key: cord-0140127-ehbl9xr0
authors: Kartik, Navin; Kleiner, Andreas; Weelden, Richard Van
title: Delegation in Veto Bargaining
date: 2020-06-11
journal: nan
DOI: nan
sha: 5076aa767f4a0dd8383428b010863c28b9b897b6
doc_id: 140127
cord_uid: ehbl9xr0

A proposer requires the approval of a veto player to change a status quo. Preferences are single peaked. Proposer is uncertain about Vetoer's ideal point. We study Proposer's optimal mechanism without transfers. Vetoer is given a menu, or a delegation set, to choose from. The optimal delegation set balances the extent of Proposer's compromise with the risk of a veto. Under reasonable conditions,"full delegation"is optimal: Vetoer can choose any action between the status quo and Proposer's ideal action. This outcome largely nullifies Proposer's bargaining power; Vetoer frequently obtains her ideal point, and there is Pareto efficiency despite asymmetric information. More generally, we identify when"interval delegation"is optimal. Optimal interval delegation can be a Pareto improvement over cheap talk. We derive comparative statics. Vetoer receives less discretion when preferences are more likely to be aligned, by contrast to expertise-based delegation. Methodologically, our analysis handles stochastic mechanisms.

Motivation. There are numerous situations in which one agent or group can make proposals 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 unconstitutional 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 house to his client, but the client must decide to put in an offer; similarly, a search committee can put forward a candidate, but the organization decides whether to hire her. Romer and Rosenthal (1978) present a seminal analysis of such veto bargaining. Their framework is one of complete information in which a proposer (Congress, government, prosecutor, 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. That is, 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 uncertain about the veto player's preferences-specifically, which proposals would actually get vetoed. 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 approach 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., one round of cheap talk in Matthews, 1989; Forges and Renault, 2020) . Instead, we consider the possible outcomes across all possible protocols by taking a mechanism design approach. Our focus is on identifying the proposer's optimum.

There are at least two reasons this mechanism design approach is of interest. First, it identifies an upper bound on the proposer's welfare. Second, as is standard in settings without transfers, any (deterministic) mechanism is readily interpreted and implemented as delegation. That is, the proposer simply offers a menu of options; the veto player can select any one or reject them all. Menus or delegation sets are observed in practice in some applications of our model. Salespeople show customers subsets of products, and search committees put forward multiple candidates for their organization to choose among. In politics, a bill authorizing at most $x of spending effectively offers an executive who controls the implementing bureaucracy the choice of any spending level in the interval [0, x] . Bills can also grant more or less discretion of how to allocate a given level of spending, as encapsulated by former Senator Russ Feingold in the context of the recent U.S. coronavirus stimulus package: "Congress has to decide how much discretion it wants to delegate to executive branch officials." (Washington Post, March 22, 2020.) Main results. We formally study a one-dimensional environment. There is status quo policy or action, 0, that obtains if there is a veto. There are two agents. Proposer's ideal action is 1. Vetoer's ideal action, v, is her private information. We assume Vetoer preferences are represented by a quadratic loss function, but allow Proposer to have any concave utility function. 1 Our first result (Proposition 1) identifies conditions under which it is optimal for Proposer to simply let Vetoer choose her preferred action in the interval [0, 1]. We call this full delegation, because Proposer only excludes options that are, from his point of view, dominated by simply offering his ideal action 1 no matter Vetoer's ideal point. Intuitively, full delegation is optimal when the specter of a veto looms large; in particular, it is sufficient that the density of Vetoer's ideal 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 who frequently gets her first best. This is a telling manifestation of private information's consequences. Unlike in most other settings that confer information rents, however, full delegation implies ex-post Pareto efficiency.

The opposite of full delegation is no compromise: Proposer only offers his ideal action, 1. Of course, Vetoer can veto and choose the status quo 0. Proposition 2 gives conditions for optimality of no compromise. It is sufficient, for example, that Proposer has a linear loss function-so, in the relevant region of actions, [0, 1] , he only cares about the mean actionand the density of Vetoer's ideal point is increasing in this region. This case juxtaposes nicely against the aforementioned decreasing-density condition for full delegation.

Both full delegation and no compromise are boundary cases of interval delegation: Proposer offers a menu of the form [c, 1] . Interval delegation is interesting for multiple reasons. Among them are that such delegation sets are simple to interpret and implement, and they turn out to be tractable for comparative statics. Proposition 3 provides conditions under which interval delegation is optimal. These are met, in particular, when Proposer has a linear or quadratic loss function and Vetoer's ideal point distribution is logconcave (Corollary 3).

We show that, under reasonable conditions, optimal interval delegation yields a Pareto improvement over singleton proposals, even when cheap talk is allowed and full delegation is not optimal (Proposition 5). We trace the intuition to Proposer being more willing to compromise when he can offer Vetoer an interval of options rather than only singletons.

We develop two comparative statics, restricting attention to interval delegation -either justified by optimality or otherwise. First, what happens when Proposer becomes more risk averse? Proposition 4(i) establishes that Vetoer is given more discretion: the optimal threshold in the interval delegation set (i.e., that denoted c above) decreases. Intuitively, Proposer offers a larger set of options to mitigate the risk of a veto. Second, what about when Vetoer becomes more ex-ante aligned with Proposer? Formally, we consider right shifts in Vetoer's ideal point distribution, in the sense of likelihood ratio dominance. Proposition 4(ii) establishes that discretion decreases: the optimal interval delegation threshold increases. Intuitively, this is because Proposer is less concerned that a veto will occur. Although these comparative statics appear natural, it is the structure of interval delegation that allows us to establish them.

Contrast with expertise-based delegation. The second comparative static mentioned above contrasts with a key theme of the expertise-based delegation literature following Holmström (1977, 1984) . In that literature, an agent is given discretion over actions because her private information is valuable to the principal; the principal limits the degree of discretion because of preference misalignment. One version of the so-called Ally Principle says that a more aligned agent receives more discretion. Holmström (1984) establishes its validity under reasonably general conditions, so long as delegation sets take the form of intervals. 2 In our setting, the reason Proposer gives Vetoer discretion is fundamentally different from that in Holmström: it is not to benefit from Vetoer's expertise; rather, Proposer trades off the risk of a veto with the extent of compromise. (In jargon, our delegator has state-independent preferences, by contrast to the state-dependent preferences in most of the literature following Holmström.) Hence we find less discretion emerging when there is, in a suitable sense, more ex-ante preference alignment.

Methodology. We hope some readers will find our analysis interesting on a methodological level. While it is convenient and economically insightful to describe our substantive results in terms of optimal delegation sets, the formal problem we study is one of mechanism design without transfers. Our analytical methodology builds on the infinite-dimensional Langrangian approach advanced by Amador, Werning, and Angeletos (2006) and Amador and Bagwell (2013) . Unlike these authors and many others, including the important contributions by Melumad and Shibano (1991) and Alonso and Matouschek (2008) , we also cover stochastic mechanisms. 3 That is, we allow for mechanisms in which Vetoer may choose among lotteries over actions. We view such mechanisms as not only theoretically important, but also relevant in 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 candidate with at least five years' work experience in that country". Both parties would view this option as a lottery over some subset of candidates.

Stochastic mechanisms can sometimes be optimal in our framework. Nevertheless, we establish that our sufficient conditions for full delegation, no compromise, and interval delegation (Propositions 1, 2, and 3) ensure optimality of these (deterministic) mechanisms even among stochastic mechanisms. Furthermore, by permitting stochastic mechanisms, our sufficient conditions are shown to also be necessary for a class of Proposer's utility functions that include linear and quadratic loss. Our approach to handling stochastic mechanisms should be useful in other delegation problems.

delegator's preferences are state independent.

Like us, Amador and Bagwell (2019) directly analyze a delegation problem with an outside option. They focus on monopoly regulation ( Baron and Myerson, 1982) without transfers. Unlike in our paper, their delegator's preferences are state dependent, and they do not consider stochastic mechanisms or necessity.

