Patchrawat Uthaisombut University of Pittsburgh Adapted from: Game Theoretic Approach in Computer Science CS3150, Fall 2002 Introduction to Game Theory Patchrawat Uthaisombut University of Pittsburgh

Outline Example: Restaurant Game Formal Definition of Games Goal: Computing outcome of a game Examples: Computing game outcomes Mixed Strategies Selfish Routing and Price of Anarchy

Restaurant Game Wendy’s or Dusty’s Malcolm Julia

Payoffs Julia Wendy’s Dusty’s Malcolm 2,1 0,0 1,2 Wendy's: Fast, cheap food Dusty's: Good, expensive food

A Play of the Restaurant Game The play Row player chooses Dusty's. Column player chooses Dusty's. The Outcome They meet at Dusty's The Payoff Row player gets 1. Column player gets 2. Wendy's Dusty's 2,1 0,0 1,2

Outline Example: Restaurant Game Formal Definition of Games Goal: Computing outcome of a game Examples: Computing game outcomes Mixed Strategies Selfish Routing and Price of Anarchy

Components of a Strategic Game Players Who is involved? Rules Who moves when? What does a player know when he/she moves? What moves are available? Outcomes For each possible combination of actions by the players, what’s the outcome of the game. Payoffs What are the players’ preferences over the possible outcomes?

Key Assumptions Common knowledge Rationality of Players Everyone is aware of all player choices and payoff functions Rationality of Players Player will move to optimize individual payoff All utility is expressed in the payoff function

Formal Definition of Strategic Game A strategic game is a 3-tuple (n,A,u) The number of players n. For 1<i<n, a set Ai of actions available for player i. For 1<i<n, a payoff function ui:A1…An  R for player i. Notice that we omit the “outcomes”. We map “the combinations of actions” directly to “payoffs”. Wendy's Dusty's 2,1 0,0 1,2

Restaurant Game as a Strategic Game Wendy's Dusty's 2,1 0,0 1,2 Players: n = 2 Player 1 = Malcolm Player 2 = Julia Actions: A1 = {Wendy's, Dusty's } A2 = {Wendy's, Dusty's } Payoffs: u1(Wendy's,Wendy's ) = 2 u1(Wendy's,Dusty's ) = 0 u1(Dusty's,Wendy's ) = 0 u1(Dusty's,Dusty's ) = 1 u2(Wendy's,Wendy's ) = 1 u2(Wendy's,Dusty's ) = 0 u2(Dusty's,Wendy's ) = 0 u2(Dusty's,Dusty's ) = 2

Outline Example: Restaurant Game Formal Definition of Games Goal: Computing outcome of a game Examples: Computing game outcomes Mixed Strategies Selfish Routing and Price of Anarchy

Goal: Compute Outcome Given a game, compute what the outcome should be Key assumption: Rationality of players Ideas Best response Nash equilibrium Dominant action or strategy Dominated action or strategy

Notations x  Ak (a) = (a1, a2,…, an)  A1A2…An = A x is an action or a strategy of player k Ak is a set of available actions for player k (a) = (a1, a2,…, an)  A1A2…An = A a profile of actions; one action from each player (a) = (X,G,H,L,S) (a-k) = (a) \ ak  A1…Ak-1Ak+1…An = A-k actions of everybody except player k (a-2) = (X,_,H,L,S) (a-k,y) = (a-k)  y (a-2,M) = (X,M,H,L,S) (a-k,ak) = (a) Variable i is a dummy variable.

Best Response Action An action x of player k is a best response to an action profile (a-k) if uk(a-k,x) > uk(a-k,y) for all y in Ak. Wendy's Dusty's 2,1 0,0 1,2 Confess Deny -5,-5 0,-10 -10,0 -1,-1

Nash Equilibrium (local optimum) An action profile (a) is a Nash equilibrium if for every player k, ak is a best response to (a-k) that is, for every player k, uk(a-k,ak) > uk(a-k,y) for all y in Ak Wendy's Dusty's 2,1 0,0 1,2 Confess Deny -5,-5 0,-10 -10,0 -1,-1

Dominant Action or Strategy An action x of player k is a dominant action if x is a best response to all (a-k) in A-k. That is, uk(a-k,x) > uk(a-k,y) for all y in Ak and any action profile (a-k) in A-k. That is, no matter what the other players do, x is a strategy for player k that is no worse than any other. Titanic Shrek 3,2 1,3 2,1 2,2 Confess Deny -5,-5 0,-10 -10,0 -1,-1

