Optimal delegation and information transmission under limited awareness

Auster, Sarah; Pavoni, Nicola Article Optimal delegation and information transmission under limited awareness Theoretical Economics Provided in Cooperation with: The Econometric Society Suggested Citation: Auster, Sarah; Pavoni, Nicola (2024) : Optimal delegation and information transmission under limited awareness, Theoretical Economics, ISSN 1555-7561, The Econometric Society, New Haven, CT, Vol. 19, Iss. 1, pp. 245-284, https://doi.org/10.3982/TE5117 This Version is available at: https://hdl.handle.net/10419/296459 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc/4.0/ Theoretical Economics 19 (2024), 245–284 1555-7561/20240245 Optimal delegation and information transmission under limited awareness Sarah Auster Department of Economics, University of Bonn Nicola Pavo n i Department of Economics, Bocconi University We study the delegation problem between a principal and an agent, who not only has better information about the performance of the available actions but also superior awareness of the set of actions that are actually feasible. We provide conditions under which the agent finds it optimal to leave the principal unaware of relevant options. By doing so, the agent increases the principal’s cost of distorting the agent’s choices and increases the principal’s willingness to grant him higher information rents. We further show that the principal may use the option of renegotiation as a tool to implement actions that are not describable to her at the contracting stage. If the agent renegotiates, his proposal signals information about the payoff state. Due to her limited awareness, the principal makes a coarse inference from the agent’s recommendations and, as a result, accepts a large number of the agent’s proposals, which ultimately benefits both. Keywords. Unawareness, optimal delegation, renegotiation. JEL classification. D82, D83, D86. 1. Introduction In many situations, economic agents delegate decisions to experts whose preferences may not be perfectly aligned with their own. Public and private organizations use procurement managers to purchase products and services for the tasks at hand; corporate headquarters rely on division managers with superior information about the profitability of new projects; small investors seek advice from financial experts with a better understanding of the risks and returns of the available portfolios. Oftentimes, the informed Sarah Auster: [email protected] Nicola Pavoni: [email protected] The authors thank Sandeep Baliga, Pierpaolo Battigalli, Sylvain Chassang, Wouter Dessein, Antonio Guarino, Yingni Guo, Johannes Hörner, Navin Kartik, Elliot Lipnowski, Alessandro Pavan, and Yuval Salant for very helpful comments. The present draft benefited from several comments received by the seminar participants at the Toulouse School of Economics, Collegio Carlo Alberto, University College London, EIEF in Rome, Bonn University and at the Max Planck Institute, at the University of Bristol, the University of Cardiff, at the Bocconi and Cattolica universities in Milan, at the University of Verona, University of Malaga, Oxford University, Stockholm School of Economics, and the Cowles Conference on Economic Theory. Nicola Pavoni acknowledges financial support from the MIUR-PRIN grant 20157NH5TP. Sarah Auster acknowledges funding from the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy—EXC 2126/1—390838866 and CRC TR 224 (Project B02). ©2024 The Authors. Licensed under the Creative Commons Attribution-NonCommercial License 4.0. Available at https://econtheory.org.https://doi.org/10.3982/TE5117 246 Auster and Pavoni Theoretical Economics 19 (2024) party not only has a better understanding of what the most suitable action is but also of the options that are actually available. Procurement managers have superior awareness of the feasible products and potential suppliers in the market where they operate; division managers have a better understanding of the projects they could pursue; financial experts are familiar with more financial instruments than retail investors, etc. This paper proposes a framework to study the implications of such asymmetry by incorporating unawareness into a canonical delegation model. More specifically, we consider the problem of a principal (she) who needs to take an action and delegates the task to an agent (he). The agent receives private information about the payoffs of each available action, and the principal’s problem is to determine a set of actions from which the agent can choose. We depart from the traditional framework of optimal delegation by considering a situation where the principal is unaware of some feasible actions and where this limits the language with which she can write a contract: the principal can only permit actions in the delegation set if she can name these actions explicitly; hence, if she is aware of them. Before the delegation stage and before receiving private information, the agent can expand the principal’s awareness by revealing additional actions and thereby enrich the set of feasible contracts for the principal. We are interested in the question if and how the agent distorts the principal’s awareness to increase his own rent. We address this question in an environment with a continuum of payoff states, a continuum of feasible actions, and an agent who prefers a higher action than the principal in each state. Given her awareness, the principal’s optimal delegation set solves the usual tradeoff between minimizing distortions and limiting the agent’s information rent. Since the agent has an upward bias, optimal delegation entails that the principal limits the agent’s choice from above. An optimal delegation set thus has a threshold above, which no action is permitted. How high this threshold is depends on the principal’s awareness set. We identify conditions under which the agent optimally leaves the principal unaware of an interval of actions around the optimal upper threshold under full awareness. By choosing the bounds of the interval appropriately, the agent makes it optimal for the principal—who still cares about the agent’s information—to permit an action above the full awareness cap, and hence an action that would be precluded if the principal was fully aware. An important assumption of our model is that the principal cannot specify actions in the delegation set of which she is unaware. An introspective principal might, however, ask herself whether there are other contracts that can improve on the optimal delegation set without giving the agent the flexibility of taking unknown, potentially harmful options. One such possibility for the principal is to forego full commitment and add a contractual clause that allows the delegation set to be adjusted when new options or contingencies come to light. Adding such a clause and thereby allowing for ex post renegotiation is indeed consistent with the principal’s sophistication and language. In the second part of the paper, we study the implications of the use of such contracts. The agent is then allowed to propose additional actions to the principal after the initial delegation set is agreed upon and he observes the state. Subsequently, the principal decides whether to permit a new action or whether to maintain the original delegation set. Theoretical Economics 19 (2024) Optimal delegation and information transmission 247 Since the disclosure of additional actions is made after the agent receives private information, the principal and agent play a signaling game at the renegotiation stage. We characterize the set of proposals the principal is willing to accept after the initial agreement is signed. Fixing the agent’s initial disclosure and focusing on the equilibrium with the maximal set of acceptable proposals, we provide conditions under which adding a renegotiation clause to the initial delegation set always benefits the principal.Thedownside of renegotiation is that it limits the agent’s disclosure incentives in the contracting phase. Indeed, we show that, depending on the principal’s initial awareness, the agent can significantly increase his flexibility by initially holding back some available actions and tailoring additional disclosures to the information he receives. From a modeler’s viewpoint, the agent’s strategy in this equilibrium is fully revealing. The unaware principal, however, cannot compare the agent’s proposal at a given state to the actions which the agent would have proposed in a different state, and hence infers strictly less information from the agent’s recommendation. An application of our setting is procurement delegation within an organization. Many organizations have procurement managers in charge of purchasing products and services