35 research outputs found
Tree Polymatrix Games Are PPAD-Hard.
We prove that it is PPAD-hard to compute a Nash equilibrium in a tree polymatrix game with twenty actions per player. This is the first PPAD hardness result for a game with a constant number of actions per player where the interaction graph is acyclic. Along the way we show PPAD-hardness for finding an -fixed point of a 2D LinearFIXP instance, when is any constant less than . This lifts the hardness regime from polynomially small approximations in -dimensions to constant approximations in two-dimensions, and our constant is substantial when compared to the trivial upper bound of
Pure-Circuit: Strong Inapproximability for PPAD
The current state-of-the-art methods for showing inapproximability in PPAD
arise from the -Generalized-Circuit (-GCircuit)
problem. Rubinstein (2018) showed that there exists a small unknown constant
for which -GCircuit is PPAD-hard, and subsequent
work has shown hardness results for other problems in PPAD by using
-GCircuit as an intermediate problem.
We introduce Pure-Circuit, a new intermediate problem for PPAD, which can be
thought of as -GCircuit pushed to the limit as , and we show that the problem is PPAD-complete. We then prove
that -GCircuit is PPAD-hard for all by a
reduction from Pure-Circuit, and thus strengthen all prior work that has used
GCircuit as an intermediate problem from the existential-constant regime to the
large-constant regime.
We show that stronger inapproximability results can be derived by reducing
directly from Pure-Circuit. In particular, we prove tight inapproximability
results for computing -well-supported Nash equilibria in
two-action polymatrix games, as well as for finding approximate equilibria in
threshold games
Tight Inapproximability for Graphical Games
We provide a complete characterization for the computational complexity of
finding approximate equilibria in two-action graphical games. We consider the
two most well-studied approximation notions: -Nash equilibria
(-NE) and -well-supported Nash equilibria
(-WSNE), where . We prove that computing an
-NE is PPAD-complete for any constant , while a
very simple algorithm (namely, letting all players mix uniformly between their
two actions) yields a -NE. On the other hand, we show that computing an
-WSNE is PPAD-complete for any constant , while a
-WSNE is trivial to achieve, because any strategy profile is a -WSNE. All
of our lower bounds immediately also apply to graphical games with more than
two actions per player
Learning Sparse Polymatrix Games in Polynomial Time and Sample Complexity
We consider the problem of learning sparse polymatrix games from observations
of strategic interactions. We show that a polynomial time method based on
-group regularized logistic regression recovers a game, whose Nash
equilibria are the -Nash equilibria of the game from which the data
was generated (true game), in samples of
strategy profiles --- where is the maximum number of pure strategies of a
player, is the number of players, and is the maximum degree of the game
graph. Under slightly more stringent separability conditions on the payoff
matrices of the true game, we show that our method learns a game with the exact
same Nash equilibria as the true game. We also show that
samples are necessary for any method to consistently recover a game, with the
same Nash-equilibria as the true game, from observations of strategic
interactions. We verify our theoretical results through simulation experiments
Pure-Circuit: Strong Inapproximability for PPAD
The current state-of-the-art methods for showing inapproximability in PPAD arise from the ε-Generalized-Circuit (ε-GCircuit) problem. Rubinstein (2018) showed that there exists a small unknown constant ε for which ε-GCircuit is PPAD-hard, and subsequent work has shown hardness results for other problems in PPAD by using ε-GCircuit as an intermediate problem.
We introduce Pure-Circuit, a new intermediate problem for PPAD, which can be thought of as ε-GCircuit pushed to the limit as ε→1, and we show that the problem is PPAD-complete. We then prove that ε-GCircuit is PPAD-hard for all ε<0.1 by a reduction from Pure-Circuit, and thus strengthen all prior work that has used GCircuit as an intermediate problem from the existential-constant regime to the large-constant regime.
We show that stronger inapproximability results can be derived by reducing directly from Pure-Circuit. In particular, we prove tight inapproximability results for computing ε-well-supported Nash equilibria in two-action polymatrix games, as well as for finding approximate equilibria in threshold games
Pure-Circuit: Strong Inapproximability for PPAD
The current state-of-the-art methods for showing inapproximability in PPAD arise from the ε-Generalized-Circuit (ε-GCircuit) problem. Rubinstein (2018) showed that there exists a small unknown constant ε for which ε-GCircuit is PPAD-hard, and subsequent work has shown hardness results for other problems in PPAD by using ε-GCircuit as an intermediate problem.
We introduce Pure-Circuit, a new intermediate problem for PPAD, which can be thought of as ε-GCircuit pushed to the limit as ε→1, and we show that the problem is PPAD-complete. We then prove that ε-GCircuit is PPAD-hard for all ε<0.1 by a reduction from Pure-Circuit, and thus strengthen all prior work that has used GCircuit as an intermediate problem from the existential-constant regime to the large-constant regime.
We show that stronger inapproximability results can be derived by reducing directly from Pure-Circuit. In particular, we prove tight inapproximability results for computing ε-well-supported Nash equilibria in two-action polymatrix games, as well as for finding approximate equilibria in threshold games