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Formal Game Theory Stphane Le Roux 1 , Pierre Lescanne 1 and Ren - PowerPoint PPT Presentation

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Motivations Simultaneous Games Sequential Games Summary Formal Game Theory Stphane Le Roux 1 , Pierre Lescanne 1 and Ren Vestergaard 2 1 LIP ENS-Lyon, France 2 JAIST Japan Workshop Chambry, Krakow, and Lyon, 2005 SLR, PL and RV Formal

  1. Motivations Basic Definitions Simultaneous Games Simultaneous Nash Equilibrium Sequential Games Properties and Extensions Summary What is a Simultaneous (Pure) Nash Equilibrium? Informally A strategy profile, Such that no agent has any incentive to change its own individual strategy. SLR, PL and RV Formal Game Theory

  2. Motivations Basic Definitions Simultaneous Games Simultaneous Nash Equilibrium Sequential Games Properties and Extensions Summary An example of a Simultaneous Nash Equilibrium ( r 2 , c 1 ) B c 1 c 2 c 3 r 1 0 1 0 0 1 2 A r 2 2 2 1 0 1 1 r 3 2 0 2 3 0 1 SLR, PL and RV Formal Game Theory

  3. Motivations Basic Definitions Simultaneous Games Simultaneous Nash Equilibrium Sequential Games Properties and Extensions Summary An example of a Simultaneous Nash Equilibrium ( r 2 , c 1 ) B c 1 c 2 c 3 r 1 0 1 0 0 1 2 A r 2 2 2 1 0 1 1 r 3 2 0 2 3 0 1 A checks whether it can increase its payoff or not. SLR, PL and RV Formal Game Theory

  4. Motivations Basic Definitions Simultaneous Games Simultaneous Nash Equilibrium Sequential Games Properties and Extensions Summary An example of a Simultaneous Nash Equilibrium ( r 2 , c 1 ) B c 1 c 2 c 3 r 1 0 1 0 0 1 2 A r 2 2 2 1 0 1 1 r 3 2 0 2 3 0 1 A checks whether it can increase its payoff or not. A has no incentive to change choices. SLR, PL and RV Formal Game Theory

  5. Motivations Basic Definitions Simultaneous Games Simultaneous Nash Equilibrium Sequential Games Properties and Extensions Summary An example of a Simultaneous Nash Equilibrium ( r 2 , c 1 ) B c 1 c 2 c 3 r 1 0 1 0 0 1 2 A r 2 2 2 1 0 1 1 r 3 2 0 2 3 0 1 A checks whether it can increase its payoff or not. A has no incentive to change choices. SLR, PL and RV Formal Game Theory

  6. Motivations Basic Definitions Simultaneous Games Simultaneous Nash Equilibrium Sequential Games Properties and Extensions Summary An example of a Simultaneous Nash Equilibrium ( r 2 , c 1 ) B c 1 c 2 c 3 r 1 0 1 0 0 1 2 A r 2 2 2 1 0 1 1 r 3 2 0 2 3 0 1 B checks whether it can increase its payoff or not. B has no incentive to change choices. Therefore, ( r 2 , c 1 ) is an equilibrium. SLR, PL and RV Formal Game Theory

  7. Motivations Basic Definitions Simultaneous Games Simultaneous Nash Equilibrium Sequential Games Properties and Extensions Summary An example of a Simultaneous Nash Equilibrium ( r 2 , c 1 ) B c 1 c 2 c 3 r 1 0 1 0 0 1 2 A r 2 2 2 1 0 1 1 r 3 2 0 2 3 0 1 B checks whether it can increase its payoff or not. B has no incentive to change choices. Therefore, ( r 2 , c 1 ) is an equilibrium. SLR, PL and RV Formal Game Theory

  8. Motivations Basic Definitions Simultaneous Games Simultaneous Nash Equilibrium Sequential Games Properties and Extensions Summary An example of a Simultaneous Nash Equilibrium ( r 2 , c 1 ) B c 1 c 2 c 3 r 1 0 1 0 0 1 2 A r 2 2 2 1 0 1 1 r 3 2 0 2 3 0 1 B checks whether it can increase its payoff or not. B has no incentive to change choices. Therefore, ( r 2 , c 1 ) is an equilibrium. SLR, PL and RV Formal Game Theory

