Introduction 1 TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAA A

 Introduction to Game Theory  Examples  Matrix form Games  Utility  Solution concepts  Dominant Strategies  Nash Equilibria  Complexity  Mechanism Design: reverse game theory 2

 The study of Game Theory in the context of Computer Science, in order to reason about problems from the perspective of computability and algorithm design. 3

 Computing involves many different selfish entities. Thus involves game theory.  The Internet, Intranet, etc. ◦ Many players (end-users, ISVs, Infrastructure Providers) ◦ Players wish to maximize their own benefit and act accordingly ◦ The trick is to design a system where it’s beneficial for the player to follow the rules 4

 Theory ◦ Algorithm design ◦ Complexity ◦ Quality of game states (Equilibrium states in particular) ◦ Study of dynamics  Industry ◦ Sponsored search ◦ Other auctions 5

 Rational Player ◦ Prioritizes possible actions according to utility or cost ◦ Strives to maximize utility or to minimize cost  Competitive Environment ◦ More than one player at the same time Game Theory analyzes how rational players behave in competitive environments 6

 Matrix representation of the game 7 Thieves honor Defect Thieves honor 3,36,2 Defect2,65,5 Row Player Column Player 2 < 3 5 < 6

 It is a dominant strategy to confess  A dominant strategy is a “solution concept” 8 Thieves honor Defect Thieves honor 3,36,2 Defect2,65,5 6,10

 Internet Service Providers (ISP) often share their physical networks for free  In some cases an ISP can either choose to route traffic in its own network or via a partner network 9

 ISP 1 needs to route traffic from s 1 to t 1  ISP 2 needs to route traffic from s 2 to t 2  The cost of routing along each edge is one 10 A B

 ISP 1 routes via B: ◦ Cost for ISP 1 : 1 ◦ Cost for ISP 2 : 4 11

 Cost matrix for the game: 12 AB A3,36,2 B2,65,5 ISP 1 ISP 2 B,A: s 1 to t 1 B,A: s 2 to t 2 Prisoners Dilemma Again

 The game consists of only one ‘turn’  All the players play simultaneously and are unaware of what the other players do  Players are selfish, seek to maximize their own benefit 13

 N = {1,…,n} players  Player i has actions We will say “action” or “strategy”  The space of all possible action vectors is  A joint action is the vector a ∈ A  Player i has a utility function If utility is negative we may call it cost 14

 A strategic game: 15 Players Actions of each player Utility of each player

 Action a i of player i is a weakly dominant strategy if: 16 Action a i of player i is a strongly dominant strategy if:

 An outcome a of a game is Pareto optimal if for every other outcome b, some player will lose by changing to b 17 Vilfredo Pareto

St. Petersburg Paradox: ◦ Toss a coin until tails, I pay you ◦ What will you pay me to play? 18 “Utility of Money”, “Bernulli Utility”

Completeness: Transitivity: Continuity: Independence: 19 Preferences over lotteries

20 Utility function over lotteries, real valued, expected utility maximization

Gamble A: 100% € 1,000,000 Gamble B: 10% € 5,000,000 89% € 1,000,000 1% Nothing Gamble C: 11% € 1,000,000 89% Nothing Gamble D: 10% € 5,000,000 90% Nothing 21 Gamble A or B? Gamble C or D? Experimental ”Fact”: Experimental “Fact”:

Gamble A: 100% € 1,000,000 Gamble B: 10% € 5,000,000 89% € 1,000,000 1% Nothing Gamble C: 11% € 1,000,000 89% Nothing Gamble D: 10% € 5,000,000 90% Nothing 22 “Fact”:

23 VNM Axioms Expected Utility Maximization Mixed Nash Equilibrium exists

 Assume there’s a shared resource (network bandwidth) and N players.  Each player “uses” the common resource, by choosing X i from [0,1]. If Otherwise, 24

25 Given that the other players are fixed, what Is the best response?

26 This is an equilibrium No player can improve

27 The case for Privatization or central control of commons

 A Nash Equilibrium is an outcome of the game in which no player can improve its utility alone:  Alternative definition: every player’s action is a best response: 28

 The payoff matrix: 29

 The payoff matrix: 30 Row player has no incentive to move up

 The payoff matrix: 31 Column player has no incentive to move left

 The payoff matrix: 32 So this is an Equilibrium state

 The payoff matrix: 33 Same thing here

 2 players need to send a packet from point O to the network.  They can send it via A (costs 1) or B (costs 2) 34

 The cost matrix: 35

 The cost matrix: 36 Equilibrium states

 2 players, each chooses Head or Tail  Row player wins if they match the column player wins if they don’t  Utility matrix: 37

 2 players, each chooses Head or Tail  Row player wins if they match the column player wins if they don’t  Utility matrix: 38 Row player is fine, but Column player wants to move left

 2 players, each chooses Head or Tail  Row player wins if they match the column player wins if they don’t  Utility matrix: 39 Column player is fine, but Row player wants to move up

 2 players, each chooses Head or Tail  Row player wins if they match the column player wins if they don’t  Utility matrix: 40 Row player is fine, but Column player wants to move right

 2 players, each chooses Head or Tail  Row player wins if they match the column player wins if they don’t  Utility matrix: 41 Column player is fine, but Row player wants to move down

 2 players, each chooses Head or Tail  Row player wins if they match the column player wins if they don’t  Utility matrix: No equilibrium state! 42

