Game Theory, 3

EC4004 Lecture 8Dr Stephen Kinsella

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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