary to our assumption that v ≤ ,then full delegation—or even the interval [ v, —would never be an optimal delegation set:Proposer would not offer any action below min { v, } . The other extreme from full delegation is no compromise : Proposer makes a take-it-or-leave-it offer of his own ideal action, not offering any other action. Of course, Vetoer can choose thestatus quo as well. When Proposer has a linear loss utility—or, a fortiori , if we had permitted u ( a ) to be convex on [0 , —then no compromise is an optimal delegation set whenever thetype density f is increasing on [0 , . This follows from reversing the previous subsection’ssecond-order stochastic dominance argument for optimality of full delegation when f is de-creasing. But neither is linear loss utility nor convexity of F on [0 , required for optimalityof no compromise. Proposition 2.
Assume Condition LQ. No compromise is optimal if and only if ( u (cid:48) (1) + κ (1 − t )) F ( t ) − F (1 / t − / ≥ ( u (cid:48) (0) − κs ) F (1 / − F ( s )1 / − s for all ≥ t > / > s ≥ . Under linear loss utility (so u (cid:48) (1) = u (cid:48) (0) = 1 and κ = 0 ), the condition in Proposition 2simplifies to f (1 / being a subgradient of F at / on the domain [0 , . This subgradientcondition is weaker than F being convex on [0 , . Remark . With linear loss utility, no compromise can be an optimal delegation set (i.e., de-terministic mechanism) even if the subgradient condition does not hold. However, there willthen be a stochastic mechanism that Proposer strictly prefers. This situation can arise, forexample, when the type density is strictly increasing except on a small interval around / ,where it is strictly decreasing. Intuitively, Proposer would like to delegate a small set ofactions around / to types close to / , but adding such actions to the no-compromise del-egation set is deleterious because it leads to many types above / choosing an action closeto / rather than . By contrast, lotteries with expected value / can be used to attract onlytypes close to / . Example E.1 in Appendix E elaborates. (cid:5) Following our general methodology discussed in Subsection 3.4, our proof of necessity uses the availabilityof stochastic mechanisms. But we can establish that under Condition LQ, (1) is necessary even for full delegationto be an optimal delegation set.
12e also note that no compromise can be an optimal delegation set even when u does notsatisfy Condition LQ. In particular, it can be shown that if no compromise is an optimaldelegation set for some u , then it is also an optimal delegation set for any utility function thatis a convex transformation of u .On the other hand, no compromise is not optimal—not even an optimal delegation set—ifProposer’s utility is differentiable at his ideal point a = 1 (which implies u (cid:48) (1) = 0 ). Thereason is that when u (cid:48) (1) = 0 , Proposer would strictly benefit from offering a small interval [1 − ε, , or even just the action − ε , instead of only offering action . For, Proposer’s decreasein utility from getting an action slightly lower than is second order, but there is a first-orderincrease in the probability of avoiding a veto. Both full delegation and no compromise are special cases of interval delegation : Proposeroffers an interval, and Vetoer chooses an action from either that interval or the status quo. Itfollows from Lemma 1 that when interval delegation is optimal, there is always an optimalinterval of the form [ c, for some c ∈ [0 , . One can thus interpret interval delegation as Pro-poser designating a minimally acceptable option; implicitly, the maximal acceptable option isProposer’s ideal point. Interval delegation, without a status quo, has been a central focus ofthe prior literature: intervals are simple, tractable, and lend themselves to comparative statics.Arguably, intervals also map more naturally into proposals likely to emerge in applications. Proposition 3.
The interval delegation set [ c ∗ , with c ∗ ∈ [0 , is optimal if(i) κF ( v ) − u (cid:48) ( v ) f ( v ) is increasing on [ c ∗ , ;(ii) ( u (cid:48) ( c ∗ ) + κ ( c ∗ − t )) F ( t ) − F ( c ∗ / t − c ∗ / ≥ u (cid:48) ( c ∗ ) F ( c ∗ ) − F ( c ∗ / c ∗ / for all t ∈ ( c ∗ / , c ∗ ] ; and(iii) u (cid:48) ( c ∗ ) F ( c ∗ ) − F ( c ∗ / c ∗ / ≥ ( u (cid:48) (0) − κs ) F ( c ∗ / − F ( s ) c ∗ / − s for all s ∈ [0 , c ∗ / .Conversely, under Condition LQ, the delegation set [ c ∗ , with c ∗ ∈ (0 , is optimal only if conditions(i), (ii), and (iii) above hold. We discuss sufficiency. The intuition for condition (i) in the proposition is analogous tothat discussed after Proposition 1; it ensures that there is no benefit to not fully delegating theinterval [ c ∗ , taking as given that Vetoer can choose c ∗ . For linear loss utility, the condition Note that when u (cid:48) (1) = 0 , the condition in Proposition 2 fails: its left-hand side is when t = 1 , while itsright-hand side is strictly positive when s = 0 . F ( v ) c ∗ / c ∗ Figure 1 – Conditions (i)—(iii) of Proposition 3 for linear loss utility. F is concave on [ c ∗ , ; f ( c ∗ / is a subgradient on [0 , c ∗ ] ; and the average density on [ c ∗ / , c ] equals f ( c ∗ / because F ( c ∗ ) intersects the subgradient.reduces to F being concave on [ c ∗ , . Linear loss utility is also helpful to interpret the otherconditions. Conditions (ii) and (iii) then simplify to the requirements that the average densityfrom c ∗ / to c ∗ be simultaneously less than that from c ∗ / to t for all t ∈ ( c ∗ / , c ∗ ] and greaterthan that from s to c ∗ / for all s ∈ [0 , c ∗ / . Equivalently, the average density from c ∗ / to c ∗ equals f ( c ∗ / and f ( c ∗ / is a subgradient of F at c ∗ / on the domain [0 , c ∗ ] . The subgradientcondition is analogous to that discussed after Proposition 2. (More generally, conditions (ii)and (iii) with c ∗ = 1 imply the condition of Proposition 2.) The additional requirement ensuresthat the threshold c ∗ is an optimal threshold. See Figure 1.Readers familiar with Amador and Bagwell (2013) will note from the discussion above thatcondition (i) in Proposition 3 plays the same role as condition (c1) in that paper. In fact, the twoconditions are identical, even though our analysis accommodates stochastic mechanisms thatcannot be reduced to their money burning. Conditions (ii) and (iii) of Proposition 3 don’thave analogs in Amador and Bagwell’s work, however, because these concern optimalityconditions that turn on our status quo.
Remark . For linear loss utility, it can be verified that if F is strictly unimodal (i.e., strictlyconvex and then strictly concave), then interval delegation is optimal. Either there will be a c ∗ ∈ [0 , satisfying the three conditions of Proposition 3 or the condition in Proposition 2 will Indeed, we conjecture that our methodology, elaborated in Subsection 3.4, can be used to show that Amadorand Bagwell’s (2013) conditions ensure optimality of their delegation sets even among stochastic mechanisms.
14e met and no compromise is optimal. (cid:5) We can extend this observation as follows:
Corollary 3.
Assume Condition LQ. Interval delegation is optimal if the type density f is logconcaveon [0 , ; if, in addition, f is strictly logconcave on [0 , or Proposer’s utility is strictly concave, thenthere is a unique optimal interval. Recall that logconcavity is stronger than unimodality, but many familiar distributions havelogconcave densities, including the uniform, normal, and exponential distributions (Bagnoliand Bergstrom, 2005). The proof of Corollary 3 also establishes that under Condition LQ andlogconcavity of the type density on [0 , , the set of optimal interval thresholds is connected:if [ c ∗ , and [ c ∗ , are both optimal interval delegation sets, then so is [ c ∗ , for all c ∗ ∈ [ c ∗ , c ∗ ] .Such multiplicity arises under the uniform distribution and linear loss utility. Either strict log-concavity of the type density or strict concavity of Proposer’s utility eliminates multiplicity. Let us outline the idea behind the proofs of Propositions 1–3. We use a Lagrangian method,as has proved fruitful in prior work on optimal delegation, notably in Amador and Bagwell(2013). However, the presence of a status quo requires some differences in our approach.In particular, while prior work has largely focussed on optimality of connected delegationsets, our Proposition 2 and Proposition 3 are effectively concerned with the optimality ofdisconnected delegation sets because of the status quo. Moreover, our approach provides asimple way to incorporate stochastic mechanisms, which, as already highlighted, are oftennot addressed in prior work.Consider the following relaxed version of the optimization problem (D) for deterministicmechanisms: max α ∈A (cid:90) (cid:18) u ( α ( v )) − κ (cid:20) vα ( v ) − α ( v ) − (cid:90) v α ( s )d s (cid:21)(cid:19) d F ( v ) (R)s.t. vα ( v ) − α ( v ) − (cid:90) v α ( s )d s ≥ ∀ v ∈ [ v, v ] . Let Mo be the (strictly positive, for simplicity) mode of F and let ∆( x ) := F (2 x ) − F ( x ) x − f ( x ) . F beingconvex-concave implies that letting c ∗ / { x > x ) = 0 } , ∆( x ) ≥ for x ∈ (0 , c ∗ / and ∆( x ) ≤ for x ∈ ( c ∗ / , . Clearly, c ∗ / ≤ Mo ≤ c ∗ and hence f is decreasing on [ c ∗ , . The convex-concave property impliesthat f ( c ∗ / is a subgradient of F at c ∗ / on the domain [0 , c ∗ ] , and if c ∗ / > / then f (1 / is a subgradient of F on the domain [0 , . κ ≡ inf a ∈ [0 , − u (cid:48)(cid:48) ( a ) , the objective is a concave functional of α . It differs from (D) in two ways.First, the constraint has been relaxed: IC requires the inequality to hold with equality. Second,the objective has been modified to incorporate a penalty for violating IC. Plainly, if α ∗ is ICand a solution to problem (R), then it is also a solution to (D). But we establish (see Lemma A.1in Appendix A) that in this case α ∗ is also a solution to problem (P), i.e., it is optimal amongstochastic mechanisms. The idea is as follows. If there were a stochastic mechanism that isstrictly better than α ∗ in problem (P), consider the corresponding deterministic mechanismthat replaces each lottery by its expected outcome. While this mechanism would not be IC ingeneral, we show that it would both be feasible for problem (R) and would obtain a strictlyhigher objective value, contradicting the optimality of α ∗ in (R). Establishing a higher valuerelies on the objective in (R) being a concave functional.The sufficiency results in Propositions 1–3 then obtain from identifying sufficient con-ditions under which the respective IC mechanisms solve (R). It is here that our approachinvolves constructing suitable Lagrange multipliers.For the necessity results, we first establish in Lemma A.4 in Appendix A that under Con-dition LQ, if a deterministic mechanism α ∗ solves problem (P) then it also solves problem(R). The idea is as follows. Suppose a solution α to problem (R) provides a strictly highervalue than α ∗ . We construct a corresponding IC mechanism m such that α ( v ) = E m ( v ) [ a ] for all v . Roughly, monotonicity of α (by definition of the set A ) implies existence of transfers thatmake α IC in a quasi-linear model; the inequality constraints in (R) mean the transfers can bechosen to be positive (i.e., they can be viewed as money burning); and, because of Vetoer’squadratic utility, positive transfers can be substituted for by the action variance of suitablelotteries. Since u (cid:48)(cid:48) ( a ) = − κ for a < under Condition LQ, the condition makes the objectivein (R) a linear functional in the relevant domain. We can thus show that mechanism m obtainsa strictly higher value than α ∗ in (P), a contradiction.We then establish necessity of the conditions in Propositions 1–3 by showing that, unlessthese conditions are satisfied, the corresponding mechanisms can be strictly improved uponin problem (R). Here we use the fact that the constraint set in (R) is convex and, therefore,first-order conditions must hold at a solution. More specifically, the (Gateaux) derivative ofthe objective in the direction of any feasible mechanism must be negative.16 . Comparative Statics and Comparisons We derive two comparative statics, restricting attention to interval delegation. This fo-cus can be justified by implicitly assuming conditions for optimality of interval delegation(Section 3), or just because such menus are simple, tractable, or relevant for applications.If Proposer proposes A = [ c, with c ∈ [0 , , then Vetoer chooses if v < c/ , c if v ∈ [ c/ , c ] , v if v ∈ [ c, , and if v > . Hence Proposer’s expected utility or welfare from A = [ c, is W ( c ) := u (0) F ( c/