Two Cases Dominant actions dictate the resulting Nash Equilibrium Dominant actions do not exist which means we need other methods

Strictly Dominated Actions An action x of player k is a never-best response or a strictly dominated action if x is not a best response to any action profile (a-k) in A-k That is, for any action profile (a-k) in A-k there exist an action y in Ak such that uk(a-k,x) < uk(a-k,y) That is, no matter what the other players do, x is a strategy for player k that she should never use. Titanic Shrek Sleep 3,1 1,3 1,2 2,3 2,1 2,2

Iterated Elimination of Dominated Actions Procedure Successively remove a strictly dominated action of a player from the game table until there are no more strictly dominated actions Removing a dominated action Reduce the size of the game May make another action dominated May make another action dominant If there is only 1 outcome remaining, the game is said to be dominant solvable. that outcome is the unique Nash equilibrium of the game

Weakly Dominated Actions An action x of player k is a weakly dominated action if for any action profile (a-k) in A-k there exists an action y in Ak such that uk(a-k,x) < uk(a-k,y) and there exists an action profile (a-k) in A-k and an action y in Ak such that uk(a-k,x) < uk(a-k,y). Titanic Shrek Sleep 3,1 1,4 2,3 2,2 2,1 1,3 Both Shrek and Sleep are weakly dominated actions for the column player while Shrek is a weakly dominated action for the row player.

Iterated Elimination of Weakly Dominated Actions Procedure Same as before except Remove weakly dominated actions instead of strictly dominated actions Undesirable properties The remaining cells may depend on the order that the actions are removed. May not yield all Nash equilibria.

Best-Response Function A set-valued function Bk Bk(a-k) = {x  Ak | x is a best response to (a-k) } called the best-response function of player k. An action profile (ai) is a Nash equilibrium if ak  Bk(a-k) for all players k. An action x of player k is a dominant action if x  Bk(a-k) for all action profiles (a-k).

Exhaustive Method Begin with a game table. We will incrementally cross out outcomes that are not Nash equilibria as follows: For each player k = 1..n For each profile (a-k) in A-k Compute v = maxxAk uk(a-k, x) Cross out all outcomes (a-k,x) such that uk(a-k, x) < v The remaining outcomes are Nash equilibria.

Example Stand Walk Run Float 62,65 38,74 34,32 Swim 68,38 55,52 31,36 Dive 33,37 32,30 22,28

Solution Stand Walk Run Float 62,65 38,74 34,32 Swim 68,38 55,52 31,36 Dive 33,37 32,30 22,28 74 52 37 68 55 34

Best-Response Table Stand Walk Run Float 62,65 38,74 34,32 Swim 68,38 55,52 31,36 Dive 33,37 32,30 22,28 Stand Walk Run Float X Swim Dive Stand Walk Run Float X Swim Dive Row player’s best-response table Column player’s best-response table

Outline Example: Restaurant Game Formal Definition of Games Goal: Computing outcome of a game Examples: Computing game outcomes Mixed Strategies Selfish Routing and Price of Anarchy

The Prisoners’ Dilemma The confession of a suspect will be used against the other. If both confess, get a reduced sentence. If neither confesses, face only minimum charge. Both players have dominant strategies. Tragedy of the Commons. Confess Deny -5,-5 0,-10 -10,0 -1,-1

Movie Game Two people go to a movie theatre. Titanic Shrek 3,2 1,3 2,1 2,2 Column has a dominant strategy. Row doesn’t. Equilibrium at Shrek, Shrek

Restaurant Game Malcolm and Julia go to a restaurant. Julia Wendy's Dusty's Malcolm 2,1 0,0 1,2 No dominant or dominated strategies. Nash equilibria at (2,1) and (1,2)

Concert Game Suppose both Malcolm and Julia are going to a concert instead of a dinner. Both like Mozart better than Mahler. Mozart Mahler 2,2 0,0 1,1 No dominant strategies or dominated strategies. Two nash equilibria. Outcome given full rationality?

