Abstract
We consider the problem of multi-robot task allocation using auctions, and study how lossy communication between the auctioneer and bidders affects solution quality. We demonstrate both analytically and experimentally that even though many auction algorithms have similar performance when communication is perfect, different auctions degrade in different ways as communication quality decreases from perfect to nonexistent. Thus, if a multi-robot system is expected to encounter lossy communication, then the auction algorithm that it uses for task allocation must be chosen carefully. We compare six auction algorithms including: standard implementations of the Sequential Auction, Parallel Auction, Combinatorial Auction; a generalization of the Prim Allocation Auction called G-Prim; and two multi-round variants of a Repeated Parallel Auction. Variants of these auctions are also considered in which award information from previous rounds is rebroadcast by the auctioneer during later rounds. We consider a variety of valuation functions used by the bidders, including: the total and maximum distance traveled (for distance based cost functions), the expected profit or cost to a robot (assuming robots’ task values are drawn from a random distribution). Different auctioneer objectives are also evaluated, and include: maximizing profit (max sum), minimizing cost (min sum), and minimizing the maximum distance traveled by any particular robot (min max). In addition to the cost value functions that are used, we are also interested in fleet performance statistics such as the expected robot utilization rate, and the expected number of items won by each robot. Experiments are performed both in simulation and on real AscTec Pelican quad-rotor aircraft. In simulation, each algorithm is considered across communication qualities ranging from perfect to nonexistent. For the case of the distance-based cost functions, the performance of the auctions is compared using two different communication models: (1) a Bernoulli model and (2) the Gilbert–Elliot model. The particular auction that performs the best changes based on the the reliability of the communication between the bidders and the auctioneer. We find that G-Prim and its repeated variant perform relatively well when communication is poor, and that re-sending winner data in later rounds is an easy way improve the performance of multi-round auctions, in general.
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Notes
Prim Allocation was originally designed to use a specific cost function that is based on a bounded approximation to the multi-agent version of the traveling salesperson problem (Lagoudakis et al. 2004).
Rising price auctions are also known as English Auctions.
Second price sealed-bid auctions are also know as a Vickrey auctions. In this auction the highest bidder wins, but pays the second-highest bid price; essentially outbidding the second highest bidder by \({\epsilon \rightarrow 0}\).
Item valuations are not independent when there are dependencies between item valuations such that there are or extra costs or values associated with owning different subsets of items.
Using the same \(f_X(x)\) for all agent-item pairs is only a requirement for the G-Prim Auction and the Repeated G-Prim Auction. For the Parallel Auction, Sequential Auction, and Repeated Parallel Auction this assumption can be relaxed such that each item j is associated with its own \(f_{X,j}(x)\) that is shared by all agents.
Prim Allocation (which uses a multi-TSP variant of Christofides TSP approximation algorithm) estimates multi-TSP length by building an incremental spanning forest from agents’ locations. A post-processing step is used to obtain a multi-TSP approximation from the spanning forest. This heuristic yields a solution no worse than 1.5 times the optimal length and is calculated in polynomial time. While the true TSP-based solution we consider provides a closer estimate of the optimal multi-TSP path, its runtime is super-polynomial with respect to item number.
If the environment is cluttered with static obstacles then Christofides TSP approximation requires a preprocessing step to find the minimum length path between different task locations. Static obstacles are beyond the scope of this work.
This statement makes the implicit assumption locations are initially chosen by a random process that would sample the environment densely, in the limit, if the number of locations were allowed to go to infinity. The “almost surely” refers to the fact that, given randomly chosen locations, the chances a new location lies along the old multi-TSP (in which case the multi-TSP length remains the same) is zero.
Note that, this assumes item locations are initially chosen by a random process that would sample the environment densely, in the limit, if the number of locations were allowed to go to infinity.
Indeed, the original Prim Allocation algorithm leveraged the fact that this approximation method was used for valuations.
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This work was performed at the Naval Research Laboratory and was funded by the Office of Naval Research under Grant Numbers N0001416WX01271 and N0001416WX01272. The views, positions and conclusions expressed herein reflect only the authors’ opinions and expressly do not reflect those of the Office of Naval Research, nor those of the Naval Research Laboratory.
