Asymmetric Price Mechanism for Symmetric Buyers: A
Two-period Model
Shraman Banerjee
Bo Chen
March 25, 2017
Abstract
We consider a two-period posted price mechanism of a seller to sell a single unit of a good
to a number of buyers before a deadline. Even though the buyers are ex-ante symmetric to
the seller, we propose a new asymmetric mechanism of di¤erential pricing and show that this
mechanism generates strictly higher revenue than the optimal mechanism (which is symmetric)
established in the literature. Instead of the tie being broken randomly, we use a deterministic
tie-breaking allocation rule to generate the asymmetry here.
JEL Classi…cation:
Keywords: Dynamic Pricing, Asymmetric Mechanism, Non-Anonymity
1
Introduction
From the standard microeconomic theory we know that when the buyers are ex-ante symmetric to
the seller, he treats them symmetrically, e.g. a single monopoly price charged by a monopolist.
Asymmetric treatment by the seller occurs only when there is some form of ex-ante heterogeneity
among the buyers, e.g. price discrimination, asymmetric auctions (Maskin and Riley, 2000). In this
paper we propose an asymmetric dynamic price-posting mechanism for a seller under the context
that the buyers are ex-ante symmetric. Our paper shows that if we allow the seller to use asymmetric
mechanisms, under a mild and payo¤ irrelevant assumption that the buyers are non-anonymous to
the seller, it strictly increases the seller’s revenue compared to a benchmark symmetric mechanism.
We show this in the context of revenue-management pricing which is the optimal pricing strategy
in many industries with some distinct features as the following: Firstly, there is only a …xed quantity
of good that the seller can sell, i.e., he has capacity constraint on the quantity of good that he can
sell. Secondly, there is a …xed deadline within which the seller has to sell the good.
Our model follows closely to that of Hörner and Samuelson (2011) which derives the optimal
posted price mechanism for symmetric buyers in the context of revenue-management pricing. We
show that the Hörner and Samuelson mechanism, which is a symmetric mechanism, is not optimal.
Corresponding Author: O.P. Jindal Global University, Sonipat-Narela Road, Sonipat, Haryana 131001, NCR of
Delhi, India Email: [email protected]
1
Symmetric treatment of symmetric buyers acts as a binding constraint to the seller, and relaxing
this constraint strictly increases the seller’s revenue.
Our model is a two-period model where the seller posts prices in each period. In the …rst period,
the buyers can either accept the price and end the game, or reject in which case the game moves to
the last period. We analyze the mechanisms under two di¤erent scenarios. The …rst case is when
the seller cannot pre-commit to any …xed price paths, so each price is sequentially rational and the
equilibrium we focus on is the perfect Bayesian equilibrium. The prices in the …rst period act as
a dynamic screening device in order to optimally price discriminate the buyers sequentially. Given
the history the seller updates his belief about the buyer types according to the Bayes’ rule. The
second case is when the seller can pre-commit to future prices. In both the cases we show that
asymmetric mechanism is more revenue-generating than the symmetric mechanism. We also derive
the two-period price paths under both the scenarios.
The di¤erence in our mechanism is in the tie-breaking allocation rule. While Hörner and Samuelson (2011) breaks the tie randomly, our mechanism allocates the good to the buyer with the highest
price charged among the set of buyers who accepted in the current period. It exploits the increased
competition among the buyers due to the di¤erential pricing and thus can potentially generate a
higher revenue than the symmetric mechanism.
Only a couple of papers in the literature have tried to address this asymmetric treatment in the
case of ex-ante symmetry. Ghosh(2008) (in case of monopoly pricing) and Kotowski (2014) (in case
of auctions) have dealt with this issue previously. We can say that our paper is a dynamic version
of the former with a more general characterization. Also, our …rst scenario where the seller cannot
pre-commit adds to the literature on dynamic mechanisms without commitment. Although with
commitment dynamic mechanism literature is extensive, but very few studies have been conducted
for dynamic mechanisms without commitment (e.g. Hörner and Samuelson (2011), Skreta(2015)
etc.). Our paper sheds light on this growing literature.
2
2.1
2.1.1
The Model
Without Commitment Solution
Anonymous Price Posting Mechanisms
This is a benchmark model similar to that in Hörner and Samuelson (2011). There is a seller who
wants to sell single unit of an indivisible good to two buyers by the end of two periods by posting
prices in each period. 1 We denote by time period t; the number of periods remaining in the game,
t = 1; 2:
In the …rst period the seller announces a price p2 : The buyers simultaneously decide whether to
accept or to reject the price. If only one buyer accepts the price, the game ends and the good is sold
to that buyer at price p2 : If no one accepts the good, the game moves to the last period. If both
buyers accept, the tie is broken randomly at the announced price.
