Learning Graphical Games from Behavioral Data: Suﬃcient and Necessary Conditions

Learning Graphical Games from Behavioral Data: Sufficient and Necessary Conditions Asish Ghoshal Jean Honorio [email protected] [email protected] Department of Computer Science, Purdue University, West Lafayette, IN, U.S.A. - 47907. Abstract identified, from congressional voting records, the most influential senators in the U.S. congress — a small In this paper we obtain sufficient and neces- set of senators whose collective behavior forces every sary conditions on the number of samples re- other senator to a unique choice of vote. Irfan and quired for exact recovery of the pure-strategy Ortiz [4] also observed that the most influential sena- Nash equilibria (PSNE) set of a graphical tors were strikingly similar to the gang-of-six senators, game from noisy observations of joint actions. formed during the national debt ceiling negotiations of We consider sparse linear influence games — 2011. Further, using graphical games, Honorio and Or- a parametric class of graphical games with tiz [6] showed that Obama’s influence on Republicans linear payoffs, and represented by directed increased in the last sessions before candidacy, while graphs of n nodes (players) and in-degree of McCain’s influence on Republicans decreased. at most k.Weshowthatonecanefficiently The problems in algorithmic game theory described recover the PSNE set of a linear influence above, i.e., computing the Nash equilibria, comput- game with k2 log n samples, under very O ing the price of anarchy or finding the most influ- general observation models. On the other ential agents, require a known graphical game which hand, we show that ⌦(k log n) samples are is not available apriori in real-world settings. There- necessary for any procedure to recover the fore, Honorio and Ortiz [6] proposed learning graphi- PSNE set from observations of joint actions. cal games from behavioral data, using maximum likelihood estimation (MLE) and sparsity-promoting meth- ods. On the other hand, Garg and Jaakkola [7] provide 1IntroductionandRelatedWork adiscriminativeapproachtolearnaclassofgraphical games called potential games. Honorio and Ortiz [6] Non-cooperative game theory is widely considered as and Irfan and Ortiz [4] have also demonstrated the use- an appropriate mathematical framework for studying fulness of learning sparse graphical games from behav- strategic behavior in multi-agent scenarios. In Non- ioral data in real-world settings, through their analysis cooperative game theory, the core solution concept of the voting records of the U.S. congress as well as the of Nash equilibrium describes the stable outcome of U.S. supreme court. the overall behavior of self-interested agents — for instance people, companies, governments, groups or In this paper, we obtain necessary and sufficient condi- autonomous systems — interacting strategically with tions for recovering the PSNE set of a graphical game each other and in distributed settings. in polynomial time. We also generalize the observation model from Ghoshal and Honorio [8], to arbitrary Over the past few years, considerable progress has distributions that satisfy certain mild conditions. Our been made in analyzing behavioral data using game- polynomial time method for recovering the PSNE set, theoretic tools, e.g. computing Nash equilibria [1, 2, 3], which was proposed by Honorio and Ortiz [6], is based most influential agents [4], price of anarchy [5] and on using logistic regression for learning the neighbor- related concepts in the context of graphical games. hood of each player in the graphical game, indepen- In political science for instance, Irfan and Ortiz [4] dently. Honorio and Ortiz [6] showed that the method of independent logistic regression is likelihood consis- Proceedings of the 20th International Conference on Artifi- cial Intelligence and Statistics (AISTATS) 2017, Fort Laud- tent; i.e., in the infinite sample limit, the likelihood erdale, Florida, USA. JMLR: W&CP volume 54. Copy- estimate converges to the best achievable likelihood. right 2017 by the author(s). In this paper we obtain the stronger guarantee of re- Sufficient and Necessary Conditions for Learning Graphical Games covering the true PSNE set exactly. among players, i.e., the payoffof the i-th player only depends on the actions of its (incoming) neighbors. Finally, we would like to draw the attention of the reader to the fact that `1-regularized logistic regression has been analyzed by Ravikumar et. al. [9] in 2.2 Linear Influence Games the context of learning sparse Ising models. Apart Irfan and Ortiz [4] and Honorio and Ortiz [6], intro- from technical differences and differences in proof tech- duced a specific form of graphical games, called Linear ` niques, our analysis