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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 Prisoner’s 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 Oskar Morgenstern, 1928 John VonNeumann London Canterbury Dover Continent

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CD C0p DP0 0 = Holmes dies p = Holmes has a fighting chance P = Holmes succeeds to escape Moriarty’s 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 payoff’s (rewards) can be displayed in a reward matrix

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Example of Zero Sum Game Colin Rose StrategyHT H3-6 T21 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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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

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Player’s Payoffs Colin Rose StrategyHT H3-6 T21 Rose’s Payoffs

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Player’s Payoffs 6T -2-3H THStrategy Colin Rose Colin’s 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) Colin Rose StrategyHT H3-6 T21 Rose’s Optimal Strategy Colin’s Optimal Strategy

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Movement Diagram (Method) Simpler way to find the saddle point 1 st - consider Rose’s point of view

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Movement Diagram (Applied) Colin Rose StrategyHT H3-6 T21

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Saddle Point Comments 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

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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 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

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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

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Indeterminate Games Moriarty’s Options Holmes's Options Canterbury (C) Dover (D) Canterbury (C) 02/3 Dover (D) 10 The Holmes Moriarty Paradox (revisited)

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Holmes and Moriarty in London 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 Information Set for Holmes Game Tree

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0 = Holmes dies 2/3 = Holmes has a fighting chance 1 = Holmes succeeds to escape Moriarty’s Options Holmes's Options Canterbury (C) Dover (D) Canterbury (C) 02/3 Dover (D) 10 No Saddle Point

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Finding Mixed Strategy Moriarty’s Options Holmes's Options Canterbury (C) Dover (D) Canterbury (C) 02/3 Dover (D) 10 p1p1 p2p2 q1q1 q2q2 Mathematical Expectation employed E = p 1 q 1 + p 2 q 2 + … + p i q i

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Mixed Strategy Holme’s Expectation Moriarty’s Options Holmes’ Options (C)(D) (C)02/3 (D)10 E Holmes : 0C+1D = 2/3C+0D D=2/3C or 1-C=2/3C C=3/5 => D=2/5 Strategy Holmes = 3/5C+2/5D

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Mixed Strategy Moriarty’s Expectation Moriarty’s Options Holmes’ Options (C)(D) (C)02/3 (D)10 E Moriarty : 0C+2/3D = 1C+0D 2/3D = C or 2/3(1-C) = C 2/3 = 5/3C C = 2/5 => D = 3/5 Strategy Moriarty = 2/5C+3/5D

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Mixed Strategy Moriarty’s Options Holmes's Options (C)(D)2/5C+3/5D (C)02/3 (D)10 3/5C+2/5D

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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 Prisoner’s 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 –Player’s 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 : Prisoner’s 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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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

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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 ? 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

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Unique, but not Pareto Optimal Unique Equilibrium The outcome (-1, -1) is not Pareto optimal –both prisoners are better off choosing (0, 0 )

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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 what’s 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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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 p>1/2. 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 player’s strategies.

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Metagame I:AA II:AB III:BA IV:BB A B (0,0) (0,0) (-2,1) (-2,1) (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

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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)

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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 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

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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 player’s 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?

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