On Optimal Bidding in Concurrent Auctions
←
→
Page content transcription
If your browser does not render page correctly, please read the page content below
On Optimal Bidding in Concurrent Auctions
Michael N. Katehakis Kartikeya S. Puranam
Rutgers University LaSalle University
Management Science and Information Systems Business Systems and Analytics
100 Rockafeller Road, Piscataway, NJ 08854 1900 Olney Ave Philadelphia PA 19141
USA USA
mnk@rutgers.edu kartys.here@gmail.com
Abstract: Online auctions have become a popular and effective tool for Internet-based E-markets. We investigate
problems and models of optimal adaptive automated bidding in an environment of concurrent online auctions,
where multiple auctions for identical items are running simultaneously. We develop new models for a firm where
its item valuation derives from the sale of the acquired items via their demand distribution, sale price, acquisition
cost, salvage value and lost sales. We discuss possible monotonicity properties for the value function and the
optimal dynamic bid strategies that can be employed in developing efficient computations.
Key–Words: Auctions, Dynamic Bidding, Simultaneous, Multi-period
1 Introduction eral functions of the market’s collective valuation of
the items, cf. [10], [13].
Online auctions have become a popular and effec- Procurement auctions have become a primary
tive tool for Internet-based E-markets. We investigate way with which firms acquire goods and services. The
problems and models of optimal adaptive automated main drivers of this phenomenon are factors such as
bidding in an environment of concurrent online auc- the belief that auctions are an impartial way of setting
tions, where multiple auctions for identical items are the price for an item, and the proliferation of internet
running simultaneously. We develop new models for which has made conducting auctions and participating
a firm where its item valuation derives from the sale of in them efficient. Few examples can be found in the
the acquired items via their demand distribution, sale following papers. (cf. [22], [24], [18], [12]). Other
price, acquisition cost, salvage value and lost sales. recent related work includes [11], [15], [13], [5], [25],
We discuss possible monotonicity properties for the [26], [6] and [21]. For a recent survey we refer to [1].
value function and the optimal dynamic bid strategies Further literature is provided in [17]. To our knowl-
that can be employed in developing efficient compu- edge this is the first study of a multi - period stochas-
tations. tic inventory problem where replenishment is done via
In the current paper we present two models which auctions.
extends our work in [13] and [17]. In the first model The paper is organized as follows. In section 2
the buyer has to acquire a fixed number of items. we present the fixed demand model. In section 3 we
These items can be acquired either at a fixed buy-it- present the variable demand model. In section 4, we
now price or by participating in sequence of concur- present concluding remarks.
rent auctions. In the second model, which is similar
to [17], there are two phases. Each cycle starts with
Phase 1 when items are bought in a sequence of con- 2 Fixed Demand Model
current auctions. This is followed by Phase 2 when
items are sold and any remaining items are salvaged 2.1 Problem Formulation
for a fixed salvage cost. In addition in the present pa- In this model we assume that the buyer has to acquire
per we the allow the demand distribution to be con- a fixed number L of items during the current cycle. To
tinuous and the set of allowable bids to be a compact that end, the buyer participates in a sequence of con-
interval. current sequential auctions of identical items. There
A key notion in both the models is that the proba- are a fixed number N of sequential bidding periods
bility of the buyer winning in an auction is a known each cycle. During each bidding period, the bidder
function of the bid and the number of opponents can bid in a random number of concurrent auctions.
present in an auction. These probabilities are in gen- The number of concurrent auctions during each bid-ding period is a Markov Chain. The transition prob- and an optimal policy is followed thereafter. Note that
ability cyy0 is the probability that number of auctions v(n, x, y) = w(n, x, y; an,x,y ). Finally, let
in the next bidding period is m0 given that the number
∞
of auctions in the current bidding period was m. The X
initial distribution of auctions at the beginning of each u(n, x, y) = cyy0 v(n, y 0 , x)
y 0 =1
period is represented using cy .