1 The model can be easily extendable for n
in this model.
2 buyers, but for simpli…cation we restrict our attention to 2 buyers
2
Each buyer has private valuation vi drawn i.i.d. from unif orm[0; 1]. A buyer of valuation vi
who obtains the good at price p derives a payo¤ of (vi p): The seller’s payo¤ is equal to the price
p at which the good is sold.2
The solution concept we adopt is the perfect Bayesian equilibrium.3 We assume that the seller
does not have any commitment power and each price is sequentially rational given the previous
history and the belief about the optimal continuation payo¤. This is an anonymous price posting
mechanism where the seller posts a single price in each period and the buyers use symmetric strategies. The strategy of a buyer depends only on his valuation but not on his identity. In Subsection
2.2, we shall consider a non-anonymous price posting mechanism where the seller o¤ers di¤erent
prices to di¤erent buyers.
A buyer with higher valuations is more anxious to accept earlier as it is possible that the other
buyers may “snatch” the good earlier, leaving him with zero payo¤. Given our focus on symmetric
perfect Bayesian equilibrium, the buyers who accept at time period t are those whose valuations
exceed a critical threshold vt and those who reject are below the threshold vt . For the rest of
this section, we shall follow Hörner and Samuelson (2011) closely in describing the buyers’and the
seller’s problems explicitly. Our next lemma, taken from Hörner and Samuelson (2011) illustrates
the seller’s posterior beliefs after the …rst-period history of no sales.
Lemma 1. (Hörner and Samuelson (2011)) Let n = 2. Fix an equilibrium, and suppose the
last period is reached without the earlier price having been accepted. Then the seller’s posterior
belief is that the buyers’ valuations are identically and independently drawn from the distribution
U (v)=U (v2 ), with support [0; v2 ], for some v2 2 (0; 1].
In the last period a buyer accepts a price if it is below or equal to his valuation. In the …rst
period for a buyer i with valuation x and if the critical threshold in the …rst period is x2 , then his
expected payo¤ from accepting p2 is :4
1
1
v22 (v p2 )
:
v2
2
On the other hand, if buyer i with type v waits for the last period to accept price p1 , his expected
payo¤ is
1
v2
1
v1
v2
v1
v2
2
(v
p1 )
2
:
The …rst term v2 is the probability that the good is still available in the last period, i.e., his
opponent has not already bought the good in the …rst period. vv12 is the conditional probability that
an opponent’s valuation is less than v1 , given that his valuation is below v2 : All other terms are
analogous to the expression in (1). If this critical threshold v2 is interior, then a v2 -type buyer is
2 Without
loss of generality, we assume that all parties discount future payo¤s using the same discount rate of 1.
of such an equilibrium in our setting is similar to that in Horner and Samuelson (2011), and follows
from standard arguments (see Chen (2012)).
1
X
1
1
4 This expression is derived from the binomial expression
(1 v2 )j v21 j (v p2 ) where j+1
is the probability
j+1
3 Existence
j=0
that buyer i receives the good when j other buyers accept the price p2 :
3
indi¤erent between accepting price p2 or waiting for the next period to accept p1 . In other words,
we have
2
v1
1
v2
(v p1 )
1 v22 (v p2 )
= v2
:
(1)
1 v2
2
1 vv12
2
Writing in the recursive structure, the seller’s optimization problem in the …rst period is to
maximize the following expected payo¤, subject to the buyer’s indi¤erence condition, :
2 (v2 )
= 1
v22 p2 + v22
1 (v1 );
where 1 v22 is the probability that at least one buyer accepts price p2 and 1 (v1 ) is the continuation payo¤. Since this is a …nite horizon problem, we apply a backward induction process to solve
for the seller’s optimal sequence of prices.
2.1.2
Non-Anonymous Price Posting Mechanisms
In this subsection we introduce a non-anonymous price posting mechanism. The key di¤erence here
is that the price o¤ered to each buyer in each period is di¤erent even though the buyers are ex-ante
symmetric. This typically requires that the seller can identify the two buyers. While this is not
a particularly strong assumption (i.e., the seller can simply assign each buyer a particular number
that will be …xed throughout the game), such mechanisms are feasible in settings where the number
of the buyers is not too large.
From the buyers’perspectives, if di¤erent buyers are treated di¤erently, the strategies adopted
by the di¤erent buyers are di¤erent. As a result, the equilibrium we shall focus on is an asymmetric
perfect Bayesian equilibrium where the buyers use asymmetric strategies where the strategy of a
buyer depends not only on the type of the buyer but also on the buyer’s identity.
