Summary of Extensive-Form Games
We have considered extensive-form games (i.e., games on trees)
� with perfect information
� with imperfect information
� with chance nodes (and both perfect and imperfect information)
We have considered pure, mixed and behavioral strategies.
We have considered Nash equilibria (NE) and subgame perfect
equilibria (SPE) in pure and behavioral strategies.
277
Summary of Extensive-Form Games (Cont.)
For perfect information we have shown that
� mixed and behavioral strategies are equivalent
� there is a pure strategy SPE in both pure as well as behavioral
strategies
� SPE can be computed using backward induction in polynomial
time
For imperfect information we have shown that
� mixed and behavioral strategies are not equivalent in general
(but they are equivalent for games with perfect recall)
� backward induction can be used to propagate values through
"perfect information nodes", but "imperfect information parts"
have to be solved by different means
� solving imperfect information games is at least as hard as
solving games in strategic-form; however, even in the zero-sum
case, most decision problems are NP-hard (for details see
the lecture).
Chance nodes do not interfere with any of the above results.
278
Summary of Extensive-Form Games (Cont.)
Finally, we discussed repeated games. We considered both, finitely
as well as infinitely repeated games.
For finitely repeated games we considered the average payoff and
discussed existence of pure strategy NE and SPE with respect to
existence of NE in the original strategic-form game.
For infinitely repeated games we considered both
� discounted payoff: We have proved that
� one-shot deviation property is equivalent to SPE
� "grim trigger" strategy profiles can be used to implement
any vector of payoffs strictly dominating payoffs for a Nash
equilibrium in the original strategic-form game (Simple Folk
Theorem)
� long-run average payoff: We have proved that all feasible and
individually rational vectors of payoffs can be achieved by Nash
equilibria (a variant of grim trigger)
279
Games of INcomplete Information
Bayesian Games
Auctions
280
Auctions
The (General) problem: How to allocate (discrete) resources among
selfish agents in a multi-agent system?
Auctions provide a general solution to this problem.
As such, auctions have been heavily used in real life, in consumer,
corporate, as well as government settings:
� eBay, art auctions, wine auctions, etc.
� advertising (Google adWords)
� governments selling public resources: electromagnetic
spectrum, oil leases, etc.
� · · ·
Auctions also provide a theoretical framework for understanding
resource allocation problems among self-interested agents: Formally,
an auction is any protocol that allows agents to indicate their interest
in one or more resources and that uses these indications to
determine both the resource allocation and payments of the agents.
281
Auctions: Taxonomy
Auctions may be used in various settings depending on
the complexity of the resource allocation problem:
� Single-item auctions: Here n bidders (players) compete for
a single indivisible item that can be allocated to just one of
them. Each bidder has his own private value of the item in
case he wins (gets zero if he loses). Typically (but not
always) the highest bid wins. How much should he pay?
� Multiunit auctions: Here a fixed number of identical units of
a homogeneous commodity are sold. Each bidder submits
both a number of units he demands and a unit price he is
willing to pay. Here also the highest bidders typically win,
but it is unclear how much they should pay (pay-as-bid vs
uniform pricing)
� Combinatorial auctions: Here bidders compete for a set of
distinct goods. Each player has a valuation function which
assigns values to subsets of the set (some goods are
useful only in groups etc.) Who wins and what he pays?
(We mostly concentrate on the single-item auctions.) 282
Single Unit Auctions
There are many single-item auctions, we consider the following
well-known versions:
� open auctions:
� The English Auction: Often occurs in movies, bidders are
sitting in a room (by computer or a phone) and the price of
the item goes up as long as someone is willing to bid it
higher. Once the last increase is no longer challenged,
the last bidder to increase the price wins the auction and
pays the price for the item.
� The Dutch Auction: Opposite of the English auction, the
price starts at a prohibitively high value and the auctioneer
gradually drops the price. Once a bidder shouts "buy",
the auction ends and the bidder gets the item at the price.
