In this chapter, we introduced four classes of anonymous games and investigated the computational complexity of pure Nash equilibrium and iterated weak dominance in these classes. We established that the former solution concept is tractable for games with a constant number of actions, but becomes intractable if the number of actions grows at least linearly in the number of players. It is worth noting that, for games with a constant number of actions, the pure equilibrium problem happens to lie in the complexity class NC1 for all types of anonymity and is thus open to parallel computation. NP-hardness also holds for games with an exponential number of players and logarithmic growth of the number of actions. For games with an exponential number of players in which the number of actions grows sub-logarithmically, the complexity remains open. Iterated dominance, on the other hand, is tractable in symmetric games with any constant number of actions, but NP-hard in anonymous and self-anonymous games with only three actions. The complexity in anonymous and self-anonymous games with two actions remains open.
In future work, it would be interesting to extend the tractability results to larger classes of games. For example, games with a certain number of player types, where indistinguishability holds only for players of the same type, can be obtained by restricting Definition 5.1 to permutations that map players from a certain subset to players of the same set. Given a game in this class, we can construct an anonymous game with the same set of players and an action set that is the Cartesian product of the original set of actions and the set of player types. By assigning a unique minimum payoff to all actions not corresponding to the type of the respective player, we can ensure that players only play actions corresponding to their type in every equilibrium of the new game, effectively allowing us to distinguish players of different types in the new game. For games with a constant number of players the size of the new game is polynomial in the size of the original game, and the tractability result of Theorem 5.3 carries over immediately. A different notion, such that players of the same type have identical payoff functions, does not seem to provide additional structure. As we have already shown, only two different payoff functions suffice to make the pure equilibrium problem TC0-hard for a constant number of actions and NP-hard for a growing number of actions. More generally, one might investigate games where payoffs are invariant under particular sets of permutations. For example, von Neumann and Morgenstern (1947) regard the number of permutations under which the payoffs of a game are invariant as a measure for the degree of anonymity. The question is in how far the computational complexity of solving a game depends on this degree.
With respect to iterated dominance, the most important open question concerns iter-ated weak dominance solvability in anonymous games with two actions, and the equiva-lent problem of matrix elimination. More generally, we looked at a problem concerning matchings on paths in a directed graph. This problem was mainly introduced as a proxy to matrix elimination, but appears to be interesting in their own right, with connections
5.5 ·Discussion 99 to ordinary matching problems, sequencing, and planning. It will therefore be worthwhile to investigate versions of this problem with restrictions on the graph structure or labeling function.
Chapter 6
Graphical Games
Another structural element commonly found in real-world interaction, besides the one considered in the previous chapter, is locality. Often a situation involves many agents, but the weal and woe of any particular agent depends only on the decisions made by a select few. Graphical games (Kearns et al., 2001) formalize this notion by assigning to each player a subset of the other players, his neighborhood, and defining his payoff as a function of the actions of these players. More formally, a graphical game is given by a (directed or undirected) graph on the set of players of a normal-form game, such that the payoff of each player depends only on the actions of his neighbors in this graph. Any graphical game with neighborhood sizes bounded by a constant can be represented using space polynomial in the number of players.
Gottlob et al. (2005) investigate the complexity of pure Nash equilibria in graphical games, and show that deciding the existence of a pure equilibrium is NP-complete already for a very restricted class, namely that where each player can choose from at most three different actions and his payoff depends on at most three other players. We begin this chapter by strengthening this result to apply to an even more restrictive setting. To be precise, we show that two actions per player, two-bounded neighborhood, and two-valued payoff functions suffice for NP-completeness. This result is tight, because deciding the existence of a pure Nash equilibrium becomes trivial in the case of a single action for each player and tractable for one-bounded neighborhood. In fact, we show the latter problem to be NL-complete in general, and thus solvable in deterministic polynomial time. Interestingly, it turns out that the number of actions in a game with one-bounded neighborhood is a sensitive parameter: restricting the number of actions for each player to a constant makes the problem even easier than NL unless L = NL. In this way, we obtain a nice alternative characterization of the determinism-nondeterminism problem for Turing machines with logarithmic space in terms of the number of actions for games with one-bounded neighborhood.
We then move on to investigate the computational complexity of the pure equilibrium problem in graphical games which additionally satisfy one of four types of anonymity within each neighborhood. Despite these additional restrictions, the question for tractable
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102 6 ·Graphical Games classes of games is answered mostly in the negative. For three of the four types of anonymity, deciding the existence of a pure equilibrium remains NP-hard for games with two actions, two payoffs, and neighborhoods of size two. Assuming the most restricted type of anonymity, the problem becomes NP-hard when either there are three different payoffs, or neighborhoods of size four. On the other hand, we use interesting connections of the latter class to the even cycle problem in directed graphs and to generalized sat-isfiability to identify tractable classes of games. One such class for example arises from a situation where each agent is faced with the decision of producing one of two types of complementary goods within a regional neighborhood. In a sense, agents are not only producers but also consumers, and thus happier when both products are available within their neighborhood.
As a corollary, we further exhibit a satisfiability problem that remains NP-hard in the presence of a matching, a result which may be of independent interest. Finally, we show that mixed equilibria in games with two of four types of anonymity can be found in polynomial time if the number of actions grows only slowly in the neighborhood size.
Quite interestingly, there exists a class of games where deciding the existence of a pure equilibrium is likely to be harder than finding a mixed one.