  9. Motivations Basic Definitions Simultaneous Games Simultaneous Nash Equilibrium Sequential Games Properties and Extensions Summary Nash Equilibrium may not be “optimal” From Prisoner’s dilemma The strategy profile ( r 2 , c 2 ) is the only Nash equilibrium in the game below. B c 1 c 2 A r 1 5 5 0 8 r 2 8 0 1 1 SLR, PL and RV Formal Game Theory

  10. Motivations Basic Definitions Simultaneous Games Simultaneous Nash Equilibrium Sequential Games Properties and Extensions Summary Nash Equilibrium may not be “optimal” From Prisoner’s dilemma The strategy profile ( r 2 , c 2 ) is the only Nash equilibrium in the game below. B c 1 c 2 A r 1 5 5 0 8 r 2 8 0 1 1 SLR, PL and RV Formal Game Theory

  11. Motivations Basic Definitions Simultaneous Games Simultaneous Nash Equilibrium Sequential Games Properties and Extensions Summary Existence of Nash Equilibrium Do Nash Equilibria always exist? The game below has no Nash equilibrium. B c 1 c 2 A r 1 0 1 1 0 r 2 1 0 0 1 SLR, PL and RV Formal Game Theory

  12. Motivations Basic Definitions Simultaneous Games Simultaneous Nash Equilibrium Sequential Games Properties and Extensions Summary Existence of Nash Equilibrium Do Nash Equilibria always exist? The game below has no Nash equilibrium. B c 1 c 2 A r 1 0 1 1 0 r 2 1 0 0 1 SLR, PL and RV Formal Game Theory

  13. Motivations Basic Definitions Simultaneous Games Simultaneous Nash Equilibrium Sequential Games Properties and Extensions Summary Existence of Nash Equilibrium Do Nash Equilibria always exist? The game below has no Nash equilibrium. B c 1 c 2 A r 1 0 1 1 0 r 2 1 0 0 1 A can increase its payoff. SLR, PL and RV Formal Game Theory

  14. Motivations Basic Definitions Simultaneous Games Simultaneous Nash Equilibrium Sequential Games Properties and Extensions Summary Existence of Nash Equilibrium Do Nash Equilibria always exist? The game below has no Nash equilibrium. B c 1 c 2 A r 1 0 1 1 0 r 2 1 0 0 1 SLR, PL and RV Formal Game Theory

  15. Motivations Basic Definitions Simultaneous Games Simultaneous Nash Equilibrium Sequential Games Properties and Extensions Summary Existence of Nash Equilibrium Do Nash Equilibria always exist? The game below has no Nash equilibrium. B c 1 c 2 A r 1 0 1 1 0 r 2 1 0 0 1 B can increase its payoff. SLR, PL and RV Formal Game Theory

  16. Motivations Basic Definitions Simultaneous Games Simultaneous Nash Equilibrium Sequential Games Properties and Extensions Summary Existence of Nash Equilibrium Do Nash Equilibria always exist? The game below has no Nash equilibrium. B c 1 c 2 A r 1 0 1 1 0 r 2 1 0 0 1 SLR, PL and RV Formal Game Theory

  17. Motivations Basic Definitions Simultaneous Games Simultaneous Nash Equilibrium Sequential Games Properties and Extensions Summary Existence of Nash Equilibrium Do Nash Equilibria always exist? The game below has no Nash equilibrium. B c 1 c 2 A r 1 0 1 1 0 r 2 1 0 0 1 Agents cycle: no Nash Equilibrium. SLR, PL and RV Formal Game Theory

  18. Motivations Basic Definitions Simultaneous Games Simultaneous Nash Equilibrium Sequential Games Properties and Extensions Summary Existence of Nash Equilibrium Do Nash Equilibria always exist? The game below has no Nash equilibrium. B c 1 c 2 A r 1 0 1 1 0 r 2 1 0 0 1 Agents cycle: no Nash Equilibrium. SLR, PL and RV Formal Game Theory

  19. Motivations Basic Definitions Simultaneous Games Simultaneous Nash Equilibrium Sequential Games Properties and Extensions Summary Existence of Nash Equilibrium One should relax the definition of equilibrium in order to guarantee existence . Two proposals: Nash: Introducing probabilities and expected payoffs. Abstract Games: Considering “cycles” as equilibria. SLR, PL and RV Formal Game Theory

