Algorithmic Game Theory
Complexity of pure Nash equilibria
We investigate the complexity of finding Nash equilibria in different kinds of congestion games. Our study is restricted to congestion games with non-decreasing delay functions.
Symmetric Network Given a directed graph G = (V,E) with delay functions d e : {1,...,n} Z, e E. Player i wants to allocate a path of minimal delay between a source s and a target t. 1,2,9 7,8,9 s 1,9,9 t 4,5,6 1,2,3 In more general asymmetric network congestion games, different players might have different source-destination pairs.
Symmetric Network It is known that there are instances of symmetric congestion games in which there are states such that every improvement sequence from this state to a Nash equilibrium has exponential length. Hence, applying improvement steps is not an efficient (i.e. polynomial time) algorithm for computing Nash equilibria in these games. However, there is another algorithm which finds Nash equilibria in polynomial time...
Complexity in Symmetric Network Polytime algorithm via a reduction to min-cost flow: (Fabrikant, Papadimitriou, Talwar 2004) Each edge is replaced by n parallel edges of capacity 1 each. The ith copy of edge e has cost d e(i), 1 i n. 1,2,9 7,8,9 s 1,9,9 t 4,5,6 1,2,3 Optimal solution minimizes Rosenthal s potential function and, hence, is a Nash equilibrium.
The relationship to local search Rosenthal s potential function allows us to interprete congestion games as local search problems: Nash equilibria are local optima wrt potential function. How difficult is it to compute local optima?
The complexity class PLS Definition (PLS (Polynomial Local Search)) PLS contains search problems with an objective function and a specified neighborhood relationship Γ. It is required that there is a poly-time algorithm that, given any solution s, computes a solution in Γ(s) with better objective value, or certifies that s is a local optimum.
The complexity class PLS Definition (PLS (Polynomial Local Search)) PLS contains search problems with an objective function and a specified neighborhood relationship Γ. It is required that there is a poly-time algorithm that, given any solution s, computes a solution in Γ(s) with better objective value, or certifies that s is a local optimum. Some examples for problems in PLS FLIP (circuit evaluation with Flip-neighborhood) TSP with 2-Opt-neighborbood Pos-NAE-kSat with Flip-neighborhood Max-Cut with Flip-neighborhood Congestion games wrt improvement steps
Positive Not-All-Equal ksat (Pos-NAE-kSAT) Definition Input: Formula over n binary variables x 1,...,x n described by m clauses c 1,...,c m each containing k positive literals, and m weights w 1,...,w m.
Positive Not-All-Equal ksat (Pos-NAE-kSAT) Definition Input: Formula over n binary variables x 1,...,x n described by m clauses c 1,...,c m each containing k positive literals, and m weights w 1,...,w m. Clause c i is satisfied by an assignment A {0,1} n if its literals are not all assigned the same value. The value of an assignment is the weighted sum of satisfied clauses. Assignments A and A are neighboring if they differ in exactly one position. Task: Find a local optimum, i.e., an assignment without neighboring assignment of higher value.
Positive Not-All-Equal 3Sat (Pos-NAE-3SAT) Example Example instance of Pos-NAE-3SAT: c 1 = x 1x 2x 3; c 2 = x 1x 2x 4; c 3 = x 1x 2x 5; c 4 = x 3x 4x 5 w 1 = 100; w 2 = 110; w 3 = 120; w 4 = 100
Positive Not-All-Equal 3Sat (Pos-NAE-3SAT) Example Example instance of Pos-NAE-3SAT: c 1 = x 1x 2x 3; c 2 = x 1x 2x 4; c 3 = x 1x 2x 5; c 4 = x 3x 4x 5 w 1 = 100; w 2 = 110; w 3 = 120; w 4 = 100 An example for a local optimum is the assignment x 1 = 0; x 2 = 0; x 3 = 1; x 4 = 1; x 5 = 1 of value 330. Each of the five neighboring assignments has a value of at most 330.
Max-Cut Definition Input: A graph G = (V,E) with edge weights w : E N. A cut partitions V into two sets Left and Right. Two cuts are neighboring if one can obtain one from the other by moving only one node from Left to Right or vice versa. The value of a cut is the weighted number of edges with one endpoint in Left and one endpoint in Right. Task: Find a local optimum, i.e., a cut without neighboring cut of higher value.
The complexity class PLS Definition (PLS-reduction) Given two PLS problems Π 1 and Π 2 find a mapping from the instances of Π 1 to the instances of Π 2 such that the mapping can be computed in polynomial time, the local optima of Π 1 are mapped to local optima of Π 2, and given any local optimum of Π 2, one can construct a local optimum of Π 1 in polynomial time.