2) + u ( c )( F ( c ) − F ( c/ (cid:90) c u ( v ) f ( v )d v. Differentiating, the first-order condition for c ∗ ∈ (0 , to be an optimal threshold amonginterval delegation sets is that it must be a zero of u (cid:48) ( c ∗ ) [ F ( c ∗ ) − F ( c ∗ / − f ( c ∗ /
2) [ u ( c ∗ ) − u (0)] . (2)In general there can be multiple optimal thresholds, even among interior thresholds. Ac-cordingly, let the set of optimal thresholds for interval delegation be C ∗ := arg max c ∈ [0 , W ( c ) . We use the strong set order to state comparative statics. Recall that for
X, Y ⊆ R , X is larger than Y in the strong set order, denoted X ≥ SSO Y , if for any x ∈ X and y ∈ Y , min { x, y } ∈ Y and max { x, y } ∈ X . We say that C ∗ increases (resp., decreases) if it gets larger(resp., smaller) in the strong set order. Since interval delegation with a lower threshold givesVetoer a superset of options to choose from, a decrease in C ∗ corresponds to offering morediscretion. It can also be interpreted as Proposer compromising more. As mentioned afterCorollary 3, under Condition LQ and a logconcave type density, C ∗ is a (closed) interval. Inthat case a decrease in C ∗ is equivalent to a decrease in both min C ∗ and max C ∗ .Our comparative statics concern Proposer’s risk aversion and the ex-ante preference align-ment between Proposer and Vetoer. We say that Proposer becomes strictly more risk averse ifthe Arrow-Prat coefficient of absolute risk aversion strictly increases in the relevant region: − u (cid:48)(cid:48) ( a ) /u (cid:48) ( a ) strictly increases for all a ∈ [0 , . As is well known, such a change can also beexpressed in terms of concave transformations of Proposer’s utility. Under Condition LQ, it17orresponds to a higher weight on the quadratic term. We say that the two players are strictlymore aligned if Vetoer’s ideal-point density changes from f to g with g strict likelihood ratiodominating f on the interval [0 , : for all ≤ v L < v H ≤ , f ( v H ) /f ( v L ) < g ( v H ) /g ( v L ) . Proposition 4.
Among interval delegation sets, there is:(i) more discretion (i.e., C ∗ decreases) if Proposer becomes strictly more risk averse; and(ii) less discretion (i.e., C ∗ increases) if Vetoer becomes strictly more aligned with Proposer. The proof uses the interval delegation structure and monotone comparative statics un-der uncertainty, specifically Karlin’s (1968) variation diminishing property for single-crossingfunctions and comparative statics from Milgrom and Shannon (1994).The intuition for part (i) of the proposition is simply that greater risk aversion makes Pro-poser more concerned about a veto, and hence she compromises more. The intuition for part(ii) is that greater ex-ante alignment makes Proposer less concerned about a veto, and henceshe compromises less. Yet, the precise conditions in the proposition are nuanced. In partic-ular, the stochastic ordering used in our notion of alignment is important: one can constructexamples in which, among interval delegation sets, Proposer optimally gives Vetoer strictlymore discretion when there is a right-shift in the type density in the sense of either hazardor reversed-hazard rate (both of which are stronger than first-order stochastic dominance butweaker than a likelihood ratio shift). Furthermore, absent the focus on interval delegation, itis not necessarily clear how to relate changes in delegation sets with the degree of discretionor compromise.It is instructive to contrast part (ii) of Proposition 4 with the expertise-based delegation lit-erature. The broad finding there is that among interval delegation, greater preference similar-ity in a suitable sense leads to more discretion (Holmstr ¨om, 1984, Theorem 3). The differenceowes to and highlights the distinct rationales for discretion. In those models, the delegatorwould like to give the agent discretion to benefit from the agent’s expertise; the degree ofdiscretion is limited by the extent of preference misalignment. In our setting, on the otherhand, the agent is given discretion only because of her veto power; greater ex-ante preferencealignment mitigates that concern.
Example . Under Condition LQ, the first-order condition for an optimal interval threshold(i.e., expression (2) equals zero) becomes γ − γc ∗ ) [ F ( c ∗ ) − F ( c ∗ / c ∗ (1 + γ − γc ∗ ) f ( c ∗ / . = μ = μ = μ = γ c * (a) σ = 1 . γ = γ = γ = γ = σ c * (b) µ = 0 . . Figure 2 – Optimal interval thresholds for Normal distributions (mean µ , variance σ ) andlinear-quadratic Proposer utility, u ( a ) = − (1 − γ ) | − a | − γ (1 − a ) .Recall from Corollary 3 that, when combined with boundary conditions, there will be aunique solution for any strictly logconcave type density; moreover, the corresponding intervalis then an (unrestricted) optimal mechanism. Given uniqueness, the implicit function theoremcan be used to affirm the general comparative statics of Proposition 4; moreover, the first-order condition can also be used to compute numerically the optimal interval threshold forstandard distributions. Figure 2 illustrates for Normal distributions. The left panel verifiescomparative statics already discussed; note that a higher mean µ is a likelihood ratio right-shift and hence more alignment. The right panel shows comparative statics in the varianceof the distribution. We see that there is less discretion when the variance is lower, with theoptimal threshold converging, as σ → , to Proposer’s optimal offer, . , to type µ = 0 . . (cid:5) While we do not have general comparative statics results in the variability of the typedistribution, it can be shown that for any strictly unimodal distribution with mode Mo ≥ ,an optimal interval delegation set has more compromise than if Proposer knew Vetoer’s typeto be Mo . That is, if c ∗ is an optimal interval threshold, then c ∗ ≤ min { , } ; the inequalityis strict if Mo ∈ (0 , / . Adding this kind of uncertainty to the complete-information modelof Romer and Rosenthal (1978) thus increases the extent of Proposer’s compromise. This subsection compares the outcome of optimal delegation with two game forms con-sidered in earlier work. When µ ≥ , a higher µ can be viewed as shifting Vetoer overall further away to the right of Proposer, butwhat is relevant is the change of the distribution on the interval [0 , .
19 natural starting point is the incomplete-information version of the Romer and Rosenthal(1978) model. Proposer makes a take-it-or-leave it proposal a ∈ R , which Vetoer can accept orveto. This can be viewed as restricting Proposer to singleton delegation sets. Clearly, Proposeris strictly worse off in this institution unless no compromise is the optimal mechanism. Weassume throughout this subsection that no compromise is not an optimal interval delegationset; as noted in Subsection 3.2 it is sufficient that Proposer’s utility u ( a ) is differentiable at hisideal point a = 1 (hence u (cid:48) (1) = 0 ). We will see below that not only does Proposer strictlybenefit from optimal delegation, but so does Vetoer under some conditions.Matthews (1989) studies cheap talk before veto bargaining: prior to Proposer making asingleton proposal, Vetoer can send a costless and nonbinding message. As usual in cheap-talk games there is an uninformative and hence noninfluential equilibrium, in which Proposermakes the same proposal, a U > , as he would absent the possibility of cheap talk. Matthewsprovides conditions under which there is also an equilibrium with informative and influentialcheap talk; it is sufficient given the support of our type density that u (cid:48) (1) = 0 (or that eventhe weaker condition in fn. 18 holds). A set of low Vetoer types pool on a “veto threat”message, while the complementary set of high types pool on an “acquiescing” message. Inresponse to the latter message, Proposer offers a = 1 ; in response to the veto threat Proposeroffers some a I ∈ (0 , . The former proposal is accepted by all types that acquiesced, whilethe latter is accepted by only a subset of types that made the veto threat; types below somestrictly positive threshold exercise the veto. An influential cheap-talk equilibrium is outcomeequivalent to the delegation set { a I , } in our framework.There can be multiple cheap-talk equilibria with distinct outcomes, both among influ-ential equilibria and among noninfluential equilibria (i.e., distinct a I and a U respectively).Matthews shows that a I < a U in any two equilibria of the respective kinds; moreover, he pro-vides conditions under which a I is unique, i.e., all influential cheap-talk equilibria have thesame outcome (Matthews, 1989, Remark 3). As elaborated in the proof of our Proposition 5,multiplicity is ruled out when the following function has a unique zero: u (cid:48) ( a ) [ F ((1 + a ) / − F ( a/ − f ( a/ u ( a ) − u (0)] . (3)By way of comparison, we recall that a zero of a similar function given in (2) is the first-order condition for optimality of an interval delegation set’s threshold. A weaker condition suffices: u (cid:48) (1)[1 − F (1 / < f (1 / u (1) − u (0)] . This ensures that is not an optimalsingleton proposal, nor is { } an optimal interval delegation set. Recall that when u is not differentiable at , u (cid:48) (1) refers to the left derivative. roposition 5. Assume no compromise is not an optimal delegation set, and that either (2) or (3) isstrictly downcrossing on (0 , . Any optimal interval delegation set [ c ∗ , has c ∗ < min { a I , a U } forany influential and noninfluential cheap-talk equilibrium a I and a U , respectively. Hence, if [ c ∗ , is anoptimal delegation set, then it strongly Pareto dominates any cheap-talk outcome, influential or not. By strong Pareto dominance, we mean that Proposer is ex-ante better off, while Vetoer isbetter off no matter his type; moreover, a set of Vetoer types that have strictly positive proba-bility are strictly better off. Proposition 5’s conclusions hold trivially when full delegation isthe optimal delegation set ( c ∗ = 0 ), even without its hypotheses. But under its hypotheses,the conclusions also apply to other optimal intervals. The function (2) is strictly downcross-ing on (0 , under Condition LQ and either strict logconcavity of the type density or strictconcavity of Proposer’s utility. Indeed, this underlies the uniqueness claim in Corollary 3; seeLemma B.1. Moreover, Corollary 3 also assures that interval delegation is then optimal.Here is the intuition behind Proposition 5. Consider Proposer’s tradeoff when marginallylowering his proposal a I ∈ (0 , in an influential cheap-talk equilibrium. The benefit is thatsome types just below a I / will accept rather than veto; the cost is that the action inducedfrom all types in the interval ( a I / , (1 + a I ) / is lower. When Proposer instead delegatesthe interval [ a I , , the benefit from lowering a I is unchanged while the cost is largely obvi-ated, as most types above a I / are unaffected by the change. Proposer is thus more willingto compromise when choosing among interval delegation sets rather than under cheap talk. Consequently, all Vetoer types benefit—at least weakly, and some strictly—from optimal in-terval delegation as compared to cheap talk. While Proposer could be harmed by a restrictionto interval delegation, there is strong Pareto dominance when intervals constitute optimaldelegation.We note that if interval delegation is not optimal, then some Vetoer types may be worse offunder optimal delegation than under cheap talk. For example, it is possible that the optimaldelegation set takes the form { a ∗ , } with a ∗ ∈ (0 , . In this case one can show that necessarily a ∗ < a I in any influential cheap-talk equilibrium; intuitively, while a I is sequentially rational,committing to a lower proposal helps ex ante by inducing action rather than a I from sometypes. Consequently, while Proposer strictly benefits from optimal delegation, some Vetoertypes would strictly prefer either cheap-talk outcome. A function h ( a ) is strictly downcrossing if for any a L < a H , h ( a L ) ≤ ⇒ h ( a H ) < . The hypotheses in Proposition 5 ensure that this local-improvement intuition extends to global optimality. . Applications We now discuss some implications and interpretations of our analysis in the context ofthree applications.
