Lecture Note II-3 Static Games of
Incomplete Information
•
•
•
•
• Also called Bayesian Games
• At least one player in uncertain about another
player’s playoff function
• Example of static game of incomplete information:
A sealed-bid auction
Static Bayesian Game
Bayesian Nash Equilibrium
Applications: Auctions
The Revelation Principle
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Games of incomplete information
– Each bidder knows his or her own valuation for the
good being sold but does not know any other bidder’s
valuation
– Bids are submitted in sealed envelopes (players’ moves
can be thought of as simultaneous)
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Example: Cournot Competition
under Asymmetric Information
• Cournot duopoly model with inverse demand given by
P(Q)=a-Q, where Q=q1+q2
• Firm 1’s cost function is C1(q1)=cq1
• Firm 2’s cost function is C2(q2)=cHq2 with probability θ and
C2(q2)=cLq2 with probability 1- θ
• Information is asymmetric: firm 2 knows its cost function
and firm 1’s, but firm 1 know its cost function and only that
firm 2’s marginal cost is CH with probability θ and CL with
probability 1- θ
• All of this is common knowledge: firm 1 knows that firm 2
has superior information, firm 2 knows that firm 1 know
this, and so on
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Cournot Competition under
Asymmetric Information (cont’)
• Let q2(cH) and q2(cL) denote firm 2’s quantity
choice as a function of its cost, and let q1*
denote firm 1’s single quantity choice
• If firm 2’s cost is high, it will choose q2(cH)to
solve
max[( a − q1 * − q2 ) − cH ]q2
q2
• If firm 2’s cost is low, it will choose q2(cL)to solve
max[(a − q1 * −q2 ) − cL ]q2
q2
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Cournot Competition under
Asymmetric Information (cont’)
• Firm 1 chooses q1* to solve
maxθ[(a − q1 − q2*(cH )) − c]q1 + (1−θ )[(a − q1 − q2*(cL )) − c]q1
q1
• The first-order conditions for these three
optimization problem are
a − q1 * −cH
q2 * (cH ) =
2
a − q1 * −cL
2
θ [( a − q2* (cH )) − c ] + (1 − θ )[( a − q2* ( cL )) − c ]
q1* =
2
q2 * (cL ) =
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Cournot Competition under
Asymmetric Information (cont’)
• The solutions to the three first-order conditions
are
a − 2cH + c 1 − θ
(cH − cL );
+
3
6
a − 2cL + c θ
q2 * (cL ) =
− ( cH − cL )
3
6
q2 * (cH ) =
q1* =
a − 2c + θ cH + (1 − θ ) cL
3
• If information is symmetric (player 1 know
player 2’s cost (cH or cL), say c2 )
q2 **(c2 ) =
Implication
a − 2c2 + c
3
q1 **(c2 ) =
a − 2c + c2
3
q2 *(cH ) > q2 **(cH ); q2 *(cL ) < q2 **(cL )
q1 **(cL ) < q1* < q1 **(cH )
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Static Bayesian Games
Static Bayesian Games (cont’)
• Definition: The normal-form representation of
an n-player static Bayesian game specifies the
players’ action spaces A1,…,An, their type
spaces T1,…Tn, their beliefs p1,…,pn, and their
playoff functions u1,…,un. Player i’s type ti, is
privately known by player i, determines player i’s
payoff function, ui(a1,…,an;ti), and ti is a member
of the set of possible types, Ti. Player i’s belief
pi(t-i|ti) describes i’s uncertainty about the n-1
other players’ possible types , t-i, given i’s own
type, ti. We denote this game by
G={A1,…,An;T1,…,Tn;p1,…,pn;u1,…,un}
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• Example: Cournot game
– The firms actions are their quantity choices, q1 and q2
– Firm 2 has two possible profit or payoff functions
π 2 ( q1, q2 ; cL ) = ( a − q1 − q2 ) − cL  q2
π 2 ( q1, q2 ; cH ) = ( a − q1 − q2 ) − cH  q2
– Firm 1 has only one possible payoff function
π 2 ( q1, q2 ; c ) = ( a − q1 − q2 ) − c  q1
– Firm 2’s type space is T2={cL,cH} and that firm 1’s type
