Informed Principal in Private-Value Environments Tymofiy Mylovanov Thomas Tröger University of Bonn June 21, 2008 1/28
Motivation 2/28
Motivation In most applications of mechanism design, the proposer of a mechanism (principal) holds relevant private information 3/28
Motivation In most applications of mechanism design, the proposer of a mechanism (principal) holds relevant private information In such environments, the mechanism proposal itself may reveal information about the principal s type 3/28
Motivation In most applications of mechanism design, the proposer of a mechanism (principal) holds relevant private information In such environments, the mechanism proposal itself may reveal information about the principal s type As a consequence, the revelation principle is inapplicable (Myerson, 1983) and computing perfect Bayesian equilibria in such signalling games is generally difficult 3/28
Motivation So far, equilibria have been computed only for a limited range of applications 4/28
Motivation So far, equilibria have been computed only for a limited range of applications For example, it is not known what is the equilibrium in the setting of Myerson optimal auction if the seller s valuation is her private information 4/28
Prior work 5/28
Prior work Myerson (1983) General preferences, discrete types, multiple agents Solution concepts, Strong Solution and Neutral Optimum, existence, and characterization 6/28
Prior work Myerson (1983) General preferences, discrete types, multiple agents Solution concepts, Strong Solution and Neutral Optimum, existence, and characterization Maskin Tirole (1990) Independent private values, discrete types, one agent A solution concept SUPO, existence, and characterization 6/28
Prior work Myerson (1983) General preferences, discrete types, multiple agents Solution concepts, Strong Solution and Neutral Optimum, existence, and characterization Maskin Tirole (1990) Independent private values, discrete types, one agent A solution concept SUPO, existence, and characterization Yilankaya (1999) Independent private values, linear payoffs, continuous types, one agent a characterization of solution 6/28
Example of environment 7/28
Example of environment partnership dissolution 8/28
Example of environment partnership dissolution principal (player 0) and agent (player 1): each owns half (one share) of a company 8/28
Example of environment partnership dissolution principal (player 0) and agent (player 1): each owns half (one share) of a company the amount of shares transferred from the principal to the agent, y [ 1, 1] 8/28
Example of environment partnership dissolution principal (player 0) and agent (player 1): each owns half (one share) of a company the amount of shares transferred from the principal to the agent, y [ 1, 1] the payment from the agent to the principal, p R 8/28
Example of environment partnership dissolution principal (player 0) and agent (player 1): each owns half (one share) of a company the amount of shares transferred from the principal to the agent, y [ 1, 1] the payment from the agent to the principal, p R let Z denote the set of feasible contracts 8/28
Example of environment partnership dissolution principal (player 0) and agent (player 1): each owns half (one share) of a company the amount of shares transferred from the principal to the agent, y [ 1, 1] the payment from the agent to the principal, p R let Z denote the set of feasible contracts disagreement outcome, z 0 = (0, 0) 8/28
Example of environment linear risk-neutral payoff functions: u 0 (y, p, t 0 ) = p yt 0, u 1 (y, p, t 1 ) = yt 1 p, where t 0 and t 1 are the parties marginal valuation of the shares (types) 9/28
Example of environment linear risk-neutral payoff functions: u 0 (y, p, t 0 ) = p yt 0, u 1 (y, p, t 1 ) = yt 1 p, where t 0 and t 1 are the parties marginal valuation of the shares (types) types are private information 9/28
Example of environment type spaces are: T 0 = {0, 3}, T 1 = {1, 2} α [0, 1] - prior probability that t 0 = 0 10/28
Example of environment type spaces are: T 0 = {0, 3}, T 1 = {1, 2} α [0, 1] - prior probability that t 0 = 0 agent types are equally likely 10/28
Full information optimum if the principal s type were common knowledge, the principal would implement { (y 0, p 0 (1, 1), if t 0 = 0; )(t 0, t 1 ) = ( 1, 2), otherwise both types of the principal obtain the payoff of 1, v 0 = v 3 = 1 11/28
Informed principal if the principal s type is her private information and the probability of t 0 = 0 is greater than or equal to 1/2, α 1/2, the following mechanism is incentive feasible { (y, p (1, 1), if t 0 = 0; )(t 0, t 1 ) = ( 1, 1), otherwise in this mechanism, v 0 = 1 and v 3 = 2 in particular, if α = 1/2, the agent is left no rent 12/28
Frontier of incentive efficient allocation rules v 3 (y 2, p ) α < 1 2 α = 1 1 (y 0, p 0 ) α > 1 2 (y, p ) 2 v 0 13/28
Main idea (Maskin and Tirole) 14/28
Main idea (Maskin and Tirole) mechanism (y, p ) is SUPO for α 1/2: 15/28
Main idea (Maskin and Tirole) mechanism (y, p ) is SUPO for α 1/2: it is incentive-feasible, and 15/28
Main idea (Maskin and Tirole) mechanism (y, p ) is SUPO for α 1/2: it is incentive-feasible, and for any prior beliefs about the principal s types, there does not exist another mechanism such that 15/28
Main idea (Maskin and Tirole) mechanism (y, p ) is SUPO for α 1/2: it is incentive-feasible, and for any prior beliefs about the principal s types, there does not exist another mechanism such that it is incentive-feasible given these beliefs 15/28
