1

Complete information games (you know the
type of every other agent, type = payoff)
◦ Nash equilibria: each players strategy is best response to the other
players strategies

Incomplete information game (you don’t know
the type of the other agents)
◦ Game G, common prior F, a strategy profile
𝑠 = 𝑠1 , 𝑠2 , … , 𝑠𝑛 , 𝑠𝑖 : 𝑇𝑖 → actions
actions – how to play game (what to bid, how to answer…)
◦ Bayes Nash equilibrium for a game G and common prior F is a
strategy profile s such that for all i and 𝑡𝑖 ∈ 𝑇𝑖 , 𝑠𝑖 𝑡𝑖 is a best
response when other agents play 𝑠−𝑖 (𝑡−𝑖 )where 𝑡−𝑖 ~𝐹−𝑖
2




There is a distribution Di on the types Ti of
Player i
It is known to everyone
The actual type of agent i, ti 2DiTi is the private
information i knows
A profile of strategis si is a Bayes Nash
Equilibrium if for i all ti and all t’i
Ed-i[ui(ti, si(ti), s-i(t-i) )] ¸ Ed-i[ui(t’i, s-i(t-i)) ]
First price auction for a single item with two
players.
 Private values (types) t1 and t2 in T1=T2=[0,1]
 Does not make sense to bid true value – utility 0.
 There are distributions D1 and D2
 Looking for s1(t1) and s2(t2) that are best replies
to each other
 Suppose both D1 and D2 are uniform.
Claim: The strategies s1(t1) = ti/2 are in Bayes Nash
Equilibrium

Win half the time
t1
Cannot win

If
◦ Other agent bids half her value (uniform [0,1])
◦ I bid b and my value is v




No point in bidding over max(1/2,v)
The probability of my winning is 2b
My Utility is 2𝑏 ⋅ (𝑣 − 𝑏) the derivative is 2𝑣 −
𝑣
4𝑏 set to zero to get 𝑏 =
This means that 𝑏 =
𝑣
2
2
maximizes my utility
5

Bayes Nash equilibria (assumes priors)
◦ Today: characterization


Special case: Dominant strategy equilibria
(VCG), problem: over “full domain” with 3
options in range (Arrow? GS? New: Roberts) –
only affine maximizers (generalization of VCG)
possible.
Implementation in undominated strategies: Not
Bayes Nash, not dominant strategy, but
assumes that agents are not totally stupid
6
7

What happens to the Bayes Nash equilibria
characterization when one deals with arbitrary
service conditions:
◦ A set 𝑆 ⊂ 0,1 𝑛 is the set of allowable characteristic
vectors
◦ The auction can choose to service any subset of
bidders for whom there exists a characteristic
vector 𝑥 ∈ 𝑆.