Menus of products.
Our framework can be applied to questions of which products to presentcustomers with, albeit in a stylized manner. For an illustration, suppose a salesperson has athis disposal a set of products indexed by a ∈ [0 , , with higher a corresponding to higherquality. The price of product a is ka , where k > . This pricing can be interpreted as emerg-ing from a constant markup on a quadratic cost. Consumers vary in how they trade off qualityand price; specifically, a consumer of type v ≥ has gross valuation va , and hence net-of-pricepayoff va − ka . If a consumer does not purchase, his payoff is ; a consumer cannot purchasea product he is not shown (perhaps because of ignorance, or because the salesperson canclaim it is unavailable). Observe that we can normalize k = 1 / , as this simply rescales theconsumer type v . The salesperson receives a higher commission on better products, reflectedby his strictly increasing and concave utility u ( a ) . Given any belief density the salespersonholds about a particular consumer’s valuation the salesperson’s problem of which products toshow the consumer is precisely that of determining the optimal delegation set in our setting.Take the case of a linear u . Propositions 1–3 imply that if the density of v is logconcave, it isoptimal to show the consumer some set of “best products” (i.e., an interval of products [ c, );if the density is strictly decreasing, then all products should be shown; and if the density isstrictly increasing, then only the highest-quality product should be shown. Proposition 4(i)implies that if the commission schedule changes to make u more concave, the salespersonshows a larger set of products. Proposition 4(ii) implies that if wealthier consumers (or, ifwealth is unobservable, some proxy thereof) have a higher distribution of v in the likelihoodratio sense, then wealthier consumers are shown a smaller set of products.What if a consumer can choose the information to disclose about her type? Specifically,suppose, as is standard in voluntary disclosure models, that any type v can send any message(a closed subset of R + ) that contains v . The salesperson decides on the product menu afterobserving the message. No matter the type distribution, there are at least two equilibria: onein which no type discloses any information, and one in which all types fully disclose. Every Ali, Lewis, and Vasserman (2019), Hidir and Vellodi (2020), and Ichihashi (2020) consider optimal consumerdisclosure in models that emphasize price discrimination. For any unused message, V ⊆ R + , let the salesperson put probability on v = max V (or, if sup V = ∞ , onsome v ∈ V with v ≥ ) and offer the correspondingly optimal singleton menu. It is then straightforward that However, when the salesperson offers all productsunder the prior (Proposition 1), the nondisclosure equilibrium is consumer optimal—not onlyex ante, but for every consumer type.
Lesser-included offenses.
The legal doctrine of lesser-included offenses in criminal cases is“the concept that a defendant may be found guilty of an uncharged lesser offense, insteadof the offenses formally charged . . . a recognized and well-established feature of the Amer-ican criminal justice system” (Adlestein, 1995). For instance, “the lesser-included offensesof first degree murder include second degree murder, voluntary manslaughter, involuntarymanslaughter, criminally negligent homicide, and aggravated assault” (Orzach and Spurr,2008).Our model views v as a jury’s (or judge’s) evaluation of the optimal penalty or true sever-ity of a crime, and assumes that the jury will convict the defendant of the closest chargeto v that is available. Verdict corresponds to a complete acquittal, which is always avail-able to the jury, while is the maximum penalty. A prosecutor can put forward any set ofcharges in [0 , , and he seeks to maximize the penalty. Assume his utility u is concave. Thelesser-included offense doctrine can be modeled as imposing a constraint that if a charge a is included, then the jury can choose any verdict in [0 , a ] . The prosecutor is ex-ante worseoff with such constraint, often strictly so. However, our full delegation result (Proposition 1)identifies when the doctrine does not hurt a prosecutor, as the outcome is the same with orwithout the doctrine. In fact, following our discussion after Corollary 1, particularly the endof fn. 12, when Proposition 1’s requirement holds, the prosecutor is not hurt, ex ante, from thedoctrine being attached to any set of charges, not only his optimum. This applies when the no consumer type does strictly better by deviating to any unused message. For example, suppose the type density is strictly decreasing on a small interval [0 , δ ] and strictly increas-ing thereafter, and u is linear. Then, under nondisclosure, the salesperson’s optimal menu is the singleton { } (Proposition 2). There is also a partial-disclosure equilibrium in which types [0 , δ ] pool on the message [0 , δ ] andall higher types pool on the message [ δ, ∞ ) ; the former message leads to the menu [0 , δ ] by the full-delegationlogic of Corollary 1, while the latter message leads to the singleton { } by the no-compromise logic of Proposi-tion 2. Every consumer type prefers this partial-disclosure equilibrium to the nondisclosure equilibrium, somestrictly. Such a prosecutorial objective is a common assumption in law and economics since Landes (1971). Ourassumption on the jury’s behavior is compatible with it acting on the basis of (expected) utility, but could alsobe viewed as a reduced form. It is distinct from an approach in which the jury only convicts the defendant of acharge if some relevant standard has been met. [0 , . As this is the additive inverse of some prosecutor utility function, the defendant is ex-ante better off when the prosecutor is constrained by the lesser-included offense doctrine. She is not assured an improvement ex post, however. For example, when no compromiseis unconstrained optimal for the prosecutor (Proposition 2), the doctrine strictly harms thedefendant when v ∈ (0 , / . More generally, the defendant is strictly harmed by the doctrinefor some realizations of v if full delegation is not unconstrained optimal for the prosecutor.The fact that a defendant can be harmed by the doctrine ex post could explain why defendantssometimes appeal verdicts on the basis that their conviction on a lesser-included offense is notlegally valid. We note that if the prosecutor were restricted to bringing a single charge, then under thecondition of Proposition 1, the lesser-included offenses doctrine would benefit him and hurta defendant with utility − u . Moreover, regardless of whether full delegation is unconstrainedoptimal for the prosecutor, the doctrine would lead him to bringing the maximum charge,whereas absent the doctrine the optimal single charge would typically not be the maximum. Legislatures, Executives, and Bureaucracies.
Legislatures write bills that can be vetoed byexecutives. But executives do more than just approve or veto: as emphasized by Epstein andO’Halloran (1996, pp. 378–379), “all laws passed by Congress are implemented by the execu-tive branch in one form or another”, and, since, “Presidents generally appoint administratorswith preferences similar to their own” the amount of discretion given is a “key variable in. . . congressional-executive relations”. One interpretation of our results is that they predictbills granting the executive more discretion when there is greater preference misalignmentbetween the executive and the legislature, in the sense of Proposition 4(ii). This flips the com-parative static emphasized in the political science literature (Epstein and O’Halloran, 1996,1999), which stems from expertise-based delegation models. Of course, in practice one ex-pects both the expertise-based rationale and our veto-based one to coexist to varying degrees. This observation holds regardless of the curvature of the defendant’s utility w ( a ) , because a prosecutor withutility − w ( a ) is ex-ante worse off with a constraint on his delegation set even if, contrary to our maintainedassumption, − w is not concave. E.g., in Shrum v. State (1999), the defendant, who was convicted for manslaughter on a charge of murder,appealed that “1) heat of passion manslaughter is not a ‘necessarily included offense’ of premeditated murder;2) a jury must acquit when the evidence supports a charge not alleged in the Information”. The appeal wasdenied. For exceptions and caveats, see, for example, Volden (2002) and Huber and McCarty (2004). Volden (2002,p. 112) notes that modeling the executive’s veto is important for his finding that “there are conditions under
6. Conclusion
We have studied Proposer’s optimal mechanism, absent transfers, in a simple model ofveto bargaining. Our main results identify sufficient and necessary conditions for the optimalmechanism to take the form of certain delegation sets, including full delegation, no compro-mise, and more generally interval delegation. While we have focussed on a quadratic lossfunction for Vetoer, our analytical methodology can be applied to deduce optimality of thesedelegation sets for a broader class of Vetoer preferences. Specifically, the methods can be read-ily applied to Vetoer utility functions of the form va + b ( a ) , for any differentiable and strictlyconcave function b . The conditions in Propositions 1–3 would be more complicated, however.Our methodology could also be used to deduce optimality of other kinds of delegation sets,for example Proposer offering his ideal point and one additional compromise option.A key assumption underlying our analysis is that of Proposer commitment. In some con- which discretion is increased upon a divergence in legislative-executive preferences”. The mechanism under-lying his findings is different from that in this paper, however; in particular, expertise-based delegation is stillessential to his analysis. References A DLESTEIN , A. L. (1995): “Conflict of the Criminal Statute of Limitations with Lesser Offensesat Trial,”
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In Appendix A we assume the support of the type distribution F is [0 , . This is withoutloss (even among stochastic mechanisms) because it is always optimal for Proposer to chooseaction for types above and, given the outside option, to choose action for types below . A.1. Sufficient Conditions
For convenience, we recall Proposer’s problem (P): max m ∈S (cid:90) E m ( v ) [ u ( a )]d F ( v ) (P)s.t. E m ( v ) (cid:2) av − a / (cid:3) − (cid:90) v E m ( x ) [ a ]d x = 0 ∀ v ∈ [0 , . (IC-env)We also recall the relaxed problem (R): max α ∈A (cid:90) (cid:18) u ( α ( v )) − κ (cid:20) vα ( v ) − α ( v ) − (cid:90) v α ( x )d x (cid:21)(cid:19) d F ( v ) (R)s.t. vα ( v ) − α ( v ) − (cid:90) v α ( x )d x ≥ ∀ v ∈ [0 , . We show first that any IC solution to the relaxed problem also solves problem (P).
