Advanced Microeconomics Notes PDF

Notes Indholdsfortegnelse {#indholdsfortegnelse.Overskrift} =================== [Lecture 1 -- Strategic Form Games 3](#lecture-1-strategic-form-games) [JR: Ch. 7, Game Theory 3](#jr-ch.-7-game-theory) [7.1, Strategic Decision Making 3](#strategic-decision-making) [7.2, Strategic Form Games 4](#...

Notes Indholdsfortegnelse {#indholdsfortegnelse.Overskrift} =================== [Lecture 1 -- Strategic Form Games 3](#lecture-1-strategic-form-games) [JR: Ch. 7, Game Theory 3](#jr-ch.-7-game-theory) [7.1, Strategic Decision Making 3](#strategic-decision-making) [7.2, Strategic Form Games 4](#strategic-form-games) [G: Ch. 1, Static Games of Complete Information 9](#g-ch.-1-static-games-of-complete-information) [1.1, Basic Theory: Normal-Form Games and Nash Equilibrium 9](#basic-theory-normal-form-games-and-nash-equilibrium) [1.2, Applications 12](#applications) [Lecture Notes 18](#lecture-notes) [General 18](#general) [Strategic Form Games 19](#strategic-form-games-1) [Rationality 22](#rationality) [Steady State 37](#steady-state) [Security 42](#security) [Mixed Strategies. 51](#mixed-strategies.) [Correlated Equilibrium: Examples 66](#correlated-equilibrium-examples) [Lecture 2 -- Strategic Form Games under Incomplete Information 69](#lecture-2-strategic-form-games-under-incomplete-information) [JR: Ch. 7.2.3, Incomplete Information 69](#jr-ch.-7.2.3-incomplete-information) [G: Ch. 3, Static Games of Incomplete Information 70](#g-ch.-3-static-games-of-incomplete-information) [3.1, Theory: Static Bayesian Games and Bayesian Nash Equilibrium 71](#theory-static-bayesian-games-and-bayesian-nash-equilibrium) [3.2, Applications 73](#applications-1) [3.3, The Revelation Principle 78](#the-revelation-principle) [Lecture notes 80](#lecture-notes-1) [Assignment 2 Notes 80](#assignment-2-notes) [Outline 81](#outline) [Introduction 81](#introduction) [Types 81](#types) [Bayesian Game 83](#bayesian-game) [Associated strategic form 83](#associated-strategic-form) [Bayesian-Nash equilibrium 84](#bayesian-nash-equilibrium) [Application: Bertrand competition 89](#application-bertrand-competition) [Bayesian-Nash equilibrium with continuous types 92](#bayesian-nash-equilibrium-with-continuous-types) [Application: first-price auction 93](#application-first-price-auction) [Purification: Mixed strategies revisited 97](#purification-mixed-strategies-revisited) [Lecture 3 -- Extensive Form Games 103](#lecture-3-extensive-form-games) [JR: Ch. 7.3.1-7.3.6 -- Extensive Form Games 103](#jr-ch.-7.3.1-7.3.6-extensive-form-games) [7.3.1, Game Trees: A Diagrammatic Representation 103](#game-trees-a-diagrammatic-representation) [7.3.2, An Informal Analysis of Take-Away 104](#an-informal-analysis-of-take-away) [7.3.3, Extensive Form Game Strategies 105](#extensive-form-game-strategies) [7.3.4, Strategies and Payoffs 105](#strategies-and-payoffs) [7.3.5, Games of Perfect Information and Backward Induction Strategies 106](#games-of-perfect-information-and-backward-induction-strategies) [7.3.6, Games of Imperfect Information and Subgame Perfect Equilibrium 109](#games-of-imperfect-information-and-subgame-perfect-equilibrium) [Gibbons: Ch. 2, Dynamic Games of Complete Information 115](#gibbons-ch.