Preference Representation with Weighted Goals: Expressivity, Succinctness, Complexity

The representation of preferences of agents is a central feature in many AI systems. In particular when the number of alternatives to be considered may become large, the use of compact preference representation languages is crucial. The framework of weighted propositional formulas can be used to define several such languages. The central idea is to associate numerical weights with goals specified in terms of propositional formulas, and to compute the utility value of an alternative as the sum of the weights of the goals it satisfies. In this paper, we analyze several properties of languages defined by weighted goals: their expressivity, the relative succinctness of different sublanguages, and the computational complexity of finding the best alternative with respect to a given utility function expressed in terms of weighted goals. Introduction Many problems in AI require the representation of and reasoning about the preferences of agents. The alternatives over which agents need to express a preference often have a combinatorial structure. This can, for example, be observed in resource allocation with indivisible goods and in combinatorial auctions (Cramton, Shoham, & Steinberg 2006). The combinatorial nature of such problems makes succinct representation of agent preferences important (Coste-Marquis et al. 2004; Lang 2004; Chevaleyre, Endriss, & Lang 2006; Nisan 2006). To see the importance of preference representation languages, consider bidding in a combinatorial auction. A combinatorial auction is one where bidders may place bids on bundles, and not just single goods. In the most general case, each bidder may submit a bid for each subset of the set of items on auction, and the items are distributed according to some criterion, such as maximizing the revenue for the auctioneer. Already, this description presents two problems: First, the number of subsets of goods is exponential in the number of goods, which is a problem for both the bidders and for the auctioneer; and second, the award criterion is a maximization problem, which is a problem for the auctioneer. In the first case, bidders are faced with having to examine each of exponentially many subsets in order to assign ∗This research was supported by a Marie Curie Early Stage Research fellowship from the GloRiClass project (MESST-CT-2005020841). values to them, and, should the bidders succeed at that, the auctioneer must then attempt to use this mountain of data to determine the outcome of the auction. It will be little consolation for the auctioneer to have an efficient winnerdetermination algorithm in this case, as his input will be exponential! Neither expressing nor computing with explicit representations is feasible for any but the smallest sets of items. In the second case, that of the award criterion, we are usually charged with accepting bids so as to maximize some quantity, be it auctioneer revenue, social welfare, or something else. Explicit representations of bids do not lend themselves to efficient computation, for reasons already stated: They are too long. Of course, the story changes radically when we no longer insist on explicit representations of bids (or, more generally, of utility functions). An agent’s total explicit bid, composed of an atomic bid for each subset, may well have structure which we can exploit for devising a shorter representation. A trivial example, but one which illustrates the point, is that of the bidder who gives a value of e1 to every nonempty set of goods. With n goods, an explicit representation of this bid—i.e., one which lists every subset with the value the bidder places on it—will of course have a length on the order of 2, but intuitively we could have a constant-length nonexplicit representation of this bid, as ∀X ⊃ ∅, u(X) = e1 (or, as we just said in words, “the bidder gives a value of e1 to every nonempty set of goods”). The insight here is that bids may have internal structure which we can be exploit in order to produce concise, and therefore more manageable, representations of those bids. The representation given in our example, while concise, is too ad hoc to be generally useful. If we permit any sort of mathematical expression for defining a bid, we may get short representations in most cases, but be faced with such syntactical variety that our computer will be unable to cope with them all. Hence, there is a need for concise representations, not just on their own, but in a well-defined language. Just as the structure of bids makes concise representations possible, the structure of the bidding language makes computation with the bids possible. For a summary of bidding languages, see (Nisan 2006); for a summary of winner determination algorithms, see (Lehmann, Muller, & Sandholm 2006).