## Bulletin of the Belgian Mathematical Society - Simon Stevin

- Bull. Belg. Math. Soc. Simon Stevin
- Volume 16, Number 1 (2009), 67-89.

### Optimal Strategies for Symmetric Matrix Games with Partitions

Bernard De Baets, Hans De Meyer, and Bart De Schuymer

#### Abstract

We introduce three variants of a symmetric matrix game corresponding to three ways of comparing two partitions of a fixed integer ($\sigma$) into a fixed number ($n$) of parts. In the random variable interpretation of the game, each variant depends on the choice of a copula that binds the marginal uniform cumulative distribution functions (cdf) into the bivariate cdf. The three copulas considered are the product copula $T_{\bf P}$ and the two extreme copulas, i.e. the minimum copula $T_{\bf M}$ and the Łukasiewicz copula $T_{\bf L}$. The associated games are denoted as the $(n,\sigma)_{\bf P}$, $(n,\sigma)_{\bf M}$ and $(n,\sigma)_{\bf L}$ games. In the present paper, we characterize the optimal strategies of the $(n,\sigma)_{\bf M}$ and $(n,\sigma)_{\bf L}$ games and compare them to the optimal strategies of the $(n,\sigma)_{\bf P}$ games. It turns out that the characterization of the optimal strategies is completely different for each game variant.

#### Article information

**Source**

Bull. Belg. Math. Soc. Simon Stevin, Volume 16, Number 1 (2009), 67-89.

**Dates**

First available in Project Euclid: 25 February 2009

**Permanent link to this document**

https://projecteuclid.org/euclid.bbms/1235574193

**Digital Object Identifier**

doi:10.36045/bbms/1235574193

**Mathematical Reviews number (MathSciNet)**

MR2498960

**Zentralblatt MATH identifier**

1159.91005

**Keywords**

Matrix game Optimal strategy Partition theory Copula Probabilistic relation

#### Citation

De Schuymer, Bart; De Meyer, Hans; De Baets, Bernard. Optimal Strategies for Symmetric Matrix Games with Partitions. Bull. Belg. Math. Soc. Simon Stevin 16 (2009), no. 1, 67--89. doi:10.36045/bbms/1235574193. https://projecteuclid.org/euclid.bbms/1235574193