according to the organization’s current needs. This choice is often limited via pre-specified lists of approved products or vendors. A possible concern rationalizing such restrictions is that the procurement manager can use his discretion to further personal goals, such as career enhancement, minimization of workload, personal enrichment, etc. (Rogerson (1994)). We can think of our agent as the procurement manager and interpret the agent’s actions as the different products (or suppliers) available in the market. The state captures the characteristics of the task at hand, determining the procurement manager’s “ideal” product. The principal can be interpreted as a high-level manager with limited awareness of the available products. Our results suggest that a biased procurement manager typically benefits from hiding certain products or suppliers. In particular, under the identified conditions, the procurement manager has incentives to propose products with relatively extreme characteristics for approval but wants to hide those with more moderate attributes. The more pronounced the awareness asymmetry, the larger the scope for the procurement manager to distort the procurement decision through strategic disclosure. Finally, if the procurement manager has the possibility to seek approval for additional options after observing the characteristics of the task, he can substantially expand his flexibility by tailoring the disclosure to the circumstances. After the literature review, the paper is organized as follows. Section 2presents the delegation model with limited awareness. In Section 3, we analyze the agent’s optimal disclosure and the resulting delegation set. Section 4analyzes the game with renegotiation and Section 5concludes. 1.1 Related literature The paper makes both applied and theoretical contributions. It introduces unawareness to the canonical delegation problem and shows how the agent can distort the principal’s delegation choice through strategic disclosure. The analysis builds on the literature on optimal delegation. Holmström (1980) first defines the delegation problem and 248 Auster and Pavoni Theoretical Economics 19 (2024) provides conditions for the existence of its solution. Following the seminal paper, the literature was further developed by Melumad and Shibano (1991), Szalay (2005), Martimort and Semenov (2006), Alonso and Matouschek (2008), Kovᡠc and Mylovanov (2009), Armstrong and Vickers (2010), Amador and Bagwell (2013), and Halac and Yared (2020), among others. None of them consider limited awareness in this framework. The paper is also related to the smaller literature that applies unawareness to games in general and contracting problems in particular. In contrast to our setting, most of the existing work considers contracting problems where contingent transfers are feasible and where the agent has limited awareness, while the principal is fully aware (Von Thadden and Zhao (2012), Zhao (2011), Filiz-Ozbay (2012), Auster (2013)). One exception is Francetich and Schipper (2020), which studies a screening model where the principal is unaware of certain cost types (but has full awareness over actions) and the agent decides which types to disclose. In Auster and Pavoni (2022), we consider a finance application of our model, interpreting the agent as a financial expert and the principal as an investor with limited awareness about the available financial products. We collect self-reported data from customers in the Italian retail investment sector and find support for the key predictions of the model: the menus offered to less knowledgeable investors contain fewer products, which are perceived to be more extreme.1Also, Lei and Zhao (2021)studyafinancial market application of our model but focus on the unawareness of contingencies (nature’s moves) rather than players’ actions. On the theoretical side, the study of the disclosure problem reveals how the agent’s information rents depend on the set of feasible actions—or the principal’s perception thereof—in delegation settings. This question is related to a recent literature looking at the determinants of agency rents in models with full awareness, initiated by Roesler and Szentes (2017). With the second part of the paper, we also contribute to the literature on incomplete contracts and unforeseen contingencies (Grossman and Hart (1986); Hart and Moore (1988)) by demonstrating the value of ex post renegotiation in settings with partial awareness. Previous papers that study the interaction between limited awareness and the possibility of renegotiation are Tirole (2009)andPiermont (2017), albeit in rather different settings. 2. Environment There is a principal and an agent. The agent has access to an interval of actions YA= [ymin,ymax]. The principal’s and agent’s payoffs depend on the action that is chosen and an unknown payoff parameter θ, which can be privately observed by the agent. Let = [0, 1]be the set of payoff states and let F(θ)denote the cumulative distribution function on , assumed to be twice differentiable on the support. The principal and the agent 1The data we collected consists of approximately 1400 investors reporting on their experience in the Italian retail investment sector. We regress both the number of offered products and a measure of perceived “extremeness” in the menu on an index of knowledge, which is based on a number of questions eliciting the investor’s background knowledge. See Auster and Pavoni (2022)fordetails. Theoretical Economics 19 (2024) Optimal delegation and information transmission 249 have twice continuously differentiable utility functions2 UP(θ,y),UA(θ,y). Fixing θ,Uifor i=P,Ais assumed to be strictly concave in ywith an interior maximum on YA. The principal’s and agent’s conditionally preferred actions are described by the functions yP(θ):=arg max y∈YAUP(θ,y),yA(θ):=arg max y∈YAUA(θ,y). We assume Ui θy >0, which implies that yP(·),yA(·)are strictly increasing functions. Furthermore, we assume that conditional on the payoff parameter θ, the agent prefers a higher action than the principal: for all θ,yP(θ)<yA(θ). Awareness Let Ydenote the set of closed subsets of [ymin,ymax]. The principal is aware of a subset of available actions, denoted by YP∈Y. Hence, unawareness in our framework does not take the form of unforeseen contingencies but concerns the set of available actions. Apart from the assumption that YPis closed, we impose no further structure on the principal’s initial awareness set. Before the principal contracts with the agent and the agent observes θ, the agent can make the principal aware of additional actions by revealing a closed set X∈Y. The principal fully understands the options that are revealed to her and accordingly updates her awareness to the union of whatever she knew initially and what the agent reveals.3 Delegation Given her updated awareness, the principal offers a contract to the agent. We rule out monetary transfers and assume that the agent’s participation constraint is always satisfied. The contracting problem of the principal then reduces to the decision over the set of actions from which the agent can choose once he observes the payoff parameter θ.4Our substantial assumption is that the principal’s unawareness restricts the language with which she can write a contract. In particular, we assume that the principal can only refer to actions in the contract, which she can name explicitly. The larger the principal’s awareness set, the richer the set of contracts she can write. Given the principal’s updated awareness set, she then has two natural options: the principal can either name the actions she allows the agent to take or she can name the actions she explicitly forbids. Under full awareness, these two options are clearly equivalent. With unawareness, on the other hand, specifying only the forbidden actions leaves 2Note that the principal does not have full access to her payoff function UPbut just to a payoff function restricted to the domain of actions of which she is aware. 3Assuming that the agent discloses actions before receiving private information avoids signaling effects in the baseline model. That is, after any disclosure by the agent, the principal’s beliefs about the payoff state are described by the prior F. This is consistent with “Reverse Bayesianism” (see Karni and Vierø (2013)), which postulates that relative beliefs on events of which the decision maker was previously aware do not change when her awareness grows. 4The standard delegation problem is equivalent to a mechanism design problem when the principal restricts herself to deterministic allocations (see Alonso and Matouschek (2008) and Kovᡠc and Mylovanov (2009)). Formally, the principal commits to a mechanism that specifies an action as a function of the agent’s message. 