Algorithmic Game Theory (Lecture 19: Pure Nash Equilibria and PLS-Completeness)

This segment introduces the core question of the course: whether players in a game reach an equilibrium, how quickly, and through what learning processes. It sets the stage for the discussion of positive and negative results regarding equilibrium prediction. The course concludes with a focus on negative results in game theory. While previous weeks showcased positive results on equilibrium convergence via learning processes (no-regret dynamics reaching correlated equilibria), this week explores limitations. The instructor introduces PLS-completeness, an analog of NP-completeness for local search problems, to explain the lack of polynomial-time algorithms for computing mixed Nash equilibria in general two-player games and pure Nash equilibria in general congestion games. The complexity of local search, exemplified by max-cut, is discussed, highlighting the difficulty of finding locally optimal solutions. A reduction from max-cut to finding pure Nash equilibria in congestion games demonstrates the PLS-completeness of the latter, implying the likely intractability of finding pure Nash equilibria in general congestion games. This segment highlights the significance of negative results in game theory. It explains that negative results, by revealing limitations of equilibrium concepts, provide a crucial counterpoint to positive results and enhance the overall understanding of game dynamics. This segment delves into the exploration of various equilibrium concepts, including correlated equilibria and Nash equilibria (both mixed and pure versions). It discusses the tractability of computing these equilibria using different learning processes and algorithms. This segment introduces the complexity of local search problems and connects it to the problem of finding pure Nash equilibria in games. It foreshadows the use of complexity theory to explain the apparent lack of efficient algorithms for certain equilibrium computation problems. This segment poses the central challenge: the lack of known polynomial-time algorithms for computing mixed strategy Nash equilibria in general two-player games or pure Nash equilibria in general routing or congestion games. It sets the stage for exploring the theoretical limitations of equilibrium computation. This segment explains the analogy between NP-completeness and a tailored version for local search problems. It highlights the shift from efficiently verifiable solutions in NP to efficiently solvable problems via local search, emphasizing the goal of proving that certain local search problems are as hard as any other, implying no significant improvement over local search itself is possible. The speaker defines the concept of a generic local search problem, outlining the three polynomial-time algorithms required: one for initialization, one for evaluating solutions, and one for finding better neighboring solutions. This formalization lays the groundwork for defining a complexity class analogous to NP for local search problems.This segment delves into the specifics of the three polynomial-time algorithms that define a generic local search problem. It explains how these algorithms work together to ensure the well-definedness of the local search process, emphasizing the polynomial time constraint and its implications for the search space. The speaker introduces the concept of PLS (Polynomial Local Search) completeness and reductions. The explanation focuses on the definition of a reduction between two PLS problems, emphasizing how the ability to solve one problem in polynomial time implies the ability to solve the other, thus establishing the notion of PLS-completeness as a measure of hardness for local search problems.This segment discusses the implications of PLS-completeness, including the existence of PLS-complete problems and the significance of max cut as a natural example. It touches upon the analogy to Cook's theorem in NP-completeness and the work of Johnson, Papadimitriou, and Yannakakis, and Schaeffer and Yannakakis in establishing PLS-completeness results. This segment focuses on the specific construction of the congestion game used in the reduction. The speaker explains the strategies available to each player and analyzes the possible loads on the resources (edges). The speaker shows that the loads are highly restricted, taking on values of only 0, 1, or 2, simplifying the analysis of the cost functions. This segment details the construction of a reduction from the Max Cut problem (known to be PLS-complete) to the problem of finding a pure Nash equilibrium in a congestion game. The speaker describes how to map an instance of Max Cut to a congestion game, establishing a correspondence between vertices, edges, and strategies. This is a key step in proving PLS-completeness. This segment introduces the significant result that finding a pure Nash equilibrium in a congestion game is not just in PLS, but is PLS-complete. The speaker highlights the implications: no polynomial-time algorithm exists unless P=PLS, and best-response dynamics requires exponential iterations in the worst case. This establishes a strong lower bound on the computational complexity of the problem. The speaker presents both conditional and unconditional results regarding the time complexity of local search algorithms for PLS-complete problems. The conditional result states that no polynomial-time algorithm can find a locally optimal solution unless P=PLS, while the unconditional result asserts that the specific local search algorithm takes exponential time in the worst case for any PLS-complete problem. The discussion clarifies the distinction between these two results and their implications. This segment explains how the problem of finding a pure Nash equilibrium in a congestion game belongs to the complexity class PLS (Polynomial Local Search) by connecting it to the Rosenthal potential function. The speaker demonstrates that pure Nash equilibria are precisely the local minima of this potential function, and best response dynamics, a local search algorithm, can be used to find them. This establishes a crucial link between the computational complexity of the game and the properties of the potential function. This segment establishes the crucial correspondence between solutions of the Max Cut problem and the pure Nash equilibria of the constructed congestion game. The speaker demonstrates how the objective function values (cut value in Max Cut and Rosenthal potential in the game) are related, showing that local optima in Max Cut correspond to local optima in the game. This completes the reduction, proving PLS-completeness.