Mechanism design for software agents with complete information☆
Introduction
With the advent of Internet computing and electronic commerce, there has been increasing interest in computational systems, referred to as multiagent systems, that involve the interaction of many different computer programs. These programs, or software agents, may be written by different people or companies with different goals in mind. In other words, the programs can be viewed as self-interested. Not surprisingly, the design and analysis of multiagent systems involves the tools of game theory and mechanism design (see [12], [18], [33], [35] for examples). Developing a clear understanding of the computational issues involved in mechanism design should facilitate its use in multiagent system design. Therefore, in this paper, we consider how placing computational limitations on the agents and the mechanism affects classic results in the mechanism design literature. In particular, we investigate the effect of restricting the agents and the mechanism to polynomial time computation. (Section 3 provides details on exactly how this is done.) To focus our investigation, we consider a particular problem which we call Multiagent MAXSAT and restrict ourselves to complete information environments. In Multiagent MAXSAT, each agent's preferences over the set of possible outcomes can be described by a disjunction over negated and unnegated Boolean variables.
For example, consider a warehouse inhabited by several robots that have different and possibly conflicting goals. (This example is based on an example from Ref. [33].) Each robot is concerned only with satisfying its own goal and does not care whether any of the other robots satisfy their goals. Rather than spending time negotiating with one another when a conflict arises, the robots rely on an outside arbitrator to resolve the conflict quickly and equitably. The arbitrator is referred to as a mechanism. The mechanism's only goal is to have the outcome of its decisions satisfy some measure of social desirability called a social choice rule. If the mechanism is successful, it is said to implement the social choice rule. In this case, let the mechanism's goal be to satisfy as many of the robots as possible. In other words, from the point of view of the mechanism, any outcome that satisfies the maximum number of simultaneously satisfiable robots is a good outcome and any other outcome is a bad outcome. Suppose in this warehouse there are n blocks B1 through Bn and one table. We can describe the state of the world using Boolean variables. Let xi=true represent Bi being on the table and let xi=false represent Bi being on the floor for i=1,…, n. If the robots' goals are restricted to those that can be represented by a disjunction over the Boolean variables and their negations, this is an instance of a multiagent MAXSAT problem.
The problem of assigning truth values to a set of variables so that the number of satisfied disjunctions is maximized is known to be a computationally difficult problem (see Ref. [6]). According to the widely held belief of computer scientists and logicians, namely that P≠NP, it would be impossible for the mechanism to maximize the number of satisfied agents in every instance if the agents and the mechanism are limited to polynomial time computation. Therefore, any polynomial time mechanism must settle for outcomes that are approximately optimal. (Readers unfamiliar with the P vs. NP question should refer to the Appendix A for an explanation.)
The main results of this paper are as follows:
- (1)
The revelation principle states that, if there is a mechanism that implements a social choice rule, then there is a truthful revelation mechanism that implements the social choice rule, i.e., there is a mechanism that asks the agents to declare their preferences and for which truthful declaration is an equilibrium strategy (see Section 4). The revelation principle allows the discussion to be restricted to social choice rules that are implementable by truthful revelation mechanisms. We show that the revelation principle applies when the mechanism and the agents are restricted to polynomial time but does not apply when the mechanism is restricted and the agents are not. This implies that, in the latter situation, we cannot restrict our attention to truthfully implementable social choice rules.
- (2)
We provide a mechanism with a non-dictatorial outcome function that implements MAXSAT in dominant strategies. The mechanism runs in polynomial time but the agents require non-polynomial time to compute their dominant strategies. (Throughout this paper, we assume that P≠NP.) This result is of interest because a classic theorem known as the Gibbard–Sattherwaite theorem states that in many situations, dominant strategy implementation with non-dictatorial outcome functions is impossible. Gibbard–Sattherwaite does not apply to Multiagent MAXSAT.
- (3)
We provide a mechanism such that all dominant strategy equilibrium outcomes satisfy at least half of the agents. In this case, the mechanism and the agents use only polynomial time.
