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171111 entrepreneurial economics 3

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In this slides deck, you will understand The basic Game Theory.
Also, you can understand how to solve it step by step.
Preparing paper and pen is recommended to solve it.

This event if organized NUS MBA Entrepreneurship Club.
Also, the speaker is Ryosuke ISHII, who is a Co-Founder of Japan Institute of Cognitive Science.

Published in: Economy & Finance
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171111 entrepreneurial economics 3

  1. 1. Entrepreneurial Economics #3 Organized by NUS MBA Entrepreneurship Club BIZ 1-0301 11th November (Sat), 9:50am – 1:00pm Topics to be covered: • Market Theory – Price Discrimination • Game Theory – from the beginning Speaker: • Kyoichiro Tamaki • Ryosuke ISHII
  2. 2. Ryo’s Part • Game Theory • Information • Pre-Nash • Nash Equilibrium
  3. 3. Game Theory from the beginning.
  4. 4. What is Game Theory? Game Theory is a simplification of the World
  5. 5. How Simplify? Only three parameters! • Player – Player A vs Player B, U.S.A vs NorthKorea • Strategy (option) – left or right? , Threat or Conciliation? • Payoff – how much gain/loss ?
  6. 6. Notation – Payoff Matrix Option α Option β Option 1 (𝐴1, 𝐵 𝛼) (𝐴1, 𝐵 𝛽) Option 2 (𝐴2, 𝐵 𝛼) (𝐴2, 𝐵 𝛽) Player A Player B
  7. 7. Notation – Payoff Matrix Option α Option β Option 1 (𝐴1, 𝐵 𝛼) (𝐴1, 𝐵 𝛽) Option 2 (𝐴2, 𝐵 𝛼) (𝐴2, 𝐵 𝛽) Player A Player B When A selected Option 1 and B selected Option α, The Payoff will be 𝑨 𝟏 𝒇𝒐𝒓 𝑨, 𝑎𝑛𝑑𝑩 𝜶 𝒇𝒐𝒓 𝑩
  8. 8. Example – The Price Wars Price ↑ Price ↓ Price ↑ + +,+ + − −,++ + Price ↓ ++ +,− − −, − Store A Store B
  9. 9. The types of Game Theory and information
  10. 10. Information of Game Theory Perfect / Imperfect information • Perfect Information • Each player knows full history of the play of the game thus far. • Dynamic game (=Sequential game) and also could observe other players’ move. • Chess, Go
  11. 11. Information of Game Theory Perfect / Imperfect information • Perfect Information • Each player knows full history of the play of the game thus far. • Dynamic game (=Sequential game) and also could observe other players’ move. • Chess, Go • Imperfect Information • Each player knows parts of history • has no information about the decisions of others • Simultaneous moving game (= static games) • Rock – Paper – Scissors, Sealed bid auction • Tennis (professional level = high speed!) • Dynamic (=Sequential) game but cannot observe other players’ move. • 麻将(麻雀, Mah-jong), Texas hold’em (don’t know other player’s private 牌/cards)
  12. 12. Information of Game Theory Complete / Incomplete information • Complete Information • Every player knows the rules and the structure of the game • Which means, every player knows • Who is the Players • Possible strategies that players can choose • Payoff • Knows the other players also knows the rule Option α Option β Option 1 (𝐴1, 𝐵 𝛼) (𝐴1, 𝐵 𝛽) Option 2 (𝐴2, 𝐵 𝛼) (𝐴2, 𝐵 𝛽)
  13. 13. Information of Game Theory Complete / Incomplete information • Complete Information • Every player knows the rules and the structure of the game • Which means, every player knows • Who is the Players • Possible strategies that players can choose • Payoff • Knows the other players also knows the rule • Incomplete information • Not complete information situation • There are information asymmetry about the rule/payoff • You are penalized from your professor on the exam, but you don’t know why you lose mark.
  14. 14. Information of Game Theory Complete / Incomplete information Information Perfect -knows full history /decision Imperfect: Simultaneous games Complete: Knows the rules, payoff Tennis, soccer (amateur-level) Not interesting game! Just select an obvious best! Rock - Paper – Scissors Tennis, soccer (pro-level) Sealed bid auction Incomplete Price negotiation of used cars Hiring talents
  15. 15. [advanced]Information of Game Theory Complete / Incomplete information Information Perfect -knows full history Imperfect: Simultaneous games Complete: Knows the rules, payoff Tennis, soccer (amateur-level) Rock - Paper – Scissors Tennis, soccer (pro-level) Sealed bid auction Incomplete Price negotiation of used cars Hiring talents Introducing probability p, any incomplete game can be re-written to complete and imperfect information game.
  16. 16. Imperfect and complete information game
  17. 17. Imperfect and complete information game Imperfect and complete information game should be Static Game (=Simultaneous Game)
  18. 18. In the Imperfect and complete information game We Know these parameters! • Player • Strategy • Payoff But we don’t know the other player’s decision before you made a decision. Also, this is non-cooperative game. So, How to analyze game and to optimize your strategy? – before you know your competitor’s move. Option α Option β Option 1 (𝐴1, 𝐵 𝛼) (𝐴1, 𝐵 𝛽) Option 2 (𝐴2, 𝐵 𝛼) (𝐴2, 𝐵 𝛽)
  19. 19. How to analyze game and to optimize your strategy? – Find an Equilibrium! The types of Equilibrium: • Pre- Nash • Dominant Strategy Equilibrium ← We will start here • Iterated Dominance Equilibrium • Maxmin Strategy Equilibrium • Nash • Nash Equilibrium • Post-Nash • Subgame-Perfect Nash Equilibrium (for perfect and incomplete information game)
  20. 20. How to analyze and optimize? – Fix the opponent. Price ↑ Price ↓ Price ↑ + +,+ + − −,++ + Price ↓ ++ +,− − −, − Store A Store B Each Player could only select a strategy. So, first of all, let fix the other player’s strategy. Step1: We are Store A! We have only 2 options: Price↑ or ↓
  21. 21. How to analyze and optimize? – Fix the opponent. Price ↑ Price ↓ Price ↑ + +,+ + − −,++ + Price ↓ ++ +,− − −, − Store A Store B Each Player could only select a strategy. So, first of all, let fix the other player’s strategy. Step1: We are Store A! Step2: Fix B’s strategy as Price ↑
  22. 22. How to analyze and optimize? – Fix the opponent. Price ↑ Price ↓ Price ↑ + +,+ + − −,++ + Price ↓ ++ +,− − −, − Store A Store B Each Player could only select a strategy. So, first of all, let fix the other player’s strategy. Step1: We are Store A! Step2: Fix B’s strategy as Price ↑ Step3: Compare only A’s Payoff. And decide which one would A choose. A would choose Price↓ If A knows B choose ↑
  23. 23. How to analyze and optimize? – Fix the opponent. Price ↑ Price ↓ Price ↑ + +,+ + − −,++ + Price ↓ ++ +,− − −, − Store A Store B Each Player could only select a strategy. So, first of all, let fix the other player’s strategy. Step1: We are Store A! Step2: Fix B’s strategy as Price ↑ Step3: Compare only A’s Pay off. And decide which one would A choose. Step4: Move to B’s next strategy. Price ↓
  24. 24. How to analyze and optimize? – Fix the opponent. Price ↑ Price ↓ Price ↑ + +,+ + − −,++ + Price ↓ ++ +,− − −, − Store A Store B Each Player could only select a strategy. So, first of all, let fix the other player’s strategy. Step4: Move to B’s next strategy. Price ↓ Step5: Compare only A’s Pay off. And decide which one would A choose. A would choose Price↓ If A knows B choose ↓
  25. 25. How to analyze and optimize? – Fix the opponent. Price ↑ Price ↓ Price ↑ + +,+ + − −,++ + Price ↓ ++ +,− − −, − Store A Store B Each Player could only select a strategy. So, first of all, let fix the other player’s strategy. Step4: Move to B’s next strategy. Price ↓ Step5: Compare only A’s Pay off. And decide which one would A choose. A would choose Price↓ If A knows B choose ↓ So, A will choose Price↓ regardless B is ↑ or ↓ This is called “Dominant Strategy”
  26. 26. How to analyze and optimize? – Fix the opponent. Price ↑ Price ↓ Price ↑ + +,+ + − −,++ + Price ↓ ++ +,− − −, − Store A Store B Each Player could only select a strategy. So, first of all, let fix the other player’s strategy. Step6: Change Player. We are Store B!