� sealed-bid-auction:
� k-th price Sealed-Bid Auction: Each bidder writes down his
bid and places it in an envelope; the envelopes are opened
simultaneously. The highest bidder wins and then pays
the k-th maximum bid. (In a reverse auction it is the k-the
minimum.) The most prominent special cases are
The First-Price Auction and The Second-Price Auction. 283
Single Unit Auctions (Cont.)
Observe that
� the English auction is essentially equivalent to the second price
auction if the increments in every round are very small.
There exists a "continuous" version, called Japanese auction, where
the price continuously increases. Each bidder may drop out at any time.
The last one who stays gets the item for the current price (which is
the dropping price of the "second highest bid").
� similarly, the Dutch auction is equivalent to the first price auction.
Note that the bidder with the highest bid stops the decrement of
the price and buys at the current price which corresponds to his
bid.
Now the question is, which type of auction is better?
284
Objectives
The goal of the bidders is clear: To get the item at as low price
as possible (i.e., they maximize the difference between their
private value and the price they pay)
We consider self-interested non-communicating bidders that
are rational and intelligent.
There are at least two goals that may be pursued by
the auctioneer (in various settings):
� Revenue maximization
This may lead to auctions that do not always sell the item to the highest
bid
� Incentive compatibility: We want the bidders to
spontaneously bid their true value of the item
This means, that such an auction cannot be strategically manipulated
by lying.
285
Auctions vs Games
Consider single-item sealed-bid auctions as strategic form games:
G = (N, (Bi)i∈N , (ui)i∈N) where
� The set of players N is the set of bidders
� Bi = [0, ∞) where each bi ∈ Bi corresponds to the bid bi
(We follow the standard notation and use bi to denote pure strategies
(bids))
� To define ui, we assume that each bidder has his own private
value vi of the item, then given bids b = (b1, . . . , bn) :
First Price: ui(b) =



vi − bi if bi > maxj�i bj
0 otherwise
Second Price: ui(b) =



vi − maxj�i bj if bi > maxj�i bj
0 otherwise
Is this model realistic? Not really, usually, the bidders are not perfectly
informed about the private values of the other bidders.
Can we use (possibly imperfect information) extensive-form games?
286
Incomplete Information Games
A (strict) incomplete information game is a tuple
G = (N, (Ai)i∈N , (Ti)i∈N , (ui)i∈N) where
� N = {1, . . . , n} is a set of players,
� Each Ai is a set of actions available to player i,
We denote by A =
�n
i=1 Ai the set of all action profiles
a = (a1, . . . , an).
� Each Ti is a set of possible types of player i,
Denote by T =
�n
i=1 Ti the set of all type profiles t = (t1, . . . , tn).
� ui is a type-dependent payoff function
ui : A1 × · · · × An × Ti → R
Given a profile of actions (a1, . . . , an) ∈ A and a type ti ∈ Ti, we
write ui(a1, . . . , an; ti) to denote the corresponding payoff.
A pure strategy of player i is a function si : Ti → Ai. As before, we
denote by Si the set of all pure strategies of player i, and by S the set
of all pure strategy profiles
�n
i=1 Si.
287
Dominant Strategies
� A pure strategy si very weakly dominates s�
i
if for every ti ∈ Ti
the following holds: For all a−i ∈ A−i we have
ui(si(ti), a−i; ti) ≥ ui(s�
i (ti), a−i; ti)
A pure strategy si weakly dominates s�
i
if for every ti ∈ Ti
the following holds: For all a−i ∈ A−i we have
ui(si(ti), a−i; ti) ≥ ui(s�
i (ti), a−i; ti)
and the inequality is strict for at least one a−i
(Such a−i may be different for different ti.)
� A pure strategy si strictly dominates s�
i
if for every ti ∈ Ti
the following holds: For all a−i ∈ A−i we have
ui(si(ti), a−i; ti) > ui(s�
i (ti), a−i; ti)
Definition 88
si is (very weakly, weakly, strictly) dominant if it (very weakly, weakly,
strictly, resp.) dominates all other pure strategies.
