On the Complexity of Nash Equilibria in Anonymous Games

We show that the problem of finding an {\epsilon}-approximate Nash equilibrium in an anonymous game with seven pure strategies is complete in PPAD, when the approximation parameter {\epsilon} is exponentially small in the number of players.Comment: full versio

On the Complexity of Nash Equilibria of Action-Graph Games

We consider the problem of computing Nash Equilibria of action-graph games (AGGs). AGGs, introduced by Bhat and Leyton-Brown, is a succinct representation of games that encapsulates both "local" dependencies as in graphical games, and partial indifference to other agents' identities as in anonymous games, which occur in many natural settings. This is achieved by specifying a graph on the set of actions, so that the payoff of an agent for selecting a strategy depends only on the number of agents playing each of the neighboring strategies in the action graph. We present a Polynomial Time Approximation Scheme for computing mixed Nash equilibria of AGGs with constant treewidth and a constant number of agent types (and an arbitrary number of strategies), together with hardness results for the cases when either the treewidth or the number of agent types is unconstrained. In particular, we show that even if the action graph is a tree, but the number of agent-types is unconstrained, it is NP-complete to decide the existence of a pure-strategy Nash equilibrium and PPAD-complete to compute a mixed Nash equilibrium (even an approximate one); similarly for symmetric AGGs (all agents belong to a single type), if we allow arbitrary treewidth. These hardness results suggest that, in some sense, our PTAS is as strong of a positive result as one can expect

Complexity and Efficiency in the Negotiation Game

This paper considers the ``negotiation game'' Busch and Wen (1995)) which combines the features of two-person alternating offers bargaining and repeated games. Despite the forces of bargaining, the negotiation game in general admits a large number of equilibria some of which involve delay and inefficiency. In order to isolate equilibria in this game, we investigate the role of complexity of implementing a strategy, introduced in the literature on repeated games played by automata. It turns out that when the players care for less complex strategies (at the margin) only efficient equilibria survive. Thus, complexity and bargaining in tandem may offer an explanation for co-operation and efficiency in repeated gamesBargaining, Repeated Game, Negotiation Game, Complexity, Automaton

Complexity and Efficiency in Repeated Games with Negotiation

This paper considers the "negotiation game" (Busch and Wen, 1995) which combines the features of two-person alternating offers bargaining and repeated games. Despite the forces of bargaining, the negotiation game in general admits a large number of equilibria some of which involve delay in agreement and inefficiency. In order to isolate equilibria in this game, we explicitly consider the complexity of implementing a strategy, introduced in the literature on repeated games played by automata. It turns out that when the players have a preference for less complex strategies (even at the margin) only efficient equilibria survive. Thus, complexity and bargaining in tandem may offer an explanation for co-operation in repeated gamesNegotiation Game, Repeated Game, Bargaining, Complexity, Bounded Rationality, Automaton

On the complexity of price equilibria

AbstractWe prove complexity, approximability, and inapproximability results for the problem of finding an exchange equilibrium in markets with indivisible (integer) goods, most notably a polynomial algorithm that approximates the market equilibrium arbitrarily close when the number of goods is bounded and the utilities linear. We also show a communication complexity lower bound in a model appropriate for markets. Our result implies that the ideal informational economy of a market with divisible goods and unique optimal allocations is unattainable in general

Complexity and Efficiency in Repeated Games with Negotiation

This paper considers the "negotiation game" (Busch and Wen, 1995) which combines the features of two-person alternating offers bargaining and repeated games. Despite the forces of bargaining, the negotiation game in general admits a large number of equilibria some of which involve delay in agreement and inefficiency. In order to isolate equilibria in this game, we explicitly consider the complexity of implementing a strategy, introduced in the literature on repeated games played by automata. It turns out that when the players have a preference for less complex strategies (even at the margin) only efficient equilibria survive. Thus, complexity and bargaining in tandem may offer an explanation for co-operation in repeated gamesNegotiation Game, Repeated Game, Bargaining, Complexity, Bounded Rationality, Automaton

Common Agency and Computational Complexity: Theory and Experimental Evidence

In a common agency game, several principals try to influence the behavior of an agent. Common agency games typically have multiple equilibria. One class of equilibria, called truthful, has been identified by Bernheim and Whinston and has found widespread use in the political economy literature. In this paper we identify another class of equilibria, which we call natural. In a natural equilibrium, each principal offers a strictly positive contribution on at most one alternative. We show that a natural equilibrium always exists and that its computational complexity is much smaller than that of a truthful equilibrium. To compare the predictive power of the two concepts, we run an experiment on a common agency game for which the two equilibria predict a different equilibrium alternative. The results strongly reject the truthful equilibrium. The alternative predicted by the natural equilibrium is chosen in 65% of the matches, while the one predicted by the truthful equilibrium is chosen in less than 5% of the matches.lobbying;experimental economics;common agency;truthful equilibrium;natural equilibrium;computational complexity

Query Complexity of Approximate Equilibria in Anonymous Games

We study the computation of equilibria of anonymous games, via algorithms that may proceed via a sequence of adaptive queries to the game's payoff function, assumed to be unknown initially. The general topic we consider is \emph{query complexity}, that is, how many queries are necessary or sufficient to compute an exact or approximate Nash equilibrium. We show that exact equilibria cannot be found via query-efficient algorithms. We also give an example of a 2-strategy, 3-player anonymous game that does not have any exact Nash equilibrium in rational numbers. However, more positive query-complexity bounds are attainable if either further symmetries of the utility functions are assumed or we focus on approximate equilibria. We investigate four sub-classes of anonymous games previously considered by \cite{bfh09, dp14}. Our main result is a new randomized query-efficient algorithm that finds a $O(n^{-1/4})$-approximate Nash equilibrium querying $\tilde{O}(n^{3/2})$ payoffs and runs in time $\tilde{O}(n^{3/2})$. This improves on the running time of pre-existing algorithms for approximate equilibria of anonymous games, and is the first one to obtain an inverse polynomial approximation in poly-time. We also show how this can be utilized as an efficient polynomial-time approximation scheme (PTAS). Furthermore, we prove that $\Omega(n \log{n})$ payoffs must be queried in order to find any $\epsilon$-well-supported Nash equilibrium, even by randomized algorithms