Outline. The rest of the paper proceeds as follows. Section 2 presents our model. Section 3 contains our main results on the conditions for optimality of full delegation, no compromise, and, more broadly, interval delegation. Section 4 develops comparative statics and makes comparisons with other mechanisms. Section 5 discusses some applications. Section 6 concludes. All proofs are in the appendices.

We consider a classic bargaining problem between two players, a proposer (he) and a veto player (she), who jointly determine a policy outcome or action a ∈ R. In a manner elaborated below, Proposer makes a proposal that Vetoer can either accept or reject. If Vetoer rejects, a status-quo action is preserved; we normalize the status quo to 0.

We assume both players have single-peaked utilities. Proposer's utility is u(a) that is concave, maximized uniquely at a = 1 (essentially a normalization), and twice differentiable at all a = 1. 5 Unless indicated explicitly, we use 'concave', 'increasing', etc., to mean 'weakly concave', 'weakly increasing', etc. We will sometimes invoke a restriction to the following subclass of Proposer preferences, which stipulates a convex combination of the widely-used linear and quadratic loss functions.

Vetoer's utility is represented by −l(|v − a|), where l(·) is strictly increasing. So her utility is symmetric around the unique ideal point v. For tractability, we assume l(|v − a|) = (v − a) 2 . A subset of our results will rely only on Vetoer's ordinal preferences, for which the choice of quadratic loss entails no loss of generality given that Vetoer's utility is symmetric around 5 Permitting a point of nondifferentiability allows the linear loss function u(a) = −|1 − a|. When we write u (1) subsequently, it refers to the left-derivative when u is not differentiable at 1. her ideal point. Specifically, Vetoer's ordinal preferences are sufficient when we consider only deterministic mechanisms.

A key ingredient of our model is that v is Vetoer's private information. We accordingly refer to v as Vetoer's type. It is drawn from a cumulative distribution F whose support is an interval [v, v] , where we permit v = −∞ and/or v = ∞. We assume F admits a continuously differentiable density f , and that f (·) > 0 on [0, 1]. All aspects of the environment except the type v are common knowledge. If v were common knowledge, this model would reduce to that of Romer and Rosenthal (1978) .

Naturally, it is in Proposer's interests to elicit information from Vetoer about v. For example, they might engage in cheap talk communication (Matthews, 1989) , possibly over multiple rounds, or Proposer might make sequential proposals, and so on. To circumvent issues about exactly how the bargaining ensues, we take a mechanism design approach. Following the revelation principle, we consider direct revelation mechanisms, hereafter simply mechanisms. A deterministic mechanism is described by a real-valued function α(v), which specifies the action when Vetoer's type is v, and must satisfy the usual incentive compatibility (IC) and individual rationality (IR) conditions. IC requires that each type v prefers α(v) to α(v ) for any v = v; IR requires that each type v prefers α(v) to the status quo 0. Notice that any deterministic mechanism is equivalent to the Proposer offering a (closed) menu or delegation set A ⊆ R, and Vetoer choosing an action from A ∪ {0}. We will also consider the more general class of stochastic mechanisms, which specify probability distributions over actions for each Vetoer type, with analogous IC and IR constraints to those aforementioned. Stochastic mechanisms are theoretically important because the revelation principle does not justify focussing only on deterministic mechanisms. As noted in the introduction, they may also be relevant for applications. A notable contribution of this paper is to establish conditions under which, despite the absence of transfers, stochastic mechanisms cannot improve upon deterministic ones. 6 We highlight that our model is one of private values: Vetoer's type does not directly affect Proposer's preferences. This is by way of contrast with the delegation literature initiated by Holmström (1984) , in which a principal gives discretion to an agent because of the agent's expertise, i.e., because they have interdependent preferences. We could extend our model and analysis to incorporate this expertise-based delegation or discretion aspect, but one of our main themes is that discretion will emerge even when that is absent.

As mentioned in the introduction, the mechanism design approach we take can be viewed as identifying an upper bound on Proposer's welfare. That said, as also mentioned there, we find 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 probability distributions on R, 7 and M 0 (R) be the subset of distributions with finite expectation and finite variance. Denote by δ a the degenerate distribution that puts probability 1 on action a. A stochastic mechanism-or simply a mechanism without qualification-is a measurable function m : [v, v] → M 0 (R), with m(v) being the probability distribution over actions for type v. 8 To reduce notation, for any deterministic mechanism α : [v, v] → R, we also denote the mechanism v → δ α(v) by α. For any integrable function g :

] denote the expectation of g(a) when a has distribution m(v). We only consider mechanisms m for which

That is, S consists of mechanisms in which type 0 gets the status quo and a higher type receives a higher expected action. The first requirement is implied by IR, since Vetoer can always choose the status quo. The second is implied by IC, since Vetoer's utility −(v − a) 2 is equivalently represented by av − a 2 /2; singlecrossing difference in (a, v) yields monotonicity of E m(v) [a] in v from standard arguments (elaborated in fn. 9 below).

Proposer's problem is:

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 the standard envelope formula. 9 Note that since IC requires that no type prefer type 0's lottery 7 We endow M (R) with the topology of weak convergence and the corresponding Borel σ-algebra. 8 There is no loss in restricting attention to M 0 (R) instead of M (R) because no type would choose a lottery with infinite mean or variance, given that the status quo is available. 9 Formally, using quadratic utility, IC requires ∀v, v , E m(v) [av − a 2 /2] ≥ E m(v ) [av − a 2 /2], and IR requires ∀v,

An IC mechanism m thus satisfies IR if and only if m(0) = δ 0 . It follows that m satisfies IC over its own, and type 0's IR constraint requires that it receive action 0 (captured in S), every type's IR constraint is implied by type 0's. An optimal mechanism is a solution to problem (P).

If we restricted attention to deterministic mechanisms, the analogous problem for Proposer would be:

Any deterministic mechanism α that is IC has a corresponding (closed) delegation set A α := v α(v). Conversely, any delegation set A has a corresponding deterministic mechanism α A where α A (v) is the action in A ∪ {0} that type v prefers the most (with ties broken in favor of Proposer). Note that α A satisfies IC and IR. While our formal analysis works with mechanisms, it is easier and more economically intuitive to describe our main results, which concern certain deterministic mechanisms, using delegation sets.

We emphasize some terminology: an optimal deterministic mechanism (or an optimal delegation set) is a solution to problem (D). But when we say that a deterministic mechanism (or delegation set) is optimal, we mean that it solves problem (P), i.e., no stochastic mechanism can strictly improve on it.

Remark 1. We capture veto power via a standard interim IR constraint. An alternative, as in Compte and Jehiel (2009) , would be to allow Vetoer to exercise her veto even after the mechanism determines an action. This is stronger than just ex-post IR because it also strengthens the IC constraint: when type v mimics type v , v may veto a different set of allocations than v would, and so the action distribution that v evaluates the deviation with is not m(v ). Which form of veto power is conceptually appropriate depends on the application. But any IC and IR deterministic mechanism also satisfies the ex-post veto constraint. Hence, the sufficient conditions we provide below for optimality of delegation sets would remain sufficient. and IR if and only if m ∈ S and the envelope formula (IC-env) holds. To confirm this, let

which holds if and only if V is convex and (Milgrom and Segal, 2002) . Consequently, m is IC and IR if and only if E m(v) [a] is increasing in v, (IC-env) holds, and m(0) = δ 0 .

Consider delegation sets. Notice first that there is no loss for Proposer in including his ideal action 1 in the delegation set: for any Vetoer type v, either it does not affect the chosen action, or it results in a preferable action. Next, there is no loss for Proposer in excluding actions outside [0, 1]: shrinking a delegation set A that contains 1 to A ∩ [0, 1] only results in each type choosing an action closer to 1. We have: Lemma 1. There is an optimal delegation set A satisfying 1 ∈ A ⊆ [0, 1].

It would also be without loss to assume that a delegation set contains the status quo, 0. We don't do so, however, because it is convenient to sometimes describe optimal delegation sets without including 0.