288
Nash Equilibrium
In order to generalize Nash equilibria to incomplete information
games, we use the following notation: Given a pure strategy profile
(s1, . . . , sn) ∈ S and a type profile (t1, . . . , tn) ∈ T, for every player i
write
s−i(t−i) = (s1(t1), . . . , si−1(ti−1), si+1(ti+1), . . . , sn(tn))
Definition 89
A strategy profile s = (s1, . . . , sn) ∈ S is an ex-post-Nash equilibrium if
for every t1, . . . , tn we have that (s1(t1), . . . , sn(tn)) is a Nash
equilibrium in the strategic-form game defined by the ti’s.
Formally, s = (s1, . . . , sn) ∈ S is an ex-post-Nash equilibrium if for all
i ∈ N and all t1, . . . , tn and all ai ∈ Ai :
ui(s1(t1), . . . , sn(tn); ti) ≥ ui(ai, s−i(t−i); ti)
289
Example: Single-Item Sealed-Bid Auctions
Consider single-item sealed-bid auctions as strict incomplete
information games: G = (N, (Bi)i∈N , (Vi)i∈N , (ui)i∈N) where
� The set of players N is the set of bidders
� Bi = [0, ∞) where each action bi ∈ Bi corresponds to the bid bi
� Vi = [0, ∞) where each type vi ∈ Vi corresponds to the private
value vi
� Let vi ∈ Vi be the type of player i (i.e. his private value), then
given an action profile b = (b1, . . . , bn) (i.e. bids) we define
First Price: ui(b; vi) =



vi − bi if bi > maxj�i bj
0 otherwise.
Second Price: ui(b; vi) =



vi − maxj�i bj if bi > maxj�i bj
0 otherwise.
Note that if there is a tie (i.e., there are k � � such that bk = b� = maxj bj),
then all players get 0.
Are there dominant strategies? Are there ex-post-Nash equilibria?
290
Second-Price Auction
For every i, we denote by vi the pure strategy si for player i defined by
si(vi) = vi.
Intuitively, such a strategy is truth telling, which means that the player bids his
own private value truthfully.
Theorem 90
Assume the Second-Price Auction. Then for every player i we have
that vi is a weakly dominant strategy. Also, v is the unique
ex-post-Nash equilibrium.
Proof. Let us fix a private value vi and a bid bi ∈ Bi such that bi � vi.
We show that for all bids of opponents b−i ∈ B−i :
ui(vi, b−i; vi) ≥ ui(bi, b−i; vi)
with the strict inequality for at least one b−i.
Intuitively, assume that player i bids bi against b−i and compare his payoff
with the payoff he obtains by playing vi against b−i.
There are two cases to consider: bi < vi and bi > vi.
291
Second-Price Auction (Cont.)
Case bi < vi : We distinguish three sub-cases depending on b−i.
A. If bi > maxj�i bj, then
ui(bi, b−i; vi) = vi − max
j�i
bj = ui(vi, b−i; vi)
Intuitively, player i wins and pays the price maxj�i bj < bi. However, then
bidding vi, player i wins and pays maxj�i bj as well.
B. If there is k � i such that bk > maxj�k bj, then
ui(bi, b−i; vi) = 0 ≤ ui(vi, b−i; vi)
Moreover, if bi < bk < vi, then we get the strict inequality
ui(bi, b−i; vi) = 0 < vi − bk = ui(vi, b−i; vi)
Intuitively, if another player k wins, then player i gets 0 and increasing bi
to vi does not hurt. Moreover, if bi < bk < vi, then increasing bi to vi
strictly increases the payoff of player i.
C. If there are k � � such that bk = b� = maxj bj, then
ui(bi, b−i; vi) = 0 ≤ ui(vi, b−i; vi)
Intuitively, there is a tie in (bi, b−i) and hence all players get 0.
292
Second-Price Auction (Cont.)
Case bi > vi : We distinguish four sub-cases depending on b−i.
A. If bi > maxj�i bj > vi, then
ui(bi, b−i; vi) = vi − max
j�i
bj < 0 = ui(vi, b−i; vi)
So in this case the inequality is strict.