  27. 27. How to analyze and optimize? – Fix the opponent. Price ↑ Price ↓ Price ↑ + +,+ + − −,++ + Price ↓ ++ +,− − −, − Store A Store B Each Player could only select a strategy. So, first of all, let fix the other player’s strategy. Step6: Change Player. We are Store B! Step 7: Fix A’s strategy as Price ↑ Step 8: Compare only B’s Payoff. B would prefer Price↓
  28. 28. How to analyze and optimize? – Fix the opponent. Price ↑ Price ↓ Price ↑ + +,+ + − −,++ + Price ↓ ++ +,− − −, − Store A Store B Each Player could only select a strategy. So, first of all, let fix the other player’s strategy. Step6: Change Player. We are Store B! Step 7: Fix A’s strategy as Price ↑ Step 8: Compare only B’s Payoff. B would prefer Price↓ Step 9: Fix A’s strategy as Price ↓
  29. 29. How to analyze and optimize? – Fix the opponent. Price ↑ Price ↓ Price ↑ + +,+ + − −,++ + Price ↓ ++ +,− − −, − Store A Store B Each Player could only select a strategy. So, first of all, let fix the other player’s strategy. Step6: Change Player. We are Store B! Step 7: Fix A’s strategy as Price ↑ Step 8: Compare only B’s Payoff. B would prefer Price↓ Step 9: Fix A’s strategy as Price ↓ Step 10: Compare only B’s Payoff. B would prefer Price↓ So, B will choose Price↓ regardless A is ↑ or ↓ This is called “Dominant Strategy”
  30. 30. How to analyze and optimize? – Fix the opponent. Price ↑ Price ↓ Price ↑ + +,+ + − −,++ + Price ↓ ++ +,− − −, − Store A Store B Each Player could only select a strategy. So, first of all, let fix the other player’s strategy. So, for both A and B, Price ↓ is a dominant strategy. This is called “dominant-strategy equilibrium” As a result, they will be worse off by playing “non-cooperative game”. “Not Efficient.” If they are playing cooperative game, They could acquire (+ +,+ +), but Keep on eyes not to cheat each other.
  31. 31. Dominant Strategy • Dominant strategy is a best strategy for a Player No matter what the other player does. • In this setting, every player has a “strictly dominant strategy” • Strategy X Strictly dominant Y means 𝑋𝑖 > 𝑌𝑖 • Strategy X Weakly dominant Y means 𝑋𝑖 ≥ 𝑌𝑖 Strategy C Strategy D Strategy A 2, 3 4, 3 Strategy B 1, 4 3, 3 Strategy A ______ly dominant B Strategy C ______ly dominant D player 2 player 1
  32. 32. Dominant Strategy • Dominant strategy is a best strategy for a Player No matter what the other player does. • In this setting, every player has a “strictly dominant strategy” • Strategy X Strictly dominant Y means 𝑋𝑖 > 𝑌𝑖 • Strategy X weakly dominant Y means 𝑋𝑖 ≥ 𝑌𝑖 Strategy C Strategy D Strategy A 2, 3 4, 3 Strategy B 1, 4 3, 3 As player 1’s perspective, Strategy A strictly dominant B If you fix the opponents strategy as C, strategy A=2 > B=1 Also, fix that as D, Strategy A=4 > B=3 player 1
  33. 33. Dominant Strategy • Dominant strategy is a best strategy for a Player No matter what the other player does. • In this setting, every player has a “strictly dominant strategy” • Strategy X Strictly dominant Y means 𝑋𝑖 > 𝑌𝑖 • Strategy X weakly dominant Y means 𝑋𝑖 ≥ 𝑌𝑖 Strategy C Strategy D Strategy A 2, 3 4, 3 Strategy B 1, 4 3, 3 As player 2’s perspective, Strategy C Weakly dominant D If you fix the opponents strategy as A, strategy C=3 = B=3 Also, fix that as B, Strategy C=4 > D=3
  34. 34. Dominated Strategy • Dominated strategy is a inferior strategy for a Player which rational player would not choose. • Strategy Y is Strictly dominated by X means 𝑋𝑖 > 𝑌𝑖 • Strategy Y is weakly dominated by X means 𝑋𝑖 ≥ 𝑌𝑖 Strategy C Strategy D Strategy A 2, 3 4, 3 Strategy B 1, 4 3, 3 Strategy B is ______ly dominated by A Strategy D is ______ly dominated by C player 2 player 1
  35. 35. Dominated Strategy • Dominated strategy is a inferior strategy for a Player which rational player would not choose. • Strategy Y is Strictly dominated by X means 𝑋𝑖 > 𝑌𝑖 • Strategy Y is weakly dominated by X means 𝑋𝑖 ≥ 𝑌𝑖 Strategy C Strategy D Strategy A 2, 3 4, 3 Strategy B 1, 4 3, 3 Strategy B is strongly dominated by A Strategy D is weakly dominated by C player 2 player 1
  36. 36. Dominant Strategy • Dominant strategy is a best strategy for a Player No matter what the other player does. • In this setting, every player has a “strictly dominant strategy” • Strategy X Strictly dominant Y means 𝑋𝑖 > 𝑌𝑖 • Strategy X weakly dominant Y means 𝑋𝑖 ≥ 𝑌𝑖 • Every player has a strictly dominant strategy, the outcome of the game is called a “dominant-strategy equilibrium” • Both Player worse off by this Dominant strategy. • Such case we call “Prisoners’ Dilemma” Price ↑ Price ↓ Price ↑ + +,+ + − −,++ + Price ↓ ++ +,− − −, −
  37. 37. Example : Find a Dominant Strategy Equilibrium : Apologize Break up Apologize 0, 0 -10, +1 Break up +1,-10 -3, -3 You Your boy/girl friend Suppose you and your boy/girl friend are relatively new relationship. You got a big fight last night. Both of you have 2 strategies: Apologize or Break up. This is non-cooperative and simultaneous move. Where is a Dominant Strategy Equilibrium ? [2min - Workshop] • Find your partner. (2 ppl 1 team) • Decide who will do first (Player1). • Explain how to solve it (process) • Player2 should do feedback to the Player1 for Player1 could explain better (on the exam paper).
