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M9302 Mathematical Models in Economics Instructor: Georgi Burlakov 3.1.Dynamic Games of Complete but Imperfect Information Lecture

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Quick Revision on Lecture 1 & 2 Static versus Dynamic Games Complete versus Incomplete Information Normal versus Extensive Form

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How to solve the GT problem? Perfect Bayesian Equilibrium (PBNE) in static games of complete information in dynamic games of complete information in static games of incomplete information in dynamic games of incomplete information Strategic Dominance Nash Equilibrium (NE) Solution Concepts: Subgame-Perfect Nash Equilibrium (SPNE) Bayesian Nash Equilibrium (BNE) Backwards Induction

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Revision: Students’ Dilemma (static game) IESDS Solution: YOU EasyHard Easy-1,-1-3,0 Hard0,-3-2,-2 Unique Nash Equilibrium: {EASY/EASY; HARD/HARD} OTHERS

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Revision: Students’ Dilemma -2 (static game) Nash Equilibrium Solution: YOU EasyHard Easy0,00,0-3,-1 Hard-1,-3-2,-2 Two Nash Equilibria: {EASY/EASY; HARD/HARD} OTHERS

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Revision: Students’ Dilemma -2 (dynamic game of perfect info) Backwards Induction Solution: YOU OTHERS (HARD, HARD)(EASY, HARD)(HARD, EASY)(EASY, EASY) EASY-3,-1 0,0 (SPNE)-3,-1 0,0 (NE) HARD -2,-2 (NE)-2,-2-1,-3 -3 EasyHard EasyHardEasyHard Three Pure-strategy Nash Equilibria, Unique Subgame-Perfect N.E.: {EASY/(EASY,HARD)}

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The aim of the third lecture is to show: 1. How to represent a static game in extensive form? 2. What is a subgame? How to use it in solving dynamic games of complete but imperfect information? Dynamic games of complete but imperfect information

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Informally, the games of this class could be described as follows: First, Players 1 and 2 simultaneously choose actions a 1 and a 2 from feasible sets A 1 and A 2, respectively Second, players 3 and 4 observe the outcome of the first stage, (a 1, a 2 ), and then simultaneously choose actions a 3 and a 4 from feasible sets A 3 and A 4, respectively. Finally, based on the resulting combination of actions chosen in total, each player receives a given payoff.

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Dynamic Games of Complete and Imperfect Information Standard assumptions: Players move at different, sequential moments – it is DYNAMIC The players’ payoff functions are common knowledge – it is COMPLETE INFORMATION At each stage of the game players move simultaneously – it is IMPERFECT INFORMATION

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How to describe a dynamic game? The extensive form representation of a game specifies: 1.Who are the PLAYERS When each player has the MOVE What each player KNOWS when she is on a move What ACTIONS each player can take. 3. What is the PAYOFF received by each player.

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Example 3.1.: Students’ Dilemma -2 (Dynamic version with imperfect info) Extensive Form Representation: 1.Reduce the players to 2 groups by 2 players First, YOU and OTHERS from your group choose SIMULTANEOUSLY an action from the set A i = {Easy, Hard} 2.2. Second, the players from the second group know what the players of the first group have chosen Both players from the second group choose an ACTION simultaneously from the set A i = {Easy, Hard} 3. Payoffs: u i (a 1, a 2, a 3, a 4 )=LEISURE(a i )-GRADE(a 1, a 2, a 3, a 4 )

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Example 3.1.: Students’ Dilemma -2 (Sequential Version) Grading Policy: the students over the average have a STRONG PASS (Grade A, or 1), the ones with average performance get a PASS (Grade B, or 2) and who is under the average FAIL (Grade F, or 5).

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Example 3.1.: Students’ Dilemma – 2 (Sequential Version) Leisure Rule: HARD study schedule devotes all the time (leisure = 0) to studying distinct from the EASY one (leisure = 2). Player i’s choice Others’ choiceLEISUREGRADEPlayer i’ payoff EasyAll Easy2-20 At least one Hard2-5-3 HardAt least one Easy0 All Hard0-2

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Example 3.1.: Students’ Dilemma -2 (Sequential Version) Standard assumptions: Students choose between HARD and EASY SIMULTANEOUSLY in each stage. Grading is announced in advance, so it is COMMON KNOWLEDGE to all the students. Before making a choice in the second stage, the players of the second group observe the first-stage choice of the players of the first group. Simplification assumptions: Performance depends on CHOICE. EQUAL EFFICIENCY of studies.

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Example 3.1.: Students’ Dilemma – 2 (Game Tree Representation) YOU1 OTHERS1 Easy Hard Easy Hard Easy Hard OTHERS1 Y2

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Dynamic games of Complete but Imperfect Information – key terms Information set – a collection of one or more decision nodes such that: only given player has a move on all the decision nodes in the information set, if more than one node is included in the information set, the player with the move does not know which of them is reached by the preceding play of the game.

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Example 3.1.: Students’ Dilemma – 2 (Game Tree Representation) YOU1 OTHERS1 Easy Hard Easy Hard H E EH E H Easy Hard OTHERS1 H E EH E H H E EH E H H E EH E H Y2 O Rule of thumb: Non-singleton information set implies imperfect information and vice versa. With imperfect information backwards induction requires solution of the whole second stage static game.

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Dynamic games of Complete but Imperfect Information – key terms Subgame – a piece of a game that remains to be played beginning at any point at which the complete history of the game thus far is common knowledge among the players, i.e.: begins at a singleton information set includes all the decision and terminal nodes following but not preceding the starting singleton decision node does not cut any (non-singleton) information sets.