B. If bi > vi ≥ maxj�i bj, then
ui(bi, b−i; vi) = vi − max
j�i
bj = ui(vi, b−i; vi)
Note that this case also covers vi = maxj�i bj where decreasing bi to vi
causes a tie with zero payoff for player i.
C. If there is k � i such that bk > maxj�k bj > vi, then
ui(bi, b−i; vi) = 0 = ui(vi, b−i; vi)
D. If there are k � k�
such that bk = bk� = maxj bj > vi, then
ui(bi, b−i; vi) = 0 = ui(vi, b−i; vi)
293
First-Price Auction
Consider the First-Price Auction.
Here the highest bidder wins and pays his bid.
Let us impose a (reasonable) assumption that no player bids more
than his private.
Question: Are there any dominant strategies?
Answer: No, to obtain a contradiction, assume that si is a very
weakly dominant strategy.
Intuitively, if player i wins against some bids of his opponents, then his bid is
strictly higher than bids of all his opponents. Thus he may slightly decrement
his bid and still win with a better payoff.
Formally, assume that all opponents bid 0, i.e., bj = 0 for all j � i, and
consider vi > 0.
If si(vi) > 0, then
ui(si(vi), b−i; vi) = vi − si(vi) < vi − si(vi)/2 = ui(si(vi)/2, b−i; vi)
If si(vi) = 0, then
ui(si(vi), b−i; vi) = 0 < vi/2 = ui(vi/2, b−i; vi)
Hence, si cannot be weakly dominant.
294
First-Price Auction (Cont.)
Question: Is there a pure strategy Nash equilibrium?
Answer: No, assume that (s1, . . . , sn) is a Nash equilibrium.
If there are v1, . . . , vn such that some player i wins, i.e., his bid si(vi)
satisfies si(vi) > maxj�i sj(vj), then
ui(si(vi), s−i(v−i); vi) = vi − si(vi)
< vi − (si(vi) − ε) = ui(si(vi) − ε, s−i(v−i); vi)
for ε > 0 small enough to satisfy si(vi) − ε > maxj�i sj(vj)
(i.e., player i may help himself by decreasing the bid a bit)
Assume that for no v1, . . . , vn there is a winner (this itself is a bit
weird). Consider 0 < v1 < · · · < vn. Since there is no winner, there are
two players i, j such that i < j satisfying
sj(vj) = si(vi) ≥ max
�
s�(v�)
But then, due to our assumption, sj(vj) = si(vi) ≤ vi < vj and thus
uj(sj(vj), s−j(v−j); vj) = 0 < vj − (sj(vj) + ε) = uj(sj(vj) + ε, s−j(v−j); vj)
for ε > 0 small enough to satisfy sj(vj) + ε < vj.
(i.e., player j can help himself by increasing his bid a bit)
295
Summary
Second Price Auction:
� There is an ex-post Nash equilibrium in weakly dominant
strategies
� It is incentive compatible (players are self-motivated to bid
their private values)
First Price Auction:
� There are neither dominant strategies, nor ex-post Nash
equilibria
Question: Can we modify the model in such a way that First
Price Auction has a solution?
Answer: Yes, give the players at least some information about
private values of other players.
296
Bayesian Games
A Bayesian Game G = (N, (Ai)i∈N , (Ti)i∈N , (ui)i∈N , P) where
(N, (Ai)i∈N , (Ti)i∈N , (ui)i∈N) is a strict incomplete information
game and P is a distribution on types, i.e.,
� N = {1, . . . , n} is a set of players,
� Ai is a set of actions available to player i,
� Ti is a set of possible types of player i,
Recall that T =
�n
i=1 Ti is the set of type profiles, and that A =
�n
i=1 Ai
is the set of action profiles.
� ui is a type-dependent payoff function
ui : A1 × · · · × An × Ti → R
� P is a (joint) probability distribution over T called common
prior.
Formally, P is a probability measure over an appropriate measurable
space on T. However, I will not go into measure theory and consider
only two special cases: finite T (in which case P : T → [0, 1] so that
�
t∈T P(t) = 1) and Ti = R for all i (in which case I assume that P is
determined by a (joint) density function p on Rn
). 297
Bayesian Games: Strategies & Payoffs
A play proceeds as follows:
� First, a type profile (t1, . . . , tn) ∈ T is randomly chosen
according to P.