  20. Motivations Basic Definitions Simultaneous Games Simultaneous Nash Equilibrium Sequential Games Properties and Extensions Summary Existence of Nash Equilibrium One should relax the definition of equilibrium in order to guarantee existence . Two proposals: Nash: Introducing probabilities and expected payoffs. Abstract Games: Considering “cycles” as equilibria. SLR, PL and RV Formal Game Theory

  21. Motivations Basic Definitions Simultaneous Games Simultaneous Nash Equilibrium Sequential Games Properties and Extensions Summary Existence of Nash Equilibrium One should relax the definition of equilibrium in order to guarantee existence . Two proposals: Nash: Introducing probabilities and expected payoffs. Abstract Games: Considering “cycles” as equilibria. SLR, PL and RV Formal Game Theory

  22. Motivations Basic Definitions Simultaneous Games Simultaneous Nash Equilibrium Sequential Games Properties and Extensions Summary Mixed Strategies and Mixed Nash Equilibira Pure setting: An agent chooses one strategy among some available strategies. Mixed setting: An agent chooses a probability distribution over the set of available strategies. B 10 % 60 % 30 % 35 % 0 1 0 0 1 2 A 20 % 2 2 1 0 1 1 45 % 2 0 2 3 0 1 Notions of payoff and equilibrium are easily extended. SLR, PL and RV Formal Game Theory

  23. Motivations Basic Definitions Simultaneous Games Simultaneous Nash Equilibrium Sequential Games Properties and Extensions Summary Mixed Strategies and Mixed Nash Equilibira Pure setting: An agent chooses one strategy among some available strategies. Mixed setting: An agent chooses a probability distribution over the set of available strategies. B 10 % 60 % 30 % 35 % 0 1 0 0 1 2 A 20 % 2 2 1 0 1 1 45 % 2 0 2 3 0 1 Notions of payoff and equilibrium are easily extended. SLR, PL and RV Formal Game Theory

  24. Motivations Basic Definitions Simultaneous Games Simultaneous Nash Equilibrium Sequential Games Properties and Extensions Summary Mixed Strategies and Mixed Nash Equilibira Pure setting: An agent chooses one strategy among some available strategies. Mixed setting: An agent chooses a probability distribution over the set of available strategies. B 10 % 60 % 30 % 35 % 0 1 0 0 1 2 A 20 % 2 2 1 0 1 1 45 % 2 0 2 3 0 1 Notions of payoff and equilibrium are easily extended. SLR, PL and RV Formal Game Theory

  25. Motivations Basic Definitions Simultaneous Games Simultaneous Nash Equilibrium Sequential Games Properties and Extensions Summary Mixed Strategies and Mixed Nash Equilibira Pure setting: An agent chooses one strategy among some available strategies. Mixed setting: An agent chooses a probability distribution over the set of available strategies. B 10 % 60 % 30 % 35 % 0 1 0 0 1 2 A 20 % 2 2 1 0 1 1 45 % 2 0 2 3 0 1 Notions of payoff and equilibrium are easily extended. SLR, PL and RV Formal Game Theory

  26. Motivations Basic Definitions Simultaneous Games Simultaneous Nash Equilibrium Sequential Games Properties and Extensions Summary Existence of (Mixed) Nash Equilibrium The game below has no pure Nash equilibrium. B c 1 c 2 A r 1 0 1 1 0 r 2 1 0 0 1 But it has a mixed Nash equilibrium: B 50 % 50 % A 50 % 0 1 1 0 50 % 1 0 0 1 SLR, PL and RV Formal Game Theory

  27. Motivations Basic Definitions Simultaneous Games Simultaneous Nash Equilibrium Sequential Games Properties and Extensions Summary Existence of (Mixed) Nash Equilibrium The game below has no pure Nash equilibrium. B c 1 c 2 A r 1 0 1 1 0 r 2 1 0 0 1 But it has a mixed Nash equilibrium: B 50 % 50 % A 50 % 0 1 1 0 50 % 1 0 0 1 SLR, PL and RV Formal Game Theory

