PPERSUASION MEETS DELEGATION
Anton Kolotilin and Andriy Zapechelnyuk
Abstract.
A principal can restrict an agent’s information (the persuasion prob-lem) or restrict an agent’s discretion (the delegation problem). We show that theseproblems are generally equivalent — solving one solves the other. We use toolsfrom the persuasion literature to generalize and extend many results in the delega-tion literature, as well as to address novel delegation problems, such as monopolyregulation with a participation constraint.
JEL Classification:
D82, D83, L43
Keywords: persuasion, delegation, regulation
Date : 8th February 2019.
Kolotilin : School of Economics, UNSW Business School, Sydney, NSW 2052, Australia.
E-mail: [email protected].
Zapechelnyuk : School of Economics and Finance, University of St Andrews, Castlecliffe, the Scores,St Andrews KY16 9AR, UK.
E-mail: [email protected] are grateful to Tymofiy Mylovanov, with whom we are working on related projects. Wethank Ricardo Alonso, Kyle Bagwell, Benjamin Brooks, Deniz Dizdar, Piotr Dworczak, AlexanderFrankel, Drew Fudenberg, Gabriele Gratton, Yingni Guo, Emir Kamenica, Navin Kartik, MingLi, Hongyi Li, Carlos Oyarzun, Alessandro Pavan, Eric Rasmusen, Philip Reny, Joel Sobel, andThomas Tr¨oger for helpful comments and suggestions. We also thank participants at variousseminars and conferences. Part of this research was carried out while Anton Kolotilin was visitingMIT Sloan School of Management, whose hospitality and support is greatly appreciated. AntonKolotilin also gratefully acknowledges support from the Australian Research Council DiscoveryEarly Career Research Award DE160100964. Andriy Zapechelnyuk gratefully acknowledges supportfrom the Economic and Social Research Council Grant ES/N01829X/1. a r X i v : . [ ec on . T H ] F e b KOLOTILIN AND ZAPECHELNYUK Introduction
There are two ways to influence decision making: delegation and persuasion. Thedelegation literature, initiated by Holmstr¨om (1984), studies the design of decisionrules. The persuasion literature, set in motion by Kamenica and Gentzkow (2011),studies the design of information disclosure rules.The delegation problem has been used to design organizational decision processes(Dessein, 2002), monopoly regulation policies (Alonso and Matouschek, 2008), andinternational trade agreements (Amador and Bagwell, 2013). The persuasion prob-lem has been used to design school grading policies (Ostrovsky and Schwarz, 2010),internet advertising strategies (Rayo and Segal, 2010), and forensic tests (Kamenicaand Gentzkow, 2011).This paper shows that, under general assumptions, the delegation and persuasionproblems are equivalent, thereby bridging the two strands of literature. The implica-tion is that the existing insights and results in one problem can be used to understandand solve the other problem.Both delegation and persuasion problems have a principal and an agent whose payoffsdepend on an agent’s decision and a state of the world. The sets of decisions and statesare intervals of the real line. The agent’s payoff function satisfies standard single-peakedness and sorting conditions. In a delegation problem, the agent privately knowsthe state and the principal commits to a set of decisions from which the agent chooses.In a persuasion problem, the principal designs the agent’s information structure andthe agent freely chooses a decision. The principal’s tradeoff is that giving morediscretion to the agent in the delegation problem and disclosing more information tothe agent in the persuasion problem allows for better use of information about thestate, but limits control over the biased agent’s decision.We consider balanced delegation and monotone persuasion problems. In the balanceddelegation problem, the principal may not be able to exclude certain indispensable de-cisions of the agent. This problem nests the standard delegation problem and includes,in particular, a novel delegation problem with an agent’s participation constraint.In the monotone persuasion problem, the principal chooses a monotone partitionalinformation structure that either reveals the state or pools it with adjacent states.This problem incorporates constraints faced by information designers in practice. Forexample, a non-monotone grading policy that gives better grades to worse performingstudents will be perceived as unfair and will be manipulated by strategic students.Moreover, in many special cases, optimal information structures are monotone parti-tions.The main result of the paper is that the balanced delegation and monotone persuasionproblems are strategically equivalent. For each primitive of one problem we explicitlyconstruct an equivalent primitive of the other problem. This construction equatesthe marginal payoffs and swaps the roles of decisions and states in the two problems.
ERSUASION MEETS DELEGATION 3
Intuitively, decisions in the delegation problem play the role of states in the persua-sion problem because the principal controls decisions in the delegation problem and(information about) states in the persuasion problem. It is worth noting that thisequivalence result is fundamentally different from the revelation principle. Specific-ally, the sets of implementable (and, therefore, optimal) decision outcomes generallydiffer in the delegation and persuasion problems with the same payoff functions.To prove the equivalence result, we show that the balanced delegation and monotonepersuasion problems are equivalent to the following discriminatory disclosure prob-lem. The principal’s and agent’s payoffs depend on an agent’s binary action, a stateof the world, and an agent’s private type. The sets of states and types are intervals ofthe real line. The agent’s payoff function is single-crossing in the state and type. Theprincipal designs a menu of cutoff tests, where a cutoff test discloses whether the stateis below or above a cutoff. The agent selects a test from the menu and chooses between inaction and action depending on his private type and the information revealed bythe test.To see why the discriminatory disclosure problem is equivalent to the balanced deleg-ation problem, observe that the agent’s essential decision is the selection of a cutofftest from the menu. Because the agent’s payoff function is single-crossing in thestate, the agent optimally chooses inaction/action if the selected test discloses thatthe state is below/above the cutoff. Thus, this problem can be interpreted as a del-egation problem in which a delegation set is identified with a menu of cutoffs, andthe agent’s decision with his selection of a cutoff from the menu.To see why the discriminatory disclosure problem is equivalent to the monotone per-suasion problem, observe that each menu of cutoff tests defines a monotone partitionof the state space. Because the agent’s payoff function is single-crossing in the state,the agent’s optimal choice between inaction and action is the same whether he ob-serves the partition element that contains the state or the result of the optimallyselected cutoff test. Moreover, because the agent’s payoff function is single-crossingin his type, the agent optimally chooses inaction/action if his type is below/above athreshold. Thus, this problem can be interpreted as a persuasion problem in which amonotone partition is identified with a menu of cutoffs, and the agent’s decision witha threshold type.We use our equivalence result to solve a monopoly regulation problem in which awelfare-maximizing regulator (principal) restricts the set of prices available to a mono-polist (agent) who privately knows his cost. This problem was first studied by Baronand Myerson (1982) as a mechanism design problem with transfers. Alonso andMatouschek (2008) pointed out that transfers between the regulator and monopolistare often forbidden, and thus, the monopoly regulation problem can be formulatedas a delegation problem. Alonso and Matouschek (2008) omitted the monopolist’sparticipation constraint, so under their optimal regulation policy, the monopolistsometimes operates at a loss. Amador and Bagwell (2016) characterized the optimalregulation policy taking the participation constraint into account.
KOLOTILIN AND ZAPECHELNYUK
The monopoly regulation problem, with and without the participation constraint,can be formulated as a balanced delegation problem. We provide an elegant methodof solving this problem, by recasting it as a monotone persuasion problem and usinga single result from the persuasion literature. When the demand function is linearand the cost distribution is unimodal, the optimal regulation policy takes a simpleform that is often used in practice. The regulator imposes a price cap and allowsthe monopolist to choose any price not exceeding the cap. The optimal price cap ishigher when the participation constraint is present; so the monopolist is given morediscretion when he has an option to exit.The literature has focused on linear delegation and linear persuasion in which themarginal payoffs are linear in the decision and the state, respectively. We showthe equivalence of linear balanced delegation and linear monotone persuasion. Wetranslate a linear delegation problem to the equivalent linear persuasion problemand solve it using methods in Kolotilin (2018) and Dworczak and Martini (2019).Specifically, we provide conditions under which a candidate delegation set is optimal.For an interval delegation set, these conditions coincide with those in Alonso andMatouschek (2008), Amador and Bagwell (2013), and Amador, Bagwell, and Frankel(2018), but we impose weaker regularity assumptions. For a two-interval delegationset, our conditions are novel and imply special cases in Melumad and Shibano (1991)and Alonso and Matouschek (2008).Our equivalence result can also be used to translate the existing results in nonlin-ear delegation problems to equivalent nonlinear persuasion problems, and vice versa.Nonlinear delegation is considered in Holmstr¨om (1984), Alonso and Matouschek(2008), Amador and Bagwell (2013), and Amador, Bagwell, and Frankel (2018). Non-linear persuasion is considered in Rayo and Segal (2010), Kamenica and Gentzkow(2011), Kolotilin (2018), Dworczak and Martini (2019), and Guo and Shmaya (2019).The rest of the paper is organized as follows. In Section 2, we define the balanceddelegation and monotone persuasion problems. In Section 3, we present and prove theequivalence result. In Section 4, we apply the equivalence result to solve a monopolyregulation problem. In Section 5, we address the linear delegation problem usingtools from the persuasion literature. In Section 6, we present the equivalence resultunder weaker assumptions. In Section 7, we make concluding remarks. The appendixcontains omitted proofs. 2.
Two Problems
Primitives.
There are a principal (she) and an agent (he). The agent’s payoff U ( θ, x ) and principal’s payoff V ( θ, x ) depend on a decision x ∈ [0 ,
1] and a state θ ∈ [0 , We relax these assumptions in Section 6.
ERSUASION MEETS DELEGATION 5 (A ) ∂∂x U ( θ, x ) and ∂∂x V ( θ, x ) are well defined and continuous in θ and x ;(A ) ∂∂x U ( θ, x ) is strictly increasing in θ and strictly decreasing in x .A pair ( U, V ) is called a primitive of the problem. Let P be the set of all primitivesthat satisfy assumptions (A ) and (A ).We now describe two problems. In a delegation problem, the agent is fully informedand the principal restricts the agent’s discretion. In a persuasion problem, the agenthas full discretion and the principal restricts the agent’s information.In both problems, the principal chooses a closed subset Π of [0 ,
1] that contains theelements 0 and 1. Let Π = { Π ⊂ [0 ,
1] : Π is closed and { , } ⊂ Π } . In the delegation problem, Π describes a set of decisions from which the agent chooses.In the persuasion problem, Π describes a partition of states that the agent observes.2.2.
Balanced Delegation Problem.
Consider a primitive ( U D , V D ) ∈ P , where weuse subscript D to refer to the delegation problem. The principal chooses a delegationset Π ∈ Π . The agent privately observes the state θ and chooses a decision x ∈ Πthat maximizes his payoff, x ∗ D ( θ, Π) = arg max x ∈ Π U D ( θ, x ) . (1a)The principal’s objective is to maximize her expected payoff,max Π ∈ Π E (cid:2) V D ( θ, x ∗ D ( θ, Π)) (cid:3) . (1b)By (A ), x ∗ D ( θ, Π) is single-valued for almost all θ , so E (cid:2) V D ( θ, x ∗ D ( θ, Π)) (cid:3) is well defined.The balanced delegation problem requires delegation sets to include the extreme de-cisions. On the one hand, this requirement can be made non-binding by defining theagent’s and principal’s payoffs on a sufficiently large interval of decisions so that theextreme decisions are never chosen (Appendix A.3). On the other hand, this require-ment allows to include indispensable decisions of the agent, such as a participationdecision (Sections 4 and 5.3).2.3.
