Game Theory Basics
The prisoner’s dilemma in real life
• The American / Soviet arms race during
the Cold War.
• Should the US disarm? Does it matter
what the Soviets plan is?
Can we build a payoff matrix?
Soviet Union
Arm
Disarm
Arm
US
Disarm
Can we build a payoff matrix?
Soviet Union
Arm
Disarm
1
Arm
-2
+2
1
US
0
+2
Disarm
-2
0
Rationale behind payoffs
• Both countries disarm = less military
spending, less chance of accidental
attack, no military superiority
• Both countries arm = more military
spending, more dangerous world
• One country disarms, but not the other =
severe military mismatch!
Dominate strategy for both: Not knowing what the
other country will do, you choose to be armed.
Soviet Union
Arm
Disarm
Dominate
strategy
1
Arm
-2
+2
1
US
0
+2
Disarm
-2
0
Nash equilibrium: Neither player would change
their strategy, even if they knew in advance what
the other player would do.
Soviet Union
Arm
Disarm
Nash
Equilibrium
1
Arm
-2
+2
1
US
0
+2
Disarm
-2
0
Example #2
• Imagine the following scenario: Your
teacher assigns a project, and you are to
work with a partner. You would like an A,
but you don’t want to do all the work.
• Imagine the following payoffs:
Joint project payoffs
• Both work hard; get a A (A=40)
• Only one works hard, get a B (B=30)
• Neither one works hard, get a D (D=10)
• The cost of hard work = 25 (After all, you
could be having fun with your friends!)
• Can you build a payoff matrix???
Homework project payoff matrix
Your decision
Work
Slack
Work
Classmate
Slack
Homework project payoff matrix
Your decision
Work
Slack
40-25=15
Work
40-25=15
Classmate
Slack
30-0=30
30-25=5
30-25=5
30-0=30
10-0=10
10-0=10
Homework project payoff matrix
Your decision
Work
Slack
40-25=15
Work
40-25=15
Classmate
Slack
30-0=30
30-25=5
30-25=5
30-0=30
10-0=10
10-0=10
Dominate strategy = slack!
Example #3
Think of rock, paper, scissors.
• In a two person game, is there a dominant
strategy?
• Is there a Nash Equilibrium?
Rock, paper, scissors payoffs
Player 1
P
R
R
0
0
Player 2
P
S
1
-1
S
1
-1
-1
0
0
1
1
-1
-1
1
-1
0
1
0
Is there a dominant strategy?
Player 1
P
R
R
0
0
Player 2
P
S
1
-1
S
1
-1
-1
0
0
1
1
-1
-1
1
-1
0
1
0
If each player chooses R, P, or S randomly
(exactly 1/3 of the time) there is no dominant
strategy.
Player 1
R
P
S
R
0
0
Player 2
P
S
1
-1
1
-1
-1
0
0
1
1
-1
-1
1
-1
0
1
0
There is no Nash equilibrium; if you knew
what the other player was going to do, you
might switch.
Player 1
R
P
S
R
0
0
Player 2
P
S
1
-1
1
-1
-1
0
0
1
1
-1
-1
1
-1
0
1
0
You would only have a dominant strategy if you
knew what the other player was likely to do. A bad
player might have a tendency to make one of the
choices more often, or in some predictable pattern.
Player 1
P
R
R
0
0
Player 2
P
S
1
-1
S
1
-1
-1
0
0
1
1
-1
-1
1
-1
0
1
0
Example #4
The only way to get to Mackinaw Island is by
ferry boat.
• Assume there are only two ferry lines, the
red fleet and the black fleet. (duopoly)
• Each company has to decide whether to
charge a low price or a high price.
Assume the following payoff matrix
per passenger
Red fleet
low price
high price
$2
Low price
Black fleet
$0
$5
$2
$4
$5
High price
$0
$4
Questions: Is there a dominant strategy for
the red fleet? For the Black fleet?
Red fleet
low price
high price
$2
Low price
Black fleet
$0
$5
$2
$4
$5
High price
$0
$4
Questions: Is there a Nash equilibrium?
Red fleet
low price
high price
$2
Low price
Black fleet
$0
$5
$2
$4
$5
High price
$0
$4
Dominant
strategy
equilibrium
Nash
equilibrium
Red fleet
low price
high price
$2
Low price
Black fleet
$0
$5
$2
$4
$5
High price
$0
$4
Repeated games
• In real life, this wouldn’t be a one-shot
game. The ferry boats would be
competing against each other all summer,
or for years to come. Both companies
would be better off if they charged the
higher price. A tit for tat strategy could
allow tacit collusion to occur.
• (Tacit collusion = more profits for both)
Main points
• The prisoner’s dilemma can appear in many
different situations
• Cooperating with the other player could lead to a
higher payoff than simply following the dominant
strategy
• In oligopoly, producers who can cooperate can
achieve monopoly pricing power
Is collusion new?
“People of the same trade seldom meet
together, even for merriment and
diversion, but the conversation ends in a
conspiracy against the public, or in some
contrivance to raise prices”
Adam Smith
The Wealth of Nations
1776