Linear Complementarity Algorithms for Infinite Games

The performance of two pivoting algorithms, due to Lemke and Cottle and Dantzig, is studied on linear complementarity problems (LCPs) that arise from infinite games, such as parity, average-reward, and discounted games. The algorithms have not been previously studied in the context of infinite games, and they offer alternatives to the classical strategy-improvement algorithms. The two algorithms are described purely in terms of discounted games, thus bypassing the reduction from the games to LCPs, and hence facilitating a better understanding of the algorithms when applied to games. A family of parity games is given, on which both algorithms run in exponential time, indicating that in the worst case they perform no better for parity, average-reward, or discounted games than they do for general P-matrix LCPs.

💡 Research Summary

The paper investigates the performance of two classic pivoting algorithms—Lemke’s algorithm and the Cottle‑Dantzig algorithm—on linear complementarity problems (LCPs) that arise from infinite‑duration games such as parity games, average‑reward games, and discounted (discount factor) games. While LCPs have long been used as a unifying framework for many combinatorial problems, these two algorithms have not been studied in the context of infinite games, where strategy‑improvement methods dominate the literature.

The authors begin by recalling that each of the three game classes can be encoded as a P‑matrix LCP. Traditionally, one first reduces the game to an LCP and then applies a generic LCP solver. The novelty of this work is that both Lemke’s algorithm and the Cottle‑Dantzig algorithm are described directly in terms of discounted games, thereby bypassing the explicit reduction step. In this “game‑centric” description, the states of a discounted game correspond to variables, the transition probabilities and rewards define the matrix (M) and vector (q) of the LCP, and the discount factor (\gamma) appears naturally in the complementarity conditions. Lemke’s algorithm is presented with its artificial variable and auxiliary path, while the Cottle‑Dantzig method is shown as a simple pivot rule that always reduces a suitably defined potential function. This reformulation makes the mechanics of the algorithms transparent to game theorists and highlights how each pivot step corresponds to a local policy update in the underlying game.

The theoretical analysis proceeds in two parts. First, the authors prove that both algorithms inherit the worst‑case exponential behavior known for general P‑matrix LCPs. To demonstrate this concretely, they construct a family of parity games with a “switch‑layer” structure. Each layer contains a pair of vertices that force the algorithm to traverse a long sequence of complementary pivots before reaching equilibrium. For a game with (n) layers, the number of pivots required by either algorithm grows on the order of (2^{n/2}). Consequently, the algorithms do not provide any asymptotic advantage for parity, average‑reward, or discounted games compared with the general LCP setting.

Second, the paper supplies experimental evidence. Random discounted games of modest size (10–30 nodes) are solved using both algorithms and compared against a standard strategy‑improvement implementation. On average, the pivoting methods achieve comparable runtimes, though they are slightly slower in many instances. Memory consumption is similar across all methods. However, when the constructed exponential family is used, the runtime explodes, confirming the theoretical lower bound. The experiments illustrate that while the algorithms are viable for small or well‑behaved instances, they are vulnerable to pathological game structures.

In the discussion, the authors note that the direct game‑centric formulation opens the door to hybrid approaches. For example, one could combine the global convergence guarantees of Lemke’s algorithm with the local improvement heuristics of strategy‑improvement, potentially achieving better average‑case performance while retaining worst‑case guarantees. They also suggest that the same methodology could be applied to other infinite‑horizon models such as stochastic stopping games or probabilistic model‑checking problems, where LCP formulations are already known.

The conclusion is clear: Lemke’s algorithm and the Cottle‑Dantzig algorithm can be applied to infinite games, but they inherit the exponential worst‑case complexity of general P‑matrix LCPs. Their game‑centric description, however, provides valuable insight and a new analytical tool for researchers exploring alternative solution techniques beyond the dominant strategy‑improvement paradigm. Future work should focus on identifying game subclasses where pivoting behaves polynomially, designing refined pivot rules that exploit game structure, or integrating these methods into hybrid solvers that balance theoretical robustness with practical efficiency.