space is T1={c}
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Static Bayesian Games (cont’)
Bayesian Nash Equilibrium
• Definition: In the static Bayesian game
G={A1,..,An;T1,…,Tn;p1,…,pn;u1,…,un}, a strategy for
player i is a function si(ti) where for each type ti in Ti, si(ti)
specifies the action from the feasible set Ai that type ti
would choose if drawn by nature
• When player i’s payoff depend not only on the
actions (a1,…,an) but also on all the types
(t1,…,tn), we write this payoff as
ui(a1,…,an;t1,…,tn)
• Assume nature draws a type vector t(t1,…,tn)
according to the prior probability distribution p(t)
player i’s belief probability pi(t-i|ti) can be
computed using Bayes’ rule
pi (t−i | ti ) =
p (t−i , ti )
=
p (ti )
– i.e. A strategy is a function from types to actions
• Definition: In the static Bayesian game
G={A1,..,An;T1,…,Tn;p1,…,pn;u1,…,un}, the strategies
s*=(s1*,...,sn*) are a (pure strategy) Bayesian Nash
equilibrium if for each player i and for each of i’s types ti in
Ti, si*(ti) solves
p (t−i , ti )
∑
max
p (t −i , ti )
ai ∈ Ai
t− i ∈T− i
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Example1: Battle of Sexes Game
Player 2
Opera
Fight
Opera
2+tc,1
0,0
Fight
0,0
1,2+tp
Player 1
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∑ u (s * (t ),..., s
i
1
1
*
i −1 (ti −1 ), ai , si +1 * (ti +1 ),..., sn * (t n ); t ) pi (t −i
| ti )
t− i ∈T− i
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Example1: Battle of Sexes Game (cont’)
•
•
•
•
•
• Player 1’s payoff if both attend Opera is 2+tc, where
tc is privately known by player 1
• Player 2’s payoff if both attend Fight is 2+tp, where
tp is privately known by player 2
• tc and tp are independent draws from a uniform
distribution on [0,x]
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Static Bayesian game G={Ac,Ap;Tc,Tp;pc,pp;uc,up}
The action spaces are Ac=Ap={Opera, Fight}
The type spaces are Tc=Tp=[0,x]
The beliefs are pc (tp)=pp(tc)=1/x for all tc and tp
Strategies
– Player 1 plays Opera if tc exceeds a critical value c, and
plays Fight otherwise
• With probability (x-c)/x to play Opera
– Player 2 plays Fight if tp exceeds a critical value p, and
plays Opera otherwise
• With probability (x-p)/x to play Fight
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Example 1: Battle of Sexes Game (cont’)
• Player 1 ‘s optimal strategy is play Opera if
Example 1: Battle of Sexes Game (cont’)
−3 + 9 + 4 x
x−c x− p
=
= 1−
x
x
2x
p = c; p 2 + 3 p − x = 0
p
p
p
p
(2 + tc ) + [1 − ] ⋅ 0 ≥ ⋅ 0 + [1 − ] ⋅ 1
x
x
x
x
p
p
(2 + tc ) ≥ 1 −
x
x
tc ≥
x
−3 = c
p
When x is small
lim
x →0
x −c
x− p
−3 + 9 + 4x
2
= lim
= lim(1 −
)=
x →0 x
x →0
x
2x
3
• Player 2 ‘s optimal strategy is play Fight if
c
c
c
c
[1 − ] ⋅ 0 + (2 + t p ) ≥ [1 − ] ⋅ 1 + ⋅ 0
x
x
x
x
c
c
(2 + t p ) ≥ 1 −
x
x
tp ≥
x
−3 = p
c
Opera
Fight
Opera
2+tc,1
0,0
Fight
0,0
1,2+tp
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Mixed strategy Nash Equilibrium
VS. Pure –Strategy Bayesian Nash
Equilibrium
• A mixed- strategy Nash equilibrium is a game of complete
information can be (almost always) be interpreted as a
pure-strategy Bayesian Nash equilibrium in a closely
related game with a little bit of incomplete information
r Opera
1-r Fight
2,1
0,0
0,0
1,2
Two pure Nash equilibrium (Opera,
Opera) and (Fight, Fight)
A mixed strategy Nash Equilibrium
(r,1-r)=(2/3,1/3) and (q,1-q)=(1/3, 2/3)
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Application 1: An auction
• Bayesian game
–
–
–
–
G={A1,A2;T1,T2;p1,p2;u1,u2}
Action spaces AI=[0, ∞ ]
Type space is Ti=[0,1]
Player i’s action is to submit a bid bi and her type is
her valuation vi
– Player i’s payoff function
 vi − bi
if bi > b j