-2-dynamic-games-of-complete-information) [2.1, Dynamic Games of Complete and Perfect Information 115](#dynamic-games-of-complete-and-perfect-information) [2.2, Two-Stage Games of Complete but Imperfect Information 124](#two-stage-games-of-complete-but-imperfect-information) [2.3, Repeated Games 127](#repeated-games) [2.4, Dynamic Games of Complete but Imperfect Information 127](#dynamic-games-of-complete-but-imperfect-information) [Lecture notes 128](#lecture-notes-2) [Introduction 128](#introduction-1) [Extensive Form Games 130](#extensive-form-games) [Game Trees 131](#game-trees) [Strategies 133](#strategies) [Backward Induction 134](#backward-induction) [Imperfect Information and Subgames 137](#imperfect-information-and-subgames) [Subgame Perfect Equilibrium 139](#subgame-perfect-equilibrium) [Mixed vs Behavioural Strategies 141](#mixed-vs-behavioural-strategies) [Perfect Recall and Subgame Perfect Equilibrium 142](#perfect-recall-and-subgame-perfect-equilibrium) [Lecture 4 -- Extensive Form Games under Incomplete Information 146](#lecture-4-extensive-form-games-under-incomplete-information) [JR: Ch. 7.3.7 -- Sequential Equilibrium 146](#jr-ch.-7.3.7-sequential-equilibrium) [Beliefs and Their Connection to Strategies 147](#beliefs-and-their-connection-to-strategies) [Sequential Rationality 152](#sequential-rationality) [Gibbons: Ch. 4 -- Sequential Equilibrium 158](#gibbons-ch.-4-sequential-equilibrium) [4.1, Introduction to Perfect Bayesian Equilibrium 158](#introduction-to-perfect-bayesian-equilibrium) [Lecture Notes 158](#lecture-notes-3) [Lecture 5 -- Repeated Games 159](#lecture-5-repeated-games) [Lecture Notes 159](#lecture-notes-4) [Lecture 6 -- Auctions 160](#lecture-6-mixed-strategies-examples) [JR: Ch. 9, 160](#_Toc180580418) Lecture 1 -- Strategic Form Games ================================= JR: Ch. 7, Game Theory ---------------------- - Game theory = Is the systematic study of how rational agents behave in strategic situations, or in games, where each agent must first know the decision of the other agents before knowing which decision is best for himself. - Solution concepts - Nash equilibrium, Bayesian-Nash equilibrium, backward induction, subgame perfection, and sequential equilibrium - Each of these concepts is more sophisticated than its predecessors - Knowing when to apply one solution rather than another is an important part of being a good applied economist ### 7.1, Strategic Decision Making - Non-strategic decisions = can be made in 'isolation', without taking into account the decisions that others might make - Given prices and income, each consumer acts entirely on his own, without regard for the behaviour of others - Cournot and Bertrand is a model where the firm understands that its optimal action depends on the action taken by the other firm - Pitcher and Batter example - - The entries in the matrix denote the players' payoffs as a result of their decisions, with the pitcher's payoff being the first number of each entry and the batter's the second - Each player seeks to maximise his payoff, and each reasons strategically - Each player must behave in a manner that is 'unpredictable' - When rational individuals make decisions strategically, each taking into account the decision the other makes, they sometimes behave in an 'unpredictable' manner ### 7.2, Strategic Form Games - Game = Elements common among games - Involves a number of participants = Players - The players has a range of possible actions = Strategies - Each of whom derives one payoff or another depending on his own strategy choice as well as the strategies chosen by each of the other players - Definition 7.1: Strategic Form Game = A strategic form game is a tuple [*G* = (*S*~*i*~, *u*~*i*~)~*i* = 1~^*N*^]{.math.inline}, where