Monotone Persuasion Problem.
Consider a primitive ( U P , V P ) ∈ P , wherewe use subscript P to refer to the persuasion problem. The principal chooses amonotone partitional information structure that partitions the state space into convexsets: separating elements and pooling intervals. A monotone partition is describedby a set Π ∈ Π of boundary points of these partition elements. Let π ( θ ) = sup { θ (cid:48) ∈ Π : θ (cid:48) ≤ θ } and π ( θ ) = inf { θ (cid:48) ∈ Π : θ (cid:48) > θ } KOLOTILIN AND ZAPECHELNYUK for θ ∈ [0 , π (1) = π (1) = 1. The partition element µ Π ( θ ) that contains θ ∈ [0 ,
1] is given by µ Π ( θ ) = (cid:40) { θ } , if π ( θ ) = π ( θ ) , [ π ( θ ) , π ( θ )) , if π ( θ ) < π ( θ ) . For example, Π = { , } is the uninformative partition that pools all states, andΠ = [0 ,
1] is the fully informative partition that separates all states.The agent observes the partition element µ Π ( θ ) that contains the state θ and choosesa decision x ∈ [0 ,
1] that maximizes his expected payoff given the posterior beliefabout θ , x ∗ P ( θ, Π) ∈ arg max x ∈ [0 , E (cid:2) U P ( θ (cid:48) , x ) (cid:12)(cid:12) θ (cid:48) ∈ µ Π ( θ ) (cid:3) . (2a)The principal’s objective is to maximize her expected payoff,max Π ∈ Π E (cid:2) V P ( θ, x ∗ P ( θ, Π)) (cid:3) . (2b)By (A ), x ∗ P ( θ, Π) is single-valued for all θ , so E (cid:2) V P ( θ, x ∗ P ( θ, Π)) (cid:3) is well defined.The monotone persuasion problem requires information structures to be monotonepartitions. On the one hand, this requirement is without loss of generality in manyspecial cases, where optimal information structures are monotone partitions (Sec-tion 5). On the other hand, this requirement may reflect incentive and legal con-straints faced by information designers. Monotone partitional information structuresare widespread and include, for example, school grades, tiered certification, creditand consumer ratings.2.4.
Persuasion versus Delegation.
We now show that implementable outcomesdiffer in the persuasion and delegation problems with the same primitive (
U, V ) ∈ P .In the persuasion problem with U ( θ, x ) = − ( θ − x ) , consider a monotone partition Π (cid:48) that reveals whether the state is above or below 1 /
3. Since θ is uniformly distributedon [0 , x ∗ P ( θ, Π (cid:48) ) = (cid:40) , if θ ∈ (cid:0) , (cid:1) , , if θ ∈ (cid:0) , (cid:1) , where 1/6 is the midpoint between 0 and 1/3, and 2/3 is the midpoint between 1/3and 1.This outcome cannot be implemented in the delegation problem with the same prim-itive U ( θ, x ) = − ( θ − x ) . To see this, consider a delegation set Π (cid:48)(cid:48) that permits onlytwo decisions, 1 / /
3. The induced decision of the agent is x ∗ D ( θ, Π (cid:48)(cid:48) ) = (cid:40) , if θ ∈ (cid:0) , (cid:1) , , if θ ∈ (cid:0) , (cid:1) , Formally, state θ = 1 is separated, but this is immaterial, because this event has zero probability. ERSUASION MEETS DELEGATION 7 where 5 /
12 is the midpoint between 1 / /
3. Thus, the induced decisions inthe persuasion and delegation problems differ on the interval of intermediate statesbetween 1 / / Note that the outcome x ∗ P ( · , Π (cid:48) ) is the first best for the principal whose payoff V ( θ, x )is maximized at x = 1 / / x = 2 / /
3. This first best is not implementable in the delegation problemwith the same primitive. Conversely, the outcome x ∗ D ( · , Π (cid:48)(cid:48) ) is the first best for theprincipal whose payoff V ( θ, x ) is maximized at x = 1 / /
12 and ismaximized at x = 2 / /
12. This first best is not implementable inthe persuasion problem with the same primitive. Equivalence
Main Result.
We use von Neumann and Morgenstern’s (1944) notion of stra-tegic equivalence. Primitives ( U D , V D ) ∈ P and ( U P , V P ) ∈ P of the balanced delega-tion and monotone persuasion problems are equivalent if there exist α > β ∈ R such that E (cid:2) V D ( θ, x ∗ D ( θ, Π)) (cid:3) = α E (cid:2) V P ( θ, x ∗ P ( θ, Π)) (cid:3) + β for all Π ∈ Π . That is, if ( U D , V D ) and ( U P , V P ) are equivalent, then, in both problems, the prin-cipal gets the same expected payoff, up to an affine transformation, for each Π;consequently, the principal’s optimal solution is also the same. Theorem 1.
For each ( U D , V D ) ∈ P , an equivalent ( U P , V P ) ∈ P is given by U P ( θ, x ) = − (cid:90) x ∂U D ( t, s ) ∂s (cid:12)(cid:12)(cid:12)(cid:12) s = θ d t and V P ( θ, x ) = − (cid:90) x ∂V D ( t, s ) ∂s (cid:12)(cid:12)(cid:12)(cid:12) s = θ d t. (3a) Conversely, for each ( U P , V P ) ∈ P , an equivalent ( U D , V D ) ∈ P is given by U D ( θ, x ) = (cid:90) x ∂U P ( s, t ) ∂t (cid:12)(cid:12)(cid:12)(cid:12) t = θ d s and V D ( θ, x ) = (cid:90) x ∂V P ( s, t ) ∂t (cid:12)(cid:12)(cid:12)(cid:12) t = θ d s. (3b)Theorem 1 explicitly connects the two problems. In fact, we prove a stronger resultthan Theorem 1. We show that if primitives ( U D , V D ) ∈ P and ( U P , V P ) ∈ P satisfy ∂U D ( θ D , x ) ∂x (cid:12)(cid:12)(cid:12)(cid:12) x = θ P = − ∂U P ( θ P , x ) ∂x (cid:12)(cid:12)(cid:12)(cid:12) x = θ D and ∂V D ( θ D , x ) ∂x (cid:12)(cid:12)(cid:12)(cid:12) x = θ P = − ∂V P ( θ P , x ) ∂x (cid:12)(cid:12)(cid:12)(cid:12) x = θ D for all θ D , θ P ∈ [0 , Since the outcome x ∗ P ( · , Π (cid:48) ) cannot be implemented in the delegation problem, it cannot beimplemented in the balanced delegation problem. Similarly, if the principal’s payoff V ( θ, x ) is maximized at decision 0 for states below 1 / /
2, then the first best is implementable by the balanceddelegation set { , } , but is not implementable by any monotone partition. KOLOTILIN AND ZAPECHELNYUK states in the two problems. In particular, the agent’s payoff is supermodular in oneproblem whenever it is concave in the decision in an equivalent problem.The rest of this section proves the equivalence of the balanced delegation and mono-tone persuasion problems, by showing that they are equivalent to a discriminatorydisclosure problem. In this problem, the agent is privately informed and has a binaryaction. The principal designs a menu of cutoff tests, where a cutoff test discloseswhether the state is below or above a cutoff. The agent selects a test from the menuand then chooses an action.3.2.
Discriminatory Disclosure Problem.
The agent chooses between actions a = 0 and a = 1. The agent’s payoff u ( s, t ) and principal’s payoff v ( s, t ) from a = 1depend on a state s ∈ [0 ,
1] and an agent’s private type t ∈ [0 , a = 0 are normalized to zero. The state and type are independently and uniformlydistributed. We assume that:(A (cid:48) ) u ( s, t ) and v ( s, t ) are continuous in s and t ;(A (cid:48) ) u ( s, t ) is strictly increasing in s and strictly decreasing in t .A pair ( u, v ) that satisfies assumptions (A (cid:48) ) and (A (cid:48) ) is a primitive of this problem.The principal designs a menu Π ∈ Π of cutoff tests. Each cutoff test y ∈ Π discloseswhether the state s is at least y . The agent knows his private type t , selects a cutofftest y from the menu Π, observes the result of the selected test, and then choosesbetween a = 0 and a = 1.3.3. Equivalence to Balanced Delegation.
Consider a discriminatory disclosureproblem with a primitive ( u, v ). A menu of cutoffs Π can be interpreted as a delegationset, and the agent’s selection of a cutoff y ∈ Π as an agent’s decision. Indeed, by(A (cid:48) ), the agent gets a higher payoff from a = 1 when the state s is higher; so eitherhe optimally chooses a = 1 whenever s ≥ y , or makes a choice irrespective of thetest result. But ignoring the test result is the same as selecting an uninformative test y ∈ { , } ⊂ Π , and then choosing a = 1 whenever s ≥ y . Therefore, without loss ofgenerality, after observing the result of the selected test, s ≥ y or s < y , the agentchooses a = 1 if s ≥ y and a = 0 if s < y .Thus, the agent selects a test y ∈ Π that maximizes his expected payoff, y ∗ ( t, Π) = arg max y ∈ Π (cid:90) y u ( s, t )d s. (4a)The principal’s objective is to maximize her expected payoff,max Π ∈ Π E (cid:20)(cid:90) y ∗ ( t, Π) v ( s, t )d s (cid:21) . (4b)By (A (cid:48) ), y ∗ ( t, Π) is single-valued for almost all t , so E (cid:2) (cid:82) y ∗ ( t, Π) v ( s, t )d s (cid:3) is well defined. ERSUASION MEETS DELEGATION 9
The discriminatory disclosure problem (4a)–(4b) is a balanced delegation problem(1a)–(1b) with U D ( θ, x ) = (cid:90) x u ( s, t ) | t = θ d s and V D ( θ, x ) = (cid:90) x v ( s, t ) | t = θ d s. Conversely, for ( U D , V D ) ∈ P , define ( ¯ U D , ¯ V D ) ∈ P by¯ U D ( θ, x ) = U D ( θ, x ) − U D ( θ,
1) and ¯ V D ( θ, x ) = V D ( θ, x ) − V D ( θ, . Note that, for each Π, the principal gets the same expected payoff in the balanceddelegation problem with ( U D , V D ) and ( ¯ U D , ¯ V D ), up to a constant, E [ V D ( θ, U D , ¯ V D ) is a discriminat-ory disclosure problem (4a)–(4b) with u ( s, t ) = − ∂U D ( t, s ) ∂s and v ( s, t ) = − ∂V D ( t, s ) ∂s . Finally, ( U D , V D ) satisfies (A )–(A ) if and only if ( u, v ) satisfies (A (cid:48) )–(A (cid:48) ).3.4. Equivalence to Monotone Persuasion.