250 Auster and Pavoni Theoretical Economics 19 (2024) the principal vulnerable to the agent taking actions that the principal does not anticipate. We will discuss this case and other options in Section 3.2 and concentrate now on the case where the principal specifies the actions which she permits. Since the principal cannot specify actions of which she is unaware, the principal’s delegation set is then a subset of her awareness set. We restrict attention to closed delegation sets. The timing of the game can be summarized as follows: 1. The agent reveals a set of actions X∈Yand the principal updates her awareness to Y=YP∪X. 2. Given awareness set Y, the principal chooses a delegation set D∈Ysuch that D⊆Y. 3. The agent observes θand chooses an action from set D. 4. Payoffs are realized. The game between the principal and the agent can be formally represented by a family of partially ordered subjective game trees. Such family includes the modeler’s view of the objectively feasible paths of play, but also the feasible paths of play as subjectively viewed by some players, or as the frame of mind attributed to a player by other players or by the same player at a later stage of the game (Heifetz, Meier, and Schipper (2021)). 5As a solution concept, we use a strong version of Perfect Bayesian Nash Equilibrium (PBE), which implies subgame perfection, adapted to generalized extensive-form games with unawareness (e.g., see Halpern and Rêgo (2014)andFeinberg (2021)). Remark. There is an alternative reading of our model as one of limited authority. We can think of a situation where the agent, rather than disclosing feasible actions to the principal, actually enables the principal to pursue them. The agent thus decides on the set of actions he makes available to the principal and, as before, the principal delegates some subset of those actions to the agent. By deciding which actions to make available, the agent is given commitment power not to take certain actions. Since such commitment limits the principal’s choice over feasible contracts, we are ultimately faced with a double delegation game between the agent and the principal. 3. Equilibrium analysis We will work backward and start the analysis by considering the last stage of the game. Given a delegation set Dand observed payoff state θ, the agent’s best response for the 5In a working paper version of this paper, available on our websites, we provide a more extensive description of the family of game trees representing the generalized game with unawareness associated to our delegation model according to the approach proposed by Heifetz, Meier, and Schipper (2013). We also describe the set of outcomes that satisfy a prudent version of extensive-form rationalizability and we show that whenever we restrict to pure strategies and assume the tie-breaking rules we adopt below to be commonly known, the PBE outcome we obtain is also the sole rationalizable outcome of the generalized game. Theoretical Economics 19 (2024) Optimal delegation and information transmission 251 last stage of the game is defined by BRA(θ,D):=arg max y∈DUA(θ,y).(1) When the agent is indifferent between two actions, let y∗(θ,D):=min BRA(θ,D)be the selection that takes the smallest value (indifference is broken in favor of the principal).6 Delegation stage Turning to the principal’s delegation choice, we first define the principal’s value of delegation set D∈Ygiven y∗: VP(D):=1 0 UPθ,y∗(θ,D)dF(θ).(2) There are typically actions that the principal could permit but the agent will not implement. Without loss of generality, we will restrict attention to delegation sets Dsuch that for any y∈D, there is some state θ∈[0, 1]such that y∗(θ,D)=y.LetD(Y)be the set of delegation sets in {D∈Y:D⊆Y}that satisfy this requirement. For each awareness set Y∈Y, the principal’s optimal delegation set solves the problem max D∈D(Y)VP(D).(3) Theorem 1 in Holmström (1980) guarantees existence for each closed Y(see also Proposition 12 in Appendix A.8). If problem (3) has multiple solutions, we assume that the principal chooses the agent-preferred set. For each Y,wedenotebyD∗(·)such selection from the set of maximizers. Furthermore, we assume that in the case where the principal is fully aware, delegation is valuable. A sufficient condition for valuable delegation is y∗ 0>y A(0),wherey∗ 0∈argmaxyVP({y}). This requires the bias to be not too large and implies that the principal prefers the delegation set [yA(0),y∗ 0]to the singleton {y∗ 0}(see also Alonso and Matouschek (2008, Corollary 2). Disclosure stage In the first stage of the game, the agent chooses an awareness set Y∈Y. Since the agent cannot make the principal unaware of actions that the principal already knows, the induced awareness set must contain the principal’s initial awareness set YP. The smaller YP, the larger the collection of awareness sets from which the agent can choose. An optimal awareness set Y∗solves the problem max Y∈Y1 0 UAθ,y∗θ,D∗(Y)dF(θ)such that YP⊆Y.(4) Since different awareness sets might induce the same delegation set, the solution to problem (4) is again typically not unique. Of course, this type of multiplicity does not affect the outcome. We assume that when two solutions of problem (4)arenested,the agent discloses the larger set. This assumption allows us to distinguish the actions that remain undisclosed for strategic reasons from those that are redundant. Let Y∗denote the set of all solutions of (4) satisfying this requirement. 6Such selection is well-defined, since the BRAcorrespondence is nonempty and upper hemicontinuous (see Holmström (1980)). In addition, for each closed D,thesetofθ’s for which BRA(·,D)is not a singleton is at most countable, and hence of F-measure zero (see Lemma 11 in Appendix A.8). 252 Auster and Pavoni Theoretical Economics 19 (2024) Our model is a sequential move game with infinite actions. This makes equilibrium existence a nontrivial issue. In Appendix A.8, Proposition 13, we show that a solution to problem (4) exists. Hence, there is an equilibrium where the agent discloses a set Y∈Y∗, the principal delegates set D∗(Y)and, after observing the state realization θ, the agent takes action y∗(θ,D∗(Y)). Equilibrium disclosure The central question of this paper is whether the agent distorts the principal’s delegation choice in his favor by leaving the principal unaware of some feasible actions. Due to the conflict of interest between the principal and the agent, a fully aware principal will not find it optimal to permit the agent his preferred action in every payoff state. Indeed, since the agent is upward biased, the principal can always improve on full delegation by excluding an interval of high actions, forcing the agent for high realizations of θto take an action closer to the principal’s conditionally preferred action. Following this argument, we define ˆ y:=maxD∗(YA)<y A(1)as the highest action, which the principal permits under the optimal delegation set in the full awareness benchmark. The following proposition shows conditions under which unawareness of ˆ yis sufficient to ensure that the agent benefits from the principal’s limited awareness.7 To this end, let s:=(yA)−1denote the inverse of yA, implicitly defined by the first-order condition UA y(s(y),y)=0. Since the agent’s utility function is twice continuously differentiable, the function s(·)is differentiable (see Lemma 8in the Appendix). Suppose now the upper threshold ˆ yis a limit point of D∗(YA). Itmustthensatisfy the following optimality conditions: 1 s(ˆ y) UP y(θ,ˆ y)dF(θ)=0(5) −UP ys(ˆ y),ˆ yfs(ˆ y)s(ˆ y)+1 s(ˆ y) UP yy(θ,ˆ y)dF(θ)≤0. (6) The first-order condition (5) says that, conditioning on the event θ≥s(ˆ y),actionˆ yis optimal for the principal in expectation. The second-order condition (6) is necessary for ˆ yto constitute a local maximizer. For the following result, we will maintain that this condition holds strictly. Proposition 1. Assume that the full awareness problem (3) has a unique maximizer D∗(YA)and that the upper threshold ˆ yis a limit point of D∗(YA),with(6)holdingas strict inequality. Then ˆ y/∈YP=⇒ YA/∈Y∗. Proposition 1shows that, under the stated conditions, if the principal is initially unaware of the highest action in the optimal delegation set under full awareness, then the agent finds it profitable to hide some of the feasible actions from the principal. To prove the result, we consider a simple perturbation of the full awareness set. The perturbation entails that the principal remains unaware of an interval (y−,y+)of actions around the 7Since YPis a closed set, unawareness of ˆ yimplies that the principal is unaware of an interval around ˆ y. Theoretical Economics 19 (2024) Optimal delegation and information transmission 259 The restriction to ruling-in contracts arises