for each player [*i* = 1, ..., *N*, *S*~*i*~]{.math.inline} is the set of strategies available to player [*i*]{.math.inline}, and [\$u\_{i}:\\ \\times\_{j = 1}\^{N}S\_{j}\\mathbb{\\rightarrow R}\$]{.math.inline} describes player [*i*′*s*]{.math.inline} payoff as a function of the strategies chosen by all players. A strategic form game is finite if each player's strategy set contains finitely many elements. - The strategic form game describing that situation, when the pitcher is designated player 1, is given by - [*S*~1~ = *S*~2~ = {*F*, *C*}]{.math.inline} - [*u*~1~(*F*, *F*) = *u*~1~(*C*, *C*) =  − 1]{.math.inline} - [*u*~1~(*F*, *C*) = *u*~1~(*C*, *F*) = 1]{.math.inline}, and - [*u*~2~(*s*~1~, *s*~2~) =  − *u*~1~(*s*~1~, *s*~2~)]{.math.inline} for all [(*s*~1~, *s*~2~) ∈ *S*~1~ × *S*~2~]{.math.inline} - Note that two-player strategic form games with finite strategy sets can always be represented in matrix form, with the rows indexing the strategies of player 1, the columns indexing the strategies of player 2, and the entries denoting their payoffs #### **7.2.1, Dominant Strategies** -  - Player 2's payoff-maximising strategy choice depends on the choice made by player 1 - If 1 chooses U (up), then it is best for 2 to choose L (left), and if 1 chooses D (down), then it is best for 2 to choose R (right) - Player 2 must make his decision strategically, and he must consider carefully the decision of player 1 before deciding what to do himself - Player 1's best choice is actually independent of the choice made by player 2. Regardless of player 2's choice, U is best for player 1. - Player 1 will surely choose U - Player 2 will therefore choose L - The only sensible outcome of this game is the strategy pair (U, L), which associated payoff vector (3, 0) - Solve a game = to deduce the outcome when it is played by rational players - The above game can be solved because player 1 possesses a strategy that is best for him regardless of the strategy chosen by player 2 - When one player possesses such a 'dominant' strategy, the outcome is rather straightforward to determine - Let [*S* = *S*~1~ × ...× *S*~*N*~]{.math.inline} denote the set of joint pure strategies - The symbol, [ − *i*]{.math.inline}, denotes 'all players except player [*i*]{.math.inline}'. - For example, [*s*~(−*i*)~]{.math.inline} denotes an element of [*S*~ − *i*~]{.math.inline}, which itself denotes the set [*S*~1~ × ... × *S*~*i* − 1~ × *S*~*i* + 1~ × ...  × *S*~*N*~]{.math.inline} - Definition 7.2: Strictly Dominant Strategies = A strategy, [\$\\widehat{s\_{i}}\$]{.math.inline}, for player [*i*]{.math.inline} is strictly dominant if [*u*~*i*~(*ŝ*~*i*~, *s*~ − *i*~) \> *u*~*i*~(*s*~*i*~, *s*~ − *i*~)]{.math.inline} for all [(*s*~*i*~, *s*~ − *i*~) ∈ *S*]{.math.inline} with [*s*~*i*~ ≠ *ŝ*~*i*~]{.math.inline} - Et billede, der indeholder bord Automatisk genereret beskrivelse - Player 1's unique best choice is U when 2 plays L, but D when 2 plays M - Player 2's unique best choice is L when 1 plays U, but R when 1 plays D - Player 1's strategy C is always outperformed by D, in the sense that 1's payoff is strictly higher when D is chosen compared to when C is chosen regardless of the strategy chosen by player 2 - Player 2's strategy M is outperformed by R - C and M have been removed - The only sensible outcome is (3, 0) - Definition 7.3: Strictly Dominated Strategies = Player [*i*′*s*]{.math.inline} strategy  [*ŝ*~*i*~]{.math.inline} strictly dominates another of his strategies [\${\\overline{s}}\_{i}\$]{.math.inline}, if [\$u\_{i}({\\widehat{s}}\_{i},\\ s\_{- i}) \> u\_{i}({\\overline{s}}\_{i},\\ s\_{- i})\$]{.math.inline} for all [*s*~ − *i*~ ∈ *S*~ − *i*~]{.math.inline}. In this case, we also say that [\${\\overline{s}}\_{i}\$]{.math.inline} is strictly dominated in S. - In the previous strategies we have *eliminated* options - Let [*S*~*i*~^0^ = *S*~*i*~]{.math.inline} for each player [*i*]{.math.inline}, and for [*n* ≥ 1]{.math.inline}, let [*S*~*i*~^*n*^]{.math.inline} denote those strategies of player [*i*]{.math.inline} surviving after the nth round of elimination. That is, [*s*~*i*~ ∈ *S*~*i*~^*n*^]{.math.inline} if [*s*~*i*~ ∈ *S*~*i*~^*n* − 1^]{.math.inline} is not strictly dominated in [*S*^*n* − 1^]{.math.inline}. - Definition 7.4: Iteratively Strictly Undominated Strategies = A strategy [*s*~*i*~]{.math.inline} for player [*i*]{.math.inline} is iteratively strictly undominated in S (or survives iterative elimination of strictly dominated strategies) if [*s*~*i*~ ∈ *S*~*i*~^*n*^]{.math.inline}, for all [*n* ≥ 1]{.math.inline} - Definition 7.5: Weakly Dominated Strategies = Player [*i*′*s*]{.math.inline} strategy  [*ŝ*~*i*~]{.math.inline} weakly dominates another of his strategies [\${\\overline{s}}\_{i}\$]{.math.inline}, if [\$u\_{i}\\left( {\\widehat{s}}\_{i},\\ s\_{- i} \\right) \\geq u\_{i}({\\overline{s}}\_{i},\\ s\_{- i})\$]{.math.inline} for all [*s*~ − *i*~ ∈ *S*~ − *i*~]{.math.inline}, with at least one strict inequality. In this case, we also say that [\${\\overline{s}}\_{i}\$]{.math.inline} is weakly dominated in S -  - Neither player has a strictly dominated strategy - Both D and R are weakly dominated by U and L - Eliminating strictly dominated strategies has no effect here, whereas eliminating weakly dominated strategies isolates the unique strategy pair (U, L) - Let [*W*~*i*~^0^ = *S*~*i*~]{.math.inline} for each player [*i*]{.math.inline}, and for [*n* ≥ 1]{.math.inline}, let [*W*~*i*~^*n*^]{.math.inline} denote those strategies of player [*i*]{.math.inline} surviving after the nth round of elimination of weakly dominated strategies. That is, [*s*~*i*~ ∈ *W*~*i*~^*n*^]{.math.inline} if [*s*~*i*~ ∈ *W*~*i*~^*n* − 1^]{.math.inline} is not weakly dominated in [*W*^*n* − 1^ = *W*~1~^*n* − 1^ × ...  × *W*~*N*~^*n* − 1^]{.math.inline}. - Definition 7.6: Iteratively Weakly Undominated Strategies = A strategy [*s*~*i*~]{.math.inline} for player [*i*]{.math.inline} is iteratively weakly undominated in S (or survives iterative elimination of weakly dominated strategies) if [*s*~*i*~ ∈ *W*~*i*~^*n*^]{.math.inline} for all [*n* ≥ 1]{.math.inline}. #### **7.2.2, Nash Equilibrium** - Market equilibrium = demand equals supply - Equilibria = Regularities in behaviour that can be 'rationally' sustained - A joint strategy  [*ŝ* ∈ *S*]{.math.inline} constitutes a Nash equilibrium as long as each individual, while fully aware of the others' behaviour, has no incentive to change his own - A Nash equilibrium describes behaviour that can be rationally sustained - Definition 7.7: Pure Strategy Nash Equilibrium = Given a strategic form game [*G* = (*S*~*i*~, *u*~*i*~)~*i* = 1~^*N*^]{.math.inline}, the joint strategy  [*ŝ* ∈ *S*]{.math.inline} is a pure strategy Nash equilibrium of G if for each player [*i*, *u*~*i*~(*ŝ*) ≥ *u*~*i*~(*s*~*i*~, *ŝ*~ − *i*~)]{.math.inline} for all [*s*~*i*~ ∈ *S*~*i*~]{.math.inline} - As seen in figs. 7.2 to 7.4 - A game may possess more than one Nash equilibrium - Ex. Fig. 7.4 (D, R) - Some