2. Static Games of Incomplete Information
Georg Nöldeke
Wirtschaftswissenschaftliche Fakultät, Universität Basel
Advanced Microeconomics (HS 10)
Static Games of Incomplete Information
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1. The Model
1.1 Motivation
Implicit in the analysis of games in strategic form is the
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.
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1. The Model
John Harsanyi
Winner of the Nobel Prize in
Economics 1994
c
The
Nobel Foundation,
http://nobelprize.org/nobel_
prizes/economics/laureates/
1994/harsanyi.jpg
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1. The Model
1.2 Description of a Static Game of Incomplete Information
A Static Game of Incomplete Information or Bayesian Game is
described by:
A (finite) set of players: i = 1, . . . , n.
A (finite) set of actions for each player: Ai with A = ∏ni=1 Ai .
A (finite) set of types for each player: Ti with T = ∏ni=1 Ti .
A probability distribution p ∈ ∆(T ) over the set of possible types.
A payoff function for each player: ui : A × T → R where
ui (a1 , . . . , an ;t1 , . . .tn ) is the payoff of player i is the action profile
a = (a1 , . . . , an ) ∈ A is chosen and the types of the players are given
by t = (t1 , . . .tn ) ∈ T .
We write G = (A1 , . . . , An ; T1 , . . . , Tn ; p; u1 , . . . , un ) for such a game.
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1. The Model
1.2 Description of a Static Game of Incomplete Information
Interpretation:
1
“Nature” draws a type vector t = (t1 , · · · ,tn ) according to the
probability distribution p.
2
Every player i is informed about his own type ti , but not about the
realized types of the other players.
3
The players simultaneously choose actions ai ∈ Ai and
4
receive the payoffs ui (a,t).
Remarks:
We restrict attention to the special case in which types are
independent, that is
p(t) = p1 (t1 )p2 (t2 ) . . . pn (tn )
holds for all t, where pi ∈ ∆(Ti ) is a probability distribution over the
types of player i.
In many applications, it is assumed that the payoff of player i
depends only on the action profile a and his own type: ui (a;ti )
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1. The Model
1.3 The Strategic Form of a Static Games of Incomplete Information
With any given static game G of incomplete information we may
associate a corresponding game Γ = (S1 , . . . Sn ; v1 , . . . , vn ) in strategic
form:
A strategy for player i is given by a function si : Ti → Ai , where si (ti )
is the action taken by player i when his type is ti . Let Si be the
corresponding strategy set of player i.
vi : S → R is given by the expected payoff of player i from the
strategy profile s ∈ S:
vi (s1 , . . . , sn ) =
∑ [p1 (t1 )p2 (t2 ) . . . pn (tn )] ui (s1 (t1 ), . . . , sn (tn );t1 , . . . ,tn ).
t∈T
The game Γ constructed in this way is the strategic form of the static
game G of incomplete information.
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2. Bayesian Nash Equilibrium
2.1 Definition
Definition (Bayesian Nash Equilibrium)
Let Γ be the strategic form of a static game G of incomplete
information. A Nash equilibrium of Γ is called a Bayesian Nash
equilibrium of G.
Remarks:
When Ai and Ti are finite for all i = 1, · · · , n, then Γi is a finite game.
Hence, it has a Nash equilibrium in mixed strategies.
However, we will not consider the extension of Γ to mixed
strategies, but will only investigate Bayesian Nash Equilibria in
pure strategies.
In many economic applications, there are some players such that
either Ai or Ti (or both) are infinite. We will consider such
examples without stating the formal definition of a Bayesian Nash
equilibrium for such games.
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2. Bayesian Nash Equilibrium
2.2 An Illustrative Example
A static game of incomplete information:
i = 1, 2.
A1 = {T, B}, A2 = {L, R}
T1 = {0, 2}, T2 = {0, 2}.
p1 (0) = p1 (2) = 0.5 = p2 (0) = p2 (1).
Payoffs as a function of actions and types given by
T
B
L
1 + t1 , 1 + t2
2,t2
R
t1 , 2
1, 1
Note: Payoffs only depend on own type.
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2. Bayesian Nash Equilibrium
2.2 An Illustrative Example
The corresponding strategic form:
A strategy for player 1 is given by (s1 (0), s1 (2)) with s1 (t1 ) ∈ {T, B}.
The corresponding strategy set is
S1 = {(T, T ), (T, B), (B, T ), (B, B)}
A strategy for player 2 is given by (s2 (0), s2 (2)) with s2 (t2 ) ∈ {L, R}.
