TY - GEN

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

AU - Bopardikar, Shaunak D.

AU - Langbort, Cedric

N1 - Publisher Copyright:
© 2014 IEEE.

PY - 2014

Y1 - 2014

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 contribution

AN - SCOPUS:84988221153

T3 - Proceedings of the IEEE Conference on Decision and Control

SP - 1936

EP - 1941

BT - 53rd IEEE Conference on Decision and Control,CDC 2014

PB - Institute of Electrical and Electronics Engineers Inc.

T2 - 2014 53rd IEEE Annual Conference on Decision and Control, CDC 2014

Y2 - 15 December 2014 through 17 December 2014

ER -