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Econ 805 Advanced Micro Theory 1PowerPoint Presentation

Econ 805 Advanced Micro Theory 1

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### Econ 805Advanced Micro Theory 1

Dan Quint

Fall 2008

Lecture 1 – Sept 2, 2007

A Quick Review of Game Theory and, in particular, Bayesian Games

Games of complete information

- A static (simultaneous-move) game is defined by:
- Players 1, 2, …, N
- Action spaces A1, A2, …, AN
- Payoff functions ui : A1 x … x AN R
all of which are assumed to be common knowledge

- In dynamic games, we talk about specifying “timing,” but what we mean is information
- What each player knows at the time he moves
- Typically represented in “extensive form” (game tree)

Solution concepts for games of complete information

- Pure-strategy Nash equilibrium: sÎA1 x … x AN s.t.
ui(si,s-i) ³ ui(s’i,s-i)

for all s’iÎAi

for all iÎ{1, 2, …, N}

- In dynamic games, we typically focus on Subgame Perfect equilibria
- Profiles where Nash equilibria are also played within each branch of the game tree
- Often solvable by backward induction

Games of incomplete information

- Example: Cournot competition between two firms, inverse demand is P = 100 – Q1 – Q2
- Firm 1 has a cost per unit of 25, but doesn’t know whether firm 2’s cost per unit is 20 or 30
- What to do when a player’s payoff function is not common knowledge?

John Harsanyi’s big idea(“Games with Incomplete Information Played By Bayesian Players”)

- Transform a game of incomplete information into a game of imperfect information – parameters of game are common knowledge, but not all players’ moves are observed
- Introduce a new player, “nature,” who determines firm 2’s marginal cost
- Nature randomizes; firm 2 observes nature’s move
- Firm 1 doesn’t observe nature’s move, so doesn’t know firm 2’s “type”

“Nature”

make 2 weak

make 2 strong

Firm 2

Firm 2

Q2

Q2

Firm 1

Q1

Q1

u1 = Q1(100 - Q1 - Q2 - 25)

u2 = Q2(100 - Q1 - Q2 - 30)

u1 = Q1(100 - Q1 - Q2 - 25)

u2 = Q2(100 - Q1 - Q2 - 20)

Bayesian Nash Equilibrium

- Assign probabilities to nature’s moves (common knowledge)
- Firm 2’s pure strategies are maps from his “type space” {Weak, Strong} to A2 = R+
- Firm 1 maximizes expected payoff
- in expectation over firm 2’s types
- given firm 2’s equilibrium strategy

“Nature”

make 2 weak

make 2 strong

Firm 2

Firm 2

p = ½

p = ½

Q2

Q2W

Q2

Q2S

Firm 1

Q1

Q1

u1 = Q1(100 - Q1 - Q2 - 25)

u2 = Q2(100 - Q1 - Q2 - 30)

u1 = Q1(100 - Q1 - Q2 - 25)

u2 = Q2(100 - Q1 - Q2 - 20)

Other players’ types can enter into a player’s payoff function

- In the Cournot example, firm 1 only cares about firm 2’s type because it affects his action
- In some games, one player’s type can directly enter into another player’s payoff function
- Poker: you don’t know what cards your opponent has, but they affect both how he’ll plays the hand and whether you’ll win at showdown

- Either way, in BNE, simply maximize expected payoff given opponent’s strategy and type distribution

Solving the Cournot example, with functionp = ½ that firm 2 is strong…

- Strong firm 2 best-responds by choosing
Q2S = arg maxqq(100-Q1-q-20)

Maximization gives Q2S = (80-Q1)/2

- Weak firm 2 sets
Q2W = arg maxqq(100-Q1-q-30)

giving Q2W = (70-Q1)/2

- Firm 1 maximizes expected profits:
Q1 = arg maxq½q(100-q-Q2S-25) + ½q(100-q-Q2W-25)

giving Q1 = (75 – Q2W/2 – Q2S/2)/2

- Solving these simultaneously gives equilibrium strategies:
Q1 = 25, (Q2W, Q2S) = (22½ , 27½)

Formally, for functionN = 2 and finite, independent types…

- A static Bayesian game is
- A set of players 1, 2
- A set of possible types T1 = {t11, t12, …, t1K} and T2 = {t21, t22, …, t2K’} for each player, and a probability for each type {p11, …, p1K, p21, …, p2K’}
- A set of possible actions Ai for each player
- A payoff function mapping actions and types to payoffs for each player
ui : A1 x A2 x T1 x T2 R

- A pure-strategy Bayesian Nash Equilibrium is a mapping si : Ti Ai for each player, such that
for each potential deviation aiÎAi

for every type tiÎ Ti

for each player i Î {1,2}

Ex-post versus ex-ante formulations function

- With a finite number of types, the following are equivalent:
- The action si(ti) maximizes “ex-post expected payoffs” for each type ti
- The mapping si : Ti Ai maximizes “ex-ante expected payoffs” among all such mappings

- I prefer the ex-post formulation for two reasons
- With a continuum of types, the equivalence breaks down, since deviating to a worse action at a particular type would not change ex-ante expected payoffs
- Ex-post optimality is almost always simpler to verify

Auctions are typically modeled as Bayesian games function

- Players don’t know how badly the other bidders want the object
- Assume nature gives each bidder a valuation for the object (or information about it) from some ex-ante probability distribution that is common knowledge

- In BNE, each bidder maximizes his expected payoffs, given
- the type distributions of his opponents
- the equilibrium bidding strategies of his opponents

- Thursday: some common auction formats and the baseline model

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