The corresponding strategy set is
S2 = {(L, L), (L, R), (R, L), (R, R)}
Payoffs are given by the following bi-matrix:
(T, T )
(T, B)
(B, T )
(B, B)
(L, L)
2, 2
1.5, 1.5
2.5, 1.5
2, 1
(L, R)
1.5, 1.5
1, 1
2, 1
1.5, 0.5
(R, L)
1.5, 2.5
1, 2
2, 2
1.5, 1.5
(R, R)
1, 2
0.5, 1.5
1.5, 1.5
1, 1
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2. Bayesian Nash Equilibrium
2.2 An Illustrative Example
(T, T )
(T, B)
(B, T )
(B, B)
(L, L)
2, 2
1.5, 1.5
2.5, 1.5
2, 1
(L, R)
1.5, 1.5
1, 1
2, 1
1.5, 0.5
(R, L)
1.5, 2.5
1, 2
2, 2
1.5, 1.5
(R, R)
1, 2
0.5, 1.5
1.5, 1.5
1, 1
This game has a dominance solution, namely s∗ = ((B, T ), (R, L)).
The dominance solution is the unique Bayesian Nash equilibrium
of the underlying game of incomplete information.
Note: There is a simpler way to see that the game has a
dominance solution by considering the payoff function of the
underlying game of incomplete information.
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2. Bayesian Nash Equilibrium
2.3 A Characterization of Bayesian Nash Equilibria
Proposition
A strategy profile s = (s1 , . . . , sn ) is a Bayesian Nash equilibrium of the
static game G of incomplete information if and only if for all players i
and all types ti ∈ Ti of player i the action si (ti ) solves
"
#
max
∑ ∏ p j (t j )
ai ∈Ai t ∈T
−i
−i
ui (ai , s−i (t−i );ti ,t−i ).
j6=i
What does that mean? And why is it true?
To determine whether a strategy si for player i is a best response
against the strategies used by the other players, it suffices to
check whether every type ti of player i is choosing an optimal
action.
A Bayesian Nash equilibrium results if and only if every type of
every player plays an action which is optimal given the strategies
used by the other players.
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2. Bayesian Nash Equilibrium
2.3 A Characterization of Bayesian Nash Equilibria
Note:
Gibbons (p. 151) defines a Bayesian Nash equilibrium as a
strategy profile satisfying the conditions in this characterization
result.
It does not matter for the analysis whether one defines Bayesian
Nash equilibrium from an ex ante perspective (as I did) or from an
interim perspective (as Gibbons does).
When trying to determine Bayesian Nash equilibria the interim
perspective is usually more intuitive and we will thus use it
throughout the following.
The (only?) advantage of the ex-ante perspective is that it
becomes clear that a Bayesian Nash equilibrium is nothing but a
Nash equilibrium in an appropriately defined game.
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3. Applications
3.1 An Auction
Players: Two bidders i = 1, 2.
Actions: Bids bi ∈ R+ .
Types: Valuations vi ∈ [0, 1].
Probability Distribution over Types: Types are independent and
uniformly distributed on [0, 1].
Payoffs:


vi − bi
ui (b1 , b2 ; vi ) = (vi − bi )/2


0
if bi > b j ,
if bi = b j ,
if bi < b j .
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3. Applications
3.1 An Auction
A strategy for player i is given by bi : [0, 1] → R+ , where bi (vi ) is her
bid if her valuation is vi .
A strategy profile (b1 , b2 ) is a Bayesian Nash equilibrium if for
i = 1, 2 the bid bi (vi ) solves
1
max [vi − bi ] Prob(v j | bi > b j (v j )) + [vi − bi ] Prob(v j | bi = b j (v j ))
bi ≥0
2
for all vi ∈ [0, 1].
Here we just show that (b∗1 , b∗2 ) given by b1 (v1 ) = v1 /2 and
b2 (v2 ) = v2 /2 is a Bayesian Nash equilibrium of this game.
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3. Applications
3.1 An Auction
Need to show that vi /2 solves
max [vi − bi ] Prob(v j | bi > v j /2)
bi ≥0
for all vi ∈ [0, 1].
Note that
(
vi − bi
[vi − bi ] Prob(v j | bi > v j /2) =
2(vi − bi )bi
if bi > 1/2,
if bi ≤ 1/2.
The solution to the maximization problem is determined by the
first order condition (Why?), so that the desired result follows:
2(vi − 2bi ) = 0 ⇒ bi = vi /2.
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3. Applications
3.2 A Double Auction
Players: A buyer b and a seller s.
Actions: A price pb ∈ [0, 1] and a price ps ∈ [0, 1].
Types: Valuations vb ∈ [0, 1] and vs ∈ [0, 1].
Probability Distribution over Types: Types are independent and
uniformly distributed on [0, 1].
