The document discusses Nash equilibrium, best responses, coordination games, and iterated elimination of dominated strategies. A Nash equilibrium exists when each player's strategy is a best response to the other players' strategies, meaning no player has an incentive to unilaterally change their strategy. Iterated elimination of dominated strategies involves repeatedly removing strategies that are strictly or weakly dominated until no further dominated strategies remain. The order of elimination matters for weakly dominated strategies but not strictly dominated strategies.

1. From Last Time
• How to solve/ make predictions for the game?
• What is Iterated Elimination of Dominated
Strategies?
• What is a Nash Equilibrium?

2. Nash Equilibrium
Nash Equilibrium:
• A set of strategies, one for each player, such
that each player’s strategy is a best response
to others’ strategies
Everybody is playing a best response
• No incentive to unilaterally change my strategy

3. BEST RESPONSE

4. Solving the game
• How to solve the game when there are no
dominated strategies?

5. Coordination Games
Opera Movie
Opera 2,1 0,0
Movie 0,0 1,2

6. Matching Pennies
L R
L 1,-1 -1, 1
R -1, 1 1,-1

7. Best Response
• Form beliefs about what others will do
• If you believe opponent will play Opera: Best
Response is to play Opera
• Similarly, if you believe opponent will play Movie:
Best Response is to play Movies

8. Coordination Games
Opera Movie
Opera 2,1 0,0
Movie 0,0 1,2

9. Best Response
• Best Response: Action that gives highest
payoff given a belief about others play
• Best Response changes with different beliefs
about opponents play.
• There may be more than one Best Response
for a belief

10. L C R
U 8, 3 0, 4 4,4
M 8,5 1,5 5,3
D 3,7 0,1 2,0

11. Forming Beliefs
• Forming one’s belief is the important part of
strategy
• Success depends upon belief formation

12. Best Response Functions
Consider best response function of i:
Bi(a-i) = {ai is element of Ai: u I (ai,a-i ) ≥ ui(ai',a-i ) for all ai’ that is
element of Ai}
• Set-valued, each member of Bi(a-i) is a best response
to a-i

13. Best Response Functions
a* is a Nash equilibrium if and only if
ai* is element of Bi(a-i*) for every i

14. Best Response Function Examples

15. Best Response Function Examples

16. Best Response Function Examples
Player 2
B1 B2 B3
Player 1 A1 10,10 14,12 14,15
A2 12,14 20,20 28,15
A3 15,14 15,28 25,25
•What is Player 1’s best response to Player 2’s
strategy of B1, B2 or B3?
•What is Player 2’s best response to Player 1’s
strategy of A1, A2 or A3?

17. Best Response in 2-player game
Using best response function to find Nash equilibrium
in a 2-player game
( s1,s2) is a Nash equilibrium if and only if
• player 1’s strategy s1 is her best response to player 2’s
strategy s2
• player 2’s strategy s2 is her best response to player 1’s
strategy s1

18. Battle of Sexes
Player2
Ball Theatre
Player1 Ball 2,1 0,0
Theatre 0,0 1,2
•Ball is Player 1’s best response to Player 2’s strategy Ball
•Ball is Player 2’s best response to Player 1’s strategy Ball
•Hence, (Ball, Ball) is a Nash equilibrium
•Theatre is Player 1’s best response to Player 2’s strategy
Theatre
•Theatre is Player 2’s best response to Player 1’s strategy
Theatre
•Hence, (Theatre, Theatre) is a Nash equilibrium

19. Matching Pennies
Player2
Head Tail
Player1 Head -1,1 1,-1
Tail 1,-1 -1,1
•Head is Player 1’s best response to Player 2’s strategy Tail
•Tail is Player 2’s best response to Player 1’s strategy Tail
•Tail is Player 1’s best response to Player 2’s strategy Head
•Head is Player 2’s best response to Player 1’s strategy Head
•Hence, NO Nash equilibrium