naturally if one views the principal’s problem as designing a direct mechanism. Under this interpretation, the principal commits to a mapping from messages to actions, where unawareness imposes restrictions on the image of such mappings. In particular, the principal cannot commit to an action that she does not know to exist. It might be interesting to consider more complex relationships between the principal’s awareness and implementable action profiles through indirect mechanisms. For instance, the principal could attempt to permit additional actions through an indirect description of those. Whether this improves the principal’s welfare or actually hurts her depends on the details of the model, of which the principal is unaware. Any aversion to such unknown possibilities might in fact call for a description of actions as specific as possible given the principal’s language. An introspective principal—one who is aware of her unawareness—might, however, wonder whether there are any contracts that can improve on the optimal delegation set without giving the agent blanket approval to take unknown actions. One such possibility is to add a contractual clause specifying that the initial contract can be adjusted when new options come to light and parties mutually agree (see also Piermont (2017)). Specifying a contractual clause of this form would not rely on the principal’s ability to describe actions outside her awareness and would hedge her against the possibility of the agent taking harmful actions without her consent. In the spirit of the incomplete contract literature, one could then view the initial delegation set as a preliminary agreement that can be renegotiated when new, mutually beneficial options appear. We explore this possibility in the following section. In contrast to the incomplete contracts literature, we will maintain the assumption that the principal has full commitment, so renegotiation is in fact fully avoidable. We ask instead whether under limited awareness the principal can actually benefit from voluntarily giving up some of that commitment. 4. Ex post renegotiation Suppose that, rather than fully committing to the initial delegation set, the principal proposes a contract that fixes a set of permitted actions but allows for an adjustment when new options appear. We thus consider contracts under which the agent can renegotiate with the principal over actions that were not disclosed (or simply not permitted) in the initial stage of the game. Crucially, we allow the agent to propose such actions after he observes the payoff state, thereby signaling information. In particular, upon receiving a proposal for a new action, the principal infers that the agent makes such a proposal only if taking the new action benefits him. However—due to the principal’s limited awareness—she cannot conceive of alternative actions the agent could have disclosed instead, and hence cannot learn from particular actions not being proposed. This asymmetry arises as a consequence of the principal’s limited awareness and plays a crucial role in the results that follow. Themodifiedgamehastwophases,the contracting phase and the renegotiation phase. The contracting phase is the same as before: the agent discloses a set of actions and the principal determines a delegation set. In the renegotiation phase, the agent first 260 Auster and Pavoni Theoretical Economics 19 (2024) observes the payoff state θand then decides between two options. Either he picks an action from the delegation set or he proposes a different action to the principal, who can then accept the proposal or keep the original delegation set. Strategies and beliefs While we return to the contracting phase at the end of the section, our main focus will lie on the renegotiation phase. To this end, we fix the principal’s interim awareness set Y∈Yand a delegation set D⊆Yas primitives. The set Yis interpreted as the principal’s updated awareness after the contracting phase and the set Das the corresponding delegation set. Next, we define the strategies of the principal and the agent. The agent’s possible moves are either “no new proposal” (let us call it N) or singleton proposals x∈YA. The set of possible proposals for the agent is thus X:=N∪YA and the agent’s strategy is a map x:[0, 1]→Xfrom the possible realizations of θto a recommendation. Upon receiving a new proposal, the principal needs to decide whether to accept or reject it. Her strategy is a mapping ρ:X→{0, 1},whereρ(x)=0 means that the principal rejects the agent’s proposal and keeps the original delegation set D, while ρ(x)=1 means that the principal accepts the agent’s proposal and the implemented action is x. To account for the fact that after “no new proposal” the original delegation set must be kept, we set ρ(N)=0. Whenever there is no new proposal or the proposal is rejected, the agent chooses an action from the original set D. The agent’s optimal choice in this case is described by y∗(θ,D), as introduced in Section 3. We will take this part of the agent’s strategy as given. We restrict strategies xand ρto be upper semicontinuous functions.12 Intuitively, this amounts to assuming that, in case of indifference, the agent breaks ties in favor of proposing a new action, and the principal breaks ties in favor of allowing new proposals. It is easy to see that this assumption generates equilibrium sets of permitted actions that are closed. We further concentrate on outcomes in pure strategies. Finally, we denote for each x∈Xthe set of conceivable proposals under the principal’s updated awareness Y∪{x}by Xx:=Y∪{x}∪{N}. Definition 1. Fix an awareness set Y∈Yand a delegation set D⊆Y.Thestrategy profile (x∗,ρ∗), together with a belief function μ∗(·|x)∈([0, 1]) for each x∈Xand a collection of strategy perceptions (x∗ x)x∈X,withx∗ x:[0, 1]→Xx, constitutes a PBE of the renegotiation game if and only if the following conditions hold: 1. Principal optimality: for all x∈X\N, ρ∗(x)∈arg max ρ∈{0,1}ρEμ∗(·|x)UP(θ,x)+(1−ρ)Eμ∗(·|x)UPθ,y∗(θ,D); 2. Agent optimality: for all θ∈[0, 1], x∗(θ)∈argmax x∈Xρ∗(x)UA(θ,x)+1−ρ∗(x)UAθ,y∗(θ,D); 12To define upper semicontinuity for the agent, associate a negative number to Nin the codomain of his strategy. Theoretical Economics 19 (2024) Optimal delegation and information transmission 261 3. Consistency of beliefs: for all x∈X,μ∗(·|x)is consistent with the perceived strategy x∗ x(·),where x∗ x(θ)∈arg max x∈Xx ρ∗xUAθ,x+1−ρ∗xUAθ,y∗(θ,D). (14) In particular, letting ∗(x)denote the preimage of xfor the function x∗ x(·),μ∗(·|x)is derived via Bayes rule whenever ∗(x)dF(θ)>0. If ∗(x)dF(θ)=0but∗(x)=∅, then μ∗(·|x)is an arbitrary distribution with support ∗(x). Finally, if ∗(x)=∅, then μ∗(·|x)is unrestricted; The main novelty in Definition 1is the awareness-adapted consistency condition, which assures that the principal’s beliefs are coherent with the agent playing optimally in the game as perceived through the principal’s awareness. To formalize this requirement, we need to account for the fact that the principal’s perception of the agent’s set of feasible strategies depends on the principal’s updated awareness set, and hence on the agent’s realized proposal. Indeed, each proposal x∈Xinduces a different subjective game in the mind of the principal. We thus define for each x∈X, a perceived strategy x∗ x, which maps the state θ∈[0, 1]to a feasible recommendation x∈Xxin the principal’s subjective game. Condition (14) then requires that in this game, strategy x∗ xis optimal against the principal’s equilibrium strategy ρ∗. The key implication of the consistency condition is that after any change of awareness following the agent’s proposal, the principal’s strategy and her beliefs are part of an equilibrium in the resulting subjective game.13 Acceptable proposals We now ask which proposals the principal is willing to accept in a renegotiation equilibrium. To answer the question, we restrict attention to initial delegation sets Dthat solve the principal’s delegation problem (3): D=D∗(Y).Giventhe principal’s awareness in the contracting phase, delegation set D∗(Y)is indeed optimal, since any action in Y, which the principal plans to permit in the renegotiation phase, can directly be included in the delegation set. Given this restriction, we can show that in any equilibrium of the renegotiation game, the principal permits an agent’s proposal x/∈D∗(Y)only if she would have preferred to add the action to the initial delegation set D∗(Y)at the contracting stage. The set of actions satisfying this requirement is defined by A(Y):=x∈YA:VPD∗(Y)∪{x}≥VPD∗(Y), where VPis the principal’s value in the “full commitment delegation” benchmark. Proposition 5. Fix an awareness set Y∈Yand a delegation set D=D∗(Y). (i) In any equilibrium (x∗,ρ∗,(μ∗(·|x),x∗ x)x∈X),ifρ∗(x)=1,thenx∈A(Y). (ii) There is an equilibrium such that ρ∗(x)=1for all x∈A(Y). 