Every bidder submits a sealed bid, which is the and
bid in all the concurrent auctions. One of the high- ∞
X
est bidders wins the auction. If there is a tie we as- u1 (n, x, y) = cm v(n, x, y).
sume that the bidder is chosen at random from the set m=1
of highest bidders. The set of bids available for each The MDP equations are:
bidder is the set A = [a0 , ap ]. We assume that a0 = 0
represents the action of not bidding and that ap i the v(n, x, y) = max{w(n, x, y; a)} (1)
a∈A
buy-it-now price. The probability of the buyer win-
ning in an auction is a function of the bid a and the where w(n, x, y; a)
number of concurrent auctions y and is represented
by the function py (a).
− a y py (a) + py (a)u(n − 1, x + y, y)
We next present the Markov Decision Process for-
+(1 − p (a))u(n − 1, x, y) if x 6= L,
y
mulation of the problem described above. =
u(n − 1, L, y) if x = L,
1. The state space X is the set of triplets (n, x, y)
where n (1 ≤ n ≤ N ) represents the number of (2)
remaining bidding periods, x (0 ≤ x ≤ L) repre-
sents the number of items already acquired by the 2.2 Structure of the Optimal Policy
buyer , and y represents the number of concurrent
auctions during the current bidding period. In this section we derive structural properties of the
optimal bidding policy under the following assump-
2. In any state (n, x, y) the bidder can takes one of tions.
the actions from the set A(n, x, y) = [a0 , ap ].
Assumption 1. The function py (a) is twice differen-
3. When an action a ∈ A(n, x, y) is taken in state tiable in a with
(n, x, y) the following transitions are possible. ∂py (a) ∂ 2 py (a)
> 0 and ≤ 0.
∂a ∂a2
i) If x = L the only possible transition is back
to the state (n − 1, L, y 0 ) with probability Assumption 2. The function py (a) is a decreasing
cyy0 ) . function of m, i.e.
ii) If x < L depending the next state is py+1 (a) ≤ py (a).
(n−1, x+y, y 0 ) with probability py (a) cyy0
or (n − 1, x, y 0 ) with probability (1 − Assumption 3. There exists a function G with
P∞
py (a)) cyy0 . G(i) = 1 such that:
i=−∞
4. The following costs are incurred.
(
G(y 0 − y) if y 0 > 1,
cyy0 = P∞ 0
(3)
i) In states (n, L, y) there is no cost. k=y−1 G(k) if y = 1.
ii) In all other states (n, x, y) a cost is in- The rationale behind assumption 1 is that when m
curred only if the item is won in the auc- is constant, the probability of winning is greater when
tion. The expected cost when action a is a is larger and there are diminishing returns for in-
taken is ay py (a). creasing values of a. Assumption 2 implies that for
the same bid a the probability of winning is lower
The MDP equations: We use the following no- when there are more concurrent auctions. Assump-
tation. Let v(n, x, y) denote the value function in tion 3 implies that the probability of there being y 0
state (n, x, y) and an,x,y the optimal action in the concurrent auctions in the next bidding period given
state (n, x, y). Let w(n, x, y; a) be the expected fu- that there are y auctions in the current bidding period
ture reward when action a is taken in state (n, x, y) depends only on the difference y 0 − y.Theorem 1. Under assumptions 1, 2, and 3 the fol- 1. The set X = (n, x, y) is the state space, where
lowing relations hold for all (n, x, y), (n, x, y + 1) ∈ n (0 ≤ n ≤ N ) represents the number of re-
X : maining bidding periods, x represents the num-
ber of items already acquired by the buyer , and
y represents the number of concurrent auctions
∂w(n, x, y; a) during the current bidding period. Note that:
≥ 0, (4)
∂x
i) If n = 0 then y = 0.
ii) State (0, x, 0) represents the state of the
system when all auctions are over.
∂v(n, x, y)
≥ 0, (5) iii) Possible states prior to the start of the N
∂x
auctions, are of the form (N, 0, y).