  38. 38. Example2 : Find a Dominant Strategy Equilibrium Ad No Ad Ad 3, 3 13, -2 No Ad -2, 13 8, 8 Suppose you are Japan Tobacco and your opponent is Phillip Morris. This is an advertisement wars. Advertisement cost you much. If only your competitor do not Ad, you gain much profit from Ad. • This is non-cooperative and simultaneous move. Where is a Dominant Strategy Equilibrium? How to get out of the Dilemma? [2min - Workshop] • Player2 Explain how to solve it (process) • Player2 should be better to explain utilizing your own feedback to Player 1. • Player1 should do feedback to the Player2 for Player2 could explain better (on the exam paper).
  39. 39. Example2 : Find a Nash Equilibrium: Ad No Ad Ad Nash 3, 3 13, -2 No Ad -2, 13 8, 8 Thanks to Government regulation for Non-Ad policy for tobacco industry, They could get out of Prisoners’ Dilemma and they are better off!!! Government regulation
  40. 40. How to analyze game and to optimize your strategy? – Find an Equilibrium! The types of Equilibrium: • Pre- Nash • Dominant Strategy Equilibrium • Iterated Dominance Equilibrium← What’s NEXT • Maxmin Strategy Equilibrium • Nash • Nash Equilibrium • Mixed Strategy • Post-Nash • Subgame-Perfect Nash Equilibrium (for perfect and incomplete information game)
  41. 41. Iterated Dominance Equilibrium • Dominant strategy was about “which strategy will be played for sure” • But NOT every time dominant strategy equilibrium exists. • So, let’s think about “which strategy will NOT be played for sure” • Which is Dominated Strategy. • If we find dominated strategy, we could eliminate the option(s). • After few times do the elimination(=iterated), we will find a equilibrium which is called “Iterated Dominance Equilibrium”
  42. 42. How to find and eliminate? Left Center Right Top 3, 6 7, 4 10, 1 Middle 5, 1 8, 2 14, 6 Bottom 6, 0 6, 2 8, 5 Player A Player B Same as before, let fix the other player’s strategy one by one and compare Your strategy. Step1: We are Player A! We have only 3 options: Select Top, Middle, or Bottom.
  43. 43. How to find and eliminate? Left Center Right Top 3, 6 7, 4 10, 1 Middle 5, 1 8, 2 14, 6 Bottom 6, 0 6, 2 8, 5 Player A Player B Same as before, let fix the other player’s strategy one by one and compare Your strategy. Step1: We are Player A! We have only 3 options: Select Top, Middle, or Bottom. So, we will see only Player A’s payoff
  44. 44. How to find and eliminate? Left Center Right Top 3, 6 7, 4 10, 1 Middle 5, 1 8, 2 14, 6 Bottom 6, 0 6, 2 8, 5 Player A Player B Same as before, let fix the other player’s strategy one by one and compare Your strategy. Step1: We are Player A! Step2: Fix your rival’s strategy. Say, Left. - Top and Middle are worse than Bottom.
  45. 45. How to find and eliminate? Left Center Right Top 3, 6 7, 4 10, 1 Middle 5, 1 8, 2 14, 6 Bottom 6, 0 6, 2 8, 5 Player A Player B Same as before, let fix the other player’s strategy one by one and compare Your strategy. Step1: We are Player A! Step2: Fix rival’s strategy. Step3: change rival’s STR. Say, Center. - Top and Bottom are worse than Middle.
  46. 46. How to find and eliminate? Left Center Right Top 3, 6 7, 4 10, 1 Middle 5, 1 8, 2 14, 6 Bottom 6, 0 6, 2 8, 5 Player A Player B Same as before, let fix the other player’s strategy one by one and compare Your strategy. Step1: We are Player A! Step2: Fix rival’s strategy. Step3: change rival’s STR. Say, Right. - Top and Bottom are worse than Middle.
  47. 47. How to find and eliminate? Left Center Right Top 3, 6 7, 4 10, 1 Middle 5, 1 8, 2 14, 6 Bottom 6, 0 6, 2 8, 5 Player A Player B Same as before, let fix the other player’s strategy one by one and compare Your strategy. Step1: We are Player A! Step2: Fix rival’s strategy. Step3: change rival’s STR. Step4: Find a dominant or Dominated strategy. In this case, No dominant STR. And the Top STR is dominated strategy.
  48. 48. How to find and eliminate? Left Center Right Top 3, 6 7, 4 10, 1 Middle 5, 1 8, 2 14, 6 Bottom 6, 0 6, 2 8, 5 Player A Player B Same as before, let fix the other player’s strategy one by one and compare Your strategy. Step1: We are Player A! Step2: Fix rival’s strategy. Step3: change rival’s STR. Step4: Find a dominant or Dominated strategy. In this case, No dominant STR. And the Top STR is dominated strategy. So Eliminate it.
  49. 49. How to find and eliminate? Left Center Right Top 3, 6 7, 4 10, 1 Middle 5, 1 8, 2 14, 6 Bottom 6, 0 6, 2 8, 5 Player A Player B Same as before, let fix the other player’s strategy one by one and compare Your strategy. Step5: Neither Middle / Bottom are Dominated/Dominant STR. So Move to Player B. As Player B, We should focus only on Player B’s payoff
  50. 50. How to find and eliminate? Left Center Right Top 3, 6 7, 4 10, 1 Middle 5, 1 8, 2 14, 6 Bottom 6, 0 6, 2 8, 5 Player A Player B Same as before, let fix the other player’s strategy one by one and compare Your strategy. Step6:We are Player B! We have only 3 options: Select Left, Center, or Right.
  51. 51. How to find and eliminate? Left Center Right Top 3, 6 7, 4 10, 1 Middle 5, 1 8, 2 14, 6 Bottom 6, 0 6, 2 8, 5 Player A Player B Same as before, let fix the other player’s strategy one by one and compare Your strategy. Step6: We are Player B! Step7: Fix Rival’s strategy. Say, Middle. -Right is better than Left and Center.
  52. 52. How to find and eliminate? Left Center Right Top 3, 6 7, 4 10, 1 Middle 5, 1 8, 2 14, 6 Bottom 6, 0 6, 2 8, 5 Player A Player B Same as before, let fix the other player’s strategy one by one and compare Your strategy. Step6: We are Player B! Step7: Fix Rival’s strategy. Step8: Change Rival’s STR. Say, Bottom. -Right is better than Left and Center.