  28. Motivations Basic Definitions Simultaneous Games Simultaneous Nash Equilibrium Sequential Games Properties and Extensions Summary Existence of (Mixed) Nash Equilibrium Guaranteed for finite games (finitely many agents and finitely many available strategies) Brouwer’s fixed point theorem (Hadamard 1910, Brouwer 1912). Non constructive proofs . Constructive proof by Scarf (1967) In Coq? SLR, PL and RV Formal Game Theory

  29. Motivations Basic Definitions Simultaneous Games Simultaneous Nash Equilibrium Sequential Games Properties and Extensions Summary Existence of (Mixed) Nash Equilibrium Guaranteed for finite games (finitely many agents and finitely many available strategies) Brouwer’s fixed point theorem (Hadamard 1910, Brouwer 1912). Non constructive proofs . Constructive proof by Scarf (1967) In Coq? SLR, PL and RV Formal Game Theory

  30. Motivations Basic Definitions Simultaneous Games Simultaneous Nash Equilibrium Sequential Games Properties and Extensions Summary Existence of (Mixed) Nash Equilibrium Guaranteed for finite games (finitely many agents and finitely many available strategies) Brouwer’s fixed point theorem (Hadamard 1910, Brouwer 1912). Non constructive proofs . Constructive proof by Scarf (1967) In Coq? SLR, PL and RV Formal Game Theory

  31. Motivations Basic Definitions Simultaneous Games Simultaneous Nash Equilibrium Sequential Games Properties and Extensions Summary Existence of (Mixed) Nash Equilibrium Guaranteed for finite games (finitely many agents and finitely many available strategies) Brouwer’s fixed point theorem (Hadamard 1910, Brouwer 1912). Non constructive proofs . Constructive proof by Scarf (1967) In Coq? SLR, PL and RV Formal Game Theory

  32. Motivations Basic Definitions Simultaneous Games Simultaneous Nash Equilibrium Sequential Games Properties and Extensions Summary Existence of (Mixed) Nash Equilibrium Guaranteed for finite games (finitely many agents and finitely many available strategies) Brouwer’s fixed point theorem (Hadamard 1910, Brouwer 1912). Non constructive proofs . Constructive proof by Scarf (1967) In Coq? SLR, PL and RV Formal Game Theory

  33. Motivations Basic Definitions Simultaneous Games Sequential Nash Equilibrium and “Backward Induction” Sequential Games Properties and Extensions Summary Outline Motivations 1 Simultaneous Games 2 Basic Definitions Simultaneous Nash Equilibrium Properties and Extensions 3 Sequential Games Basic Definitions Sequential Nash Equilibrium and “Backward Induction” Properties and Extensions SLR, PL and RV Formal Game Theory

  34. Motivations Basic Definitions Simultaneous Games Sequential Nash Equilibrium and “Backward Induction” Sequential Games Properties and Extensions Summary What is a (Finite) Sequential Game? Informaly A system, With some agents, Who play in turn, By choosing among finitely many options. The game stops after finitely many steps. And payoffs are disclosed. SLR, PL and RV Formal Game Theory

  35. Motivations Basic Definitions Simultaneous Games Sequential Nash Equilibrium and “Backward Induction” Sequential Games Properties and Extensions Summary What is a (Finite) Sequential Game? Informaly A system, With some agents, Who play in turn, By choosing among finitely many options. The game stops after finitely many steps. And payoffs are disclosed. SLR, PL and RV Formal Game Theory

  36. Motivations Basic Definitions Simultaneous Games Sequential Nash Equilibrium and “Backward Induction” Sequential Games Properties and Extensions Summary What is a (Finite) Sequential Game? Informaly A system, With some agents, Who play in turn, By choosing among finitely many options. The game stops after finitely many steps. And payoffs are disclosed. SLR, PL and RV Formal Game Theory

  37. Motivations Basic Definitions Simultaneous Games Sequential Nash Equilibrium and “Backward Induction” Sequential Games Properties and Extensions Summary What is a (Finite) Sequential Game? Informaly A system, With some agents, Who play in turn, By choosing among finitely many options. The game stops after finitely many steps. And payoffs are disclosed. SLR, PL and RV Formal Game Theory

  38. Motivations Basic Definitions Simultaneous Games Sequential Nash Equilibrium and “Backward Induction” Sequential Games Properties and Extensions Summary What is a (Finite) Sequential Game? Informaly A system, With some agents, Who play in turn, By choosing among finitely many options. The game stops after finitely many steps. And payoffs are disclosed. SLR, PL and RV Formal Game Theory