of 1-penalized logistic regres- Influential Games,characterizedbybinaryactions,or sion for learning sparse graphical games differs from pure strategies, and linear payofffunctions. We as- Ravikumar et. al. [9] conceptually — in the sense sume, without loss of generality, that the joint ac- that we are not interested in recovering the edges of tion space = 1, +1 n.Alinearinfluencegame X {− } the true game graph, but only the PSNE set. There- between n players, (n)=(W, b),ischaracterized fore, we are able to avoid some stronger conditions G n n by (i) a matrix of weights W R ⇥ , where the required by Ravikumar et. al. [9], such as mutual 2 entry wij indicates the amount of influence (signed) incoherence. that the j-th player has on the i-th player and (ii) n abiasvectorb R , where bi captures the prior 2 2Preliminaries preference of the i-th player for a particular action x 1, +1 . The payoffof the i-th player is a lin- i 2{− } In this section we provide some background informa- ear function of the actions of the remaining players: T tion on graphical games introduced by Kearns et. al. ui(xi, x i)=xi(w ix i bi),andthePSNEsetis − − − − [10]. defined as follows: T ( (n)) = x ( i) xi(w ix i bi) 0 , (2) 2.1 Graphical Games NE G | 8 − − − ≥ where w i denotes the i-th row of W without the i- A normal-form game in classical game theory is de- − G th entry, i.e. w i = wij j = i .Notethatwehave fined by the triple =(V, , ) of players, actions and − { | 6 } G X U diag(W)=0. Thus, for linear influence games, the payoffs. V is the set of players, and is given by the set weight matrix W and the bias vector b,completely V = 1,...,n ,iftherearen players. is the set of { } X specify the game and the PSNE set induced by the actions or pure-strategies and is given by the Cartesian def game. Finally, let (n, k) denote a sparse game over product = i V i, where i is the set of pure- G 2 n players where the in-degree of any vertex is at most X ⇥ X X def n strategies of the i-th player. Finally, = ui ,is k. U { }i=1 the set of payoffs, where ui : i j V i j R spec- X ⇥ 2 \ X ! ifies the payofffor the i-th player given its action and 3ProblemFormulation the joint actions of the all the remaining players. An important solution concept in the theory of non- Now we turn our attention to the problem of learn- cooperative games is that of Nash equilibrium.Fora ing graphical games from observations of joint actions def non-cooperative game, a joint action x⇤ is a pure- only. Let ⇤ = ( ⇤(n, k)).Weassumethat 2X NE NE G strategy Nash equilibrium (PSNE) if, for each player there exists a game ⇤(n, k)=(W⇤, b⇤) from which i x argmax u (x , x ) x = x j = G (l) m , i⇤ xi i i i ⇤ i , where ⇤ i j⇤ a“noisy”dataset = x l=1 of m observations is 2 2X − − { | 6 D { } (l) i . In other words, x⇤ constitutes the mutual best- generated, where each observation x is sampled in- } response for all players and no player has any incentive dependently and identically from some distribution . P to unilaterally deviate from their optimal action x⇤ We will use two specific distributions and , which i Pg Pl given the joint actions of the remaining players x⇤ i. we refer to as the global and local noise model, to pro- − The set of all pure-strategy Nash equilibrium (PSNE) vide further intuition behind our results. In the global for a game is defined as follows: noise model, we assume that a joint action is observed G from the PSNE set with probability q ( ⇤ /2n, 1), g 2 |N E | ( )= x⇤ ( i V ) xi⇤ argmax ui(xi, x⇤ i) . i.e. NE G 8 2 2 x − ⇢ i2Xi qg1 [x ⇤] (1 qg)1 [x / ⇤] (1) g(x)= 2NE + − 2NE . (3) P ⇤ 2n ⇤ |N E | −|NE | Graphical games, introduced by Kearns et al. [10], In the above distribution, qg can be thought of as the are game-theoretic analogues of graphical models. A “signal” level in the data set, while 1 q can be − g graphical game G is defined by the directed graph, G = thought of as the “noise” level in the data set. In (V,E), of vertices and directed edges (arcs), where ver- the local noise model we assume that the joint ac- tices correspond to players and arcs encode “influence” tions are drawn from the PSNE set with the action of Ghoshal, Honorio each player corrupted independently by some Bernoulli To get some intuition for the above assumption, con- noise. Then in the local noise model the distribution sider the global noise model. In this case we have that q over joint actions is given as follows: p˜ =˜p =(1 q ), p = g/ ⇤ ,and x min max − g max |N E | 8 2 ⇤, (x)=pmax.Forthelocalnoisemodel,con- 1 n NE P 1[xi=yi] 1[xi=yi] sider, for simplicity, the case when there are only two l(x)= qi (1 qi) 6 , P ⇤ − 1 2 y i=1 joint actions in the PSNE set: ⇤ = x , x ,such |N E | 2XNE⇤ Y 1 2 1 NE2 { } that x1 =+1,x1 = 1 and xi = xi =+1for all i =1.

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