[Paper Review] Strategic Decompositions of Normal Form Games: Zero-sum Games and Potential Games

Sung‐Ha Hwang, Luc Rey-Bellet|arXiv (Cornell University)|Feb 22, 2016

Game Theory and Applications39 references3 citations

TL;DR

This paper introduces zero-sum equivalent games and zero-sum equivalent potential games—classes of normal-form games strategically equivalent to zero-sum or potential games, respectively. It proves a unique decomposition of any normal-form game into three components: a zero-sum normalized game, a zero-sum equivalent potential game, and an identical interest normalized game, each with distinct equilibrium properties, enabling systematic equilibrium analysis via decomposition.

ABSTRACT

We study new classes of games, called zero-sum equivalent games and zero-sum equivalent potential games, and prove decomposition theorems involving these classes of games. We say that two games are "strategically equivalent" if, for every player, the payoff differences between two strategies (holding other players' strategies fixed) are identical. A zero-sum equivalent game is a game that is strategically equivalent to a zero-sum game; a zero-sum equivalent potential game is a zero-sum equivalent game that is strategically equivalent to a common interest game. We also call a game "normalized" if the sum of one player's payoffs, given the other players' strategies, is always zero. We show that any normal form game can be uniquely decomposed into either (i) a zero-sum equivalent game and a normalized common interest game, or (ii) a zero-sum equivalent potential game, a normalized zero-sum game, and a normalized common interest game, each with distinctive equilibrium properties. For example, we show that two-player zero-sum equivalent games with finite strategy sets generically have a unique Nash equilibrium and that two-player zero-sum equivalent potential games with finite strategy sets generically have a strictly dominant Nash equilibrium.

Motivation & Objective

• To develop a systematic framework for decomposing normal-form games into structurally distinct components based on strategic equivalence to zero-sum and potential games.
• To define and analyze new game classes: zero-sum equivalent games and zero-sum equivalent potential games, which preserve key equilibrium properties of their reference classes.
• To establish a unique decomposition of any normal-form game (finite or continuous strategy sets) into three orthogonal components with distinct equilibrium characteristics.
• To apply the decomposition to analyze Nash equilibria in two-player finite games and contest games, demonstrating its utility in proving uniqueness and characterizing equilibrium behavior.
• To provide a theoretical foundation for isolating strategic effects—such as conflict, coordination, and identical interest—through component games in equilibrium analysis.

Proposed method

• Define strategic equivalence: two games are strategically equivalent if payoff differences between strategies (holding others fixed) are identical for all players.
• Introduce 'normalized' games where the sum of one player’s payoffs over their strategies is zero, enabling decomposition into conflict and coordination components.
• Define zero-sum equivalent games as those strategically equivalent to a zero-sum game, and zero-sum equivalent potential games as potential games that are strategically equivalent to a zero-sum game.
• Use linear algebraic decomposition in the space of payoff functions, decomposing the payoff vector field into components in subspaces: $χ$ (zero-sum normalized), $ε$ (identical interest normalized), and $Π$ (zero-sum equivalent potential).
• Leverage operator-theoretic tools: $T_i$ (fixing player $i$'s strategy), $I - T_i$ (difference operator), and $f_M^{(i)}$ (marginal contribution over subset $M$), to characterize equilibrium properties and derive decomposition identities.
• Prove that any game $f$ can be uniquely written as $f = m + z + e$, where $m \in \mathcal{I}$ (identical interest), $z \in \mathcal{Z} \cap \mathcal{N}$ (zero-sum normalized), and $e \in \mathcal{E}$ (zero-sum equivalent potential), with $\mathcal{Z} \cap \mathcal{N} \subset \mathcal{E}$.

Experimental results

Research questions

• RQ1 Can any normal-form game be uniquely decomposed into components representing conflict (zero-sum), coordination (potential), and identical interest, with distinct equilibrium properties?
• RQ2 What are the equilibrium properties of zero-sum equivalent games and zero-sum equivalent potential games, especially in two-player finite games?
• RQ3 How does the decomposition isolate the strategic effects of conflict, coordination, and alignment on the Nash equilibrium of a game?
• RQ4 Can the decomposition be used to prove uniqueness of Nash equilibria in specific classes of games, such as rent-seeking or contest games?
• RQ5 What conditions on the decomposition components determine whether the total number of Nash equilibria is maximal or minimal?

Key findings

• Any normal-form game, whether with finite or continuous strategy sets, admits a unique decomposition into three orthogonal components: a zero-sum normalized game, a zero-sum equivalent potential game, and an identical interest normalized game.
• Two-player zero-sum equivalent games with finite strategy sets generically have a unique Nash equilibrium.
• Two-player zero-sum equivalent potential games with finite strategy sets generically have a strictly dominant Nash equilibrium.
• The decomposition isolates the strategic effect of each component on the Nash equilibrium, enabling analysis of equilibrium multiplicity or uniqueness based on component structure.
• The decomposition facilitates equilibrium analysis in contest games: the uniqueness of Nash equilibria in rent-seeking games is proven using the special structure of zero-sum equivalent games.
• The decomposition is algebraically exact: every payoff function $f$ can be written as $f = m + z + e$, where $m \in \mathcal{I}$, $z \in \mathcal{Z} \cap \mathcal{N}$, and $e \in \mathcal{E}$, with $\mathcal{Z} \cap \mathcal{N} \subset \mathcal{E}$, and the decomposition is unique.