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

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?