Game Theory: The Mathematics
of Competition
6th Edition = Chapter 16
5th Edition = Chapter 15
Game Theory - definitions
• Strategies – courses of action a player might
choose
– Pure Strategy – a course of action which does
not involve randomized choices – pick a
strategy and stay with it
– Mixed Strategy – randomizes the strategies to
get the best outcome
• Outcomes – the consequences of the course
of action
Game Theory
• Game Theory – using mathematical tools to study
situations involving conflict and co-operation
• Game Theory - analyzes the rational choice of
strategies How players select strategies to obtain
preferred outcomes
• Game Theory – analyzing situations in which
there are at least 2 players in conflict because of
different goals
Applications of Game Theory
•
•
•
•
Labor – Management Disputes
Resource Allocation Decisions
Military Choices in international conflict
Threats by animals
Definitions - Continued
• Saddlepoint – When MaxMin and MiniMax
Values are the same (=), Same Result(outcome)
– Complex Games have saddlepoints e.g. Chess just don’t
know where it is.
– No Saddlepoint Games - Poker
• Value of the Game Where the two strategies
intersect
• Zero Sum Game – payoff to one player is the
negative payoff to the other player
Definitions - continued
• Conflict between players
– Total – One player WINS while the other loses.
– Partial – Players can benefit from some kind or
form of co-operation
Henry and Lisa- Strategy
• MaxMin Strategy – the Maximum value of
the minimum choices
• MiniMax Strategy– the minimum value of
the maximum choices
• Both – Worst Case analysis
• Each player is guaranteed at Least the value
of their MaxMin and MiniMax strategies
Two Person – Total Conflict –
Mixed Strategy
• Baseball !!!
Pitcher
Fast
Batter
Fast
Curve
Curve
Two Person – Total Conflict –
Mixed Strategy
Batter
Fast
Curve
EF=(.300)(1-P)+.200P
= .300-.300P + .200P
= .300-.100P
EC = .100(1-P) + .500P
= .100 -.100P + .500P
= .100 + .400P
Pitcher
Fast
0.300
0.100
1-P
Curve
0.200
0.500
P
1-Q
Q
EF = EC
.300 - .100P = .100 + .400P
.300 - .100 = .400P + 100P
.200 = .500P
P = .200/.500 P=2/5
1-P = 3/5
Two Person – Total Conflict –
Mixed Strategy
Batter
Fast
Curve
EF=(.300)(1-Q)+.100Q
= .300-.300Q + .100Q
= .300-.200Q
EC = .200(1-Q) + .500Q
= .200 -.200Q + .500Q
= .200 + .300Q
Pitcher
Fast
0.300
0.100
1-P
Curve
0.200
0.500
P
1-Q
Q
EF = EC
.300 - .200Q = .200 + .300Q
.300 - .200 = .300Q + 200Q
.100 = .500Q
Q = .100/.500 Q=1/5
1-Q = 4/5
Partial Conflict Games
• Partial Conflict – Variable Sum Games.
Different payoffs as the outcome changes
• Non-Cooperative – No binding agreement is
possible or can be enforced
• Ordinal Games – Players rank the outcomes
from best to worst
Prisoner’s Dilemma
• 2 people accused of a crime – both held
incommunicado (Harry and Joe)
• Each have two choices:
– Stay quiet
– Tell on your partner
Prisoner’s Dilemma – cont.
•
Harry needs to rank the possible outcomes from
low to high
4.
Harry tells on Joe and Joe stays quiet – Harry might
get to go home!! (Joe’s going to Jail)
3. Harry remains quiet and so does Joe – possible both
get off
2. Harry tells on Joe and Joe tells on Harry – both going
to Jail
1. Harry is quiet and Joe tells on him – Joe gets off and
Harry goes to jail for a long time
Prisoner’s Dilemma
Harry
JOE
Confess Silent
Confess
(2,2)
(4,1)
Silent
(1,4)
(3,3)
John Nash
• Nash Equilibrium –
When no player can
benefit by departing
unilaterally from the
strategy associated
with an outcome
John Nash
• Nash Equilibrium –
When no player can
benefit by departing
unilaterally from the
strategy associated
with an outcome
Chicken – Partial Conflict
•
Each Player has 2 choices:
1. Keep going
2. Swerve out of the way
Chicken - continued
•
•
Frank vs Mustang Sally
Frank’s ordinal Choices
4. Frank keeps going – Sally swerves – Sally is the
chicken – Frank “wins”
3. Frank swerves - Sally Swerves – both chicken – both
alive
2. Frank swerves – Sally keeps going – Frank is the
chicken and Sally “wins”
1. Frank keeps going – Sally keeps going (disaster – both
dead)
Chicken - continued
Frank
Swerve
Don't
Sally
Swerve
(3,3)
(4,2)
Don't
(2,4)
(1,1)
Chicken - continued
• Nash Equilibrium at (4,2) and (2,4)
• There is no dominate strategy in Chicken
making it a very dangerous game – can’t
tell what you opponent will do
• “Best” outcome at (3,3) but no way to get
there – until T.O.M.