For any a ∈ (0, 1), the delegation set A = [a, 1] strictly dominates the singleton {a} because A results in preferable actions for Proposer when v > a. This simple observation highlights the significance of giving Vetoer discretion, despite our model shutting down the expertisebased rationale that the literature initiated by Holmström (1984) has focused on.

While Proposer always wants to include action 1 in the delegation set, he faces a tradeoff when including any action a ∈ (0, 1). Allowing Vetoer to choose such an action a reduces the probability of a veto (or any action less than a) but also reduces the probability that Vetoer chooses an action even higher than a, which Proposer would prefer to a.

In light of Lemma 1, we refer to the delegation set [0, 1] as full delegation. Note that full delegation does impose some constraints on Vetoer. But the constraints are minimal: only actions outside the convex hull of the status quo and Proposer's ideal point are excluded. Given the veto-bargaining institution, an outcome of full delegation starkly captures how Vetoer's private information can corrode Proposer's bargaining or agenda-setting power. Indeed, if the type distribution's support is [0, 1], then Vetoer gets her first best. Regardless of the support, however, full delegation results in ex-post Pareto efficiency, unlike in most other settings that confer information rents. Full delegation thus contrasts sharply with the outcome under complete information (Romer and Rosenthal, 1978) , in which case Proposer would make Vetoer with ideal point v ∈ (0, 1/2] indifferent with exercising the veto while getting his own ideal action 1 from types v ∈ (1/2, 1). It also contrasts with the outcome under incomplete information were Proposer restricted to making a singleton proposal. In that case the proposal would lead to a veto by some subinterval of types v ∈ [0, 1], hence to ex-post Pareto inefficiency, and all Vetoer types would be weakly worse off, many strictly. 10 It is thus of interest to know when full delegation is optimal. The following quantity concerning the concavity of Proposer's utility will play a key role in our analysis:

(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 mechanisms and assumed that Proposer's utility is a quadratic loss function, then the result would follow from a result in Alonso and Matouschek (2008) , even though their model does not have a veto constraint and, as such, highlights expertise-based delegation. To make the connection, we observe that when u(a) = −(1 − a) 2 , condition (1) is equivalent to Alonso and Matouschek's "backward bias" being convex on [0, 1]. Their Proposition 2 then implies that if {0, 1} is contained in the optimal delegation set, then the interval [0, 1] is contained in the optimal delegation set. But recall from Lemma 1 that in choosing among delegation sets, Proposer need not offer any action outside [0, 1] and can offer his ideal point 1; moreover, he may as well offer the status quo 0. It follows that full delegation is an optimal delegation set.

We therefore emphasize that Proposition 1 establishes optimality among more general Proposer preferences and stochastic mechanisms. 11 The proposition directly implies: Corollary 1. Full delegation is optimal if the type density is decreasing on [0, 1].

In particular, it is sufficient that the type density is unimodal with a negative mode. To obtain intuition for the corollary, consider removing any interval (a, a) from a delegation set 10 Note that vetoes can have positive probability even under full delegation; this is the case if and only if Prob(v < 0) > 0. 11 In a model without an outside option, Kováč and Mylovanov (2009) provide sufficient conditions for certain delegation sets to be optimal when stochastic mechanisms are allowed and Proposer has a quadratic loss function.

A 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 ∈ a, a+a 2 , which harms Proposer, while Vetoer will choose a when v ∈ a+a 2 , a , which benefits Proposer. When the type density is decreasing on [a, a], the former possibility is more likely; in fact, the pruned delegation set induces an action distribution that is second-order stochastically dominated. 12 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 an optimal delegation set. While this explanation applies among delegation sets, Proposition 1 implies 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, 1]. In general, removing an interval 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.

is sufficiently large.

Consider now necessity in Proposition 1. If Proposer has a linear loss utility, then our preceding discussion explains why a delegation set A containing [a, a] ⊆ [0, 1] should be pruned to 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 (and with additional smoothness assumptions), Alonso and Matouschek's (2008) Proposition 2 implies that that a weaker condition,

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, but Proposer is hurt by pruning (a, a) if the expression in condition (1) is increasing on [a, a]. delegation to be optimal. Proposition 1 subsumes these two cases by deducing necessity of condition (1) for the linear-quadratic family of utilities (Condition LQ). 13 We note that were Vetoer's lowest type v ∈ (0, 1), contrary to our assumption that v ≤ 0, then full delegation-or even the interval [v, 1]-would never be an optimal delegation set: Proposer would not offer any action below min{2v, 1}.

The other extreme from full delegation is no compromise: Proposer makes a take-it-or-leaveit offer of his own ideal action, not offering any other action. Of course, Vetoer can choose the status quo as well. When Proposer has a linear loss utility-or, a fortiori, if we had permitted u(a) to be convex on [0, 1]-then no compromise is an optimal delegation set whenever the type density f is increasing on [0, 1]. This follows from reversing the previous subsection's second-order stochastic dominance argument for optimality of full delegation when f is decreasing. But neither is linear loss utility nor convexity of F on [0, 1] required for optimality of no compromise.

Under linear loss utility (so u (1) = u (0) = 1 and κ = 0), the condition in Proposition 2 simplifies to f (1/2) being a subgradient of F at 1/2 on the domain [0, 1]. This subgradient condition is weaker than F being convex on [0, 1].

Remark 2. With linear loss utility, no compromise can be an optimal delegation set (i.e., deterministic mechanism) even if the subgradient condition does not hold. However, there will then be a stochastic mechanism that Proposer strictly prefers. This situation can arise, for example, when the type density is strictly increasing except on a small interval around 1/2, where it is strictly decreasing. Intuitively, Proposer would like to delegate a small set of actions around 1/2 to types close to 1/2, but adding such actions to the no-compromise delegation set is deleterious because it leads to many types above 1/2 choosing an action close to 1/2 rather than 1. By contrast, lotteries with expected value 1/2 can be used to attract only types close to 1/2. Example E.1 in Appendix E elaborates.

We also note that no compromise can be an optimal delegation set even when u does not satisfy Condition LQ. In particular, it can be shown that if no compromise is an optimal delegation set for some u, then it is also an optimal delegation set for any utility function that is a convex transformation of u.

On the other hand, no compromise is not optimal-not even an optimal delegation set-if Proposer's utility is differentiable at his ideal point a = 1 (which implies u (1) = 0). 14 The reason is that when u (1) = 0, Proposer would strictly benefit from offering a small interval [1−ε, 1], or even just the action 1−ε, instead of only offering action 1. For, Proposer's decrease in utility from getting an action slightly lower than 1 is second order, but there is a first-order increase in the probability of avoiding a veto.

Both full delegation and no compromise are special cases of interval delegation: Proposer offers an interval, and Vetoer chooses an action from either that interval or the status quo. It follows from Lemma 1 that when interval delegation is optimal, there is always an optimal interval of the form [c, 1] for some c ∈ [0, 1]. One can thus interpret interval delegation as Proposer designating a minimally acceptable option; implicitly, the maximal acceptable option is Proposer's ideal point. Interval delegation, without a status quo, has been a central focus of the 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.

Conversely, under Condition LQ, the delegation set [c * , 1] with c * ∈ (0, 1) is optimal only if conditions (i), (ii), and (iii) above hold.

We discuss sufficiency. The intuition for condition (i) in the proposition is analogous to that discussed after Proposition 1; it ensures that there is no benefit to not fully delegating the interval [c * , 1] taking as given that Vetoer can choose c * . For linear loss utility, the condition

reduces to F being concave on [c * , 1]. Linear loss utility is also helpful to interpret the other conditions. Conditions (ii) and (iii) then simplify to the requirements that the average density from c * /2 to c * be simultaneously less than that from c * /2 to t for all t ∈ (c * /2, c * ] and greater than that from s to c * /2 for all s ∈ [0, c * /2). Equivalently, the average density from c * /2 to c * equals f (c * /2) and f (c * /2) is a subgradient of F at c * /2 on the domain [0, c * ]. The subgradient condition 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 ensures that the threshold c * is an optimal threshold. See Figure 1 .