∂an,x,y 2. In any state (n, x, y) the following action sets
≤ 0. (6)
∂x A(n, x, y) are available.
i) A(0, x, 0) = {a0 }.
ii) A(n, x, y) ∈ [a0 , ap ] for n > 0.
v(n, x, y) ≤ v(n + 1, x, y) (7)
3. When an action a ∈ A(n, x, y) is taken in state
(n, x, y) the following transitions are possible.
an,x,y ≥ an+1,x,y (8) i) If n = 0, then starting from state (0, x, 0)
the next state is (0, x, 0) with probability 1.
ii) If n > 0 then depending on whether or not
the buyer wins the current set of concurrent
v(n, x, y) ≥ v(n, x, y + 1) (9) auctions the next state is (n − 1, x + y, y 0 )
with probability py (a) cyy0 or state (n −
1, x, y 0 ) with (1 − py (a)) cyy0 .
an,x,y ≤ an,x,y+1 (10)
4. When an action a ∈ A(n, x, y) is taken in state
(n, x, y) the expected reward ra (n, x, y) is as fol-
lows.
In the next theorem we establish monotonicity of
ra (0, x, 0) = xd=0 (rd+s(x−d)) fD (d)+
P
the optimal value function and the optimal bid with i) P
∞
respect to the number of opposing bidders m. d=x+1 (rx − δ(d − x))fD (d)
ii) ra (n, x, y) = −a y pm (a) if n > 0.
3 Random Demand Model
3.2 Structure of Optimal Policy
3.1 Problem Formulation Theorem 2. Under assumptions 1, 2, and 3 the fol-
We assume there are two phases in each cycle. Phase lowing relations hold for all (n, x, y), (n, x, y + 1) ∈
1 is identical to what was described in the previous X :
model. During Phase 2, the items bought in the above
auctions are sold in the firm’s market where the de- ∂w(n, x, y; a)
mand distribution D is assumed to be i.i.d. per pe- ≥ 0, (11)
∂x
riod with a continuous probability distribution func-
tion f (·) and a cumulative distribution function F (·).
The sales price is r. Excess demand is lost with a
penalty and unsold items at the end of the period have ∂v(n, x, y)
≥ 0, (12)
same salvage value. Let δ(x) denote the penalty asso- ∂x
ciated with x units of excess demand and let s be the
unit salvage value. We assume that s < r. ∂an,x,y
≤ 0. (13)
We next present the Markov Decision Process for- ∂x
mulation of the problem described above.[3] Burnetas, A. and Katehakis, M.: Efficient esti-
v(n, x, y) ≤ v(n + 1, x, y) (14) mation and control for Markov processes. In:
Decision and Control, Proceedings of the 34th
IEEE Conference on, 2, 1995, pp. 1402–1407.
an,x,y ≥ an+1,x,y (15)
[4] Burnetas, A. and Katehakis, M.: Optimal
adaptive policies for Markov decision pro-
cesses. Mathematics of Operations Research
22(1),1997, pp. 222–255
v(n, x, y) ≥ v(n, x, y + 1) (16)
[5] Chen, Y., Seshadri, S. and Zemel, E.: Sourcing
through auctions and audits. Production and Op-
an,x,y ≤ an,x,y+1 (17) erations Management 17(2), 2008, pp. 121–138
[6] Chen, X., Ghate, A. and Tripathi, A.: Dy-
namic lot-sizing in sequential online retail auc-
tions, European Journal of Operational Re-
4 Conclusion search, 215(1), 2011, pp. 257–267.
In this paper we presented two models of a firm that [7] Day, R.W. and Cramton, Peter.: Quadratic
participates in sequential bidding periods with concur- core-selecting payment rules for combinatorial
rent auctions. In the first model the objective was to auctions. Operations Research 60(3), 2012,
acquire a fixed number of items. In the second model pp. 588–603.
the objective was to acquire items that were to be sold [8] Day, R.W. and Raghavan, S.: Matrix bidding
in a secondary market with a known demand distribu- in combinatorial auctions. Operations Research
tion. In each auction the buyer may acquire a discrete 57(4), 2009, pp. 916–933.
amount of new inventory. [9] Dynkin, E. and Yushkevich, A.: Controlled
We formulated both problems as a Markov De- Markov processes. 235, 1979
cision Processes and established, under pertinent as-
[10] Easley, D. and Kleinberg, J.: Networks, Crowds,
sumptions, that the optimal value function is a de-
and Markets: Reasoning About a Highly Con-
creasing function of the number of remaining auc-
nected World. Cambridge University Press,
tions, increasing function of the number of opponents
2010.
