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Pricing combinatorial markets for tournaments
 In Proc. of STOC
, 2008
"... 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 ..."
Abstract

Cited by 20 (15 self)
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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 singleelimination tournaments. With n competing teams, the outcome space is of size 2 n−1. We show that the general pricing problem for tournaments is #Phard. We derive a polynomialtime 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.
Gaming Dynamic Parimutuel Markets
"... Abstract. We study the strategic behavior of riskneutral nonmyopic agents in Dynamic Parimutuel Markets (DPM). In a DPM, agents buy or sell shares of contracts, whose future payoff in a particular state depends on aggregated trades of all agents. A forwardlooking agent hence takes into considerat ..."
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Abstract. We study the strategic behavior of riskneutral nonmyopic agents in Dynamic Parimutuel Markets (DPM). In a DPM, agents buy or sell shares of contracts, whose future payoff in a particular state depends on aggregated trades of all agents. A forwardlooking agent hence takes into consideration of possible future trades of other agents when making its trading decision. In this paper, we analyze nonmyopic strategies in a twooutcome DPM under a simple model of incomplete information and examine whether an agent will truthfully reveal its information in the market. Specifically, we first characterize a single agent’s optimal trading strategy given the payoff uncertainty. Then, we use a twoplayer game to examine whether an agent will truthfully reveal its information when it only participates in the market once. We prove that truthful betting is a Nash equilibrium of the twostage game in our simple setting for uniform initial market probabilities. However, we show that there exists some initial market probabilities at which the first player has incentives to mislead the other agent in the twostage game. Finally, we briefly discuss when an agent can participate more than once in the market whether it will truthfully reveal its information at its first play in a threestage game. We find that in some occasions truthful betting is not a Nash equilibrium of the threestage game even for uniform initial market probabilities. 1