### Abstract

A one-sided limit order book is modeled as a noncooperative game for several players. Agents offer various quantities of an asset at different prices, competing to fulfill an incoming order, whose size is not known a priori. Players can have different payoff functions, reflecting different beliefs about the fundamental value of the asset and probability distribution of the random incoming order. In a previous paper, the existence of a Nash equilibrium was established by means of a fixed point argument. The main issue discussed in the present paper is whether this equilibrium can be obtained from the unique solution to a two-point boundary value problem, for a suitable system of discontinuous ordinary differential equations. Some additional assumptions are introduced, which yield a positive answer. In particular, this is true when there are exactly two players, or when all players assign the same exponential probability distribution to the incoming order. In both of these cases, we also prove that the Nash equilibrium is unique. A counterexample shows that these assumptions cannot be removed, in general.

Original language | English (US) |
---|---|

Pages (from-to) | 1018-1048 |

Number of pages | 31 |

Journal | Journal of Optimization Theory and Applications |

Volume | 163 |

Issue number | 3 |

DOIs | |

State | Published - Oct 25 2014 |

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### All Science Journal Classification (ASJC) codes

- Control and Optimization
- Management Science and Operations Research
- Applied Mathematics

### Cite this

*Journal of Optimization Theory and Applications*,

*163*(3), 1018-1048. https://doi.org/10.1007/s10957-014-0551-5

}

*Journal of Optimization Theory and Applications*, vol. 163, no. 3, pp. 1018-1048. https://doi.org/10.1007/s10957-014-0551-5

**A Bidding Game with Heterogeneous Players.** / Bressan, Alberto; Wei, Deling.

Research output: Contribution to journal › Article

TY - JOUR

T1 - A Bidding Game with Heterogeneous Players

AU - Bressan, Alberto

AU - Wei, Deling

PY - 2014/10/25

Y1 - 2014/10/25

N2 - A one-sided limit order book is modeled as a noncooperative game for several players. Agents offer various quantities of an asset at different prices, competing to fulfill an incoming order, whose size is not known a priori. Players can have different payoff functions, reflecting different beliefs about the fundamental value of the asset and probability distribution of the random incoming order. In a previous paper, the existence of a Nash equilibrium was established by means of a fixed point argument. The main issue discussed in the present paper is whether this equilibrium can be obtained from the unique solution to a two-point boundary value problem, for a suitable system of discontinuous ordinary differential equations. Some additional assumptions are introduced, which yield a positive answer. In particular, this is true when there are exactly two players, or when all players assign the same exponential probability distribution to the incoming order. In both of these cases, we also prove that the Nash equilibrium is unique. A counterexample shows that these assumptions cannot be removed, in general.

AB - A one-sided limit order book is modeled as a noncooperative game for several players. Agents offer various quantities of an asset at different prices, competing to fulfill an incoming order, whose size is not known a priori. Players can have different payoff functions, reflecting different beliefs about the fundamental value of the asset and probability distribution of the random incoming order. In a previous paper, the existence of a Nash equilibrium was established by means of a fixed point argument. The main issue discussed in the present paper is whether this equilibrium can be obtained from the unique solution to a two-point boundary value problem, for a suitable system of discontinuous ordinary differential equations. Some additional assumptions are introduced, which yield a positive answer. In particular, this is true when there are exactly two players, or when all players assign the same exponential probability distribution to the incoming order. In both of these cases, we also prove that the Nash equilibrium is unique. A counterexample shows that these assumptions cannot be removed, in general.

UR - http://www.scopus.com/inward/record.url?scp=84912045168&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84912045168&partnerID=8YFLogxK

U2 - 10.1007/s10957-014-0551-5

DO - 10.1007/s10957-014-0551-5

M3 - Article

AN - SCOPUS:84912045168

VL - 163

SP - 1018

EP - 1048

JO - Journal of Optimization Theory and Applications

JF - Journal of Optimization Theory and Applications

SN - 0022-3239

IS - 3

ER -