  53. 53. How to find and eliminate? Left Center Right Top 3, 6 7, 4 10, 1 Middle 5, 1 8, 2 14, 6 Bottom 6, 0 6, 2 8, 5 Player A Player B Same as before, let fix the other player’s strategy one by one and compare Your strategy. Step6: We are Player B! Step7: Fix Rival’s strategy. Step8: Change Rival’s STR. Step9: Find a dominant or Dominated strategy. In this case, Right is a Dominant STR For player B.
  54. 54. How to find and eliminate? Left Center Right Top 3, 6 7, 4 10, 1 Middle 5, 1 8, 2 14, 6 Bottom 6, 0 6, 2 8, 5 Player A Player B Same as before, let fix the other player’s strategy one by one and compare Your strategy. Step6: We are Player B! Step7: Fix Rival’s strategy. Step8: Change Rival’s STR. Step9: Find a dominant or Dominated strategy. In this case, Right is a Dominant STR For player B. So, eliminate others.
  55. 55. How to find and eliminate? Left Center Right Top 3, 6 7, 4 10, 1 Middle 5, 1 8, 2 14, 6 Bottom 6, 0 6, 2 8, 5 Player A Player B Same as before, let fix the other player’s strategy one by one and compare Your strategy. Step6: We are Player B! Step7: Fix Rival’s strategy. Step8: Change Rival’s STR. Step9: Find a STR. Step10: As a Player A, Select the Best: Middle.
  56. 56. How to find and eliminate? Left Center Right Top 3, 6 7, 4 10, 1 Middle 5, 1 8, 2 14, 6 Bottom 6, 0 6, 2 8, 5 Player A Player B Same as before, let fix the other player’s strategy one by one and compare Your strategy. Step11: Finally we find iterated dominance equilibrium
  57. 57. Exercise: Find an iterated dominance equilibrium 東 east 南 south 西 west 北 north Top 1,1 1,2 5,0 1,1 Middle 2,3 1,2 3,0 5,1 Bottom 1,1 0,5 1,7 0,1 Player A Player B Same as before, let fix the other player’s strategy one by one and compare Your strategy. [5min - Workshop] • Find your partner. (2 ppl 1 team) • Decide who will do first (Player1). • Both player solve it individually. (2 min) • Explain how to solve it (process) (2 min) • Player2 should do feedback to the Player1 for Player1 could explain better (on the exam paper).
  58. 58. Exercise: Find an iterated dominance equilibrium 東 east 南 south 西 west 北 north Top 1,1 1,2 5,0 1,1 Middle 2,3 1,2 3,0 5,1 Bottom 1,1 0,5 1,7 0,1 Player A Player B Same as before, let fix the other player’s strategy one by one and compare Your strategy. [+3min - Workshop] • Change your role! • Player2 Explain how to solve it (process – 2min) • Player2 should be better to explain utilizing your own feedback to Player 1. • Player1 give Player2 a quality feedback. • High challenge! High support!
  59. 59. Exercise: Find an iterated dominance equilibrium 東 east 南 south 西 west 北 north Top 1, 1 1, 2 5, 0 1, 1 Middle 2, 3 1, 2 3, 0 5, 1 Bottom 1, 1 0, 5 1, 7 0, 1 Player A Player B Same as before, let fix the other player’s strategy one by one and compare Your strategy. let’s turn in Player A’s shoes.
  60. 60. Exercise: Find an iterated dominance equilibrium 東 east 南 south 西 west 北 north Top 1, 1 1, 2 5, 0 1, 1 Middle 2, 3 1, 2 3, 0 5, 1 Bottom 1, 1 0, 5 1, 7 0, 1 Player A Player B Same as before, let fix the other player’s strategy one by one and compare Your strategy. Bottom is dominated STR.
  61. 61. Exercise: Find an iterated dominance equilibrium 東 east 南 south 西 west 北 north Top 1, 1 1, 2 5, 0 1, 1 Middle 2, 3 1, 2 3, 0 5, 1 Bottom 1, 1 0, 5 1, 7 0, 1 Player A Player B Same as before, let fix the other player’s strategy one by one and compare Your strategy. Bottom is dominated STR. So, Eliminate it.
  62. 62. Exercise: Find an iterated dominance equilibrium 東 east 南 south 西 west 北 north Top 1, 1 1, 2 5, 0 1, 1 Middle 2, 3 1, 2 3, 0 5, 1 Bottom 1, 1 0, 5 1, 7 0, 1 Player A Player B Same as before, let fix the other player’s strategy one by one and compare Your strategy. As Player B, 西 – west and 北 – north are also dominated. so,
  63. 63. Exercise: Find an iterated dominance equilibrium 東 east 南 south 西 west 北 north Top 1, 1 1, 2 5, 0 1, 1 Middle 2, 3 1, 2 3, 0 5, 1 Bottom 1, 1 0, 5 1, 7 0, 1 Player A Player B Same as before, let fix the other player’s strategy one by one and compare Your strategy. As Player B, 西 – west and 北 – north are also dominated. so, delete it. (But we still cannot say east or south is dominant. So left it.)
  64. 64. Exercise: Find an iterated dominance equilibrium 東 east 南 south 西 west 北 north Top 1, 1 1, 2 5, 0 1, 1 Middle 2, 3 1, 2 3, 0 5, 1 Bottom 1, 1 0, 5 1, 7 0, 1 Player A Player B Same as before, let fix the other player’s strategy one by one and compare Your strategy. As Player A again, Top is weakly dominated.
  65. 65. Exercise: Find an iterated dominance equilibrium 東 east 南 south 西 west 北 north Top 1, 1 1, 2 5, 0 1, 1 Middle 2, 3 1, 2 3, 0 5, 1 Bottom 1, 1 0, 5 1, 7 0, 1 Player A Player B Same as before, let fix the other player’s strategy one by one and compare Your strategy. As Player A again, Top is weakly dominated. So eliminate it.
  66. 66. Exercise: Find an iterated dominance equilibrium 東 east 南 south 西 west 北 north Top 1, 1 1, 2 5, 0 1, 1 Middle 2, 3 1, 2 3, 0 5, 1 Bottom 1, 1 0, 5 1, 7 0, 1 Player A Player B Same as before, let fix the other player’s strategy one by one and compare Your strategy. In this situation, A has no choice but Middle STR. So, as Player B, you could choose Proper one. Comparing 東east = 3 南south = 2 And choose 3. So, (Middle, East) is Iterated dominance equilibrium.
  67. 67. Tips… (another way to get answer) 東 east 南 south 西 west 北 north Top 1, 1 1, 2 5, 0 1, 1 Middle 2, 3 1, 2 3, 0 5, 1 Bottom 1, 1 0, 5 1, 7 0, 1 Player A Player B You can start from Player B’s viewpoint, and eliminate north at first, because north is inferior strategy which means north is always dominated by west. We are comparing north , 1 , 1 , 1 And others. Especially south. south , 2 , 2 , 5 And we know north is dominated by south.