Lemma A.1.
Suppose α ∗ ∈ A solves problem (R) and is incentive compatible. Then α ∗ also solves (P) . Proof.
To obtain a contradiction, let m ∈ S be feasible for (P) and suppose it achieves a strictlyhigher objective value in (P) than α ∗ . Define α ∈ A by setting α ( v ) := E m ( v ) [ a ] for each v .Then, Jensen’s inequality implies E m ( v ) [ av − a / ≥ vα ( v ) − α ( v ) / ; whereas (cid:82) v E m ( x ) [ a ]d x = (cid:82) v α ( x )d s . Hence, feasibility of m in (P) implies feasibility of α in (R). Moreover, (cid:90) (cid:18) u ( α ( v )) − κ (cid:20) vα ( v ) − α ( v ) − (cid:90) v α ( x )d x (cid:21)(cid:19) d F ( v ) ≥ (cid:90) (cid:18) E m ( v ) (cid:20) u ( a ) − κ (cid:18) va − a (cid:19)(cid:21) + κ (cid:90) v E m ( x ) [ a ]d x (cid:19) d F ( v )= (cid:90) E m ( v ) [ u ( a )]d F ( v ) (cid:90) u ( α ∗ ( v ))d F ( v )= (cid:90) (cid:18) u ( α ∗ ( v )) − κ (cid:20) vα ∗ ( v ) − α ∗ ( v ) − (cid:90) v α ∗ ( x )d x (cid:21)(cid:19) d F ( v ) , where the first inequality holds because the first line is a concave functional (by the defini-tion of κ ≡ inf a ∈ [0 , − u (cid:48)(cid:48) ( a ) ), the first equality holds because m is feasible in (P), the secondinequality holds because of our assumption that m achieves a strictly higher value than α ∗ ,and the final equality holds because α ∗ being IC implies it is feasible in (P). Therefore, α ∗ isnot optimal in (R), a contradiction.To show that a given delegation set solves the relaxed problem, we construct Lagrangemultipliers and show that the induced action rule maximizes the following Lagrangean. Given α ∈ A and an increasing and right-continuous function Λ( v ) , let the Lagrangean be given by L ( α, Λ) := (cid:90) (cid:18) u ( α ( v )) f ( v ) − κf ( v ) (cid:20) vα ( v ) − α ( v ) − (cid:90) v α ( x )d x (cid:21)(cid:19) d v + (cid:90) (cid:18) vα ( v ) − α ( v ) − (cid:90) v α ( x )d x (cid:19) dΛ( v )= (cid:90) (cid:18) u ( α ( v )) f ( v ) − α ( v )[ κF ( v ) − Λ( v )] − κf ( v ) (cid:20) vα ( v ) − α ( v ) (cid:21)(cid:19) d v + (cid:90) (cid:18) vα ( v ) − α ( v ) (cid:19) dΛ( v ) + (cid:90) α ( v )d v [ κF (1) − Λ(1)] , (A.1)where the second equality follows from integration by parts. Lemma A.2.
Let α ∗ be induced by a delegation set. Suppose there is an increasing and right-continuous function Λ such that L ( α ∗ , Λ) ≥ L ( α, Λ) for all α ∈ A . Then α solves problem (R) . Proof.
Under the conditions stated in the lemma, Theorem 1 in Amador and Bagwell (2013)implies that α ∗ solves the relaxed problem. To see this, let their f be the negative of theobjective function in (R), X and Z be the vector space of bounded measurable functions, and Ω = A ; let P = { z ∈ Z |∀ v ∈ [0 ,
1] : z ( v ) ≥ } and G be the negative of the left-hand side of theconstraint in (R). Define the linear function T : Z → R by T ( z ) = (cid:82) z ( v )dΛ( v ) (which is well-defined since Λ corresponds to a finite measure (Royden and Fitzpatrick, 2010, p. 437) and (cid:82) v α ( s )d s is continuous and κF ( v ) − Λ( v ) has bounded variation as the difference of two increasing func-tions. Hence, the Riemann-Stieltjes integral (cid:82) (cid:82) v α ( s )d s d[ κF ( v ) − Λ( v )] exists and integration by parts is valid. That is, there is some delegation set A such that α ∗ ( v ) is an action in A ∪ { } that type v prefers the most. z is bounded and measurable (Royden and Fitzpatrick,2010, p. 375)) and observe that, for all z ∈ P , T ( z ) ≥ , and hence condition (i) holds. Since α ∗ is incentive compatible, − G ( α ∗ ) ∈ P (condition (iii) holds) and T ( G ( α ∗ )) = 0 (condition(iv) holds). Since α ∗ maximizes the Lagrangean by assumption, condition (ii) holds and weconclude that α ∗ solves problem (R).The follow observation will allow us to establish that the Lagrangean is a concave func-tional of α if Λ is increasing. Lemma A.3.
Suppose K is right-continuous and increasing and h : R → R is bounded, measurable,and for each value of its second argument concave in its first argument. Then the function S : L ∞ → R defined by S ( α ) := (cid:82) h ( α ( v ) , v )d K ( v ) is concave. Proof.
Fix α , α ∈ L ∞ , c ∈ (0 , and let α c = cα + (1 − c ) α . Then S ( α c ) − cS ( α ) − (1 − c ) S ( α ) = (cid:90) (cid:0) h ( α c ( v ) , v ) − ch ( α ( v ) , v ) − (1 − c ) h ( α ( v ) , v ) (cid:1) d K ( v ) ≥ because concavity of h implies that the integrand is positive for each v and because K isincreasing.Note that for each v , u ( α ( v )) f ( v ) + κf ( v ) α ( v ) is concave in α ( v ) since its second derivativeis given by f ( v )[ u (cid:48)(cid:48) ( α ( v )) + κ ] , which is negative by definition of κ . This implies that, for each v , each integrand in (A.1) is a concave function of α ( v ) . Since Λ is increasing, Lemma A.3implies that the Lagrangean is concave in α .This implies that the Lagrangean is maximized at α if the Gateaux differential satisfies ∂ L ( α, α − α, Λ) ≤ for all α ∈ A .If Λ(1) = κF (1) , (A.1) simplifies to L ( α, Λ) = (cid:90) (cid:18) u ( α ( v )) f ( v ) − α ( v )[ κF ( v ) − Λ( v )] − κf ( v ) (cid:20) vα ( v ) − α ( v ) (cid:21)(cid:19) d v + (cid:90) (cid:18) vα ( v ) − α ( v ) (cid:19) dΛ( v ) . The Gateaux differential is ∂ L ( α, α, Λ) = (cid:90) (cid:16)(cid:104) u (cid:48) ( α ( v )) f ( v ) − κF ( v ) + Λ( v ) (cid:105) α ( v ) (cid:17) d v + (cid:90) ([ v − α ( v )] α ( v )) d[Λ( v ) − κF ( v )] (A.2) (cid:90) (cid:18)(cid:90) v u (cid:48) ( α ( x )) f ( x ) − κF ( x ) + Λ( x )d x (cid:19) d α ( v ) + (cid:90) (cid:18)(cid:90) v [ x − α ( x )]d[Λ( x ) − κF ( x )] (cid:19) d α ( v ) , (A.3) where the second equality obtains using integration by parts.Below, we construct increasing and right-continuous Lagrange multipliers that satisfy Λ(1) = κF (1) such that ∂ L ( α, α − α, Λ) ≤ .We begin with the optimality of full delegation. Proof of sufficiency part of Proposition 1.
Note that the action function induced by full del-egation is α ( v ) = v . We claim that α maximizes the Lagrangean for the multiplier Λ( v ) = κF ( v ) − u (cid:48) ( v ) f ( v ) for v < and Λ(1) = κF (1) . Note that the multiplier is increasing since κF ( v ) − u (cid:48) ( v ) f ( v ) is increasing by assumption and u (cid:48) ( v ) ≥ . The Lagrangean is thereforemaximized at α if ∂ L ( α, α − α, Λ) ≤ for all α ∈ A . Note that the integrand of the first integralin (A.2) is 0 for almost every v by choice of Λ and the second integral is 0 since α ( v ) = v .We next consider the optimality of no compromise. Proof of sufficiency part of Proposition 2.