13We could impose the consistency condition only on beliefs following on-path proposals. None of our results would be affected. While this is obvious for Propositions 5(ii), 6, and 7, also Proposition 5(i) remains valid, as the argument proving it does not rely on the specification of off-path beliefs (see Appendix A.5). 262 Auster and Pavoni Theoretical Economics 19 (2024) Proposition 5characterizes the set of proposals that can be accepted by the principal in equilibrium. By definition of D∗(Y),thesetA(Y)does not include any actions that belong to Yother than those already in the delegation set D∗(Y). Hence, in equilibrium we have ρ∗(x)=0forallx∈Y\D∗(Y). This means that, given delegation set D∗(Y),the agent can only gain from renegotiation if he discloses new actions of which the principal was previously unaware. Consider then an equilibrium where proposal x/∈Yis accepted in the renegotiation phase. By the consistency condition, the principal’s beliefs after this proposal have support ∗(x)=θ∈[0, 1]:UA(θ,x)≥max y∈D∗(Y)UA(θ,y). (15) According to the principal’s awareness, xis the only new action that the agent can propose. The principal thus believes that the agent proposes xwhenever he prefers it over his best alternative in D∗(Y). The question is then whether conditional on the agent preferring xover the actions belonging to D∗(Y), the principal prefers xas well. The answer to this question is yes if and only if the principal would have preferred to add x to the delegation set D∗(Y), i.e., if and only if x∈A(Y). This is because adding an action to the delegation set changes the outcome only in those states where the agent prefers the action to the alternatives in the delegation set. In the renegotiation phase, the same consideration applies. The benefit of partial commitment The previous result demonstrates that by adding a renegotiation option to the optimal delegation set the principal can keep some flexibility to implement additional actions should her awareness grow while generating the same outcome as under full commitment in case her awareness remains unchanged. We show next that, if Assumption 1holds, partial commitment indeed dominates full commitment. Proposition 6. Let Assumption 1be satisfied. Fix an awareness set Y∈Y, a delegation set D=D∗(Y)and consider a renegotiation equilibrium (x∗,ρ∗,(μ∗(·|x),x∗ x)x∈X)where ρ∗(x)=1for all x∈A(Y)and ρ∗(x)=0otherwise. The principal’s expected payoff in this equilibrium is VP(A(Y)) and satisfies VPA(Y)≥VPD∗(Y). (16) Proposition 6shows that, focusing on the equilibrium where the set of accepted proposals is maximal, the principal benefits from partially forgoing her commitment if Assumption 1is satisfied. In equilibrium, the set of implementable actions for the agent is A(Y)and the principal’s equilibrium payoff (as viewed from the perspective of a fully aware outside observer) is given by VP(D∗(Y)∪A(Y)). By Assumption 1,theset A(Y)includes all actions in [yA(0),maxD∗(Y)]. Intuitively, this means that the agent can “close potential gaps” of the original delegation set D∗(Y)through renegotiation. By the same assumption, convexifying the set not only benefits the agent but also the Theoretical Economics 19 (2024) Optimal delegation and information transmission 263 principal. The set A(Y)may also include actions strictly higher than max D∗(Y).Inthe proof, we show that their inclusion benefits the principal as well.14 Information transmission A striking feature of the described equilibrium is that the implemented action is strictly increasing in the state for all θsuch that yA(θ)≤maxD∗(Y), even when the initial delegation set D∗(Y)has gaps. This would not be possible under full awareness: in any candidate equilibrium where types perfectly separate themselves through their announcement, the fully aware principal learns the payoff state and has incentives to deviate to a strictly lower action, at least in some states. In the case of limited awareness, however, the principal cannot contemplate the agent’s moves of which she remains unaware and this limits the extent to which she infers information from the agent’s recommendation. In particular, if the realized value is θand the agent proposes an action yA(θ)/∈Y such that yA(θ)∈[yA(0),max D∗(Y)], the subjective game tree that represents the principal’s frame of mind after updating does not include moves of the agent involving a proposal just below or above yA(θ). As a consequence, the principal cannot conceive of the fact that she would have permitted such actions if the agent had proposed them instead. In the principal’s subjective game following proposal yA(θ), there is an equilibrium where the agent reveals yA(θ)in all states where the agent prefers yA(θ)over the actions in the initial delegation set. Each of the agent’s equilibrium proposals is thus perceived to be consistent with an interval of states and these intervals overlap; that is, the principal’s information can no longer be represented by a partition of the state space into pairwise disjoint sets. The discrepancy between the agent’s true strategy and the principal’s perception of it is exactly what allows for a continuum of on-path proposals. Sometimes the principal’s coarse inference leads her to accept proposals that she should reject, e.g., when the agent proposes an action close to the lower boundary of any potential gap in D∗(Y). In expectation, however, she gains from the additional flexibility that she grants in equilibrium. Disclosure in the contracting phase While for a fixed awareness set renegotiation unambiguously benefits the principal, the prospect of being able to renegotiate after the arrival of information affects the agent’s disclosure incentives in the contracting phase. The agent’s initial disclosure determines the principal’s delegation set and with that the set of proposals the principal is willing to accept in the renegotiation phase. His goal is to maximize the final set of permitted actions, whether permission is given in the contracting phase or in the renegotiation phase. Focusing again on the case where, upon inducing awareness Y, the agent expects the principal to delegate D∗(Y)and accept any additional proposal in A(Y), the agent’s optimal disclosure in the contracting phase solves the problem max Y1 0 UAθ,y∗θ,A(Y)dF(θ)(17) 14To see why Assumption 1is needed, suppose it is not satisfied. Then we cannot rule out the possibility to have an awareness Y,adelegationsetD∗(Y), and two actions y,y/∈D∗(Y)such that VP(D∗(Y)∪ {y}),VP(D∗(Y)∪{y})≥VP(D∗(Y)) and VP(D∗(Y)∪{y,y})<VP(D∗(Y)). The principal may thus be worse off after renegotiation. 264 Auster and Pavoni Theoretical Economics 19 (2024) subject to YP⊆Y⊆YA. When Assumption 1is satisfied, this problem has a simple solution. Proposition 7. Let Assumption 1be satisfied. A solution to problem (17)existsandis given by the awareness set Ythat solves max YP⊆Y⊆YAmaxD∗(Y). The proposition shows that the agent’s highest equilibrium payoff is attained by having the agent disclose a set of actions in the contracting phase that maximizes the upper threshold of the principal’s resulting delegation set. Since through renegotiation, the agent is able to implement all actions below the upper threshold, this maximizes the agent’s flexibility in equilibrium, and thus his equilibrium payoff. For concreteness, consider case (ii) of Proposition 2, where the principal’s value VP([yA(0),y]) is single-peaked in yand the agent’s optimal disclosure policy introduces a single gap around ˆ y.Let(ˆ y−¯ 1,ˆ y+¯ 2)be the largest feasible awareness gap such that the corresponding delegation set is D∗(Y)=[ymin,ˆ y−¯ 1]∪{ˆ y+¯ 2}. The highest action the principal is willing to delegate in the contracting phase is thus max YP⊆Y⊆YAmaxD∗(Y)=ˆ y+¯ 2. Given awareness set Y=[ymin,ˆ y−¯ 1]∩[ˆ y+¯ 2,ymax]and delegation set D∗(Y)= [ymin,ˆ y−¯ 1]∪{ˆ y+¯ 2}, there is an equilibrium in the renegotiation phase where the agent can implement any action in the interval [yA(0),ˆ y+¯ 2]. If the realized state θis such that yA(θ)∈D∗(Y), the agent does not renegotiate and takes his preferred action yA(θ). If the realized θis such that yA(θ)>ˆ y+¯ 2, the agent does not renegotiate either, because conditioning on the event that the agent prefers some action x>ˆ y+¯ 2over ˆ y+¯ 2, the principal strictly prefers ˆ y+¯ 2. If instead the state θis such that yA(θ)∈(ˆ y−¯ 1,ˆ y+¯ 2), the agent renegotiates and proposes his preferred action yA(θ). The principal is unaware of other actions in the interval (ˆ y−¯ 1,ˆ y+¯ 2),and thus only infers that the agent prefers the proposed action to all other actions in D∗(Y). Conditioning on this information, she prefers xas well. The model with renegotiation highlights an important aspect concerning the