## Rational Market Making with Probabilistic Knowledge

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@MISC{Othman_rationalmarket,

author = {Abraham Othman and Tuomas Sandholm},

title = {Rational Market Making with Probabilistic Knowledge},

year = {}

}

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### Abstract

A market maker sets prices over time for wagers that pay out contingent on the future state of the world. The market maker has knowledge of the probability of realizing each state of the world, and of how the price of a bet affects the probability that traders will accept it. We compare the optimal policy for risk-neutral (expected utility maximizing) and Kelly criterion (expected log-utility maximizing) market makers. Computing the optimal policy for a risk-neutral market maker is relatively simple, while computing the optimal policy for a Kelly criterion market maker is challenging, requiring advanced techniques adapted from the computational economics literature to run efficiently. We show that while a riskneutral market maker has an optimal policy that does not depend on the market maker’s state, a Kelly criterion market maker’s optimal policy has an intricate dependence on both time and state. Counterintuitively, a Kelly criterion market maker may offer bets that are myopically irrational with respect to the market maker’s beliefs for the entire trading period. In contrast, a risk-neutral market maker never offers a myopically irrational bet.

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(Show Context)
Citation Context ...e (or no bet), and then exit. The traders do not learn from historical prices or strategize about their behavior. Myopic traders (also known as noise traders) are a feature of much of the literature [=-=Glosten and Milgrom, 1985-=-, Kyle, 1985, Othman and Sandholm, 2010]. Empirical studies of market microstructure have shown that the behavior of these agents is qualitatively very similar to behavior observed in real markets wit... |

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Citation Context ...t. The traders do not learn from historical prices or strategize about their behavior. Myopic traders (also known as noise traders) are a feature of much of the literature [Glosten and Milgrom, 1985, =-=Kyle, 1985-=-, Othman and Sandholm, 2010]. Empirical studies of market microstructure have shown that the behavior of these agents is qualitatively very similar to behavior observed in real markets with human trad... |

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Citation Context ...e mean that the approximated function precisely matches the actual function at a set of interpolating points. While shape-preserving interpolation is wellknown in the scientific computing literature [=-=Judd, 1998-=-], this specific technique has been featured rarely. Perhaps the most practical example is Wang and Judd [2000], who study a tax planning problem with stochastic stocks and bonds. Because the theory o... |

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Citation Context ...he value function for an arbitrary time step may have a complex, non-analytic form, we know a great deal about its shape from the properties it inherits from the log utility of the terminating state [=-=Stokey et al., 1989-=-]. In particular: (1) it is increasing in wealth, (2) it is concave, and (3) it goes to minus infinity as the wealth in any state goes to zero. Since these properties are intrinsically linked to the l... |

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Citation Context ...man and Sandholm, 2010]. Empirical studies of market microstructure have shown that the behavior of these agents is qualitatively very similar to behavior observed in real markets with human traders [=-=Gode and Sunder, 1993-=-, Othman, 2008]. However, in some settings the simple behavior of these agents may be an unrealistic model [Chen et al., 2007, Dimitrov and Sami, 2008, Chen et al., 2010]. • The number of trading peri... |

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Citation Context ...that maximizes the deviation from sum of squares at a set of relevant nodes can create aberrant behavior and an approximation that deviates significantly from the actual value function [Gordon, 1995, =-=Guestrin et al., 2001-=-, Stachurski, 2008]. The correct optimization to use to determine the weights is to minimize the sup norm (that is, L∞, rather than L2), which is a significantly more challenging problem to solve nume... |

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Citation Context ...wo prior literatures that deal with the sequential interaction of a market maker with traders: artificial intelligence, and finance. In the AI literature, there are cost function based market makers [=-=Chen and Pennock, 2007-=-, 2010]—agents which price bets so that they are neutral between them being accepted and them being not accepted [Ben-Tal and Teboulle, 2007, Agrawal et al., 2009]. Another closely-related branch of t... |

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Citation Context ...arkets with human traders [Gode and Sunder, 1993, Othman, 2008]. However, in some settings the simple behavior of these agents may be an unrealistic model [Chen et al., 2007, Dimitrov and Sami, 2008, =-=Chen et al., 2010-=-]. • The number of trading periods is drawn independently of the market maker’s policy. Since traders have the ability to decline to place a bet with the market maker if they do not find the offered b... |

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Citation Context ...-related branch of the AI literature involves Bayesian market makers which attempt to learn the correct value of a security by applying Bayes Rule to a series of interactions with traders [Das, 2008, =-=Das and Magdon-Ismail, 2009-=-, Chakraborty et al., 2011]. In contrast to these agents, the market makers we consider here behave rationally in the classical sense: they maximize utility given knowledge of the future and of how th... |

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15 | The effects of market-making on price dynamics
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Citation Context ...her closely-related branch of the AI literature involves Bayesian market makers which attempt to learn the correct value of a security by applying Bayes Rule to a series of interactions with traders [=-=Das, 2008-=-, Das and Magdon-Ismail, 2009, Chakraborty et al., 2011]. In contrast to these agents, the market makers we consider here behave rationally in the classical sense: they maximize utility given knowledg... |

10 |
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Citation Context ...t event in order to guarantee a small but certain profit. In this paper, we compute the policy of a Kelly criterion market maker over a series of interactions with traders. The Kelly criterion [Kelly =-=Jr, 1956-=-] is a way to make bets that mandates maximizing the expected log utility of a setting. While simple as a guiding precept, the Kelly criterion accomplishes a broad range of objectives: over a series o... |

7 | When do markets with simple agents fail
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(Show Context)
Citation Context ...rs do not learn from historical prices or strategize about their behavior. Myopic traders (also known as noise traders) are a feature of much of the literature [Glosten and Milgrom, 1985, Kyle, 1985, =-=Othman and Sandholm, 2010-=-]. Empirical studies of market microstructure have shown that the behavior of these agents is qualitatively very similar to behavior observed in real markets with human traders [Gode and Sunder, 1993,... |

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Citation Context ...erature involves Bayesian market makers which attempt to learn the correct value of a security by applying Bayes Rule to a series of interactions with traders [Das, 2008, Das and Magdon-Ismail, 2009, =-=Chakraborty et al., 2011-=-]. In contrast to these agents, the market makers we consider here behave rationally in the classical sense: they maximize utility given knowledge of the future and of how their prices affect trader a... |

4 | Continuous State Dynamic Programming via Nonexpansive Approximation
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(Show Context)
Citation Context ...ation from sum of squares at a set of relevant nodes can create aberrant behavior and an approximation that deviates significantly from the actual value function [Gordon, 1995, Guestrin et al., 2001, =-=Stachurski, 2008-=-]. The correct optimization to use to determine the weights is to minimize the sup norm (that is, L∞, rather than L2), which is a significantly more challenging problem to solve numerically [Judd, 199... |

2 |
Zero-intelligence agents in prediction markets
- Othman
- 2008
(Show Context)
Citation Context .... Empirical studies of market microstructure have shown that the behavior of these agents is qualitatively very similar to behavior observed in real markets with human traders [Gode and Sunder, 1993, =-=Othman, 2008-=-]. However, in some settings the simple behavior of these agents may be an unrealistic model [Chen et al., 2007, Dimitrov and Sami, 2008, Chen et al., 2010]. • The number of trading periods is drawn i... |