- (4)
We provide a polynomial time mechanism that guarantees that each Nash equilibrium outcome satisfies at least half of the agents. This mechanism is in many ways superior to the mechanism we developed for dominant strategy implementation.
- (5)
We show that in the case of strong implementation in dominant strategy, Nash, undominated Nash or subgame perfect equilibrium, it is impossible to guarantee that the equilibrium outcomes will satisfy more than half of the maximum number of simultaneously satisfiable agents. In contrast, there are approximation algorithms for the non-multiagent version of MAXSAT that guarantee that 3/4 of the maximum number of simultaneously satisfiable agents will be satisfied (see [1], [8], [40]). This result suggests that we will be much less successful using approximation to overcome computational complexity in self-interested multiagent environments than in traditional computational environments.
The Multiagent MAXSAT problem is simplistic in that each agent is limited to preferences defined by simple unweighted disjunctions. However, since most of our results demonstrate the difficulty of designing mechanisms for such a restricted version of the problem, these difficulties will carry over to more realistic models. The environment we study is also somewhat unrealistic since we assume the agents have complete information, i.e., we assume each agent knows every other agent's goal. However, understanding the problems that arise in the complete information environment should help to provide a foundation for future work in incomplete information environments (see [16], [29] for surveys of mechanism design in incomplete information environments).
A computational formulation of the mechanism design problem is presented by Nisan and Ronen in Ref. [23] (see also Ref. [25]). They studied dominant strategy implementation for a task scheduling problem in which there is a set of tasks to be distributed among a group of agents in such a way that the time at which the last task is completed is minimized. The agents, who prefer to do no work at all, have different times in which they can perform each task. These times are unknown to the mechanism. The task scheduling problem is a computationally difficult problem. Therefore, even if the agents truthfully reveal their times to the mechanism, a mechanism restricted to polynomial time computation cannot always find an optimal task distribution. The best the mechanism can hope for is to find an approximately optimal task distribution. An “approximation” mechanism is provided in Ref. [23] such that each agent's unique dominant strategy is to truthfully inform the mechanism of their times. In Ref. [24], it is shown that, under rather mild assumptions of what constitutes a reasonable social rule, dominant strategy implementation of reasonable social choice rules that approximately maximize the sum of the agents' utilities is impossible. Restricted conditions under which such an approximation mechanism can be found are provided in Ref. [38]. Approximation in the context of combinatorial auctions is studied in Ref. [13].
All of the papers mentioned above consider only dominant strategy implementation. A dominant strategy is a strategy that gives the agent his best outcome regardless of what the other agents do. This makes dominant strategy implementation desirable since we can be very confident that agents will play dominant strategies when they can. However, because of the Gibbard–Sattherwaite theorem, dominant strategy implementation in general environments is often impossible (see Section 5). Therefore, the papers above, like much of the work in dominant strategy implementation, restrict the environment to be quasilinear. A quasilinear environment is one in which there is money (or another good) that can be transferred among the agents. An agent's utility function in such an environment is simply the value he places on the outcome plus the amount of money he receives in transfer.
Since there may be problems of interests for which the environments are not quasilinear, understanding the implications of computational limitations for mechanism design in non-quasilinear environments is important. Without the quasilinear assumption, dominant strategy implementation often must be abandoned in favor of other forms of implementation. For complete information environments, the most widely studied of these is Nash implementation. We consider both dominant strategy and Nash implementation for Multiagent MAXSAT. We also briefly consider undominated Nash and subgame perfect implementation.
Another difference between our work and the work cited above is that the latter studies only truthful revelation mechanisms. For these mechanisms, the agents have no computation to perform. They simply pass their preferences unchanged to the mechanism. In our work, the agents' computation time plays a significant role in the results. (In Ref. [24], the agents' computational abilities are considered in trying to overcome their impossibility results.)
