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

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

3. Restaurant Game
Wendy’s or
Dusty’s
Malcolm
Julia

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

5. 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

6. 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

7. 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?

8. 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

9. 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

10. 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

11. 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

12. 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

13. 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.

14. 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

15. 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

16. 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

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

18. 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

19. 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

20. 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.

21. 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.

22. 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).

23. 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.

24. 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

25. 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

26. 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

27. 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

28. 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

29. 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

30. 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)

31. 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?

32. 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

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

34. 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

35. 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

36. 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.

37. 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.

38. 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* * *0.2
= = 4.3
In general, if X = vi with probability pi
Then E[X] = Σ vi pi

39. 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

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

41. 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.

42. 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.

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

44. 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

45. 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?

46. 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?

47. 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?

48. Braess’ Paradox L(x) = x v L(x) = 1 s t L(x) = x L(x) = 1 ws 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?

49. 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