Readers familiar with Amador and Bagwell (2013) will note from the discussion above that condition (i) in Proposition 3 plays the same role as condition (c1) in that paper. In fact, the two conditions are identical, even though our analysis accommodates stochastic mechanisms that cannot be reduced to their money burning. 15 Conditions (ii) and (iii) of Proposition 3 don't have analogs in Amador and Bagwell's work, however, because these concern optimality conditions that turn on our status quo.

Remark 3. For linear loss utility, it can be verified that if F is strictly unimodal (i.e., strictly convex and then strictly concave), then interval delegation is optimal. Either there will be a c * ∈ [0, 1] satisfying the three conditions of Proposition 3 or the condition in Proposition 2 will be met and no compromise is optimal. 16 We can extend this observation as follows:

Corollary 3. Assume Condition LQ. Interval delegation is optimal if the type density f is logconcave on [0, 1]; if, in addition, f is strictly logconcave on [0, 1] or Proposer's utility is strictly concave, then there is a unique optimal interval.

Recall that logconcavity is stronger than unimodality, but many familiar distributions have logconcave densities, including the uniform, normal, and exponential distributions (Bagnoli and Bergstrom, 2005) . The proof of Corollary 3 also establishes that under Condition LQ and logconcavity of the type density on [0, 1], the set of optimal interval thresholds is connected:

. Such multiplicity arises under the uniform distribution and linear loss utility. Either strict logconcavity 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 delegation sets, our Proposition 2 and Proposition 3 are effectively concerned with the optimality of disconnected delegation sets because of the status quo. Moreover, our approach provides a simple way to incorporate stochastic mechanisms, which, as already highlighted, are often not addressed in prior work.

Consider the following relaxed version of the optimization problem (D) for deterministic mechanisms:

16 Let Mo be the (strictly positive, for simplicity) mode of F and let ∆(

Problem (R) is well-behaved because the constraint set is convex and, owing to κ ≡ inf a∈[0,1) −u (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 IC and a solution to problem (R), then it is also a solution to (D). But we establish (see Lemma A.1 in Appendix A) that in this case α * is also a solution to problem (P), i.e., it is optimal among stochastic mechanisms. The idea is as follows. If there were a stochastic mechanism that is strictly better than α * in problem (P), consider the corresponding deterministic mechanism that replaces each lottery by its expected outcome. While this mechanism would not be IC in general, we show that it would both be feasible for problem (R) and would obtain a strictly higher objective value, contradicting the optimality of α * in (R). Establishing a higher value relies on the objective in (R) being a concave functional.

The sufficiency results in Propositions 1-3 then obtain from identifying sufficient conditions under which the respective IC mechanisms solve (R). It is here that our approach involves constructing suitable Lagrange multipliers.

For the necessity results, we first establish in Lemma A.4 in Appendix A that under Condition 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 higher value than α * . We construct a corresponding IC mechanism m such that α

Roughly, monotonicity of α (by definition of the set A) implies existence of transfers that make α IC in a quasi-linear model; the inequality constraints in (R) mean the transfers can be chosen to be positive (i.e., they can be viewed as money burning); and, because of Vetoer's quadratic utility, positive transfers can be substituted for by the action variance of suitable lotteries. Since u (a) = −κ for a < 1 under Condition LQ, the condition makes the objective in (R) a linear functional in the relevant domain. We can thus show that mechanism m obtains a strictly higher value than α * in (P), a contradiction.

We then establish necessity of the conditions in Propositions 1-3 by showing that, unless these conditions are satisfied, the corresponding mechanisms can be strictly improved upon in 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 of the objective in the direction of any feasible mechanism must be negative.

We derive two comparative statics, restricting attention to interval delegation. This focus 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.

Differentiating, the first-order condition for c * ∈ (0, 1) to be an optimal threshold among interval delegation sets is that it must be a zero of

( 2) In general there can be multiple optimal thresholds, even among interior thresholds. Accordingly, let the set of optimal thresholds for interval delegation be

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 gives Vetoer a superset of options to choose from, a decrease in C * corresponds to offering more discretion. It can also be interpreted as Proposer compromising more. As mentioned after Corollary 3, under Condition LQ and a logconcave type density, C * is a (closed) interval. In that 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 alignment between Proposer and Vetoer. We say that Proposer becomes strictly more risk averse if the Arrow-Prat coefficient of absolute risk aversion strictly increases in the relevant region: −u (a)/u (a) strictly increases for all a ∈ [0, 1). As is well known, such a change can also be expressed in terms of concave transformations of Proposer's utility. Under Condition LQ, it corresponds to a higher weight on the quadratic term. We say that the two players are strictly more aligned if Vetoer's ideal-point density changes from f to g with g strict likelihood ratio

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 under uncertainty, specifically Karlin's (1968) variation diminishing property for single-crossing functions and comparative statics from Milgrom and Shannon (1994).

The intuition for part (i) of the proposition is simply that greater risk aversion makes Proposer 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 hence she compromises less. Yet, the precise conditions in the proposition are nuanced. In particular, the stochastic ordering used in our notion of alignment is important: one can construct examples in which, among interval delegation sets, Proposer optimally gives Vetoer strictly more discretion when there is a right-shift in the type density in the sense of either hazard or reversed-hazard rate (both of which are stronger than first-order stochastic dominance but weaker than a likelihood ratio shift). Furthermore, absent the focus on interval delegation, it is not necessarily clear how to relate changes in delegation sets with the degree of discretion or compromise.

It is instructive to contrast part (ii) of Proposition 4 with the expertise-based delegation literature. The broad finding there is that among interval delegation, greater preference similarity in a suitable sense leads to more discretion (Holmström, 1984, Theorem 3) . The difference owes to and highlights the distinct rationales for discretion. In those models, the delegator would like to give the agent discretion to benefit from the agent's expertise; the degree of discretion is limited by the extent of preference misalignment. In our setting, on the other hand, the agent is given discretion only because of her veto power; greater ex-ante preference alignment mitigates that concern.

Example 1. Under Condition LQ, the first-order condition for an optimal interval threshold (i.e., expression (2) equals zero) becomes Recall from Corollary 3 that, when combined with boundary conditions, there will be a unique solution for any strictly logconcave type density; moreover, the corresponding interval is then an (unrestricted) optimal mechanism. Given uniqueness, the implicit function theorem can be used to affirm the general comparative statics of Proposition 4; moreover, the firstorder condition can also be used to compute numerically the optimal interval threshold for standard distributions. Figure 2 illustrates for Normal distributions. The left panel verifies comparative statics already discussed; note that a higher mean µ is a likelihood ratio rightshift and hence more alignment. 17 The right panel shows comparative statics in the variance of the distribution. We see that there is less discretion when the variance is lower, with the optimal threshold converging, as σ → 0, to Proposer's optimal offer, 0.9, to type µ = 0.45.

While we do not have general comparative statics results in the variability of the type distribution, it can be shown that for any strictly unimodal distribution with mode Mo ≥ 0, an optimal interval delegation set has more compromise than if Proposer knew Vetoer's type to be Mo. That is, if c * is an optimal interval threshold, then c * ≤ min{2Mo, 1}; the inequality is strict if Mo ∈ (0, 1/2). Adding this kind of uncertainty to the complete-information model of Romer and Rosenthal (1978) thus increases the extent of Proposer's compromise.

This subsection compares the outcome of optimal delegation with two game forms considered in earlier work.

A 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 or veto. This can be viewed as restricting Proposer to singleton delegation sets. Clearly, Proposer is strictly worse off in this institution unless no compromise is the optimal mechanism. We assume throughout this subsection that no compromise is not an optimal interval delegation set; as noted in Subsection 3.2 it is sufficient that Proposer's utility u(a) is differentiable at his ideal point a = 1 (hence u (1) = 0). 18 We will see below that not only does Proposer strictly benefit from optimal delegation, but so does Vetoer under some conditions. Matthews (1989) studies cheap talk before veto bargaining: prior to Proposer making a singleton proposal, Vetoer can send a costless and nonbinding message. As usual in cheaptalk games there is an uninformative and hence noninfluential equilibrium, in which Proposer makes the same proposal, a U > 0, as he would absent the possibility of cheap talk. Matthews provides conditions under which there is also an equilibrium with informative and influential cheap talk; it is sufficient given the support of our type density that u (1) = 0 (or that even the 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. In response to the latter message, Proposer offers a = 1; in response to the veto threat Proposer offers some a I ∈ (0, 1). The former proposal is accepted by all types that acquiesced, while the latter is accepted by only a subset of types that made the veto threat; types below some strictly positive threshold exercise the veto. An influential cheap-talk equilibrium is outcome equivalent to the delegation set {a I , 1} in our framework.

