EC487 Advanced Microeconomics, Part I: Lecture 8 Leonardo Felli - PowerPoint PPT Presentation

EC487 Advanced Microeconomics, Part I: Lecture 8 Leonardo Felli 32L.LG.04 17 November 2017

Extensive Form Game Consider the following extensive form game: 1 ★ ❝❝❝❝❝❝❝❝❝❝❝ ❜ ★ ★ ★ U D ★ ★ ★ ★ ★ 2 2 ★ ★ r r ✔ ❚ ✁ ❙ ✔ ❚ ✁ ❙ L 1 R 1 L 2 R 2 ✔ ❚ ✁ ❙ ✔ ❚ ✁ ❙ ✔ ❚ ✁ ❙ r r r r (4 , 1) (0 , 0) ( − 1 , 1) (3 , 2) Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 17 November 2017 2 / 61

Formal description of an extensive form game ◮ The set of players: N = { 1 , 2 } ◮ The set of histories of the game: H � ∅ , U , D , [ U , L 1 ] , [ U , R 1 ] , [ D , L 2 ] , [ D , R 2 ] � = the (sub)set of terminal histories: Z � � = [ U , L 1 ] , [ U , R 1 ] , [ D , L 2 ] , [ D , R 2 ] ◮ The player function that determines which player chooses an action after each non-terminal history: P ( ∅ ) = 1 , P ( U ) = P ( D ) = 2 . Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 17 November 2017 3 / 61

Formal description of an extensive form game (cont’d) ◮ The players’ payoffs associated to each terminal history: u 1 ([ U , L 1 ]) = 4 u 2 ([ U , L 1 ]) = 1 u 1 ([ U , R 1 ]) = 0 u 2 ([ U , R 1 ]) = 0 u 1 ([ D , L 2 ]) = − 1 u 2 ([ D , L 2 ]) = 1 u 1 ([ D , R 2 ]) = 3 u 2 ([ D , R 2 ]) = 2 ◮ The strategies for each player: s 1 = a 1 ( ∅ ) ∈ { U , D } s 2 = ( a 2 ( U ) , a 2 ( D )) ∈ { ( L 1 , L 2 ) , ( L 1 , R 2 ) , ( R 1 , L 2 ) , ( R 1 , R 2 ) } Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 17 November 2017 4 / 61

Nash Equilibrium Definition A Nash Equilibrium is a strategy profile s ∗ such that for every player i ∈ N : u i ( o ( s ∗ i , s ∗ − i )) ≥ u i ( o ( s i , s ∗ − i )) ∀ s i The strategy s ∗ i is the best reply to s ∗ − i for every player i ∈ N . Where o ( s ) denotes the outcome of the game, that is the terminal history that results when each player follows strategy s i : o ( s ) ∈ Z . Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 17 November 2017 5 / 61

Associated Normal Form ◮ Notice that the set of Nash equilibria of the extensive form game can be identified by looking at the Nash equilibria of the normal form game associated with the extensive form game: Γ = { N , S i , u i ( o ( · )) } ◮ In the previous example this associated normal form game is: 1/2 L 1 , L 2 L 1 , R 2 R 1 , L 2 R 1 , R 2 U 4 , 1 4 , 1 0 , 0 0 , 0 D − 1 , 1 3 , 2 − 1 , 1 3 , 2 ◮ The Nash equilibria of this game are: { ( U , ( L 1 , L 2 )); ( U , ( L 1 , R 2 )); ( D , ( R 1 , R 2 )) } Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 17 November 2017 6 / 61

Credible Threats ◮ Clearly there exist several extensive form games with the same normal form representation. ◮ A serious problem with Nash equilibria of extensive form games is that there exist outcomes o ( s ) that are not affected by the actions that the strategy s i specifies after contingencies that are inconsistent with the history that leads to the terminal node. ◮ In other words, there exists Nash Equilibria that are built on non-credible threats. ◮ Example: the outcome o ( D , ( · , R 2 )) is independent of the action chosen following player 1’s choice U . Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 17 November 2017 7 / 61

Non-Credible Threat 1 ★ ❝❝❝❝❝❝❝❝❝❝❝ ❜ ★ ★ ★ U D ★ ⑦ ★ ★ ★ ★ 2 2 ★ ★ r r ✔ ❚ ✁ ❙ L 1 R 1 ✔ ❚ ✁ ❙ L 2 R 2 ✔ ❚ ✁ ❙ ✇ ✇ ✔ ❚ ✁ ❙ ✔ ❚ ✁ ❙ r r r r (4 , 1) (0 , 0) ( − 1 , 1) (3 , 2) Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 17 November 2017 8 / 61

Subgame Perfect Equilibrium (Selten 1965): We need a stronger equilibrium concept to make predictions that are credible . Define a proper subgame of a given extensive form game with perfect information as the subgame that follows any non-terminal history. Definition A Nash equilibrium s ∗ i of a game Γ is a subgame perfect equilibrium if and only if it is a Nash equilibrium for every proper subgame. Notice that this way to proceed eliminates any Nash equilibrium sustained by a non-credible threat. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 17 November 2017 9 / 61