Main idea (Maskin and Tirole) mechanism (y, p ) is SUPO for α 1/2: it is incentive-feasible, and for any prior beliefs about the principal s types, there does not exist another mechanism such that it is incentive-feasible given these beliefs every type of the principal is better off 15/28
Main idea (Maskin and Tirole) mechanism (y, p ) is SUPO for α 1/2: it is incentive-feasible, and for any prior beliefs about the principal s types, there does not exist another mechanism such that it is incentive-feasible given these beliefs every type of the principal is better off at least one type of the principal is strictly better off 15/28
Main idea (Maskin and Tirole) mechanism (y, p ) is SUPO for α 1/2: it is incentive-feasible, and for any prior beliefs about the principal s types, there does not exist another mechanism such that it is incentive-feasible given these beliefs every type of the principal is better off at least one type of the principal is strictly better off we can show that SUPO, if it exists, is an equilibrium outcome 15/28
Main idea (Maskin and Tirole) mechanism (y, p ) is SUPO for α 1/2: it is incentive-feasible, and for any prior beliefs about the principal s types, there does not exist another mechanism such that it is incentive-feasible given these beliefs every type of the principal is better off at least one type of the principal is strictly better off we can show that SUPO, if it exists, is an equilibrium outcome main question: existence! 15/28
Main idea (Maskin and Tirole) mechanism (y, p ) is SUPO for α 1/2: it is incentive-feasible, and for any prior beliefs about the principal s types, there does not exist another mechanism such that it is incentive-feasible given these beliefs every type of the principal is better off at least one type of the principal is strictly better off we can show that SUPO, if it exists, is an equilibrium outcome main question: existence! approach: slack-exchange equilibrium 15/28
General environment 16/28
Environment principal (player 0) and n agents 17/28
Environment principal (player 0) and n agents players must collectively choose an outcome from Z 17/28
Environment principal (player 0) and n agents players must collectively choose an outcome from Z every player i has a type t i that belongs to a type space T i in this talk, T i R is compact (results extend to compact metric spaces) 17/28
Environment principal (player 0) and n agents players must collectively choose an outcome from Z every player i has a type t i that belongs to a type space T i in this talk, T i R is compact (results extend to compact metric spaces) player i s payoff function u i : Z T IR, where T = T 0 T n 17/28
Environment principal (player 0) and n agents players must collectively choose an outcome from Z every player i has a type t i that belongs to a type space T i in this talk, T i R is compact (results extend to compact metric spaces) player i s payoff function u i : Z T IR, where T = T 0 T n the types t 0,..., t n are realizations of stochastically independent random variables with c.d.fs F 0,..., F n, where the support of F i equals T i. 17/28
Environment the interaction leads to a probability distribution over outcomes 18/28
Environment the interaction leads to a probability distribution over outcomes let Z denote the set of probability measures on Z 18/28
Environment the interaction leads to a probability distribution over outcomes let Z denote the set of probability measures on Z a disagreement outcome outcome z 0 Z 18/28
Informed principal game 19/28
Informed principal game the interaction is described by the following informed-principal game 20/28
Informed principal game the interaction is described by the following informed-principal game 1. each player privately observes her type t i 20/28
Informed principal game the interaction is described by the following informed-principal game 1. each player privately observes her type t i 2. principal offers a (possibly, direct) mechanism M 20/28
Informed principal game the interaction is described by the following informed-principal game 1. each player privately observes her type t i 2. principal offers a (possibly, direct) mechanism M 3. agent decides whether to accept M 20/28
Informed principal game the interaction is described by the following informed-principal game 1. each player privately observes her type t i 2. principal offers a (possibly, direct) mechanism M 3. agent decides whether to accept M 4. if M is accepted, the agent and the principal choose messages in M, and the outcome specified by M is implemented 20/28
Informed principal game the interaction is described by the following informed-principal game 1. each player privately observes her type t i 2. principal offers a (possibly, direct) mechanism M 3. agent decides whether to accept M 4. if M is accepted, the agent and the principal choose messages in M, and the outcome specified by M is implemented 4. if the agent rejects M, the disagreement outcome z 0 is implemented 20/28
Existence 21/28
Existence of SUPO We say that any best outcome for the principal is a worst outcome for all agents if arg max z Z u 0(z, t) arg min z Z u i(z, t) for all t 22/28
Existence of SUPO We say that any best outcome for the principal is a worst outcome for all agents if arg max z Z u 0(z, t) arg min z Z u i(z, t) for all t An environment has semi-private values if the agents payoff functions are independent of the principal s type, that is, for all i 1, u i (z, (t 0, t 0 )) = u i (z, (t 0, t 0 )) for all z, t 0, t 0, t 0. 22/28