There can be multiple cheap-talk equilibria with distinct outcomes, both among influential 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 provides conditions under which a I is unique, i.e., all influential cheap-talk equilibria have the same 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:

By way of comparison, we recall that a zero of a similar function given in (2) is the firstorder condition for optimality of an interval delegation set's threshold.

Proposition 5. Assume no compromise is not an optimal delegation set, and that either (2) or (3) is strictly downcrossing on (0, 1). 19 Any optimal interval delegation set [c * , 1] has c * < min{a I , a U } for any influential and noninfluential cheap-talk equilibrium a I and a U , respectively. Hence, if [c * , 1] is an optimal 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 is better off no matter his type; moreover, a set of Vetoer types that have strictly positive probability are strictly better off. Proposition 5's conclusions hold trivially when full delegation is the 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 downcrossing on (0, 1) under Condition LQ and either strict logconcavity of the type density or strict concavity of Proposer's utility. Indeed, this underlies the uniqueness claim in Corollary 3; see Lemma 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 marginally lowering his proposal a I ∈ (0, 1) in an influential cheap-talk equilibrium. The benefit is that some types just below a I /2 will accept rather than veto; the cost is that the action induced from all types in the interval (a I /2, (1 + a I )/2) is lower. When Proposer instead delegates the interval [a I , 1], the benefit from lowering a I is unchanged while the cost is largely obviated, as most types above a I /2 are unaffected by the change. Proposer is thus more willing to compromise when choosing among interval delegation sets rather than under cheap talk. 20 Consequently, all Vetoer types benefit-at least weakly, and some strictly-from optimal interval delegation as compared to cheap talk. While Proposer could be harmed by a restriction to interval delegation, there is strong Pareto dominance when intervals constitute optimal delegation.

We note that if interval delegation is not optimal, then some Vetoer types may be worse off under optimal delegation than under cheap talk. For example, it is possible that the optimal delegation set takes the form {a * , 1} with a * ∈ (0, 1). 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 1 rather than a I from some types. Consequently, while Proposer strictly benefits from optimal delegation, some Vetoer types would strictly prefer either cheap-talk outcome.

We now discuss some implications and interpretations of our analysis in the context of three applications.

Our framework can be applied to questions of which products to present customers with, albeit in a stylized manner. For an illustration, suppose a salesperson has at his disposal a set of products indexed by a ∈ [0, 1], with higher a corresponding to higher quality. The price of product a is ka 2 , where k > 0. This pricing can be interpreted as emerging from a constant markup on a quadratic cost. Consumers vary in how they trade off quality and price; specifically, a consumer of type v ≥ 0 has gross valuation va, and hence net-of-price payoff va − ka 2 . If a consumer does not purchase, his payoff is 0; a consumer cannot purchase a product he is not shown (perhaps because of ignorance, or because the salesperson can claim it is unavailable). Observe that we can normalize k = 1/2, as this simply rescales the consumer type v. The salesperson receives a higher commission on better products, reflected by his strictly increasing and concave utility u(a). Given any belief density the salesperson holds about a particular consumer's valuation the salesperson's problem of which products to show 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 is optimal to show the consumer some set of "best products" (i.e., an interval of products [c, 1]); if the density is strictly decreasing, then all products should be shown; and if the density is strictly 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 salesperson shows a larger set of products. Proposition 4(ii) implies that if wealthier consumers (or, if wealth is unobservable, some proxy thereof) have a higher distribution of v in the likelihood ratio sense, then wealthier consumers are shown a smaller set of products.

What if a consumer can choose the information to disclose about her type? 21 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 after observing the message. No matter the type distribution, there are at least two equilibria: one in which no type discloses any information, and one in which all types fully disclose. 22 Every consumer type prefers the former equilibrium to the latter; some types have a strict preference unless nondisclosure results in only a single product being shown (Proposition 2). In general there can be other equilibria, some of which may dominate the nondisclosure equilibrium in terms of ex-ante consumer welfare. 23 However, when the salesperson offers all products under the prior (Proposition 1), the nondisclosure equilibrium is consumer optimal-not only ex 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, instead of the offenses formally charged . . . a recognized and well-established feature of the American criminal justice system" (Adlestein, 1995) . For instance, "the lesser-included offenses of first degree murder include second degree murder, voluntary manslaughter, involuntary manslaughter, 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 severity of a crime, and assumes that the jury will convict the defendant of the closest charge to v that is available. Verdict 0 corresponds to a complete acquittal, which is always available to the jury, while 1 is the maximum penalty. A prosecutor can put forward any set of charges in [0, 1], and he seeks to maximize the penalty. 24 Assume his utility u is concave. The lesser-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 worse off 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 or without the doctrine. In fact, following our discussion after Corollary 1, particularly the end of fn. 12, when Proposition 1's requirement holds, the prosecutor is not hurt, ex ante, from the doctrine 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. 23 For example, suppose the type density is strictly decreasing on a small interval [0, δ] and strictly increasing thereafter, and u is linear. Then, under nondisclosure, the salesperson's optimal menu is the singleton {1} (Proposition 2). There is also a partial-disclosure equilibrium in which types [0, δ] pool on the message [0, δ] and all higher types pool on the message [δ, ∞); the former message leads to the menu [0, δ] by the full-delegation logic of Corollary 1, while the latter message leads to the singleton {1} by the no-compromise logic of Proposition 2. Every consumer type prefers this partial-disclosure equilibrium to the nondisclosure equilibrium, some strictly.

24 Such a prosecutorial objective is a common assumption in law and economics since Landes (1971). Our assumption on the jury's behavior is compatible with it acting on the basis of (expected) utility, but could also be viewed as a reduced form. It is distinct from an approach in which the jury only convicts the defendant of a charge if some relevant standard has been met. prosecutor is sufficiently risk averse (Corollary 2).

Conversely, consider the expected utility of a defendant whose utility is strictly decreasing on [0, 1]. As this is the additive inverse of some prosecutor utility function, the defendant is exante better off when the prosecutor is constrained by the lesser-included offense doctrine. 25 She is not assured an improvement ex post, however. For example, when no compromise is unconstrained optimal for the prosecutor (Proposition 2), the doctrine strictly harms the defendant when v ∈ (0, 1/2). More generally, the defendant is strictly harmed by the doctrine for 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 defendants sometimes appeal verdicts on the basis that their conviction on a lesser-included offense is not legally valid. 26 We note that if the prosecutor were restricted to bringing a single charge, then under the condition of Proposition 1, the lesser-included offenses doctrine would benefit him and hurt a defendant with utility −u. Moreover, regardless of whether full delegation is unconstrained optimal 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 write bills that can be vetoed by executives. But executives do more than just approve or veto: as emphasized by Epstein and O'Halloran (1996, pp. 378-379) , "all laws passed by Congress are implemented by the executive branch in one form or another", and, since, "Presidents generally appoint administrators with 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 predict bills granting the executive more discretion when there is greater preference misalignment between the executive and the legislature, in the sense of Proposition 4(ii). This flips the comparative static emphasized in the political science literature (Epstein and O'Halloran, 1996, 1999) , which stems from expertise-based delegation models. 27 Of course, in practice one expects both the expertise-based rationale and our veto-based one to coexist to varying degrees.

