Introduction Course basics Course basics Ruben Hoeksma MZH 3320 - PowerPoint PPT Presentation

Algorithmic game theory Ruben Hoeksma October 15, 2019 Introduction Course basics

Course basics Ruben Hoeksma MZH 3320 Webpage: https://www.cslog.uni-bremen.de/teaching/winter19/agt/ Lectures: Monday 12:00 – 14:00 12:15 – 13:45 Tuesday 12:00 – 14:00 12:15 – 13:45 Exercises: ◮ 1 set per week ◮ 1 week to finish each ◮ 60%: +1/3 point ◮ All exercises treated during the lectures are part of the exam

Course basics Ruben Hoeksma MZH 3320 Webpage: https://www.cslog.uni-bremen.de/teaching/winter19/agt/ Lectures: Monday 12:00 – 14:00 12:15 – 13:45 Tuesday 12:00 – 14:00 12:15 – 13:45 Examination: ◮ Oral exam (around 30 minutes) ◮ First question: Say something about your favorite topic/game from the course. ◮ Questions will include proofs and intu¨ ıtion ◮ Anything spoken about during lectures + any material on webpage

Introduction Games, selfish behavior, and equilibria

What is a game?

What is a game? Situations with multiple actors who make their own decisions. ◮ Situations of conflicting interests ◮ Situations of mutual interests ◮ Actors are called players ◮ Each player has some objective ◮ Each player has choices that influence both their own objective and that of others ◮ Each player is rational, i.e., they optimize for their objective ◮ If a player improves their objective by changing their strategy, we say that they have an incentive to do so

Simultaneous game In a simultaneous game, all players, at the same time, choose a strategy from their own strategy space without knowledge about what the other players have done. Definition (Simultaneous game) A simultaneous game is defined by N : Set of n players S i : Set of strategies for each player i ∈ N S = S 1 × S 2 × . . . × S n : set of strategy vectors u i : S → R Utility function for each player i ∈ N

Example: a routing game 10 x 0 o d x 10 ◮ Given this directed graph with origin o and sink d ◮ 10 players want to go from o to d ◮ Cost, c ( x ), for each arc depends on number of players that use it ◮ Cost of each player is the sum of cost of arcs they chose

Example: a routing game 10 x 0 o d x 10 Question: What do the players do? ◮ There are three routes { U , L , Z } ◮ If there are n players:  10 + # U ( s ) + # Z ( s ) if s i = U ,    c i ( s ) = 10 + # L ( s ) + # Z ( s ) if s i = L ,  # L ( s ) + # U ( s ) + 2# Z ( s ) if s i = Z .  

Equilibrium Definition (Equilibrium) An equilibrium is a state in which no player has an incentive to change their strategy. Definition (Dominant strategy equilibrium (DSE)) A strategy vector s ∈ S is a DSE if for each player i ∈ N , all alternative strategies x i ∈ S i , and all strategies of the other players x − i ∈ S − i , we have u i ( s i , x − i ) ≥ u i ( x i , x − i ) . The strategy s i is called a dominant strategy for player i . s − i is the strategy vector s with player i ’s strategy omitted. S − i = S 1 × . . . × S i − 1 × S i +1 × . . . × S n .

Example: a routing game 10 x 0 o d x 10 Question: Does this game have a DSE?  10 + # U ( s ) + # Z ( s ) if s i = U ,    c i ( s ) = 10 + # L ( s ) + # Z ( s ) if s i = L ,  # L ( s ) + # U ( s ) + 2# Z ( s ) if s i = Z .  

Example: Battle of the sexes ◮ Two players N = { Man , Woman } ◮ Two strategies: F : go to the football match; T : go to the theater ◮ Woman prefers going to football and Man prefers going to theater ◮ Both prefer to go anywhere together over going anywhere alone ◮ Normal form: explicit Woman description of utility for all F T strategy combinations (5 , 6) (1 , 1) F Man ◮ For two-player game: matrix T (2 , 2) (6 , 5) ◮ Row/column player Question: Does this game have a dominant strategy equilibrium?

Pure Nash equilibrium Answer: The battle of the sexes game does not have a DSE. Proof. Man and Woman have two strategies F and T . If Woman plays F , Man prefers to play F . If Woman plays T , Man prefers to play T . So neither strategies is dominant for Man and no DSE exists. Definition ((Pure) Nash equilibrium (NE)) A strategy vector s ∈ S is a NE if for each player i ∈ N and all alternative strategies of that player x i ∈ S i , we have u i ( s i , s − i ) ≥ u i ( x i , s − i ) . s i is a best response of player i to s − i .

