Adventures of Sherlock Holmes
• The story...
Adventures of Sherlock Holmes
London
Canterbury
Dover
Continent
"Sherlock Holmes,
Criminal Interrogations and
Aspects of Non-cooperative
Game Theory"
• Brandi Ahlers
• Jennifer Lohmann
• Madoka Miyata
• Soo-Bong Park
• Rae-San Ryu
• Jill Schlosser
Index
•
•
•
•
Holmes Moriarty paradox
Zero sum games
The Prisoner’s dilemma
F-scale
The Holmes Moriarty Paradox
• Introduction to
solving the problem
using some principles
of game theory
The Adventures of
Sherlock Holmes
London
Canterbury
Dover
• Oskar Morgenstern, 1928
• John VonNeumann
Continent
Moriarty’s Options
Holmes’
Options
C
D
C
0
p
D
P
0
• 0
= Holmes dies
• p
= Holmes has a fighting chance
• P = Holmes succeeds to escape
Zero-sum Games
• Definition of zero-sum
game
• Example of a zero sum
game
• Assumptions of games
• Important concepts of
game theory
• Determinate games
• Indeterminate games
What Is a Zero Sum Game?
• Competitive game
• Players either win or lose
Example of Zero Sum Game
• Two players play a game where a coin is
flipped (call the players rose & Colin)
• Each player chooses heads or tails
independent of the other player
• The payoff’s (rewards) can be displayed in a
reward matrix
Example of Zero Sum Game
Reward Matrix
Colin
Rose
Strategy
H
T
H
3
-6
T
2
1
Assumptions of the Game
• Games are non-cooperative
• There is no communication between players
• Rational play is used by each player to determine
the strategy he should play
– Each player does what is in his own best interest
– I.E. Player does whatever possible to earn the highest
payoff (within the rules of the game)
Key Concepts of Game Theory
• Payoff
• Saddle point
Player’s Payoffs
• The reward (or deficit) a player earns from a given
play in a game
• Row player’s payoffs are shown in matrix
• Column player’s payoffs are the negatives of the
row player’s payoffs
Player’s Payoffs
Rose’s Payoffs
Colin
Rose
Strategy
H
T
H
3
-6
T
2
1
Player’s Payoffs
Colin’s Payoffs
Rose
Colin
Strategy
H
T
H
-3
-2
T
6
-1
Saddlepoint
• Pair of strategies (one for each player)
which the game will evolve to when each
player uses rational play
• This is the optimal strategy for both players
• Two ways to find saddle point
– Minimax & Maximin principles
– Movement diagram
Minimax/Maximin (Method)
• Maximin: row player's strategy
– Find minimum row entry in each row
– Take the maximum of these
• Minimax: column player's strategy
– Find the maximum column entry in each
column
– Take the minimum of these
Minimax/Maximin (Applied)
Colin’s Optimal Strategy
Colin
Rose
Strategy
H
T
H
3
-6
T
2
1
Rose’s Optimal Strategy
Movement Diagram (Method)
• Simpler way to find the saddle point
• 1st - consider Rose’s point of view
Movement Diagram (Applied)
Colin
Rose
Strategy
H
T
H
3
-6
T
2
1
Saddle Point
Comments
Rose
Colin
H
T
H
3
-6
T
2
1
• Saddlepoint = 0 fair game
• Saddlepoint  0 biased game
– Game biased toward Rose
• This game has a saddlepoint
– It is a “determinate” game
Determinate Games
• Rose/Colin game is “determinate”
– There is a saddle point
• The saddle point indicates
– There is a clear set of strategies which the
players ought to use to attain the highest
payoff in the long run
• When there is no saddle point
– The game is called “indeterminate”
Game Tree
Diagram showing the progression of
moves in the game
Information Set
When a player makes a choice, he/she
knows he/she is at a node in a particular
information set, but he/she does not know
which node
Decision Node
•A moment in the game at which a player must act
Indeterminate Games
• No saddle point
• Rationalization of the other player’s moves
used
– Players look out for own best interest
– Each player can take advantage of the other
Indeterminate
Games
The Holmes Moriarty Paradox (revisited)
Moriarty’s Options
Holmes's
Options
Canterbury
(C)
Dover
(D)
Canterbury
(C)
0
2/3
Dover
(D)
1
0
Game Tree
Holmes and Moriarty in London Information Set
for Holmes
Moriarty detrains
at Canterbury
Moriarty detrains
at Dover
Holmes
detrains at
Canterbury
Holmes
detrains at
Dover
Holmes
detrains at
Canterbury
Holmes
detrains at
Dover
Holmes
dies
Holmes
escapes
Fighting
chance
Holmes
dies
No Saddle Point
Moriarty’s Options
Holmes's
Options
Canterbury
(C)
Dover
(D)
Canterbury
(C)
0
2/3
Dover
(D)
1
0
•0 = Holmes dies
•2/3 = Holmes has a fighting chance
•1 = Holmes succeeds to escape
Finding Mixed Strategy
Holmes's
Options
Moriarty’s Options
Canterbury
Dover
(C)
(D)
Canterbury
0
2/3
(C)
Dover
1
0
(D)
p1
p2
q1
Mathematical Expectation employed
E = p1q1 + p2q2 + … + piqi
q2
Mixed Strategy
Moriarty’s Options
Holme’s
Expectation
EHolmes :
Holmes’
Options
(C)
(D)
(C)
0
2/3
(D)
1
0
0C+1D = 2/3C+0D
D=2/3C or 1-C=2/3C
C=3/5 => D=2/5
StrategyHolmes = 3/5C+2/5D
Mixed Strategy
Moriarty’s
Expectation
EMoriarty :
Moriarty’s Options
Holmes’
Options
(C)
(D)
(C)
0
2/3
(D)
1
0
0C+2/3D = 1C+0D
2/3D = C or 2/3(1-C) = C
2/3 = 5/3C
C = 2/5 => D = 3/5
StrategyMoriarty= 2/5C+3/5D
Mixed Strategy
Moriarty’s Options
Holmes's
Options
(C)
(D)
(C)
0
2/3
(D)
1
0
3/5C+2/5D
2/5C+3/5D
Imagine…
• You & a cohort have been arrested
• Separate rooms in the police station
• You are questioned by the district attorney
Imagine...