games do not possess any pure strategy Nash equilibria - Et billede, der indeholder bord Automatisk genereret beskrivelse ##### *Mixed Strategies and Nash Equilibrium* - To make a choice in a manner that others cannot predict is to make it in a manner that you yourself cannot predict -- randomize among choices - This qualifies as an equilibrium - Batter-Pitcher example: Expected utility: [\$\\frac{1}{2}\\left( - 1 \\right) + \\frac{1}{2}\\left( 1 \\right) = 0\$]{.math.inline} for both players - The players' randomized choices form an equilibrium: each is aware of the (randomized) manner in which the other makes his choice, and neither can improve his expected payoff by unilaterally changing the manner in which his choice is made - Definition 7.8: Mixed Strategies = Fix a finite strategic form game [*G* = (*S*~*i*~, *u*~*i*~)~*i* = 1~^*N*^]{.math.inline}. A mixed strategy, [*m*~*i*~]{.math.inline}, for player [*i*]{.math.inline} is a probability distribution over [*S*~*i*~]{.math.inline}. That is, [*m*~*i*~ : *S*~*i*~ → \[0, 1\]]{.math.inline} assigns to each [*s*~*i*~ ∈ *S*~*i*~]{.math.inline} the probability, [*m*~*i*~(*s*~*i*~)]{.math.inline}, that [*s*~*i*~]{.math.inline} will be played. We shall denote the set of mixed strategies for player [*i*]{.math.inline} by [*M*~*i*~]{.math.inline}. Consequently, [\$M\_{i} = \\left\\{ m\_{i}:S\_{i} \\rightarrow \\left\\lbrack 0,\\ 1 \\right\\rbrack\\left\| \\sum\_{s\_{i} \\in S\_{i}}\^{}{m\_{i}\\left( s\_{i} \\right) = 1} \\right.\\ \\right\\}\$]{.math.inline}. From now on, we shall call [*S*~*i*~]{.math.inline} player [*i*′*s*]{.math.inline} set of pure strategies. - A mixed strategy is the means by which players randomize their choices - Each player [*i*]{.math.inline} is now allowed to choose from the set of mixed strategies [*M*~*i*~]{.math.inline} rather than [*S*~*i*~]{.math.inline} - This gives each player [*i*]{.math.inline} strictly more choices than before, because every pure strategy [\${\\overline{s}}\_{i} \\in S\_{i}\$]{.math.inline} is represented in [*M*~*i*~]{.math.inline} by the (degenerate) probability distribution assigning probability one to [\${\\overline{s}}\_{i}\$]{.math.inline} - Let [\$M = \\times\_{i = 1}\^{N}M\_{i}\$]{.math.inline} denote the set of joint mixed strategies - Joint strategy = [*m* ∈ *M*]{.math.inline} - Strategy for player [*i*]{.math.inline} = [*m*~*i*~ ∈ *M*~*i*~]{.math.inline} - If [*u*~*i*~]{.math.inline} is a von Neumann-Morgenstern utility function on S, and the strategy [*m* ∈ *M*]{.math.inline} is played, then player [*i*′*s*]{.math.inline} expected utility is \ [\$\$u\_{i}\\left( m \\right) = \\sum\_{s \\in S}\^{}{m\_{1}\\left( s\_{1} \\right)\\ldots m\_{N}\\left( s\_{N} \\right)u\_{i}(s)}\$\$]{.math.display}\ - Follows from the fact that the players choose their strategies independently - Definition 7.9: Nash Equilibrium = Given a finite strategic form game [*G* = (*S*~*i*~, *u*~*i*~)~*i* = 1~^*N*^]{.math.inline}, a joint strategy  [*m̂* ∈ *M*]{.math.inline} is a Nash equilibrium of G if for each player [*i*]{.math.inline}, [*u*~*i*~(*m̂*) ≥ *u*~*i*~(*m*~*i*~, *m̂*~ − *i*~)]{.math.inline} for all [*m*~*i*~ ∈ *M*~*i*~]{.math.inline} - Theorem 7.1: Simplified Nash Equilibrium Tests = The following statements are equivalent: - \(a)  [*m̂* ∈ *M*]{.math.inline} is a Nash equilibrium - \(b) For every player [*i*]{.math.inline}, [*u*~*i*~(*m̂*) = *u*~*i*~(*s*~*i*~, *m̂*~ − *i*~)]{.math.inline} for every [*s*~*i*~ ∈ *S*~*i*~]{.math.inline} given positive