One important issue is the tie-breaking rule when both buyers accept the o¤ers. In the anonymous
price posting mechanism, the tie was broken randomly. But here the seller can modify this to a
deterministic allocation rule. For example, he can specify that among all the accepting buyers,
the buyer with the highest price o¤er is allocated the good with probability 1. It should be noted
however that there are many other tie-breaking rules the seller can adopt and the above tie-breaking
rule is not necessarily the revenue-maximizing rule in the entire dynamic game. We shall however
restrict our attention to such an intuitive tie-breaking rule and show that such a rule su¢ ces for the
non-anonymous price posting mechanism to generate strictly higher expected payo¤ for the seller
than an anonymous price posting mechanism.
Suppose at each period t = 1; 2; the seller sets two di¤erent prices pt (buyer 1) and qt (buyer 2)
to the two buyers, and without loss of generality, we assume that pt > qt : We denote ut (buyer 1)
and vt (buyer 2) to be the critical valuations at time t for the two buyers respectively.
Here the indi¤erence conditions that pin down the corresponding threshold types will be di¤erent
for the two buyers thus generating two di¤erent threshold types.
In the …rst period; the incentives for a u2 -type of buyer 1 is given by the indi¤erence condition
(u2
p2 ) = v2 (u2
4
p1 ):
Notice that buyer 1 can get the good with certainty if he accepts the o¤er. If he rejects, the
game goes to the last period only in the event that buyer 2 has also rejected his own price o¤er in
period 1.
Similarly the incentives for a v2 -type of buyer 2 is given by the indi¤erence condition
(v2
u1
(v2
u2
q2 ) =
q1 ):
The seller’s optimization problem in the …rst period is to choose p2 and q2 to maximize
given the buyers’indi¤erence conditions and the continuation payo¤ 1 (v1 ).
maxp2 ;q2
=
[(1
2 (u2 ; v2 )
2 (u2 ; v2 )
u2 )p2 + u2 (1
v2 ) q2 + u2 v2
1 (u1 ; v1 )]
The seller gets the lower price q2 if buyer 1 rejects the o¤er, i.e., buyers 1’s valuation is lower
than his own threshold level, while buyer 2’s valuation is above his threshold level in period t. Also,
the seller gets the higher price p2 if buyer 1’s valuation is higher than his own threshold level no
matter what the valuation of buyer 2 is. If both have their valuations below their own threshold
levels, the game moves on to the next period.
As we shall see in the next section, if the seller is allowed to treat di¤erent buyers di¤erently, the
seller might be tempted to do so to increase his expected payo¤.
2.1.3
Revenue Comparison
In this section we compare the revenues between the symmetric and the asymmetric mechanisms. It
is clear that posting di¤erential pricing makes the seller weakly better-o¤, since pt = qt is trivially
possible. But what we show is that moving from the symmetric to an asymmetric mechanism can
strictly increase the seller’s expected revenue even for ex-ante symmetric buyers. The following table
shows the prices and the revenues of the seller under both the mechanisms for a two-period case.
Let A and N A be the revenues to the seller under anonymous and non-anonymous mechanisms
respectively.
There are some interesting observations to be noted here. Firstly, N A = 0:404 > A = 0:4,
/ i.e.
the di¤erential pricing strictly increases the revenue of the seller.
Anonymous Mechanism
Non-Anonymous Mechanism
Price in period 2
0:58 (p2 )
0:62 (p12 ); 0:56 (p22 )
Price in period 1
0:48 (p1 )
0:55 (p11 ); 0:40 (p21 )
Expected payo¤
0:4
0:404
This is formally stated in the following proposition.
Proposition 1: Under dynamic price mechanisms without commitment, if N A and A are the
expected revenues of the seller in cases of non-anonymous and anonymous mechanisms respectively,
then N A > A :
5
Figure 1: Single Price Path for Symmetric Mechanism and Two Price Paths for Asymmetric Mechanism for buyers 1 and 2 respectively
Denote pt as the optimal price in period t under the anonymous mechanism, t 2 f1; 2g and
denote pit as the optimal price in period t for buyer i 2 f1; 2g, t 2 f1; 2g under the non-anonymous
mechanism. A …rst observation from the table is that the two prices in each period under the nonanonymous mechanism are a “spread” from the corresponding price under the anonymous mechanism, i.e., p1t > pt > p2t for each t: One possible explanation maybe a risk sharing motive for the
seller: The seller charges a higher price to buyer 1, while at the same time, a lower price to buyer 2
as a safer option in case the high price gamble does not work out.