Note that the action rule induced by no compro-mise satisfies α ( v ) = 0 for v ∈ [0 , ) and α ( v ) = 1 for v ∈ [ , . Now suppose for all s ∈ [0 , / and t ∈ (1 / , we have ( u (cid:48) (1) + κ (1 − t )) F ( t ) − F (1 / t − / ≥ ( u (cid:48) (0) − κs ) F (1 / − F ( s )1 / − s . and let ψ := inf t ∈ (1 / , ( u (cid:48) (1) + κ (1 − t )) F ( t ) − F (1 / t − / . Define Λ( v ) = κF (1 / − ψ for v ∈ [0 , and Λ(1) = κF (1) .Let s ∈ (1 / , . Note that integration by parts implies (cid:82) / s v d F ( v ) = 1 / F (1 / − sF ( s ) − (cid:82) / s F ( v )d v . Since Λ( v ) is constant on [0 , , the definition of ψ implies that, for any s ∈ [0 , / , (cid:90) / s u (cid:48) ( α ( v )) f ( v ) − κF ( v ) + Λ( v )d v + (cid:90) / s v d[Λ( v ) − κF ( v )]= u (cid:48) (0)[ F (1 / − F ( s )] + 1 / / − κF (1 / − s [Λ( s ) − κF ( s )]=[ u (cid:48) (0) + κs ][ F (1 / − F ( s )] − (1 / − s ) ψ ≤ . (A.4)32imilarly, for any t ∈ (1 / , , (cid:90) t / u (cid:48) ( α ( v )) f ( v ) − κF ( v ) + Λ( v )d v + (cid:90) t / [ v − α ( v )]d[Λ( v ) − κF ( v )]= u (cid:48) (1)[ F ( t ) − F (1 / t − t ) − κF ( t )] + 1 / / − κF (1 / u (cid:48) (1) + κ (1 − t )][ F ( t ) − F (1 / − ( t − / ψ ≥ . (A.5)Fix arbitrary α ∈ A that satisfies α (1) = 1 . It follows from (A.3) and the definition of α that ∂ L ( α, α − α, Λ)= (cid:90) (cid:20)(cid:90) v ( u (cid:48) ( α ( x )) f ( x ) − κF ( x ) + Λ( x ))d x + (cid:90) v ([ x − α ( x )]d[Λ( x ) − κF ( x )]) (cid:21) d[ α ( v ) − α ( v )]= (cid:90) (cid:34)(cid:90) / v ( u (cid:48) ( α ( x )) f ( x ) − κF ( x ) + Λ( x ))d x + (cid:90) / v ([ x − α ( x )]d[Λ( x ) − κF ( x )]) (cid:35) d α ( v ) . Since α is increasing, (A.4) and (A.5) imply that ∂ L ( α, α − α, Λ) ≤ . Since the optimal actionrule chooses action 1 for type 1, we conclude that α is optimal.Lastly, we consider optimality of interval delegation. Proof of sufficiency part of Proposition 3.
The induced action function is α ( v ) = 0 for v
2) + 1 c ∗ − c ∗ / (cid:90) c ∗ c ∗ / u (cid:48) ( c ∗ )d F ( x ) (cid:21)(cid:19) d v − κ (cid:90) c ∗ t ( v − c ∗ )d F ( v )= u (cid:48) ( c ∗ )[ F ( c ∗ ) − F ( t )] − u (cid:48) ( c ∗ )[ F ( c ∗ ) − F ( c ∗ / c ∗ − tc ∗ − c ∗ / κ ( c ∗ − t )[ F ( c ∗ / − F ( t )]= − [ u (cid:48) ( c ∗ ) + κ ( c ∗ − t )][ F ( t ) − F ( c ∗ / t − c ∗ / u (cid:48) ( c ∗ ) F ( c ∗ ) − F ( c ∗ / c ∗ − c ∗ / ≤ , where the inequality is by Proposition 3’s condition (ii), and holds with equality for t = c ∗ / .Analogously, note that (cid:82) c ∗ / s F ( v ) + vf ( v )d v = c ∗ / F ( c ∗ / − sF ( s ) and Λ is constant on34 , c ∗ / . Hence, for s ∈ [0 , c ∗ / , (cid:90) c ∗ / s ( u (cid:48) (0) f ( v ) − [ κF ( v ) − Λ( v )]) d v + (cid:90) c ∗ / s v d[Λ( v ) − κF ( v )]= u (cid:48) (0)[ F ( c ∗ / − F ( s )] + Λ( s )( c ∗ / − s ) − κ [ c ∗ / F ( c ∗ / − sF ( s )]=[ u (cid:48) (0) − κs ][ F ( c ∗ / − F ( s )] − u (cid:48) ( c ∗ ) F ( c ∗ ) − F ( c ∗ / c ∗ − c ∗ / c ∗ / − s ) ≤ , where the inequality is by Proposition 3’s condition (iii). We conclude that α is optimal. A.2. Necessary Conditions
Lemma A.4.
Suppose Condition LQ holds. If α ∗ is deterministic and solves problem (P) then it alsosolves problem (R) . Proof.
The proof is by contraposition: assuming there exists α ∈ A that is feasible for (R) andachieves a strictly higher objective value in (R) than α ∗ , we will construct a solution to (P) thatachieves a strictly higher objective value than α ∗ . Claim 1 : There exists ˜ α ∈ A that is feasible for (R), satisfies ˜ α ( v ) ≤ for all v , v ˜ α ( v ) − ˜ α ( v ) − (cid:82) v ˜ α ( s )d s = 0 for all v such that ˜ α ( v ) = 1 , and achieves a weakly higher objective valuein problem (R) than α .We can assume α ( v ) ≤ since u is decreasing above . Now suppose instead that vα ( v ) − α ( v ) − (cid:82) v α ( s )d s > for some v such that α ( v ) = 1 . Consider an auxiliary setting in whicha principal chooses a pair of functions ( α, t ) and an agent with type v gets utility vα ( v ) − α ( v ) − t ( v ) . Since α is monotonic, it follows from standard arguments that there exist transfers t : [0 , → R such that ( α, t ) is incentive compatible in the auxiliary setting (e.g., Amador andBagwell, 2013). For all v , these transfers satisfy t ( v ) − t (0) = vα ( v ) − α ( v ) − (cid:82) v α ( s )d s ≥ ,where the inequality holds because α is feasible for (R). Define ( ˜ α, ˜ t ) by setting ( ˜ α ( v ) , ˜ t ( v )) =( α ( v ) , t ( v )) or ( ˜ α ( v ) , ˜ t ( v )) = (1 , t (0)) , whichever gives an agent with type v higher expectedutility (and choosing the latter if type v is indifferent). Note that ˜ t (0) = t (0) , which togetherwith t ( v ) ≥ t (0) implies ˜ t ( v ) − ˜ t (0) ≥ , with equality for any v such that ˜ α ( v ) = 1 .Observe that ( ˜ α, ˜ t ) corresponds to an incentive compatible direct mechanism: indeed, iftype v strictly prefers ( ˜ α ( v (cid:48) ) , ˜ t ( v (cid:48) )) to ( ˜ α ( v ) , ˜ t ( v )) then v also strictly prefers ( α ( v (cid:48) ) , t ( v (cid:48) )) to ( α ( v ) , t ( v )) , contradicting the assumption that ( α, t ) is incentive compatible. It follows from35he standard characterization of incentive compatible mechanisms that ˜ α is increasing, v ˜ α ( v ) − ˜ α ( v ) − (cid:90) v ˜ α ( s )d s = ˜ t ( v ) − ˜ t (0) ≥ , with the inequality holding as equality for v such that ˜ α ( v ) = 1 .Finally, note that α ( v ) ≤ ˜ α ( v ) ≤ for all v . Also, ˜ t ( v ) − ˜ t (0) ≤ t ( v ) − t (0) , which implies v ˜ α ( v ) − ˜ α ( v ) − (cid:90) v ˜ α ( s )d s ≤ vα ( v ) − α ( v ) − (cid:90) v α ( s )d s. It follows that ˜ α achieves a weakly higher objective value in problem (R). (cid:5) Claim 2 : Let ˜ α ∈ A be feasible for (R) and satisfy ˜ α ( v ) ≤ and v ˜ α ( v ) − ˜ α ( v ) / − (cid:82) v ˜ α ( s )d s = 0 for all v such that ˜ α ( v ) = 1 . There is a stochastic mechanism m such that,for all v , Prob m ( v ) ( a ≤
1) = 1 , E m ( v ) [ a ] = ˜ α ( v ) , and E m ( v ) (cid:104) va − a (cid:105) − (cid:82) v E m ( s ) [ a ]d s = 0 .Intuitively, this is because Vetoer’s utility function is quadratic and we can use noise as asubstitute for transfers. We provide an explicit construction of the mechanism m below.For any v such that ˜ α ( v ) = 1 , define m ( v ) to put mass 1 on action 1. Now fix arbitrary v such that ˜ α ( v ) < and arbitrary d ∈ ( −∞ , and let t ( d ) = − ˜ α ( v )1 − d . Then t ( d ) d + (1 − t ( d ))1 =˜ α ( v ) for all d . Moreover, for any real number r we can choose d ∈ ( −∞ , small enough suchthat − − ˜ α ( v )1 − d d − (cid:18) − − ˜ α ( v )1 − d (cid:19) ≤ r because the LHS → −∞ as d → −∞ . Hence, by choosing d small enough we get v ˜ α ( v ) − t ( d ) d − (1 − t ( d )) 12 − (cid:90) v ˜ α ( s )d s ≤ . Given v ∈ [0 , and t ∈ [0 , , we define m ( v ) to put probability t on action ˜ α ( v ) , probability (1 − t ) t ( d ) on action d , and probability (1 − t )(1 − t ( d )) on action . It follows from theabove that m ( v ) satisfies E m ( v ) [ a ] = ˜ α ( v ) , Prob m ( v ) ( a ≤
1) = 1 , and we can choose t ∈ [0 , such that E m ( v ) (cid:20) va − a (cid:21) − (cid:90) v E m ( s ) [ a ]d s = 0 . Defining m ( v ) in this way for all v such that ˜ α ( v ) < , the claim follows. (cid:5)
36e conclude that m is feasible for (P). Therefore, (cid:90) u ( α ∗ ( v ))d F ( v ) = (cid:90) (cid:18) u ( α ∗ ( v )) − κ (cid:20) vα ∗ ( v ) − α ∗ ( v ) − (cid:90) v α ∗ ( s )d s (cid:21)(cid:19) d F ( v ) < (cid:90) (cid:18) u ( α ( v )) − κ (cid:20) vα ( v ) − α ( v ) − (cid:90) v α ( s )d s (cid:21)(cid:19) d F ( v ) ≤ (cid:90) (cid:18) u ( ˜ α ( v )) − κ (cid:20) v ˜ α ( v ) − ˜ α ( v ) − (cid:90) v ˜ α ( s )d s (cid:21)(cid:19) d F ( v ) (A.6)where the equality holds because α ∗ is feasible for (P), the first inequality holds because weassume that α achieves a strictly higher value than α ∗ , and the second inequality holds byClaim 1.Under Condition LQ, κ ≡ inf v ∈ [0 , − u (cid:48)(cid:48) ( v ) = 2 γ . Hence, for any a, b ≤ and λ ∈ [0 , ,some algebra shows that u ( λa + (1 − λ ) b ) + κ [ λa + (1 − λ ) b ] λ (cid:20) u ( a ) + κ a (cid:21) + (1 − λ ) (cid:20) u ( b ) + κ b (cid:21) . Since
Prob m ( v ) ( a ≤
1) = 1 and E m ( v ) [ a ] = ˜ α ( v ) for all v , expression (A.6) therefore equals (cid:90) (cid:18) E m ( v ) (cid:20) u ( a ) − κ (cid:18) va − a (cid:19)(cid:21) + κ (cid:90) v E m ( s ) [ a ]d s (cid:19) d F ( v ) . Since m is feasible for (P), this expression equals (cid:82) E m ( v ) [ u ( a )]d F ( v ) . This contradicts theassumption that α ∗ solves (P), and we conclude that α ∗ solves (R).Let φ denote the objective function in (R). The set of feasible solutions for (R) is convex,and optimality of α therefore implies ∂φ ( α, α − α ) ≤ for any α ∈ A that is feasible for (R).Recall the assumption F (1) = 1 . φ ( α ) = (cid:90) (cid:18) u ( α ( v )) − κ (cid:20) vα ( v ) − α ( v ) − (cid:90) v α ( s )d s (cid:21)(cid:19) d F ( v ) ,∂φ ( α, α − α ) = (cid:90) (cid:18) [ u (cid:48) ( α ( v )) − κ [ v − α ( v )]]( α ( v ) − α ( v )) + κ (cid:90) v α ( s ) − α ( s )d s (cid:19) d F ( v )= (cid:90) (cid:20) u (cid:48) ( α ( v )) − κ (cid:20) v − α ( v ) − − F ( v ) f ( v ) (cid:21)(cid:21) ( α ( v ) − α ( v ))d F ( v ) . (A.7) Lemma A.5.
