We analyze the computational complexity of market maker pricing algorithms for combinatorial prediction markets. We focus on Hanson's popular logarithmic market scoring rule market maker (LMSR). Our goal is to implicitly maintain correct LMSR prices across an exponentially large outcome space. We examine both permutation combinatorics, where outcomes are permutations of objects, and Boolean combinatorics, where outcomes are combinations of binary events. We look at three restrictive languages that limit what traders can bet on. Even with severely limited languages, we find that LMSR pricing is @math -hard, even when the same language admits polynomial-time matching without the market maker. We then propose an approximation technique for pricing permutation markets based on a recent algorithm for online permutation learning. The connections we draw between LMSR pricing and the vast literature on online learning with expert advice may be of independent interest.
@cite_17 study the computational complexity of finding acceptable trades among a set of bids in a Boolean combinatorial market. In their setting, the center is an who takes no risk, only matching together willing traders. They study a call market setting in which bids are collected together and processed once en masse. They show that the auctioneer matching problem is co-NP-complete when orders are divisible and @math -complete when orders are indivisible, but identify a tractable special case in which participants are restricted to bet on disjunctions of positive events or single negative events.
@cite_17
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We consider a permutation betting scenario, where people wager on the final ordering of n candidates: for example, the outcome of a horse race. We examine the auctioneer problem of risklessly matching up wagers or, equivalently, finding arbitrage opportunities among the proposed wagers. Requiring bidders to explicitly list the orderings that they'd like to bet on is both unnatural and intractable, because the number of orderings is n! and the number of subsets of orderings is 2n!. We propose two expressive betting languages that seem natural for bidders, and examine the computational complexity of the auctioneer problem in each case. Subset betting allows traders to bet either that a candidate will end up ranked among some subset of positions in the final ordering, for example, "horse A will finish in positions 4, 9, or 13-21", or that a position will be taken by some subset of candidates, for example "horse A, B, or D will finish in position 2". For subset betting, we show that the auctioneer problem can be solved in polynomial time if orders are divisible. Pair betting allows traders to bet on whether one candidate will end up ranked higher than another candidate, for example "horse A will beat horse B". We prove that the auctioneer problem becomes NP-hard for pair betting. We identify a sufficient condition for the existence of a pair betting match that can be verified in polynomial time. We also show that a natural greedy algorithm gives a poor approximation for indivisible orders. We study the computational challenge faced by a intermediary who attempts to profit from trade with a small number of buyers and sellers of some item. In the version of the problem that we study, the number of buyers and sellers is constant, but their joint distribution of the item’s value may be complicated. We consider discretized distributions, where the complexity parameter is the support size, or number of different prices that may occur. We show that maximizing the expected revenue is computationally tractable (via an LP) if we are allowed to use randomized mechanisms. For the deterministic case, we show how an optimal mechanism can be efficiently computed for the one-seller one-buyer case, but give a contrasting NP-completeness result for the one-seller two-buyer case. This paper deals with the design of efficiently computable incentive compatible, or truthful, mechanisms for combinatorial optimization problems with multi-parameter agents. We focus on approximation algorithms for NP-hard mechanism design problems. These algorithms need to satisfy certain monotonicity properties to ensure truthfulness. Since most of the known approximation techniques do not fulfill these properties, we study alternative techniques.Our first contribution is a quite general method to transform a pseudopolynomial algorithm into a monotone FPTAS. This can be applied to various problems like, e.g., knapsack, constrained shortest path, or job scheduling with deadlines. For example, the monotone FPTAS for the knapsack problem gives a very efficient, truthful mechanism for single-minded multi-unit auctions. The best previous result for such auctions was a 2-approximation. In addition, we present a monotone PTAS for the generalized assignment problem with any bounded number of parameters per agent.The most efficient way to solve packing integer programs (PIPs) is LP-based randomized rounding, which also is in general not monotone. We show that primal-dual greedy algorithms achieve almost the same approximation ratios for PIPs as randomized rounding. The advantage is that these algorithms are inherently monotone. This way, we can significantly improve the approximation ratios of truthful mechanisms for various fundamental mechanism design problems like single-minded combinatorial auctions (CAs), unsplittable flow routing and multicast routing. Our approximation algorithms can also be used for the winner determination in CAs with general bidders specifying their bids through an oracle. In this article, we study the problem of online market clearing where there is one commodity in the market being bought and sold by multiple buyers and sellers whose bids arrive and expire at different times. The auctioneer is faced with an online clearing problem of deciding which buy and sell bids to match without knowing what bids will arrive in the future. For maximizing profit, we present a (randomized) online algorithm with a competitive ratio of ln(pmax − pmin) p 1, when bids are in a range [pmin, pmax], which we show is the best possible. A simpler algorithm has a ratio twice this, and can be used even if expiration times are not known. For maximizing the number of trades, we present a simple greedy algorithm that achieves a factor of 2 competitive ratio if no money-losing trades are allowed. We also show that if the online algorithm is allowed to subsidize matches---match money-losing pairs if it has already collected enough money from previous pairs to pay for them---then it can actually be 1-competitive with respect to the optimal offline algorithm that is not allowed subsidy. That is, for maximizing the number of trades, the ability to subsidize is at least as valuable as knowing the future. We also consider objectives of maximizing buy or sell volume and social welfare. We present all of these results as corollaries of theorems on online matching in an incomplete interval graph.We also consider the issue of incentive compatibility, and develop a nearly optimal incentive-compatible algorithm for maximizing social welfare. For maximizing profit, we show that no incentive-compatible algorithm can achieve a sublinear competitive ratio, even if only one buy bid and one sell bid are alive at a time. However, we provide an algorithm that, under certain mild assumptions on the bids, performs nearly as well as the best fixed pair of buy and sell prices, a weaker but still natural performance measure. This latter result uses online learning methods, and we also show how such methods can be used to improve our “optimal” algorithms to a broader notion of optimality. Finally, we show how some of our results can be generalized to settings in which the buyers and sellers themselves have online bidding strategies, rather than just each having individual bids.