Adventures of Sherlock Holmes

1. Adventures of Sherlock Holmes • The story...

2. Adventures of Sherlock Holmes London Canterbury Dover Continent

3. "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

4. Index • Holmes Moriarty paradox • Zero sum games • The Prisoner’s dilemma • F-scale

5. The Holmes Moriarty Paradox • Introduction to solving the problem using some principles of game theory

6. The Adventures of Sherlock Holmes London Canterbury Dover Continent • Oskar Morgenstern, 1928 • John VonNeumann

7. 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

8. 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

9. What Is a Zero Sum Game? • Competitive game • Players either win or lose

10. 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

11. Example of Zero Sum Game Reward Matrix

12. 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)

13. Key Concepts of Game Theory • Payoff • Saddle point

14. 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

15. Player’s Payoffs Rose’s Payoffs

16. Rose Colin Strategy H T H -3 -2 T 6 -1 Player’s Payoffs Colin’s Payoffs

17. 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

18. 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

19. Minimax/Maximin (Applied) Saddle point Colin’s Optimal Strategy Rose’s Optimal Strategy

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

21. Movement Diagram (Applied) Saddle point

22. 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

23. 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”

24. 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

25. 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

26. Indeterminate Games The Holmes Moriarty Paradox (revisited)

27. 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

28. No Saddle Point • 0 = Holmes dies • 2/3 = Holmes has a fighting chance • 1 = Holmes succeeds to escape

29. Finding Mixed Strategy q1 p1 p2 q2 Mathematical Expectation employed E = p1q1 + p2q2 + … + piqi

30. 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

31. 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

32. Mixed Strategy

33. Imagine… • You & a cohort have been arrested • Separate rooms in the police station • You are questioned by the district attorney

34. 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

35. Imagine...

36. The Prisoner’s Dilemma • Non-zero-sum games • Nash equilibrium • Pareto efficiency and inefficiency • Non-cooperative solutions

37. 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

38. 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

39. Games without Equilibrium Colin H T H (2, 4) (1, 0) Rose T (3, 1) (0, 4) Example

40. 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

41. Two Different Equilibrium Colin H T H (1, 1) (2, 5) Rose T (5, 2) (-1, -1) Example

42. 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

43. Unique Equilibrium Outcome Equilibrium Point

44. 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

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

46. 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

47. Non-Cooperative Solutions • Repeated Play-theory • Metagames argument

48. Repeated Play -Theory • Definition • Assumption • Formal approach

49. 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.

50. 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)