Suppose Condition LQ holds. If a delegation set containing the interval [ a, b ] ⊆ [0 , isoptimal, then κF ( v ) − u (cid:48) ( v ) f ( v ) is increasing on [ a, b ] . roof. Suppose a delegation set containing the interval [ a, b ] is optimal, and let α denote thecorresponding allocation rule. Suppose to the contrary that κF ( v ) − u (cid:48) ( v ) f ( v ) is strictly de-creasing for some v ∈ [ a, b ] . Then u (cid:48) ( v ) f ( v ) − κf ( v )[ − − F ( v ) f ( v ) ] is strictly increasing on someinterval [ d, e ] ⊂ [ a, b ] with d (cid:54) = e that contains v .Set α ( v ) = α ( v ) for v / ∈ [ d, e ] and α ( v ) = d for v ∈ [ d, e ] . Then α ∈ A and it is feasible for (R).Since α ( v ) = v for v ∈ [ d, e ] , it follows from (A.7) that ∂φ ( α, α − α ) > , a contradiction. Proof of the necessity part of Proposition 1.
Suppose Condition LQ holds. The result fol-lows directly from Lemma A.5.
Proof of the necessity part of Proposition 2.
Suppose Condition LQ holds and let α ∈ A bethe action rule induced by the delegation set { , } . Fix s ∈ [0 , / and t ∈ (1 / , , ε ∈ (0 , and define α ε ( v ) := if v ∈ [0 , s ) ε if v ∈ [ s, / − / − st − / ε if v ∈ [1 / , t )1 if v ∈ [ t, . Note that (cid:90) / s (cid:20) v − − F ( v ) f ( v ) (cid:21) d F ( θ ) = s [1 − F ( s )] − / − F (1 / , and (cid:90) t / (cid:20) v − − − F ( v ) f ( v ) (cid:21) d F ( θ ) = F ( t )( t − − F (1 / / − − t + 1 / . It follows that ∂φ ( α, α ε − α ) = ε (cid:90) / s (cid:18) u (cid:48) (0) − κ (cid:20) v − − F ( v ) f ( v ) (cid:21)(cid:19) d F ( v ) − / − st − / ε (cid:90) t / (cid:18) u (cid:48) (1) − κ (cid:20) v − − − F ( v ) f ( v ) (cid:21)(cid:19) d F ( v )= ε (1 / − s ) (cid:18) [ u (cid:48) (0) − κs ] F (1 / − F ( s )1 / − s + κ [1 − F (1 / (cid:19) − ε / − st − / u (cid:48) (1) + κ (1 − t )] ( F ( t ) − F (1 /
2) + κ [(1 / − t ) F (1 /
2) + t − / ε (1 / − s ) (cid:18) [ u (cid:48) (0) − κs ] F (1 / − F ( s )1 / − s − [ u (cid:48) (1) + κ (1 − t )] F ( t ) − F (1 / t − / (cid:19) ∂φ ( α, α ε − α ) ≤ for all ε > , s ∈ [0 , / and t ∈ (1 / , if and only if thecondition in Proposition 2 holds. Proof of necessity part of Proposition 3.
Suppose Condition LQ holds and c ∗ ∈ (0 , . Let α ∈ A be the action rule induced by the delegation set [ c ∗ , . We prove necessity of eachcondition in Proposition 3 in order.Condition (i): This follows from Lemma A.5.Condition (ii): Fix t ∈ ( c ∗ / , c ∗ ) and ε > . Let a ( ε ) be the positive solution to ( c ∗ − t ) a + a / ε ( t − c ∗ / and define α ε by α ε ( v ) = α ( v ) if v / ∈ [ c ∗ / , c ∗ + a ( ε )] c ∗ − ε if v ∈ [ c ∗ / , t ) c ∗ + a ( ε ) if v ∈ [ t, c ∗ + a ( ε )] . Note that for any ε > small enough (so that c ∗ − ε > c ∗ / and c ∗ + a ( ε ) < ), α ε ∈ A . Bydefinition of a ( ε ) , for any v ∈ [0 , , vα ε ( v ) − [ α ε ( v )] − (cid:82) v α ε ( s )d s ≥ and, hence, α ε is feasiblefor (R). Therefore, if α is optimal, ∂φ ( α, α ε − α ) ≤ .Note that ∂φ ( α, α ε − α ) = − ε (cid:90) tc ∗ / (cid:18) u (cid:48) ( α ( v )) f ( v ) − κf ( v ) (cid:20) v − c ∗ − − F ( v ) f ( v ) (cid:21)(cid:19) d v + a ( ε ) (cid:90) c ∗ t (cid:18) u (cid:48) ( α ( v )) f ( v ) − κf ( v ) (cid:20) v − c ∗ − − F ( v ) f ( v ) (cid:21)(cid:19) d v + (cid:90) c ∗ + a ( ε ) c ∗ ( c ∗ + a ( ε ) − v ) (cid:18) u (cid:48) ( α ( v )) f ( v ) − κf ( v ) (cid:20) − − F ( v ) f ( v ) (cid:21)(cid:19) d v. By the implicit function theorem, lim ε → a ( ε ) ε = t − c ∗ / c ∗ − t . It follows that the last integral is of order o ( ε ) . Also, integration by parts implies that for x, y ∈ R , (cid:82) yx ( f ( v )( v − c ∗ ) − [1 − F ( v )])d v = F ( y )( y − c ∗ ) − F ( x )( x − c ∗ ) − y + x . We conclude lim ε → + ∂φ ( α, α ε − α ) ε = − u (cid:48) ( c ∗ )[ F ( t ) − F ( c ∗ / κ [ F ( t )( t − c ∗ ) + F ( c ∗ / c ∗ − c ∗ / − t + c ∗ / t − c ∗ / c ∗ − t [ u (cid:48) ( c ∗ )( F ( c ∗ ) − F ( t )) − κ ( c ∗ − t )( F ( t ) − − c ∗ − c ∗ / c ∗ − t u (cid:48) ( c ∗ )[ F ( t ) − F ( c ∗ / κ ( c ∗ − c ∗ / F ( c ∗ / − F ( t )] t − c ∗ / c ∗ − t u (cid:48) ( c ∗ )( F ( c ∗ ) − F ( c ∗ / c ∗ − c ∗ / t − c ∗ / c ∗ − t × (cid:26) − [ u (cid:48) ( c ∗ ) + κ ( c ∗ − t )] F ( t ) − F ( c ∗ / t − c ∗ / u (cid:48) ( c ∗ ) F ( c ∗ ) − F ( c ∗ / c ∗ − c ∗ / (cid:27) . Since ∂φ ( α, α ε − α ) ≤ for all ε > , the last expression is negative for all t ∈ ( c ∗ / , c ∗ ) , whichimplies condition (ii).Condition (iii): Fix s ∈ [0 , c ∗ / and ε > . Let α ε ( v ) = for v < sε if v ∈ [ s, c ∗ / c ∗ − a ( ε ) if v ∈ [ c ∗ / , c ∗ − a ( ε )) v if v ≥ c ∗ − a ( ε ) , where a ( ε ) ≥ satisfies c ∗ / c ∗ − a ( ε )) − ( c ∗ − a ( ε )) / − ( c ∗ / − s ) ε = 0 ⇐⇒ ( c ∗ − c ∗ / a ( ε ) − ( a ( ε )) = ( c ∗ / − s ) ε. For ε small enough, there is a real solution a ( ε ) ≥ , and a simple calculation shows that α ε isfeasible for (R). Also, note that lim ε → a ( ε ) ε = c ∗ / − sc ∗ − c ∗ / .It follows from (A.7) that ∂φ ( α, α ε − α ) = ε (cid:34)(cid:90) c ∗ / s u (cid:48) ( α ( v )) − κ (cid:20) v − − F ( v ) f ( v ) (cid:21) d F ( v ) (cid:35) − a ( ε ) (cid:20)(cid:90) c ∗ c ∗ / u (cid:48) ( α ( v )) − κ (cid:20) v − c ∗ − − F ( v ) f ( v ) (cid:21) d F ( v ) (cid:21) + o ( ε ) . Using integration by parts, we conclude lim ε → + ∂φ ( α, α ε − α ) ε = u (cid:48) (0)[ F ( c ∗ / − F ( s )] − κ [ c ∗ / F ( c ∗ / − sF ( s ) − c ∗ / s ] − c ∗ / − sc ∗ − c ∗ / u (cid:48) ( c ∗ )[ F ( c ∗ ) − F ( c ∗ / κ ( c ∗ − c ∗ / − F ( c ∗ / c ∗ / − s ) (cid:26) [ u (cid:48) (0) − κs ] F ( c ∗ / − F ( s ) c ∗ / − s − u (cid:48) ( c ∗ ) F ( c ∗ ) − F ( c ∗ / c ∗ − c ∗ / (cid:27) ≤ , B. Proofs of Corollaries 1, 2, and 3
B.1. Proof of Corollary 1
Since u is concave, u (cid:48) is decreasing on [0 , . Recall κ ≥ . Hence, if the type density f isdecreasing on [0 , , then κF − u (cid:48) f is increasing on [0 , . The result follows from Proposition 1. B.2. Proof of Corollary 2 As κF ( v ) − u (cid:48) ( v ) f ( v ) is continuous on [0 , , it is increasing on [0 , if its derivative ispositive for all v ∈ [0 , . The derivative is ( κ − u (cid:48)(cid:48) ( v )) f ( v ) − u (cid:48) ( v ) f (cid:48) ( v ) , which is larger than u (cid:48)(cid:48) ( v ) f ( v ) − u (cid:48) ( v ) f (cid:48) ( v ) . The latter function is positive for all v ∈ [0 , if inf v ∈ [0 , − u (cid:48)(cid:48) ( v ) u (cid:48) ( v ) ≥ sup v ∈ [0 , f (cid:48) ( v ) f ( v ) . The RHS above is finite since f is continuously differentiable and strictly positive on [0 , .Therefore, κF ( v ) − u (cid:48) ( v ) f ( v ) is increasing on [0 , when the LHS above is sufficiently large.The result follows from Proposition 1. B.3. Proof of Corollary 3
Assume Condition LQ. We prove the result by establishing that (i) logconcavity of f on [0 , ensures that the conditions of either Proposition 2 or Proposition 3 are satisfied, and (ii)if γ > (equivalently, given Condition LQ, u is strictly concave) or f is strictly logconcave on [0 , , then among interval delegation sets there is a unique optimum.As introduced in Section 4, Proposer’s expected utility from delegating the interval [ c, with c ∈ [0 , is: W ( c ) ≡ u (0) F ( c/
2) + u ( c )( F ( c ) − F ( c/ (cid:90) c u ( v ) f ( v )d v. (B.1)As shorthand for the function in condition (i) of Proposition 3, define G ( v ) := κF ( v ) − u (cid:48) ( v ) f ( v ) . (B.2)We establish some properties of the W and G functions.41 emma B.1. Assume Condition LQ and f is logconcave on [0 , . The functions W and G defined by (B.1) and (B.2) are respectively quasiconcave and quasiconvex on [0 , , both strictly so if either γ > or f is strictly logconcave on [0 , . Furthermore, for any c ∗ ∈ arg max c ∈ [0 , W ( c ) , G (cid:48) ( c ∗ / ≤ if c ∗ > and G (cid:48) ( c ∗ ) ≥ if c ∗ < . Proof.
