Slide 1Slide 2The Lemke-Howson Algorithm (1964)The Lemke-Howson AlgorithmThe Lemke-Howson AlgorithmThe Lemke-Howson AlgorithmLemke-Howson ExamplePost MortemSlide 9Approximability of Nash Equilibrium[ Normalization AssumptionAlgorithms for Fixed Values of ApproximationA simple algorithm for .5 approximationThe trouble with approximationThe trouble with approximation (cont.)Bottom lineSlide 17No FTPAS ExistsFrom PPAD to Polymatrix the [DGP ’06] machineryA-to-D unhealthy setA-to-D unhealthy setFrom PPAD to Polymatrix the [DGP ’06] machineryRelaxing the Approximation RequirementSlide 24Special Classes of GamesSpecial Classes of Graphical GamesIdea of these algorithms[ Total Variation DistanceThe TV BoundThe TV BoundIdea of these algorithms6.896: Topics in Algorithmic Game TheoryLecture 12Constantinos DaskalakisThe Lemke-Howson AlgorithmThe Lemke-Howson Algorithm (1964)Problem: Find an exact equilibrium of a 2-player game.Since there exists a rational equilibrium this task is feasible.Cannot get the exact equilibrium (directly) from a simplicial algorithm, but I can get it from the support enumeration algorithm.Idea of LH: Instead of pivoting between simplices of a subdivision, perform pivoting steps between the corners of a polytope related to the game.Assumption (w.l.o.g.): The game given in the input is a symmetric game, i.e.Polytope of Interest:Assumption 2 (w.l.o.g): At every corner of the polytope exactly n out of the 2n inequalities are tight.(perturb original game entries with exponentially small noise to achieve this; equilibria of the new game are approximate eq. of original game of very high accuracy, and these can be converted to exact equilibria (exercise of past lecture) )The Lemke-Howson AlgorithmAt corner (0,0,…,0) all pure strategies are present. Call any corner of the polytope where this happens a democracy.Lemma: If a vertex z≠0 of the polytope is a democracy, then is a Nash eq.Proof:Hence:Def: Pure strategy i is represented at a corner z of the polytope if at least one of the following is tight: At a democracy we have the following implication:The Lemke-Howson AlgorithmStart at the corner (0,0,…,0).By non-degeneracy there are exactly n edges of the polytope adjacent to the (0,0,…,0) corner. Each of these edges corresponds to un-tigthening one of the inequalities.Select an arbitrary pure strategy, say pure strategy n, and un-tighten . This corresponds to an edge of the polytope adjacent to 0. Jump to the other endpoint of this edge.If the obtained vertex z is a democracy, then a Nash equilibrium has been found because z≠0.Otherwise, one of the strategies 1,…, n-1, say strategy j, is represented twice, by both was already tightjust became tightQuestion:I will untighten one of the above. What happens if I require ?A: I am going to return to (0,0,…,0), since I would be walking on the edge of the polytope that brought me here.So let me untighten the other one, requiring .The Lemke-Howson AlgorithmIf the obtained vertex is a democracy, then a Nash equilibrium has been found.Otherwise, one of the strategies 1,…, n-1, is represented twice. This strategy is doubly represented because one of its inequalities was tight before the step, and the other one became tight after the step was taken. To proceed, un-tighten the former.This defines a directed walk on the polytope, starting at the democracy (0,0,…,0), and with every intermediate node having all of 1,…, n-1 represented, and exactly one of them represented twice. The two neighbors of that node are obtained by un-tightening one of the two inequalities of the doubly represented strategy.The walk cannot have a rho-shape, since every intermediate vertex has two neighbors.Since there is a finite number of corners, the walk has to settle at a democracy that is different from (0,0,…,0).Moreover, it cannot return to (0,0,…,0) since that vertex has exactly one neighbor. (If we try to un-tighten , for any j ≠ n, we will transition to a vertex that is either a democracy or will not have j represented.)Lemke-Howson Example3 0 02 2 20 3 0Post MortemThe Lemke-Howson algorithm:- provides an alternative proof that a Nash equilibrium exists in 2-player games;- moreover, it shows that there always exists a rational equilibrium in 2-player games;- it works by virtue of the same parity argument justifying the correctness of the simplicial approximation algorithms (for solving SPERNER and BROUWER); in fact, it preceded and inspired the development of these algorithms, ultimately leading to the definition of the class PPAD.-there are analogs of the Lemke-Howson algorithm for multi-player games working with manifolds instead of polytopes (see [Rosenmuller ’71] and [Wilson ’71])ApproximationsApproximability of Nash EquilibriumOn the other hand, if is any constant, we know the algorithm of Lipton, Markakis and Mehta, running in quasi-polynomial time for two-player n strategy games.Two obvious questions:- what about functions that are inverse polynomial in the size of the game? - are there polynomial time algorithms for fixed values of ?(assuming all payoffs in [0,1])From the definition of the problem NASH (defined in terms of finding an - Nash equilibrium for given in the input) and the PPAD-completeness of NASH it follows that computing an - well supported Nash equilibrium (and hence also an -approximate Nash equilibrium) of a game is PPAD-complete for a function[ Normalization Assumption Recall the definition of additive notions of approximation:In order to fairly compare the approximation achieved by our algorithms, we asume that the payoffs of the game are normalized to the set [0,1].Additive approximation guarantees are not scale invariant!( e.g. ( for 2-player games ))If the game is un-normalized, then there is an implicit loss of a factor of umax in the approximation guarantee, where umax is the difference between the maximum and the minimum payoff in the payoff tables of the game. I.e. our guarantee from an approximation algorithm is]Algorithms for Fixed Values of Approximation[Kontogiannis, Panagopoulou, Spirakis ’06], [Feder, Nazerzadeh-Saberi ’06], [Daskalakis, Mehta, Papadimitriou ’06, ’07], [Bosse, Byrka, Markakis ’07], [Spirakis, Tsaknakis ’07]- A long line of research has been trying to improve