In this paper we introduce a novel flow representation for finite games in strategic form. This representation allows us to develop a canonical direct sum decomposition of an arbitrary game into three components, which we refer to as the potential, harmonic and nonstrategic components. We analyze natural classes of games that are induced by this decomposition, and in particular, focus on games with no harmonic component and games with no potential component. We show that the first class corresponds to the well-known potential games. We refer to the second class of games as harmonic games, and study the structural and equilibrium properties of this new class of games. Intuitively, the potential component of a game captures interactions that can equivalently be represented as a common interest game, while the harmonic part represents the conflicts between the interests of the players. We make this intuition precise, by studying the properties of these two classes, and show that indeed they have quite distinct and remarkable characteristics. For instance, while finite potential games always have pure Nash equilibria, harmonic games generically never do. Moreover, we show that the nonstrategic component does not affect the equilibria of a game, but plays a fundamental role in their efficiency properties, thus decoupling the location of equilibria and their payoff-related properties. Exploiting the properties of the decomposition framework, we obtain explicit expressions for the projections of games onto the subspaces of potential and harmonic games. This enables an extension of the properties of potential and harmonic games to "nearby" games. We exemplify this point by showing that the set of approximate equilibria of an arbitrary game can be characterized through the equilibria of its projection onto the set of potential games.
Deep Dive into Flows and Decompositions of Games: Harmonic and Potential Games.
In this paper we introduce a novel flow representation for finite games in strategic form. This representation allows us to develop a canonical direct sum decomposition of an arbitrary game into three components, which we refer to as the potential, harmonic and nonstrategic components. We analyze natural classes of games that are induced by this decomposition, and in particular, focus on games with no harmonic component and games with no potential component. We show that the first class corresponds to the well-known potential games. We refer to the second class of games as harmonic games, and study the structural and equilibrium properties of this new class of games. Intuitively, the potential component of a game captures interactions that can equivalently be represented as a common interest game, while the harmonic part represents the conflicts between the interests of the players. We make this intuition precise, by studying the properties of these two classes, and show that indeed th
Potential games play an important role in game-theoretic analysis due to their desirable static properties (e.g., existence of a pure strategy Nash equilibrium) and tractable dynamics (e.g., convergence of simple user dynamics to a Nash equilibrium); see [32,31,35]. However, many multi-agent strategic interactions in economics and engineering cannot be modeled as a potential game.
This paper provides a novel flow representation of the preference structure in strategic-form finite games, which allows for delineating the fundamental characteristics in preferences that lead to potential games. This representation enables us to develop a canonical orthogonal decomposition of an arbitrary game into a potential component, a harmonic component, and a nonstrategic component, each with its distinct properties. The decomposition can be used to define the “distance” of an arbitrary game to the set of potential games. We use this fact to describe the approximate equilibria of the original game in terms of the equilibria of the closest potential game.
The starting point is to associate to a given finite game a game graph, where the set of nodes corresponds to the strategy profiles and the edges represent the “comparable strategy profiles” i.e., strategy profiles that differ in the strategy of a single player. The utility differences for the deviating players along the edges define a flow on the game graph. Although this graph contains strictly less information than the original description of the game in terms of utility functions, all relevant strategic aspects (e.g., equilibria) are captured.
Our first result provides a canonical decomposition of an arbitrary game using tools from the study of flows on graphs (which can be viewed as combinatorial analogues of the study of vector fields). In particular, we use the Helmholtz decomposition theorem (e.g., [21]), which enables the decomposition of a flow on a graph into three components: globally consistent, locally consistent (but globally inconsistent), and locally inconsistent component (see Theorem 3.1). The globally consistent component represents a gradient flow while the locally consistent flow corresponds to flows around global cycles. The locally inconsistent component represents local cycles (or circulations) around 3-cliques of the graph.
Our game decomposition has three components: nonstrategic, potential and harmonic. The first component represents the “nonstrategic interactions” in a game. Consider two games in which, given the strategies of the other players, each player’s utility function differs by an additive constant. These two games have the same utility differences, and therefore they have the same flow representation. Moreover, since equilibria are defined in terms of utility differences, the two games have the same equilibrium set. We refer to such games as strategically equivalent. We normalize the utilities, and refer to the utility differences between a game and its normalization as the nonstrategic component of the game. Our next step is to remove the nonstrategic component and apply the Helmholtz decomposition to the remainder. The flow representation of a game defined in terms of utility functions (as opposed to preferences) does not exhibit local cycles, therefore the Helmholtz decomposition yields the two remaining components of a game: the potential component (gradient flow) and the harmonic component (global cycles). The decomposition result is particularly insightful for bimatrix games (i.e., finite games with two players, see Section 4.3), where the potential component represents the “team part” of the utilities (suitably perturbed to capture the utility matrix differences), and the harmonic component corresponds to a zero-sum game.
The canonical decomposition we introduce is illustrated in the following example.
Example 1.1 (Road-sharing game). Consider a three-player game, where each player has to choose one of the two roads {0, 1}. We denote the players by d 1 , d 2 and s. The player s tries to avoid sharing the road with other players: its payoff decreases by 2 with each player d 1 and d 2 who shares the same road with it. The player d 1 receives a payoff -1, if d 2 shares the road with it and 0 (0, 0, 0)
(1, 0, 0)
(1, 1, 0) (0, 1, 0)
(1, 1, 1)
(1, 0, 1) (0, 1, 1) (0, 0, 1) (0, 0, 0)
(1, 0, 0)
(1, 1, 0) (0, 1, 0)
(1, 1, 1)
(1, 0, 1) (0, 1, 1) (0, 0, 1) In Figure 1a we present the flow representation for this game (described in detail in Section 2.2), where the nonstrategic component has been removed. Figures 1b and1c show the decomposition of this flow into its potential and harmonic components. In the figure, each tuple (a, b, c) denotes a strategy profile, where player s uses strategy a and players d 1 and d 2 use strategies b and c respectively.
These components induce a direct sum decomposition of the space of games into three respective subspaces, which we refer to as the nonstrategic, potential and harmonic subsp
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