Payoffs:
(
vb − (pb + ps )/2
ub (pb , ps ; vb ) =
0
if pb ≥ ps ,
if pb < ps .
(
(pb + ps )/2 − vs
us (pb , ps ; vs ) =
0
if pb ≥ ps ,
if pb < ps .
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3. Applications
3.2 A Double Auction
A strategy for the buyer is given by pb : [0, 1] → [0, 1] where pb (vb )
is her bid if her valuation is vb .
A strategy for the seller is given by ps : [0, 1] → [0, 1] where ps (vs ) is
her ask if her valuations in vs .
A strategy profile is a Bayesian Nash equilibrium if the following
two conditions hold:
1
For all vb ∈ [0, 1], pb (vb ) solves
pb + E[ps (vs ) | pb ≥ ps (vs )]
· Prob(vs | pb ≥ ps (vs ))
max vb −
0≤pb ≤1
2
2
For all vs ∈ [0, 1], ps (vs ) solves
pb + E[ps (vs ) | pb ≥ ps (vs )]
max
− vs · Prob(vs | pb ≥ ps (vs ))
0≤ps ≤1
2
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3. Applications
3.2 A Double Auction
This game has many Bayesian Nash equilibria.
Among those are the so-called one-price equilibria: Let
(
x if vb ≥ x,
pb (vb ) =
0 if vb < x,
(
x if vs ≤ x,
ps (vs ) =
0 if vs > x,
where x ∈ [0, 1]
For any x ∈ [0, 1] the above strategy profile is a Bayesian Nash
equilibrium. (Why?)
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3. Applications
3.2 A Double Auction
There also exists a linear equilibrium given by the strategy profile
1
8
pb (vb ) = vb + ,
12
12
3
8
ps (vs ) = vs + .
12
12
Why is this an equilibrium?
Given the strategy of the seller, we may assume that the buyer’s
bid satisfies pb ∈ [3/12, 11/12] (Why?)
For bids in this range, the expected payoff of a buyer of type vb
who bids pb = a/12 is:
a/12 + (a + 3)/24 a − 3
a−3
vb −
·
= [vb − (a + 1)/16] ·
2
8
8
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3. Applications
3.2 A Double Auction
Maximizing
a−3
[vb − (a + 1)/16] ·
8
with respect to a yields the first order condition
1 a−3
1
−
+ [vb − (a + 1)/16] = 0.
16 8
8
Solving the first order condition yields a = 1 + 8vb , showing that the
buyer’s strategy is a best response.
The argument for the optimality of the seller’s strategy is
analogous.
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4. The Revelation Principle
Roger Myerson
Winner of the Nobel Prize in
Economics 2007
Photo: University of Chicago,
http://experts.uchicago.edu/
experts.php?id=111
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4. The Revelation Principle
4.1 Mechanisms
Consider an economic environment characterized by
The set of players: i = 1, . . . , n.
A set of possible types for each player: Ti .
A probability distribution over the set of possible types: p ∈ ∆(T ).
A set of possible outcomes: X.
A payoff function for each player: Ui : X × T → R
Definition (Mechanism)
A mechanism defines a static game G of incomplete information for
an environment by specifying
A set of possible actions Ai for every player i and
a function f : A → X mapping action profiles into outcomes.
The payoff function for the game G is given by
ui (a,t) = Ui ( f (a),t).
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4. The Revelation Principle
4.2 Direct Mechanisms and the Revelation Principle
Definition (Direct mechanism)
A mechanism is direct if Ai = Ti holds for all i.
In a direct mechanism
1
each player is asked to report his type
2
an outcome x = f (τ) ∈ X is determined as a function of the
players’ reported type profile τ = (τ1 , . . . , τn ).
Proposition (Revelation Principle)
Let s∗ be a Bayesian Nash equilibrium of some mechanism G for a
given environment. Then there exists a direct mechanism such that
truthful revelation, that is the strategy profile given by τi (ti ) = ti for all i
and ti ∈ Ti , is a Bayesian Nash equilibrium of the direct mechanism.
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4. The Revelation Principle
4.2 Direct Mechanisms and the Revelation Principle
The revelation principle is the key result underlying the theory of
mechanism design:
Consider any function g : T → X specifying an outcome for every
possible type profile.
Such a function is called implementable if there is some
mechanism G possessing a Bayesian Nash equilibrium s∗ such
that g(t) = f (s∗ (t)) holds.
The revelation principle implies that rather than considering all
mechanism when trying to determine the set of implementable
functions, it suffices to consider those functions g that are
incentive compatible, i.e., truth-telling is a Bayesian Nash
equilibrium in the corresponding direct mechanism.
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