  39. Motivations Basic Definitions Simultaneous Games Sequential Nash Equilibrium and “Backward Induction” Sequential Games Properties and Extensions Summary What is a (Finite) Sequential Game? Informaly A system, With some agents, Who play in turn, By choosing among finitely many options. The game stops after finitely many steps. And payoffs are disclosed. SLR, PL and RV Formal Game Theory

  40. Motivations Basic Definitions Simultaneous Games Sequential Nash Equilibrium and “Backward Induction” Sequential Games Properties and Extensions Summary What is a (Finite) Sequential Game? Semi-formaly A set of agents, A finite rooted tree, Every internal node is owned by some agent, Leaves are labeled with payoff functions: Agents → Reals SLR, PL and RV Formal Game Theory

  41. Motivations Basic Definitions Simultaneous Games Sequential Nash Equilibrium and “Backward Induction” Sequential Games Properties and Extensions Summary What is a (Finite) Sequential Game? Semi-formaly A set of agents, A finite rooted tree, Every internal node is owned by some agent, Leaves are labeled with payoff functions: Agents → Reals SLR, PL and RV Formal Game Theory

  42. Motivations Basic Definitions Simultaneous Games Sequential Nash Equilibrium and “Backward Induction” Sequential Games Properties and Extensions Summary What is a (Finite) Sequential Game? Semi-formaly A set of agents, A finite rooted tree, Every internal node is owned by some agent, Leaves are labeled with payoff functions: Agents → Reals SLR, PL and RV Formal Game Theory

  43. Motivations Basic Definitions Simultaneous Games Sequential Nash Equilibrium and “Backward Induction” Sequential Games Properties and Extensions Summary What is a (Finite) Sequential Game? Semi-formaly A set of agents, A finite rooted tree, Every internal node is owned by some agent, Leaves are labeled with payoff functions: Agents → Reals SLR, PL and RV Formal Game Theory

  44. Motivations Basic Definitions Simultaneous Games Sequential Nash Equilibrium and “Backward Induction” Sequential Games Properties and Extensions Summary A play in a Sequential Game a � � �������� � � � � � � � 1 , 2 b � � � � � � � � � � � � � � � � 0 , 1 3 , 0 SLR, PL and RV Formal Game Theory

  45. Motivations Basic Definitions Simultaneous Games Sequential Nash Equilibrium and “Backward Induction” Sequential Games Properties and Extensions Summary A play in a Sequential Game a � � � � � � � � � � � � � � � � � � � � � � � � � � 1 , 2 b � � � � � � � � � � � � � � � � 0 , 1 3 , 0 SLR, PL and RV Formal Game Theory

  46. Motivations Basic Definitions Simultaneous Games Sequential Nash Equilibrium and “Backward Induction” Sequential Games Properties and Extensions Summary A play in a Sequential Game b � � � � � � � � � � � � � � � � 0 , 1 3 , 0 SLR, PL and RV Formal Game Theory

  47. � Motivations Basic Definitions Simultaneous Games Sequential Nash Equilibrium and “Backward Induction” Sequential Games Properties and Extensions Summary A play in a Sequential Game b � � � � � � � � � � � � � � � � � � � � � � 0 , 1 3 , 0 SLR, PL and RV Formal Game Theory

  48. Motivations Basic Definitions Simultaneous Games Sequential Nash Equilibrium and “Backward Induction” Sequential Games Properties and Extensions Summary A play in a Sequential Game 3 , 0 SLR, PL and RV Formal Game Theory

  49. Motivations Basic Definitions Simultaneous Games Sequential Nash Equilibrium and “Backward Induction” Sequential Games Properties and Extensions Summary What is a (Binary and Finite) Sequential Game? Inductive definition bG ::= bgL Payoffs Payoffs : Agents → Reals | bgN Agents bG bG SLR, PL and RV Formal Game Theory

  50. Motivations Basic Definitions Simultaneous Games Sequential Nash Equilibrium and “Backward Induction” Sequential Games Properties and Extensions Summary What is a Sequential Game in COQ? Parameter Agent : Set. Definition POF := Agent -> Reals. Inductive bG : Set := | bgL : POF -> bG | bgN : Agent -> bG -> bG -> bG. SLR, PL and RV Formal Game Theory

