Game Theory, Continued: From
Zero-Sum to Non-Zero-Sum
Problem Set 3 due on FRIDAY!
Blue
Cooperate
Cooperate
Defect
3
3
0
5
5
0
1
1
Red
Defect
Game Theory: Basic Taxonomy
• 
• 
• 
• 
• 
Zero- vs. non-zero sum
Two- vs. N-person games
Finite vs. infinite number of choices
Iterated vs. non-iterated games
Games of perfect information
Two-Person Zero Sum Games
•  The notion of a dominant choice
•  The solution of a zero-sum game
–  The value of a game
–  The minimax theorem
•  Pure vs. Mixed strategies
Blue
A
A
5
B
4
B
3
Red
1
Worst I
can do
Blue
Red
Worst I
can do
4
5
10
3
8
4
3
7
7
6
8
9
7
2
4
6
Scissors
Paper
Stone
Scissors
0
1
-1
Paper
-1
0
1
1
-1
Stone
0
The Minimax Theorem in Game
Theory
•  Applies to zero-sum, two-person finite games.
•  The minimax theorem says that in such a game, there is a
value V for the game (the same value V for both players).
Given an optimal strategy (possibly a mixed strategy),
each player can be assured (on average) of obtaining at
least V for the game regardless of what the other player
does.
•  What this means, essentially, is that both players can
examine such a matrix and determine beforehand (and
regardless of the other player s plan) what they need to do
to ensure receiving an average of V for the game.
A Non-Zero-Sum Game
Blue
8
4
5
7
6
2
4
8
Red
Blue
5
6
7
2
3
7
9
8
Red
Clyde
Clam up
Clam up
Stool Pigeon
1 year
1 year
20 years
0 years
0 years
20 years
10 years
10 years
Bonnie
Stool Pigeon
Blue
Cooperate
Cooperate
Defect
3
3
0
5
5
0
1
1
Red
Defect
Blue
6
6
4
7
7
4
-3
-3
Red
Blue
3
3
1
3
3
1
0
0
Red
Blue
4
4
1
3
3
1
0
0
Red
The Ultimatum Game : adding
intangibles to a utility function
•  Two players, Red and Blue. Red is given
$1000 and is told that he can choose an
amount to be taken out of this total to offer
Blue. Blue then chooses whether to accept
this amount or not.
•  If Blue rejects Red s offer, both get nothing.
Axelrod s idea: hold a computer
tournament in which each
contestant is a program that will
play a series of rounds of the
Prisoner s Dilemma game.
Researchers can send in a program
to enter in the tournament: each
program will play all the others
in a series of about 100 successive
rounds of the PD game .
Some Simple AxelrodTournament-like strategies
•  All-Defect simply defects on every round
•  Poor-Trusting-Fool simply cooperates on
every round
•  Random is a test strategy that simply
cooperates or defects randomly
•  Unforgiving cooperates initially until the
first time that the other player defects; after
that, Unforgiving defects forever
The Tit-for-Tat Strategy
•  Tit-for-Tat cooperates on the first round.
Thereafter, on every subsequent round, it
simply imitates what the other player did on
the previous round. (That is, Tit-for-Tat
does at round N what the other player did at
round N-1.)
A British staff office on a tour of the trenches
remarked that he was
astonished to observe German soldiers walking about within
rifle range behind their own line. Our men appeared to take
no notice. I privately made up my mind to do away with that
sort of thing when we took over; such things should not be
allowed. These people evidently did not know there was a
war on. Both sides apparently believed in the policy of live
and let live .
Dugdale 1932, quoted in Axelrod, p. 74
The high commands of the two sides did not
share the view of the common soldier who
said:
The real reason for the quietness of some sections of
the line was that neither side had any intention of
advancing in that particular district... If the British
shelled the Germans, the Germans replied, and the
damage was equal: if the Germans bombed an
advanced piece of trench and killed five
Englishmen, an answering fusillade killed five
Germans. Belton Cobb 1916, quoted in Axelrod p. 76
The ethics that developed are illustrated in this
incident, related by a British officer recalling
his experience while facing a Saxon unit of the
German Army.
I was having tea with A Company when we heard a
lot of shouting and went out to investigate. We found
our men and the Germans standing on their respective
parapets. Suddenly a salvo arrived but did no damage.
Naturally both sides got down and our men started
swearing at the Germans, when all at once a brave
German got on to his parapet and shouted out We are
very sorry about that; we hope no one was hurt. It is
not our fault, it is that damned Prussian artillery. Rutter 1934, quoted in Axelrod, p. 85