• The clever district attorney tells each of you that:
– If one of you confesses & the other does not
• The confessor will get a reward
• His/her partner will get a heavy sentence
– If both confess
• Each will receive a light sentence
• You have good reason to believe that
– If neither of you confess
• You will both go free
Imagine...
Partner’s
Options
Options
Your
A
(do not
confess)
B
(confess)
A
B
(do not confess)
(confess)
(0,0)
both go free
(1,-2)
you get reward,
partner get heavy
sentence
(-2,1)
you get heavy
sentence,
partner gets reward
(-1,-1)
both get light
sentence
The Prisoner’s Dilemma
• Non-zero-sum games
• Nash equilibrium
• Pareto efficiency and
inefficiency
• Non-cooperative
solutions
Non Zero Sum Game
• Zero sum game
– The interest of players are strictly opposed
• Non zero sum game
– The interest of players are not strictly opposed
– Player’s payoffs do not add to zero
Equilibrium : Non Zero Sum Game
• Equilibrium outcomes in non zero sum games
correspond to saddle points in zero sum games
• Non Zero Sum Game
– No Equilibrium Outcome
– Two different Equilibrium Outcome
– Unique Equilibrium Outcome
• Pareto Optimal
• Non Pareto Optimal : Prisoner’s Dilemma
Games without Equilibrium
Colin
Example
H
T
H
(2, 4)
(1, 0)
T
(3, 1)
(0, 4)
Rose
Games without Equilibrium
• No equilibrium = No saddle point in zero
sum game
• No pure strategy
How to solve
• Suppose this game as zero sum game
• Solve this game by using mixed strategy
Two Different Equilibrium
Colin
Example
H
T
H
(1, 1)
(2, 5)
T
(5, 2)
(-1, -1)
Rose
Two Different Equilibrium
Zero Sum Game
• Multiple saddle points are equivalent and
interchangeable
• Optimal Strategy : always saddle point
Non Zero Sum Game
• Players may end up with their worst outcome
• Not clear which equilibrium the players should
try for, because games may have non equivalent
and non interchangeable equilibrium
Unique Equilibrium Outcome
Partner’s
Options
Options
Your
A
(do not
confess)
B
(confess)
A
B
(do not confess)
(confess)
(0,0)
(-2,1)
(1,-2)
(-1,-1)
Equilibrium Point
What is Pareto Optimal ?
Definition
Non Pareto Optimal : if there is another outcome
which would give both players higher payoffs,
or one player the same payoff, but the other player a
higher payoff.
Pareto Optimal : if there is no such other outcome
Note
In zero sum game every outcome is Pareto optimal
since every gain to one player means a loss to other
player
Unique, but not Pareto Optimal
The outcome (-1, -1) is not Pareto optimal Partner’s
–both prisoners are better off
Options
choosing (0, 0)
Options
Your
A
(do not
confess)
B
(confess)
A
B
(do not confess)
(confess)
(0,0)
(-2,1)
(1,-2)
(-1,-1)
Unique Equilibrium
When are Non Zero Sum Games
Pareto Optimally solvable ?
• If there is at least one equilibrium outcome
which is Pareto optimal
• If there is more than one Pareto optimal
equilibrium, all of them are equivalent and
interchangeable
Non-Cooperative Solutions
• Repeated Play-theory
• Metagames argument
Repeated Play -Theory
• Definition
• Assumption
• Formal approach
Definition
• Game is played not just once, but
repeated
• In repeated play theory people may be
willing to cooperate in the beginning,
but when it comes to the final play each
player will logically chooses what’s
best for them.
