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PERSUASION 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 these problems are generally equivalent — solving one solves the other. We use tools from the persuasion literature to generalize and extend many results in the delega- tion literature, as well as to address novel delegation problems, such as monopoly regulation with a participation constraint.

JEL Classification: D82, D83, L43

Keywords: persuasion, delegation, regulation

Date: 7th February 2019.

Kolotilin: School of Economics, UNSW Business School, Sydney, NSW 2052, Australia. E-mail: akolotilin@gmail.com. Zapechelnyuk : School of Economics and Finance, University of St Andrews, Castlecli↵e, the Scores, St Andrews KY16 9AR, UK. E-mail: az48@st-andrews.ac.uk.

We are grateful to Tymofiy Mylovanov, with whom we are working on related projects. We thank Ricardo Alonso, Kyle Bagwell, Benjamin Brooks, Deniz Dizdar, Piotr Dworczak, Alexander Frankel, Drew Fudenberg, Gabriele Gratton, Yingni Guo, Emir Kamenica, Navin Kartik, Ming Li, Hongyi Li, Carlos Oyarzun, Alessandro Pavan, Eric Rasmusen, Philip Reny, Joel Sobel, and Thomas Tröger for helpful comments and suggestions. We also thank participants at various seminars and conferences. Part of this research was carried out while Anton Kolotilin was visiting MIT Sloan School of Management, whose hospitality and support is greatly appreciated. Anton Kolotilin also gratefully acknowledges support from the Australian Research Council Discovery Early Career Research Award DE160100964. Andriy Zapechelnyuk gratefully acknowledges support from the Economic and Social Research Council Grant ES/N01829X/1.

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2 KOLOTILIN AND ZAPECHELNYUK

1. Introduction

There are two ways to influence decision making: delegation and persuasion. The delegation literature, initiated by Holmström (1984), studies the design of decision rules. 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), and international 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 (Kamenica and Gentzkow, 2011).

This paper shows that, under general assumptions, the delegation and persuasion problems 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 understand and solve the other problem.

Both delegation and persuasion problems have a principal and an agent whose payo↵s depend on an agent’s decision and a state of the world. The sets of decisions and states are intervals of the real line. The agent’s payo↵ function satisfies standard single- peakedness and sorting conditions. In a delegation problem, the agent privately knows the 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 and the agent freely chooses a decision. The principal’s tradeo↵ is that giving more discretion to the agent in the delegation problem and disclosing more information to the agent in the persuasion problem allows for better use of information about the state, but limits control over the biased agent’s decision.

We consider balanced delegation and monotone persuasion problems. In the balanced delegation 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 partitional information structure that either reveals the state or pools it with adjacent states. This problem incorporates constraints faced by information designers in practice. For example, a non-monotone grading policy that gives better grades to worse performing students 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 persuasion problems are strategically equivalent. For each primitive of one problem we explicitly construct an equivalent primitive of the other problem. This construction equates the marginal payo↵s and swaps the roles of decisions and states in the two problems.

PERSUASION 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 this equivalence result is fundamentally di↵erent from the revelation principle. Specific- ally, the sets of implementable (and, therefore, optimal) decision outcomes generally di↵er in the delegation and persuasion problems with the same payo↵ functions.

To prove the equivalence result, we show that the balanced delegation and monotone persuasion problems are equivalent to the following discriminatory disclosure prob- lem. The principal’s and agent’s payo↵s depend on an agent’s binary action, a state of the world, and an agent’s private type. The sets of states and types are intervals of the real line. The agent’s payo↵ function is single-crossing in the state and type. The principal designs a menu of cuto↵ tests, where a cuto↵ test discloses whether the state is below or above a cuto↵. The agent selects a test from the menu and chooses between inaction and action depending on his private type and the information revealed by the 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 cuto↵ test from the menu. Because the agent’s payo↵ function is single-crossing in the state, the agent optimally chooses inaction/action if the selected test discloses that the state is below/above the cuto↵. Thus, this problem can be interpreted as a del- egation problem in which a delegation set is identified with a menu of cuto↵s, and the agent’s decision with his selection of a cuto↵ from the menu.

To see why the discriminatory disclosure problem is equivalent to the monotone per- suasion problem, observe that each menu of cuto↵ tests defines a monotone partition of the state space. Because the agent’s payo↵ 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 optimally selected cuto↵ test. Moreover, because the agent’s payo↵ function is single-crossing in his type, the agent optimally chooses inaction/action if his type is below/above a threshold. Thus, this problem can be interpreted as a persuasion problem in which a monotone partition is identified with a menu of cuto↵s, and the agent’s decision with a threshold type.

We use our equivalence result to solve a monopoly regulation problem in which a welfare-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 Baron and Myerson (1982) as a mechanism design problem with transfers. Alonso and Matouschek (2008) pointed out that transfers between the regulator and monopolist are often forbidden, and thus, the monopoly regulation problem can be formulated as a delegation problem. Alonso and Matouschek (2008) omitted the monopolist’s participation constraint, so under their optimal regulation policy, the monopolist sometimes operates at a loss. Amador and Bagwell (2016) characterized the optimal regulation policy taking the participation constraint into account.

4 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 method of solving this problem, by recasting it as a monotone persuasion problem and using a single result from the persuasion literature. When the demand function is linear and the cost distribution is unimodal, the optimal regulation policy takes a simple form that is often used in practice. The regulator imposes a price cap and allows the monopolist to choose any price not exceeding the cap. The optimal price cap is higher when the participation constraint is present; so the monopolist is given more discretion when he has an option to exit.

The literature has focused on linear delegation and linear persuasion in which the marginal payo↵s are linear in the decision and the state, respectively. We show the equivalence of linear balanced delegation and linear monotone persuasion. We translate a linear delegation problem to the equivalent linear persuasion problem and 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 and Matouschek (2008), Amador and Bagwell (2013), and Amador, Bagwel