Decision SupportSample average approximation for the continuous type principal-agent problem
Introduction
In the principal-agent problem, the principal optimizes the terms of an exchange with an agent who may have some private characteristic θ that is unknown to the principal. For example, the agent may have private demand θ for a product produced by the principal. While the exact value of θ may be unknown to the principal, both parties know the distribution of θ across different agents. The principal seeks to maximize her expected profit given uncertainty in θ by offering the agent quantity q units of the product at total price t.
This paper studies the principal-agent problem when the distribution of θ is continuous over a bounded range with density f(θ). We refer to this setting as the “continuous problem” and the setting where θ is a discrete random variable as the “discrete problem.” In the continuous problem, the principal offers the agent contract functions q(θ) and t(θ), so the agent chooses the quantity and price option depending on his private demand θ. In the discrete problem, the principal offers an option (qm, tm) for each possible realization of the random variable according to the revelation principle.
Many principal-agent results rely on the ability to derive analytical solutions for the contract options. This paper explores the case where such analytical solutions are intractable either because the formulation is too complex, or the space of possible values for θ is too large. For the continuous problem, the solution can be found analytically for some specific functions f(θ). See, Laffont and Martimort (2009) for the foundations behind the continuous problem, and Singham and Cai (2017) for a specific solution example. Performing optimization over a function space for q(θ) and t(θ) is generally a difficult problem. This paper provides a method for bounding the optimal profit and finding solution estimates for the continuous problem. This method can be used when the density f(θ) is too complex to yield analytical solutions, or when f(θ) may not be available but data samples of θ are present to estimate the distribution.
We refer to the continuous problem formulation as Φ, and its optimal objective value as Φ*. We approximate the continuous problem using an empirical distribution with M discrete samples from f(θ) when M is very large. We call this discrete formulation ΦM and its optimal objective value . ΦM can be solved numerically, but for arbitrarily large M becomes computationally intractable due to the large number of decision variables and constraints. We show how a sample average approximation (SAA) to ΦM, based on a smaller sample size N bootstrapped from the M samples, yields an upper bound in expectation on . Call this SAA problem and its optimal objective value . The solution to can be interpolated to provide a feasible solution which yields a lower bound to the continuous problem optimal objective Φ*. Without needing to solve ΦM, we show how a series of smaller problems can be used to bound and closely approximate the continuous objective profit Φ*.
As advanced numerical computing methods become more popular, the number and type of contracting problems that can be solved has increased drastically. Many recent computing methods can be used to solve contracting problems numerically, and we present relevant examples where numerical optimization is used to solve pricing problems that are traditionally solved analytically. Bertsimas and Thiele (2005) show how historical data can be used to formulate linear programs to solve newsvendor formulations. Ehtamo, Berg, and Kitti (2010) use an interative online method to solve a nonlinear pricing scheme in a monopolistic setting with two buyers. Of particular interest is the work by Cecchini, Ecker, Kupferschmid, and Leitch (2013) which formulates and solves bilevel nonlinear programs to solve principal-agent problems numerically. The bilevel structure is apparent from the principal’s problem incorporating the agent’s optimization over his set of possible actions. The authors motivate the work by citing the limitations of the assumptions of the traditional principal-agent model, quoting Lambert (2006) who argues that relying on closed-form results limit the type and complexity of models that can be solved. Dempe (1995) originally suggested modeling the principal-agent problem as a bilevel program, and Cecchini et al. (2013) construct a version of the ellipsoid method for obtaining numerical solutions. Traditionally, the difficulties with solving bilevel programs are managed using a first-order approach which allows for replacing the inner agent optimization using assumptions on the probability distribution of possible outcomes. While we adopt these standard assumptions here, these assumptions can be relaxed and a polynomial optimization approach used to find solutions in a moral hazard setting (Renner & Schmedders, 2015).
Most recently, there have been attempts to use numerical methods to expand the types of agent distributions that can be solved in principal-agent problems. In particular, the size and structure of the discrete agent-type space could be expanded to allow for more complex heterogeneous agent settings. Cai and Singham (2018) developed a nonlinear programming formulation to solve principal-agent problems when agents were subject to one of multiple discrete demand distributions. The principal faces the adverse selection problem with regards to the possible demand distribution and both the principal and the agent face stochastic uncertainty within the distribution. Singham, Cai, and Fügenschuh (2019) expands the types of contracts solved to include nonparticipation options and aggregation across discrete agent types, meaning a large agent space could be reduced to allow for feasible implementation of contracts through shutdown or pooling options. Finally, Singham and Cai (2017) initially present the idea of using sample average approximation for principal-agent models with continuous demand distributions through a single numerical example. This present paper provides mathematical justification for this idea, and shows concretely how sample average approximation can be used to form a discrete problem that closely approximates the continuous problem. Thus, results of the discrete problem can be leveraged to solve and derive bounds for the continuous problem using sample average approximation, which is often used in stochastic optimization.
Straightforward sample average approximation methods cannot be directly applied to the continuous problem Φ in the usual way because the solution space of the discrete approximation ΦM is fundamentally different from that of the continuous problem. The solution to Φ lies in a continuous function space, while any discrete problem has a finite-sized solution space which depends on the sampled values used. As N and M increase, the number of decision variables and constraints increases in the principal-agent problem, and so the feasible space also changes and is different from that of Φ. We construct ΦM as a way of compiling all discrete problems on the same space, allowing us to invoke known SAA convergence properties. The main result is that we can use an easily solvable discrete problem to obtain useful information about an intractable continuous problem.
Section 2 details the formulations for the continuous and discrete problems. Section 3 presents the approximation problem and bounds. Section 4 presents numerical examples that demonstrate convergence of the optimal objective value as the number of bootstrap samples increases. Section 5 concludes.
Section snippets
Formulation
Maskin and Riley (1984) establish the conditions for adverse selection to be studied in the principal-agent setting, and show the nature of nonlinear pricing schemes for the principal using quantity discounts. Burnetas, Gilbert, and Smith (2007) study asymmetric information between a supplier and a retailer where there is uncertainty in the demand distribution and quantity discounts can be used to improve the supplier’s profits. Babich, Li, Ritchken, and Wang (2012) study contracting options
Large M-problem and bootstrap
Recalling the assumptions on boundedness for the terms in the objective functions of Φ and ΦM, we note the objective function of ΦM is a Monte Carlo approximation of the integral objective function in Φ. The error, for a fixed q(θ), Δ(θ), between the objective in Φ and that ofis . The objective profit function (3.1) converges uniformly to the objective function in Φ because of a.e. differentiability and boundedness of the included terms. Establishing
Numerical results
We demonstrate the performance of the algorithm with two examples. The first uses a past implementation of the principal-agent problem where an analytical solution to the continuous demand problem has been derived. The second example is a different implementation where the true solution is not known. We construct the formulation using Pyomo (Hart, Laird, Watson, Woodruff, 2012, Hart, Watson, Woodruff, 2011) and employ the nonlinear solver IPOPT (Wächter & Biegler, 2006) for generating
Conclusion
Most continuous principal-agent problems rely on analytical solutions in order to obtain structural results. We present a method for computing approximate solutions to the continuous principal-agent problem when analytical solutions are not available using a sample average approximation approach towards solving this problem. The method relies on sampling values from the continuous demand distribution and using them to solve a discrete version of the problem. The discrete problem can be solved
Acknowledgments
We acknowledge the partial support of National Science Foundation grant CMMI-1535831 for this research, and many useful discussions with Wenbo Cai.
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