Partial Conflict – important
Points
• Dominant Strategy – the strategy that will
give the highest average result
• (x,y) = x+y = value of the game
• (1,1) = disaster
• (3,3) = compromise
• (4,x) = best for row player – won’t change
• (1,x) = worst for row player – nash
equilibrium not possible
TOM – Theory of Moves
• John Neumann
• Based on Game
Theory
• Postulate – players
will think AHEAD
• Elucidates on different
kinds of Power
Tom - Continued
• Oskar Morgenstern
• Games in extended
form – sequential
choice for players.
• Many games only
depend on the final
state reached
• Payoffs only if you
stay
TOM
• Backward induction – reasoning process in
which players working backward from the
last possible move in a game, anticipate
each other’s rational choices
• Survivor – payoff selected at each state as a
result of backward induction
• Block(age) – when it is not rational to move
beyond this point in a game
TOM - Outcomes
• Non-myopic Equilibria (NME) regardless of
who moves first the same outcome is
reached. The consequence of both players
looking ahead and anticipating where the
move – countermove process will end up
• Indeterminate – the result of the game
depends on who moves first – the outcome
is different depending on who goes first
Samson
• Great Warrior
4. Samson Don’t tell – Delilah Don’t nag
(party all the time)
3. Samson Tell – Delilah Nag
2. Samson Tell – Delilah Don’t Nag
1. Samson Don’t Tell – Delilah Nag
Delilah
• Paid for Info
4. Delilah Don’t Nag - Samson Tells (no
work involved)
3. Delilah Nag – Samson Tells (have to work
but get results)
2. Delilah Don’t Nag - Samson Don’t Tell
1. Delilah Nag – Sampson Don’t Tell
(disaster)
Samson vs Delilah
Delilah
Samson
Don't Tell
Don't Nag
(2,4)
Nag
(1,1)
Tell
(4,2)
(3,3)
Samson vs Delilah
Delilah
Samson
Don't Tell
Don't Nag
(2,4)
Nag
(1,1)
Tell
(4,2)
(3,3)
Delilah Starts (4,2)->(3,3)->(1,1)->(2,4)->(4,2)
Samson Starts (4,2)->(2,4)->(1,1)->(3,3)->(4,2)
Larger Games
•
•
•
Truel – Duel with Three People
Each Player has a gun with One bullet –
everyone is a perfect shot – no
communications between players
Goals
1. Survive
2. Survive with as few opponents left as possible
Larger Games with TOM
• Modify Rules
1. Take Turns firing – One Player at a time
“moves”
• Now must “think ahead”
• Two choices
1. Shot
2. Don’t shot
Order Power
• A player has order power – if that player
can force the other player to move first
• Only beneficial when the outcome is
indeterminate
Samson vs Delilah
Delilah
Samson
Don't Tell
Don't Nag
(2,4)
Nag
(1,1)
Tell
(4,2)
(3,3)
Delilah Starts (1,1)->(2,4)->(4,2)->(3,3)->(1,1)
Samson Starts (1,1)->(3,3)->(4,2)->(2,4)->(1,1)
Cycling
•
TOM Rule changes
5’ If at any state a player whoes turn it is to
move has received his best payoff (4) that
player will not move!
–
Moving Power – one player has the ability
to force the other player to STOP! Then
6’ at some point in cycling the player must stop
Row vs Column
Row
S1
S2
Column
S1
(2,4)
(1,2)
S2
(4,1)
(3,3)
Rows turn (2,4)->(1,2)->(3,3)->(4,1)->(2,4)
Column has moving power!! Tell Row has to
stop!!