Adventures of Sherlock Holmes

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Adventures of Sherlock Holmes

• The story...

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Adventures of Sherlock Holmes
London Canterbury
Dover Continent
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"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

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Index

• Holmes Moriarty paradox
• Zero sum games
• The Prisoners dilemma
• F-scale

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The Holmes Moriarty Paradox

• Introduction to solving the problem using some
  principles of game theory

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The Adventures of Sherlock Holmes
London Canterbury
Dover
Continent

• Oskar Morgenstern, 1928
• John VonNeumann

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• C D
• C 0 p
• D P 0
• 0 Holmes dies
• p Holmes has a fighting chance
• P Holmes succeeds to escape

Moriartys Options
Holmes Options
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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

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What Is a Zero Sum Game?

• Competitive game
• Players either win or lose

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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 payoffs (rewards) can be displayed in a
  reward matrix

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Example of Zero Sum Game
Reward Matrix
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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)

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Key Concepts of Game Theory

• Payoff
• Saddle point

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Players Payoffs

• The reward (or deficit) a player earns from a
  given play in a game
• Row players payoffs are shown in matrix
• Column players payoffs are the negatives of the
  row players payoffs

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Players Payoffs
Roses Payoffs
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Players Payoffs
Colins Payoffs
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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

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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

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Minimax/Maximin (Applied)
Colins Optimal Strategy
Saddle point
Roses Optimal Strategy
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Movement Diagram (Method)

• Simpler way to find the saddle point
• 1st - consider Roses point of view

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Movement Diagram (Applied)
Saddle point
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Saddle PointComments

• Saddlepoint 0 fair game
• Saddlepoint 0 biased game
• Game biased toward Rose
• This game has a saddlepoint
• It is a determinate game

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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

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Game Tree

• Diagram showing the progression of moves in
  the game
• 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
• A moment in the game at which a player must act

Information Set
Decision Node
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Indeterminate Games

• No saddle point
• Rationalization of the other players moves used
• Players look out for own best interest
• Each player can take advantage of the other

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Indeterminate Games
The Holmes Moriarty Paradox (revisited)
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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 escapes
Fighting chance
Holmes dies
Holmes dies
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No Saddle Point

• 0 Holmes dies
• 2/3 Holmes has a fighting chance
• 1 Holmes succeeds to escape

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Finding Mixed Strategy
p1
p2
q2

• q1

Mathematical Expectation employed E p1q1 p2q2
piqi
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Mixed Strategy

• Holmes
• Expectation

EHolmes 0C1D 2/3C0D D2/3C or
1-C2/3C C3/5 gt D2/5 StrategyHolmes
3/5C2/5D
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Mixed Strategy

• Moriartys
• Expectation

EMoriarty 0C2/3D 1C0D 2/3D C or
2/3(1-C) C 2/3 5/3C C 2/5 gt D 3/5
StrategyMoriarty 2/5C3/5D
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Mixed Strategy
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Imagine

• You a cohort have been arrested
• Separate rooms in the police station
• You are questioned by the district attorney

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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

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Imagine...
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The Prisoners Dilemma

• Non-zero-sum games
• Nash equilibrium
• Pareto efficiency and inefficiency
• Non-cooperative solutions

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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
• Players payoffs do not add to zero

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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 Prisoners Dilemma

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Games without Equilibrium

• Colin
• H T
• H (2, 4) (1, 0)
• Rose
• T (3, 1) (0, 4)

Example
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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

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Two Different Equilibrium

• Colin
• H T
• H (1, 1) (2, 5)
• Rose
• T (5, 2) (-1, -1)

Example
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Two Different Equilibrium

• Multiple saddle points are equivalent and
  interchangeable
• Optimal Strategy always saddle point

Zero Sum Game
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

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Unique Equilibrium Outcome
Equilibrium Point
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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
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Unique, but not Pareto Optimal

• The outcome (-1, -1) is not Pareto optimal
• both prisoners are better off
• choosing (0, 0)

Unique Equilibrium
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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

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Non-Cooperative Solutions

• Repeated Play-theory
• Metagames argument

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Repeated Play -Theory

• Definition
• Assumption
• Formal approach

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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
  whats best for them.

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Assumption
Assume your opponent will start by choosing C
(cooperate), and continue to choose C(cooperate)
until you choose D (defect).
R reward (0) S sucker payoff (-2) T Temptation
(-1) U Uncooperative (0)
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Formal Approach
With cooperation the payoff would be
Without cooperation the payoff would be
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Formal Approach
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Formal Approach
R Reward for cooperation (0) S Sucker payoff
(-2) T Temptation payoff (1) U Uncooperative
payoff (-1)
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Formal Approach
Under the assumption it makes sense for both
players to cooperate (C) when pgt1/2. This will
lead us to a Pareto Optimal solution
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Metagame Approach

• Will lead to an equilibrium which is cooperative.
• This argument depends on both players being able
  to predict the other players strategies.

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Metagame
Your partners choice is contingent on your
choice. Your partner has four strategies
Partner
IAA IIAB IIIBA IVBB
A B
(0,0) (0,0) (-2,1) (-2,1) (1,-2)
(-1,-1) (1,-2) (-1,-1)
You
I Choose A regardless III Choose
opposite of partner II Choose same as partner
IV Choose B regardless
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I AA IIAB IIIBA IVBB
IAAAA (0,0) (0,0)
(-2,1) (-2,1) IIAAAB (0,0)
(0,0) (-2,1) (-1,-1) IIIAABA
(0,0) (0,0) (1,-2)
(-2,1) IVAABB (0,0) (0,0)
(1,-2) (-1,-1) VABAA (0,0)
(-1,-1) (-2,1) (-2,1) VIABAB
(0,0) (-1,-1) (-2,1)
(-1,-1) VIIABBA (0,0) (-1,-1)
(1,-2) (-2,1) VIIIABBB (0,0)
(-1,-1) (1,-2) (-1,-1) IXBAAA
(1,-2) (0,0) (-2,1) (-2,1)
XBAAB (1,-2) (0,0) (-2,1)
(-1,-1) XIBABB (1,-2) (0,0)
(1,-2) (-2,1) XIIBABB
(1,-2) (0,0) (1,-2)
(-1,-1) XIIIBBAA (1,-2) (-1,-1)
(-2,1) (-2,1) XIVBBAB (1,-2)
(-1,-1) (-2,1) (-1,-1) XVBBBA
(1,-2) (-1,-1) (1,-2)
(-2,1) XVIBBBB (1,-2) (-1,-1)
(1,-2) (-1,-1)
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F-scale

• Practical applications

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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

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Applications in Social Psychology

• T.W. Adorno
  The Authoritarian Personality
• Test personality variables
• Controversial
• Research
• trust, suspicion, trustworthiness

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Research on Trustworthiness

• Morton Deutsch
• Experimentation
• F-Scale Questionnaire
• Subjects played the prisoners dilemma
• Strong Correlation between
• Suspicion
• Untrustworthiness
• Scoring high on the F-Scale (Adornos
  Authoritarian Personality)
• High F-scale scorers play the Prisoners dilemma
  differently

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Results of F-Scale Research

• Discrepancy between interpretations
• Experimental Games
• Previously vague concepts precise operational
• Provide measurable results

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Conclusion

• Many uses of game theory
• Zero sum games / non zero sum games
• Cooperative / non-cooperative
• Applications of game theory

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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 players at any given moment
• Ability to analyze complex problems

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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.

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Thank you

• Dr. Steve Deckelman

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Questions?