Prove the characterization of dominant
truthful equilibria.
8
Claim1: If (¯1 ; ¯2 ; : : : ; ¯n ) is a Bayes-Nash equilibrium, (agent i bids ¯i (vi )
when vi is her value), t hen, for all i :
1. T he probability of allocat ion ai (vi ) is monot one increasing in vi .
2. T he expect ed ut ility ui (vi ) (expect ed ut ility of agent i when agent s wit h
value vj bid bj (vj )) is a convex funct ion of vi ,
Z vi
ui (vi ) =
ai (z)dz:
0
3. T he expect ed payment
Z
pi (vi ) = vi ai (vi ) ¡
Z
vi
ai (z)dz =
0
vi
za0i(z)dz:
0
Claim2: If (¯1 ; ¯2 ; : : : ; ¯n ) are such t hat eit her (1) and (2) hold or (1) and (3)
hold t hen (¯1 ; ¯2 ; : : : ; ¯n ) are a Bayes-Nash equilibria.
9
Let ui (w; v) be t he (expect ed) ut ility of agent i when she bids ¯i (w) and her
value is v. Let vi be t he t rue value of agent i .
Choose some i , ¯x all ¯j , j 6
= i , u = ui , v = vi , a = ai .
If ¯1 ; : : : ; ¯n is a Bayes Nash Equilibrium t hen
u(v; v) = va(v) ¡ p(v) ¸ va(w) ¡ p(w) = u(w; v):
But , if t he t rue value of agent i was w we also get t hat
u(w; w) = wa(w) ¡ p(w) ¸ wa(v) ¡ p(v) = u(v; w):
Adding t hese two
(v ¡ w) (a(v) ¡ a(w)) ¸ 0:
If v ¸ w t hen a(v) ¸ a(w). I.e., ai is monot onic for all i .
10
f convex:
f (®x + (1 ¡ ®)z) · ®f (x) + (1 ¡ ®)f (z):
The supremum of a family of convex functions is convex
u(v) = u(v; v) = sup u(w; v) = sup f va(w) ¡ p(w)g:
w
w
Ergo, ui (v) is convex
If f : [a; b] 7
! < is convex t hen it is t he int egral of it 's (right ) derivat ive
Z
f (t) = f (a) +
t
f + (x)dx:
a
where f + (x) is t he right derivat ive at x
11
For every v and w:
u(v) ¸ u(w; v) ¸ va(w)¡ p(w) = (wa(w) ¡ p(w))+ (v¡ w)a(w) = u(w)+ (v¡ w)a(w)
Or,
u(v) ¡ u(w)
¸ a(w):
v¡ w
If v approaches w from above, t he left derivat ive u0(w) ¸ a(w). If v approaches w from below t he right derivat ive u0(w) · a(w). If u is di®erent iable
at w t hen
u0(w) = a(w):
Since a convex funct ion is t he int egral of it 's right derivat ive we have t hat
Zv
u(v) ¡ u(0) =
a(z)dz:
0
12

Since
u(v) = va(v) ¡ p(v)
p(v) = va(v) ¡ u(v)
Z
pi (vi ) = vi ai (vi ) ¡
Z
vi
ai (z)dz =
0
vi
za0i(z)dz:
0
13
From (condit ion 3)
Z
Z
vi
pi (vi ) = vi ai (vi ) ¡
vi
ai (z)dz =
0
za0i(z)dz:
0
it follows t hat
Z
u(v) =
v
a(z)dz:
0
Z
u(w; v) = va(w) ¡ p(w) = (v ¡ w)a(w) +
w
a(z)dz:
0
As ai is monot onic (condit ion 1) t his implies t hat
u(v) ¸ u(w; v):
14

If bidding truthfully (𝛽𝑖 𝑣 = 𝑣 for all i) is a
Bayes Nash equilibrium for auction A then A is
said to be Bayes Nash incentive compatible
15


For every auction A with a Bayes Nash
equilibrium, there is another “equivalent”
auction A’ which is Bayes-Nash incentive
compatible (in which bidding truthfully is a
Bayes-Nash equilibrium )
A’ simply simulates A with inputs 𝛽𝑖 𝑣𝑖
◦ A’ for first price auctions when all agents are U[0,1]
runs a first price auction with inputs 𝑣𝑖 /2

The Big? Lie: not all “auctions” have a single
input.
16

Dominant strategy truthful: Bidding truthfully
maximizes utility irrespective of what other
bids are. Special case of Bayes Nash incentive
compatible.
17


The probability of 𝑎𝑖 (𝑣𝑖 , 𝑏−𝑖 ) is weakly
increasing in 𝑣𝑖 - must hold for any distribution
including the distribution that gives all mass on
𝑏−𝑖
The expected payment of bidder i is 𝑝𝑖 𝑣𝑖 , 𝑏−𝑖
𝑣𝑖
= 𝑣𝑖 ⋅ 𝑎𝑖 𝑣𝑖 , 𝑣−𝑖 − 0 𝑎𝑖 𝑧, 𝑏−𝑖 𝑑𝑧
over internal randomization
Z
pi (vi ) = vi ai (vi ) ¡
Z
vi
ai (z)dz =
0
vi
za0i(z)dz:
0
18