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Example 3.1.: Students’ Dilemma – 2 (pure-strategy N.E./mechanical solution of stage 2) YOU1 OTHERS1 Easy Hard Easy Hard Easy Hard OTHERS1 EasyHard E0,00,0-3,-1 H-1,-3-2,-2 EasyHard E-3,-3-3,-2 H-2,-3-2,-2 EasyHard E-3,-3-3,-2 H-2,-3-2,-2 EasyHard E-2,-2-3,-2 H-2,-3-2,-2 O2 Y2 O2 Y2 O2 Y2 O2 Y2 The pure-strategy N.E. in the second stage game is not unique. Y2 and O2 would cooperate for sure but their choice is uncertain if O1 chooses Easy.

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Dynamic games of Complete but Imperfect Information – key terms Strategy – a complete plan of action – it specifies a feasible action which the player will take in each stage, for every possible history of play through the previous stage.

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Example 3.1.: Students’ Dilemma – 2 (pure-strategy N.E./mechanical solution of stage 1) YOU1 OTHERS1 Easy Hard Easy Hard H E EH E H Easy Hard OTHERS1 H E EH E H H E EH E H H E EH E H Y2 O2 EasyHard E0,00,0-3,-1 H-1,-3-2,-2 EasyHard E-3,-3-3,-2 H-2,-3-2,-2 EasyHard E-3,-3-3,-2 H-2,-3-2,-2 EasyHard E-2,-2-3,-2 H-2,-3-2,-2 O2 Y2 O2 Y2 O2 Y2 O2 Y2 4 possible strategies (pairs of actions) appear as solutions of the second-stage static game: [(E,E),(H,H),(E,E),(H,H)];[(E,E),(H,H),(H,H),(H,H)]; [(H,H),(H,H),(E,E),(H,H)];[(H,H),(H,H),(H,H),(H,H)];

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Example 3.1.: Students’ Dilemma – 2 (pure-strategy N.E./mechanical solution of stage 1) Is it reasonable to play Easy if YOU1 chose Hard and Hard if all from group 1 chose Easy ? EasyHard E0,00,0-3,-2 H -2,-2 EasyHard E0,00,0-3,-2 H -2,-2 O1 Y1 O1 Y1 Possible strategies of group 2 and best responses of group 1: Strategy 1: [(E,E)(H,H)(E,E)(H,H)] Strategy 2: [(E,E)(H,H)(H,H)(H,H)] EasyHard E-3,-3-3,-2 H-2,-3-2,-2 EasyHard E-3,-3-3,-2 H-2,-3-2,-2 O1 Y1 O1 Y1 Strategy 3: [(H,H)(H,H)(E,E)(H,H)] Strategy 4: [(H,H)(H,H)(H,H)(H,H) ]

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Example 3.1.: Students’ Dilemma – 2 (pure-strategy N.E./mechanical solution of stage 1) Is it reasonable to play Easy if YOU1 chose Hard and Hard if all from group 1 chose Easy ? There are six pure-strategy N.E.: N.E. 1:{[(E,E)[(E,E)(H,H)(E,E)(H,H)]} N.E. 2:{[(H,H)[(E,E)(H,H)(E,E)(H,H)]} N.E. 3:{[(E,E)[(E,E)(H,H)(H,H)(H,H)]} N.E. 6:{[(H,H)[(H,H)(H,H)(H,H)(H,H)]} N.E. 5:{[(H,H)[(E,E)(H,H)(E,E)(H,H)]} N.E. 4:{[(H,H)[(E,E)(H,H)(H,H)(H,H)]}

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Example 3.1.: Students’ Dilemma – 2 (subgame-perfect N.E./non-mechanical solution) YOU1 OTHERS1 Easy Hard Easy Hard H E E H Easy Hard OTHERS1 H H H E E H H H Y2 O2 Is it reasonable to expect any from group 2 to play Hard if all from group 1 chose Easy and to play Easy if YOU1 chose Hard? Suggested Best Strategy of group 2 (Strategy 2): Play Hard if at least one from group 1 played Hard, otherwise play Easy.

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Example 3.1.: Students’ Dilemma – 2 (subgame-perfect N.E./non-mechanical solution) EasyHard E0,00,0-3,-2 H -2,-2 O1 Y1 The best response of group 1: Strategy 2: [(E,E)(H,H)(H,H)(H,H)] There are two subgame-perfect N.E.: SGPNE 1:{[(E,E)[(E,E)(H,H)(H,H)(H,H)]} SGPNE 1:{[(H,H)[(E,E)(H,H)(H,H)(H,H)]}

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Dynamic games of Complete but Imperfect Information – SGPNE (Selten 1965) Subgame-perfect Nash Equilibrium (SGPNE)– a Nash equilibrium is subgame perfect if the players’ strategies constitute a Nash equilibrium in every subgame.

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Relation between Information Set and Imperfect Information? Subgame versus Stage Game? Subgame-perfect Outcome versus Equilibrium? Dynamic games of Complete but Imperfect Information – Summary

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Example 3.2.: Students’ Dilemma (two-stage repeated game) EasyHard E-1,-1-3,0 H0,-3-2,-2 OTHERS YOU The original static Student Dilemma game is played twice in a row: There is a unique N.E. of the static second-stage game: {Hard, Hard}

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Example 3.2.: Students’ Dilemma (two-stage repeated game) EasyHard E-3,-3-5,-2 H-2,-5-4,-4 OTHERS YOU Adding the N.E. payoffs to the original payoff matrix yields the first-stage payoff matrix : The subgame-perfect outcome is {(Hard,Hard),(Hard,Hard)}

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Finitely Repeated Games Let G(T) denote the finitely repeated game in which stage game G is played T times, with the outcomes of all preceding plays observed before the next play begins. The payoffs for G(T) are the sum of the payoffs from the T stage games If G has a unique N.E. then, for any finite T, G(T) has a unique subgame-perfect outcome: N.E. is played in every stage.

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