Chicken Game Malcolm and Julia dare one another to drive their cars straight into one another. Julia Swerve Straight Malcolm 0,0 -1,1 1,-1 -3,-3 No dominant or dominated strategies. (Swerve, Straight) and (Straight, Swerve) are 2 nash eq

Matching Pennies Head Tail 1,-1 -1,1 No dominant or dominated strategies. There is no Nash equilibrium

Outline Example: Restaurant Game Formal Definition of Games Goal: Computing outcome of a game Examples: Computing game outcomes Mixed Strategies Selfish Routing and Price of Anarchy

Randomness in Payoff Functions 2002 US open Final match. Serena is about to return the ball. She can either hit the ball down the line (DL) or crosscourt (CC) Venus must prepare to cover one side or the other Venus Williams DL CC Serena Williams 50,50 80,20 90,10 20,80

Mixed Strategies What is a mixed strategy? Example: Suppose Ak is the set of pure strategies for player k. A mixed strategy for player k is a probability distribution over Ak. An actual move is chosen randomly according to the probability distribution. Example: Ak = { DL, CC } “DL 60%, CC 40%” is a mixed strategy for k.

Need for Mixed Strategies Multiple pure-strategy Nash equilibria No pure-strategy Nash equilibria Games where players prefer opposite outcomes Matching Pennies Chicken Sports Attack and Defense Each player does very badly if her action is revealed to the other, because the other can respond accordingly. Want to keep the other guessing. Mixed strategy Nash equilibrium always exists.

Expectation Suppose X is a random variable. Suppose X = 5 with probability 0.5 Suppose X = 6 with probability 0.3 Suppose X = 0 with probability 0.2 Then E[X] = 5*0.5 + 6*0.3 + 0*0.2 = 2.5 + 1.8 + 0 = 4.3 In general, if X = vi with probability pi Then E[X] = Σ vi pi

Mixed Strategies in the Chicken Games Mixing 2 pure strategies Swerve with probability p and Straight with probability (1-p) A continuous range of mixed strategies. Julia Swerve Straight Malcolm 0, 0 -1, 1 1, -1 -2, -2 p-mix

Mixed Strategies in the Chicken Games Julia Swerve Straight q-mix Malcolm 0, 0 -1, 1 1, -1 -2, -2 p-mix

Finding Mixed Strategy Nash Equilibrium Compute Row’s payoffs as a function of q. Find q that make Row’s payoffs indifferent no matter what pure strategy she chooses. Plot Row’s best-response curve. Do steps 1-3 for the Column player and p. Plot Row’s and Column’s best-response curves together. Points where the 2 curves meet are Nash equilibria.

Why it is an equilibrium? It is a Nash equilibrium because Malcolm can’t change his strategy to do better and Julia can’t change her strategy to do better Why can’t Malcolm do better? Julia chooses a mix such that it doesn’t matter what Malcolm does. Why can’t Julia do better? Malcolm chooses a mix such that it doesn’t matter what Julia does.

Exercise Find mixed strategy Nash equilibrium in the following game. Tennis match Venus DL CC Serena 50,50 80,20 90,10 20,80

Outline Example: Restaurant Game Formal Definition of Games Goal: Computing outcome of a game Examples: Computing game outcomes Mixed Strategies Selfish Routing and Price of Anarchy

Selfish Routing s t L(x) = x L(x) = 1 Input Questions: A directed graph G = (V,E) Set of source-destination pairs {(si,ti)} where ri units of flow must be transmitted from si to ti Each infinitesimal unit of flow is controlled by a selfish agent seeking to minimize its own latency. Latency functions L on each edge e Le(x) is latency of edge e given load x on e Questions: Identify the Nash Equilibria of the system Price of Anarchy: How bad can the total latency of a Nash Equilibrium be compared to that of a socially optimal solution?

Simple Example 1 L(x) = 1 s t L(x) = x (s,t) demand is 1 unit What is optimal flow to minimize total latency? What is Nash equilibrium? Price of Anarchy in this example?

Simple Example 2 L(x) = 1 s t L(x) = xp for some integer p > 0 (p+1)-1/p to lower link, remainder to upper link Latency of 1 – p (p+1)-(p+1)/p (s,t) demand is 1 unit What is optimal flow to minimize total latency? What is Nash equilibrium? Price of Anarchy in this example?

Braess’ Paradox L(x) = x v L(x) = 1 s t L(x) = x L(x) = 1 w s t L(x) = x L(x) = 1 w (s,t) demand is 1 unit What is optimal flow to minimize total latency? What is Nash equilibrium? Price of Anarchy in this example?

Price of Anarchy Approximation Algorithms Online Algorithms Lack of unbounded computing power leads to loss of optimality Online Algorithms Lack of complete information leads to loss of optimality Noncooperative Games Lack of coordination leads to loss of optimality