dynamics of unawareness. Much like information, awareness is not reversible. This means that if a player becomes aware of an action today, he remains aware of that action in the future (similarly for outcomes, events, etc.). Hence, the more a player reveals at an early stage of the game, the smaller the collection of the opponent’s awareness sets from which he can choose later on. When there is uncertainty about the future, this creates incentives to hide feasible actions from the other player until the later stages of the game. In our environment, this principle is reflected in the fact that the agent reveals fewer actions in the contracting phase when renegotiating is possible than when it is not. In the case discussed above, the optimal awareness gap in the contracting phase is maximal when renegotiating is possible. Notice that, even without renegotiation, the agent could implement any single action below ˆ y+¯ 2by revealing the “right” set of actions. Theoretical Economics 19 (2024) Optimal delegation and information transmission 265 He cannot, however, implement all actions below ˆ y+¯ 2because some actions crowd out others. The agent has to make a choice based on the expected value of the feasible awareness sets and the resulting delegation sets. When renegotiation is possible, instead, the agent can condition the principal’s awareness on the realization of θ. Ex ante welfare Since the possibility to renegotiate limits the agent’s disclosure incentives in the contracting phase, the ranking of the principal’s expected payoff between the two cases, with and without renegotiation, depends on the principal’s initial level of awareness. To illustrate this, consider the following example based on the model analyzed in Section 3.1. Example. Assume UP(y,θ)=−(y−(θ−β))2,UA=−(y−θ)2.Letθbe uniformly distributed on [0, 1]and assume β<1/2, so that delegation is valuable. The optimal full awareness cap in this example is ˆ y=1−2β, while the parameter characterizing the agent’s unconstrained solution of the disclosure problem is ∗=2(√2−1)β.Denoteby ¯ (YP)the action in the principal’s initial awareness set closest to ˆ y, where for ease of notation, we will suppress the argument YP. A calculation shows that under this specification, the principal’s expected payoff in the case with renegotiation is higher than in the case without it if and only if ¯  β≤√2−1 23(4√2−5)−1≈1.21. (18) ♦ In words, accounting for the agent’s initial disclosure incentives, the principal is better off with renegotiation if and only if ¯ is sufficiently small with respect to the agent’s bias β. To understand this property, note first that if ¯ ≤∗, the agent’s optimal disclosure in the contracting phase is not affected by the possibility to renegotiate later: in either case, the optimal awareness gap for the agent is (ˆ y−¯ ,ˆ y+¯ )and the resulting delegation set is D∗(Y∗)=[0, ˆ y−¯ ]∪{ˆ y+¯ }. Given that the principal prefers to have the gap in D∗(Y∗)closed, she is strictly better off when renegotiation is allowed. Moreover, the larger the bias β, the larger ∗, and hence the gain from renegotiation. If instead ¯ > ∗, the possibility to renegotiate gives the agent incentives to leave the principal unaware of more actions in the contracting phase than in the case of full commitment, thus a tradeoff arises. The principal’s expected payoff in the case of pure delegation is now VP([0, ˆ y−∗]∪{ˆ y+∗})(independent of ¯ ), while her expected payoff in the case of renegotiation is VP([0, ˆ y+¯ ]).15 The ranking of these two payoffs depends on how high ˆ y+¯ is, i.e., how much additional flexibility the agent gains in the case of renegotiation. Condition (18)isequivalenttoVP([0, ˆ y+¯ ]) ≥VP([0, ˆ y−∗]∪{ˆ y+∗}). 15The closed-form expressions are: VP0, ˆ y−∗∪ˆ y+∗=−β2+8 3(3−2√2)β3 VP[0, ˆ y+¯ ]=− (1−2β+¯ )β2+1 3(¯ −β)3−β3. 266 Auster and Pavoni Theoretical Economics 19 (2024) 5. Conclusion This paper formulates a flexible delegation model with limited awareness and derives several properties of the optimal solution. The solution shows that by leaving the principal unaware of moderate options, the agent makes it optimal for the principal to permit actions closer to his own preferences. As argued in the Introduction, our framework has interesting implications for applications. We however believe that a key component of the contribution is to provide at least three general insights that apply to games with a principal-agent structure where the agent has superior awareness over feasible actions. First, the paper illustrates that limited awareness can impose natural constraints on the language of contracts and that such limits may be exploited by the contracting party with superior awareness. This principle is not restricted to delegation problems but applies to other contracting problems. A principal facing a privately informed agent must resolve a tradeoff between exploiting the agent’s private information and limiting the agent’s information rents. The distortions solving this tradeoff are optimal for the principal but not for the agent. By manipulating the principal’s awareness set, and hence the set of feasible contracts, the agent can increase the principal’s cost of such distortions, thereby increasing the principal’s willingness to grant the agent higher information rents. The unconstrained solution to the agent’s disclosure problem determines the maximal information rents he can get by modifying the set of feasible actions. Second, the paper shows how in contracting situations with limited awareness, the option of renegotiation may be used as a tool to implement outcomes that are not describable at the contracting stage, without giving the other party blanket approval for unknown actions. While the existing literature largely focuses on the costs caused by the impossibility of avoiding ex post renegotiations in the presence of ex ante specific investments, we thus see renegotiation as an opportunity for the principal to improve the outcome. The downside of ex post renegotiation for the principal in our setting is the reduction in the agent’s incentives to disclose actions ex ante, giving rise to an interesting tradeoff. A clever design of the renegotiation process may shift this tradeoff further in favor of renegotiation (see Aghion, Dewatripont, and Rey (1994)andHart and Moore (2004)). An intriguing general question is indeed what a designer with limited awareness can achieve with mechanisms that are expressible in her language. The current paper may be viewed as a step in that direction. Third, our modification of the game with renegotiation exemplifies how unawareness changes the ways in which agents infer information. If a player is unaware of the set of possible signals and only becomes aware of the signal s/he observes, the player cannot infer information from the fact that a different signal did not realize. This asymmetry gives rise to nonstandard information structures, and hence to rather different equilibrium outcomes with respect to the full awareness benchmark. Appendix A A.1 Proof of Proposition 1 Let t:(YA)2→[0, 1]be a symmetric function, indicating the state at which the agent is indifferent between any two actions yand y.It is specified as follows. For y=y,set t(y,y)=s(y)(recall that s(·)is the inverse of yA(·)). For y<y ,t(y,y)is defined by Theoretical Economics 19 (2024) Optimal delegation and information transmission 267 –ifUA(θ,y)<UA(θ,y)for all θ∈[0, 1],thent(y,y)=0; –ifUA(θ,y)>UA(θ,y)for all θ∈[0, 1],thent(y,y)=1; –otherwiset(y,y)is such that UAty,y,y=UAty,y,y. (19) Due to the single-crossing condition, the solution of (19) is unique. For y>y ,t(y,y) is pinned down by the symmetry condition t(y,y)=t(y,y). The following lemma links the slope of swith a partial derivative of t. Lemma 8. Consider y0such that s(y0)∈(0, 1),then lim y→y0 dt(y,y0) dy=1 2s(y0) Proof.Forthecasewheretis determined by (19), we apply the implicit function theorem to derive dt(y,y0) dy=UA yt(y,y0),y UA θt(y,y0),y0−UA θt(y,y0),y Taking the limit, we have lim y→y0 dt(y,y0) dy=lim y→y0 UA yt(y,y0),y UA θt(y,y0),y0−UA θt(y,y0),y =lim y→y0 UA θy t(y,y0),ydt(y,y0) dy+UA yyt(y,y0),y UA θθt(y,y0),y0−UA θθt(y,y0),ydt(y,y0) dy−UA θy t(y,y0),y = UA θy s(y0),y0lim y→y0 dt(y,y0) dy+UA yys(y0),y0 −UA θy s(y0),y0 where the second equality follows from L’Hôspital’s rule. Also, recall t(y0,y0)=s(y0).We can solve the above equality for limy→y0 dt(y,y0) dyand obtain lim y→y0 dt(y,y0) dy=−1 2·UA yys(y0),y0 UA y0s(y0),y0 From UA y(s(y),y)=0, we derive via the implicit function theorem: s(y)=−UA yys(y),y UA θy s(y),y which is well-defined given our assumption that UA(θ,y)is in C2and UA θy >0. The two results together establish the claim. 