There are examples of legislatures apparently providing misaligned executives greater discretion because of veto power. A case in point is the "U.S. Troop Readiness, Veterans' Care, Katrina Recovery, and Iraq Accountability Appropriations Act" (House Resolution 2206) enacted in May 2007 that provided funding for the United States' war on Iraq. Congress passed an earlier version, House Resolution 1591, that set a deadline of April 2008 for U.S. troops to withdraw from Iraq. This suggests that, even after accounting for any expertise-based delegation rationale, Congress preferred a relatively tight deadline. But the bill was vetoed by President George W. Bush. Although Democrats controlled both chambers of Congress, they did not have the requisite supermajority to override the veto. To secure the President's approval, the eventual Act replaced the withdrawal deadline with vague metrics that gave the President more discretion.

We must also stress an alternative perspective on our results: rather than passing bills that grant ex-post discretion, discretion can manifest in the executive effectively selecting which bill (from some subset, none of which grant ex-post discretion) the legislature passes. For example, the President may be consulted by Congress about different versions of legislation. These two forms of discretion are equivalent within our model. An empirical test of our model's predictions in the political arena would have to overcome this challenge, and that of the coexistence of expertise-and veto-based delegation rationales.

We have studied Proposer's optimal mechanism, absent transfers, in a simple model of veto bargaining. Our main results identify sufficient and necessary conditions for the optimal mechanism to take the form of certain delegation sets, including full delegation, no compromise, and more generally interval delegation. While we have focussed on a quadratic loss function for Vetoer, our analytical methodology can be applied to deduce optimality of these delegation sets for a broader class of Vetoer preferences. Specifically, the methods can be readily applied to Vetoer utility functions of the form va + b(a), for any differentiable and strictly concave 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 conwhich discretion is increased upon a divergence in legislative-executive preferences". The mechanism underlying his findings is different from that in this paper, however; in particular, expertise-based delegation is still essential to his analysis. texts Proposer may be unable to preclude reapproaching Vetoer with another proposal (or menu of proposals) following a veto. When the optimal mechanism in our setting is full delegation, we believe that such lack of commitment is not problematic. By offering the fulldelegation menu to begin with, bargaining will effectively conclude at the first opportunity.

When full delegation is not optimal, however, matters are considerable more nuanced. Sequential veto bargaining without commitment has received only limited theoretical attention, largely in finite-horizon models with particular type distributions (e.g., Cameron, 2000) . In ongoing research, we are studying an infinite-horizon model. Our preliminary results suggest that, owing to single-peaked preferences, non-Coasian dynamics can emerge that allow Proposer to obtain her commitment solution when players are patient.

In Appendix A we assume the support of the type distribution F is [0, 1]. This is without loss (even among stochastic mechanisms) because it is always optimal for Proposer to choose action 1 for types above 1 and, given the outside option, to choose action 0 for types below 0.

For convenience, we recall Proposer's problem (P):

We also recall the relaxed problem (R):

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).

To obtain a contradiction, let m ∈ S be feasible for (P) and suppose it achieves a strictly higher objective value in (P) than α * . Define α ∈ A by setting α(v)

Hence, feasibility of m in (P) implies feasibility of α in (R). Moreover,

where the first inequality holds because the first line is a concave functional (by the definition of κ ≡ inf a∈[0,1) −u (a)), the first equality holds because m is feasible in (P), the second inequality 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 * is not optimal in (R), a contradiction.

To show that a given delegation set solves the relaxed problem, we construct Lagrange multipliers 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

where the second equality follows from integration by parts. 28

Lemma A.2. Let α * be induced by a delegation set. 29 Suppose there is an increasing and rightcontinuous 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 the objective 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) ≥ 0} and G be the negative of the left-hand side of the constraint in (R). Define the linear function T : Z → R by T (z) = 1 0 z(v)dΛ(v) (which is welldefined since Λ corresponds to a finite measure (Royden and Fitzpatrick, 2010, p. 437 ) and

28 v 0 α(s)ds is continuous and κF (v) − Λ(v) has bounded variation as the difference of two increasing functions. Hence, the Riemann-Stieltjes integral

exists and integration by parts is valid. 29 That is, there is some delegation set A such that α * (v) is an action in A ∪ {0} that type v prefers the most. the integral therefore exists because z is bounded and measurable (Royden and Fitzpatrick, 2010, p. 375) ) and observe that, for all z ∈ P , T (z) ≥ 0, 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 we conclude that α * solves problem (R).

The follow observation will allow us to establish that the Lagrangean is a concave functional of α if Λ is increasing. Lemma A.3. Suppose K is right-continuous and increasing and h : R 2 → R is bounded, measurable, and for each value of its second argument concave in its first argument. Then the function S :

because concavity of h implies that the integrand is positive for each v and because K is increasing.

is concave in α(v) since its second derivative is given by f (v)[u (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.3 implies 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) ≤ 0 for all α ∈ A.

If Λ(1) = κF (1), (A.1) simplifies to

The Gateaux differential is

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) ≤ 0.

We begin with the optimality of full delegation.

Proof of sufficiency part of Proposition 1. Note that the action function induced by full delegation is α(v) = v. We claim that α maximizes the Lagrangean for the multiplier

for v < 1 and Λ(1) = κF (1). Note that the multiplier is increasing since

is increasing by assumption and u (v) ≥ 0. The Lagrangean is therefore maximized at α if ∂L(α, α −α, Λ) ≤ 0 for all α ∈ A. Note that the integrand of the first integral in (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 compromise satisfies α(v) = 0 for v ∈ [0, 1 2 ) and α(v) = 1 for v ∈ [ 1 2 , 1]. Now suppose for all s ∈ [0, 1/2) and t ∈ (1/2, 1] we have

and let ψ := inf t∈(1/2,1] (u (1) + κ(1 − t)) F (t)−F (1/2) t−1/2 . Define Λ(v) = κF (1/2) − ψ for v ∈ [0, 1) and Λ(1) = κF (1).

Let s ∈ (1/2, 1]. Note that integration by parts implies constant on [0, 1) , the definition of ψ implies that, for any s ∈ [0, 1/2),

Similarly, for any t ∈ (1/2, 1],

Fix arbitrary α ∈ A that satisfies α(1) = 1. It follows from (A.3) and the definition of α that

Since α is increasing, (A.4) and (A.5) imply that ∂L(α, α − α, Λ) ≤ 0. Since the optimal action rule chooses action 1 for type 1, we conclude that α is optimal.

Lastly, we consider optimality of interval delegation.

We propose the following multiplier:

Λ is constant on [0, c * ) and it follows from Proposition 3's condition (i) that Λ is increasing on (c * , 1]. To see that Λ is increasing at c * , note that condition (ii) holds as an equality for t = c * and hence the derivative of the LHS of condition (ii) with respect to t must be negative at t = c * , which yields

Hence, Λ is increasing at c * . It is thus sufficient to show ∂L(α, α − α, Λ) ≤ 0 for all α ∈ A.

Note that, for v ∈ [c * , 1], v − α(v) = 0 and the definition of Λ implies that for v ∈ [c * , 1),

Therefore,

Since α is constant on [0, c * /2) and [c * /2, c * ], α−α is increasing on [0, c * /2) and [c * /2, c * ]. Hence, the following conditions are sufficient for α to maximize the Lagrangian:

for t ∈ [c * /2, c * ] with equality at t = c * /2 and

Note that

and Λ is constant on [0, c * ). Hence, using the definition of Λ, we get that for t ∈ [c * /2, c * ],

where the inequality is by Proposition 3's condition (ii), and holds with equality for t = c * /2.

Analogously, note that

where the inequality is by Proposition 3's condition (iii). We conclude that α is optimal.

Lemma A.4. Suppose Condition LQ holds. If α * is deterministic and solves problem (P) then it also solves problem (R).

Proof. The proof is by contraposition: assuming there exists α ∈ A that is feasible for (R) and achieves a strictly higher objective value in (R) than α * , we will construct a solution to (P) that achieves a strictly higher objective value than α * .

(s)ds = 0 for all v such thatα(v) = 1, and achieves a weakly higher objective value in problem (R) than α.