A second useful observation is that jpit pt j is decreasing in t for each t = 1; 2 and i = 1; 2. In
other words, in the earlier period the spread of the prices is less than that in the …nal period. In
the …nal period, the seller tends to diversify even more (i.e., reducing the “risk”) so that it is more
likely that at least one of the buyers accepts the good in the …nal period. To be more explicit, the
price path for buyer 2 in the non-anonymous mechanism is steeper than that of buyer 1 in the nonanonymous mechanism, while the slope of the line of the price path for the anonymous mechanism
lies in the mid-way. In addition, we can see that the price di¤erence (between the two mechanisms)
for buyer 1 is relatively higher in the …rst period than that of buyer 2, i.e., jp12 p2 j > jp22 p2 j;
while in the …nal period the price di¤erence for buyer 2 is higher, i.e. jp11 p1 j < jp21 p1 j:
2.2
Commitment Solution
In this section we assume that the seller can pre-commit to the entire price path. In the commitment case we again examine the e¤ect of non-anonymous over anonymous mechanism on revenue
generation. In the case of anonymous mechanism, the seller’s maximization problem is
maxp2
2 (v2 )
v22 )p2 + (v22
= (1
6
v12 )p1
s:t:
1
1
v22
(v2
v2
p2 ) =
v22
v2
v12
(v2
v1
p1 )
Similarly, in the case of non-anonymous mechanism, the seller’s problem is
maxp2 ;q2
2 (u2 ; v2 )
= (1
u2 )p2 + u2 (1
v2 )q2 + u2 v2 (1
u1
)p1 + u2 v2 u1 (1
u2
v1
)q1
v2
s:t:
(u2
p2 )
(v2
q2 )
= v2 (u2
u1
=
(v2
u2
p1 )
q1 )
The following table compares the revenues and the price paths of the two-period commitment
case.
Anonymous Mechanism Non-Anonymous Mechanism
Price in period 2
0:59 (p2 )
0:642 (p12 ); 0:584 (p22 )
Price in period 1
0:527 (p1 )
0:516 (p11 ); 0:518 (p21 )
Expected payo¤
0:407
0:409
The …rst observation is that even in the commitment case, the non-anonymous mechanism genC
/ Figure 2 shows the
erates more revenue than the anonymous one, i:e: C
N A = 0:409 > A = 0:407:
price paths for the buyers in the commitment case. This is stated formally in Proposition 2:
C
Proposition 2: Under dynamic price mechanisms with commitment, if C
N A and
A are the
expected revenues of the seller in cases of non-anonymous and anonymous mechanisms respectively,
C
then C
NA > A :
Similar to the non-commitment case, there is a single price path for the anonymous mechanism
and two price paths for the non-anonymous mechanism.
But the di¤erence is that the two asymmetric price paths cross each other, i.e. buyer 1, who was
o¤ered the higher price in the …rst period, is o¤ered the lower price in the second, and vice versa
for buyer 2: By this mechanism the seller pre-commits to a punishment threat to buyer 2 that if
he rejects the …rst period price, he will be charged relatively higher price than buyer 1 in the last
period. For buyer 1; since he is already o¤ered a higher price in the …rst period, a rejection from him
makes the seller’s posterior belief that his valuation is truly below the threshold is su¢ ciently high.
Finally the terminal price for both the symmetric and the asymmetric price paths are well above
their respective counterparts in non-commitment case. In the non-commitment case, the seller in
the …rst period cannot pre-commit to keeping the terminal prices high enough and his future self is
forced to lower down the prices in the last period.
References
7
Figure 2: Single Price Path for Symmetric Mechanism and Two Price Paths for Asymmetric Mechanism for buyers 1 and 2 respectively
Ghosh, Parikshit (2008), "Price Discrimination As Portfolio Diversi…cation." Economics Bulletin,
4(5), 1-9
Horner, Johannes, and Larry Samuelson (2011). "Managing Strategic Buyers," Journal of Political Economy, 119, 379-425.
Kotowski, Maciej H. (2014). "On Asymmetric Reserve Prices" Mimeo
Maskin, Eric and John Riley (2000). "Asymmetric Auctions", Review of Economic Studies, 67,
413-438.
Myerson, Roger (1981). "Optimal Auction Design", Mathematics of Operations Research, 6(1),
58-73.
Skreta, Vasiliki (2015). "Optimal Auction Design under Non-commitment", Journal of Economic
Theory, 159(B), 854-890.
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