1. Econ 805 Advanced Micro Theory 1
Dan Quint
Fall 2008
Lecture 1 – Sept 2, 2007
A Quick Review of Game Theory and, in particular, Bayesian Games

2. 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)

3. Solution concepts for games of complete information
Pure-strategy Nash equilibrium: s Î A1 x … x AN s.t.
ui(si,s-i) ³ ui(s’i,s-i)
for all s’i Î Ai
for all i Î {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

4. 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?

5. 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)

6. 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)

7. 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

8. Solving the Cournot example, with p = ½ that firm 2 is strong…
Strong firm 2 best-responds by choosing
Q2S = arg maxq q(100-Q1-q-20)
Maximization gives Q2S = (80-Q1)/2
Weak firm 2 sets
Q2W = arg maxq q(100-Q1-q-30)
giving Q2W = (70-Q1)/2
Firm 1 maximizes expected profits:
Q1 = arg maxq ½q(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)

9. Formally, for N = 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 ai Î Ai
for every type ti Î Ti
for each player i Î {1,2}

10. Ex-post versus ex-ante formulations
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

11. Auctions are typically modeled as Bayesian games
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