20. ITERATED ELIMINATION OF
STRICTLY DOMINATED STRATEGY

21. Iterated Elimination of Strictly
Dominated Strategies
X Y Z
A 3, 3 0, 5 0,4
B 0,0 3,1 1,2

22. Common Knowledge
• First Level:
You and Opponent know the matrix
• Second Level:
-You know that Opponent knows the matrix
-Opponent knows that you know the matrix
• Third Level
- You know that opponent knows that you know
the matrix
- Opponent knows that you know that opponent
knows the matrix

23. Iterated Elimination of Strictly
Dominated Strategies
X Y Z
A 3, 3 0, 5 0,4
B 0,0 3,1 1,2

24. Iterated Elimination of Strictly
Dominated Strategies
• If a strategy is strictly dominated for some
player, eliminate it
• Repeat, eliminating any strictly dominated
strategies in reduced game

25. Dominance example

26. Dominance example
B is dominated for Player 1
After eliminating B,R is dominated for Player 2

27. IEDS Example

28. IEDS Example

29. IEDS Example

30. IEDS Example

31. L C R
T 2,3 2,2 5,0
Y 3,2 5,3 3,1
Z 4,3 1,1 2,2
B 1,2 0,1 4,4

32. L C R
T 2,3 2,2 5,0
Y 3,2 5,3 3,1
Z 4,3 1,1 2,2
B 1,2 0,1 4,4

33. L C R
T 2,3 2,2 5,0
Y 3,2 5,3 3,1
Z 4,3 1,1 2,2
B 1,2 0,1 4,4

34. L C R
T 2,3 2,2 5,0
Y 3,2 5,3 3,1
Z 4,3 1,1 2,2
B 1,2 0,1 4,4

35. L C R
T 2,3 2,2 5,0
Y 3,2 5,3 3,1
Z 4,3 1,1 2,2
B 1,2 0,1 4,4

36. Another IEDS Example

37. Order of Elimination
Question: Does the order of elimination
matter?
Answer: Although it is not obvious, the end
result of iterated strict dominance is always the
same regardless of the sequence of eliminations.

38. Payoff Matrix for Bottled Water Game
Firm Pepsi Co
Raise Decrease
Raise +1, +1 -1, +2
Firm Coca Cola
Decrease
+2, -1 0, 0

39. ITERATED ELIMINATION OF WEAKLY
DOMINATED STRATEGY

40. Weakly Dominated Strategies
ai weakly dominates a’i if for all strategy profiles a-i of the other
players
ui(ai, a-i) ≥ui(a’i, a-i) and there is at least one a-i'
such that ui(ai, a-i') >ui(a’i, a-i')

41. Example
A B C
I -1, 3 0, 3 3, 7
II -1, 4 2, 7 6, 5

42. Weakly Dominated Strategies
• A weakly dominated strategy can be chosen in
a Nash equilibrium

43. Firm Pepsi Co
Raise Decrease
Raise +1, +1 0, 0
Firm Coca Cola
Decrease
0,0 0, 0

44. Weakly Dominated Strategies
• A weakly dominated strategy can be chosen in
a Nash equilibrium
• Order of eliminating weakly dominated
strategies matters

45. Example
A B C
I -1, 3 0, 3 3, 7
II -1, 4 2, 7 6, 5

46. Example
A B C
I -1, 3 0, 3 3, 7
II -1, 4 2, 7 6, 5
• Eliminate I,
• Eliminate A, I

47. A B
I 0,0 2,5
II 5,5 100,5
III 5,5 0,0

48. A B
I 0,0 2,5
II 5,5 100,5
III 5,5 0,0
• I is weakly dominated by II
• B is weakly dominated by A
• (II,A) AND (III,A) are two possible outcomes

49. A B
I 0,0 2,5
II 5,5 100,5
III 5,5 0,0
• III is weakly dominated by II
• A is weakly dominated by B
• (II,B) is a possible outcome

50. Order of Elimination
• If you eliminate a strategy when there is some other strategy
that yields payoffs that are higher or equal no matter what the
other players do, you are doing iterated weak dominance
• In this case you will not always get the same answer regardless
of the sequence of eliminations.
•This is a serious problem, and this is the reason why iterated
strict dominance is mostly used.