Consider a discriminatory disclosureproblem with a primitive ( u, v ). A menu Π ∈ Π defines a monotone partition of [0 , a = 0 and a = 1 is the same whether he observesthe partition element µ Π ( s ) or the result of the optimally selected cutoff test y ∈ Π.Indeed, by (A (cid:48) ), the agent gets a higher payoff from a = 1 when a partition elementis higher; so he optimally chooses a = 1 whenever the partition element is at least y ∈ Π. Therefore, the agent behaves as if he observes the partition element µ Π ( s )that contains the state s .Furthermore, by (A (cid:48) ), the agent gets a higher expected payoff from a = 1 whenhis type t is lower; so he optimally chooses a = 1 whenever t ≤ z for some z ∈ [0 , µ Π ( s ), the agent choosesa threshold type z ∈ [0 , a = 1 if t ≤ z and a = 0 if t > z .Thus, the agent chooses a threshold type z ∈ [0 ,
1] that maximizes his expected payoff z ∗ ( s, Π) = arg max z ∈ [0 , E (cid:20)(cid:90) z u ( s (cid:48) , t )d t (cid:12)(cid:12)(cid:12)(cid:12) s (cid:48) ∈ µ Π ( s ) (cid:21) . (5a)The principal’s objective is to maximize her expected payoff,max Π ∈ Π E (cid:34)(cid:90) z ∗ ( s, Π)0 v ( s, t )d t (cid:35) . (5b)By (A (cid:48) ), z ∗ ( s, Π) is single-valued for all s , so E (cid:2) (cid:82) z ∗ ( s, Π)0 v ( s, t )d t (cid:3) is well defined.The discriminatory disclosure problem (5a)–(5b) is a monotone persuasion problem(2a)–(2b) with U P ( θ, x ) = (cid:90) x u ( s, t ) | s = θ d t and V P ( θ, x ) = (cid:90) x v ( s, t ) | s = θ d t. Conversely, for ( U P , V P ) ∈ P , define ( ¯ U P , ¯ V P ) ∈ P by¯ U P ( θ, x ) = U P ( θ, x ) − U P ( θ,
0) and ¯ V P ( θ, x ) = V P ( θ, x ) − V P ( θ, . Note that, for each Π, the principal gets the same expected payoff in the monotonepersuasion problem with ( U P , V P ) and ( ¯ U P , ¯ V P ), up to a constant, E [ V P ( θ, U P , ¯ V P ) is a discrimin-atory disclosure problem (5a)–(5b) with u ( s, t ) = ∂U P ( s, t ) ∂t and v ( s, t ) = ∂V P ( s, t ) ∂t . Finally, ( U P , V P ) satisfies (A )–(A ) if and only if ( u, v ) satisfies (A (cid:48) )–(A (cid:48) ).4. Application to Monopoly Regulation
We consider the classical problem of monopoly regulation as in Baron and Myerson(1982). The monopolist privately knows his cost and chooses a price to maximizeprofit. The welfare-maximizing regulator can restrict the set of prices the monopol-ist can choose from, for example, by imposing a price cap. Following Alonso andMatouschek (2008), we assume that the demand function is linear and the marginalcost has a unimodal distribution. Importantly, unlike in Baron and Myerson (1982),transfers between the monopolist and regulator are prohibited.We study two versions of this problem: (i) with the monopolist’s participation con-straint, as in Baron and Myerson (1982) and Amador and Bagwell (2016), and (ii)without any participation constraint, as in Alonso and Matouschek (2008). We for-mulate both versions as balanced delegation problems. We then find the equivalentmonotone persuasion problems and solve them using a single result from the per-suasion literature. We show that, in both versions, the welfare-maximizing regulatorimposes a price cap, which is higher when the participation constraint is present.4.1.
Setup.
The demand function is q = 1 − x where x is the price and q is thequantity demanded at this price. The cost of producing quantity q is γq . The marginalcost γ ∈ [0 ,
1] has a distribution F that admits a strictly positive, continuous, andunimodal density f .The monopolist’s (agent’s) payoff is the profit U D ( γ, x ) = ( x − γ )(1 − x ) . The regulator’s (principal’s) payoff is the sum of the profit and consumer surplus, V D ( γ, x ) = U D ( γ, x ) + (1 − x ) . The regulator chooses a set of prices Π ⊂ [0 ,
1] available to the monopolist. Themonopolist privately observes the marginal cost γ and chooses a price x from Π tomaximize profit U D ( γ, x ). ERSUASION MEETS DELEGATION 11
We first assume that the monopolist cannot be forced to operate at a loss. Formally,the monopolist can always choose to produce zero quantity, which is the same assetting price x = 1; so 1 ∈ Π.Notice that selling at zero price gives a lower profit than not producing at all, re-gardless of the value of the marginal cost. Thus, allowing the price x = 0 does notaffect the monopolist’s behavior; so, without loss of generality, 0 ∈ Π. To sum up,the regulator chooses a closed set of prices Π ⊂ [0 ,
1] that contains 0 and 1; so Π ∈ Π .To interpret this problem as a balanced delegation problem defined in Section 2.2,we change the variable θ = F ( γ ), so that θ is uniformly distributed on [0 , U D ( θ, x ) = ( x − F − ( θ ))(1 − x ) and V D ( θ, x ) = U D ( θ, x ) + (1 − x ) . (6)4.2. Analysis.
By Theorem 1, an equivalent primitive ( U P , V P ) of the monotonepersuasion problem is given by U P ( θ, x ) = − (cid:90) x (1 + F − ( t ) − θ )d t = (cid:90) F − ( x )0 (2 θ − − γ ) f ( γ )d γ,V P ( θ, x ) = − (cid:90) x ( F − ( t ) − θ )d t = (cid:90) F − ( x )0 ( θ − γ ) f ( γ )d γ, where the set of decisions is [0 , , m Π ( θ ) = E [ θ (cid:48) | θ (cid:48) ∈ µ Π ( θ )] be the posterior mean state induced by a partitionelement µ Π ( θ ) of a monotone partition Π. Since U P is linear in θ , the agent’s optimaldecision depends on µ Π ( θ ) only through m Π ( θ ); so x ∗ P ( θ, Π) = ¯ x ∗ ( m Π ( θ )) with¯ x ∗ ( m ) = arg max x ∈ [0 , (cid:90) F − ( x )0 (2 m − − γ ) f ( γ )d γ = F (2 m − , where, by convention, F (2 m −
1) = 0 if 2 m − ≤ F (2 m −
1) = 1 if 2 m − ≥ V P is linear in θ , the principal’s expected payoff given µ Π ( θ ) is a function ν thatdepends on µ Π ( θ ) only through m Π ( θ ): ν ( m ) = (cid:90) F − (¯ x ∗ ( m ))0 ( m − γ ) f ( γ )d γ = (cid:90) m − ( m − γ ) f ( γ )d γ. (7)Thus, the principal’s objective is to maximize the expectation of ν ,max Π ∈ Π E (cid:2) ν ( m Π ( θ )) (cid:3) . In Appendix A.1, we provide an interpretation of this persuasion problem. θ ∗ ν ( m )separating pooling θ ∗ θ ∗∗ ν ( m )separating pooling θ ∗∗ (a) With Participation Constraint (b) Without Participation Constraint Figure 1.
Optimal Monotone PartitionsThe curvature of ν determines the form of the optimal monotone partition. Becausethe density f is unimodal, ν is S -shaped (see Figure 1(a)). Thus, the optimal mono-tone partition is an upper-censorship : the states below a cutoff θ ∗ are separated, andthe states above θ ∗ are pooled and induce the posterior mean state equal to (1+ θ ∗ ) / Proposition 1.
Let γ m ∈ (0 , be the mode of the density f . The set Π ∗ = [0 , θ ∗ ] ∪{ } is optimal, where θ ∗ ∈ ( γ m , (1 + γ m ) / is the unique solution to ν (cid:18) θ ∗ (cid:19) − ν ( θ ∗ ) = (cid:18) θ ∗ − θ ∗ (cid:19) ν (cid:48) (cid:18) θ ∗ (cid:19) . (8)Since upper censorship Π ∗ = [0 , θ ∗ ] ∪ { } is optimal in the monotone persuasionproblem, the same delegation set is optimal in the monopoly regulation problem.That is, the regulator imposes the price cap θ ∗ , thus implementing the price function x ∗ ( γ ) = γ , if γ < θ ∗ − ,θ ∗ , if 2 γ ∈ ( θ ∗ − , θ ∗ ) , , if γ > θ ∗ . In words, the monopolist chooses not to participate if his marginal cost γ is abovethe price cap θ ∗ . The participating monopolist chooses his preferred price (1 + γ ) / Analysis without Participation Constraint.
We now assume that the reg-ulator can force the monopolist to operate even when making a loss. That is, the
ERSUASION MEETS DELEGATION 13 regulator can choose any set of prices, without an additional constraint to include theprice x = 1.To interpret this problem as a balanced delegation problem defined in Section 2.2,we observe that when the price x is sufficiently high or sufficiently low, both themonopolist and regulator prefer intermediate prices. Thus, the requirement to includesufficiently extreme prices into the delegation set is not binding.Specifically, consider U D and V D given by (6) and defined on the domain of prices [0 , ⊂ [0 , γ ∈ [0 , x > x = 1. Therefore, an optimal delegation set Π ∗ must contain a price x ∈ [0 , x to 0 and to any price x ≥
2. Therefore, Π ∗ ∪{ , } implements the same price function as Π ∗ .We thus obtain a balanced delegation problem, up to rescaling of the monopolist’s de-cision. Using Theorem 1, we find the equivalent primitive of the monotone persuasionproblem.For comparability, it is convenient to rescale the state in the monotone persuasionproblem, so that it is uniformly distributed on [0 , ⊂ [0 ,
2] such that { , } ∈ Π to maximize the expectation of ν given by (7) defined on [0 , ν is still S -shaped (see Figure 1(b)), the optimal monotone partition is an upper-censorship: the states below a cutoff θ ∗∗ are separated, and the states above θ ∗∗ arepooled and induce the posterior mean state equal to (2 + θ ∗∗ ) / Proposition 1 (cid:48) . Let γ m ∈ (0 , be the mode of the density f . The set Π ∗∗ =[0 , θ ∗∗ ] ∪ { } is optimal, where θ ∗∗ ∈ (0 , (1 + γ m ) / is the unique solution to ν (cid:18) θ ∗∗ (cid:19) − ν ( θ ∗∗ ) = (cid:18) θ ∗∗ − θ ∗∗ (cid:19) ν (cid:48) (cid:18) θ ∗∗ (cid:19) . (9)Proposition 1 (cid:48) implies that [0 , θ ∗∗ ] is the optimal delegation set in the monopoly reg-ulation problem without the participation constraint. That is, the regulator imposesthe price cap θ ∗∗ , thus implementing the price function x ∗∗ ( γ ) = (cid:40) γ , if γ < θ ∗∗ − ,θ ∗∗ , if γ > θ ∗∗ − . In words, the monopolist chooses his preferred price (1 + γ ) / Discussion.