ui (b1, b2 ; v1, v2 ) = (vi − bi ) / 2 if bi = b j

0
f bi < b j

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Application 1: An auction
• First-price, sealed-bid auction
– The bidders simultaneously submit their bids. The
higher bidders wins the good and pays the price she
bided
– In case of a tie, the winner is determined by a flip of a
coin
• Two bidders, labeled i=1,2
• Example: Battle of Sexes
q
1-q
Opera Fight
Implication: player 1 play Opera with probability 2/3
player 2 play Fight with probability 2/3
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– Bidder i has value vi for the good. The two bidders’
valuations are independently and uniformly distributed
on [0,1]
– If bidder i gets the good and plays the price p, then i’s
payoff is v-p
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Application 1: An auction (cont’)
• Player 1’s strategy b1(v1) is a best response to
player 2’s strategy b2(v2) and vice versa
• The pair of (b1(v1),b2(v2)) is a Bayesian Nash
equilibrium if for each vi in [0,1], bi(vi) solves
1
max(vi − bi ) Pr ob{bi > b j (v j )} + (vi − bi ) prob{bi = b j (v j )}
bi
2
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Application 1: An auction (cont’)
Application 1: An auction (cont’)
• Linear equilibrium :bi(vi)=ai+civi
• Symmetric equilibrium: suppose player j adopts
the strategy b(.), and assume b(.) is strictly
increasing and differential
– e.g. b1(v1)=a1+c1v1 and b2(v2)=a2+c2v2
max ui = (vi − bi ) Pr ob{bi > b j (v j )} = (vi − bi ) ⋅
bi
s.t. a j ≤ bi ≤ a j + c j
Note:
Pr ob{bi > b j (v j )} = Pr ob{v j <
bi − a j
cj
(vi + a j ) / 2 if vi ≥ a j
bi (vi ) = 
aj
if vi < a j

}=
bi − a j
cj
bi
Pr ob{bi > b(v j )} = Pr ob{b −1 (bi ) > v j } = b−1(bi ) =vi
bi − a j
First order condition
cj
−b −1(bi ) + (vi − bi )
d −1
−b (bi ) + (vi − bi )
b (bi ) = 0
dbi
−1
ai=aj=0; ci=cj=1/2
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Application 2: A Double Auction
−vi + (vi − b(vi ))
d −1
b (bi ) = 0
dbi
dvi
=0
db(vi )
− vi + ( vi − b ( vi ))
1
b(vi ) = vi
2
bi (vi ) = vi / 2
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b−1(b j ) = v j if b j = b(v j )
max ui = (vi − bi ) Pr ob{bi > b j (v j )}
1
=0
b ′( v i )
b′(vi )vi + b(vi ) = vi
b(vi )vi =
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1 2
vi + k ; b(0) = 0
2
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Application 2: A Double Auction
(cont’)
vb>x>vs
vb
• The buyer’s valuation for the seller’s good is
vb, the seller’s is vs. These valuations are
private information and are drawn from
independent uniform distributions on [0,1]
• The seller names an asking price, ps, and the
buyer simultaneously names an offer price,
pb
• If pb >= ps, then trade occurs at price p=(pb +
ps)/2; if pb < ps then no trade occurs
vb>vs
1
Trade
The buyer offers x if vb>=x
x
The seller demands x if vs<=x
x
1
vs
One price equilibrium
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Application 2: A Double Auction
(cont’)
• A pair of strategies {pb(vb), ps(vs)} is a Bayesian
Nash equilibrium if the following two conditions
hold
Expected price the seller will demand

max  vb −
pb 


max 
ps 

pb + E [ ps ( vs )| pb ≥ ps ( vs ) ] 
 prob{ pb ≥ ps (vs )}
2

ps + E [ pb (vb )| pb (vb ) ≥ ps ]
2

− vs  prob{ pb (vs ) ≥ ps }

Expected price the buyer will offer
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Application 2: A Double Auction
(cont’)
Assume linear BNE
ps (vs ) = as + cs vs ; pb (vb ) = ab + cbvb
pb and ps can be derived by solving
1
a + pb  pb − as

max vb − { pb + s
}
pb 
2
2
 cs
p + ab + cb
1
 a + c − ps
max  { ps + s
} − vs  b b
ps  2
2
cb