First example for a PLS-reduction Theorem: Pos-NAE-3Sat PLS Pos-NAE-2Sat
First example for a PLS-reduction Theorem: Pos-NAE-3Sat PLS Pos-NAE-2Sat Proof: For each 3-clause (x 1,x 2,x 3) of weight w introduce the three 2-clauses (x 1,x 2),(x 1,x 3),(x 2,x 3) each of weight w/2.
First example for a PLS-reduction Theorem: Pos-NAE-3Sat PLS Pos-NAE-2Sat Proof: For each 3-clause (x 1,x 2,x 3) of weight w introduce the three 2-clauses (x 1,x 2),(x 1,x 3),(x 2,x 3) each of weight w/2. The value of an assignment in the 2SAT instance is identical to its value in the 3SAT instance as a 3-clause is satisfied if and only if exactly two of the three corresponding 2-clauses are satisfied. Hence, the local optima of both instances coincide, and the conditions of a PLS-reduction are fulfilled.
Second example for a PLS-reduction Theorem: Pos-NAE-2Sat PLS Max-Cut
Second example for a PLS-reduction Theorem: Pos-NAE-2Sat PLS Max-Cut Proof: Each variable is represented by a node. Each clause and its weight is represented by a weighted edge. Multi-edges are merged to single edges by adding their weight. Given an assignment for the variables, 0 is interpreted as Left and 1 is interpreted as Right. This way, the local optima of both instances coincide, and the conditions of a PLS-reduction are fulfilled.
The complexity class PLS Definition (PLS-completeness) A problem Π in PLS is called PLS-complete if, for every problem Π in PLS, it holds Π PLS Π. Examples for PLS-complete problem: A master reduction shows that FLIP is PLS-complete. A PLS-reduction from FLIP to POS-NAE-3SAT shows that the latter problem is PLS-complete as well. Hence, POS-NAE-2SAT and Max-Cut are PLS-complete, too.
Complexity of congestion games Results from [Fabrikant, Papadimitriou, Talwar 2004] network games general games symmetric poly-time Algo PLS-complete asymmetric PLS-complete PLS-complete We present only one of these PLS-completeness proof, the one for general, asymmetric congestion games (the simplest one).
PLS-hardness proof for general congestion games We prove a PLS-reduction from Max-Cut to congestion games.
PLS-hardness proof for general congestion games We prove a PLS-reduction from Max-Cut to congestion games. First of all, we observe that Max-Cut can be represented as a game: Party Affiliation Game (Max-Cut) Players correspond to nodes in a weighted graph G = (V,E). Every player has 2 strategies: left or right. A state of the game yields a cut, i.e., a partition of V into left and right nodes. Edge weights represent antisympathy. Players maximize the sum of weights of incident edges crossing the cut. Nash equilibria correspond to local optima of Max-Cut.
PLS-hardness proof for general congestion games Minimization Variant of the Party Affiliation Game The strategies of a node are left: choose the left hand side of the cut right: choose the right hand side of the cut The costs for these strategies are left: sum of the weights of the incident edges to the left right: sum of the weights of the incident edges to the right Both games have the same transition graph as, for each player, minimizing the weights of incident edges on her side, is equivalent to maximizing the sum of edges leading to the other side.
PLS-hardness proof for general congestion games Now the minimization variant can be described in terms of a congestion game. Party Affiliation Congestion Game: Represent each edge e by two resources e left,e right with delay functions d(1) = 0 and d(2) = w e. For each player the strategy S left contains resources e left for all incident edges; strategy S right contains resources r right for all incident edges.
PLS-hardness proof for general congestion games Now the minimization variant can be described in terms of a congestion game. Party Affiliation Congestion Game: Represent each edge e by two resources e left,e right with delay functions d(1) = 0 and d(2) = w e. For each player the strategy S left contains resources e left for all incident edges; strategy S right contains resources r right for all incident edges. Players in this congestion game have exactly the same cost as players in the minimization variant of the party affiliation game. Hence, the Nash equilibria of this congestion game coincide with local optima of the Max-Cut instance, which yields a PLS-reduction from Max-Cut to congestion games. (PLS completeness)
Recommended Literature D. Monderer, L. Shapley. Potential Games. Games and Economic Behavior, 14:1124 1143, 1996. (Equivalence Congestion and Potential Games) M. Yannakakis. Computational complexity. Chapter 2 in E. Aarts and J. Lenstra (Eds), Local Search in Combinatorial Optimization, pages 19 55, 1997. (Survey of the class PLS) A. Fabrikant, C. Papadimitriou, K. Talwar. The complexity of pure Nash equilibria. STOC 2004. (PLS-Completeness in ) H. Ackermann, H. Röglin, B. Vöcking. On the impact of combinatorial structure on congestion games. Journal of the ACM, 55(6), 2008. (Matroid Games, Exponential Convergence, PLS-Completeness)