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
IBM
DOS
UNIX
DOS
(600,200)
(100,100)
UNIX
(100, 100)
(200,600)
Common Games.

Prisoner’s Dilemma
PRISONER’s DILEMMA.
Prisoner 2
Stay Silent
Prisoner 1
Betray
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
Husband
Wife
Hockey Game
Ballet
Performance
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
Player 1
(U)
(D)
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
Player 1
(U)
(D)
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
Player 1
(U,U)
(U,D)
(D,U)
(D,D)
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.
SPNE

Normal Form Presentation – Payoff Matrix.
Player 2
Player 1
(U,U)
(U,D)
(D,U)
(D,D)
R
(3,6)
(3,6)
(0,2)
(0,2)
L
(0,2)
(6,3)
(0,2)
(6,3)
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
Firm1
PB
PM
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.