Decision SupportRobust pooling for contracting models with asymmetric information
Introduction
In principal-agent contracting problems, a principal wants to persuade an agent to perform a certain action and uses financial incentives to do so. Both parties are individually rational and only want to improve their own situation. We consider contracting problems where the principal is a seller of a certain product and the agent is a potential buyer. Thus, the seller desires either to initiate new trade with the buyer or to change the existing buyer’s order quantity. In order to do so, the seller offers a contract to the buyer, describing the order quantity (the action) and a side payment (the incentive). The contract design must balance the value of the contract for both parties, since the buyer can refuse a disadvantageous contract.
The complexity of the contracting problem increases significantly when the buyer has private information on his valuation of contracts, i.e., when there is information asymmetry. In terms of Mechanism Design, the buyer’s private information is represented by so-called types. That is, the buyer’s identity is an element of a known set of types and specified by a probability distribution on . The distribution of types is assumed to be common knowledge, in particular also to the seller. We consider the case where the buyer has single-dimensional private information, represented by the type .
In case of information asymmetry, the seller offers a menu of contracts, typically one contract for each of the possible buyer types. First, the optimal menu is determined by solving a certain optimisation problem, which we will discuss in later sections. Second, this menu is offered to the buyer. Finally, the buyer either chooses to accept a contract of the menu or refuses the offer, depending on what is most beneficial for the buyer. Note that the buyer can lie about his true type and choose any contract, which complicates the seller’s optimisation process.
The modelling of the buyer types is crucial for the contracting problem. In the Mechanism Design literature there are two typical choices. First, we have the classical discrete model: a finite discrete set for some (discrete distribution). Here, the menu consists of K contracts, one for each type. Hence, the buyer chooses from a finite number of contracts. Second, we have the classical continuous model: a bounded interval with (continuous distribution). Here, the menu is a function that maps every type to a contract. In other words, infinitely many contracts are offered to the buyer.
Our goal is to design and analyse a model that combines aspects of both the discrete and continuous models. For this model, the buyer’s type is continuously distributed, but only finitely many contracts are offered. The main motivation for this approach is that offering finitely many contracts is often preferred in practice, as such menus are easier to communicate and implement. We present a modelling approach which we call robust pooling in order to achieve the stated goal. First, the seller chooses the number of contracts that will be offered. Second, he partitions the set of types into K subintervals denoted by for . Third, he designs a menu of K contracts with a single contract intended for each subinterval . Finally, he offers the menu to the buyer, as usual.
Our modelling approach has two fundamental properties: pooling of types and robustness. First, the (discrete) pooling property refers to offering finitely many contracts, and thus offering the same contract to multiple types, by design. Second, the (continuous) robustness property means that each type accepts a contract from the menu and that this choice is correctly reflected in the model (for example in the objective function). In other words, the menu specifies an intended contract for each type and each type chooses its intended contract. Consequently, the buyer always accepts a contract from the menu, making the menu robust to the buyer’s private information. In our case, for each it is for all types in most beneficial to choose the kth contract.
In our approach, the seller must decide on a partition scheme, i.e., the number of contracts and an appropriately corresponding partition of . The robust pooling model enables us to determine the effect of different partition schemes, since our model handles an arbitrary number of contracts and any partition in a natural way. Due to the robustness property, we can evaluate the use of different schemes in a fair way by directly comparing the resulting objective values of the model.
Such a fair comparison is not possible with the classical discrete approach, since varying the number of contracts also implies changing the distribution of the buyer’s type, effectively changing which scenarios could happen. Moreover, if the discrete distribution is actually an approximation of a continuous distribution, then the discrete approach is generally not robust. The classical continuous approach does not pool types by design and is therefore also unsuitable.
As already hinted, there are several aspects of the robust pooling model to analyse. First, what is the complexity of the model? In particular, can we identify conditions under which the model can be solved efficiently? Second, can we quantify the performance of partition schemes? A natural choice for a partition is the equiquantile partition, where is partitioned into subintervals of equal probability. However, is the equiquantile partition the best possible partition and if not, how much performance is lost? Also, offering infinitely many contracts (the continuous approach) results in the best possible objective value and is partition independent. When using our approach, how many contracts should be offered to guarantee, say, 95% of this best possible value?
The presented combination of the discrete and continuous approaches has received limited attention in the literature, which we will review in the next section.
For a general reference for the classical discrete and continuous modelling approaches, see for example (Laffont & Martimort, 2002). To our knowledge, a combination of the discrete and continuous approaches, such as our robust pooling model, has received limited attention in the literature. Bergemann, Shen, Xu, and Yeh (2011) consider a linear-quadratic model based on Mussa and Rosen (1978), but with limited communication between the seller and the buyer. The limited communication implies that only a menu with a limited number of contracts can be offered. Their approach is effectively a restricted form of the classical continuous approach, where the menu is restricted to have finitely many contracts. The resulting model satisfies our desired pooling and robustness properties. They are able to reformulate the problem into a mean square minimisation problem and apply Quantisation theory (Lloyd-Max conditions) to determine the optimal menu of contracts and the optimal partition scheme. In particular, they show that compared to offering infinitely many contracts the loss in performance is of the tight order Θ(1/K2) when using K optimal contracts.