� Then each player i learns his type ti.
(It is a common knowledge that every player knows his own type but not
the types of other players.)
� Each player i chooses his action based on ti.
� Each player receives his payoff ui(a1, . . . , an; ti).
A pure strategy for player i is a function si : Ti → Ai.
As before, we use S to denote the set of pure strategy profiles.
298
Properties
� We assume that ui depends only on ti and not on t−i. This
is called private values model and can be used to model
auctions. This model can be extended to common values
by using ui(a1, . . . , an; t1, . . . , tn).
� We assume the common prior P. This means that all
players have the same beliefs about the type profile. This
assumption is rather strong. More general models allow
each player to have
� his own individual beliefs about types
� ... his own beliefs about beliefs about types
� .... beliefs about beliefs about beliefs about types
� .....
� (we get an infinite hierarchy)
There is a generic result of Harsanyi saying that
the hierarchy is not necessary: It is possible to extend
the type space in such a way that each player’s "extended
type" describes his original type as well as all his beliefs.
(This does not mean that common prior suffices.)
299
Example: Battle of Sexes
Assume that player 1 may suspect that player 2 is angry with him/her
(the choice is yours) but cannot be sure.
In other words, there are two types of player 2 giving two different
games.
Formally we have a Bayesian Game
G = (N, (Ai)i∈N , (Ti)i∈N , (ui)i∈N , P) where
� N = {1, 2}
� A1 = A2 = {F, O}
� T1 = {t1} and T2 = {t1
2
, t2
2
}
� The payoffs are given by
t1
2
t2
2
t1 :
F O
F 2, 1 0, 0
O 0, 0 1, 2
F O
F 2, 0 0, 2
O 0, 1 1, 0
� P(t1
2
) = P(t2
2
) = 1
2
300
Example: Single-Item Sealed-Bid Auctions
Consider single-item sealed-bid auctions as Bayesian games:
G = (N, (Bi)i∈N , (Vi)i∈N , (ui)i∈N , P) where
� The set of players N = {1, . . . , n} is the set of bidders
� Bi = [0, ∞) where each action bi ∈ Bi corresponds to the bid
� Vi = R where each type vi corresponds to the private value
� Let vi ∈ Vi be the type of player i (i.e. his private value), then
given an action profile b = (b1, . . . , bn) (i.e. bids) we define
First Price: ui(b; vi) =



vi − bi if bi > maxj�i bj
0 otherwise.
Second Price: ui(b; vi) =



vi − maxj�i bj if bi > maxj�i bj
0 otherwise.
� P is a probability distribution of the private values such that
P(v ∈ [0, ∞)n
) = 1. For example, we may (and will) assume that
each vi is chosen independently and uniformly from [0, vmax]
where vmax is a given number. Then P is uniform on [0, vmax]n
.
301
Finite-Type Bayesian Games: Payoffs
For now, let us assume that each player has only finitely many
types, i.e., T is finite.
Given a type profile t = (t1, . . . , tn), we denote by P(t−i | ti)
the conditional probability that the opponents of player i have
the type profile t−i conditioned on player i having ti, i.e.,
P(t−i | ti) :=
P(ti, t−i)
�
t�
−i
P(ti, t�
−i
)
Intuitively, P(t−i | ti) is the maximum information player i may squeeze out of
P about possible types of other players once he learns his own type ti.