The optimal regulation policy takes the form of a price cap, re-gardless of whether the monopolist’s participation constraint is present. However, the optimal price cap is higher when the participation constraint is present, as fol-lows from Propositions 1 and 1 (cid:48) . Indeed, since ν is concave on [(1 + γ m ) / ,
2] and1 + γ m < θ ∗ < < θ ∗∗ < , the slope of ν is higher at (1 + θ ∗ ) / θ ∗∗ ) /
2; so (8) and (9) imply that θ ∗ > θ ∗∗ (see Figures 1(a) and 1(b)).We now build the intuition for why the optimal price cap is higher when the parti-cipation constraint is present. The first-order condition (8) for the optimal price cap θ ∗ can be written as (cid:90) θ ∗ θ ∗ − ( θ ∗ − γ ) f ( γ )d γ = 12 (1 − θ ∗ ) f ( θ ∗ ) , (8 (cid:48) )where the left-hand side and right-hand side correspond to the regulator’s marginalgain and marginal loss of decreasing the price cap by d θ . The gain is that themonopolist with the cost γ ∈ (2 θ ∗ − , θ ∗ ) now chooses the decreased price cap θ ∗ − d θ , which is closer to his cost γ . The loss is that the monopolist with the cost γ ∈ ( θ ∗ − d θ, θ ∗ ) now chooses to exit.Instead, if the regulator does not take into account that the monopolist with the costhigher than the price cap exits, then the first-order condition (9) for the price cap θ ∗∗ can be written as (cid:90) θ ∗∗ θ ∗∗ − ( θ ∗∗ − γ ) f ( γ )d γ = (cid:90) θ ∗∗ ( γ − θ ∗∗ ) f ( γ )d γ. (9 (cid:48) )The regulator’s marginal gain here is the same. But the marginal loss is that themonopolist with the cost γ ∈ ( θ ∗∗ ,
1) chooses the decreased price cap θ ∗∗ − d θ , whichis further from his cost γ .Intuitively, the marginal loss in (8 (cid:48) ) is higher than in (9 (cid:48) ), because all surplus is lostif the monopolist exits, but only a part of surplus is lost if the monopolist sets theprice further away from his cost. This suggests that the regulator should give morediscretion to the monopolist when she is concerned that the monopolist can exit.5.
Linear Delegation and Linear Persuasion
Setup.
Consider a primitive ( U D , V D ) ∈ P of the balanced delegation problemthat satisfies ∂U D ( θ, x ) ∂x = b ( θ ) − c ( x ) and ∂V D ( θ, x ) ∂x = d ( θ ) − c ( x ) , (10) The right-hand side of (8 (cid:48) ) can be expressed as (cid:82) θ ∗ ( γ − θ ∗ ) f ( θ ∗ )d γ . This marginal loss is higherthan in (9 (cid:48) ), because f ( θ ∗ ) > f ( γ ) for γ > θ ∗ > γ m by the unimodality of f . ERSUASION MEETS DELEGATION 15 where b , c , and d are continuous, and b and c are strictly increasing. By Theorem 1,for each ( U D , V D ) ∈ P that satisfies (10), an equivalent primitive ( U P , V P ) ∈ P of themonotone persuasion problem satisfies ∂U P ( θ, x ) ∂x = c ( θ ) − b ( x ) and ∂V P ( θ, x ) ∂x = c ( θ ) − d ( x ) . (11)Conversely, for each ( U P , V P ) ∈ P that satisfies (11), an equivalent primitive ( U D , V D ) ∈P of the balanced delegation problem satisfies (10).We call ( U D , V D ) and ( U P , V P ) that satisfy (10) and (11) linear , because the marginalpayoffs from a decision are linear, respectively, in a transformation of the decision, c ( x ), and in a transformation of the state, c ( θ ).Linear delegation (albeit without the inclusion of the extreme decisions) has beenstudied by Holmstr¨om (1984), Melumad and Shibano (1991), Martimort and Semenov(2006), Alonso and Matouschek (2008), Goltsman, H¨orner, Pavlov, and Squintani(2009), Kov´aˇc and Mylovanov (2009), Amador and Bagwell (2013), and Amador,Bagwell, and Frankel (2018). Linear persuasion (albeit without the restriction to monotone partitions) has beenstudied by Kamenica and Gentzkow (2011), Gentzkow and Kamenica (2016), Kolo-tilin, Mylovanov, Zapechelnyuk, and Li (2017), Kolotilin (2018), and Dworczak andMartini (2019). It is convenient to represent a linear monotone persuasion problem asmax Π ∈ Π E (cid:2) ν ( m Π ( θ )) (cid:3) , (12)for some function ν , where m Π ( θ ) = E [ c ( θ (cid:48) ) | θ (cid:48) ∈ µ Π ( θ )]. We can derive ν from( U P , V P ) as follows. Since U P is linear in c ( θ ), the agent’s optimal decision dependson µ Π ( θ ) only through m Π ( θ ); so x ∗ P ( θ, Π) = ¯ x ∗ ( m Π ( θ )). Moreover, since V P is linearin c ( θ ), the principal’s expected payoff is a function ν that depends on µ Π ( θ ) onlythrough m Π ( θ ), ν ( m ) = V P ( m, ¯ x ∗ ( m )) . Conversely, for each function ν , a monotone persuasion problem reduces to (12) if( U P , V P ) satisfies (11) with b ( x ) = x , c ( θ ) = θ , and d ( x ) = − ν (cid:48) ( x ). Amador and Bagwell (2016) study linear delegation with a participation constraint. Despitethe same assumptions on the payoffs, linear delegation is conceptually different from veto-baseddelegation of Krishna and Morgan (2001), Dessein (2002), and Mylovanov (2008) and from limited-commitment delegation of Kolotilin, Li, and Li (2013). Dworczak and Martini (2019) and Kolotilin and Li (2019) study linear monotone persuasion.Dworczak and Martini (2019) provide conditions under which monotone partitions are optimalamong all information structures. Kolotilin and Li (2019) characterize optimal monotone parti-tions when they differ from optimal information structures.
Optimal Linear Delegation.
We now generalize and extend the existing res-ults in the literature on linear delegation, using the tools from the literature on linearpersuasion (Kolotilin, 2018, and Dworczak and Martini, 2019).We consider a linear delegation problem where the agent’s and principal’s payoffs aregiven by (10), the set of states is a compact interval, and the set of decisions is the realline. Without loss of generality, we rescale the state and decision so that b ( θ ) = θ ,where the state θ ∈ [0 ,
1] has a distribution F that admits a strictly positive andcontinuous density f . For the problem to be well defined, we assume that there existthe agent’s and principal’s preferred decisions for each state. Specifically, we assumethat there exist x (cid:48) , x (cid:48)(cid:48) ∈ R such that c ( x (cid:48) ) ≤ θ ≤ c ( x (cid:48)(cid:48) ) and c ( x (cid:48) ) ≤ d ( θ ) ≤ c ( x (cid:48)(cid:48) ) forall θ .In this problem, the principal chooses a compact subset Π ⊂ R to maximize herexpected payoff, max Π ∈ Π ( R ) E (cid:2) V D ( θ, x ∗ D ( θ, Π)) (cid:3) , where Π ( R ) is the set of all compact subsets of R . As we show in Appendix A.3,this problem can be formulated as a balanced delegation problem with a sufficientlylarge compact set of decisions [ y, y ]. Notice that the decision is rescaled so that theprincipal chooses Π ∈ Π ([ y, y ]) where Π ([ y, y ]) = { Π ⊂ [ y, y ] : Π is closed and { y, y } ⊂ Π } . (13)By Theorem 1, an equivalent primitive ( U P , V P ) of the monotone persuasion problemis given by U P ( θ, x ) = (cid:90) F − ( x )0 ( c ( θ ) − t ) f ( t )d t and V P ( θ, x ) = (cid:90) F − ( x )0 ( c ( θ ) − d ( t )) f ( t )d t, where the set of decisions is [0 , y, y ], and the state is uniformlydistributed. Notice that the state is rescaled so that the principal chooses a monotonepartition Π ∈ Π ([ y, y ]).Since U P is linear in c ( θ ), the agent’s optimal decision depends on µ Π ( θ ) only through m Π ( θ ) = E [ c ( θ (cid:48) ) | θ (cid:48) ∈ µ Π ( θ )] = { c ( θ ) } , if π ( θ ) = π ( θ ) , (cid:82) π ( θ ) π ( θ ) c ( θ (cid:48) )d θ (cid:48) π ( θ ) − π ( θ ) , if π ( θ ) < π ( θ ) , where π ( θ ) and π ( θ ) are defined in Section 2.3; so x ∗ P ( θ, Π) = ¯ x ∗ ( m Π ( θ )) with¯ x ∗ ( m ) = arg max x ∈ [0 , (cid:90) F − ( x )0 ( m − t ) f ( t )d t = F ( m ) , where, by convention, F ( m ) = 0 if m ≤ F ( m ) = 1 if m ≥ ERSUASION MEETS DELEGATION 17 c ( x ∗ )0 ν ( m ) p Π ∗ ( m ) c ( x ∗ H )0 ν ( m ) p Π ∗ ( m ) c ( x ∗ L ) c ( x ∗ L )0 ν ( m ) p Π ∗ ( m ) c ( x ∗ H ) 1 m ∗ (a) Part 1 (b) Part 2 (c) Part 3 Figure 2.
Three Parts in Proposition 2Since V P is linear in c ( θ ), the principal’s expected payoff given µ Π ( θ ) is a function ν that depends on µ Π ( θ ) only through m Π ( θ ), ν ( m ) = (cid:90) F − (¯ x ∗ ( m ))0 ( m − d ( t )) f ( t )d t = (cid:90) m ( m − d ( t )) f ( t )d t. (14)The next theorem verifies whether a candidate delegation set is optimal. Theorem 2. Π ∗ ∈ Π ([ y, y ]) is optimal if p Π ∗ ( m ) is convex and p Π ∗ ( m ) ≥ ν ( m ) forall m ∈ c ([ y, y ]) , where, for all s ∈ [ y, y ] , p Π ∗ ( c ( s )) = ν ( m Π ∗ ( s )) + ν (cid:48) ( m Π ∗ ( s ))( c ( s ) − m Π ∗ ( s )) . Proof.
Consider any Π ∈ Π ([ y, y ]). The theorem follows from E [ ν ( m Π ∗ ( s ))] − E [ ν ( m Π ( s ))] = E [ p Π ∗ ( c ( s ))] − E [ ν ( m Π ( s ))] ≥ E [ p Π ∗ ( c ( s ))] − E [ p Π ∗ ( m Π ( s ))]= E [ E [ p Π ∗ ( c ( s (cid:48) ))] − p Π ∗ ( m Π ( s )) | s (cid:48) ∈ µ Π ( s )]] ≥ , where the first equality holds by definition of p Π ∗ , the first inequality holds by p Π ∗ ≥ ν , the second equality holds by the law of iterated expectations, and the secondinequality holds by Jensen’s inequality applied to convex p Π ∗ . (cid:3) Using Theorem 2, we find sufficient conditions under which one- or two-interval del-egation sets are optimal (see Figure 2).