The proof proceeds in four steps. Throughout, we restrict attention to the domain [0 , for the type density. Step 1 shows that G is (strictly) quasiconvex and that { v : G (cid:48) ( v ) = 0 } isconnected. Step 2 shows that W can be expressed in terms of G (cid:48) . Step 3 establishes that givenany maximizer c ∗ of W , G is decreasing on [0 , c ∗ / and increasing on [ c ∗ , . Step 4 establishesthe (strict) quasiconcavity of W . Note that under Condition LQ, κ ≡ inf v ∈ [0 , − u (cid:48)(cid:48) ( v ) = 2 γ , u (cid:48) ( v ) = 1 − γ + 2 γ (1 − v ) , and hence G ( v ) = 2 γF ( v ) − (1 − γ + 2 γ (1 − v )) f ( v ) .Step 1: We first establish that G is (strictly) quasiconvex and that { v : G (cid:48) ( v ) = 0 } is con-nected. Logconcavity of f implies that its modes (i.e., maximizers) are connected, and more-over f (cid:48) ( v ) = 0 = ⇒ v is a mode. Denote by Mo the smallest mode. Since G (cid:48) ( v ) = 4 γf ( v ) − (1 − γ + 2 γ (1 − v )) f (cid:48) ( v ) , (B.3)it holds that sign G (cid:48) ( v ) = sign β ( v ) , where β ( v ) := 4 γ − f (cid:48) ( v ) f ( v ) (1 − γ + 2 γ (1 − v )) . On the domain [0 , Mo) , f (cid:48) /f is positive and decreasing by logconcavity. Furthermore, − γ + 2 γ (1 − v ) is positive and decreasing. As the product of positive decreasing functions isdecreasing, β is increasing on the domain [0 , Mo) . Since β ( v ) ≥ when v ≥ Mo , it follows that β is upcrossing (once strictly positive, it stays positive), and hence G is quasiconvex.We claim { v : β ( v ) = 0 } is connected, which implies the same about { v : G (cid:48) ( v ) = 0 } . If γ = 0 then β ( v ) = 0 ⇐⇒ f (cid:48) ( v ) = 0 , which is a connected set, as noted earlier. If γ > , thenthe conclusion follows because β is increasing on [0 , Mo) , β ( v ) > for v > Mo (as f (cid:48) ( v ) ≤ ),and β is continuous. Furthermore, analogous observations imply that if either f is strictlylogconcave or γ > , then |{ v : G (cid:48) ( v ) = 0 }| ≤ and so G is strictly quasiconvex.Step 2: We now show that W (cid:48) ( c ) = (cid:90) cc/ ( v − c ) G (cid:48) ( v )d v. (B.4)42he derivation is as follows: W (cid:48) ( c ) =( F ( c ) − F ( c/ γ − γc ) − c f ( c/ γ − γc )=(1 + γ − γc ) (cid:20)(cid:90) cc/ f ( v )d v − c f ( c/ (cid:21) − γ c f ( c/ − (1 + γ − γc ) (cid:90) cc/ ( v − c ) f (cid:48) ( v )d v − γ c f ( c/ − (cid:90) cc/ ( v − c )(1 + γ − γv ) f (cid:48) ( v )d v + 2 γ (cid:20) − (cid:90) cc/ ( v − c ) f (cid:48) ( v )d v − (cid:16) c (cid:17) f ( c/ (cid:21) = − (cid:90) cc/ ( v − c )(1 + γ − γv ) f (cid:48) ( v )d v + 2 γ (cid:90) cc/ v − c ) f ( v )d v = (cid:90) cc/ ( v − c ) G (cid:48) ( v )d v. The first equality above is obtained by differentiating Equation B.1 and using u (cid:48) ( c ) = 1 + γ − γc and u ( c ) − u (0) = c (1 + γ − γc ) ; the third and fifth equalities use integration by parts;the last equality involves substitution from (B.3); and the remaining equalities follow fromalgebraic manipulations.Step 3: We now establish that for any c ∗ ∈ arg max c ∈ [0 , W ( c ) , c ∗ > ⇒ G (cid:48) ( c ∗ / ≤ and c ∗ < ⇒ G (cid:48) ( c ∗ ) ≥ .By Step 1, there exist v ∗ and v ∗ with ≤ v ∗ ≤ v ∗ ≤ such that G (cid:48) ( v ) < on [0 , v ∗ ) , G (cid:48) ( v ) = 0 on [ v ∗ , v ∗ ] , and G (cid:48) ( v ) > on ( v ∗ , . By (B.4), c ∈ (0 , v ∗ ) = ⇒ W (cid:48) ( c ) > , and c/ ∈ ( v ∗ ,
1) = ⇒ W (cid:48) ( c ) < . Since c ∗ is optimal, c ∗ > ⇒ W (cid:48) ( c ∗ ) ≥ ⇒ c ∗ / ≤ v ∗ = ⇒ G (cid:48) ( c ∗ / ≤ .Similarly, c ∗ < ⇒ W (cid:48) ( c ∗ ) ≤ ⇒ c ∗ ≥ v ∗ = ⇒ G (cid:48) ( c ∗ ) ≥ .Step 4: Finally we establish that W is quasiconcave, strictly if γ > or f is strictly logcon-cave. For this it is sufficient to establish that if c > and W (cid:48) ( c ) = 0 , then W (cid:48)(cid:48) ( c ) ≤ , with astrict inequality if γ > or f is strictly logconcave.Differentiating (B.4), W (cid:48)(cid:48) ( c ) = c G (cid:48) ( c/ − ( G ( c ) − G ( c/ . (B.5)Integrating by parts, (cid:90) cc/ [( v − c ) G (cid:48) ( v ) + G ( v )]d v = [( v − c ) G ( v )] cc/ = c G ( c/ . c > such that W (cid:48) ( c ) = 0 . By (B.4) and the above integration by parts, G ( c/
2) =(2 /c ) (cid:82) cc/ G ( v )d v , which, because G is quasiconvex by Step 1, implies G ( c/ ≤ G ( c ) , with astrict inequality if γ > or f strictly logconcave. Similarly G (cid:48) ( c/ ≤ , and hence from (B.5)we conclude that W (cid:48)(cid:48) ( c ) ≤ , with a strict inequality if γ > or f is strictly logconcave.We build on Lemma B.1 to establish Corollary 3 by verifying the conditions of Proposi-tion 2 and Proposition 3. Proof of Corollary 3.