Given a pure strategy profile s = (s1, . . . , sn) and a type ti ∈ Ti
of player i the expected payoff for player i is
ui(s; ti) =
�
t−i∈T−i
P(t−i | ti) · ui(s1(t1), . . . , sn(tn); ti)
(this is the conditional expectation of ui assuming the type ti of player i)
302
Example: Battle of Sexes
t1
2
t2
2
t1 :
F O
F 2, 1 0, 0
O 0, 0 1, 2
F O
F 2, 0 0, 2
O 0, 1 1, 0
P(t1
2
) = P(t2
2
) = 1
2
Consider strategies s1 of player 1 and s2 of player 2 defined by
� s1(t1) = F
� s2(t1
2
) = F and s2(t2
2
) = O
Then
� u1(s1, s2; t1) = 1
2 · 2 + 1
2 · 0 = 1
� u2(s1, s2; t1
2
) = 1 and u2(s1, s2; t2
2
) = 2
303
Infinite-Type Bayesian Games: Payoffs
Now assume that for each player i we have Ti = R and thus that
T = Rn
. The concrete type is randomly chosen according to P,
denote by t = (t1, . . . , tn) the corresponding random vector with
distribution P (each ti is a random variable giving a type of player i).
Assume that the type t is absolutely continuous which means that
there is a (joint) density function p such that for all rectangles
R = [a1, b1] × · · · × [an, bn]
P[t ∈ R] =
� b1
a1
· · ·
� bn
an
p(t1, . . . , tn)dtn · · · dt1
Let pi be the marginal density function of ti, i.e.,
pi(ti) =
�
T−i
p(ti, t−i)dt−i
The conditional density of t−i = (t1, . . . , ti−1, ti+1, . . . , tn) conditioned
on ti = ti where pi(ti) > 0 is
p(t−i | ti) = p(t)/pi(ti)
(Here t = (t1, . . . , tn) is a type profile.)
304
Infinite-Type Bayesian Games: Payoffs
Given a pure strategy profile s = (s1, . . . , sn) and a type ti ∈ Ti of
player i, the expected payoff for player i is
ui(s; ti) =
�
T−i
ui(s1(t1), . . . , sn(tn); ti) p(t−i | ti) dt−i
Example: First-Price Auction
Consider the first-price auction as a Bayesian game where the types
of players are chosen uniformly and independently from [0, vmax].
Consider a pure strategy profile v = (v1/2, . . . , vn/2) (i.e., each player
i plays vi/2). What is ui(v; vi) ?
ui(v; vi) = P(player i wins) · vi/2 + P(player i loses) · 0
= P(all players except i bid less than vi/2) · vi/2
=
�
vi
2vmax
�n−1
· vi/2
=
vn
i
2nvn−1
max 305
Risk Aversion
We assume that players maximize their expected payoff. Such
players are called risk neutral.
In general, there are three kinds of players that can be described
using the following experiment. A player can choose between two
possibilities: Either get $50 surely, or get $100 with probability 1
2 and
0 with probability 1
2 .
� risk neutral person has no preference
� risk averse person prefers the first alternative
� risk seeking person prefers the second one
306
Dominance and Nash Equilibria
A pure strategy si weakly dominates s�
i
if for every ti ∈ Ti the following
holds: For all s−i ∈ S−i we have
ui(si, s−i; ti) ≥ ui(s�
i , s−i; ti)
and the inequality is strict for at least one s−i.
The other modes of dominance are defined analogously. Dominant strategies
are defined as usual.
Definition 91
A pure strategy profile s = (s1, . . . , sn) ∈ S in the Bayesian game is
a pure strategy Bayesian Nash equilibrium if for each player i and
each type ti ∈ Ti of player i and every strategy s�
i
∈ Si we have that
ui(si, s−i; ti) ≥ ui(s�
i , s−i; ti)
307
Example: Battle of Sexes
t1
2
t2
2
t1 :
F O
F 2, 1 0, 0
O 0, 0 1, 2
F O
F 2, 0 0, 2
O 0, 1 1, 0
P(t1
2
) = P(t2
2
) = 1
2
Use the following notation: (X, (Y, Z)) means that player 1 plays X ∈ {F, O},
and player 2 plays Y ∈ {F, O} if his/her type is t1
2
and Z ∈ {F, O} otherwise.
Are there pure strategy Bayesian Nash equilibria?
(F, (F, O)) is a Bayesian NE.
Even though O is preferred by player 2, the outcome (O, O) cannot
occur with a positive probability in any BNE.
� To ever meet at the opera, player 1 needs to play O.