  51. Motivations Basic Definitions Simultaneous Games Sequential Nash Equilibrium and “Backward Induction” Sequential Games Properties and Extensions Summary What is a Sequential Game in COQ? Parameter Agent : Set. Definition POF := Agent -> Reals. Inductive bG : Set := | bgL : POF -> bG | bgN : Agent -> bG -> bG -> bG. SLR, PL and RV Formal Game Theory

  52. � Motivations Basic Definitions Simultaneous Games Sequential Nash Equilibrium and “Backward Induction” Sequential Games Properties and Extensions Summary What is a Sequential Strategy Profile? Graphicaly a � ���������������� � � ���������������� � � � � � � � � � � � � � � b b � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � po 1 po 2 po 3 po 4 SLR, PL and RV Formal Game Theory

  53. Motivations Basic Definitions Simultaneous Games Sequential Nash Equilibrium and “Backward Induction” Sequential Games Properties and Extensions Summary What is a Sequential Strategy Profile? Inductive definition bS ::= bsL Payoffs Choice ::= l | r | bsN Agents Choice bS bS No big changes in Coq. SLR, PL and RV Formal Game Theory

  54. Motivations Basic Definitions Simultaneous Games Sequential Nash Equilibrium and “Backward Induction” Sequential Games Properties and Extensions Summary What is a Sequential Nash Equilibrium? Informally Same intuitive definition as for simultaneous games. A strategy profile, Such that no agent has any incentive to change its own individual strategy, SLR, PL and RV Formal Game Theory

  55. Motivations Basic Definitions Simultaneous Games Sequential Nash Equilibrium and “Backward Induction” Sequential Games Properties and Extensions Summary What is a Sequential Nash Equilibrium? Informally Same intuitive definition as for simultaneous games. A strategy profile, Such that no agent has any incentive to change its own individual strategy, SLR, PL and RV Formal Game Theory

  56. � Motivations Basic Definitions Simultaneous Games Sequential Nash Equilibrium and “Backward Induction” Sequential Games Properties and Extensions Summary An example of a Nash Sequential Equilibrium a � � � � �������� � � � � � � � � � � � � 1 , 2 b � � � � � � � � � � � � � � � � � � � � � � � � � 0 , 1 3 , 3 If a changes its mind then the associated payoff decreases. If b changes its mind then the associated payoff does not change. SLR, PL and RV Formal Game Theory

  57. � Motivations Basic Definitions Simultaneous Games Sequential Nash Equilibrium and “Backward Induction” Sequential Games Properties and Extensions Summary An example of a Nash Sequential Equilibrium a � � � � �������� � � � � � � � � � � � � 1 , 2 b � � � � � � � � � � � � � � � � � � � � � � � � � 0 , 1 3 , 3 If a changes its mind then the associated payoff decreases. If b changes its mind then the associated payoff does not change. SLR, PL and RV Formal Game Theory

  58. � Motivations Basic Definitions Simultaneous Games Sequential Nash Equilibrium and “Backward Induction” Sequential Games Properties and Extensions Summary An example of a Nash Sequential Equilibrium a � � � � �������� � � � � � � � � � � � � 1 , 2 b � � � � � � � � � � � � � � � � � � � � � � � � � 0 , 1 3 , 3 If a changes its mind then the associated payoff decreases. If b changes its mind then the associated payoff does not change. SLR, PL and RV Formal Game Theory

  59. � Motivations Basic Definitions Simultaneous Games Sequential Nash Equilibrium and “Backward Induction” Sequential Games Properties and Extensions Summary Another Nash Sequential Equilibrium in the same game a � � � � � � � � � � � � � � � � � � � � � � � � � � 1 , 2 b � � � � � � � � � � � � � � � � � � � � � � � � 0 , 1 3 , 3 If a changes its mind then the associated payoff decreases. If b changes its mind then the associated payoff decreases. SLR, PL and RV Formal Game Theory

  60. � Motivations Basic Definitions Simultaneous Games Sequential Nash Equilibrium and “Backward Induction” Sequential Games Properties and Extensions Summary Another Nash Sequential Equilibrium in the same game a � � � � � � � � � � � � � � � � � � � � � � � � � � 1 , 2 b � � � � � � � � � � � � � � � � � � � � � � � � 0 , 1 3 , 3 If a changes its mind then the associated payoff decreases. If b changes its mind then the associated payoff decreases. SLR, PL and RV Formal Game Theory

