### Abstract

We consider the problem of approximately and efficiently computing saddle-point values for zero-sum matrix games. This problem arises in scenarios where the game's exact value is hard to compute, either because the columns of the matrix are revealed incrementally in time, or because the game's strategy space is too large for traditional methods (e.g., linear programming) to be effective in practice. We lever-age the established adaptive multiplicative weights algorithm but introduce a novel simple criterion to determine whether the minimizer's best strategy needs to be approximately re-computed as a new column of the matrix is introduced. Our main results are two-fold. First, we show that our proposed incremental approach achieves the same accuracy as applying the adaptive multiplicative weights algorithm on the entire matrix, if known a priori. Secondly, we argue that our approach can be computationally more efficient than simply re-computing the minimizer's best strategy upon addition of every new column of the matrix. Specifically, for the case when the columns of the matrix are generated independently and from the same distribution, we characterize the probability that the expected number of times the best response is re-computed exceeds a given fraction of the total number of columns in the matrix. Numerical simulations indicate even more significant computational improvement as compared to the analytic result.

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

Article number | 7039681 |

Pages (from-to) | 1936-1941 |

Number of pages | 6 |

Journal | Proceedings of the IEEE Conference on Decision and Control |

Volume | 2015-February |

Issue number | February |

DOIs | |

State | Published - Jan 1 2014 |

Event | 2014 53rd IEEE Annual Conference on Decision and Control, CDC 2014 - Los Angeles, United States Duration: Dec 15 2014 → Dec 17 2014 |

### Fingerprint

### Keywords

- Computational Methods
- Game theory
- Matrix Games
- Probabilistic Methods

### ASJC Scopus subject areas

- Control and Systems Engineering
- Modeling and Simulation
- Control and Optimization

### Cite this

*Proceedings of the IEEE Conference on Decision and Control*,

*2015-February*(February), 1936-1941. [7039681]. https://doi.org/10.1109/CDC.2014.7039681

**Incremental approximate saddle-point computation in zero-sum matrix games.** / Bopardikar, Shaunak D.; Langbort, Cedric.

Research output: Contribution to journal › Conference article

*Proceedings of the IEEE Conference on Decision and Control*, vol. 2015-February, no. February, 7039681, pp. 1936-1941. https://doi.org/10.1109/CDC.2014.7039681

}

TY - JOUR

T1 - Incremental approximate saddle-point computation in zero-sum matrix games

AU - Bopardikar, Shaunak D.

AU - Langbort, Cedric

PY - 2014/1/1

Y1 - 2014/1/1

N2 - We consider the problem of approximately and efficiently computing saddle-point values for zero-sum matrix games. This problem arises in scenarios where the game's exact value is hard to compute, either because the columns of the matrix are revealed incrementally in time, or because the game's strategy space is too large for traditional methods (e.g., linear programming) to be effective in practice. We lever-age the established adaptive multiplicative weights algorithm but introduce a novel simple criterion to determine whether the minimizer's best strategy needs to be approximately re-computed as a new column of the matrix is introduced. Our main results are two-fold. First, we show that our proposed incremental approach achieves the same accuracy as applying the adaptive multiplicative weights algorithm on the entire matrix, if known a priori. Secondly, we argue that our approach can be computationally more efficient than simply re-computing the minimizer's best strategy upon addition of every new column of the matrix. Specifically, for the case when the columns of the matrix are generated independently and from the same distribution, we characterize the probability that the expected number of times the best response is re-computed exceeds a given fraction of the total number of columns in the matrix. Numerical simulations indicate even more significant computational improvement as compared to the analytic result.

AB - We consider the problem of approximately and efficiently computing saddle-point values for zero-sum matrix games. This problem arises in scenarios where the game's exact value is hard to compute, either because the columns of the matrix are revealed incrementally in time, or because the game's strategy space is too large for traditional methods (e.g., linear programming) to be effective in practice. We lever-age the established adaptive multiplicative weights algorithm but introduce a novel simple criterion to determine whether the minimizer's best strategy needs to be approximately re-computed as a new column of the matrix is introduced. Our main results are two-fold. First, we show that our proposed incremental approach achieves the same accuracy as applying the adaptive multiplicative weights algorithm on the entire matrix, if known a priori. Secondly, we argue that our approach can be computationally more efficient than simply re-computing the minimizer's best strategy upon addition of every new column of the matrix. Specifically, for the case when the columns of the matrix are generated independently and from the same distribution, we characterize the probability that the expected number of times the best response is re-computed exceeds a given fraction of the total number of columns in the matrix. Numerical simulations indicate even more significant computational improvement as compared to the analytic result.

KW - Computational Methods

KW - Game theory

KW - Matrix Games

KW - Probabilistic Methods

UR - http://www.scopus.com/inward/record.url?scp=84988221153&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84988221153&partnerID=8YFLogxK

U2 - 10.1109/CDC.2014.7039681

DO - 10.1109/CDC.2014.7039681

M3 - Conference article

AN - SCOPUS:84988221153

VL - 2015-February

SP - 1936

EP - 1941

JO - Proceedings of the IEEE Conference on Decision and Control

JF - Proceedings of the IEEE Conference on Decision and Control

SN - 0191-2216

IS - February

M1 - 7039681

ER -