Basins of Attraction in Two-Player Random Ordinal Potential Games

Abstract

We consider the class of two-person ordinal potential games where each player has the same number of actions $K$. Each game in this class admits at least one pure Nash equilibrium and the best-response dynamics converges to one of these pure Nash equilibria; which one depends on the starting point. So, each pure Nash equilibrium has a basin of attraction. We pick uniformly at random one game from this class and we study the joint distribution of the sizes of the basins of attraction. We provide an asymptotic exact value for the expected basin of attraction of each pure Nash equilibrium, when the number of actions $K$ goes to infinity.

Figures (2)

• Figure 1: Plot of the functions $\varphi(\,\cdot\,)$ (left) and $\Phi(\,\cdot\,)$ (right).
• Figure 2: In this example $\varepsilon K=4$, $\tau_{\varepsilon K}=6$, $R_{\tau_{\varepsilon K}}=4$ and $C_{\tau_{\varepsilon K}}=5$. Green dots represent PNE. When the BRD reaches an action profile on a green dotted line, it reaches the PNE on the same line. If the BRD reaches an action profile on a blue dotted line, then it reaches the $\varepsilon K$-th equilibrium. We start the process $\mathop{\mathrm{\mathsf{BRD}}}\nolimits(\,\cdot\,)$ at $\boldsymbol{x}_{0}=(R_{\tau_{\varepsilon K}}+1,C_{\tau_{\varepsilon K}}+1)=(5,6)$; the profiles $(\boldsymbol{x}_{0},\boldsymbol{x}_{1},\dots,\boldsymbol{x}_{6})$ represent the trajectory of the BRD. The best-response dynamics reaches the row of the $\varepsilon K$-th equilibrium in $6$ steps. Hence it reaches the $\varepsilon K$-th equilibrium in $7$ steps. Strategy profiles on red dotted lines are explored by the best-response dynamics starting at $\boldsymbol{x}_{0}$.

Theorems & Definitions (24)

• Theorem 3.1: MimQuaSca:GEB2024
• Remark 4.1
• Remark 4.2
• Example 4.3
• Theorem 4.4
• Corollary 4.5
• Theorem 4.6
• Remark 5.1
• Definition 6.1
• Proposition 6.2