  61. � Motivations Basic Definitions Simultaneous Games Sequential Nash Equilibrium and “Backward Induction” Sequential Games Properties and Extensions Summary Another Nash Sequential Equilibrium in the same game a � � � � � � � � � � � � � � � � � � � � � � � � � � 1 , 2 b � � � � � � � � � � � � � � � � � � � � � � � � 0 , 1 3 , 3 If a changes its mind then the associated payoff decreases. If b changes its mind then the associated payoff decreases. SLR, PL and RV Formal Game Theory

  62. � � Motivations Basic Definitions Simultaneous Games Sequential Nash Equilibrium and “Backward Induction” Sequential Games Properties and Extensions Summary Nash Equilibria may not be “optimal” The only Nash equilibrium of the game below is not “optimal”. a a � � � � � �������� �������� � � � � � � � � � � � � � � � � � � � � 1 , 0 1 , 0 b b � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 2 , 3 0 , 4 2 , 3 0 , 4 SLR, PL and RV Formal Game Theory

  63. � � Motivations Basic Definitions Simultaneous Games Sequential Nash Equilibrium and “Backward Induction” Sequential Games Properties and Extensions Summary Nash Equilibria may not be “optimal” The only Nash equilibrium of the game below is not “optimal”. a a � � � � � �������� �������� � � � � � � � � � � � � � � � � � � � � 1 , 0 1 , 0 b b � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 2 , 3 0 , 4 2 , 3 0 , 4 SLR, PL and RV Formal Game Theory

  64. Motivations Basic Definitions Simultaneous Games Sequential Nash Equilibrium and “Backward Induction” Sequential Games Properties and Extensions Summary Construction of a “Backward Induction” strategy profile a � ���������������� � � � � � � � � � � � � � � � b b � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 3 , 0 1 , 2 3 , 3 0 , 4 SLR, PL and RV Formal Game Theory

  65. � Motivations Basic Definitions Simultaneous Games Sequential Nash Equilibrium and “Backward Induction” Sequential Games Properties and Extensions Summary Construction of a “Backward Induction” strategy profile a � ���������������� � � � � � � � � � � � � � � � b b � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 3 , 0 1 , 2 3 , 3 0 , 4 SLR, PL and RV Formal Game Theory

  66. � � Motivations Basic Definitions Simultaneous Games Sequential Nash Equilibrium and “Backward Induction” Sequential Games Properties and Extensions Summary Construction of a “Backward Induction” strategy profile a � ���������������� � � � � � � � � � � � � � � � b b � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 3 , 0 1 , 2 3 , 3 0 , 4 SLR, PL and RV Formal Game Theory

  67. � � Motivations Basic Definitions Simultaneous Games Sequential Nash Equilibrium and “Backward Induction” Sequential Games Properties and Extensions Summary Construction of a “Backward Induction” strategy profile a � ���������������� � � ���������������� � � � � � � � � � � � � � � b b � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 3 , 0 1 , 2 3 , 3 0 , 4 SLR, PL and RV Formal Game Theory

  68. � � Motivations Basic Definitions Simultaneous Games Sequential Nash Equilibrium and “Backward Induction” Sequential Games Properties and Extensions Summary Construction of a “Backward Induction” strategy profile “BI” may not be “optimal”. a � ���������������� � � ���������������� � � � � � � � � � � � � � � b b � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 3 , 0 1 , 2 3 , 3 0 , 4 SLR, PL and RV Formal Game Theory

  69. Motivations Basic Definitions Simultaneous Games Sequential Nash Equilibrium and “Backward Induction” Sequential Games Properties and Extensions Summary Properties of “Backward Induction” Proved in Coq for binary games “BI” ⇒ Nash Equilibrium. Guaranteed existence for “BI”. SLR, PL and RV Formal Game Theory

  70. Motivations Basic Definitions Simultaneous Games Sequential Nash Equilibrium and “Backward Induction” Sequential Games Properties and Extensions Summary Properties of “Backward Induction” Proved in Coq for binary games “BI” ⇒ Nash Equilibrium. Guaranteed existence for “BI”. SLR, PL and RV Formal Game Theory

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