We can assume α(v) ≤ 1 since u is decreasing above 1. Now suppose instead that vα(v) − α(v) 2 2 − v 0 α(s)ds > 0 for some v such that α(v) = 1. Consider an auxiliary setting in which a principal chooses a pair of functions (α, t) and an agent with type v gets utility vα(v) − α(v) 2 2 − t(v). Since α is monotonic, it follows from standard arguments that there exist transfers t : [0, 1] → R such that (α, t) is incentive compatible in the auxiliary setting (e.g., Amador and Bagwell, 2013) . For all v, these transfers satisfy

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 expected utility (and choosing the latter if type v is indifferent). Note thatt(0) = t(0), which together with t(v) ≥ t(0) impliest(v) −t(0) ≥ 0, with equality for any v such thatα(v) = 1.

Observe that (α,t) corresponds to an incentive compatible direct mechanism: indeed, if type v strictly prefers (α(v ),t(v )) to (α(v),t(v)) then v also strictly prefers (α(v ), t(v )) to (α(v), t(v)), contradicting the assumption that (α, t) is incentive compatible. It follows from the standard characterization of incentive compatible mechanisms thatα is increasing,

with the inequality holding as equality for v such thatα(v) = 1.

It follows thatα achieves a weakly higher objective value in problem (R).

Claim 2: Letα ∈ A be feasible for (R) and satisfyα(v) ≤ 1 and vα (

Intuitively, this is because Vetoer's utility function is quadratic and we can use noise as a substitute for transfers. We provide an explicit construction of the mechanism m below.

For any v such thatα(v) = 1, define m(v) to put mass 1 on action 1. Now fix arbitrary v such thatα(v) < 1 and arbitrary d ∈ (−∞, 0] and let t 1 (d)

Moreover, for any real number r we can choose d ∈ (−∞, 0] small enough such

Given v ∈ [0, 1] and t 2 ∈ [0, 1], we define m(v) to put probability t 2 on actionα(v), probability (1 − t 2 )t 1 (d) on action d, and probability (1 − t 2 )(1 − t 1 (d)) on action 1. It follows from the

Defining m(v) in this way for all v such thatα(v) < 1, the claim follows.

We conclude that m is feasible for (P). Therefore,

where the equality holds because α * is feasible for (P), the first inequality holds because we assume that α achieves a strictly higher value than α * , and the second inequality holds by Claim 1.

Hence, for any a, b ≤ 1 and λ ∈ [0, 1], some algebra shows that

Since m is feasible for (P), this expression equals

. This contradicts the assumption 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) ≤ 0 for any α ∈ A that is feasible for (R). Recall the assumption F (1) = 1. 

Proof. Suppose a delegation set containing the interval [a, b] is optimal, and let α denote the corresponding allocation rule. Suppose to the contrary that κF d, e] . Then α ∈ A and it is feasible for (R).

Since d, e] , it follows from (A.7) that ∂φ(α, α − α) > 0, a contradiction.

Proof of the necessity part of Proposition 1. Suppose Condition LQ holds. The result follows directly from Lemma A.5.

Proof of the necessity part of Proposition 2. Suppose Condition LQ holds and let α ∈ A be the action rule induced by the delegation set {0, 1}. Fix s ∈ [0, 1/2) and t ∈ (1/2, 1], ε ∈ (0, 1) and define

Note that

Therefore, ∂φ(α, α ε − α) ≤ 0 for all ε > 0, s ∈ [0, 1/2) and t ∈ (1/2, 1] if and only if the condition in Proposition 2 holds.

Proof of necessity part of Proposition 3. Suppose Condition LQ holds and c * ∈ (0, 1). Let α ∈ A be the action rule induced by the delegation set [c * , 1]. We prove necessity of each condition in Proposition 3 in order.

Condition (i): This follows from Lemma A.5.

Condition (ii): Fix t ∈ (c * /2, c * ) and ε > 0. Let a(ε) be the positive solution to (c * − t)a + a 2 /2 = ε(t − c * /2) and define α ε by

Note that for any ε > 0 small enough (so that c * − ε > c * /2 and c * + a(ε) < 1), α ε ∈ A. By definition of a(ε), for any

By the implicit function theorem, lim ε→0 a(ε) ε = t−c * /2 c * −t . It follows that the last integral is of order o(ε). Also, integration by parts implies that for x, y ∈ R,

Since ∂φ(α, α ε − α) ≤ 0 for all ε > 0, the last expression is negative for all t ∈ (c * /2, c * ), which implies condition (ii).

Condition (iii): Fix s ∈ [0, c * /2) and ε > 0. Let

where a(ε) ≥ 0 satisfies

For ε small enough, there is a real solution a(ε) ≥ 0, and a simple calculation shows that α ε is feasible for (R). Also, note that lim ε→0 a(ε) ε = c * /2−s c * −c * /2 . It follows from (A.7) that

Using integration by parts, we conclude

which yields condition (iii).

Since u is concave, u is decreasing on [0, 1]. Recall κ ≥ 0. Hence, if the type density f is decreasing on [0, 1], then κF − u f is increasing on [0, 1]. The result follows from Proposition 1.

The RHS above is finite since f is continuously differentiable and strictly positive on [0, 1].

is increasing on [0, 1] when the LHS above is sufficiently large. The result follows from Proposition 1.

Assume Condition LQ. We prove the result by establishing that (i) logconcavity of f on [0, 1] ensures that the conditions of either Proposition 2 or Proposition 3 are satisfied, and (ii) if γ > 0 (equivalently, given Condition LQ, u is strictly concave) or f is strictly logconcave on [0, 1], then among interval delegation sets there is a unique optimum.

As introduced in Section 4, Proposer's expected utility from delegating the interval [c, 1] with c ∈ [0, 1] is:

As shorthand for the function in condition (i) of Proposition 3, define

We establish some properties of the W and G functions. 

Proof. The proof proceeds in four steps. Throughout, we restrict attention to the domain [0, 1] for the type density.

Step 1 shows that G is (strictly) quasiconvex and that {v : G (v) = 0} is connected.

Step 2 shows that W can be expressed in terms of G .

Step 3 establishes that given any maximizer c * of W , G is decreasing on [0, c * /2] and increasing on [c * , 1].

Step 4 establishes the (strict) quasiconcavity of W . Note that under Condition LQ,

Step 1: We first establish that G is (strictly) quasiconvex and that {v : G (v) = 0} is connected. Logconcavity of f implies that its modes (i.e., maximizers) are connected, and moreover f (v) = 0 =⇒ v is a mode. Denote by Mo the smallest mode. Since

On the domain [0, Mo), f /f is positive and decreasing by logconcavity. Furthermore, 1 − γ + 2γ(1 − v) is positive and decreasing. As the product of positive decreasing functions is decreasing, β is increasing on the domain [0, Mo). Since β(v) ≥ 0 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 (v) = 0}. If γ = 0 then β(v) = 0 ⇐⇒ f (v) = 0, which is a connected set, as noted earlier. If γ > 0, then the conclusion follows because β is increasing on [0, Mo), β(v) > 0 for v > Mo (as f (v) ≤ 0), and β is continuous. Furthermore, analogous observations imply that if either f is strictly logconcave or γ > 0, then |{v : G (v) = 0}| ≤ 1 and so G is strictly quasiconvex.

Step 2: We now show that

Now fix any c > 0 such that W (c) = 0. By (B.4) and the above integration by parts, G(c/2) = (2/c) c c/2 G(v)dv, which, because G is quasiconvex by Step 1, implies G(c/2) ≤ G(c), with a strict inequality if γ > 0 or f strictly logconcave. Similarly G (c/2) ≤ 0, and hence from (B.5) we conclude that W (c) ≤ 0, with a strict inequality if γ > 0 or f is strictly logconcave.

We build on Lemma B.1 to establish Corollary 3 by verifying the conditions of Proposition 2 and Proposition 3.

Proof of Corollary 3. If the interval delegation set [c * , 1] is optimal then c * must maximize W (c) defined in (B.1). Hence if W is strictly quasiconcave-as is the case if γ > 0 or f is strictly logconcave on [0, 1], 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∈[0,1] W (c) then [c * , 1] is optimal.

