Design of Incentive Compatible Mechanisms for Stackelberg Problems

This paper takes the first steps towards designing incentive compatible mechanisms for hierarchical decision making problems involving selfish agents. We call these Stackelberg problems. These are problems where the decisions or actions in successive layers of the hierarchy are taken in a sequential way while decisions or actions within each layer are taken in a simultaneous manner. There are many immediate applications of these problems in distributed computing, grid computing, network routing, ad hoc networks, electronic commerce, and distributed artificial intelligence. We consider a special class of Stackelberg problems called SLRF (Single Leader Rest Followers) problems and investigate the design of incentive compatible mechanisms for these problems. In developing our approach, we are guided by the classical theory of mechanism design. To illustrate the design of incentive compatible mechanisms for Stackelberg problems, we consider first-price and second-price electronic procurement auctions with reserve prices. Using the proposed framework, we derive some interesting results regarding incentive compatibility of these two mechanisms. 1 Mechanism Design and Stackelberg Problems The Theory of Mechanism Design is an important discipline in the area of Welfare Economics. The area of Welfare Economics is concerned with settings where a policy maker faces the problem of aggregating the individual preferences into a collective (or social) decision and the individuals’ actual preferences are not publicly known. The theory of mechanism design aims at studying how this privately held information can be elicited [2, 4, 7]. The state-of-the-art literature on mechanism design theory deals with situations where individuals are symmetric, that is to say, no single individual dominates the decision process. However, there are situations arising in Welfare Economics, Sociology, Engineering, Operations Research, Control Theory, and Computer Science where individuals take decisions in a hierarchical manner. A simple example is that of a situation wherein one of the individuals (or a group of individuals), called leader(s), has the ability ? corresponding author to enforce his preference on the other individual(s), called follower(s). In such problems, the policy maker first invites the leaders to reveal their privately held information in a simultaneous manner. After receiving this information the policy maker broadcasts it among the followers and the followers respond to this by revealing their preferences in a simultaneous manner. After receiving the preferences from all the individuals, the policy maker aggregates the information into a social decision. The problem faced by the individuals in such a situation can be naturally modeled as a Stackelberg game following the seminal work of Stackelberg [15]. Following are some interesting examples where one can see these problems arising naturally. – Task allocation in parallel/distributed systems – Scheduling in grids – Internet routing – Admission, routing, and scheduling in telecom networks – Flow control, routing, and sequencing in manufacturing systems – Auctions in electronic commerce In a recent work, Roughgarden [14] considered the problem of job shop scheduling, where fraction of the jobs are scheduled by a centralized authority (leader) and the remaining jobs are scheduled by selfish users (followers). He modeled this scheduling problem as Stackelberg game and showed that it is NP-hard to compute Stackelberg strategies. The underlying Stackelberg game in this problem is complete information game and there is no privately held information by the players which we require to elicit truthfully. Thus, the problem considered in this paper is about computing the optimal strategies of the leader and the followers and not about designing a mechanism for Stackelberg problem. 1.1 Contributions and Outline of the Paper The major contributions of this paper are as follows. – We investigate the mechanism design problem for SLRF games . In this new framework, we define the notion of Bayesian Stackelberg Incentive Compatible (BaSIC) social choice functions. To the best of our knowledge, this is the first time mechanisms are being investigated in the context of Stackelberg problems. – To illustrate our approach, we investigate the Bayesian Stackelberg incentive compatibility of first-price and second-price procurement auctions with reserve prices. We obtain two key results in this regard. The first result shows that in the first-price auction with reserve prices, the social choice function is BaSIC for the buyer but not for the sellers. The second result shows that in the second-price auction with reserve prices, the social choice function is BaSIC for the sellers but not for the buyer. 3 We keep using the phrases SLRF games and SLRF problems interchangeably. The organization of the paper is as follows. Section 2 presents a crisp review of relevant concepts in Stackelberg games. A more detailed treatment can be found in [1]. In Section 3, we motivate the Stackelberg mechanism design problems by means of two examples first-price procurement auction with reserve prices (F-PAR) and second-price procurement auction with reserve prices (S-PAR). We then describe the problem of designing incentive compatible mechanisms for SLRF problems in Section 4. In Section 5, we state two important results concerning the Bayesian Stackelberg incentive compatibility of the two mechanisms F-PAR and S-PAR. Due to paucity of space, we are unable to provide the proofs for these results. Interested readers are urged to look into our recent technical report [3].