Chapter 13 – Introduction to

Game Theory.Goals:

+ Set Up a Game.

+ Solve a Game:

Dominant Strategy Equilibrium.

Nash Equilibrium.

Sub-game Perfect Nash Equilibrium.

Describing the Game.

Elements of a game:

◦ The players

◦ Timing of a game: Simultaneous or Sequential.

◦ Information Availability: Perfect or Imperfect.

◦ The list of possible strategies for each player.

◦ The payoffs associated with each combination

of strategies.

◦ Repetition.

◦ The decision rule.

Presentation of a Game.

Extensive-form presentation (a game tree) of

a game:

◦ The players

◦ Their possible actions

The set of possible actions

The sequence of actions

The information the players have when they decide

◦ The outcome following the actions, i.e. payoffs for

all players

Presentation of a Game – extensive

form. Example 1: A sequential game.

Presentation of a Game – extensive

form. Example 2: A simultaneous-move game.

Presentation of a Game.

Normal-form or strategic-form or pay-off

matrix presentation.

◦ The players

◦ Their possible strategies

◦ The outcome following these strategies

Presentation of a Game – Normal

Form Example 3: The normal-form representation

TOSHIBA

DOS UNIX

IBM DOS (600,200) (100,100)

UNIX (100, 100) (200,600)

Common Games.

Prisoner’s Dilemma

PRISONER’s DILEMMA. Prisoner 2

Stay Silent Betray

Prisoner 1 Stay Silent Each serves 1

year

Prisoner 1:

Serves 20 years.

Prisoner 2: Go

free.

Betray Prisoner 1: Go

free.

Prisoner 2:

Serves 20 years

Each serves 10

years.

Common Games.

Battle of Sexes

Battle of Sexes Wife

Hockey Game Ballet

Performance

Husband Hockey Game (100,50) (20,20)

Ballet

Performance

(0,0) (50,100)

Equilibrium of a Game.

Equilibrium of a game: a situation in which no

player wishes to change his/her strategy.

We study three equilibrium concepts in

Econ 203:

◦ Dominant strategy equilibrium.

◦ Nash equilibrium.

◦ Sub-game perfect Nash equilibrium.

Equilibrium of a Game – Dominant

Strategy Equilibrium. Dominant strategy equilibrium.

◦ Consider the following game:

◦ The players: Player 1 and Player 2

◦ Timing of a game: Simultaneous.

◦ Information Availability: Perfect.

◦ Possible strategies: Player 1 (L;R) and Player 2(U;D)

◦ The payoffs associated with each combination of strategies (table

below)

◦ Repetition: Non - repeated.

◦ The decision rule: Max own payoff.

◦ Equilibrium: (Player 1, Player 2) = (R;D) with payoff (6,3)

Player 2

(U) (D)

Player 1 L (2,2) (4,4)

R (0,1) (6,3)

Equilibrium of a Game – Nash

Equilibrium. Dominant Strategy does not always exist.

The Nash equilibrium (NE)◦ Set of strategies (one for each player) such that

no player wishes to change her strategy given the strategies of the other players The strategy of each player is a so-called best response

to the given strategies of the other.

Equilibrium of a Game – Nash

Equilibrium. Nash equilibrium.

◦ Consider the following game:

◦ The players: Player 1 and Player 2

◦ Timing of a game: Simultaneous.

◦ Information Availability: Perfect.

◦ Possible strategies: Player 1 (L;R) and Player 2(U;D)

◦ The payoffs associated with each combination of strategies (table

below)

◦ Repetition: Non - repeated.

◦ The decision rule: Max own payoff.

◦ Nash Equilibria: (Player 1, Player 2) = (R;U) and (L;D) with

payoff (3;6) and (6;3)

Player 2

(U) (D)

Player 1 L (0,2) (6,3)

R (3,6) (0,2)

Equilibrium of a Game – Nash

Equilibrium.◦ Consider the following game:

◦ The players: Player 1 and Player 2

◦ Timing of a game: Sequential with Player 1 moves first.

◦ Information Availability: Perfect.

◦ Possible strategies: Player 1 (L,R) and Player 2(UU,UD,DU,DD)

◦ The payoffs associated with each combination of strategies (table

below)

◦ Repetition: Non - repeated.

◦ The decision rule: Max own payoff.

◦ Equilibria: (Player 1, Player 2) = (R;U,U), (L;U,D), (L;D,D)

with payoff (3;6), (6;3) and (6;3)

◦ 1 equilibrium is not valid…which one?

