Adventures of Sherlock Holmes• The story...
Adventures of Sherlock HolmesLondon
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 The Holmes Moriarty ParadoxParadox
• Introduction to solving the problem using some principles of game theory
The Adventures of Sherlock Holmes
• Oskar Morgenstern, 1928• John VonNeumann
London
Canterbury
Dover
Continent
C D
C 0 p
D P 0
• 0 = Holmes dies
• p = Holmes has a fighting chance
• P = Holmes succeeds to escape
Moriarty’s Options
Holmes’Options
Zero-sum GamesZero-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
Colin
Rose
Strategy H T
H 3 -6
T 2 1
Reward Matrix
Assumptions of the GameAssumptions 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 TheoryKey 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
Colin
Rose
Strategy H T
H 3 -6
T 2 1
Rose’s Payoffs
Player’s Payoffs
-16T
-2-3H
THStrategy
Colin
Rose
Colin’s Payoffs
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
Rose
Strategy H T
H 3 -6
T 2 1
Rose’s Optimal Strategy
Colin’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 PointComments
• Saddlepoint = 0 fair game
• Saddlepoint 0 biased game
– Game biased toward Rose
• This game has a saddlepoint
– It is a “determinate” game
12T
-63H
TH
Rose
Colin
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
Information Set
Decision Node
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
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
Moriarty’s Options
Holmes'sOptions
Canterbury(C)
Dover(D)
Canterbury(C)
0 2/3
Dover(D)
1 0
The Holmes Moriarty Paradox (revisited)
Holmes and Moriarty in London
Moriarty detrains at Canterbury
Moriarty detrainsat Dover
Holmes detrains atCanterbury
Holmes detrains at
Dover
Holmes detrains atCanterbury
Holmes detrains at
Dover
Holmesdies
Holmesescapes
Fightingchance
Holmesdies
Information Setfor Holmes
Game Tree
•0 = Holmes dies
•2/3 = Holmes has a fighting chance
•1 = Holmes succeeds to escape
Moriarty’s Options
Holmes'sOptions
Canterbury(C)
Dover(D)
Canterbury(C)
0 2/3
Dover(D)
1 0
No Saddle Point
Finding Mixed StrategyMoriarty’s Options
Holmes'sOptions
Canterbury(C)
Dover(D)
Canterbury(C)
0 2/3
Dover(D)
1 0
p1
p2
q1q2
Mathematical Expectation employed
E = p1q1 + p2q2 + … + piqi
Mixed Strategy
Holme’s Expectation
Moriarty’s Options
Holmes’Options
(C) (D)
(C) 0 2/3
(D) 1 0
EHolmes : 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
Moriarty’s Options
Holmes’Options
(C) (D)
(C) 0 2/3
(D) 1 0
EMoriarty : 0C+2/3D = 1C+0D 2/3D = C or 2/3(1-C) = C2/3 = 5/3C C = 2/5 => D = 3/5
StrategyMoriarty= 2/5C+3/5D
Mixed Strategy
Moriarty’s Options
Holmes'sOptions
(C) (D) 2/5C+3/5D
(C) 0 2/3
(D) 1 0
3/5C+2/5D
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’sOptions
A(do not confess)
B(confess)
A(do notconfess)
(0,0)both go free
(-2,1)you get heavy
sentence,partner gets reward
You
rO
ptio
ns
B(confess)
(1,-2)you get reward,
partner get heavysentence
(-1,-1)both get light
sentence
The Prisoner’s DilemmaThe 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 H T
H (2, 4) (1, 0)Rose
T (3, 1) (0, 4)
Example
• No equilibrium = No saddle point in zero sum game
• No pure strategy
Games without Equilibrium
How to solve
• Suppose this game as zero sum game• Solve this game by using mixed strategy
Two Different Equilibrium
Colin H T
H (1, 1) (2, 5) Rose
T (5, 2) (-1, -1)
Example
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
Unique Equilibrium Outcome
Partner’sOptions
A(do not confess)
B(confess)
A(do notconfess)
(0,0) (-2,1)
You
rO
ptio
ns
B(confess)
(1,-2) (-1,-1)
Equilibrium Point
What is Pareto Optimal ?
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
Definition
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
Partner’s Options
A(do not confess)
B(confess)
A(do notconfess)
(0,0) (-2,1)
You
rO
ptio
ns
B(confess)
(1,-2) (-1,-1)
Unique Equilibrium
The outcome (-1, -1) is not Pareto optimal –both prisoners are better off choosing (0, 0)
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.
AssumptionAssume your opponent will start by choosing C (cooperate), and continue to choose C(cooperate) until you choose D (defect).
C D
C (R,R) (S,T)
D (T,S) (U,U)
R: reward (0)
S: sucker payoff (-2)
T: Temptation (-1)
U: Uncooperative (0)
Formal Approach
)1(...........32
p
RRpRppRR
With cooperation the payoff would be:
Without cooperation the payoff would be:
)1(
)1( 1
p
UpTppRpR mmm
Formal Approach
)1()1(
11
p
UpTpTpRpR
p
R mmmm
UT
RTp
Formal Approach
R: Reward for cooperation (0)
S: Sucker payoff (-2)
T: Temptation payoff (1)
U: Uncooperative payoff (-1)
2
1
)1(1
01
p
Formal Approach
C D
C (R,R) (S,T)
D (T,S) (U,U)
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
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
I:AA II:AB III:BA IV:BB
A
B
(0,0) (0,0) (-2,1) (-2,1)
(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
Your partner’s choice is contingent on your choice.
Your partner has four strategies:
Partner
You
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
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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?