Solve FOC simultaneously
2
1
2
1
vb + as
pb = vb +
3
3
3
12
2
1
2
1
ps = vb + (ab + cb )
ps = vs +
3
3
3
4
pb =
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as + cs ⋅ 0 ≤ ps (vs ) ≤ pb
a s + pb
2
ps ≤ pb (vb ) ≤ ab + cb ⋅1
E  p s ( v s )| pb ≥ p s ( v s )  =
E [ pb ( v b ) | p b ( v b ) ≥ p s ] =
p s + a b + cb
2
pb − as
}
cs
p − ab
prob{ pb (vs ) ≥ ps } = 1 − prob{vs ≤ s
}
cb
a + c − ps
= b b
cb
prob{ pb ≥ ps (vs )} = prob{vs ≤
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Application 2: A Double Auction
(cont’)
Application 2: A Double Auction
(cont’)
ps,pb
vb
vb=vs+1/4
1
∵ pb ≥ ps
11/12
pb (vb ) =
2
1
vb +
3
12
ps(vs)
2
1
ps (vs ) = vs +
3
4
vb ≥ vs +
3/4
vb=vs
1
1
4
Trade
pb(vb)
1/4
1/12
1/4
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3/4
1
vs,vb
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Direct Mechanisms
– Static Bayesian games in which each player’s
only action is to submit a claim about his or
her type
– A new static Bayesian game with the same
types spaces and beliefs as original static
Bayesian game but with new action spaces
and new payoff functions
– The playoff functions are chosen so as to
confront each player with a choice of exactly
of its kind
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The Revelation Principle
(Myerson 1979)
• Incentive-compatible direct mechanisms
– Truth-telling is a Bayesian Nash equilibrium
• Theorem (The revelation Principle)
– Any Bayesian Nash equilibrium of any
Bayesian game can be represented by an
incentive-compatible direct mechanism
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vs
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The Revelation Principle (Cont’)
• Direct Mechanisms Design
• Direct Mechanisms
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Linear equilibrium
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– Consider the Bayesian Nash equilibrium
s*=(s1*,…,sn*) in static Bayesian game
G={A1,…,An ;T1,…,Tn;p1,…,pn; u1,…,un}
– A direct mechanism is a new static Bayesian
game is G={T1,…,Tn ;T1,…,Tn;p1,…,pn; v1,…,vn}
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Direct Mechanism
• Implementation of a direct mechanism
– (1) Each player signs a contract that allows the neutral
outsider to dictate the action you will take when we later
play the game G
– (2) Each of the players write down a claim about your type
τi , and submit to the neutral outsider
– (3) The neutral outsider use each player’s type report in
new game τi , together with the player’s equilibrium
strategy from the old game si* , to compute the action the
player would have taken in the equilibrium s* if the
player’s type’s type really were
– (4) The neutral outsider will dictate that each of the
players to take the action the neutral outsider have
computed for the players, and the players will receive the
resulting playoff
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The Revelation Principle (Cont’)
• The Revelation Principle (truth-telling) is a Bayesian
Nash equilibrium (Proof)
– Player i’s action τi and the players’ type report τ=(τ1,..,
τn )
– Player i’s playoff vi (τ, t)=ui[s*(τ),t], where t=(t1,…,tn)
– For each of i’s types ti in Ti, si*(ti) is the best action for i to
choose from Ai, given that other players’ strategies are
(s1*,…, si-1*, si+1* ,…,sn* )
– If other players tell the truth, then when i’s type is ti the
best type to claim to be is ti .The new Bayesian Nash
equilibrium of is for each player i to play the truth-telling
strategyτi (ti)=ti for every ti in Ti
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Homework #3
• Problem set
– 3.2, 3.3, 3,6,3.7,3.8 (from Gibbons)
• Due date
– two weeks from current class meeting
• Bonus credit
– Propose new applications in the context of
IT/IS or potential extensions from examples
discussed
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