268 Auster and Pavoni Theoretical Economics 19 (2024) Define ¯ D(y):=D∗(YA)∩[yA(0),y]as the set obtained by capping the optimal delegation set under full awareness at y. Lemma 9. If (6) holds as a strict inequality, there exists some y<ˆ ysuch that for all y∈ (y,ˆ y)∩D∗(YA), VP¯ D(y)<VP¯ D(y)∪{ˆ y}. Proof. We define the difference between the principal’s expected payoffs when adding action ˆ yto a delegation set whose highest action is y<ˆ y. Given the strict monotonicity of the agent’s preferred action in θ, adding action ˆ yto a delegation set Dwith maxD= y<ˆ yonly changes the outcome in the states where the agent optimally switches from y to ˆ y. The set of states where this happens is (t(y,ˆ y),1 ], so the payoff difference is  W(y):=1 t(y,ˆ y) UP(θ,ˆ y)dF(θ)−1 t(y,ˆ y) UP(θ,y)dF(θ) Note that for all y∈(y,ˆ y)∩D∗(YA),wehave VP¯ D(y)∪{ˆ y}−VP¯ D(y)= W(y) We calculate the first derivative of  W(·)and evaluate it at ˆ y:  W(y)=−1 t(y,ˆ y) UP y(θ,y)dF(θ)−UPt(y,ˆ y),ˆ y−UPt(y,ˆ y),yft(y,ˆ y)dt(y,ˆ y) dy  W(ˆ y)=−1 s(ˆ y) UP y(θ,ˆ y)dF(θ) By (5), the above term is equal to zero. We must therefore consider the second derivative:  W(y)=−1 t(y,ˆ y) UP yy(θ,y)dF(θ)+2UP yt(y,ˆ y),yft(y,ˆ y)dt(y,ˆ y) dy −UP θt(y,ˆ y),ˆ y−UP θt(y,ˆ y),yft(y,ˆ y)dt(y,ˆ y) dy 2 −UPt(y,ˆ y),ˆ y−UPt(y,ˆ y),yft(y,ˆ y)dt(y,ˆ y) dy +ft(y,ˆ y)d2t(y,ˆ y) dy2  W(ˆ y)=−1 s(ˆ y) UP yy(θ,ˆ y)dF(θ)+2UP ys(ˆ y),ˆ yfs(ˆ y)dt(y,ˆ y) dy y=ˆ y Since, by Lemma 8,wehavedt(y,ˆ y) dy |y=ˆ y=1 2s(ˆ y), condition (6) holding as a strict inequality implies  W(ˆ y)>0. Remembering  W(ˆ y)=0, there is then an interval for yto the left of ˆ y,where W(y)<0. With  W(ˆ y)=0, this property implies, in turn, that there is some y<ˆ ysuch that  W(y)>0forally∈(y,ˆ y). Hence, for all y∈(y,ˆ y)∩D∗(YA),we have VP(¯ D(y)∪{ˆ y})−VP(¯ D(y)) = W(y)>0. Theoretical Economics 19 (2024) Optimal delegation and information transmission 275 is strictly concave with the interior solution characterized by (11). The conditions for applying the implicit function theorem are again satisfied, hence there is a function ∗(β) describing the unconstrained solution for the agent that solves the first-order condition U(∗(β);β)=0, which becomes an identity when seen as a function of β,and ∗(β)=−Uβ∗(β);β U∗(β);β. To prove the statement of the proposition, we must then show Uβ(∗(β);β)>0. Differentiating the expression of the first-order condition (11) with respect to βkeeping ∗ fixed, after some rearrangement, delivers Uβ∗(β);β=−ˆ y(β)1+Fˆ y(β)−∗(β)−2Fˆ y(β). Given ˆ y(β)<0(see(24)),wearedoneif1+F(ˆ y(β)−∗(β))−2F(ˆ y(β)) >0, or equivalently 21−Fˆ y(β)>1−Fˆ y(β)−∗(β). Using ˆ y(β)=E[θ−β|θ≥ˆ y(β)], the first-order condition (11) can be written as 1−Fˆ y(β)−∗(β)Eθ|θ≥ˆ y(β)−∗(β)−ˆ y(β)−∗(β) =21−Fˆ y(β)β. (25) Since ˆ y(β)−∗(β)is strictly smaller than ˆ y(β), the following condition holds: Eθ−β|θ≥ˆ y(β)−∗(β)−ˆ y(β)−∗(β)>0. Equivalently, we can write Eθ|θ≥ˆ y(β)−∗(β)−ˆ y(β)−∗(β)>β. Given this inequality, (25)requires2 (1−F(ˆ y(β))) >1−F(ˆ y(β)−∗(β)), as desired. A.5 Proof of Proposition 5 Proof. We begin with part (i). Consider a renegotiation equilibrium and let ˆ X:={x∈ Y:ρ∗(x)=1}be the set of actions in the principal’s initial awareness set which are allowed in equilibrium. By upper semicontinuity of ρ∗,thisset ˆ Xis closed. Since also D∗(Y)is closed, the set D∗(Y)∪ˆ Xis closed as well. We first want to show that VPD∗(Y)∪ˆ X≥VPD∗(Y). Since the agent can always guarantee choices in D∗(Y)by proposing N,agent’soptimality in the eyes of the principal implies x∗ x(θ)=y∗(θ,D∗(Y)∪ˆ X)for all x∈ˆ Xand θ∈x∈ˆ X∗(x), with the exclusion of points where the agent is indifferent between two actions in D∗(Y)∪ˆ X. For each x∈D∗(Y)∪ˆ X,wethushave cl∗(x)=θ∈[0, 1]:UA(θ,x)≥max y∈D∗(Y)∪ˆ X UA(θ,y) 276 Auster and Pavoni Theoretical Economics 19 (2024) Usingthefactthatforallx∈ˆ Xand all θ∈∗(x),wehavex=y∗(θ,D∗(Y)∪ˆ X)and that for each B⊆∗(x)we have ∗(x)μ∗(B|x)f(θ)dθ=BdF(θ),17 the principal’s optimality condition yields ∗(x)∗(x) UP(θ,x)dμ∗(θ|x)dFθ≥∗(x)∗(x) UPθ,y∗θ,D∗(Y)dμ∗(θ|x)dFθ ⇐⇒ ∗(x) UPθ,y∗θ|D∗(Y)∪XdF(θ)≥∗(x) UPθ,y∗θ,D∗(Y)dF(θ) for all x∈ˆ X. Next, for θ/∈x∈ˆ X∗(x),wehavex∗ x(θ)=Nor ρ∗(x∗ x(θ)) =0. In either case, the agent will take y∗(θ,D∗(Y)).Wefurtherhavey∗(θ,D∗(Y)) =y∗(θ,D∗(Y)∪ˆ X)for all θ/∈x∈ˆ X∗(x), since the agent is free to take any action in D∗(Y)∪ˆ X. Setting C(ˆ X):=[0, 1]\(x∈ˆ X∗(x)), we can now integrate over ˆ Xand obtain C(ˆ X) UPθ,y∗θ,D∗(Y)∪ˆ XdF(θ)+ˆ X∗(x) UPθ,y∗θ,D∗(Y)∪ˆ XdF(θ)dx ≥C(ˆ X) UPθ,y∗θ,D∗(Y)dF(θ)+ˆ X∗(x) UPθ,y∗θ,D∗(Y)dF(θ)dx or equivalently VPD∗(Y)∪ˆ X≥VPD∗(Y). Recall that D∗(Y)∪ˆ X⊆Yis a closed set. Since D∗(Y)is the largest closed optimal awareness set with respect to Ythat includes actions that will actually be taken by the agent under some contingency, this inequality yields a contradiction unless D∗(Y)∪ˆ X= D∗(Y). Having shown that the principal only accepts additional actions in the renegotiation phase if they do not belong to Y, consider proposal x∈YA\Ysuch that ρ∗(x)= 1. Perceived agent optimality then requires x∗ x(θ)=xfor all θsuch that UA(θ,x)> maxy∈D∗(Y)UA(θ,y). Principal optimality in turn requires that conditioning on the event UA(θ,x)>U A(θ,y∗(θ,D∗(Y))), the principal prefers xover y∗(θ,D∗(Y)) in expectation. This is the case only if x∈A(Y). To show part (ii), we need to construct an equilibrium of the renegotiation game where proposal xis accepted by the principal whenever x∈A(Y).Tothisend,we set ρ∗(x)=1forallx∈A(Y)and ρ∗(x)=0 otherwise. Recall that ρ∗(N)=0. For the agent, we set x∗(θ)=argmaxx∈A(Y)UA(θ,x)if maxx∈A(Y)U(θ,x)≥maxy∈D∗(Y)UA(θ,y) and x∗(θ)=Notherwise.18 Similarly, for the agent’s strategy as perceived by the principal when receiving proposal x,wesetx∗ x(θ)=xif U(θ,x)≥maxy∈D∗(Y)UA(θ,y)and 17Note that monotonicity of the agent’s optimal policy y∗in θimplies that ∗(x)is either of positive measure or a singleton. 18Recall in case of indifference the agent brakes ties in favor of the principal. Theoretical Economics 19 (2024) Optimal delegation and information transmission 277 x∗ x(θ)=Notherwise. The principal beliefs system μ∗is defined as follows. For all x∈X, μ∗(B|x)=B dF(θ ∗(x) dF(θ)∀B⊆∗(x), whenever ∗(x)=(x∗ x)−1(x)is of positive measure, and μ∗({θ}|x)=1if∗(x)={θ}.It can be checked directly that this strategy and belief profile satisfy principal optimality, agent optimality, and consistency of beliefs, and thus constitute a PBE of the renegotiation game, as specified in Definition 1. A.6 Proof of Proposition 6 Proof. Note that, under Assumption 1,foranyx∈(minD∗(Y),maxD∗(Y)),wehave VP(D∗(Y)∪{x})≥VP(D∗(Y)),soxbelongs to the set of acceptable proposals. The same is true for all x<minD∗(Y), since conditioning on the fact that the (upward biased) agent prefers xover minD∗(Y), the principal prefers xas well. The set of implementable action A(Y)thus includes all action in [yA(0),max D∗(Y)]. Under Assumption 1,we clearly have VP([yA(0),maxD∗(Y)]) ≥VP(D∗(Y)).Ifmax D∗(Y)=max A(Y),thisconcludes the argument. For the other case, let ¯ y:=maxA(Y)assume ¯ y>max D∗(Y).By definition of A,wehaveVP(D∗(Y)∪{¯ y})≥VP(D∗(Y)). Monotonicity of the agent’s action in θthen implies VPyA(0),maxD∗(Y)∪{¯ y}≥VPyA(0),maxD∗(Y) But given that permitting action ¯ yweakly increases the principal’s expected payoff, permitting any additional action in (max D∗(Y),¯ y)benefits the principal as well (again by Assumption 1), so (16) is satisfied. A.7 Proof of Proposition 7 Proof. We start by showing the existence of a solution of maxYD∗(Y). Recall that BRP(·)denotes the principal’s solution correspondence for problem (3). Let us then define ˇ y(Y):=sup D∈BRP(Y) maxD. Since any D∈BRP(Y)is compact, the last max is well-defined. From the proof of Proposition 12, we know that BRPis upper hemicontinuous. If we show that the function m(D):=maxDis continuous in D, the generalized version of the maximum theorem (e.g., Theorem 17.30 in Aliprantis and Border (2006)) implies that the max exists for each Yand ˇ y(Y)is upper semicontinuous in Y. This in turn implies that the following object is well-defined: ¯ y∗:=max YP⊆Y⊆YAˇ y(Y). 