Algorithmic Game Theory. Alexander Skopalik

Transcription

1 Algorithmic Game Theory Alexander Skopalik

2 Today Course Mechanics & Overview Introduction into game theory and some examples Chapter 1: Selfish routing

3 Alexander Skopalik Office hours: By appointment Room F Phone:

4 Course Mechanics 2h lecture, 1h tutorial lectures: Thu 09:15 10 :45 Room F tutorial groups Tue 13:15-14:00 Room F0 560 Thu 11:00-11:45 Room F1 110 Starting next week.

5 Exercises and Exams Weekly exercise sheets Solutions will be discussed in class (by you). You do not have to hand in solutions. However: You have to solve 50% of the exercises and present at least one solution during the semester. There will be a list of attendants and you mark which problem you solved. This should be 50% at the end of the semester. You will be asked (randomly) to present a solution that you have (marked as) solved. Unless someone volunteers (hint!)

6 Textbook & Website Book: Algorithmic Game Theory, Edited by Nisan, Roughgarden, Tardos, and Vazirani. Available online: ds/nisan_non-printable.pdf Website (exercises, links to papers and further material):

7 What Does Game Theory Study? Interactions of rational decision-makers (agents, players) Decision-makers: humans, robots, computer programs, firms in the market, political parties Interactions: 2 or more agents act simultaneously or consequently Rational: each agent has preferences over outcomes and chooses an action that is most likely to lead to the best feasible outcome

8 Why Study Game Theory? To understand the behavior of others in strategic situations To know how to alter our own behavior in such situations to gain advantage To predict the outcome of strategic situations To be able to design systems such that the desired outcome is ensured

9 Why Algorithmic Game Theory Many games are now being played in the internet. Auctions (e.g. google) Behavior can now be measured by computers even in the offline world. Car traffic One may want to predict or influence the outcome of a game. Therefore we need to be able to compute the outcome. One may want to design and implement mechanism that deal with strategic behavior.

10 Game Theory vs. Optimization Think of traffic routing for example. Optimization Global view optimizer controls all variables Global objective function Result: globally optimal solution Games Each agent controls/influences a part of the environment Agents each have their own objective functions Individual success in making choices depends on the choices of others Result:?

11 What do we study in this lecture? Basics of game theory. Part I: Games and Equilibria Existence and Efficiency of equilibria. Complexity and computation of equilibria Zero sum games, Potential games, congestion games, stable matching, repeated games (?) Connection to current research Part II: Social choice and mechanism design Impossibility theorems The famous VCG-mechanism

12 Required Background Basic math Complexity and algorithms NP-hardness Reductions O-notation Linear programming Convex optimization Flow problems Warning: This is a theoretical computer science lecture

13 Introduction into game theory and some examples

14 Strategic Games 1 Set of players N For each player i N a set of strategies S i Utility or cost function u i : S 1 S 2 R maps a pure strategy profile to a utility/cost value for each player. Example: Prisoner s Dilemma (cost) Silent Confess Silent 1 / 1 10 / 0 Confess 0 / 10 6 / 6

15 Strategic Games 2 All players are rational, they are only interested in their own utility. One shot simultaneous move games What would a player choose in our Example? Confess is a dominant strategy Both choose Confess (cost) Silent Confess Silent 1 / 1 10 / 0 Confess 0 / 10 6 / 6

16 2nd Example Battle of the sexes A couple (Man & Woman) decides on how to spend the evening together. She prefers to watch soccer, he wants to go to the movies. Which pure strategy profiles are plausible outcomes? (utilities) Soccer Movie Soccer 3 / 5 0 / 0 Movie 0 / 1 5 / 3

17 Basic Solution Concepts Pure strategy profile = Each player chooses exactly one strategy. Pure Nash equilibrium: A pure strategy profile is a Nash equilibrium if no player can do better by unilaterally changing his or her strategy. A PNE is optimal for each player, given that the strategies of the other players are fixed.

18 What are the pure Nash equilibria? (utilities) Soccer Movie Soccer 3 / 5 0 / 0 Movie 0 / 1 5 / 3 (cost) Silent Confess Silent 1 / 1 10 / 0 Confess 0 / 10 6 / 6

19 Is there always a pure Nash equilibrium? (Utility) Rock Paper Scissors Rock 0 / 0 0 / 1 1 / 0 Paper 1 / 0 0 / 0 0 / 1 Scissors 0 / 1 1 / 0 0 / 0

20 Mixed strategies A mixed strategy is a probability distribution of the set of strategies. For a player i with strategy set S i = S 1 i,, S k i a mixed strategy is a vector x i = (x 1 i,, x k i ) with k i=1 x i k = 1 x i j denotes the probability of player i playing strategy S i j. A mixed strategy profile is a vector of mixed strategies, one for each player. (i.e., a vector of vectors).

21 Example Set of players N = 1,2 Set of strategies: S 1 = {Rock, Paper, Scissors} S 2 = {Rock, Paper, Scissors} Utility functions u 1 and u 2 : see table Examples: A mixed strategy: ( 1 2, 1 3, 1 6) A mixed strategy profile: (( 1 2, 1 3, 1 6),( 1 3, 1 3, 1 3)) (Utility) Rock Paper Scissors Rock 0 / 0 0 / 1 1 / 0 Paper 1 / 0 0 / 0 0 / 1 Scissors 0 / 1 1 / 0 0 / 0

22 John F. Nash Movie: A Beautiful Mind Non-Cooperative Games, PhD Thesis, Princeton University, pages. Theorem: Nash equilibrium (in mixed strategies) always exists in finite strategic games. Proof later in the lecture

23 What is the mixed Nash equilibrium? (Utility) Rock Paper Scissors Rock 0 / 0 0 / 1 1 / 0 Paper 1 / 0 0 / 0 0 / 1 Scissors 0 / 1 1 / 0 0 / 0 (( 1 3, 1 3, 1 3),( 1 3, 1 3, 1 3))

24 Another example: Auction Set of strategies of player i is S i = R 0 + (his bid) Player i values the item with v i. Utility of player i if he wins the auction is price v i and 0 otherwise. What is the price on ebay? Which strategy should a player choose?

25 Chapter 1 Selfish Routing and the Price of Anarchy

26 Wardrop's Traffic Model (1952) Graph G = V, E with latency functions l e (x) on the edges. k commodities with demand r i Flow of volume r i is sent from source s i to target s i (can be split and send via several paths). Latency function maps the maps the amount of flow on an edge to a cost (latency) value

27 Some notation

28 Equilibrium Flow Intuitively, the flow consists of (infinitely many) particles of infinitesimal size. A flow is in equilibrium if no flow particle can choose a cheaper path. This model is a good approximation of real-world street traffic.

29 Example/Braess Paradox

30 Cost of a flow The average cost of a flow in equilibrium can be higher than the cost of an optimal flow.

31 Price of Anarchy

32 Price of Anarchy How good is a Wardrop equilibrium compared to an optimal flow? Definition: Price of anarchy = cost in equilibrium optimal cost

33 Preparation Lemma (Variational inequality) A flow f is an equilibrium flow if and only if for every flow g the following holds: P P f p g p l p f p 0 Proof Follows from equilibrium inequality Exercise

34 Proof (Theorem)

35 Proof

36 Proof

37 Proof

38 Polynomial latency functions