� The unique best response of player 2 to O is (O, F)
� But (O, (O, F)) is not a BNE:
� The expected payoff of player 1 at (O, (O, F)) is 1
2
� The expected payoff of player 1 at (F, (O, F)) is 1 308
Second Price Auction
Consider the second-price sealed-bid auction as a Bayesian
game where the types of players are chosen according to
an arbitrary distribution.
Proposition 7
In a second-price sealed-bid auction, with any probability
distribution P, the truth revealing profile of bids, i.e.,
v = (v1, . . . , vn), is a weakly dominant strategy profile.
Proof.
The exact same proof as for the strict incomplete information
games. Indeed, we do not need to assume that the players
have a common prior for this! �
309
First Price Auction
Consider the first-price sealed-bid auction as a Bayesian game with
some prior distribution P.
Note that bidding truthfully does not have to be a dominant strategy.
For example, if player i knows that (with high probability) his value vi
is much larger than maxj�i vj, he will not waste money and bid less
than vi.
So is there a pure strategy Bayesian Nash equilibrium?
Proposition 8
Assume that for all players i the type of player i is chosen
independently and uniformly from [0, vmax]. Consider a pure strategy
profile s = (s1, . . . , sn) where si(vi) = n−1
n vi for every player i and
every value vi. Then s is a Bayesian Nash equilibrium.
Proof. We show that si(vi) = n−1
n vi is the best response to s−i for all i.
Let us fix i and consider a pure strategy s�
i
of player i.
Fix vi and define bi = s�
i
(vi). We show (see the greenboard) that
bi = n−1
n vi maximizes ui(bi, s−i; vi). This holds for all vi, and thus
s�
i
= si is the best response to s−i. �
310
First Price Auction (Cont.)
More generally, assume only that the private values vi are identically
and independently distributed on [vmin, vmax] (this is called
independent private values model). Let F(x) be the cumulative
distribution function of the private value (for each player).
Let us restrict to strictly increasing strategies.
Note that this restriction is quite reasonable, intuitively it means, that
the higher the private value, the higher is the bid.
Then one may show that there is a symmetric Bayesian Nash
equilibrium (s1, . . . , sn) where each si is defined by
si(vi) = vi −
� vmax
vmin
[F(vi)]n−1
dx
[F(vi)]n−1
That is, in particular, the bid is always smaller than the private value.
311
Expected Revenue
Consider the first and second price sealed-bid auctions. For
simplicity, assume that the type of each player is chosen
independently and uniformly from [0, 1].
What is the expected revenue of the auctioneer from these two
auctions when the players play the corresponding Bayesian NE?
� In the first-price auction, players bid n−1
n vi. Thus the probability
distribution of the revenue is
F(x) = P(max
j
n − 1
n
vj ≤ x) = P(max
j
vj ≤
nx
n − 1
) =
�
nx
n − 1
�n
It is straightforward to show that then the expected maximum bid
in the first-price auction (i.e., the revenue) is n−1
n+1 .
� In the second-price auction, players bid vi. However, the revenue
is the expected second largest value. Thus the distribution of the
revenue is
F(x) = P(max
j
vj ≤ x) +
n�
i=1
P(vi > x and for all j � i, vj ≤ x)
Amazingly, this also gives the expectation n−1
n+1 . 312
Revenue Equivalence (Cont.)
The result from the previous slide is a special case of a rather
general revenue equivalence theorem, first proved by Vickrey
(1961) and then generalized by Myerson (1981).
Both Vickrey and Myerson were awarded Nobel Prize in economics for their
contribution to the auction theory.
Theorem 92 (Revenue Equivalence)
Assume that each of n risk-neutral players has independent
private values drawn from a common cumulative distribution
function F(x) which is continuous and strictly increasing on
an interval [vmin, vmax] (the probability of vi � [vmin, vmax] is zero).
Then any efficient auction mechanism in which any player with
value vmin has an expected payoff zero yields the same
expected revenue.
Here efficient means that the auction has a symmetric and
increasing Bayesian Nash equilibrium and always allocates
the item to the player with the highest bid.
313