If the interval delegation set [ c ∗ , is optimal then c ∗ must maximize W ( c ) defined in (B.1). Hence if W is strictly quasiconcave—as is the case if γ > or f isstrictly logconcave on [0 , , by Lemma B.1—there can be at most one interval that is optimal.So it suffices to establish that if c ∗ ∈ arg max c ∈ [0 , W ( c ) then [ c ∗ , is optimal.To that end, we verify that if c ∗ = 1 the conditions of Proposition 2 are satisfied and,if c ∗ < , then conditions (i)–(iii) of Proposition 3 are satisfied. Note that condition (i) isimmediate from Lemma B.1. As conditions (ii) and (iii) are vacuous for c ∗ = 0 we need onlyconsider c ∗ ∈ (0 , . For any c ∗ ∈ (0 , conditions (ii) and (iii) are jointly equivalent to ( u (cid:48) (0) − κs ) F ( c ∗ / − F ( s ) c ∗ / − s ≤ u (cid:48) ( c ∗ ) F ( c ∗ ) − F ( c ∗ / c ∗ / ≤ ( u (cid:48) ( c ∗ ) + κ ( c ∗ − t )) F ( t ) − F ( c ∗ / t − c ∗ / for all s ∈ [0 , c ∗ / and t ∈ ( c ∗ / , c ∗ ] . Substituting into the middle expression from the first-order condition W (cid:48) ( c ∗ ) = 0 (i.e., setting expression (2) equal to zero and rearranging) yields ( u (cid:48) (0) − κs ) F ( c ∗ / − F ( s ) c ∗ / − s ≤ ( u ( c ∗ ) − u (0)) f ( c ∗ / c ∗ ≤ ( u (cid:48) ( c ∗ ) + κ ( c ∗ − t )) F ( t ) − F ( c ∗ / t − c ∗ / (B.6)for all s ∈ [0 , c ∗ / and t ∈ ( c ∗ / , c ∗ ] . So if (B.6) holds for c ∗ ∈ (0 , then the conditions inProposition 3 are verified. On the other hand, since the condition in Proposition 2 is equiv-alent to the right-most term in (B.6) being larger than the left-most term for all s ∈ [0 , c ∗ / and t ∈ ( c ∗ / , c ∗ ] when c ∗ = 1 , (B.6) holding for c ∗ = 1 implies the condition in Proposition 2.Accordingly, we fix a c ∗ > and verify the two inequalities of (B.6) in turn.First inequality of (B.6): Using u (cid:48) ( a ) = 1 + γ − γa , κ = 2 γ , and u ( a ) − u (0) a = 1 + γ − γa , thefirst inequality of (B.6) reduces to (1 + γ − γs ) F ( c ∗ / − F ( s ) c ∗ / − s ≤ (1 + γ − γc ∗ ) f ( c ∗ / ∀ s ∈ [0 , c ∗ / . It follows from L’Hopital’s rule that the above inequality holds with equality in the limit44s s → c ∗ / . Hence it is sufficient to demonstrate that the LHS of the inequality is increasingfor all s ∈ [0 , c ∗ / . For any s ∈ [0 , let D ( s ) := (1 + γ − γc ∗ )( F ( c ∗ / − F ( s )) − ( c ∗ / − s )(1 + γ − γs ) f ( s ) , (B.7)and observe that ∂∂s (cid:20) (1 + γ − γs ) F ( c ∗ / − F ( s ) c ∗ / − s (cid:21) = 1( c ∗ / − s ) D ( s ) . So it is sufficient to show that, for all s ∈ [0 , c ∗ / , D ( s ) ≥ . This holds because D ( c ∗ /
2) = 0 and, for all s < c ∗ / , D (cid:48) ( s ) = ( c ∗ / − s )[4 γf ( s ) − (1 + γ − γs ) f (cid:48) ( s )] differentiating (B.7) and simplifying = ( c ∗ / − s ) G (cid:48) ( s ) substituting from (B.3) (B.8) ≤ by Lemma B.1 . Second inequality of (B.6): Using u (cid:48) ( a ) = 1 + γ − γa , κ = 2 γ , and u ( a ) − u (0) a = 1 + γ − γa ,the second inequality of (B.6) reduces to (1 + γ − γc ∗ ) f ( c ∗ / ≤ (1 + γ − γt ) F ( t ) − F ( c ∗ / t − c ∗ / ∀ t ∈ ( c ∗ / , c ∗ ] . Using L’Hopital’s rule for the limit as t → c ∗ / and the fact that W (cid:48) ( c ∗ ) ≥ by optimality of c ∗ > , it follows that lim t → c ∗ / (1 + γ − γt ) F ( t ) − F ( c ∗ / t − c ∗ / γ − γc ∗ ) f ( c ∗ / ≤ (1 + γ − γc ∗ ) F ( c ∗ ) − F ( c ∗ / c ∗ / . Hence it is sufficient to show that (1 + γ − γt ) F ( t ) − F ( c ∗ / t − c ∗ / is quasiconcave for t ∈ ( c ∗ / , c ∗ ] .Note that ∂∂t (cid:20) (1 + γ − γt ) F ( t ) − F ( c ∗ / t − c ∗ / (cid:21) = 1( t − c ∗ / D ( t ) , where D is defined in (B.7), and so sign ∂∂t (cid:20) (1 + γ − γt ) F ( t ) − F ( c/ t − c/ (cid:21) = sign D ( t ) . Since D ( c ∗ /
2) = 0 , it follows that (1 + γ − γt ) F ( t ) − F ( c ∗ / t − c ∗ / is quasiconcave for t ∈ ( c ∗ / , c ∗ ] if D is quasiconcave. D is quasiconcave because, as was shown in (B.8), D (cid:48) ( t ) = ( c ∗ / − t ) G (cid:48) ( t ) ,45hich is positive then negative on ( c ∗ / , c ∗ ] by the quasiconvexity of G (Lemma B.1). C. Proof of Proposition 4
Proof of Proposition 4(i).
Let H ( a, c ) denote the cumulative distribution function of the ac-tion implemented under the interval delegation set [ c, . That is, H ( a, c ) = if a < F ( c/ if ≤ a < cF ( a ) if c ≤ a < if ≤ a. For any ≤ c L < c H ≤ the difference H ( · , c L ) − H ( · , c H ) is upcrossing: once strictlypositive, it stays positive. It follows from the variation diminishing property (Karlin, 1968,Theorem 3.1 on p. 21) that (cid:90) ˆ u (cid:48) ( a ) [ H ( a, c L ) − H ( a, c H )] d a ≥ ( > )0 = ⇒ (cid:90) u (cid:48) ( a ) [ H ( a, c L ) − H ( a, c H )] d a ≥ ( > )0 when ˆ u is strictly more risk averse than u . Integrating by parts, we obtain (cid:90) ˆ u ( a ) [ H (d a, c L ) − H (d a, c H )] ≤ ( < )0 = ⇒ (cid:90) u ( a ) [ H (d a, c L ) − H (d a, c H )] ≤ ( < )0 . A standard monotone comparative statics argument (Milgrom and Shannon, 1994) then im-plies that C ∗ ( u ) ≥ SSO C ∗ (ˆ u ) . Proof of Proposition 4(ii).
Let density f ( v ) strictly dominate density g ( v ) in likelihood ratioon the unit interval: i.e., for all ≤ v L < v H ≤ , f ( v L ) g ( v H ) < f ( v H ) g ( v L ) . Let w ( c, v ) denoteProposer’s payoff under the interval delegation set [ c, when Vetoer’s type is v . We have w ( c, v ) = u (0) if v ∈ [0 , c/ u ( c ) if v ∈ ( c/ , c ) u ( v ) if v > ∈ ( c, . Consider any ≤ c L < c H ≤ . The difference w ( c H , · ) − w ( c L , · ) is upcrossing: once strictlypositive, it stays positive. It follows from the variation diminishing property (Karlin, 1968,46heorem 3.1 on p. 21) that (cid:90) [ w ( c H , v ) − w ( c L , v )] g ( v )d v ≥ ( > )0 = ⇒ (cid:90) [ w ( c H , v ) − w ( c L , v )] f ( v )d v ≥ ( > )0 . A standard monotone comparative statics argument (Milgrom and Shannon, 1994) then im-plies that C ∗ ( f ) ≥ SSO C ∗ ( g ) . D. Proof of Proposition 5
Let a U and a I denote proposals in some noninfluential and influential cheap-talk equilib-ria, respectively (the latter may not exist). It is straightforward that a U > and, if it exists, a I ∈ (0 , . Since Proposition 5’s conclusion is trivial for full delegation ( c ∗ = 0 ), it suffices toestablish that any optimal interval delegation set [ c ∗ , with c ∗ ∈ (0 , has c ∗ < min { a I , a U } .(By convention, min { a I , a U } = a U if a I does not exist.)Plainly, a U is a noninfluential equilibrium proposal if and only if a U ∈ arg max a [ u (0) F ( a/
2) + u ( a )(1 − F ( a/ , and so if a U < then it solves the first-order condition u (cid:48) ( a ) [1 − F ( a/ − f ( a/
2) [ u ( a ) − u (0)] = 0 . (D.1)Any influential cheap-talk equilibrium outcome can be characterized by a threshold type v I ∈ (0 , such that types v < v I pool on the “veto threat” message, and types v > v I poolon the “acquiesce” message. Since type v I must be indifferent between sending the two mes-sages, and she will accept either proposal from the Proposer, it holds that v I = 1 + a I . It follows that a I is an influential equilibrium proposal if and only if a I ∈ arg max a (cid:20) u (0) F ( a/ F ((1 + a I ) /
2) + u ( a ) (cid:18) − F ( a/ F ((1 + a I ) / (cid:19)(cid:21) . The first-order condition is that function (3) in the main text equals zero, i.e., a I ∈ (0 , solves u (cid:48) ( a ) [ F ((1 + a ) / − F ( a/ − f ( a/
2) [ u ( a ) − u (0)] = 0 . (D.2)47ote that at a = 0 , the LHS is strictly positive. Hence, if the LHS is strictly downcrossingon (0 , , then Equation D.2 has at most one solution in that domain; if there is a solution,then Equation D.2’s LHS is strictly positive (resp., strictly negative) to its left (resp., right);furthermore, it can be verified that the solution then identifies an influential equilibrium. Notethat if there is no solution to Equation D.2 on (0 , then there is no influential equilibrium.Turning to optimal interval delegation, recall from Section 4 that the threshold is a zero ofthe function (2), i.e., c ∗ ∈ (0 , solves u (cid:48) ( a ) [ F ( a ) − F ( a/ − f ( a/
2) [ u ( a ) − u (0)] = 0 . (D.3)If the LHS is strictly downcrossing on (0 , , then on that domain c ∗ is the unique solutionto Equation D.3 and Equation D.3’s LHS is strictly positive (resp., strictly negative) to thesolution’s left (resp., right).For any a ∈ (0 , the LHS of Equation D.1 is strictly larger than the LHS of Equation D.2,which in turn is strictly larger than the LHS of Equation D.3. If there is no solution in (0 , to Equation D.2, then its LHS is always strictly positive, and hence there are neither anyinfluential equilibria nor any noninfluential equilibria with a U < , and we are done. Soassume at least one solution in (0 , to Equation D.2. Let a := inf { a ∈ (0 ,
1) :
Equation D.2’s LHS ≤ } ,a := sup { a ∈ (0 ,
1) :
Equation D.2’s LHS ≥ } , and analogously define a and a using Equation D.3’s LHS. The aforementioned ordering ofthe equations’ LHS, Equation D.2’s LHS being strictly positive at 0, and continuity combineto imply ≤ a < a < a U , and a ≤ a with a strict inequality if either a < or a < .Furthermore, a I ∈ [ a , a ] and c ∗ ∈ [ a , a ] .If the LHS of Equation D.2 is strictly downcrossing on (0 , , then by the properties notedright after Equation D.2, a I = a = a < and hence c ∗ < min { a I , a U } . If the LHS ofEquation D.3 is strictly downcrossing on (0 , , then by the properties noted right after Equa-tion D.3, c ∗ = a = a < and hence c ∗ < min { a I , a U } . E. Stochastic Mechanisms can be Optimal
Example