To that end, we verify that if c * = 1 the conditions of Proposition 2 are satisfied and, if c * < 1, then conditions (i)-(iii) of Proposition 3 are satisfied. Note that condition (i) is immediate from Lemma B.1. As conditions (ii) and (iii) are vacuous for c * = 0 we need only consider c * ∈ (0, 1]. For any c * ∈ (0, 1) conditions (ii) and (iii) are jointly equivalent to

for all s ∈ [0, c * /2) and t ∈ (c * /2, c * ]. Substituting into the middle expression from the firstorder condition W (c * ) = 0 (i.e., setting expression (2) equal to zero and rearranging) yields

for all s ∈ [0, c * /2) and t ∈ (c * /2, c * ]. So if (B.6) holds for c * ∈ (0, 1) then the conditions in Proposition 3 are verified. On the other hand, since the condition in Proposition 2 is equivalent to the right-most term in (B.6) being larger than the left-most term for all s ∈ [0, c * /2) and t ∈ (c * /2, c * ] when c * = 1, (B.6) holding for c * = 1 implies the condition in Proposition 2. Accordingly, we fix a c * > 0 and verify the two inequalities of (B.6) in turn.

First inequality of (B.6): Using u (a) = 1 + γ − 2γa, κ = 2γ, and u(a)−u(0) a = 1 + γ − γa, the first inequality of (B.6) reduces to

It follows from L'Hopital's rule that the above inequality holds with equality in the limit as s → c * /2. Hence it is sufficient to demonstrate that the LHS of the inequality is increasing for all s ∈ [0, c * /2). For any s ∈ [0, 1] let

and observe that Second inequality of (B.6): Using u (a) = 1 + γ − 2γa, κ = 2γ, and u(a)−u(0) a = 1 + γ − γa, the second inequality of (B.6) reduces to

(1 + γ − γc * )f (c * /2) ≤ (1 + γ − 2γt) F (t) − F (c * /2) t − c * /2 ∀t ∈ (c * /2, c * ].

Using L'Hopital's rule for the limit as t → c * /2 and the fact that W (c * ) ≥ 0 by optimality of c * > 0, it follows that lim t→c * /2

(1 + γ − 2γt) F (t) − F (c * /2) t − c * /2 = (1 + γ − γc * )f (c * /2) ≤ (1 + γ − 2γc * ) F (c * ) − F (c * /2) c * /2 .

Hence it is sufficient to show that (1 + γ − 2γt) F (t)−F (c * /2) t−c * /2 is quasiconcave for t ∈ (c * /2, c * ].

where D is defined in (B.7), and so

Since D(c * /2) = 0, it follows that (1 + γ − 2γt) F (t)−F (c * /2) t−c * /2 is quasiconcave for t ∈ (c * /2, c * ] if D is quasiconcave. D is quasiconcave because, as was shown in (B.8), D (t) = (c * /2 − t)G (t), which is positive then negative on (c * /2, c * ] by the quasiconvexity of G (Lemma B.1).

Proof of Proposition 4(i). Let H(a, c) denote the cumulative distribution function of the action implemented under the interval delegation set [c, 1]. That is, A standard monotone comparative statics argument (Milgrom and Shannon, 1994) then implies that C * (u) ≥ SSO C * (û).

. Let density f (v) strictly dominate density g(v) in likelihood ratio on the unit interval: i.e., for all 0 ≤

Let w(c, v) denote Proposer's payoff under the interval delegation set [c, 1] when Vetoer's type is v. We have

Consider any 0 ≤ c L < c H ≤ 1. The difference w(c H , ·) − w(c L , ·) is upcrossing: once strictly positive, it stays positive. It follows from the variation diminishing property (Karlin, 1968, Theorem 3.1 on p. 21) that A standard monotone comparative statics argument (Milgrom and Shannon, 1994) then implies that C * (f ) ≥ SSO C * (g).

Let a U and a I denote proposals in some noninfluential and influential cheap-talk equilibria, respectively (the latter may not exist). It is straightforward that a U > 0 and, if it exists, a I ∈ (0, 1). Since Proposition 5's conclusion is trivial for full delegation (c * = 0), it suffices to establish that any optimal interval delegation set [c * , 1] with c * ∈ (0, 1) has c * < min{a I , a U }. Any influential cheap-talk equilibrium outcome can be characterized by a threshold type v I ∈ (0, 1) such that types v < v I pool on the "veto threat" message, and types v > v I pool on the "acquiesce" message. Since type v I must be indifferent between sending the two messages, and she will accept either proposal from the Proposer, it holds that v I = 1 + a I 2 .

It follows that a I is an influential equilibrium proposal if and only if a I ∈ arg max a u(0) F (a/2) F ((1 + a I )/2) + u(a) 1 − F (a/2) F ((1 + a I )/2) .

The first-order condition is that function (3) Note that at a = 0, the LHS is strictly positive. Hence, if the LHS is strictly downcrossing on (0, 1), 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. Note that if there is no solution to Equation D.2 on (0, 1) then there is no influential equilibrium.

Turning to optimal interval delegation, recall from Section 4 that the threshold is a zero of the function (2) 2's LHS being strictly positive at 0, and continuity combine to imply 0 ≤ a 3 < a 2 < a U , and a 3 ≤ a 2 with a strict inequality if either a 2 < 1 or a 3 < 1. Furthermore, a I ∈ [a 2 , a 2 ] and c * ∈ [a 3 , a 3 ].

If the LHS of Equation D.2 is strictly downcrossing on (0, 1), then by the properties noted right after Equation D.2, a I = a 2 = a 2 < 1 and hence c * < min{a I , a U }. If the LHS of Equation D.3 is strictly downcrossing on (0, 1), then by the properties noted right after Equation D.3, c * = a 3 = a 3 < 1 and hence c * < min{a I , a U }.

Example E.1. Suppose Proposer has a linear loss function, v = 0, v = 1, and f (v) is strictly increasing except on (1/2−δ, 1/2+δ), where it is strictly decreasing. Assume |f (v)| is constant v 1 0 1 2 f Figure 3 -A density under which no compromise is the optimal delegation set when Proposer has a linear loss function, but it is worse than some stochastic mechanism.

(on [0, 1]). 30 Take δ > 0 to be small. See Figure 3 .

Recall that if δ were 0, then no compromise (i.e., the singleton menu {1}) would be optimal by Proposition 2 or the discussion preceding it. It can be verified that no compromise remains an optimal delegation set for small δ > 0. We argue below that Proposer can obtain a strictly higher payoff, however, by adding a stochastic option that has expected value 1/2 and is chosen only by types in (1/2 − δ, 1/2 + δ).

The stochastic option provides action 1 − 1 2p with probability p and action 1 with probability 1 − p. For any p ∈ (0, 1), this lottery has expected value 1/2. Moreover, when p = 1 2−4δ , quadratic loss implies that type 1/2 − δ is indifferent between and action 0 while type 1/2 + δ is indifferent between and 1. Consequently, any type in [0, 1/2 − δ) strictly prefers 0 to both and 1; any type in (1/2 − δ, 1/2 + δ) strictly prefers to both 0 and 1; and any type in (1/2 + δ, 1] strictly prefers 1 to both and 0.

Therefore, offering the menu { , 1} rather than {1} changes the induced expected action from 0 to 1/2 when v ∈ (1/2 − δ, 1/2) and from 1 to 1/2 when v ∈ (1/2, 1/2 + δ). Since f (v) is strictly decreasing on (1/2 − δ, 1/2 + δ), Proposer is strictly better off.

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The derivation is as follows:The first equality above is obtained by differentiating Equation B.1 and using u (c) = 1 + γ − 2γc 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 from algebraic manipulations.Step 3: We now establish that for any c * ∈ arg maxStep 4: Finally we establish that W is quasiconcave, strictly if γ > 0 or f is strictly logconcave. For this it is sufficient to establish that if c > 0 and W (c) = 0, then W (c) ≤ 0, with a strict inequality if γ > 0 or f is strictly logconcave.