Unique Subgame Perfect Equilibrium The unique subgame perfect equilibrium in the example above is : [ U , ( L 1 , R 2 )] 1 ★ ❜ ❝❝❝❝❝❝❝❝❝❝❝ ★ ★ ★ U D ★ ★ ✴ ★ ★ ★ 2 2 ★ ★ r r ✔ ❚ ✁ ❙ L 1 R 1 ✔ ❚ ✁ ❙ L 2 R 2 ✔ ✢ ❚ ✁ ❙ ✇ ✔ ❚ ✁ ❙ ✔ ❚ ✁ ❙ r r r r (4 , 1) (0 , 0) ( − 1 , 1) (3 , 2) Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 17 November 2017 10 / 61

Properties of SPE: ◮ How to construct Subgame Perfect equilibria: ◮ Do subgame perfect equilibria exist? ◮ The answer to both questions for finite horizon games is provided by the following result. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 17 November 2017 11 / 61

Kuhn Theorem: Theorem (Kuhn Theorem) Every finite extensive form game with perfect information has a subgame perfect equilibrium. We will go through the construction and the use of the algorithm that proves this theorem — known as backward induction — in an example. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 17 November 2017 12 / 61

Example of the use of Backward Induction 1 ✱ ❜ ❅ ✱ A ❅ B ✱ ❅ ✱ ❅ 2 ✱ 2 ✱ ❅ ✱ r r ❅ � ❅ C 1 D 1 C 2 D 2 ✱ ❅ ❅ � ✱ ❅ ❅ � 1 ✱ ❅ 1 r r ✱ ❅ (1 , − 1) (1 , 3) ◗ ★ r r ◗ ★ ◗ ★ ◗ F 1 E 2 ★ ◗ E 1 F 2 ★ ◗ ★ ◗ ◗ ★ r r r r (2 , 0) (0 , 2) (4 , 1) (3 , 7) Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 17 November 2017 13 / 61

Example of the use of Backward Induction (cont’d) ◮ Notice first that the strategies for the two players are: s 1 = ( a 1 ( ∅ ) , a 1 ( A , C 1 ) , a 1 ( B , D 2 )) , s 2 = ( a 2 ( A ) , a 2 ( B )) ◮ Start from the last two subgames in which player 1 chooses between E 1 and F 1 at one node and E 2 and F 2 at the other node. ◮ Compute the Nash equilibrium of these two extremely simple subgames. ◮ This Nash equilibrium is: E 1 and E 2 respectively. ◮ Hence, all Subgame Perfect strategy of player 1 have the two components: s ∗ 1 = ( · , E 1 , E 2 ) Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 17 November 2017 14 / 61

Example of the use of Backward Induction (cont’d) We can now transform the game tree from: 1 ✱ ❜ ❅ ✱ A ❅ B ✱ ❅ ✱ ❅ 2 ✱ 2 ✱ ❅ ✱ r r ❅ � ❅ C 1 D 1 C 2 D 2 ✱ ❅ ❅ � ✱ ❅ ❅ � 1 ✱ ❅ 1 r r ✱ ❅ (1 , − 1) (1 , 3) ◗ ★ r r ◗ ★ ◗ ★ ◗ F 1 E 2 ★ ◗ E 1 F 2 ★ ❄ ◗ ✴ ★ ◗ ◗ ★ r r r r (2 , 0) (0 , 2) (4 , 1) (3 , 7) Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 17 November 2017 15 / 61

Example of the use of Backward Induction (cont’d) To: 1 ✱ ❜ ❅ ✱ A ❅ B ✱ ❅ ✱ ❅ 2 ✱ 2 ✱ ❅ ✱ r r ❏❏❏❏❏ ✡ ❅ ✱ ✡ ❅ ✱ C 1 D 1 C 2 ✡ ❅ D 2 ✱ ✡ ❅ ✱ ✡ ❅ r r r r (2 , 0) (1 , − 1) (4 , 1) (1 , 3) Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 17 November 2017 16 / 61

Example of the use of Backward Induction (cont’d) ◮ Consider the last two subgames of this new game tree in which player 2 chooses between C 1 and D 1 at one node and C 2 and D 2 at the other node. ◮ The Nash equilibrium of these two subgames is: C 1 and C 2 respectively. ◮ The Subgame Perfect strategy of player 2 is then: s ∗ 2 = ( C 1 , C 2 ) Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 17 November 2017 17 / 61

Example of the use of Backward Induction (cont’d) We can once again transform the game tree: 1 ✱ ❜ ❅ ✱ A ❅ B ✱ ❅ ✱ ❅ 2 ✱ 2 ✱ ❅ ✱ r r ❏❏❏❏❏ ✡ ❅ ✱ ✡ ❅ ✴ ✴ ✱ C 1 D 1 C 2 ✡ ❅ D 2 ✱ ✡ ❅ ✱ ✡ ❅ r r r r (2 , 0) (1 , − 1) (4 , 1) (1 , 3) Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 17 November 2017 18 / 61

Example of the use of Backward Induction (cont’d) In the following one: 1 ✱ ❜ ❅ ✱ A ❅ B ✱ ✴ ❅ ✱ ❅ ✱ ✱ ❅ r r (2 , 0) (1 , 3) Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 17 November 2017 19 / 61

Example of the use of Backward Induction (cont’d) ◮ The Nash equilibrium of this subgame is for player 1 to choose A . ◮ This completes the description of the Subgame Perfect equilibrium strategy for player 1: s ∗ 1 = ( A , E 1 , E 2 ). ◮ The unique Subgame Perfect equilibrium of this game is then: s ∗ = ( s ∗ 1 , s ∗ � � 2 ) = ( A , E 1 , E 2 ) , ( C 1 , C 2 ) . Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 17 November 2017 20 / 61