and decreasing function of the inventory on hand. We
also prove that the optimal bid is an increasing func- [11] Federgruen, A. and Wang, M.: Monotonicity
tion of the number of remaining auctions, a decreasing properties of a class of stochastic inventory sys-
function of the number of opponents and an increas- tems. Annals of Operations Research 194, 2012,
ing function of the inventory on hand. These prop- pp. 1–32.
erties can be used to obtain efficient computational [12] Heezen, J. and Baets, W.: The impact of elec-
methods. This model can be extended in several di- tronic markets: the case of the dutch flower auc-
rections. Some are discussed in [17]. We note that the tions. The Journal of Strategic Information Sys-
current model does not capture aspects of “collusion” tems 5 (4), 1996, pp. 317–333.
between bidders such as the bidders placing coordi- [13] Katehakis, M. and Puranam, K.: On bidding for
nated bids in order to acquire the items at lower cost. a fixed number of items in a sequence of auc-
tions. European Journal of Operational Research
Acknowledgements: The research was supported by
222 (1), 2012, pp. 76–84.
the University of ABC and in the case of the first au-
thor, it was also supported by the Grant Agency of [14] Katehakis, M. and Robbins, H.: Sequential
DEF (grant No. 000/05/ 0000). choice from several populations. Proceedings of
the National Academy of Sciences of the United
States of America 92(19), 1995, pp. 84–85.
References: [15] Katehakis, M. and Smit, L.: On computing opti-
[1] Bergemann, D. and Said, M.: Dynamic auctions: mal (Q, r) replenishment policies under quantity
A survey. Cowles Foundation Discussion Paper discounts. Annals of Operations Research 200
No. 1757R, 2010. (1), 2012, pp. 1–20.
[2] Boros, E., Lei, L., Zhao, Y. and Zhong, H.: [16] Katehakis, M. and Smit, L.: A successive lump-
Scheduling vessels and container-yard opera- ing procedure for a class of Markov chains.
tions with conflicting objectives. Annals of Op- Probability in the Engineering and Informational
erations Research 161, 2008, pp. 149–170. Sciences 26(4), 2012, pp. 483–508.[17] Katehakis, M.N. and Puranam, K.S.: On optimal
bidding in sequential procurement auctions. Op-
erations Research Letters 40 (4), 2012, pp. 244–
249.
[18] Kleijnen, J. and van Schaik, F.: Sealed-bid auc-
tion of dutch mussels: statistical analysis. Inter-
national Journal of Production Economics, 132
(1), 2011, pp. 154-161.
[19] Lippman, S.A., Ross, S.M. and Seshadri, S.: A
weakest link marked stopping problem. Journal
of Applied Probability 44 (4), 2007, pp. 843-
851.
[20] Nahmias, Steven.: Perishable inventory theory:
A review. Operations Research 30 (4), 1982,
pp. 680–708.
[21] Pekec, A. and Rothkopf, M.H.: Combinatorial
auction design. Management Science 49 (11),
2003, pp. 1485–1503.
[22] Scholer, M.: The world’s first internet coffee
auction: Design, implementation and lessons
learned. Electronic commerce for development,
2002, p. 121.
[23] Shi, J., Katehakis, M.N. and Melamed, B.: Mar-
tingale methods for pricing inventory penalties
under continuous replenishment and compound
renewal demands. Annals of Operations Re-
search, 208 (1), 2013, pp. 593-612.
[24] Temu, A., Winter-Nelson, A. and Garcia, P.:
Market liberalisation, vertical integration and
price behaviour in tanzania’s coffee auction. De-
velopment Policy Review 19 (2), 2001, pp. 205–
222.
[25] Yang, S., Yang, J. and Abdel-Malek, L.: Sourc-
ing with random yields and stochastic demand:
A newsvendor approach. Computers & opera-
tions research 34 (12), 2007, pp. 3682–3690.
[26] Zhou, S., Tao, Z., Zhang, N. and Cai, G.:
Procurement with reverse auction and flexible
noncompetitive contracts. Available at SSRN
2012130, 2012.You can also read
