Incomplete Information and
Bayesian-Nash-Equilibrium
Georg Nöldeke
Wirtschaftswissenschaftliche Fakultät, Universität Basel
Advanced Microeconomics, HS 11
Lecture 4
1/12
Introduction
Our analysis of strategic form games rests on the implicit
assumption that the structure of the game is common
knowledge:
Every player knows the strategy sets and payoff function of
all other players . . .
. . . and knows that all other players know the structure of
the game . . .
. . . and knows that all other players know that all other
players know the structure of the game . . .
. . . and so on ad infinitum.
Here we extend the previous model of a game in strategic
form to incorporate incomplete information about players’
payoff functions.
Advanced Microeconomics, HS 11
Lecture 4
2/12
Introduction
John Harsanyi
Winner of the Nobel Prize in
Economics 1994
c
The
Nobel Foundation,
http://nobelprize.org/nobel_
prizes/economics/laureates/
1994/harsanyi.jpg
Advanced Microeconomics, HS 11
Lecture 4
3/12
Types and Beliefs
To model uncertainty, two additional elements are added to
the description of a strategic form game:
1. For each player i there is a finite set of possible types Ti .
2. For each player i there is a belief pi specifying the
probability that player i attaches to the possible types of the
other players.
Let T = ∏Ni=1 Ti denote the set of possible type profiles.
Payoff functions are now given by ui : S × T → R.
This models that players not only need to know s to
determine their own payoff and the payoff of other players –
as long as (some) players do not know t ∈ T there is
incomplete information about payoff functions.
Observe: Throughout it is assumed that each player knows
his own type.
Advanced Microeconomics, HS 11
Lecture 4
4/12
Types and Beliefs
The textbook introduces a very general model of beliefs.
We simplify by assuming that beliefs are given by
pi (t−i | ti ) = ∏ p j (t j ),
j6=i
where p j is a probability distribution over T j .
This means that there is a common prior and types are
independent.
Advanced Microeconomics, HS 11
Lecture 4
5/12
Static Game of Incomplete Information
A (Static) Game of Incomplete Information or Bayesian Game
is given by
A (finite) set of players: i = 1, . . . , N.
A (finite) set of strategies Si for each player.
A (finite) set of types Ti for each player.
A probability distribution pi over the set of possibly types of
every player.
A payoff function ui : S × T → R for each player i.
We write G = (p1 , . . . , pN ; T1 , . . . , TN ; S1 , . . . , SN ; u1 , . . . , uN ) for such
a game.
Advanced Microeconomics, HS 11
Lecture 4
6/12
Strategy Profiles and Payoff Functions
A pure strategy profile s∗ for a game of incomplete
information specifies a choice of pure strategy for every type
of every player:
s∗ = (s∗1 , . . . , s∗N ),
where s∗i : Ti → Si .
Given a pure strategy profile s∗ the payoff for type ti of player
i is:
!
∑
t−i ∈T−i
∏ p j (t j )
ui (s∗1 (t1 ), . . . , s∗N (tN );t1 , . . . ,tN )
j6=i
This captures the idea that player i knows his own type ti , but
is uncertain about the other player’s types and takes into
account that the choices of the other players may depend on
what their types are.
Observe: The payoff to a given type of player i does not
depend on the strategies chosen by other types of the same
player.
Advanced Microeconomics, HS 11
Lecture 4
7/12
Example: Two Players
Consider the case in which there are only two players:
Belief of player 1 is independent of his own type and given
by p2 (t2 ).
Belief of player 2 is independent of his own type and given
by p1 (t1 ).
A pure strategy for player 1 is given by s∗1 : T1 → S1 .
A pure strategy for player 2 is given by s∗2 : T2 → S2 .
The payoff for type t1 of player 1 is:
∑
p2 (t2 )u1 (s∗1 (t1 ), s∗2 (t2 );t1 ,t2 ).
t2 ∈T2
The payoff for type t2 of player 2 is:
∑
p1 (t1 )u2 (s∗1 (t1 ), s∗2 (t2 );t1 ,t2 ).
t1 ∈T1
Advanced Microeconomics, HS 11
Lecture 4
8/12
Example: Two Players
If for one of the players, say player 1, there is only one possible
type, notation may be simplified:
Strategy for player 1 is simply given by s1 ∈ S1 .
There is no need to introduce a belief for player 2.
Hence, payoffs may be written as:
∑
p2 (t2 )u1 (s1 , s∗2 (t2 );t2 )
t2 ∈T2
for player 1 and
u2 (s1 , s∗2 (t2 );t2 ).
for player 2.
Advanced Microeconomics, HS 11
Lecture 4
9/12
Bayesian Nash Equilibrium
Definition (Pure Strategy Bayesian-Nash Equilibrium)
A strategy profile s∗ is a pure strategy Bayesian-Nash
equilibrium for a game of incomplete information if and only if
for all types ti of all players i the strategy s∗i (ti ) solves the
problem
!
max
∑
si ∈Si t ∈T
−i
−i
∏ p j (t j )
ui (si , s∗−i (t−i );t1 , . . . ,tN ).
j6=i
As explained in the textbook, this definition is identical to
requiring that s∗ is a pure strategy Nash equilibrium in a
game in which each type of each player is treated as a
separate player.
Advanced Microeconomics, HS 11
Lecture 4
10/12
Remarks
Mixed Strategies:
Allowing each type of each player to choose a probability
distribution mi over Si in the above constructions yields the
definition of a mixed strategy Bayesian-Nash equilibrium
for a game of incomplete information.
This is what is meant by a Bayesian-Nash equilibrium in
Definition 7.12 from the textbook. In particular, the
existence result in Theorem 7.3 requires the use of mixed
strategies.
To simplify, we will only consider pure strategies when
considering examples of games with incomplete
information.
Advanced Microeconomics, HS 11
Lecture 4
11/12
Remarks
Continuous Type Distributions:
In may economic applications the set of possible types of a
player is modeled as an interval, say Ti = [0, 1], and beliefs
are described by a probability density function.
The summations over types appearing in our formulas then
have to be replaced by integrals.
Examples of such games appear on Problem Set 4.
We will consider more such games when studying auctions
later on.
Advanced Microeconomics, HS 11
Lecture 4
12/12