  68. 68. How to analyze game and to optimize your strategy? – Find an Equilibrium! The types of Equilibrium: • Pre- Nash • Dominant Strategy Equilibrium • Iterated Dominance Equilibrium • Maxmin Strategy Equilibrium ← Let’s talk about it! • Nash • Nash Equilibrium • Mixed Strategy • Post-Nash • Subgame-Perfect Nash Equilibrium (for perfect and incomplete information game)
  69. 69. Maxmin Strategy Equilibrium • What we already done: • We dealt with 2 strategy: Dominant and Dominated. • Mainly focus on how much payoff we could earn, how to maximize your profit. • Remember we eliminate dominated as Dominated STR has lower profit always. • Also suppose all players are Rational and No mistakes. • Remember we eliminate dominated and we adopt dominant for sure. • What is MAXMIN? • Sometimes people want to avoid risk, rather Gain or Payoff. • So, MAXMIN STR. is a strategy which select a strategy: • Max(Min(Strategy1), Min(Strategy2), Min(Strategy3)… Min(Strategy 𝑛), )
  70. 70. Maxmin Strategy Equilibrium • So, MAXMIN STR. is a strategy which select a strategy: • Max(Min(Strategy1), Min(Strategy2), Min(Strategy3)… Min(Strategy 𝑛), ) a b c d Min of A Top 1, 1 1, 2 5, 0 1, 1 1 Middle 2, 3 -100, 2 3, 0 5, 1 -100 Bottom 1, 1 0, 5 1, 7 0, 1 0 Player A Player B This Top STR. has Maximum payoff among each strategy’s Minimum payoff. So, MAXMIN.
  71. 71. How to do MAXMIN? And how different? Let’s compare 2 strategies: Dominant STR and MAXMIN STR. STR B1 STR B2 STR A1 10, 4 8, 15 STR A2 -10,5 20, 10 5 min for ① Dominant (or iterated dominance) Strategy: ② MAXMIN Strategy: • Both player solve it individually. (2 min) • Player1 explain how to solve it (process) (2 min) • Player2 should do feedback to the Player1 for Player1 could explain better (on the exam paper).
  72. 72. How to do MAXMIN? And how different? Let’s compare 2 strategies: Dominant STR and MAXMIN STR. STR B1 STR B2 STR A1 10, 4 8, 15 STR A2 -10,5 20, 10 ① Dominant(or iterated dominance) Strategy: Player A has no dominant Strategy. For Player B, STR B2 is dominant STR. Given the Player B’s STR as B2, Player A will choose A2. So, Dominant STR is (A2, B2)
  73. 73. How to do MAXMIN? And how different? Let’s compare 2 strategies: Dominant STR and MAXMIN STR. STR B1 STR B2 STR A1 10, 4 8, 15 STR A2 -10,5 20, 10 ② MAXMIN Strategy:
  74. 74. How to do MAXMIN? And how different? Let’s compare 2 strategies: Dominant STR and MAXMIN STR. ② MAXMIN Strategy: As the Graph Shows, For A: MIN(A1)=8 MIN(A2)= –10 MAX(MIN(A1), MIN(A2))→ A1 For B: MIN(B1)=4 MIN(B2)=10 MAX(MIN(B1), MIN(B2))→B2 So, MAXMIN STR.= (A1,B2) STR B1 STR B2 Min of A STR A1 10, 4 8, 15 8 STR A2 -10,5 20, 10 -10 Min of B 4 10
  75. 75. How to analyze game and to optimize your strategy? – Find an Equilibrium! The types of Equilibrium: • Pre- Nash • Dominant Strategy Equilibrium • Iterated Dominance Equilibrium • Maxmin Strategy Equilibrium • Nash • Nash Equilibrium • Mixed Strategy • Post-Nash • Subgame-Perfect Nash Equilibrium (for perfect and incomplete information game) Before taking break: • Form a team of 4 people • Every player recall the “Pre-Nash” part and remember you thought “it’s important”(2min) • Each player share 1 point above to the team. Player 1 → 2 → 3 →4. • Repeat the share 3 times. So you will get 12 points from this 10 min.
  76. 76. How to analyze game and to optimize your strategy? – Find an Equilibrium! The types of Equilibrium: • Pre- Nash • Dominant Strategy Equilibrium • Iterated Dominance Equilibrium • Maxmin Strategy Equilibrium • Nash • Nash Equilibrium ← So finally, “Beautiful Mind” • Mixed Strategy • Post-Nash • Subgame-Perfect Nash Equilibrium (for perfect and incomplete information game)
  77. 77. “Nash Equilibrium” What is a Nash equilibrium? • No firm wants to change its strategy, given what the other firm is doing. • So a Nash equilibrium is when no firm wants to change/deviate, given what the other firms are doing. • It's basically when, given what all the other firms are doing, you are happy with what you're doing. • If every players’ Dominant Strategy intersects, it is Nash. In Nash Equilibrium, You could think: • This is a best response to others’ strategy. • The game is stable • Everybody's satisfied and the market can roll forward.
  78. 78. “Nash Equilibrium” What is a Difference? • Dominance STR. • Search out equilibrium by “take dominance” or “eliminate dominated” • Nash Equilibrium • You are already given the strategy 𝐴𝑖 , 𝐵𝑗, 𝐶 𝑘, … . . • Your strategy is the best strategy for everyone if you fix others’ strategy. • So, some payoff matrix could have more than 1 Nash Equilibrium. • Nash does not mean “the best” strategy bundle.
  79. 79. Nash Equilibrium Price ↑ Price ↓ Price ↑ + +,+ + − −,++ + Price ↓ ++ +,− − −, − Store A Store B Equilibrium = the point at which all of the players are satisfied. Remember the example of Dominance Strategy. We concluded (Price↓, Price↓) is “dominant-strategy equilibrium”. This is Also called “Nash Equilibrium” Because Given the situation, No player want to change Each of strategies.
  80. 80. Nash Equilibrium Price ↑ Price ↓ Price ↑ + +,+ + − −,++ + Price ↓ ++ +,− − −, − Store A Store B Here is not a Nash For Player A, Given the Player B’s strategy (P↑) Player A want to change strategy. So, this is not a Nash Equilibrium.
  81. 81. Nash Equilibrium finding • So, How to find a Nash Equilibrium? 東 east 南 south 西 west 北 north Top Middle Bottom Player A Player B If you do not want to use your brain resource, but have a lot of time, You can check Whether Nash or Not Cell by Cell Remember, each player can only select his/her STR.
