17 research outputs found
Decompositions of two player games: potential, zero-sum, and stable games
We introduce several methods of decomposition for two player normal form
games. Viewing the set of all games as a vector space, we exhibit explicit
orthonormal bases for the subspaces of potential games, zero-sum games, and
their orthogonal complements which we call anti-potential games and
anti-zero-sum games, respectively. Perhaps surprisingly, every anti-potential
game comes either from the Rock-Paper-Scissors type games (in the case of
symmetric games) or from the Matching Pennies type games (in the case of
asymmetric games). Using these decompositions, we prove old (and some new)
cycle criteria for potential and zero-sum games (as orthogonality relations
between subspaces). We illustrate the usefulness of our decomposition by (a)
analyzing the generalized Rock-Paper-Scissors game, (b) completely
characterizing the set of all null-stable games, (c) providing a large class of
strict stable games, (d) relating the game decomposition to the decomposition
of vector fields for the replicator equations, (e) constructing Lyapunov
functions for some replicator dynamics, and (f) constructing Zeeman games
-games with an interior asymptotically stable Nash equilibrium and a pure
strategy ESS
Nested potentials and robust equilibria
This paper introduces the notion of nested best-response potentials for complete in- formation games. It is shown that a unique maximizer of such a potential is a Nash equilibrium that is robust to incomplete information in the sense of Kajii and Morris (1997, mimeo).incomplete information, potential games, robustness, refinements
Rosenthal's potential and a discrete version of the Debreu--Gorman Theorem
The acyclicity of individual improvements in a generalized congestion game (where the sums of local utilities are replaced with arbitrary aggregation rules) can be established with a Rosenthal-style construction if aggregation rules of all players are "quasi-separable." Every universal separable ordering on a finite set can be represented as a combination of addition and lexicography
Rosenthal's potential and a discrete version of the Debreu--Gorman Theorem
The acyclicity of individual improvements in a generalized congestion game (where the sums of local utilities are replaced with arbitrary aggregation rules) can be established with a Rosenthal-style construction if aggregation rules of all players are "quasi-separable." Every universal separable ordering on a finite set can be represented as a combination of addition and lexicography
Rosenthal's potential and a discrete version of the Debreu--Gorman Theorem
The acyclicity of individual improvements in a generalized congestion game (where the sums of local utilities are replaced with arbitrary aggregation rules) can be established with a Rosenthal-style construction if aggregation rules of all players are "quasi-separable." Every universal separable ordering on a finite set can be represented as a combination of addition and lexicography
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 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