This is one of the several papers published in Autonomous Robots comprising the Special Issue on Multi-Robot and Multi-Agent Systems.
Appendices
A Iterative calculations for Repeated Parallel Auction with re-sends in Scenario 1
In this section we present the iterative calculation of quantities for the Repeated Parallel Auction with re-sends in Scenario 1, assuming a Bernoulli communication model. Intermediate quantities are denoted with capital letters, and variations of them.
B Additional auction performance curves for Scenario 1
In this section we present additional figures from the first series of experiments. Each figure in this section pertains to a single auction, and shows how agent utilization changes over various swarm sizes and item counts (Figs. 16, 17, 18, 19, 20, 21, 22, 23, 24).
Parallel Auction agent utilization performance curves. The average number of items each agent visits over various communication qualities in Scenario 1 (in which bids are realizations of random variables), assuming a Bernoulli communication model. Items are visited twice if an agent receives an award message from the auctioneer but fails to send an acknowledgment message back to the auctioneer (and so the auctioneer also visits the item). In the special case that no symbiosis exist between items (which happens in Scenario 1) the performance of the Parallel Auction and the Combinatorial Auction are identical
Sequential Auction agent utilization performance curves. The average number of items each agent visits over various communication qualities in Scenario 1 (in which bids are realizations of random variables), assuming a Bernoulli communication model. Items are visited twice if an agent receives an award message from the auctioneer but fails to send an acknowledgment message back to the auctioneer (and so the auctioneer also visits the item)
G-Prim Auction agent utilization performance curves. The average number of items each agent visits over various communication qualities in Scenario 1 (in which bids are realizations of random variables), assuming a Bernoulli communication model. Items are visited twice if an agent receives an award message from the auctioneer but fails to send an acknowledgment message back to the auctioneer (and so the auctioneer also visits the item)
Repeated Parallel Auction agent utilization performance curves. The average number of items each agent visits over various communication qualities in Scenario 1 (in which bids are realizations of random variables), assuming a Bernoulli communication model. Items are visited twice if an agent receives an award message from the auctioneer but fails to send an acknowledgment message back to the auctioneer (and so the auctioneer also visits the item)
Repeated G-Prim Auction agent utilization performance curves. The average number of items each agent visits over various communication qualities in Scenario 1 (in which bids are realizations of random variables), assuming a Bernoulli communication model. Items are visited twice if an agent receives an award message from the auctioneer but fails to send an acknowledgment message back to the auctioneer (and so the auctioneer also visits the item)
Sequential Auction With Winner Rebroadcasts agent utilization performance curves. The average number of items each agent visits over various communication qualities in Scenario 1 (in which bids are realizations of random variables), assuming a Bernoulli communication model. Items are visited twice if an agent receives an award message from the auctioneer but fails to send an acknowledgment message back to the auctioneer (and so the auctioneer also visits the item)
G-Prim Auction With Winner Rebroadcasts agent utilization performance curves. the average number of items each agent visits over various communication qualities in Scenario 1 (in which bids are realizations of random variables), assuming a Bernoulli communication model. Items are visited twice if an agent receives an award message from the auctioneer but fails to send an acknowledgment message back to the auctioneer (and so the auctioneer also visits the item)
Repeated Parallel Auction With Winner Rebroadcasts agent utilization performance curves. the average number of items each agent visits over various communication qualities in Scenario 1 (in which bids are realizations of random variables), assuming a Bernoulli communication model. Items are visited twice if an agent receives an award message from the auctioneer but fails to send an acknowledgment message back to the auctioneer (and so the auctioneer also visits the item)
Repeated G-Prim Auction With Winner Rebroadcasts agent utilization performance curves. The average number of items each agent visits over various communication qualities in Scenario 1 (in which bids are realizations of random variables), assuming a Bernoulli communication model. Items are visited twice if an agent receives an award message from the auctioneer but fails to send an acknowledgment message back to the auctioneer (and so the auctioneer also visits the item)
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Otte, M., Kuhlman, M.J. & Sofge, D. Auctions for multi-robot task allocation in communication limited environments. Auton Robot 44, 547–584 (2020). https://doi.org/10.1007/s10514-019-09828-5
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DOI: https://doi.org/10.1007/s10514-019-09828-5