  82. 82. Nash Equilibrium finding • So, How to find a Nash Equilibrium? More efficient? 東 east 南 south 西 west 北 north Top 1,1 1,2 5,0 1,1 Middle 2,3 1,2 3,0 5,1 Bottom 1,1 0,5 1,7 0,1 Player B Step1: Find out “The Best Response” given the other players’ strategy. Suppose you are Player A, and Player B select STR 東(east). Player A
  83. 83. Nash Equilibrium finding • So, How to find a Nash Equilibrium? More efficient? 東 east 南 south 西 west 北 north Top 1,1 1,2 5,0 1,1 Middle 2,3 1,2 3,0 5,1 Bottom 1,1 0,5 1,7 0,1 Player B Step1: Find out “The Best Response” given the other players’ strategy. Suppose you are Player A, and Player B select STR 東(east). Given, the situation, “2” is the best strategy. Next, Change B’s STR. Player A
  84. 84. Nash Equilibrium finding • So, How to find a Nash Equilibrium? More efficient? 東 east 南 south 西 west 北 north Top 1,1 1,2 5,0 1,1 Middle 2,3 1,2 3,0 5,1 Bottom 1,1 0,5 1,7 0,1 Player B Step1: Find out “The Best Response” given the other players’ strategy. Suppose you are Player A, and Player B select STR 東(east). Given, the situation, “2” is the best strategy. Next, Change B’s STR. Player A
  85. 85. Nash Equilibrium finding • So, How to find a Nash Equilibrium? More efficient? 東 east 南 south 西 west 北 north Top 1,1 1,2 5,0 1,1 Middle 2,3 1,2 3,0 5,1 Bottom 1,1 0,5 1,7 0,1 Player A Player B Step1: Find out “The Best Response” given the other players’ strategy. So , these are A’ s best responses given the each of B’s STR.
  86. 86. Nash Equilibrium finding • So, How to find a Nash Equilibrium? More efficient? 東 east 南 south 西 west 北 north Top 1,1 1,2 5,0 1,1 Middle 2,3 1,2 3,0 5,1 Bottom 1,1 0,5 1,7 0,1 Player A Player B Step2: Find out B’s “The Best Response” given the A’s strategy. Suppose you are a Player B, and Player A select “Top” STR. Given the situation, (Top, south) is a best response for B.
  87. 87. Nash Equilibrium finding • So, How to find a Nash Equilibrium? More efficient? 東 east 南 south 西 west 北 north Top 1,1 1,2 5,0 1,1 Middle 2,3 1,2 3,0 5,1 Bottom 1,1 0,5 1,7 0,1 Player A Step2: Find out B’s “The Best Response” given the A’s strategy. We could do same thing. Player B
  88. 88. Nash Equilibrium finding • So, How to find a Nash Equilibrium? More efficient? 東 east 南 south 西 west 北 north Top 1,1 1,2 5,0 1,1 Middle 2,3 1,2 3,0 5,1 Bottom 1,1 0,5 1,7 0,1 Player A Step3: Find Nash. So, if the cell becomes (Player A, Player B) This is a Nash. In this case, 2 Nash EQs. Player B
  89. 89. Exercise on Nash Equilibrium finding eat run study Drink Top 3,5 7,5 3,6 1,-3 Bottom 2,6 8,7 2,5 5,-5 Right 6,9 0,5 1,4 0,-3 Left 1,9 1,3 2,5 2,1 Player A Player B • So, How to find a Nash Equilibrium? More efficient? [5 min - Workshop] • Both player solve it individually. (2 min) • Player2 explain how to solve it (process) (2 min) • Player1 should do feedback to the Player2 for Player1 could explain better (on the exam paper).
  90. 90. Nash Equilibrium finding eat run study Drink Top 3,5 7,5 3,6 1,-3 Bottom 2,6 8,7 2,5 5,-5 Right 6,9 0,5 1,4 0,-3 Left 1,9 1,3 2,5 2,1 Player B • So, How to find a Nash Equilibrium? More efficient? As Player A, let’s fix B’s STR one by one. And check it red. We can ignore B’s payoff now.Player A
  91. 91. Nash Equilibrium finding eat run study Drink Top 3,5 7,5 3,6 1,-3 Bottom 2,6 8,7 2,5 5,-5 Right 6,9 0,5 1,4 0,-3 Left 1,9 1,3 2,5 2,1 • So, How to find a Nash Equilibrium? More efficient? Next, as Player B, Deal A’s STR as given. And select a best response We can ignore A’s payoff now. Player A Player B
  92. 92. Nash Equilibrium finding eat run study Drink Top 3,5 7,5 3,6 1,-3 Bottom 2,6 8,7 2,5 5,-5 Right 6,9 0,5 1,4 0,-3 Left 1,9 1,3 2,5 2,1 • So, How to find a Nash Equilibrium? More efficient? These three are (pure STR) Nash Equilibrium. Player A Player B
  93. 93. Nash Equilibrium finding eat run study Drink Top 3,5 7,5 3,6 1,-3 Bottom 2,6 8,7 2,5 5,-5 Right 6,9 0,5 1,4 0,-3 Left 1,9 1,3 2,5 2,1 Player A Player B • So, How to find a Nash Equilibrium? More efficient? If one strategy is dominated, this strategy must not Nash Equilibrium Because this cannot be a best response. So you can start from eliminate dominated strategy. As Player B, Drink is dominated.
  94. 94. Nash Equilibrium finding eat run study Drink Top 3,5 7,5 3,6 1,-3 Bottom 2,6 8,7 2,5 5,-5 Right 6,9 0,5 1,4 0,-3 Left 1,9 1,3 2,5 2,1 Player A Player B • So, How to find a Nash Equilibrium? More efficient? If one strategy is dominated, this strategy must not Nash Equilibrium Because this cannot be a best response. So you can start from eliminate dominated strategy. As Player B, Drink is dominated. After eliminated Drink, Player A also could eliminate Left as a dominated STR. But after this, we could not eliminate. So, start same as before.
  95. 95. Nash Equilibrium finding eat run study Drink Top 3,5 7,5 3,6 1,-3 Bottom 2,6 8,7 2,5 5,-5 Right 6,9 0,5 1,4 0,-3 Left 1,9 1,3 2,5 2,1 Player A Player B • So, How to find a Nash Equilibrium? More efficient? If one strategy is dominated, this strategy must not Nash Equilibrium Because this cannot be a best response. So you can start from eliminate dominated strategy. As Player B, Drink is dominated. After eliminated Drink, Player A also could eliminate Left as a dominated STR. But after this, we could not eliminate. So, start same as before.
  96. 96. Nash Equilibrium finding eat run study Drink Top 3,5 7,5 3,6 1,-3 Bottom 2,6 8,7 2,5 5,-5 Right 6,9 0,5 1,4 0,-3 Left 1,9 1,3 2,5 2,1 Player A Player B • So, How to find a Nash Equilibrium? More efficient? And finally, same as before.