Equilibrium of a Game – Nash

Equilibrium. Extensive Presentation – Game Tree.

Normal Form Presentation – Payoff Matrix.

Player 2

(U,U) (U,D) (D,U) (D,D)

Player 1 R (3,6) (3,6) (0,2) (0,2)

L (0,2) (6,3) (0,2) (6,3)

Equilibrium of a Game – Sub-Game

Perfect Nash Equilibrium Sub-game Perfect Nash equilibrium

◦ A player’s best response to a given strategy played by another

player.

Q: How do I find a sub-game perfect Nash equilibrium?

A: Take game tree and use method called backward

induction

◦ Remember, that the SPNE is a set of strategies, not an outcome

or a sequence of actions

The SPNE: (Player 1, Player 2) = (L;U,D)

What are the other two NE?

◦ They are non-credible threats.

Equilibrium of a Game – Sub Game

Perfect Nash Equilibrium. Extensive Presentation – Game Tree.

Normal Form Presentation – Payoff Matrix.

Player 2

(U,U) (U,D) (D,U) (D,D)

Player 1 R (3,6) (3,6) (0,2) (0,2)

L (0,2) (6,3) (0,2) (6,3)

SPNE

Game Theory

What can we conclude?

◦ Always use normal form/pay-off matrix to solve a

simultaneous game.

First identify if there exists dominant strategy for any of the

players.

If there is, the equilibrium must contain such dominant strategy.

If no dominant strategy exists, solve the game using the very

definition of Nash Equilibrium: my best response given your

action.

◦ Always use the game tree to solve a sequential game

using backward induction.

The equilibrium is a sub-game perfect nash equilibrium.

Eliminate all the non-credible threats.

Duopoly - Revisited

Consider the following game:

◦ The players: Firm 1 and Firm 2 with same cost functions.

◦ Timing of a game: Simulteneous

◦ Information Availability: Perfect.

◦ Possible strategies: Firm1 (PB,PM) and Firm 2(PB,PM)

◦ The payoffs associated with each combination of strategies

(table below)

◦ Repetition: Non - repeated.

◦ The decision rule: Max own payoff.

Nash Equilibrium: (Player 1, Player 2) = (PB;PB) with

payoff (1,1) Non-cooperate the lowest

welfare outcome.

How to induce cooperation?

Duopoly - Revisited

Penalty:

◦ Suppose there is a enforceable penalty of 5 charged to the firm

who does not comply with the agreement (always play PM) and is

given to the one that suffers from that noncompliance.

Result: Cooperative outcome.

Firm 2

PB PM

Firm1 PB (1,1) (10-5=5,0+5=5)

PM (0+5=5,10-5=5) (6,6)

Duopoly - Revisited

What if the penalty of 5 is not enforceable?

◦ Punishment imposed by the other firm.

Tit-for-Tat: My action in this game depends on your action in the

previous game.

Grim Strategy: once a cheater always a cheater. Punishment sustains

for the rest of the game.

Consider the previous game with number of repetition (round) = 10.

Firm 1 chooses to cooperate or not at round 10.

If cooperate = 6 (and the other cooperates)

If non-cooperate = 10 (and the other cooperates)

Non-cooperate at round 10 for both firms.

Move backward. Firm 1 chooses to cooperate or not at round 9

If cooperate = 6 + 0 = 6

Non cooperate = 10 + 0 = 10

Non-cooperate at round 9 and 10 for both firms.

Keep moving backward and we will see that two firms will not cooperate in any of the round under grim strategy with 10 rounds.

How to get a cooperative outcome?

Duopoly - Revisited◦ Grim Strategy: once a cheater always a cheater.

Punishment sustains for the rest of the game.

Consider the previous game with number of repetition (round) =

infinity.

Firm 1 chooses to cooperate or not at round 1.

If cooperate = 6 + 6 + 6 + 6 + … (and the other cooperates)

If non-cooperate = 10 + 0 + 0 + 0 + … (and the other cooperates)

Cooperate at round 1 for both firms.

Move forward. Firm 1 chooses to cooperate or not at round 2 If cooperate = 6 + 6 + 6 + … = infinity

Non cooperate = 10 + 0 + 0 + … = 10

Cooperate at round 1 and 2for both firms.

Keep moving forward and we will see that two firms will always cooperate under grim strategy with infinity number of rounds.