The same modelling approach is used in Wong (2014), who analyses a more general version of the non-linear pricing problem in Bergemann et al. (2011). He determines qualitative convergence rates on the loss of performance when offering K optimal contracts. Under more general assumptions than (Bergemann et al., 2011), he proves that the loss in performance is of the order when comparing an optimal menu with K contracts to using infinitely many contracts. Furthermore, the marginal benefit of using contracts compared to K contracts is decreasing and of the order . Finally, he considers a linear-quadratic model, where the buyer’s utility is linear in the order quantity and the seller’s utility is quadratic. Under certain regularity conditions on the distribution of the types, he shows that an optimal menu with contracts achieves at least 2/3 of the maximum utility when using infinitely many contracts.
We shall refer to the modelling approach used in Bergemann et al. (2011) and Wong (2014) as the limited variety model. In general, our robust pooling model is more restrictive than the limited variety model, since we partition (pool) types into subintervals a priori. We note that under the considered assumptions in Bergemann et al. (2011) and Wong (2014), the limited variety model effectively also partitions the types. We show in Section 2.2 that under our considered assumptions, both modelling approaches are equivalent provided that the optimal partition scheme is used. We use our robust pooling approach for the following reasons.
First, the robust pooling model has an added benefit regarding information extraction. The seller can extract private information from the buyer by observing the buyer’s chosen contract. Recall that by design the kth contract is chosen by all types in . Thus, after observing the buyer’s choice, the seller can narrow down the buyer’s type to one of the subintervals of the partition. Since the used partition is a decision made by the seller, he is able to control the accuracy of said identification in a natural and intuitive way. In general, the limited variety model cannot guarantee such structured information extraction.
Second, an implicit goal of offering a limited number of contracts is to have a simple mechanism. Partitioning using a certain heuristic (e.g., equidistantly or according to some ‘square-root’ rule) is simple and intuitive, and could have a decent performance. That is, the formulation promotes the experimentation with partition schemes. Moreover, it could be that the additional loss in performance by restricting to a partition scheme a priori is negligible.
The robust pooling model is also related to Robust Optimisation (see Ben-Tal, El Ghaoui, & Nemirovski, 2009). Our model can be interpreted as a Robust Optimisation variant for the discrete model, where each subinterval is the so-called uncertainty set of type pk. This will be further discussed when we have formalised the model.
In the recent years, there has been an increase in the application of Robust Optimisation to Mechanism Design models in the literature. For examples, see Aghassi and Bertsimas (2006); Bandi and Bertsimas (2014); Bergemann and Morris (2005); Pinar and Kizilkale (2016). The main focus lies on making contracting models robust to the distribution of the buyer’s type, i.e., it only depends on which types can occur and not on any probabilities. To our knowledge, Robust Optimisation has not been applied to obtain a model similar to our robust pooling model.
Our contributions are as follows. We present a new modelling approach for contracting problems called robust pooling. Our approach distinguishes itself from the classical discrete and continuous models by having a continuous distribution for the buyer’s type and offering a menu with finitely many contracts. Its two fundamental properties are pooling of types by design and robustness. Compared to the limited variety model, we use a partition to pool types a priori. Consequently, our modelling approach promotes the experimentation of heuristic partition schemes leading to simple mechanisms. We restrict the analysis to single-dimensional types, but the robust pooling principle can be applied to more general settings. We show that under certain assumptions robust pooling models have a simplified reformulation and can be solved efficiently.
We apply robust pooling to the Decreasing Marginal Utility (DMU) problem. It is the robust pooling variant and a generalisation of the linear-quadratic-uniform model considered in Wong (2014). This problem assumes a uniform distribution of the buyer’s type, which although restrictive allows for closed-form formulas. Consequently, the equiquantile partition is equidistant, which we use as a simple and intuitive benchmark partition.
For the DMU problem, we first derive closed-form formulas for the optimal menu and corresponding optimal objective value for any number of contracts and any partition. Second, we show that structural results and performance measures can be expressed by functions of a single new parameter based on the instance. Remarkably, this new parameter does not depend on all parameters of the instance, implying families of instances with the same structure. Third, we determine the optimal partition scheme, either analytically or numerically, depending on the setting. In particular, this leads to new insights into the (sub)optimality of the equidistant/equiquantile partition. Finally, we give performance guarantees for the equidistant and optimal partitions. As a special case, this extends and completes the analysis of the linear-quadratic-uniform model in Wong (2014).
The remainder of this paper is organised as follows. In Section 2 we consider robust pooling models in the context of utility maximisation. We apply the concept to the mentioned DMU problem in Section 3. Finally, we conclude our findings in Section 4.
Section snippets
Robust pooling
In this section we consider principal-agent contracting models in the setting of utility maximisation. We formalise the robust pooling model in Section 2.1, which we reformulate and analyse under certain assumptions in Section 2.2. All proofs are given in Appendix A.
Decreasing marginal utility problem
In this section, we consider the DMU contracting model that fits our robust pooling setting of Section 2 and can be analysed in detail. The model is based around the concept that the marginal utility of a product decreases for the buyer as the order quantity increases. First, we introduce and analyse the general DMU model in Sections 3.1–3.4. Second, we consider two cases of the DMU model in more detail, namely the cases with a quadratic buyer’s utility function in Section 3.5 and with a cubic
Conclusion
We have presented and analysed a new modelling approach for principal-agent contracting models, called robust pooling. This approach considers a buyer whose type follows a continuous distribution on the interval . The seller wants to offer a menu with a finite number of contracts . In our approach, the seller partitions into K subintervals and designs a menu with a single contract for each subinterval. The menu is constructed such that each type will choose its intended
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