This paper can also be considered a contribution to the mechanism design literature on bounded rationality. In Ref. [19], it is pointed out that bounded rationality has received little attention in the mechanism design community. We believe that computer science provides a rich set of tools for modeling bounded rationality. Ideas from computer science have been used previously by game theorists and computer scientists studying bounded rationality in repeated games, see [3], [21], [22], [27], [31], [37].
Section 2 provides a short review of mechanism design and formally defines the Multiagent MAXSAT problem as a computational problem. Section 3 formalizes the idea of a polynomial time mechanism. Section 4 discusses polynomial time revelation mechanisms. 5 Dominant strategy implementation, 6 Nash implementation look at designing polynomial time mechanisms using dominant strategy and Nash equilibrium, respectively, for Multiagent MAXSAT. Section 7 provides a proof that the best a mechanism can guarantee is that half of the maximum number of simultaneously satisfiable agents will be satisfied. Since some readers may not be familiar with the relevant complexity concepts from computer science, we have included an appendix that briefly explains these concepts.
Section snippets
Motivation and definitions
The mechanism design problem is intended to model situations in which a set of self-interested agents must come to a collective decision. The designer of the decision making process would like the decision to be good for the society as a whole as defined by a social choice rule. The way the collective decision is made is that each agent sends a message to a decision making procedure. This decision making procedure then selects an outcome based on the messages sent by the agents. The intent of
Polynomial time mechanisms
An algorithm is said to run in polynomial time if there is a polynomial q such that, for every possible input, the algorithm produces an output in no more than q(n) primitive computational steps where n is the size of the input. (For more details on this concept, see [5], [6], [9].) There are two computational processes to consider for Multiagent MAXSAT.
- (1)
The mechanism must compute the outcome function which, as described in the previous section, takes the description of the problem instance and
Revelation mechanisms
Mechanisms which require the agents to declare their preferences (truthfully or falsely) form an important class of mechanisms known as revelation mechanisms. A social choice rule F is said to be truthfully implementable if there is a revelation mechanism for which truthful preference declarations by all the agents constitutes an equilibrium with outcome in F(p, θ) for all problem instances p and all preference profiles θ. The following result is known as the Revelation Principle. This
Dominant strategy implementation
When dealing with dominant strategy implementation, one must contend with an impossibility result known as the Gibbard–Satterthwaite Theorem [7], [36] which restricts the set of implementable social choice functions to those that are dictatorial. (A social choice function maps a problem instance and a preference profile to a single outcome.) Definition 5.1 A social choice function f is dictatorial for problem instance p if there is a single agent i such that, for all preference profiles θ, f(p, θ) is agent i's
Nash implementation
We have seen that (1/2)-MAXSAT is strongly implementable in dominant strategies using the Complement Mechanism. However, the Complement Mechanism is not a particularly attractive mechanism since, for a problem instance p, the outcome is always either tp or t̄p for a fixed tp. An unfortunate choice of tp eliminates many outcomes that would be more desirable from a social viewpoint. For example, consider a problem instance with four agents and two variables x1 and x2. Let tp=x1x2, i.e., tp sets x1
Upper bounds on approximability
Existing work in mechanism design shows that it is possible to implement a wider range of social choice rules in undominated Nash equilibrium4 [10], [30] and subgame perfect equilibrium5 [20] than in Nash equilibrium. In
Conclusion
Using a multiagent version of MAXSAT, we have investigated the difficulties that arise in applying classic results from the mechanism design literature to computationally complex optimization problems. Table 1 summarizes the results presented in this paper.
We have demonstrated that, despite the impossibility results regarding dominant strategy implementation, it is possible to implement an approximate social choice rule for Multiagent MAXSAT in dominant strategy equilibrium. Our results suggest
Acknowledgements
We would like to thank Larry Kranich for the many helpful comments and suggestions.
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See Ref. [26] for a preliminary version of this paper. Material from that paper is used with the permission of Kluwer Academic Publishers.
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Supported in part by National Science Foundation Grant CCR-97-34936.