Proposition 2. . A delegation set { x ∗ } is optimal if ν ( m ) ≤ for all m ≤ c ( x ∗ ) ,ν ( m ) ≤ ν (1) + m − for all m ≥ c ( x ∗ ) ,ν (1) + c ( x ∗ ) − . . A delegation set [ x ∗ L , x ∗ H ] with x ∗ L < x ∗ H is optimal if ν ( m ) is convex on [ c ( x ∗ L ) , c ( x ∗ H )] ,ν ( m ) ≤ for all m ≤ c ( x ∗ L ) with equality at m = c ( x ∗ L ) , ν ( m ) ≤ ν (1) + m − for all m ≥ c ( x ∗ H ) with equality at m = c ( x ∗ H ) ,ν (cid:48) (0+) ≥ if c ( x ∗ L ) = 0 and ν (cid:48) (1 − ) ≤ if c ( x ∗ H ) = 1 . . A delegation set ( −∞ , x ∗ L ] ∪ [ x ∗ H , ∞ ) with x ∗ L < x ∗ H is optimal if ν ( m ) is convex on ( −∞ , c ( x ∗ L )] and on [ c ( x ∗ H ) , ∞ ) ,ν ( m ) ≤ ν ( m ∗ ) + ν (cid:48) ( m ∗ )( m − m ∗ ) for all m ∈ [ c ( x ∗ L ) , c ( x ∗ H )] with equality at m = c ( x ∗ L ) and m = c ( x ∗ H ) ,ν (cid:48) ( m ∗ ) ≥ if c ( x ∗ L ) = 0 and ν (cid:48) ( m ∗ ) ≤ if c ( x ∗ H ) = 1 . where m ∗ = 1 x ∗ H − x ∗ L (cid:90) x ∗ H x ∗ L c ( s )d s. The literature on linear delegation has focused on characterizing sufficient conditionsfor interval delegation to be optimal. The conditions in Amador and Bagwell (2013,Proposition 1) coincide with those in Proposition 2 (part 2), but we impose weakerregularity assumptions on the primitives. Amador and Bagwell (2013, Proposition 2)show that these conditions are also necessary. The sufficient conditions in Amador, Bagwell, and Frankel (2018, Proposition 1) fordegenerate interval delegation to be optimal coincide with those in Proposition 2 (part1), but again we impose weaker regularity assumptions on the primitives. Alonso andMatouschek (2008, Proposition 1) show that these conditions are also necessary. The set of the agent’s preferred decisions is [ c − (0) , c − (1)]. Thus, delegation set ( −∞ , x ∗ L ] ∪ [ x ∗ H , ∞ ) implements the same outcome as set [ y ∗ L , x ∗ L ] ∪ [ x ∗ H , y ∗ H ] does where y ∗ L = min { x ∗ L , c − (0) } and y ∗ H = max { x ∗ H , c − (1) } . Alonso and Matouschek (2008, Propositions 3, 6, 7) also show the necessity and sufficiency ofthese conditions but only for quadratic payoffs ( c ( x ) = x ). They show the necessity only for quadratic payoffs ( c ( x ) = x ), but we can use their argument toshow the necessity for non-quadratic payoffs. To this end, suppose first that there exists m (cid:48) < c ( x ∗ )such that ν ( m (cid:48) ) > x (cid:48) < x ∗ be such that (cid:82) x ∗ x (cid:48) c ( s )d s = ( x ∗ − x (cid:48) ) m (cid:48) . The principal’s expectedpayoff is larger by ( x ∗ − x (cid:48) ) ν ( m (cid:48) ) > { x (cid:48) , x ∗ } than for { x ∗ } . The case in whichthere exists m (cid:48) > c ( x ∗ ) such that ν ( m (cid:48) ) > ν (1) + m (cid:48) − ERSUASION MEETS DELEGATION 19
The sufficient conditions in Proposition 2 (part 3) for two-interval delegation to beoptimal are novel. This result implies Melumad and Shibano (1991, Proposition 4)and Alonso and Matouschek (2008, Result 6).Using Proposition 2, we now characterize optimal delegation sets in prominent cases.
Proposition 3. (convex). If ν ( m ) is convex on [0 , , then there exist x ∗ L ≤ x ∗ H such that delegationset [ x ∗ L , x ∗ H ] is optimal. (concave). If ν ( m ) is concave on [0 , , then there exist x ∗ L ≤ x ∗ H such that delegationset { x ∗ L , x ∗ H } is optimal. (convex-concave). If ν ( m ) is convex on [0 , ˜ m ] and concave on [ ˜ m, for some < ˜ m < , then there exist x ∗ L ≤ x ∗ M ≤ x ∗ H such that delegation set [ x ∗ L , x ∗ M ] ∪ { x ∗ H } is optimal. (concave-convex). If ν ( m ) is concave on [0 , ˜ m ] and convex on [ ˜ m, for some < ˜ m < , then there exist x ∗ L ≤ x ∗ M ≤ x ∗ H such that delegation set { x ∗ L } ∪ [ x ∗ M , x ∗ H ] is optimal. Amador, Bagwell, and Frankel (2018, Proposition 2) coincides with Proposition 3(part 1), but again we impose weaker regularity assumptions on the primitives. Mar-timort and Semenov (2006) assume that d ( x ) = x + δ and f ( θ ) − δf (cid:48) ( θ ) ≥
0, so that ν is convex on [0 , f ( θ ) = 1 and constant slope d (cid:48) ( x ) = k >
0, so that ν is convex on [0 ,
1] if k < ,
1] if k >
2. In the monopoly regulation problem of Section 4, ν is convex-concave on [0 ,
1] because f is unimodal, but ν would be concave-convexon [0 ,
1] if f was uniantimodal.Finally, we comment on the limits of the tools from the persuasion literature for thedelegation problem. The persuasion literature studies optimal information structureswithout the restriction to monotone partitions. Under the conditions in Theorem 2,the set Π is optimal among all information structures. These conditions are not ne-cessary, because Π may be optimal among monotone partitions, but not among allinformation structures. However, as shown by Dworczak and Martini (2019, The-orem 3), these conditions become necessary if ν is affine-closed (intuitively, if it hasat most one interior peak). Even when optimal information structures are not mono-tone partitions, it may be possible to characterize optimal monotone partitions usingtools from Kolotilin and Li (2019).5.3. Optimal Linear Delegation with Participation Constraint.
We now con-sider a linear delegation problem from Section 5.2 with the only difference that theset of decisions is [ x , ∞ ) and the decision x must always be permitted by the prin-cipal. We assume that c ( x ) ≤ x (cid:48)(cid:48) > x such that θ ≤ c ( x (cid:48)(cid:48) ) and d ( θ ) ≤ c ( x (cid:48)(cid:48) ) for all θ ∈ [0 , x , y ]. That is, the principal chooses Π ∈ Π ([ x , y ]) to maximize theexpectation of ν given by (14). Thus, Theorem 2 continues to hold with y = x .As an illustration, we use Theorem 2 to find sufficient conditions under which anoptimal delegation set takes the form of a floor on the decisions. Proposition 2 (cid:48) . A delegation set { x } ∪ [ x ∗ , ∞ ) with x ∗ ≥ x is optimal if ν ( m ) is convex on [ c ( x ∗ ) , ∞ ) ,ν ( m ) ≤ ν ( m ∗ ) + ν (cid:48) ( m ∗ )( m − m ∗ ) for all m ∈ [ c ( x ) , c ( x ∗ )] with equality at m = c ( x ∗ ) ,ν (cid:48) (0+) ≥ if c ( x ∗ ) = 0 and ν (cid:48) (1 − ) ≤ if c ( x ∗ ) = 1 , where m ∗ = (cid:40) c ( x ) , if x ∗ = x , x ∗ − x (cid:82) x ∗ x c ( s )d s, if x ∗ > x . Proposition 2 (cid:48) can be used to derive conditions for a price cap (equivalently, a quantityfloor) to be optimal in a monopoly regulation problem of Section 4 with more generalprimitives specified as follows. The inverse demand function is P ( q ). The marginalcost γ ∈ [ γ, γ ], with 0 ≤ γ < γ < ∞ , has a strictly positive and continuous density.The regulator’s payoff is a weighted sum, with weights λ ∈ (0 ,
1) and 1 − λ , of theprofit and consumer surplus, V ( γ, q ) = λ ( P ( q ) − γ ) q + (1 − λ ) (cid:18)(cid:90) q P ( s )d s − P ( q ) q (cid:19) . We assume that the monopolist can always choose to produce zero quantity (exit).The regulator’s problem reduces to a balanced delegation problem in which the reg-ulator chooses a set of quantities Π ∈ Π ([0 , q ]), where P ( q ) ≤ γ , available to themonopolist. Up to rescaling of the state and decision, all assumptions imposed inthis section are satisfied in each of the following three cases:1 . P ( q ) = A − B ln q with A ∈ R and B > . P ( q ) = A − Bq C with A > B >
0, and
C > . P ( q ) = A − Bq C with A < γ , B <
0, and C ∈ ( − , (cid:48) gives sufficient conditions on the cost distribu-tion, demand parameters, and payoff weights for price-cap regulation to be optimal.These conditions can be compared with those in Amador and Bagwell (2016) whoconsider a similar setting but focus on delegation sets under which the monopolistnever chooses to exit. ERSUASION MEETS DELEGATION 21 General Equivalence
We now generalize Theorem 1. We assume that the marginal payoffs are integrableand single crossing, rather than continuous and monotone. We then discuss how thisresult can be used to deal with arbitrary distributions of the state, which may haveatoms and zero density.6.1.
Primitives.
We first define single crossing properties. A function φ ( y i , y − i )satisfies(i) upcrossing in y i if, for each y − i , φ ( y (cid:48) i , y − i ) ≥ ( > ) 0 = ⇒ φ ( y (cid:48)(cid:48) i , y − i ) ≥ ( > ) 0 whenever y (cid:48)(cid:48) i > y (cid:48) i ;(ii) aggregate upcrossing in y i if, for each distribution H of y − i , (cid:90) φ ( y (cid:48) i , y − i )d H ( y − i ) ≥ ( > ) 0 = ⇒ (cid:90) φ ( y (cid:48)(cid:48) i , y − i )d H ( y − i ) ≥ ( > ) 0 whenever y (cid:48)(cid:48) i > y (cid:48) i ;(iii) downcrossing (aggregate downcrossing) in y i if − φ ( y i , y − i ) satisfies upcrossing(aggregate upcrossing) in y i . The agent’s payoff U ( θ, x ) and the principal’s payoff V ( θ, x ) depend on a decision x ∈ [0 ,
1] and a state θ ∈ [0 , ) U ( θ, x ) and V ( θ, x ) are absolutely continuous in x ; ∂∂x U ( θ, x ) and ∂∂x V ( θ, x ) areintegrable in θ .In addition, for the balanced delegation problem, we assume that( ¯A D ) ∂∂x U ( θ, x ) satisfies downcrossing in x and aggregate upcrossing in θ ;for the monotone persuasion problem, we assume that( ¯A P ) ∂∂x U ( θ, x ) satisfies upcrossing in θ and aggregate downcrossing in x .The balanced delegation and monotone persuasion problems are defined as in Section2. But, unlike in Section 2, the agent’s optimal correspondences x ∗ D ( θ, Π) and x ∗ P ( θ, Π)may not be single-valued. Depending on which optimal decisions are selected by theagent, the principal can obtain different expected payoffs. Denote by E (cid:2) V D ( θ, x ∗ D ( θ, Π)) (cid:3) and E (cid:2) V P ( θ, x ∗ P ( θ, Π)) (cid:3) the sets of the principal’s expected payoffs resulting from all integrable selections fromthe correspondences x ∗ D ( θ, Π) and x ∗ P ( θ, Π) (Aumann, 1965). Quah and Strulovici (2012) characterize conditions for aggregate single crossing. In particular, φ ( y i , y − i ) satisfies aggregate single crossing in y i if φ ( y i , y − i ) is monotone in y i . Equivalence.
