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.
Hanson highlights the use of LMSR for Boolean combinatorial markets, noting that the subsidy required to run a combinatorial market on @math outcomes is no greater than that required to run @math independent one-dimensional markets @cite_4 @cite_3 . Hanson discusses the computational difficulty of maintaining LMSR prices on a combinatorial space, and proposes some solutions, including running market makers on overlapping subsets of events, allowing traders to synchronize the markets via arbitrage.
@cite_4 @cite_3
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In a prediction market, agents trade assets whose value is tied to a future event, for example the outcome of the next presidential election. Asset prices determine a probability distribution over the set of possible outcomes. Typically, the outcome space is small, allowing agents to directly trade in each outcome, and allowing a market maker to explicitly update asset prices. Combinatorial markets, in contrast, work to estimate a full joint distribution of dependent observations, in which case the outcome space grows exponentially. In this paper, we consider the problem of pricing combinatorial markets for single-elimination tournaments. With @math competing teams, the outcome space is of size 2n-1. We show that the general pricing problem for tournaments is P-hard. We derive a polynomial-time algorithm for a restricted betting language based on a Bayesian network representation of the probability distribution. The language is fairly natural in the context of tournaments, allowing for example bets of the form "team i wins game k". We believe that our betting language is the first for combinatorial market makers that is both useful and tractable. We briefly discuss a heuristic approximation technique for the general case. Prediction markets are economic mechanisms for aggregating information about future events through sequential interactions with traders. The pricing mechanisms in these markets are known to be related to optimization algorithms in machine learning and through these connections we have some understanding of how equilibrium market prices relate to the beliefs of the traders in a market. However, little is known about rates and guarantees for the convergence of these sequential mechanisms, and two recent papers cite this as an important open question. In this paper we show how some previously studied prediction market trading models can be understood as a natural generalization of randomized coordinate descent which we call randomized subspace descent (RSD). We establish convergence rates for RSD and leverage them to prove rates for the two prediction market models above, answering the open questions. Our results extend beyond standard centralized markets to arbitrary trade networks. In this paper, we study the Nash dynamics of strategic interplays of n buyers in a matching market setup by a seller, the market maker. Taking the standard market equilibrium approach, upon receiving submitted bid vectors from the buyers, the market maker will decide on a price vector to clear the market in such a way that each buyer is allocated an item for which he desires the most (a.k.a., a market equilibrium solution). While such equilibrium outcomes are not unique, the market maker chooses one (maxeq) that optimizes its own objective --- revenue maximization. The buyers in turn change bids to their best interests in order to obtain higher utilities in the next round's market equilibrium solution. This is an (n+1)-person game where buyers place strategic bids to gain the most from the market maker's equilibrium mechanism. The incentives of buyers in deciding their bids and the market maker's choice of using the maxeq mechanism create a wave of Nash dynamics involved in the market. We characterize Nash equilibria in the dynamics in terms of the relationship between maxeq and mineq (i.e., minimum revenue equilibrium), and develop convergence results for Nash dynamics from the maxeq policy to a mineq solution, resulting an outcome equivalent to the truthful VCG mechanism. Our results imply revenue equivalence between maxeq and mineq, and address the question that why short-term revenue maximization is a poor long run strategy, in a deterministic and dynamic setting. We consider a nonlinear extension of the generalized network flow model, with the flow leaving an arc being an increasing concave function of the flow entering it, as proposed by Truemper and Shigeno. We give a polynomial time combinatorial algorithm for solving corresponding flow maximization problems, finding an @math -approximate solution in @math arithmetic operations and value oracle queries, where @math and @math are upper bounds on simple parameters. This also gives a new algorithm for linear generalized flows, an efficient, purely scaling variant of the Fat-Path algorithm by Goldberg, Plot kin and Tardos, not using any cycle cancellations. We show that this general convex programming model serves as a common framework for several market equilibrium problems, including the linear Fisher market model and its various extensions. Our result immediately provides combinatorial algorithms for various extensions of these market models. This includes nonsymmetric Arrow-Debreu Nash bargaining, settling an open question by Vazirani [4].