weight by  [*m̂*~*i*~]{.math.inline}, and [*u*~*i*~(*m̂*) ≥ *u*~*i*~(*s*~*i*~, *m̂*~ − *i*~)]{.math.inline} for every [*s*~*i*~ ∈ *S*~*i*~]{.math.inline} given zero weight by  [*m̂*~*i*~]{.math.inline} - (c ) For every player [*i*]{.math.inline}, [*u*~*i*~(*m̂*) ≥ *u*~*i*~(*s*~*i*~, *m̂*~ − *i*~)]{.math.inline} for every [*s*~*i*~ ∈ *S*~*i*~]{.math.inline} - Theorem 7.2: (Nash) Existence of Nash Equilibrium = Every finite strategic form game possesses at least one Nash equilibrium - Prof pp. 317-319 - No matter how many players are involved, as long as each possesses finitely many pure strategies there will be at least on Nash equilibrium G: Ch. 1, Static Games of Complete Information ---------------------------------------------- - Complete information = Each player's payoff function is common knowledge among all the players - Payoff function = The function that determines the player's payoff from the combination of actions chosen by the players - Incomplete information = Games in which some player is uncertain about another player's payoff function -- as in an auction where each bidder's willingness to pay for the good being sold is unknown to the other bidders ### 1.1, Basic Theory: Normal-Form Games and Nash Equilibrium #### **1.1.A, Normal-Form Representation of Games** - Normal-form representation of a game = Each player simultaneously chooses a strategy, and the combination of strategies chosen by the players determines a payoff for each player - Specify: - \(1) The players in the game - \(2) The strategies available for each player - \(3) The payoff received by each player for each combination of strategies that could be chosen by the players - The Prisoners' Dilemma -  - Let [*S*~*i*~]{.math.inline} denote the set of strategies available to player [*i*]{.math.inline} - Let [*s*~*i*~]{.math.inline} denote an arbitrary member of this set - When written [*s*~*i*~ ∈ *S*~*i*~]{.math.inline} it's to indicate that the strategy [*s*~*i*~]{.math.inline} is a member of the set of strategies [*S*~*i*~]{.math.inline} - Let [(*s*~1~, ..., *s*~*n*~)]{.math.inline} denote a combination of strategies, one for each player - Let [*u*~*i*~]{.math.inline} denote player [*i*′*s*]{.math.inline} payoff function: [*u*~*i*~(*s*~1~, ..., *s*~*n*~)]{.math.inline} is the payoff to player [*i*]{.math.inline} if the players choose the strategies [(*s*~1~, ..., *s*~*n*~)]{.math.inline} - Definition: The normal-form representation = of an n-player game specifies the players' strategy spaces [*S*~1~, ..., *S*~*n*~]{.math.inline} and their payoff functions [*u*~1~, ..., *u*~*n*~]{.math.inline}. We denote this game by [*G* = {*S*~1~, ..., *S*~*n*~; *u*~1~, ..., *u*~*n*~}]{.math.inline}. #### **1.1.B, Iterated Elimination of Strictly Dominated Strategies** - Definition: In the normal-form game [*G* = {*S*~1~, ..., *S*~*n*~; *u*~1~, ..., *u*~*n*~}]{.math.inline}, let [*s*~*i*~′]{.math.inline} and [*s*~*i*~″]{.math.inline} be feasible strategies for player [*i*]{.math.inline} (i.e., [*s*~*i*~′]{.math.inline} and [*s*~*i*~″]{.math.inline} are members of [*S*~*i*~]{.math.inline}). Strategy [*s*~*i*~′]{.math.inline} is strictly dominated by strategy [*s*~*i*~″]{.math.inline} if for each feasible combination of the other players' strategies, [*i*′*s*]{.math.inline} payoff from playing [*s*~*i*~′]{.math.inline} is strictly less than [*i*′*s*]{.math.inline} payoff from playing [*s*~*i*~″]{.math.inline}: \ [*u*~*i*~(*s*~1~, ..., *s*~*i* − 1~, *s*~*i*~^′^, *s*~*i* + 1~, ..., *s*~*n*~) \