Existence of SUPO We say that any best outcome for the principal is a worst outcome for all agents if arg max z Z u 0(z, t) arg min z Z u i(z, t) for all t An environment has semi-private values if the agents payoff functions are independent of the principal s type, that is, for all i 1, u i (z, (t 0, t 0 )) = u i (z, (t 0, t 0 )) for all z, t 0, t 0, t 0. An environment with semi-private values is constraint-non-degenerate if there exists an allocation rule such that 22/28
Existence of SUPO We say that any best outcome for the principal is a worst outcome for all agents if arg max z Z u 0(z, t) arg min z Z u i(z, t) for all t An environment has semi-private values if the agents payoff functions are independent of the principal s type, that is, for all i 1, u i (z, (t 0, t 0 )) = u i (z, (t 0, t 0 )) for all z, t 0, t 0, t 0. An environment with semi-private values is constraint-non-degenerate if there exists an allocation rule such that for any beliefs Q0 = 1 t0 the incentive constraints and participation constraints are satisfied with strict inequality for all agents i 1, ˆt i, and t i 22/28
Existence of SUPO Proposition Suppose that the type spaces T 0,..., T n are finite. Consider any semi-private constraint-non-degenerate environment, where any best outcome for the principal is a worst outcome for all agents. Then an SUPO exists. 23/28
Equilibrium Existence Proposition Any SUPO is a perfect Bayesian equilibrium outcome of the informed-principal game 24/28
Existence of SUPO Comments on the proof Our basic line of proof is analogous to Maskin and Tirole (1986, 1990), who in turn follow Debreu (1959) 25/28
Existence of SUPO Comments on the proof Our basic line of proof is analogous to Maskin and Tirole (1986, 1990), who in turn follow Debreu (1959) We consider a fictitious Walrasian economy where the various types of the principal trade slack in the agents incentive and participation constraints; a competitive equilibrium of this economy exists and is an SUPO 25/28
Existence of SUPO Comments on the proof Our basic line of proof is analogous to Maskin and Tirole (1986, 1990), who in turn follow Debreu (1959) We consider a fictitious Walrasian economy where the various types of the principal trade slack in the agents incentive and participation constraints; a competitive equilibrium of this economy exists and is an SUPO We consider trade in all constraints, whereas Maskin and Tirole consider trade in just two constraints in a specific class of economic environments with one agent and two types, where the other two constraints are automatically satisfied 25/28
Existence of SUPO Comments on the proof The focus on semi-private environments is essential to make the Walrasian-economy-technique applicable: it guarantees that the form of the agents incentive compatibility and participation constraints is independent of the type of the principal 26/28
Existence of SUPO Comments on the proof The focus on semi-private environments is essential to make the Walrasian-economy-technique applicable: it guarantees that the form of the agents incentive compatibility and participation constraints is independent of the type of the principal The assumption of constraint non-degeneracy is essential towards showing that the demand correspondence in the fictitious economy is upper hemicontinuous 26/28
Existence of SUPO Comments on the proof The focus on semi-private environments is essential to make the Walrasian-economy-technique applicable: it guarantees that the form of the agents incentive compatibility and participation constraints is independent of the type of the principal The assumption of constraint non-degeneracy is essential towards showing that the demand correspondence in the fictitious economy is upper hemicontinuous The assumption that any best outcome for the principal is a worst outcome for all agents guarantees that Walras law holds for the fictitious economy 26/28
Existence of SUPO Comments on the proof The focus on semi-private environments is essential to make the Walrasian-economy-technique applicable: it guarantees that the form of the agents incentive compatibility and participation constraints is independent of the type of the principal The assumption of constraint non-degeneracy is essential towards showing that the demand correspondence in the fictitious economy is upper hemicontinuous The assumption that any best outcome for the principal is a worst outcome for all agents guarantees that Walras law holds for the fictitious economy Because we consider trade in all constraints, the Walrasian equilibrium price of some constraints may be 0 26/28
Conclusions This paper: extends Maskin and Tirole s SUPO solution to general environments with semi-private values 27/28
Conclusions This paper: extends Maskin and Tirole s SUPO solution to general environments with semi-private values SUPO is an equilibrium of the informed principal game 27/28
Conclusions This paper: extends Maskin and Tirole s SUPO solution to general environments with semi-private values SUPO is an equilibrium of the informed principal game proves existence of SUPO 27/28
Conclusions This paper: extends Maskin and Tirole s SUPO solution to general environments with semi-private values SUPO is an equilibrium of the informed principal game proves existence of SUPO characterizes SUPO in quasi-linear settings 27/28
Conclusions This paper: extends Maskin and Tirole s SUPO solution to general environments with semi-private values SUPO is an equilibrium of the informed principal game proves existence of SUPO characterizes SUPO in quasi-linear settings constructs an example of a quasi-linear setting in which private information of the principal is important 27/28
Thank you! 28/28