The probability of 𝑎𝑖 (𝑣𝑖 , 𝑏−𝑖 ) is weakly
increasing in 𝑣𝑖 - must take values 0,1 only
The expected payment of bidder i is 𝑝𝑖 𝑣𝑖 , 𝑏−𝑖
𝑣𝑖
= 𝑣𝑖 ⋅ 𝑎𝑖 𝑣𝑖 , 𝑣−𝑖 − 0 𝑎𝑖 𝑧, 𝑏−𝑖 𝑑𝑧
◦ There is a threshold value 𝜃 𝑏−𝑖 such that the item is
allocated to bidder i if 𝑣𝑖 > 𝜃 𝑏−𝑖 but not if 𝑣𝑖
< 𝜃 𝑏−𝑖 .
◦ If i gets item then payment is 𝑣𝑖 ⋅ 𝑎𝑖 𝑣𝑖 , 𝑣−𝑖
𝑣𝑖
− 0 𝑎𝑖 𝑧, 𝑏−𝑖 𝑑𝑧 = 𝑣𝑖 ⋅ 1 − 𝑣𝑖 − 𝜃 𝑣𝑖 = 𝜃 𝑣𝑖
19
Expected Revenue:
◦ For first price auction: max(T1/2, T2/2) where T1 and T2 uniform
in [0,1]
◦ For second price auction min(T1, T2)
◦ Which is better?
◦ Both are 1/3.
◦ Coincidence?
Theorem [Revenue Equivalence]: under very general conditions, every
two Bayesian Nash implementations of the same social choice
function
if for some player and some type they have the same expected
payment then
◦ All types have the same expected payment to the player
◦ If all player have the same expected payment: the expected
revenues are the same

If A and A’ are two auctions with the same
allocation rule in Bayes Nash equilibrium 𝑎𝑖𝐴 𝑣𝑖
𝐴′
= 𝑎𝑖 𝑣𝑖 then for all bidders i and values 𝑣𝑖 we
𝐴
𝐴′
have that 𝑝𝑖 𝑣𝑖 = 𝑝𝑖 𝑣𝑖 .
21


F strictly increasing
If 𝛽𝑖 = 𝛽 is a symmetric Bayes-Nash
equilibrium and strictly increasing in [0,h] then
◦ 𝑎 𝑣 = 𝐹 𝑣 𝑛−1
𝑣
◦ 𝑝 𝑣 = 0 𝑎 𝑣 − 𝑎 𝑤 𝑑𝑤
Z
pi (vi ) = vi ai (vi ) ¡
Z
vi
ai (z)dz =
0
◦ 𝑝 𝑣 =𝐹 𝑣
𝑛−1 𝐸[ max 𝑉
𝑖
𝑖≤𝑛−1
vi
za0i(z)dz:
0
| max 𝑉𝑖 ≤ 𝑣]
𝑖≤𝑛−1
 This is the revenue from the 2nd price auction
22

𝑎 𝑣 =F v
𝑣
0
𝑣
0
n−1

𝑝 𝑣 =

𝑝 𝑣 =

𝑝 𝑣 =𝐹 𝑣
𝑛−1

𝑝 𝑣 =𝐹 𝑣
𝑛−1

𝑎 𝑣 − 𝑎 𝑤 𝑑w
𝐹 𝑣
𝑛−1
−𝐹 𝑤
𝑛−1
𝐸[ max 𝑉𝑖 | max 𝑉𝑖 ≤ 𝑣]
𝑖≤𝑛−1
𝑖≤𝑛−1
𝛽 𝑣
𝛽 𝑣 = 𝐸[ max 𝑉𝑖 | max 𝑉𝑖 ≤ 𝑣] =
𝑖≤𝑛−1
𝑑𝑤
𝑖≤𝑛−1
𝑣
1
0
−
𝐹 𝑤
𝐹 𝑣
𝑛−1
𝑑𝑤
23

n bidders U[0,1]

𝑢 𝑣 = 𝑣𝑎 𝑣 − 𝑝 𝑣 = 𝑣𝑎 𝑣 − 𝑣𝑎 𝑣 −
𝑣
𝑎
0


𝑣
𝑎
0
𝑧 𝑑𝑧 =
𝑧 𝑑𝑧
𝑎 𝑧 = F z n−1 = z n−1
𝑎 𝑧 = 𝑧 𝑛 /𝑛
24