Let ¯ P D be the set of all primitives ( U D , V D ) that satisfy assump-tions ( ¯A ) and ( ¯A D ), and let ¯ P P be the set of all primitives ( U P , V P ) that satisfyassumptions ( ¯A ) and ( ¯A P ).Primitives ( U D , V D ) ∈ ¯ P D and ( U P , V P ) ∈ ¯ P P are equivalent if there exist α > β ∈ R such that E (cid:2) V D ( θ, x ∗ D ( θ, Π)) (cid:3) = α E (cid:2) V P ( θ, x ∗ P ( θ, Π)) (cid:3) + β for all Π ∈ Π . That is, if ( U D , V D ) and ( U P , V P ) are equivalent, then, in both problems, the principalgets the same sets of expected payoffs, up to an affine transformation, for each Π. Theorem 1 (cid:48) . For each ( U D , V D ) ∈ ¯ P D , an equivalent ( U P , V P ) ∈ ¯ P P is given by (3a) .Conversely, for each ( U P , V P ) ∈ ¯ P P , an equivalent ( U D , V D ) ∈ ¯ P D is given by (3b) . The principal’s set of expected payoffs is a singleton for each Π (and thus the prin-cipal’s maximization problem is well defined) if the optimal correspondence is single-valued for almost all θ . This property holds in the delegation problem if ∂∂x U D ( θ, x )satisfies strict aggregate upcrossing in θ . Similarly, this property holds in the persua-sion problem if ∂∂x U P ( θ, x ) satisfies strict aggregate downcrossing in x .Alternatively, the principal’s maximization problem can be defined by specifying aselection rule from the agent’s optimal correspondence, such as the max rule (wherethe agent chooses the principal’s most preferred decision) or the min rule (where theagent chooses the principal’s least preferred decision).6.3. General Distributions.
Consider a balanced delegation or monotone persua-sion problem with the state ω ∈ [ ω, ω ] that has a distribution F (possibly, with atomsand zero density). To apply Theorem 1 (cid:48) , we redefine the state to be θ uniformly dis-tributed on [0 , ω = F − ( θ ), where F − ( θ ) = inf { ω ∈ [ ω, ω ] : θ ≤ F ( ω ) } is the quantile function.With this change of variable, atoms in F translate into intervals where ∂∂x U ( F − ( θ ) , x )and ∂∂x V ( F − ( θ ) , x ) are constant in θ ; zero-density intervals in F translate into pointswhere ∂∂x U D ( F − ( θ ) , x ) and ∂∂x V D ( F − ( θ ) , x ) have simple discontinuities in θ . Thischange of variable preserves both integrability in θ assumed in ( ¯A ) and single-crossingin θ assumed in ( ¯A D )/( ¯A P ).For the case of atomless distributions with a strictly positive density, Theorem 1 (cid:48) canbe conveniently expressed as follows. Let U D ( ω D , x ) and V D ( ω D , x ) satisfy assump-tions ( ¯A ) and ( ¯A D ), and let U P ( ω P , x ) and V P ( ω P , x ) satisfy assumptions ( ¯A ) and( ¯A P ). Suppose that ω D ∈ [0 ,
1] and ω P ∈ [0 ,
1] are distributed with strictly positivedensities f D and f P . Applying Theorem 1 (cid:48) to the primitives with redefined states,and changing states back to ω D and ω P , we obtain that ( U D , V D ) and ( U P , V P ) are ERSUASION MEETS DELEGATION 23 equivalent if, for all ω D , ω P ∈ [0 , ∂U D ( ω D , x ) ∂x (cid:12)(cid:12)(cid:12)(cid:12) x = ω P f D ( ω D ) = − ∂U P ( ω P , x ) ∂x (cid:12)(cid:12)(cid:12)(cid:12) x = ω D f P ( ω P ) ,∂V D ( ω D , x ) ∂x (cid:12)(cid:12)(cid:12)(cid:12) x = ω P f D ( ω D ) = − ∂V P ( ω P , x ) ∂x (cid:12)(cid:12)(cid:12)(cid:12) x = ω D f P ( ω P ) . To illustrate how Theorem 1 (cid:48) applies when F has atoms, consider the example inKamenica and Gentzkow (2011, p. 2591) in which a prosecutor (principal) persuadesa judge (agent) to convict a suspect. The suspect is innocent ( ω = 0) with probability0.7 and guilty ( ω = 1) with probability 0.3, so that the distribution of ω is F ( ω ) = (cid:40) . , if ω ∈ [0 , , if ω = 1.The prosecutor’s preferred decision is to convict the suspect irrespective of the state,whereas the judge’s preferred decision is to convict the suspect whenever his posteriorthat the suspect is guilty is at least 1 / V P ( ω, x ) = x and U P ( ω, x ) = ( ω − / x, where ω, x ∈ [0 , ω = F − ( θ ) yields V P ( θ, x ) = x and U P ( θ, x ) = (cid:40) − x/ , if θ ∈ [0 , . ,x/ , if θ ∈ [0 . , . Clearly, ( U P , V P ) ∈ ¯ P P . By Theorem 1 (cid:48) , an equivalent primitive ( U D , V D ) ∈ ¯ P D ofthe balanced delegation problem is given by V D ( θ, x ) = 1 − x and U D ( θ, x ) = (cid:40) ( x − . / , if x ∈ [0 , . , (1 − x ) / , if x ∈ [0 . , . Assume that the agent breaks ties in favor of the principal. It is easy to see that theoptimal balanced delegation set is Π ∗ = { , . , } , so that the agent is indifferentbetween x = 0 . x = 1 and thus chooses the principal’s preferred decision, x ∗ = 0 .
4. The principal’s optimal expected payoff is 1 − x ∗ = 0 . ∗ = { , . , } within the persuasion problem. Whenthe state θ belongs to the partition element [0 , . ω = F − ( θ ) =0 with probability one, so the judge acquits the suspect. When the state θ belongs tothe partition element [0 . , ω = F − ( θ ) = 0 with probability1 /
2, when θ ∈ [0 . , . ω = F − ( θ ) = 1 with probability 1 /
2, when θ ∈ [0 . , .
6, and Π ∗ is the optimal informationstructure derived in Kamenica and Gentzkow (2011, p. 2591). Concluding Remarks
We have shown the equivalence of balanced delegation and monotone persuasion,with the upshot that insights in delegation can be used for better understanding ofpersuasion, and vice versa. For instance, persuasion as the design of a distributionof posterior beliefs is notoriously hard to explain to a non-specialized audience. Theconnection to delegation can thus be instrumental in relaying technical results fromthe persuasion literature to practitioners and policy makers.We have used the tools from the literature on linear persuasion to obtain new res-ults on linear delegation. Amador and Bagwell (2013) have developed a Lagrangianmethod to derive sufficient conditions for the optimality of interval delegation in anonlinear delegation problem. This method may be useful for deriving conditions forthe optimality of interval persuasion in an equivalent nonlinear persuasion problem.The classical delegation and persuasion problems have numerous extensions, which in-clude a privately informed principal, competing principals, multiple agents, repeatedinteractions, and multidimensional state and decision spaces. We hope that our equi-valence result will be a starting point for studying the connection between delegationand persuasion in these extensions.It may be interesting to compare the values of delegation and persuasion in a givenproblem. This comparison can be made by recasting the persuasion problem as anequivalent delegation problem and then directly comparing the solutions and valuesof these two delegation problems.Naturally, a principal may wish to influence an agent’s decision by a combination ofpersuasion and delegation instruments. How to optimally control both informationand decisions of the agent, how these instruments interact, and whether they aresubstitutes or complements are important questions that are left for future research.
Appendix
A.1.
Interpretation of Persuasion Problem.
In Section 4, we have expressedthe monopoly regulation problem as a balanced delegation problem and derived anequivalent monotone persuasion problem. The primitive of this problem, up to amultiplicative constant, is U P ( θ, x ) = 12 (cid:90) x (2 θ − − F − ( γ ))d γ and V P ( θ, x ) = (cid:90) x ( θ − F − ( γ ))d γ. (15)We now provide an interpretation of this problem.A producer (agent) chooses a quantity x to produce. He faces uncertainty about anexogenous price θ that is uniformly distributed on [0 , U P ( θ, x ) = θx − C ( x ) , (16) ERSUASION MEETS DELEGATION 25 where C ( x ) is the producer’s cost function. A government agency (principal) candisclose information about the price to the producer. The agency’s payoff is V D ( θ, x ) = θx − D ( x ) , (17)where is D ( x ) is the social cost function. The difference, C ( x ) − D ( x ), is the producer’sexternality. The agency chooses Π ∈ Π . The producer observes the partition element µ Π ( θ ) that contains the price θ and chooses a quantity x that maximizes his expectedpayoff given the posterior belief about the price.Observe that the payoffs (16)–(17) are the same as (15) if C ( x ) = (cid:90) x F − ( γ )2 d γ and D ( x ) = (cid:90) x F − ( γ )d γ. Both cost functions C and D are convex, since F − is increasing. Notice that C ( x ) − D ( x ) > x >
0. So, in this problem, we have a positive externality, where thesocial cost is smaller than the producer’s cost.A.2.
Proof of Propositions 1 and 1 (cid:48) . The derivative of ν ( m ) is ν (cid:48) ( m ) = 2(1 − m ) f (2 m −
1) + (cid:90) m − f ( γ )d γ. First, we show that ν ( m ) is S -shaped (that is, ν (cid:48) is single-peaked with an interiorpeak) when the density f is unimodal. Lemma 1.
Let γ m ∈ (0 , be the mode of the density f . Then, ν ( m ) is convex on ( −∞ , (1 + γ m ) / (strictly so on [1 / , (1 + γ m ) / ) and concave on [(1 + γ m ) / , ∞ ) (strictly so on [(1 + γ m ) / , ).Proof. For any m , m , ν (cid:48) ( m ) − ν (cid:48) ( m ) = 2(1 − m )[ f (2 m − − f (2 m − (cid:90) m − m − [ f ( γ ) − f (2 m − γ. Thus, if m < m ≤ (1 + γ m ) /
2, then ν (cid:48) ( m ) ≥ ν (cid:48) ( m ), because f (2 m −
1) isincreasing for m ∈ ( −∞ , (1 + γ m ) / / ≤ m < m ≤ (1 + γ m ) /
2, then ν (cid:48) ( m ) > ν (cid:48) ( m ), because f (2 m −
1) is strictly increasing for m ∈ [1 / , (1 + γ m ) / γ m ) / ≤ m < m , then ν (cid:48) ( m ) ≤ ν (cid:48) ( m ), because f (2 m −
1) isdecreasing for m ∈ [(1 + γ m ) / , ∞ ). Moreover, if (1 + γ m ) / ≤ m < m ≤
1, then ν (cid:48) ( m ) < ν (cid:48) ( m ), because f (2 m −
1) is strictly decreasing for m ∈ [(1 + γ m ) / , (cid:3) Propositions 1 and 1 (cid:48) are special cases of Lemma 2.
Lemma 2.
Let γ m ∈ (0 , be the mode of the density f , and let θ be uniformlydistributed on [0 , θ ] with θ ≥ . The set Π (cid:63) = [0 , θ (cid:63) ] ∪ { θ } is optimal, where θ (cid:63) ∈ (cid:0) max (cid:8) , γ m − θ (cid:9) , (1 + γ m ) / (cid:1) is the unique solution to ν (cid:18) θ + θ (cid:63) (cid:19) − ν ( θ (cid:63) ) = (cid:18) θ + θ (cid:63) − θ (cid:63) (cid:19) ν (cid:48) (cid:18) θ + θ (cid:63) (cid:19) . (18) Proof.