Example: Battle of the sexes ◮ Two players N = { Man , Woman } ◮ Two strategies: F : go to the football match; T : go to the theater ◮ Woman prefers going to football and Man prefers going to theater ◮ Both prefer to go anywhere together over going anywhere alone Woman F T F (5 , 6) (1 , 1) Man (2 , 2) (6 , 5) T Question: What are the Pure Nash equilibria?

Example: Rock-paper-scissors Definition (Zero-sum game) A zero-sum game is a game where for any strategy vector s ∈ S the sum of the utilities of the players for that strategy vector is zero. � u i ( s ) = 0 i ∈ N Rock-paper-scissors Q: Does RPS have a NE? A: No R P S Proof. For any strategy of the row R (0 , 0) ( − 1 , 1) (1 , − 1) player, there is a strategy for the (1 , − 1) (0 , 0) ( − 1 , 1) P column player that wins. Same the S ( − 1 , 1) (1 , − 1) (0 , 0) other way around, so no pair of strategies are each a best response to each other.

Mixed strategies Definition (Mixed strategy) A mixed strategy of player i ∈ N is a probability distribution over their strategy space S i . A mixed strategy vector is a vector of mixed strategies. Definition (Mixed Nash equilibrium (MNE)) A mixed strategy vector s ∈ S is a MNE if for each player i ∈ N and all alternative strategies of that player x i ∈ S i , we have E [ u i ( s i , s − i )] ≥ E [ u i ( x i , s − i )] .

Example: Rock-paper-scissors (cont.) R P S R (0 , 0) ( − 1 , 1) (1 , − 1) (1 , − 1) (0 , 0) ( − 1 , 1) P S ( − 1 , 1) (1 , − 1) (0 , 0) Claim. Both players playing each strategy with probability 1 3 is a MNE. Proof. Let s be the mixed strategy 1 E [ u r ( s r , s c )] = 9 (3 · − 1 + 3 · 1 + 3 · 0) = 0 1 E [ u r ( x r , s c )] = 3 ( − 1 + 1 + 0) = 0 ∀ x r ∈ S r

Two more. . . Definition (Correlated equilibrium (CorEq)) Let p be a probability distribution over S . p is a CorEq if for each player i ∈ N and all strategies of that player s i , x i ∈ S i , we have � � p ( s i , s − i ) u i ( s i , s − i ) ≥ p ( s i , s − i ) u i ( x i , s − i ) . s − i ∈ S − i s − i ∈ S − i Definition (Coarse correlated equilibrium (CCE)) Let p be a probability distribution over S . p is a CCE if for each player i ∈ N and all alternative strategies of that player x i ∈ S i , we have � � p ( s ) u i ( s ) ≥ p ( s ) u i ( x i , s − i ) . s ∈ S s ∈ S

CorEq and CCE CorEq: � � p ( s i , s − i ) u i ( s i , s − i ) ≥ p ( s i , s − i ) u i ( x i , s − i ) . s − i ∈ S − i s − i ∈ S − i Let player i be the row player in the following representation s 1 s 2 · · · − i − i s 1 i s 2 i . . .

CorEq and CCE CCE: � � p ( s ) u i ( s ) ≥ p ( s ) u i ( x i , s − i ) . s ∈ S s ∈ S Let player i be the row player in the following representation s 1 s 2 · · · − i − i s 1 i s 2 i . . .

Example: Game of chicken (the traffic light) D S D ( − 10 , − 10) (1 , − 1) ( − 1 , 1) (0 , 0) S DSE? No NE? ( D , S ) or ( S , D ) MNE? p i ( D ) = 1 10 , p i ( S ) = 9 10 for i ∈ { 1 , 2 } CorEq? Traffic light { ( S , S ) , ( D , S ) , ( S , D ) }

Example: Rock-paper-scissors (again) R P S R (0 , 0) ( − 1 , 1) (1 , − 1) (1 , − 1) (0 , 0) ( − 1 , 1) P S ( − 1 , 1) (1 , − 1) (0 , 0) CCE: ( R , P ) , ( P , R ) , ( R , S ) , ( S , R ) , ( P , S ) , ( S , P ) all with probability 1 6 . Claim: The above probability distribution is not a CorEq. Proof. We consider the row player playing Rock. Given that the row player plays Rock, the column player plays Paper and Scissors with probability 1 2 each and expected utility equal to 0. If the row player plays Scissors instead their expected utility is 1 2 .

Next time ◮ Existence of equilibria (Nash’s theorem) Exercise set 1 available today. Deadline 21.10.