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## Adventures of Sherlock Holmes

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**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 Continent • Oskar Morgenstern, 1928 • John VonNeumann**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 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**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**Rose**Colin Strategy H T H -3 -2 T 6 -1 Player’s Payoffs 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)**Saddle point Colin’s Optimal Strategy Rose’s Optimal Strategy**Movement Diagram (Method)**• Simpler way to find the saddle point • 1st - consider Rose’s point of view**Movement Diagram (Applied)**Saddle point**Colin**Rose 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**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 • 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**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)**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**No Saddle Point**• 0 = Holmes dies • 2/3 = Holmes has a fighting chance • 1 = Holmes succeeds to escape**Finding Mixed Strategy**q1 p1 p2 q2 Mathematical Expectation employed E = p1q1 + p2q2 + … + piqi**Mixed Strategy**Holme’s Expectation 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 EMoriarty : 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**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**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 H T H (2, 4) (1, 0) Rose T (3, 1) (0, 4) Example**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 H T H (1, 1) (2, 5) Rose T (5, 2) (-1, -1) Example**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**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 • both prisoners are better off • choosing (0, 0) 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). R: reward (0) S: sucker payoff (-2) T: Temptation (-1) U: Uncooperative (0)