It is straightforward to show (see Figures 1(a) and 1(b)) that there exists aunique solution θ (cid:63) to (18) with1 − E [ θ ] ≤ θ (cid:63) < γ m < θ + θ (cid:63) . We now use Theorem 2 in Section 5 to verify that Π (cid:63) is optimal. For Π (cid:63) = [0 , θ (cid:63) ] ∪{ θ } ,we have p Π (cid:63) ( m ) = (cid:40) ν ( m ) , if m < θ (cid:63) ,ν (cid:16) θ + θ (cid:63) (cid:17) + (cid:16) m − θ + θ (cid:63) (cid:17) ν (cid:48) (cid:16) θ + θ (cid:63) (cid:17) , if m ≥ θ (cid:63) . Figures 1(a) and 1(b) show p Π (cid:63) as a dashed red curve. It is straightforward to verifythat p Π (cid:63) ( m ) is convex and p Π (cid:63) ( m ) ≥ ν ( m ) for all m ∈ [0 , θ ]. Thus, Π (cid:63) is optimal byTheorem 2. (cid:3) A.3.
Standard Delegation.
Consider a delegation problem in which the set ofstates is Θ = [ θ, θ ] and the set of decisions is the real line. The principal choosesΠ ∈ Π ( R ), where Π ( R ) is the set of all compact subsets of R . Payoffs U D and V D satisfy assumptions ( ¯A ) and ( ¯A D ). In addition, we assume that( ¯A ) sup θ ∈ Θ U D ( θ, x ) → −∞ and sup θ ∈ Θ V D ( θ, x ) → −∞ as x → ±∞ .Note that ( ¯A ), ( ¯A D ), and ( ¯A ) are satisfied in the linear delegation problem inSection 5.We now show that this problem can be formulated as a balanced delegation problem,up to rescaling of the decision, in which the principal chooses Π ∈ Π ([ y, y ]) given by(13) for a sufficiently large compact set [ y, y ]. Lemma 3.
There exists an interval [ x, x ] such that, for each y < x and each y > x , max Π ∈ Π ([ y,y ]) E (cid:2) V D ( θ, x ∗ D ( θ, Π)) (cid:3) = max Π ∈ Π ( R ) E (cid:2) V D ( θ, x ∗ D ( θ, Π)) (cid:3) . Proof.
Consider z ∈ R . Let V be the principal’s expected payoff if Π = { z } , V = E [ V D ( θ, z )] . Let Z = (cid:26) x ∈ R : sup θ ∈ Θ V D ( θ, x ) ≥ V (cid:27) . ERSUASION MEETS DELEGATION 27
Note that Z is nonempty because z ∈ Z , and it is bounded by ( ¯A ). Let z = inf Z and z = sup Z . Clearly, for each Π ∈ Π ( R ),Π ∩ [ z, z ] = ∅ = ⇒ E (cid:2) V D ( θ, x ∗ D ( θ, Π)) (cid:3) < V , so an optimal Π ∗ ∈ Π ( R ) must have a nonempty intersection with [ z, z ].Next, we say that y ∈ R is dominated by [ z, z ] if the agent strictly prefers any decisionin [ z, z ] to y , min z ∈ [ z,z ] U D ( θ, z ) − U D ( θ, y ) > θ ∈ Θ . By ( ¯A ) and ( ¯A D , downcrossing in x ),min z ∈ [ z,z ] U D ( θ, z ) = min { U D ( θ, z ) , U D ( θ, z ) } . (19)By ( ¯A ) and ( ¯A D , aggregate upcrossing in θ ), U D ( θ, z ) − U D ( θ, y ) = (cid:90) zy ∂∂x U D ( θ, x )d x > θ ∈ [ θ, θ ]if and only if (cid:40) U D ( θ, z ) − U D ( θ, y ) > z > y,U D ( θ, z ) − U D ( θ, y ) > z < y. (20)Let X be the set of all x ∈ R that are not dominated by [ z, z ]. If x ∈ [ z, z ], then,trivially, x ∈ X . If x < z , then, by (19) and (20), x ∈ X ⇐⇒ U D ( θ, x ) ≥ min { U D ( θ, z ) , U D ( θ, z ) } . If x > z , then, by (19) and (20), x ∈ X ⇐⇒ U D ( θ, x ) ≥ min { U D ( θ, z ) , U D ( θ, z ) } . Thus, [ z, z ] ⊂ X and, by ( ¯A ), X is bounded.So, we have obtained that (i) if Π ∗ is optimal, then it has a nonempty intersection with[ z, z ], and (ii) any decision x (cid:54)∈ X is dominated by [ z, z ]. Given that Π ∗ ∩ [ z, z ] (cid:54) = ∅ ,adding or removing any decisions outside of X does not affect the agent’s behavior, x ∗ D ( θ, Π ∗ ) = x ∗ D ( θ, Π ∗ ∪ Y ) = x ∗ D ( θ, Π ∗ \ Y ) for all θ and Y ⊂ R \ X. Hence, for each y < min X and each y > max X , E (cid:2) V D ( θ, x ∗ D ( θ, Π ∗ )) (cid:3) = E (cid:2) V D ( θ, x ∗ D ( θ, Π ∗ ∩ [ y, y ] ∪ { y, y } )) (cid:3) . (cid:3) A.4.
Proof of Proposition 2. . For Π ∗ = { x ∗ , y, y } , we have p Π ∗ ( m ) = (cid:40) ν ( m ∗ L ) + ν (cid:48) ( m ∗ L )( m − m ∗ L ) , if m < c ( x ∗ ) ,ν ( m ∗ H ) + ν (cid:48) ( m ∗ H )( m − m ∗ H ) , if m ≥ c ( x ∗ ) , where m ∗ L = E [ c ( s ) | s < x ∗ ] and m ∗ H = E [ c ( s ) | s ≥ x ∗ ] . As shown in Appendix A.3, y is sufficiently small and y is sufficiently large, such that m ∗ L = (cid:82) x ∗ y c ( s )d sx ∗ − y = (cid:82) x (cid:48) y c ( s )d s + (cid:82) x ∗ x (cid:48) c ( s )d sx (cid:48) − y + x ∗ − x (cid:48) < ,m ∗ H = (cid:82) yx ∗ c ( s )d sy − x ∗ = (cid:82) x (cid:48)(cid:48) x ∗ c ( s )d s + (cid:82) yx (cid:48)(cid:48) c ( s )d sx (cid:48)(cid:48) − x ∗ + y − x (cid:48)(cid:48) > , where the first inequality holds because c ( x ) < x < x (cid:48) and y (cid:28) x (cid:48) and thesecond inequality holds because c ( x ) > x > x (cid:48)(cid:48) and y (cid:29) x (cid:48)(cid:48) .Thus, taking into account (14), we have p Π ∗ ( m ) = (cid:40) , if m < c ( x ∗ ) ,ν (1) + m − , if m ≥ c ( x ∗ ) . Figure 2(a) shows p Π ∗ as a dashed red curve. The conditions imply that p Π ∗ is convexand p Π ∗ ≥ ν ; so Π ∗ is optimal by Theorem 2.2 . For Π ∗ = [ x ∗ L , x ∗ H ] ∪ { y, y } , we have p Π ∗ ( m ) = ν ( m ∗ L ) + ν (cid:48) ( m ∗ L )( m − m ∗ L ) , if m < c ( x ∗ L ) ,ν ( m ) , if m ∈ [ c ( x ∗ L ) , c ( x ∗ H )) ,ν ( m ∗ H ) + ν (cid:48) ( m ∗ H )( m − m ∗ H ) , if m ≥ c ( x ∗ H ) , where m ∗ L = E [ c ( s ) | s < x ∗ L ] and m ∗ H = E [ c ( s ) | s ≥ x ∗ H ] . As shown in Appendix A.3, y is sufficiently small and y is sufficiently large, such that m ∗ L = (cid:82) x ∗ L y c ( s )d sx ∗ L − y = (cid:82) x (cid:48) y c ( s )d s + (cid:82) x ∗ L x (cid:48) c ( s )d sx (cid:48) − y + x ∗ L − x (cid:48) < ,m ∗ H = (cid:82) yx ∗ H c ( s )d sy − x ∗ H = (cid:82) x (cid:48)(cid:48) x ∗ H c ( s )d s + (cid:82) yx (cid:48)(cid:48) c ( s )d sx (cid:48)(cid:48) − x ∗ H + y − x (cid:48)(cid:48) > , where the first inequality holds because c ( x ) < x < x (cid:48) and y (cid:28) x (cid:48) and thesecond inequality holds because c ( x ) > x > x (cid:48)(cid:48) and y (cid:29) x (cid:48)(cid:48) . ERSUASION MEETS DELEGATION 29
Thus, taking into account (14), we have p Π ∗ ( m ) = , if m < c ( x ∗ L ) ,ν ( m ) , if m ∈ [ c ( x ∗ L ) , c ( x ∗ H )) ,ν (1) + m − , if m ≥ c ( x ∗ L ) . Figure 2(b) shows p Π ∗ as a dashed red curve. The conditions imply that p Π ∗ is convexand p Π ∗ ≥ ν ; so Π ∗ is optimal by Theorem 2. In particular, if c ( x ∗ L ) ∈ (0 , p Π ∗ ( m ) is convex at m = c ( x ∗ L ) because ν ( m ) is differentiable at all m ∈ (0 , c ( x ∗ L ) = 0, then p Π ∗ ( m ) is convex at m = 0 because ν ( m ) is convexat m = 0, by the last line in the conditions. By the same argument, p Π ∗ ( m ) is convexat m = c ( x ∗ H ).3 . For Π ∗ = [ y, x ∗ L ] ∪ [ x ∗ H , y ], we have p Π ∗ ( m ) = ν ( m ) , if m < c ( x ∗ L ) ,ν ( m ∗ ) + ν (cid:48) ( m ∗ )( m − m ∗ ) , if m ∈ [ c ( x ∗ L ) , c ( x ∗ H )) ,ν ( m ) , if m ≥ c ( x ∗ H ) . Figure 2(c) shows p Π ∗ as a dashed red curve. The conditions imply that p Π ∗ is convexand p Π ∗ ≥ ν ; so Π ∗ is optimal by Theorem 2. As in part 2, the last line in theconditions verifies that p Π ∗ ( m ) is convex at m = c ( x ∗ L ) and at m = c ( x ∗ H ) if ν ( m ) isnot differentiable at these points.A.5. Proof of Proposition 3.
The proof is straightforward but tedious. We onlysummarize possible cases. The reader may refer to the corresponding figures forguidance.1 . There are 4 cases (see Figure 3).( a ) If ν (cid:48) (0+) ≥ ν (cid:48) (1 − ) ≤
1, then Π ∗ = [ y, y ].( b ) If ν (cid:48) (0+) ≥ ν (cid:48) (1 − ) >
1, then Π ∗ = [ y, x ∗ H ] ∪ { y } with c ( x ∗ H ) ≤ c ) If ν (cid:48) (0+) < ν (cid:48) (1 − ) ≤
1, then Π ∗ = [ x ∗ L , y ] ∪ { y } with c ( x ∗ L ) ≥ d ) If ν (cid:48) (0+) < ν (cid:48) (1 − ) >
1, then Π ∗ = [ x ∗ L , x ∗ H ] ∪ { y, y } with 0 ≤ c ( x ∗ L ) ≤ c ( x ∗ H ) ≤ . There are 3 cases (see Figure 4).( a ) If ν (cid:48) (0+) > ν (cid:48) (1 − ) <
1, then Π ∗ = { x ∗ L , x ∗ H , y, y } with c ( x ∗ L ) ≤ c ( x ∗ H ) ≥ b ) If ν (cid:48) (0+) ≤
0, then Π ∗ = { x ∗ H , y, y } with c ( x ∗ H ) ≥ c ) If ν (cid:48) (1 − ) ≥
1, then Π ∗ = { x ∗ L , y, y } with c ( x ∗ L ) ≤ . There are 4 cases (see Figure 5).( a ) If ν (cid:48) (0+) ≥ ν (cid:48) (1 − ) ≥
1, then Π ∗ = [ y, x ∗ M ] ∪ { y } with c ( x ∗ M ) ≤ ˜ m . ν ( m ) p Π ∗ ( m ) 1 c ( x ∗ H )0 ν ( m ) p Π ∗ ( m ) (a) (b) c ( x ∗ L )0 ν ( m ) p Π ∗ ( m ) c ( x ∗ H )0 ν ( m ) p Π ∗ ( m ) c ( x ∗ L ) (c) (d) Figure 3.
