1. EC4004 Lecture 8
Dr Stephen Kinsella
Game Theory, 3

2. Last Time:

3. Any Game

4. Described by
{Players, Strategies, Payoffs}

5. Nash
Equilibrium

6. Normal & Extensive Forms
.
A
. .
Confess Silent
B B
Confess Silent Confess Silent
-3, -3 -1, -10 -10, -1 -2, -2

7. Find Nash Eqm by
Underlining

8. Step 1

9. Step 2

10. Step 3

11. Step 4

12. Step 5

13. Stuff to Remember:

14. 1. Nash Eqm
2. Underlining
3. Normal vs. Extensive Form

15. Dominance Today
Mixed & Pure Strategies
Subgame Perfection
Backward Induction

16. A dominant strategy refers to the best
response to any strategy chosen by the other
player. Dominance
When a player has a dominant strategy in a game,
there is good reason to predict that this is how the
player will play the game.

17. A mixed strategy refers to when the player
randomly selects from several possible actions.
By contrast, the strategies in which a player
chooses one action or another with certainty
are called pure strategies.

18. Matching Pennies

19. Story

21. .
A
. .
Heads Tails
B B
Heads Tails Heads Tails
1, -1 -1, 1 -1, 1 1, -1

22. Solving for Pure-Strategy Nash Equilibrium
in Matching Pennies Game

23. Battle of the Sexes

24. Story

26. .
A (Wife)
. .
Ballet Boxing
B (Husband)
B (Husband)
Ballet Boxing Ballet Boxing
2, 1 0, 0 0, 0 1, 2

27. Solving for Pure-Strategy Nash
Equilibria in Battle of the Sexes

28. Best-Response Function
The function which gives the payoff-maximizing
choice for one player in each of a continuum of
actions of the other player is referred to as the
best-response function.

29. Computing the Wife’s Best Response to the
Husband’s Mixed Strategy
(h)(2) + (1 – h)(0) =
2h
(h)(0) + (1 – h)(1) =
1-h

30. Best-Response Functions Allowing Mixed
Strategies in the Battle of the Sexes
.
h
Husband’s best- Pure-strategy
1 response Nash equilibrium
function (both play Ballet)
Wife’s best-
response function
1/3 .
.
Mixed-strategy
Nash equilibrium
w
2/3 1
Pure-strategy Nash equilibrium (both play Boxing)

31. Subgame Perfection

32. Subgame-Perfect Equilibrium
Game theory offers a formal way of selecting the
reasonable Nash equilibria in sequential games
using the concept of subgame-perfect
equilibrium.
A proper subgame consists of the part of the
game tree including an initial decision not
connected to another in an oval and everything
branching out below it.

33. Proper Subgames in the Battle of the
Sexes
Simultaneous
Version .
A (Wife)
. .
Ballet Boxing
B (Husband) B (Husband)
Ballet Boxing Ballet Boxing
2, 1 0, 0 0, 0 1, 2

34. Proper Subgames in the Battle of the
Sexes
Sequential
Version .
A (Wife)
. .
Ballet Boxing
B (Husband) B (Husband)
Ballet Boxing Ballet Boxing
2, 1 0, 0 0, 0 1, 2

35. Finally: Backward
Induction

36. Backward Induction
A shortcut to finding the subgame-perfect
equlibrium directly is to use backward
induction.
Backward induction solves for the equilibrium
by working backwards from the end of the
game to the beginning.

37. Backward Induction in the Sequential
Battle of the Sexes
.
A (Wife)
. .
Ballet Boxing
B (Husband) B (Husband)
Ballet Boxing Ballet Boxing
2, 1 0, 0 0, 0 1, 2

38. Backward Induction in the Sequential Battle
of the Sexes
.
A (Wife)
Ballet Boxing
.
B (Husband)
.
B (Husband)
plays Ballet plays Boxing
2, 1 1, 2

39. Next Time: Production
Read Chapter 7

40. EC4004 Lecture 8
Dr Stephen Kinsella
Game Theory, 3