278 Auster and Pavoni Theoretical Economics 19 (2024) Lemma 10. The function m(D):=max Dis continuous with respect to the Hausdorff metric. Proof. Recall again that we are working with metric spaces. Take a sequence Dn→HD. Consider now the sequence of real numbers dn:=max Dn∀n. We need to show that the sequence converges to d:=maxD∈R. Since Dnconverges, it is Cauchy. We want to show that also dnis Cauchy. For any δ,letNδbe such that dH(Dn,Dm)≤δ∀n,m≥ Nδ.Now,ifdn=dmthere is nothing to prove. Suppose dn= dm. Then, without loss of generality, assume dn>d m.Wehave |dn−dm|=|max Dn−max Dm|= inf y∈Dm|dn−y|≤dH(Dn,Dm)≤δ. Given that δis generic, dnis Cauchy, and since Ris complete, the sequence dnmust converge. Let d∗be the converging point of the sequence. Again, without loss of generality assume d∗>d. But then, by the definition of convergence, it must be that for Nlarge enough, dn>dfor all n≥N. Delivering dH(D,Dn)≥inf y∈D|dn−y|=|dn−d|>0∀n≥N, which contradicts the fact that Dnconverges to D.Hence,itmustbethatd∗=d. Next, we want to show that for each Y,ˇ y(Y)is equal to max D∗(Y). Suppose this is not true. Then there exists an awareness set ˜ Ysuch that ˇ y(˜ Y)>maxD∗(˜ Y)and a delegation set ˜ D∈BRP(˜ Y)such that max ˜ D>maxD∗(˜ Y). Since it is never optimal to curtail the agent’s flexibility from below and Assumption 1holds, we have min ˜ D=minD∗(˜ Y)=y∗(0, ˜ Y) Given min ˜ D=minD∗(˜ Y)≤maxD∗(˜ Y)<max ˜ D, Assumption 1implies VP(˜ D∪D∗(˜ Y)) ≥ VP(˜ D), and hence (˜ D∪D∗(˜ Y)) ∈BRP(˜ Y). But since D∗selects the agent-preferred delegation set from BRP(˜ Y)and since D∗(˜ Y)⊆(˜ D∪D∗(˜ Y)),wemusthaveD∗(Y)=˜ D,and hence maxD∗(Y)=max ˜ D, a contradiction. Combining these results, we have shown that sup D∈BRP(Y) maxD=maxD∗(Y), and hence ¯ y∗=maxYP⊆Y⊆YAmaxD∗(Y). Let ¯ Y∗be an awareness set that maximizes max D∗(Y)over Ysubject to YP⊆Y⊆ YA. We now want to show that ¯ Y∗solves (17). Suppose not. Since A¯ Y∗=yA(0),¯ y∗ there must then exist an awareness set Ysuch that max(A(Y)) >¯ y∗. By the definition of ¯ y∗, there is no awareness set Ysuch that YP⊆Y⊆YAand ¯ y∗∈D∗(Y).Hence, theremustbeaproposalx> ¯ y∗that the principal accepts in the renegotiation phase. Theoretical Economics 19 (2024) Optimal delegation and information transmission 279 By Proposition 5,thisrequiresx∈A(Y), or equivalently VPD∗(Y)∪{x}≥VPD∗(Y). This in turn implies x∈D∗(Y∪{x}), and hence ¯ y∗≥x, a contradiction. A.8 Existence results Consider the following properties of our setup: (a) The set Y⊆YA=[ymin,ymax]is a compact subset of the complete and separable metric space (R,|·|). (b) Recall Ydenotes the set of closed subsets of [ymin,ymax],and ˆ D(Y):={D∈Y:D⊆Y}. Then ˆ D(Y)is a closed subset of 2[Y]with respect to the Hausdorff-metric dHD,D=maxsup y∈D inf y∈Dy−y,sup y∈D inf y∈Dy−y, and hence compact in the topology generated by the Hausdorff metric dH(see point 3 of Theorem 3.85 in Aliprantis and Border (2006)). (c) Recall that UAand UPare continuous and uniformly bounded on their domains [0, 1]×[ymin,ymax],andFadmits a density. (d) Recall that UA θy >0, UA y(θ,y)>0, UA yy(θ,y)<0. Hence, if we fix a closed set D⊆ YAand an open interval O⊆YA\D, there is at most one value of θsuch that yA(θ)∈Oand BRA(θ,D)is not single-valued. (e) The set (ymin,ymax )\Dis an open set, and hence it can be uniquely defined as a countable union of disjoint open intervals (e.g., Theorem 6, p. 51, in Kolmogorov and Fomin (1975)). Recall the agent chooses according to BRA(θ,D):=arg maxy∈DUA(y,θ).Notethatby continuity BRA(θ,D)is nonempty for each θ, since D⊆YAis compact from (a) and UA is continuous from (c). In addition, since the feasibility set Dchanges continuously with Din the Hausdorff norm (and UAis continuous in (D,y,θ)), the maximum theorem implies that BRAis an upper hemicontinuous correspondence when seen as a function of (D,θ). In addition, combining (d) with (e), and noticing that for all θsuch that yA(θ)∈ Dwe have BRA(θ,D)={yA(θ)},weconcludethatforeachDthe set of θ’s for which BRA(θ,D)is not single-valued is countable. Since Fadmits a density, the set of values for which the agent is indifferent is then of F-measure zero. In summary, we have the following. Lemma 11. BRAis a nonempty upper hemicontinuous correspondence in (θ,D).Moreover, for each D, the set A(D):={θ∈[0, 1]|BRA(θ,D)is not a singleton}has measure zero according to F. 280 Auster and Pavoni Theoretical Economics 19 (2024) Recall as well, we denoted with y∗the selection that resolves ties in favor of the principal. Due to Lemma 11, for each awareness set Y∈Y, the principal optimally selects a delegation set D⊆Yto solve max D∈ˆ D(Y) VP(D)where VP(D)=1 0 UPθ,y∗(θ,D)dF(θ). (26) From Lemma 4 in Holmström (1980), VPis upper semicontinuous in Dfor each closed Y⊆YA(where distances in Dare defined according to the Hausdorff-metric). Since according to this metric the feasibility set ˆ D(Y)is compact, we have the following. Proposition 12. An optimal solution to the principal’s problem in ˆ D(Y)—and hence in D(Y)—exists. In addition, VPis continuous in D. The existence of an optimal solution in D(Y)is guaranteed by the fact that for any solution to (26)in ˆ D(Y)we obtain a solution in D(Y)by eliminating actions that the agent does not take in equilibrium. The continuity of Vis implied by the fact that whenever an upper hemicontinuous correspondence is single-valued it is continuous. Hence, the second part of Lemma 11 implies that any selection hfrom BRAwill have discontinuities at a set of points that have probability zero according to F.Thatis, VP(D)=1 0 UPθ,y∗(θ,D)dF(θ)=1 0 UPθ,h(θ,D)dF(θ), and the latter varies continuously with Dby the continuity property of the selection h and the continuity of UPsummarized in (c). Denote by BRP(Y)the solution correspondence for the principal’s problem. If we can show that the correspondence from Yto D(Y)is continuous, thanks to properties (a)–(e), we can apply the theorem of the maximum to show that BRP(Y)is upper hemicontinuous. Now recall, we indicate with D∗(Y)the selection from BRP(Y)that resolves ties in favor of the agent. The problem of the agent at the initial disclosure stage solves max Y∈YVA(Y)where VA(Y)=1 0 UAθ,y∗θ,D∗(Y)dF(θ)such that YP⊆Y⊆YA. If we can show that the value VA(Y)is upper semicontinuous by the property of the selection D∗, we have a solution. Hence, let us show the following result. Proposition 13. BRP(Y)is upper hemicontinuous and the problem of the agent has at least one solution. Proof. As argued above, there are two crucial steps. First, the upper semicontinuity of VAand then the continuity of the correspondence D(Y). Theoretical Economics 19 (2024) Optimal delegation and information transmission 281 Lemma 14. The function VAis upper semicontinuous in Yunder the Hausdorff metric. Proof. Recall that D∗has the following property: D∗(Y)=arg max D∈BRP(Y)1 0 UAθ,y∗(θ,D)dF(θ). To show upper semicontinuity, take a converging sequence Yn→H¯ Yand suppose there is a sequence (in the real numbers) Vnconverging to ¯ V(in the |·|metric), where for each n, Vn=VA(Yn)=1 0 UAθ,y∗θ,D∗(Yn)dF(θ). We need to show that ¯ V≤VA(¯ Y). To simplify notation, let ˆ Dn=D∗(Yn).Wehence haveasequence ˆ Dnsuch that ˆ Vn=1 0UA(θ,y∗(θ,ˆ Dn))dF(θ)→¯ V. Since Yis compact, there is a converging subsequence, ˆ Dn→Hˆ D,andrecallthat ˆ Vn→¯ V.Notenowthat the function T(D):=1 0UA(θ,y∗(θ,D))dF(θ)is continuous in Dbased on the same arguments as in the proof of Proposition 12. It must hence be the case that T(ˆ Dn)→ T(ˆ D). This implies that ¯ V=1 0UA(θ,y∗(θ,ˆ D))dF(θ). Now, recall that BRPis upper hemicontinuous, i.e., it has a closed graph Gr. We have shown that ˆ Dis the limit of a sequence Dnsuch that ˆ Dn∈BRP(Yn)for all n, i.e., (ˆ Dn,Yn)∈Gr for all n. The limit point must be in the graph as well: (ˆ D,¯ Y)∈Gr. This is equivalent to saying that ˆ D∈BRP(¯ Y), and hence, from the definition of D∗, we have the desired inequality ¯ V=1 0 UAθ,y∗(θ,ˆ D)dF(θ)≤1 0 UAθ,y∗θ,D∗(¯ Y)dF(θ). Lemma 15. The correspondence mapping each set Yfrom the metric space (Y,dH)to ˆ D(Y)is both upper and lower hemicontinuous Proof. First of all, note that from ˆ D(Y),wehave D∈ˆ D(Y)⇐⇒ D⊆Y. Since we are working with metric spaces (and hence first countable topological spaces), we can prove our statement using sequences. (i) Upper hemicontinuity: take any Y∈Y and a generic converging sequence Yn→HY. Now, take a sequence Dnsuch that Dn⊆ Ynfor all n. We want to show that there is a subsequence Dnsconverging to D⊆Y.The existence of a converging sequence is implied by the compactness of the space. So, let Dbe such a point. We need to show that D⊆Y. This is implied by the convergence condition dH(Dn,D)→0 in the Hausdorff metric. Suppose D⊃Y.Itmusthencebe that dH(D,Y)=ε>0, i.e., the distance between the two sets is positive. Now, since both sequences converge, for each δthere is a Nδsuch that for all n≥Nδwe have both dH(Dn,D)≤δand dH(Yn,Y)≤δ. This indicates that Dncannot be smaller than the maximal reduction of Dcompatible with the distance and Ycannot be larger than the maximal extension of Ycompatible with the distance. Such reductions and extensions 282 Auster and Pavoni Theoretical Economics 19 (2024) can be made arbitrarily small, so if we have D⊃Y,wemusthaveforNlarge enough DN⊃YN, which is a contradiction. (ii) Lower hemicontinuity: take any D⊆Yand a converging sequence Yn→HY.We need to show that there is a sequence Dn→HDsuch that Dn⊆Ynfor all n.IfD=Y, we can take the sequence Dn=Yn. Alternatively, suppose D⊂Y. Since Ynconverges to Y,forNsufficiently large, we have D⊂Ynfor all n≥N. Hence, consider the following sequence: Dn=Ynfor n<Nand Dn=Dfor n≥N. This concludes the proof. 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