Four Cases in Proposition 3 (Part 1) c ( x ∗ L ) c ( x ∗ H ) c ( x ∗ H ) c ( x ∗ L ) (a) (b) (c) Figure 4.
Three Cases in Proposition 3 (Part 2)
ERSUASION MEETS DELEGATION 31 c ( x ∗ M ) c ( x ∗ M ) c ( x ∗ H ) (a) (b) c ( x ∗ L ) c ( x ∗ M ) c ( x ∗ L ) c ( x ∗ H ) c ( x ∗ M ) (c) (d) Figure 5.
Four Cases in Proposition 3 (Part 3)( b ) If ν (cid:48) (0+) ≥ ν (cid:48) (1 − ) <
1, then Π ∗ = [ y, x ∗ M ] ∪ { x ∗ H , y } with c ( x ∗ M ) ≤ ˜ m and c ( x ∗ H ) ≥ c ) If ν (cid:48) (0+) < ν (cid:48) (1 − ) ≥
1, then Π ∗ = [ x ∗ L , x ∗ M ] ∪ { y, y } with 0 ≤ c ( x ∗ L ) ≤ c ( x ∗ M ) ≤ ˜ m .( d ) If ν (cid:48) (0+) < ν (cid:48) (1 − ) <
1, then Π ∗ = [ x ∗ L , x ∗ M ] ∪ { x ∗ H , y, y } with 0 ≤ c ( x ∗ L ) ≤ c ( x ∗ M ) ≤ ˜ m and c ( x ∗ H ) ≥ . There are 4 cases analogous to those in part 3.( a ) If ν (cid:48) (0+) ≤ ν (cid:48) (1 − ) ≥
1, then Π ∗ = [ x ∗ M , y ] ∪ { y } with c ( x ∗ M ) ≥ ˜ m .( b ) If ν (cid:48) (0+) > ν (cid:48) (1 − ) ≥
1, then Π ∗ = [ x ∗ M , y ] ∪ { x ∗ L , y } with c ( x ∗ M ) ≥ ˜ m and c ( x ∗ L ) ≤ c ) If ν (cid:48) (0+) ≤ ν (cid:48) (1 − ) >
1, then Π ∗ = [ x ∗ M , x ∗ H ] ∪ { y, y } with 1 ≥ c ( x ∗ H ) ≥ c ( x ∗ M ) ≥ ˜ m . ( d ) If ν (cid:48) (0+) > ν (cid:48) (1 − ) <
1, then Π ∗ = [ x ∗ M , x ∗ H ] ∪ { x ∗ L , y, y } with 1 ≥ c ( x ∗ H ) ≥ c ( x ∗ M ) ≥ ˜ m and c ( x ∗ L ) ≤ Proof of Proposition 2 (cid:48) . For Π ∗ = { x } ∪ [ x ∗ , y ], we have p Π ∗ ( m ) = (cid:40) ν ( m ∗ ) + ν (cid:48) ( m ∗ )( m − m ∗ ) , if m ∈ [ c ( x ) , c ( x ∗ )) ,ν ( m ) , if m ≥ c ( x ∗ ) . The conditions imply that p Π ∗ is convex and p Π ∗ ≥ ν ; so Π ∗ is optimal by Theorem 2.As in Proposition 2, the last line in the conditions verifies that p Π ∗ ( m ) is convex at m = c ( x ∗ ) if ν ( m ) is not differentiable at this point.A.7. Proof of Theorem 1 (cid:48) . Consider ( U D , V D ) ∈ ¯ P D and ( U P , V P ) ∈ ¯ P P that satisfy ∂U D ( t, x ) ∂x (cid:12)(cid:12)(cid:12)(cid:12) x = s = − ∂U P ( s, x ) ∂x (cid:12)(cid:12)(cid:12)(cid:12) x = t and ∂V D ( t, x ) ∂x (cid:12)(cid:12)(cid:12)(cid:12) x = s = − ∂V P ( s, x ) ∂x (cid:12)(cid:12)(cid:12)(cid:12) x = t (21)for all s, t ∈ [0 , β ∈ R such that E (cid:2) V D ( θ, x ∗ D ( θ, Π)) (cid:3) = E (cid:2) V P ( θ, x ∗ P ( θ, Π)) (cid:3) + β for all Π ∈ Π . Consider Π ∈ Π and let s be uniformly distributed on [0 , u Π ( s, t ) = E (cid:20) ∂U P ( s (cid:48) , t ) ∂t (cid:12)(cid:12)(cid:12)(cid:12) s (cid:48) ∈ µ Π ( s ) (cid:21) = E (cid:20) − ∂U D ( t, s (cid:48) ) ∂s (cid:48) (cid:12)(cid:12)(cid:12)(cid:12) s (cid:48) ∈ µ Π ( s ) (cid:21) ,v Π ( s, t ) = E (cid:20) ∂V P ( s (cid:48) , t ) ∂t (cid:12)(cid:12)(cid:12)(cid:12) s (cid:48) ∈ µ Π ( s ) (cid:21) = E (cid:20) − ∂V D ( t, s (cid:48) ) ∂s (cid:48) (cid:12)(cid:12)(cid:12)(cid:12) s (cid:48) ∈ µ Π ( s ) (cid:21) . Note that u Π ( s, t ) is integrable in s and t by ( ¯A ) and satisfies upcrossing in s anddowncrossing in t by ( ¯A D )/( ¯A P ).First, consider the balanced delegation problem. By (21), we have, for s ∈ Π, U D ( t, s ) = U D ( t, − (cid:90) s ∂U D ( t, x ) ∂x (cid:12)(cid:12)(cid:12)(cid:12) x = s (cid:48) d s (cid:48) = U D ( t,
1) + (cid:90) s u Π ( s (cid:48) , t )d s (cid:48) ,V D ( t, s ) = V D ( t, − (cid:90) s ∂V D ( t, x ) ∂x (cid:12)(cid:12)(cid:12)(cid:12) x = s (cid:48) d s (cid:48) = V D ( t,
1) + (cid:90) s v Π ( s (cid:48) , t )d s (cid:48) . Since u Π ( s, t ) satisfies upcrossing in s , we have s ∈ y ∗ ( t, Π) = arg max y ∈ Π (cid:90) y u Π ( s, t )d s if and only if for all s (cid:48) , s (cid:48)(cid:48) ∈ Π such that s (cid:48) ≤ s ≤ s (cid:48)(cid:48) we have u Π ( s (cid:48) , t ) ≤ ≤ u Π ( s (cid:48)(cid:48) , t ). ERSUASION MEETS DELEGATION 33
The principal’s expected payoff is E (cid:2) V D ( t, x ∗ D ( t, Π)) (cid:3) = E [ V D ( t, E [ V D ( t, y ∗ ( t, Π))]= E [ V D ( t, (cid:90) (cid:90) y ∗ ( t, Π) v Π ( s, t )d s d t. Define J Π = { ( s, t ) ∈ [0 , : u Π ( s, t ) ≥ } and J +Π = { ( s, t ) ∈ [0 , : u Π ( s, t ) > } , J Π = { J ⊂ [0 , : J +Π ⊂ J ⊂ J Π , and J is Lebesgue measurable } . Using Aumann (1965, Theorem 1), we have E (cid:2) V D ( t, x ∗ D ( t, Π)) (cid:3) = E [ V D ( t, (cid:26)(cid:90) ( s,t ) ∈ J v Π ( s, t )d s d t : J ∈ J Π (cid:27) . Next, consider the monotone persuasion problem. By (21), we have E (cid:2) U P ( s (cid:48) , t ) (cid:12)(cid:12) s (cid:48) ∈ µ Π ( s ) (cid:3) = E (cid:2) U P ( s (cid:48) , (cid:12)(cid:12) s (cid:48) ∈ µ Π ( s ) (cid:3) + (cid:90) t E (cid:20) ∂U P ( s (cid:48) , t ) ∂t (cid:12)(cid:12)(cid:12)(cid:12) s (cid:48) ∈ µ Π ( s ) (cid:21) d t (cid:48) = E (cid:2) U P ( s (cid:48) , (cid:12)(cid:12) s (cid:48) ∈ µ Π ( s ) (cid:3) + (cid:90) t u Π ( s, t (cid:48) )d t (cid:48) and E (cid:2) V P ( s (cid:48) , t ) (cid:12)(cid:12) s (cid:48) ∈ µ Π ( s ) (cid:3) = E (cid:2) V P ( s (cid:48) , (cid:12)(cid:12) s (cid:48) ∈ µ Π ( s ) (cid:3) + (cid:90) t E (cid:20) ∂V P ( s (cid:48) , t ) ∂t (cid:12)(cid:12)(cid:12)(cid:12) s (cid:48) ∈ µ Π ( s ) (cid:21) d t (cid:48) = E (cid:2) V P ( s (cid:48) , (cid:12)(cid:12) s (cid:48) ∈ µ Π ( s ) (cid:3) + (cid:90) t v Π ( s, t (cid:48) )d t (cid:48) . Since u Π ( s, t ) satisfies downcrossing in t , we have t ∈ z ∗ ( s, Π) = arg max z ∈ [0 , (cid:90) z u Π ( s, t )d t if and only if for all t (cid:48) , t (cid:48)(cid:48) ∈ [0 ,
1] such that t (cid:48) ≤ t ≤ t (cid:48)(cid:48) we have u Π ( s, t (cid:48) ) ≥ ≥ u Π ( s, t (cid:48)(cid:48) ).The principal’s expected payoff is E [ V P ( s, x ∗ P ( s, Π))] = E (cid:2) V P ( s, (cid:3) + E [ V P ( s, z ∗ ( s, Π))]= E [ V P ( s, (cid:90) (cid:90) z ∗ ( s, Π)0 v Π ( s, t (cid:48) )d t (cid:48) d s = E [ V P ( s, (cid:26)(cid:90) ( s,t ) ∈ J v Π ( s, t )d s d t : J ∈ J Π (cid:27) . We thus have shown that E (cid:2) V D ( θ, x ∗ D ( θ, Π)) (cid:3) = E [ V P ( θ, x ∗ P ( θ, Π))]+ β for each Π ∈ Π ,where β = E [ V D ( t, − E [ V P ( s, Finally, it is straightforward to verify that if ( U D , V D ) ∈ ¯ P D and ( U P , V P ) is givenby (3a), then ( U P , V P ) satisfies ( ¯A ) and ( ¯A P ), and thus it is in ¯ P P . Conversely, if( U P , V P ) ∈ ¯ P P and ( U D , V D ) is given by (3b), then ( U D , V D ) satisfies ( ¯A ) and ( ¯A D ),and thus it is in ¯ P D . ReferencesAlonso, R., and
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