Winner determination for mixed multi-unit combinatorial auctions via petri nets

Abstract

Mixed Multi-Unit Combinatorial Auctions (MMUCAs) allow agents to bid for bundles of goods to buy, goods to sell, and transformations of goods. In particular, MMUCAs offer a high potential to be employed for the automated assembly of supply chains of agents offering goods and services, and in general MMUCAs extend and generalise several types of combinatorial auctions. Here we provide a formalism, based on an extension of Petri Nets, with which MMUCAs, and therefore all auction types subsumed by MMUCAs -and in particular combinatorial auctions for supply chain formation (SCF)-, can be formally analysed. As a second direct benefit, consequence of the provided mapping to Petri Nets, we manage to dramatically reduce the number of decision variables involved in the optimisation problem posed by MMUCAs from quadratic to linear for a wide class of MMUCA Winner Determination Problems (WDPs). Hence, we also make headway in the practical application of MMUCAs, and in particular to SCF.

Empirical Hardness for Mixed Auctions

“…However, there are major differences between CAs and MMUCAs that lead to the list of features in table 2. First of all, as formerly argued in [5] and [7], the topological features of the search space handled by the WDP matter. Indeed, while in [5] Giovannucci et al observed that the order of the number of variables required by CCIP, the IP, is directly related to the size of the largest strongly connected component (SCC), in [7] Vinyals et al empirically observed that large SCCs (cycles) lead to high solving times.…”

“…Notice that hereafter we limit to outlining the intuitions underlying CCIP. We refer the reader to [5] for a detailed IP formulation.…”

“…As discussed in section 2, MMUCA WDPs can be solved via integer linear programming (ILP). We select an ILOG CPLEX implementation of the integer program outlined in section 2 (fully described in [5]), because it is the fastest algorithm reported in the literature, largely outperforming the solver in [1].…”

“…In order to generate instance distributions we resort to the artificial data set generator for MMUCA WDPs introduced in [7]. The algorithm takes inspiration on the structure of supply chains, whose formation has been identified as a very promising application domain for MMUCA [5,1]. A supply chain is composed of levels (or tiers).…”

“…In section 2 we outline an integer program introduced in [5] to efficiently solve the WDP for MMUCAs. In section 3 we outline the methodology introduced in [4] to subsequently employ it to build a model that predicts the time required by the integer program to solve a WDP instance.…”