Advanced Microeconomics Instructors: Wojtek Dorabialski & Olga Kiuila

Advanced Microeconomics

• Instructors: Wojtek Dorabialski & Olga Kiuila

• Lectures: Mon. & Wed. 9:45 – 11:20 room 201

• Office hours: Mon. & Wed. 9:15 – 9:45 room 201

Contents

• Game Theory

• General Equilibrium Theory

• Computable General Equilibrium modeling

How to complete successfuly

• Attend classes

• Read lecture notes at: www.wne.uw.edu.pl/~kiuila/am

• Read textbooks

• Solve problem sets

• Ask for help (office hours)

• Prepare for and pass the exam

Game Theory - Intro

• What is Game Theory?

• A branch of mathematics (decision theory), which formalizes games and defines solutions to them

• What is a Game?

• It is a decision problem, where decision-maker’s payoff (profit) may depend not only on his own decision, but also on the decisions made by other decision makers

Defining a game

• Formally, a game is a set of 4 elements:

• a set of players (can even be infinite)
• a set of rules (allowable actions and sequencing of actions by each player)
• a payoff function (which assigns payoffs for each player as a function of strategies chosen)
• informational structure (what players know at each point in the game)

General Assumptions

• Standard GT assumes that players are:

• selfish: maximize their own payoffs and do not care about the opponent’s payoffs
• rational: they understand the game and can determine the optimal strategy
• expected-utility maximizers: in uncertain situations players they base their choices on (von Neumann-Morgenstern) expected utility
• share common knowledge about all aspects of the game
• in addition, it is often assumed that players do not communicate, cooperate or negotiate, unless the game allows it explicitly

More on Assumptions

• All of the above are simplifying assumptions, i.e. they rarely hold in reality

• There is a lot of research on games with altruistic players, players with bounded-rationality or non-expected-utility maximizers or even non-decision makers (e.g. Evolutionary Game Theory)

• A whole separate branch of decision theory deals with cooperative games (Cooperative Game Theory)

More on Common Knowledge

• “As we know, there are known knowns. These are things we know we know.

• We also know, there are known unknowns. That is to say we know there are some things we do not know.

• But there are also unknown unknowns, the ones we don't know we don't know”.

• D.H. Rumsfeld, Feb. 12, 2002, Department of Defense news briefing

• Common knowledge means that there are no unknown unknowns.

Incomplete Information vs Asymmetric Knowledge

• Modeling asymmetric knowledge (unknown unknowns) is difficult

• Instead, Game theorists assume that if a player doesn’t know something, she has some initial beliefs about it and these beliefs are commonly known (there are only known unknowns)

• Games with known unknowns are called games with incomplete (imperfect) information.

History of Game Theory

• Cournot (1838) - quantity-setting duopoly model

• Bertrand (1883) – price-setting duopoly model

• Zermelo (1913) – the game of chess

• von Neumann & Morgenstern (1944) – defined games, min-max solution for 0-sum games

• Nash (1950) – defined a the equilibrium and the solution to a cooperative bargaining problem

‘Nobel’ prize winners for Game Theory (Economics)

• 1994 – John Nash, John Harsanyi, Reinhard Selten

• 1996 – James Mirrlees, William Vickrey

• 2005 – Robert Aumann, Thomas Shelling

Normal Form Games

• Simple games without „timing”, i.e. where players make decisions simultaneously. Dynamic games can be reduced into a normal form.

• The set of strategies is simply the set of possible choices for each player.

• Normal (Strategic) Form Game consists of the following elements:

• - N={1,.., n} the finite set of players

• - S={S1,.., Sn} the set of strategies, including a (possibly infinite) set for each player

• - U(s1,.., sn) the vector of payoff functions, where si  Si for each player

• If the set of strategies is small and countable (typically 2-5), then we can use a game matrix to represent a normal-form game

Game Matrix

• Example 1: Advertising game

• N={1, 2}

• S={S1, S2} and S1 = S2 = {A, N}

• U(s1, s2) = { u1(s1, s2); u2(s1, s2)}

• u1(A, A) = 40; u1(A, N) = 60; u1(N, A) = 30; u1(N, N) = 50

• u2(A, A) = 40; u2(A, N) = 30; u2(N, A) = 60; u2(N, N) = 50

Mixed Strategies

• In any game, but especially in games such as above (with countable strategies), it is often useful to consider mixed strategies

• Mixed strategies are a probability distribution over the set of (pure) strategies S, a convex extension of that set.

• The set of mixed strategies is denoted by ∑={∑1, ∑2}, a single strategy is of player i is denoted by σi  ∑ i

• We simply allow the players to make a random choice, with any possible probability distribution over the set of choices.

Dominance

• σ-i = vector of mixed strategies of players other than i

• Def: Pure strategy si is strictly dominated (never-best-response), if for every σ-i there is a strategy zi  ∑ i of player i s.t. ui(zi, σ-i) > ui(si, σ-i)

• There is also a notion of weak dominance, where it is enough that the strategy zi is never worse (but doesn’t have to be always better) than si

• Iterated elimination of dominated strategies (IEDS) is a simple procedure that provides a solution to many normal-form games

• Step 1: Identify all dominated strategies
• Step 2: Eliminate them to obtain a reduced game
• Step 3: Go to Step 1

Iterated Elimination of Dominated Strategies

• In the Advertising game, the IEDS solution is (A, A)

• What about the game below?

• D is dominated by a mixed strategy (e.g. 50-50 mix od U-M), then L is dominated by R, then U by M, solution: M-R