CS 294 Topics in Algorithmic Game Theory September 27, 2011Lecture 5Lecturer: Christos Papadimitriou Scribe: Seung Woo Shin, Patrick Ghannage1 Approximate Nash equilibria1.1 IntroductionThe solution concept of Nash equilibrium is vastly influential because intellectually it is useful andneat, and philosophically the fact that every game has a Nash eq ui l i br i um is reassuring. However,we have also seen that it is intractible unless PPAD ⊆ P, and this certainly poses a problem tous.Traditionally, there are three types of remedies to the hardness of a problem. The first is tosolve interesting special cas es. In fact, we have already touched some work in this domain throughexamples of zero-sum games and separab l e network games. In this section, we discuss the secondremedy, which is to try to find an approximate solution to th e problem. We start with the followingdefinition.Definition 1. A strategy profile (x1,...,xn) is an �-approximate Nash equilibrium if for everyplayer i and every action a in the support of xi,E[ui(a, x−i)] ≥ E[ui(b, x−i)] − � for any actionb ∈ Ai.Certainly, for this definition to make sense, we have to assume that the utility function isnormalized (ui(�x) ∈ [0, 1]). Note that � acts additively, i n contrast to the common practice in thefield of approximation algorithms where � usually acts multiplicatively. In fact, i t is known thatthe multiplicative version is PPAD-complete [1].An unsatisfactory aspect of this concept is that unlike approximate solutions in most optimiza-tion problems, an approximate Nash equilibrium is hardly useful because once it is found, eachplayer immediately knows how they can achieve better payoffs by deviating from it. Thus, an �-approximate Nash equilibrium is not a compelling proposal unless � is very small. Thus it becomesan important question whether there is an efficient scheme for approximating Nash equilibria. Anoptimization problem is said to have a PTAS (polynomial-time approximation scheme) if thereexists an algorithm that computes the �-approximate solution in time O(nf(1/�)). It is said to havea FPTAS (fully polynomial-t i me approximation scheme) if ther e exists an algorithm with timecomplexity O((1/�)cnd).In 2006, Chen et al. [2] proved that a FPTAS for Nash equilibria is impossible unless PPAD ⊆P. Whether there exists a PTAS for this problem st i l l remains to be a major open question.Currently, the smallest value of � for which we have a polynomial algorithm is 0.3393 [3].1.2 A quasi-p o ly no mi al time approximation scheme for Nash equilibriumIn t hi s section, we introduce a quasi-polynomial time algorithm by Lipton, Markakis and Mehta [4]which computes an �-approximate Nash equilibrium in time O(nlog n/�2). The algorithm employsthe f ol lowing simple idea.1Suppose (X, Y ) is a Nash equilibrium of a two-player game. Assuming we know (X, Y ), it is easyto sample e ach player’s action using the distributions X and Y . Thus, we can repeat this samplingprocedure many times to obtain a discret e approximation of the original distribution. Here, t =8 log n/�2samples should suffice for our purpose. Let (ˆX,ˆY ) be the approximate distribution weobtained by sampling. Then , the following lemma holds.Lemma 1. (ˆX,ˆY ) is an �-approximate Nash equilibrium with high probability.Clearly, all probabilit ie s inˆX andˆY have a denominator t. Thus, even if we do not know what(X, Y ) is, we can search exh aus ti vely on all probability distributions with denominator t to find the�-approximate Nash equilibrium in time O(nt)=O(nlog n/�2). (Naively, it might appear that thetime complexity should be O(tn), because each of the n probabilities can take values in {0, 1,...,t}.However, we can think of it as distributing t objects into n bins, in which case the number of waysbecomes�t+n−1n−1�∼ nt.) The final step is to prove the above lemma.Proof. The proof consists of two par t s. In the first part, we use a Chernoff-style argument to provethatˆX is close to X with high probability. In this lecture, we will omit this step and directlyproceed to showing that ifˆX is close to X, it must be an �-approximate Nash equilibrium . Forthis, we fi rs t introduce the following definition.2468100.20.40.60.8Figure 1: TV distanceDefinition 2. The TV distance (total variation distance) between two (discrete) distributions D1and D2is de fin ed as follows:||D1− D2||TV=�i|D1(i) − D2(i)|.Now, suppose (X1,...,Xm) and (X�1,...,X�m) are two strategy p rofi l es (m is the number ofplayers). What is the difference in the utilities between the two strategy profiles? Observe that forany player i,|ui(X) − ui(X�)| ≤ umax�a∈A| PrX(a) − PrX�(a)|= umax||X − X�||TV≤ umax�i||Xi,X�i||TV,2where umaxis the biggest payoff that appears in the utility table. This mean s that if two profilesare pairwise close in the TV distance, the payoff of each player will be close. This completes ourproof of the lemma.1.3 Approximate Nash equilibria in anonymous gamesNow we shall study the approximate Nash equilibria in anonymous games, which is a type of asuccinct game. Anonymous games are games in which once you fix a player, the utility functionbecomes symmetric in other players’ actions. In other words, this means that our payoff is com-pletely determined by “how many” of the other players choose each action. In anonymous games,it is usually assumed that there are on l y two possible actions (e.g. do I take BART to San Franciscoor drive?). If n is the numb er of players and s is the number of (pure) strategies (n � s), the rep-resentation size of an anonymous game is sns(using the same argument as in the time complexityanalysis of the LMM algorithm), as opposed to snof an ordinary game.How do we find approximate Nash equilibria in such games? We can use the following algorithmdue to Daskalakis and Papadimitriou [5]. First, pick a positive integer k. As we did i n th e LMMalgorithm, we will disc re t iz e probabilitie s to multiples of 1/k, i.e., we will exhaustively search onall strategies that have probabilities that are multiples of 1/k. Then, we can pick the one thatis closest to the equilibrium. This algorithm has time complexity O( nk+1), because we need toexplore all possible ways of distributing n people into k + 1 bins. In order to fix the value of k,weappeal to the following theorem.Theorem 1. Let X1,...,Xnbe independent Bernoulli random variables
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