  97. 97. Exercise Nash You and your rival could select 0~4 integer. And if only $𝑁 − $𝑀 = 1, you could earn the $N where N is a integer you selected and M is your rival’s. How many Nash Equilibrium do we have? 0 1 2 3 4 0 1 2 3 4
  98. 98. Exercise Nash You and your rival could select 0~4 integer. And if only $𝑁 − $𝑀 = 1, you could earn the $100 where N is a integer you selected and M is your rival’s. How many Nash Equilibrium do we have? 0 1 2 3 4 0 0, 0 0, 1 0, 0 0, 0 0, 0 1 1, 0 0, 0 1, 2 0, 0 0, 0 2 0, 0 2, 1 0, 0 2, 3 0, 0 3 0, 0 0, 0 3, 2 0, 0 3, 4 4 0, 0 0, 0 0, 0 4, 3 0, 0 Suppose, you are Player A. Fix, your opponent’s strategy. Given the B’s strategy above, Search out the best response, among your strategies. In this case, when B select STR B=1, Your best response will be STR A=2 So, color it.
  99. 99. Exercise Nash You and your rival could select 0~4 integer. And if only $𝑁 − $𝑀 = 1, you could earn the $100 where N is a integer you selected and M is your rival’s. How many Nash Equilibrium do we have? 0 1 2 3 4 0 0, 0 0, 1 0, 0 0, 0 0, 0 1 1, 0 0, 0 1, 2 0, 0 0, 0 2 0, 0 2, 1 0, 0 2, 3 0, 0 3 0, 0 0, 0 3, 2 0, 0 3, 4 4 0, 0 0, 0 0, 0 4, 3 0, 0 2 Nash Equilibriums.
  100. 100. How to analyze game and to optimize your strategy? – Find an Equilibrium! The types of Equilibrium: • Pre- Nash • Dominant Strategy Equilibrium • Iterated Dominance Equilibrium • Maxmin Strategy Equilibrium • Nash • Nash Equilibrium • Mixed Strategy ← Every payoff matrix has at least one Nash! • Post-Nash • Subgame-Perfect Nash Equilibrium (for perfect and incomplete information game)
  101. 101. Nash Equilibrium in Mixed Strategy What is an Nash Equilibrium in Mixed Strategy? Pure strategy Paper Scissors Rock Paper 0, 0 -1, 1 1, -1 Scissors 1, -1 0, 0 -1, 1 Rock -1, 1 1, -1 0, 0 In the repeated game, if you fix your strategy, such as “you always use paper”. You surely lose. So we need to “Mix Strategies.” and find a proper probability to mix the strategies. Mixed Strategy Paper q1 Scissors q2 Rock 1-q1-q2 Paper p1 0, 0 -1, 1 1, -1 Scissors P2 1, -1 0, 0 -1, 1 Rock 1-p1-p2 -1, 1 1, -1 0, 0
  102. 102. Mixed Strategy This payoff matrix below represents the probability (%) of successfully – get a score for kicker, and keep a goal for goalkeeper, on a penalty kick. Guard Left Guard Right Kick Left 58%, 42% 95%, 5% Kick Right 93%, 7% 70%, 30% In this case, There are no (Pure Strategy) Nash Equilibrium. But there are MAXMIN strategy.
  103. 103. Mixed Strategy This payoff matrix below represents the probability (%) of successfully – get a score for kicker, and keep a goal for goalkeeper, on a penalty kick. Guard Left Guard Right Min of kicker Kick Left 58%, 42% 95%, 5% 58% Kick Right 93%, 7% 70%, 30% 70% Min of keeper 7% 5% From the MAXMIN strategy, it is better -for kicker to select “kick right” -for keeper to select “guard left” So the MAXMIN Equilibrium is (Right, Left). But if do so, kicker always win. So keeper changes to right and your MAXMIN payoff will be 70%. Or, if you fix your strategy, your rival always gain from your inflexibility.
  104. 104. Mixed Strategy So, let’s “mix” your strategy. To mix your strategy it is important (i) randomize, and (ii) maximize your payoff or minimize rival’s payoff. Guard Left Guard Right Kick Left 58%, 42% 95%, 5% Kick Right 93%, 7% 70%, 30% Left q Right 1-q Left p 58%, 42% 95%, 5% Right 1-p 93%, 7% 70%, 30% There are no (Pure Strategy) Nash Eq.
  105. 105. Mixed Strategy Exercise for understanding: Suppose you are kicker, calculate your “minimum payoff” if 𝑝 = 0.5, 0.4 Left q Right 1-q Left p 58%, 42% 95%, 5% Right 1-p 93%, 7% 70%, 30% kicker keeper [5 min - Workshop] • Both player solve it individually. (2 min) • Player1 explain how to solve it (process) (2 min) • Player2 should do feedback to the Player1 for Player2 could explain better (on the exam paper).
  106. 106. Mixed Strategy Exercise for understanding: Suppose you are kicker, calculate your “minimum payoff” if 𝑝 = 0.5, 0.4 Left q Right 1-q Left p 58%, 42% 95%, 5% Right 1-p 93%, 7% 70%, 30% kicker keeper If p = 0.5 how should we think? 1. Fix the rival’s strategy again. So the keeper select left. 2. Kicker’s payoff will be 58% × 0.5 + 93% × 0.5=75.5% 3. Next, change rival’s strategy.
  107. 107. Mixed Strategy Exercise for understanding: Suppose you are kicker, calculate your “minimum payoff” if 𝑝 = 0.5, 0.4 Left q Right 1-q Left p 58%, 42% 95%, 5% Right 1-p 93%, 7% 70%, 30% kicker keeper If p = 0.5 how should we think? 1. Fix the rival’s strategy again. So the keeper select left. 2. Kicker’s payoff will be 58% × 0.5 + 93% × 0.5=75.5% 3. Next, change rival’s strategy. Say keeper → Right. 95% × 0.5 + 70% × 0.5=82.5% 4. So if keeper select left always, your MIX strategy will result in 75.5%. 5. This is better than MAXMIN Strategy(70%) Mix Strategy Left Right 𝑃 = 0.5 75.5% 82.5%
  108. 108. Mixed Strategy Exercise for understanding: Suppose you are kicker, calculate your “minimum payoff” if 𝑝 = 0.5, 0.4 Left q Right 1-q Left p 58%, 42% 95%, 5% Right 1-p 93%, 7% 70%, 30% kicker keeper If p = 0.4 how should we think? 1. Fix the rival’s strategy again. So the keeper select left. 2. Kicker’s payoff will be 58% × 0.4 + 93% × 0.6=79% 3. Next, change rival’s strategy. Say keeper → Right. 95% × 0.4 + 70% × 0.6=80% 4. So if keeper select left always, your MIX strategy will result in 79%. Mix Strategy Left Right 𝑃 = 0.5 75.5% 82.5% 𝑃 = 0.4 79% 80%
  109. 109. Mixed Strategy In this case, the most awful thing is your opponent could predict your action. So, Find a P where: make the opponent feel indifferent whether s/he select Left or Right Left q Right 1-q Left p 58%, 42% 95%, 5% Right 1-p 93%, 7% 70%, 30% kicker keeper Mix Strategy Left Right 𝑃 = 0.5 75.5% 82.5% 𝑃 = 0.4 79% 80% 𝑃 = 𝑋 =Right =Left 1. If Keeper’s strategy is Left, your payoff is: 58% × 𝑝 + 93% × (1 − 𝑝) 2. If Keeper’s strategy is Right, your payoff is: 95% × 𝑝 + 70% × (1 − 𝑝) 3. So, the best mix strategy is when 58% × 𝑝 + 93% × 1 − 𝑝 = 95% × 𝑝 + 70% × 1 − 𝑝 ↔ 𝑝 = 23 60
  110. 110. Mixed Strategy In this case, the most awful thing is your opponent could predict your action. So, Find a P where: make the opponent feel indifferent whether s/he select Left or Right Left q Right 1-q Left p 58%, 42% 95%, 5% Right 1-p 93%, 7% 70%, 30% kicker keeper Mix Strategy Left Right 𝑃 = 0.5 75.5% 82.5% 𝑃 = 0.4 79% 80% 𝑝 = 23 60 79.6% 79.6%
  111. 111. Mixed Strategy Suppose you are keeper. Find your “q” Left q Right 1-q Left p 58%, 42% 95%, 5% Right 1-p 93%, 7% 70%, 30% kicker keeper Mix Strategy 𝑞 = ? Left Right [5 min - Workshop] • Both player solve it individually.(2 min) • Player2 explain how to solve it (process) (2 min) • Player1 should do feedback to the Player2 for Player1 could explain better (on the exam paper).
