Game Theory, 3
EC4004 Lecture 8Dr Stephen Kinsella
Last Time:
Any Game
Described by{Players, Strategies, Payoffs}
Nash Equilibrium
Normal & Extensive Forms
.. .Confess
Confess Confess
Silent
Silent Silent
-3, -3 -1, -10 -10, -1 -2, -2
B B
A
Find Nash Eqm by Underlining
Step 1
Step 2
Step 3
Step 4
Step 5
Stuff to Remember:
1. Nash Eqm2. Underlining3. Normal vs. Extensive Form
TodayDominance
Mixed & Pure Strategies
Subgame Perfection
Backward Induction
DominanceA dominant strategy refers to the best response to any strategy chosen by the other player.
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.
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.
Matching Pennies
Story
.. .Heads
Heads Heads
Tails
Tails Tails
1, -1 -1, 1 -1, 1 1, -1
A
B B
Solving for Pure-Strategy Nash Equilibrium in Matching Pennies Game
Battle of the Sexes
Story
.. .Ballet
Ballet Ballet
Boxing
Boxing Boxing
2, 1 0, 0 0, 0 1, 2
A (Wife)
B (Husband)B (Husband)
Solving for Pure-Strategy Nash Equilibria in Battle of the Sexes
Best-Response FunctionThe 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.
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
Best-Response Functions Allowing Mixed Strategies in the Battle of the Sexes
w
h
12/3
1
1/3
.
.
Pure-strategy Nash equilibrium (both play Ballet)
Mixed-strategy Nash equilibrium
Pure-strategy Nash equilibrium (both play Boxing)
.
Wife’s best-response function
Husband’s best-response function
Subgame Perfection
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.
Proper Subgames in the Battle of the Sexes
.. .Ballet
Ballet Ballet
Boxing
Boxing Boxing
2, 1 0, 0 0, 0 1, 2
A (Wife)
B (Husband) B (Husband)
Simultaneous Version
Proper Subgames in the Battle of the Sexes
.. .Ballet
Ballet Ballet
Boxing
Boxing Boxing
2, 1 0, 0 0, 0 1, 2
A (Wife)
B (Husband) B (Husband)
Sequential Version
Finally: Backward Induction
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.
Backward Induction in the Sequential Battle of the Sexes.
. .Ballet
Ballet Ballet
Boxing
Boxing Boxing
2, 1 0, 0 0, 0 1, 2
A (Wife)
B (Husband) B (Husband)
Backward Induction in the Sequential Battle of the Sexes
.
. .Ballet
plays Ballet
Boxing
plays Boxing2, 1 1, 2
A (Wife)
B (Husband) B (Husband)
Next Time: ProductionRead Chapter 7
Game Theory, 3
EC4004 Lecture 8Dr Stephen Kinsella