Structural relations of symmetry among players in strategic games

ABSTRACT

The notions of symmetry and anonymity in strategic games have been formalized in different ways in the literature. We propose a combinatorial framework to analyze these notions, using group actions. Then, the same framework is used to define partial symmetries in payoff matrices. With this purpose, we introduce the notion of the role a player plays with respect to another one, and combinatorial relations between roles are studied. Building on them, we define relations directly between players, which provide yet another characterization of structural symmetries in the payoff matrices of strategic games.

KEYWORDS:

• Symmetry
• anonymity
• strategic games
• combinatorics
• group action

Acknowledgments

We thank Alfredo Álzaga and Rodrigo Iglesias for fruitful discussion on these topics.

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

Fernando A. Tohmé http://orcid.org/0000-0003-2988-4519

Ignacio D. Viglizzo http://orcid.org/0000-0002-5303-623X

Notes

1 Similar, equivalent notation and results can be found in Stein (2011) and Ham (2018).

2 Nash (1951) also defines symmetric games in terms of permutations of actions, which in turn lead to permutations of the name of players. Depending on how one interprets his notation, this may lead either to Definition 2.3 or to our characterization (Definition 3.1) below.

3 This is equivalent (via curryfication) to saying that $u_{\cdot }^i:A\to {\mathbb{R}}^{P_{n-1}}$, given that $u_a^i:P_{n-1}\to \mathbb{R}$.

4 In Bouyer, Markey, and Vester (2017) this characterization of symmetry is presented as a fix to Definition 2.3. Hefti (2017) uses an alternative characterization: ${\pi }_i(a_1,\dots ,a_n)={\pi }_{\sigma \left(i\right)}(a_{{\sigma }^{-1}\left(1\right)},\dots ,a_{{\sigma }^{-1}\left(n\right)})$.

5 Alternatively, we can say that the role $r_i^j$ is invariant under the subgroup of permutations that leave i and j fixed.

Additional information

Notes on contributors

Fernando A. Tohmé

Fernando Tohmé is a principal researcher of CONICET (National Research Council of Argentina) and full professor at the Department of Economics of the Universidad Nacional del Sur, in Baha Blanca, Argentina. He holds an undergraduate degree in Mathematics and a PhD in Economics (under the late Rolf Mantel). A former Fulbright Scholar he held visiting positions at U.C. Berkeley, Washington University in St. Louis and Endicott College. His research has focused on decision problems, game theory and optimization in socio-economic settings. He is currently working on a category-theoretical reformulation of game theory. Tohm has been the thesis advisor of more than fifteen doctoral students in Economics, Engineering and Mathematics. He has published research articles in peer-reviewed journals like Theory and Decision, Mathematics of Social Sciences, Journal of Applied Logic, Physica A, Artificial Intelligence, Mathematical and Computational Modeling, and Annals of Operations Research among others.

Ignacio D. Viglizzo

Ignacio Viglizzo is a researcher of CONICET (National Research Council of Argentina) and full professor at the Department of Mathematics of the Universidad Nacional del Sur, in Baha Blanca, Argentina. He holds a Ph.D. in Mathematics from Indiana University. His research interests include mathematical logic and game theory. He has published research articles in peer-reviewed journals like Information and Computation, Studia Logica, Journal of Applied Logic, Annals of Mathematics and Artificial Intelligence and Order, among others.