  112. 112. Mixed Strategy Suppose you are keeper. Find your “q” Left q Right 1-q Left p 58%, 42% 95%, 5% Right 1-p 93%, 7% 70%, 30% kicker keeper 1. Fix the rival’s strategy again. So the kicker select left. 2. Keeper’s payoff will be 42% × 𝑞 + 5% × (1 − 𝑞) 3. Next, change rival’s strategy. Say kicker → Right. 7% × 𝑞 + 30% × (1 − 𝑞) 4. The mix probability that Kicker feel indifferent is: 42% × 𝑞 + 5% × 1 − 𝑞 = 7% × 𝑞 + 30% × (1 − 𝑞) 5. Solve for 𝑞 and we will get Mix Strategy 𝑞 = 5 12 Left 20.42% Right 20.42%
  113. 113. Mixed Strategy The result is: Mix Strategy 𝑞 = 5 12 Kicker Left 20.42% Kicker Right 20.42% Mix Strategy Keeper Left Keeper Right 𝑝 = 23 60 79.6% 79.6% Keeper’s payoff Kicker’s payoff This is zero-sum game, so kicker’s payoff + keeper’s payoff = 100%
  114. 114. Game Theory in dynamic(=sequential) setting and Strategic Move: Perfect but incomplete information game
  115. 115. [advanced]Information of Game Theory Complete / Incomplete information Information Perfect -knows full history Imperfect: Simultaneous games Complete: Knows the rules, payoff Tennis, soccer (amateur-level) Rock - Paper – Scissors Tennis, soccer (pro-level) Sealed bid auction Incomplete Price negotiation of used cars Hiring talents
  116. 116. “Dynamic Game” (Sequential Game) What is a Dynamic Game? • Like Chess, first, Player A move • And then, Player B move • (then, Player A move again) So, your action today influence other players’ behavior tomorrow. • Remember the topic we already dealt was “Static” = Simultaneous game.
  117. 117. “Dynamic Game” Also called, extensive form. Different timing of strategic choice Multiple decision points
  118. 118. Notation – Game Trees / extensive form Morning You Wake up! Sleep forever Wake up! Sleep forever (before, morning) (10, 0) (0, 10)(-2, -5)(8, -1) Nodes: decision point Actions: you could choose Strategies: whole game plan. In this case, 4 strategies we have. Path: a sequence from start to end. Not every path is an equilibrium path. You before sleep Set Alarm Don’t set Alarm
  119. 119. Notation – Game Trees / extensive form Morning You Wake up! Sleep forever Wake up! Sleep forever (before, morning) (10, 0) (0, 10)(-2, -5)(8, -1) Information set You before sleep Set Alarm Don’t set Alarm
  120. 120. Notation – Game Trees / extensive form What is a difference of these two “information set”? Morning You should select your action, Without knowing what option You Before sleep select. Say, “I forgot set alarm or not. Should I wake up now…?” Morning You Wake up! Sleep forever Wake up! Sleep forever (10, 0) (0, 10)(-2, -5)(8, -1) You before sleep Set Alarm Don’t set Alarm Morning You Wake up! Sleep forever Wake up! Sleep forever (10, 0) (0, 10(-2, -5)(8, -1) You before sleep Set Alarm Don’t set Alarm You know you set alarm or not. And you will decide wake up or not.
  121. 121. Notation – Game Trees / extensive form So, what does it mean? Player n+1 will explain to Player n. You Your Rival Rock PaperScissors R PS R PSR PS
  122. 122. Notation – Game Trees / extensive form This is cheating Rock Paper Scissors. So you will always win!! You Your Rival Rock PaperScissors R PS R PSR PS You Your Rival Rock PaperScissors R PS R PSR PS This is normal Rock Paper Scissors.
  123. 123. Notation – Game Trees / extensive form Can you tell why this is prohibited? If this is information set, we need same action each node. (if you select S, it means you ignore the possibility of the right node.) You Your Rival Rock PaperScissors R PS R PR PS
  124. 124. Subgame Subgame: A smaller part of the whole game starting from any one node and continuing to the end of the entire game, with the qualification that no information set is subdivided. (That is, it consists of a singleton decision node owned by a player and all subsequent parts of the original game.) So, How many subgame do we have? You Your Rival
  125. 125. Subgame Subgame: A smaller part of the whole game starting from any one node and continuing to the end of the entire game, with the qualification that no information set is subdivided. (That is, it consists of a singleton decision node owned by a player and all subsequent parts of the original game.) You Your Rival So we are all set, about notation!
  126. 126. How to find an Equilibrium on Dynamic Game? HIJACKER Success! Give in Explode (100, -50) (-∞, -∞)(-100, 1M) PILOT Obey Hijacker Resist Hijacker Think Backward.
  127. 127. How to find an Equilibrium on Dynamic Game? HIJACKER Success! Give in Explode (100, -50) (-∞, -∞)(-100, 1M) PILOT Obey Hijacker Resist Hijacker Think Backward. Focus this subgame first, (We assume Hijacker is enough rational.) And we will think it “Backward” which means, We will think the subgame first. So, Give in = (100, −50) > Explode = (−∞, −∞) So, pilot should only compare the result of the subgame Success (−100, −1𝑀) and, Give in (100, −50) So, PILOT should resist. This way of thinking is called Backward induction: ruling out the actions that players would not play if they were actually given a chance to choose. subgame
  128. 128. Reference NUS MBA BMA5001 Lecture Note 8-13 by Professor Jo Principles of Microeconomics (Mankiw's Principles of Economics) MITx: 14.100x Microeconomics https://en.wikiquote.org/wiki/Greg_Mankiw#Ch._1._Ten_Principles_of_Economics Amazon.com https://www.youtube.com/watch?v=ErJNYh8ejSA The Art of Strategy http://ykamijo.web.fc2.com/lecture1.html

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