CS 599: Algorithmic Game Theory August 25, 2010Lecture 1 - Overview of Algorithmic Game TheoryProf. Shang-hua Teng Scribes: Joseph Bebel and Henry Yuen1 Our starting point: game theoryThis class is going to be about Algorithmic Game Theory (AGT). That requires us to understandgame theory a bit. What are the characteristics of game theory?In game theory, the fundamental objects of consideration are games, which involve a number ofplayers and their strategies. In particular, we assume players to be selfishly motivated, or at leasthave an objective each player wishes to optimize. An important concept in game theory is the ideaof equilibrium; intuitively, equilibrium in a game is a situation in which no player has an incentiveto change her strategy. One of the most important results in game theory is what is called Nashequilibrium.One of the main applications of game theory is to model economic situations, ranging from simpletwo player exchanges to huge online marketplaces. Of course, now this begs the question, what iseconomics? An instinctual definition of economics: it is the study of how we all use money.But economics need not be about money. As we shall see later, we can study an idealized marketmodel (albeit one that has been realized at one point in time) that does not use money whatsoever.A better definition, and more textbook-like: economics is the study of scarce resources, each ofwhich have different uses.Most people agree that game theory really took off with von Neumann’s papers and book Theoryof Games and Economic Behavior in the early 20th century. von Neumann, man of many talents,also played a seminal role in the development of computer algorithms. Game theory and computerscience took off on their own paths, but the advent of the internet and electronic commerce hasbrought the two fields together in a meaningful way, giving rise to algorithmic game theory, whichsits at the intersection of economics, theoretical computer science, operations management, andindustrial systems engineering.2 Computer science, and what it can do for you!We all love computer science. We love its algorithms, its theory, and its widespread applicability.In particular, we all love its most important question: P vs NP. The question is about our burningdesire to understand the intrinsic difficulty of computational problems. Answering this questionwill net you one million dollars - there’s definitely some interesting economics in that!Computer science is also heavily concerned with constraint optimization problems, which are for-mulated as follows:1Optimization problem: Maximize/minimize an objective function f(x), subject to the con-straints x ∈ C.Examples of objective functions and constraints can vary wildly. For example, we may want tominimize the cost of building a toy factory, but subject to the constraint that we actually build atleast 100,000 toys.One of the big players in developing the computer science of optimization is AT&T labs, whowished to use this knowledge to solve a very practical problem: improving the reliability of itsphone networks. Because of their efforts, as well as of computer scientists around the world, bythe 1990’s, lots and lots was known about single-objective optimization. Computer scientists thenstudied multiobjective optimization:Multiobjective optimization problem: Optimize f1(x), f2(x), . . . , fk(x) subject to the con-straints x ∈ C.By itself, multiobjective optimization already has many uses in economics. However, multiobjectiveoptimization in itself is still too simple to capture the dazzling complexity in real world economicsituations. For example, optimizing an objective function (or an array of objective functions)is doable and well understood, but what happens when you have many people all competing tooptimize objective functions? One thing we can do is to view these situations as games.Here’s where everything comes together: traditional game theory can answer things about the exis-tence of equilibria for these kinds of games. However, being the devoted computer scientists we are,we also are dying to know equally important things: how complicated is computing equilibrium? Isit possible to reach equilibrium? This is a starting point of algorithmic game theory: applying ideasand models from computer science to game theory. The tools of theoretical computer science arewell-suited towards the study of games, mostly because of our deep understanding of optimizationproblems.Nowadays, the confluence of algorithms and game theory has a very pragmatic application: theinternet, e-commerce, and social networks have realized fantastically complex networks and games.Google AdWords is a prime example of algorithmic game theory in action. So answering questionsin algorithmic game theory has a definite real-world impact.Just as we marked John von Neumann as the father of game theory and computer algorithms, wecan identify Christos Papadimitriou with writing the paper that put together the vision of AGT aswe know it today: in 1991, he wrote The Complexity of the Parity Argument and Other Inefficientproofs of Existence.3 A Brave New WorldSo what is the setup of our new paradigm? Whereas in both single- and multi-objective optimiza-tion, the problem is essentially about one person’s interests, we have to consider multiple playersnow, and their interests, which may - and generally will - be conflicting. For simplicity, we will onlyconsider a single objective function now: that is, every player seeks to optimize their own objectivefunction, but only one of them.2So formally, let’s say we have players p1, . . . , pn, and they each have objective functions u1, . . . , unthey want to optimize, respectively.p1u1(x1, . . . , xn)......pnun(x1, . . . , xn)Player picontrols variable xionly (note that these variables themselves may represent vectors orsome other kind of structure - not just single numbers). One can see where action comes from:Player 1 wishes to optimize his objective function u1, but not all the variables are under his control!He can only choose to change x1in response to what others have chosen for x2, . . . , xn. Of course,each of the xi’s have to satisfy constraints: ximust belong to some constraint set Ci.This is the basic setup of a game. It is ridiculously general, but now we have some language towork with. Before we continue, we will be entertained by a quiz:Quiz 1 Suppose G = (V, E) is a finite directed acyclic graph (DAG). Prove that there
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