 Players do not choose a pure strategy (one specific strategy)  Players choose a distribution over their possible pure strategies  For example: with probability p choose Heads, and with probability 1-p choose Tails 43

 Row player chooses Heads with probability p and Tails with probability 1-p  Column player chooses Heads with probability q and Tails with probability 1-q  Row plays Heads:  Row plays Tails: 44

 Each player selects where is the set of all possible distributions over A i  An outcome of the game is the Joint Mixed Strategy  An outcome of the game is a Mixed Nash Equilibrium if for every player 45

 2 nd definition of Mixed Nash Equilibrium:  Definition:  Property of Mixed Nash Equilibrium: 46

 No pure strategy Nash Equilibrium, only Mixed Nash Equilibrium, for mixed strategy (1/3, 1/3, 1/3). 47

 N ice cream vendors are spread on the beach  Assume that the beach is the line [0,1]  Each vendor chooses a location X i, which affects its utility (sales volume).  The utility for player i : X 0 = 0, X n+1 = 1 48

 For N=2 we have a pure Nash Equilibrium: No player wants to move since it will lose space  For N=3 no pure Nash Equilibrium: The player in the middle always wants to move to improve its utility /2 0 1

 If instead of a line we will assume a circle, we will always have a pure Nash Equilibrium where every player is evenly distanced from each other: 50

  ame/ ame/ 51

 N companies are producing the same product  Company I needs to choose its production volume, x i ≥ 0  The price is determined based on the overall production volume,  Each company has a production cost:  The utility of company i is: 52

 Case 1: Linear price, no production cost ◦ Utility: ◦ Pure Nash Equilibrium is reached at: 53

 Case 2: Harmonic price, no production cost ◦ Company i’s utility: ◦ Companies have incentive to produce as much as they can – no pure or mixed Nash Equilibrium 54

 n players wants to buy a single item which is on sale  Each player has a valuation for the product,  Assume WLOG that  Each player submits its bid,, all players submit simultaneously. 55

 Case 1: First price auction ◦ The player with the highest bid wins ◦ The price equals the bid ◦ 1 st Equilibrium is:  The first player needs to know the valuation of the second player – not practical ◦ 2 nd Equilibrium is: 56

 Case 2: Second price auction: Vickrey Auction ◦ The player with the highest bid wins ◦ The price equals the second highest bid  No incentive to bid higher than one’s valuation - a player’s utility when it bids its valuation is at least as high than when it bids any other value  This mechanism encourages players to bid truthfully  Mechanism Design: reverse game theory – set up a game so that the equilibria has a desired property 57

58 pure Nash mixed Nash correlated eq no regret best- response dynamics

59  a directed graph G = (V,E)  k source-destination pairs (s 1,t 1 ), …, (s k,t k )  a rate (amount) r i of traffic from s i to t i  for each edge e, a cost function c e () ◦ assumed nonnegative, continuous, nondecreasing s1s1 t1t1 c(x)=x Flow = ½ c(x)=1 Example: (k,r=1)

60 Traffic and Flows:  f P = amount of traffic routed on s i -t i path P  flow vector f routing of traffic Selfish routing: what are the equilibria? st

61 Some assumptions:  agents small relative to network (nonatomic game)  want to minimize cost of their path Def: A flow is at Nash equilibrium (or is a Nash flow) if all flow is routed on min-cost paths [given current edge congestion] x st 1 Flow =.5 st 1 Flow = 0 Flow = 1 x Example:

62  model, defn of Nash flows by [Wardrop 52]  Nash flows exist, are (essentially) unique ◦ due to [Beckmann et al. 56] ◦ general nonatomic games: [Schmeidler 73]  congestion game (payoffs fn of # of players) ◦ defined for atomic games by [Rosenthal 73] ◦ previous focus: Nash eq in pure strategies exist  potential game (equilibria as optima) ◦ defined by [Monderer/Shapley 96]

63 Def: the cost C(f) of flow f = sum of all costs incurred by traffic (avg cost × traffic rate) st x 1 ½ ½ Cost = ½½ +½1 = ¾

64 Def: the cost C(f) of flow f = sum of all costs incurred by traffic (avg cost × traffic rate) Formally: if c P (f) = sum of costs of edges of P (w.r.t. the flow f), then: C(f) =  P f P c P (f) st st x 1 ½ ½ Cost = ½½ +½1 = ¾

65 Note: Nash flows do not minimize the cost  observed informally by [Pigou 1920]  Cost of Nash flow = = 1  Cost of optimal (min-cost) flow = ½½ +½1 = ¾  Price of anarchy := Nash/OPT ratio = 4/3 st x ½ ½

66 Initial Network: st x1 ½ x 1 ½ ½ ½ cost = 1.5

67 Initial Network: Augmented Network: st x1 ½ x 1 ½ ½ ½ cost = 1.5 st x1 ½ x 1 ½ ½ ½ 0 Now what?

68 Initial Network: Augmented Network: st x1 ½ x 1 ½ ½ ½ cost = 1.5 cost = 2 st x 1 x 1 0

69 Initial Network: Augmented Network: All traffic incurs more cost! [Braess 68]  see also [Cohen/Horowitz 91], [Roughgarden 01] st x1 ½ x 1 ½ ½ ½ cost = 1.5 cost = 2 st x 1 x 1 0