Assumption
Assume your opponent will start by choosing C
(cooperate), and continue to choose C(cooperate)
until you choose D (defect).
C
D
C
D
(R,R)
(S,T)
(T,S)
(U,U)
R: reward (0)
S: sucker payoff (-2)
T: Temptation (-1)
U: Uncooperative (0)
Formal Approach
With cooperation the payoff would be:
R
R  pR  p R  p R  ........... 
(1  p )
2
3
Without cooperation the payoff would be:
R  p m R  (1  p) p mT  p m 1U
(1  p)
Formal Approach
R
R  p m R  p mT  p m1T  p m1U

(1  p)
(1  p)
T  R
p 
T U
Formal Approach
R: Reward for cooperation (0)
S: Sucker payoff (-2)
T: Temptation payoff (1)
U: Uncooperative payoff (-1)
1 0
1
p

1  (1) 2
Formal Approach
Under the assumption it makes sense for
both players to cooperate (C) when p>1/2.
This will lead us to a Pareto Optimal solution
C
D
C
(R,R)
(S,T)
D
(T,S)
(U,U)
Metagame Approach
• Will lead to an equilibrium which is
cooperative.
• This argument depends on both players
being able to predict the other player’s
strategies.
Metagame
Your partner’s choice is contingent on your choice.
Your partner has four strategies:
Partner
I:AA II:AB III:BA IV:BB
You
A
(0,0)
(0,0)
(-2,1)
(-2,1)
B
(1,-2)
(-1,-1)
(1,-2)
(-1,-1)
I: Choose A regardless
III: Choose opposite of partner
II: Choose same as partner
IV: Choose B regardless
I: AA
II:AB
III:BA
IV:BB
I:AAAA
(0,0)
(0,0)
(-2,1)
(-2,1)
II:AAAB
(0,0)
(0,0)
(-2,1)
(-1,-1)
III:AABA
(0,0)
(0,0)
(1,-2)
(-2,1)
IV:AABB
(0,0)
(0,0)
(1,-2)
(-1,-1)
V:ABAA
(0,0)
(-1,-1)
(-2,1)
(-2,1)
VI:ABAB
(0,0)
(-1,-1)
(-2,1)
(-1,-1)
VII:ABBA
(0,0)
(-1,-1)
(1,-2)
(-2,1)
VIII:ABBB
(0,0)
(-1,-1)
(1,-2)
(-1,-1)
IX:BAAA
(1,-2)
(0,0)
(-2,1)
(-2,1)
X:BAAB
(1,-2)
(0,0)
(-2,1)
(-1,-1)
XI:BABB
(1,-2)
(0,0)
(1,-2)
(-2,1)
XII:BABB
(1,-2)
(0,0)
(1,-2)
(-1,-1)
XIII:BBAA
(1,-2)
(-1,-1)
(-2,1)
(-2,1)
XIV:BBAB
(1,-2)
(-1,-1)
(-2,1)
(-1,-1)
XV:BBBA
(1,-2)
(-1,-1)
(1,-2)
(-2,1)
XVI:BBBB
(1,-2)
(-1,-1)
(1,-2)
(-1,-1)
F-scale
• Practical applications
Have you ever seen this?
Rate on a scale from 1 to 7 (1 is high)
for the following:
 How satisfied are you with …
 How sure are you that …
Applications in Social
Psychology
 T.W. Adorno:
“The Authoritarian Personality”
 Test personality variables
 Controversial
 Research
– trust, suspicion, trustworthiness
Research on Trustworthiness
 Morton Deutsch
– Experimentation
• F-Scale Questionnaire
• Subject’s played the prisoner’s dilemma
– Strong Correlation between
• Suspicion
• Untrustworthiness
• Scoring high on the F-Scale (Adorno’s Authoritarian
Personality)
• High F-scale scorers play the Prisoner’s dilemma
differently
Results of F-Scale Research
 Discrepancy between interpretations
 Experimental Games
 Previously vague concepts precise &
operational
 Provide measurable results
Conclusion
• Many uses of game theory
– Zero sum games / non zero sum games
– Cooperative / non-cooperative
• Applications of game theory
Conclusion
• Why is game Theory a successful model?
– Wide variety of applications
– Concrete map of
• Rules of the game
• How the game is played
• Knowledge of player’s at any given moment
– Ability to analyze complex problems
References
• Eatweel, Milgate, Newman. The new Palgrave, game theory: W.W.
Norton &company inc; New York, NY 1989.
• Case, James. Paradoxes involving conflicts of interest. Mathematical
association of America; 33-38, January 2000.
• Straffin, Philip D. Game Theory and strategy: The Mathematical
Association of America; 1993.
Thank you
• Dr. Steve Deckelman
Questions?