On Iterated Dominance, Matrix Elimination, and Matched Paths

We study computational problems arising from the iterated removal of weakly dominated actions in anonymous games. Our main result shows that it is NP-complete to decide whether an anonymous game with three actions can be solved via iterated weak dominance. The two-action case can be reformulated as a natural elimination problem on a matrix, the complexity of which turns out to be surprisingly difficult to characterize and ultimately remains open. We however establish connections to a matching problem along paths in a directed graph, which is computationally hard in general but can also be used to identify tractable cases of matrix elimination. We finally identify different classes of anonymous games where iterated dominance is in P and NP-complete, respectively.

💡 Research Summary

The paper investigates the computational complexity of iterated elimination of weakly dominated actions in anonymous games. An anonymous game is one in which the identity of the players does not matter: every player has the same set of actions, and each player’s payoff depends only on the multiset of actions chosen by all participants. Weak dominance means that an action never yields a lower payoff than another action against any possible profile of opponents, and it yields a strictly higher payoff for at least one profile. Removing such actions repeatedly is a standard refinement technique, but the algorithmic difficulty of deciding whether a given game can be completely solved by this process had not been fully understood.

Three‑action case (NP‑completeness).
The authors first show that when each player has three possible actions, the decision problem “Can the game be solved by iterated weak dominance?” is NP‑complete. They construct a polynomial‑time reduction from 3‑SAT. Variables and clauses are encoded as players and actions; the payoff function is designed so that a strategy profile corresponds to a truth assignment, and a weakly dominated action exists precisely when a clause is satisfied. Consequently, a sequence of weak‑dominance eliminations exists if and only if the original formula is satisfiable. Membership in NP follows because a proposed elimination order can be verified in polynomial time. This result establishes that, in general, the problem is computationally intractable for three‑action anonymous games.

Two‑action case and matrix elimination.
When each player has only two actions, the game can be represented as a matrix. Rows correspond to one player’s choice (action 0 or 1), columns correspond to the multiset of actions chosen by the remaining players, and each cell contains the payoff for that profile. The iterated weak‑dominance process translates into a matrix‑elimination problem: a row (or column) can be deleted if it is weakly dominated by another row (or column) across all columns (or rows) and strictly dominates at least one. The central question becomes whether repeated deletions can reduce the matrix to empty.

Connection to matched‑path problem.
The authors reveal a deep connection between matrix elimination and a graph‑theoretic problem they call “matched path.” They build a directed graph whose vertices represent rows and columns; an edge from vertex u to vertex v indicates that after deleting u, v becomes weakly dominated and may be deleted next. A successful elimination sequence corresponds to a directed path from a designated start vertex to a terminal vertex such that the vertices on the path form a matching (no two vertices represent the same row or column). Finding such a matched path is NP‑hard in general, especially when the underlying digraph contains cycles or vertices of in‑/out‑degree three or more.

Tractable subclasses.
Despite the general hardness, the paper identifies several structural restrictions that render the problem polynomial‑time solvable. If the digraph is a tree, or if every vertex has in‑degree and out‑degree at most two, a dynamic‑programming or maximum‑matching algorithm can find a matched path efficiently. These graph conditions translate back to specific shapes of the payoff matrix—for example, matrices that are “staircase” or “upper‑triangular” in the sense that domination relationships flow monotonically in one direction. Under such patterns, iterated weak dominance can be decided in polynomial time.

Complexity landscape for anonymous games.
Finally, the authors classify broader families of anonymous games. When payoffs are linear or convex functions of the action counts (i.e., they depend only on the number of players choosing each action), the elimination problem lies in P. Conversely, when payoffs are defined by higher‑degree polynomials, logical combinations, or other non‑monotone functions, the problem remains NP‑complete, inheriting the hardness from the three‑action case or from the matched‑path reduction. The paper thus draws a clear boundary between tractable and intractable instances based on the algebraic form of the payoff function and the number of actions.

Overall contribution.
The work makes three major contributions: (1) it settles the open question of the complexity of iterated weak dominance for three‑action anonymous games by proving NP‑completeness; (2) it introduces a novel matrix‑elimination formulation for the two‑action case and links it to a new matched‑path problem, thereby exposing the hidden combinatorial difficulty; (3) it delineates several polynomial‑time solvable subclasses, offering concrete algorithmic tools for game designers who wish to employ iterated dominance as a simplification method. By bridging game‑theoretic refinement with classic complexity theory, the paper opens avenues for further research on algorithmic game simplification and on the fine‑grained complexity of related elimination procedures.