Transcript Auctions On The Cognitive Packet Network

Incertezza e Sistemi Dinamici in Rete
Erol Gelenbe
Professor in the Dennis Gabor Chair
Head of Intelligent Systems and Networks
Department of Electrical and Electronic Engineering
Imperial College London
http://www.ee.ic.ac.uk/gelenbe
Vision: The World’s Economy is governed by Electronic
Economic Transactions in Cognitive Networks
Auction: Formal Mechanism that Governs Decisions for
Economic Transactions or Resource Allocation and Exchange
Cognitive Network: A Computer-Communication Network
where Resource Allocation including Routing is Achieved by
Adaptive Procedures that Optimise QoS, Profit or Other
Criteria
Vision:
The World’s Economy is a “Chain Reaction” of Electronic Economic
Transactions
Users and Services, Buyers and Sellers, are Agents with
Interchangeable Roles
Computer Networks are the Infrastructure of the World Economy
Networks are Becoming Autonomic and Cognitive
Auctions – Economic mechanisms which have been studied
both in Economic Theory and Computer Science
- Guo, X. 2002. An optimal strategy for sellers in an online auction. ACM Trans. Internet Tech. 2 (1): 1–13.
- Hajiaghayi, M. T., Kleinberg, R., and Parkes, D. C. 2004. Adaptive limited-supply online auctions. Proc. 5th
ACM Conference on Electronic Commerce, May 17-20, 71–90.
- McAfee, R. P. and McMillan, J. 1987. Auctions and bidding. J. Economic Literature 25: 699–738.
- Milgrom, P. R. and Weber, R. 1982. A theory of auctions and competitive bidding. Econometrica 50: 1089–
1122.
- Shehory, O. 2002. Optimal bidding in multiple concurrent auctions. International Journal of Cooperative
Information Systems 11 (3-4): 315–327.
- Gelenbe, E. 2008. Networked Auctions. In press ACM Trans. Internet Tech.
Cognitive Networks – Systems studied in recent EU projects
- Gelenbe, E., R. Lent, Z. Xu. 2001. Measurement and performance of a cognitive packet network”, Computer
Networks, 37: 691-791.
- Gelenbe, E. 2003. Sensible decisions based on QoS. Computational Management Science 1 (1): 1–14.
- Dobson, S. et al. 2006. Autonomic Communications. ACM Trans. on Autonomous and Adaptive Systems, 1
(2): 223-259.
Auctions
Cognitive Networks
Research carried out in the
EU FP6 CASCADAS Project with Telecom Italia, Univ. of Modena,
Fokus-Fraunhofer and other partners
and the UK ALADDIN Project with
Southampton University, Imperial College, Oxford, Bristol
UK Engineering and Physical Sciences Research Council
BAE Systems Ltd & Selex
• Study the Dynamics of Agent Interactions in Deterministic or Stochastic, Collaborative or
Adversarial Environments
• Compute and Predict the Performance, Stability and Robustness of Resulting (Emergent)
Behaviours
• Obtain Parameters or Design Algorithms that Optimise Desired Outcomes
• When Information is Unavailable or Develops over Time, Design Self-Adaptive and Learning
Schemes to Approach or Approximate Desired Outcomes
Auctions have Many Buyers (Bidders) and
Sellers – Leading to Interesting Research
Questions
-
Mechanisms that create incentives (encouraging certain behaviours)
Outcome of collective behaviours
Coordinated buying and selling (cartels, trusts)
Rational Bidders
Adversarial Behaviours (competitors)
Learning from collective behaviour (observing the buyers and sellers)
- Modelling collective behaviour
- Effect of Network QoS on Economic Considerations
- Malicious Behaviours & Protecting the Information Infrastructure
Formalisation of an Auction
•Decision Framework: An auction System
•The value of the good for the bidders is a r.v. V, whose prob. distribution
p(v)=P[V=v], which may be unknown to the seller
•Buyers will not bid above the value they associate with the good, but
V=infinity is possible ( .. I am willing to buy it at any price .. )
•The seller observes the bids, and after each bid waits for some time before
accepting the bid; a new bid may arrive in the meanwhile, and the process
repeats itself
•How and when should the seller accept the bid?
•What is the probabilistic outcome of such a system, for instance in terms of
the expected price that the good brings in, or the time it takes to sell the
good, or the income generated per unit time?
•How can the seller learn the value of the good and act accordingly?
•How can bidders also adapt their behaviour to get the best price?
•How can buyers take advantage of multiple Networked Auctions?
The Secretary or Sultan’s Dowry Problem .. Related but Different
Martin Gardner, Scientific American, 1960:
•A sequence of candidates show up, each of value or quality C1, C2, .. , Cn, .. which
are r.v.’s
•The buyer’s purpose is to select one of these whose quality is close to the
maximum quality
•The buyer observes the sequence for a finite time, hoping to wait long enough to
select the best .. and selects the k-th, after which the decision is irrevocable
•What is the probability that the one selected is the optimum?
•It has been shown that the optimum outcome is to select the best with probability
1/e
•Y.S. Chow et al., Israel J. Math. 2, 81-90, 1964
•S.R. Finch “Optimal stopping constants”, in Mathematical Constants, Cambridge
Univ. Press, 361-363, 2003
System Description
• Ascending auction with n+1 bidders
• Item for sale with maximum valuation V>0. This may be
a random variable with some known probability
distribution p(v)=Prob[V=v]
• Auction proceeds with unit price minimum increment
• Bidding rate  for “other bidders” and  for the the SB
• The seller’s decision delay after a bid is made is
exponentially distributed with parameter 
• After concluding a sale, the auction “rests” for some
random period with average r-1, or equivalently it waits
for the good or resource to be available again and then
restarts with another sale of the good
An Auction
For a fixed value v of the “good” we have a state space
{0,1, … ,v,A1, … , Av} where i represents the value that
is attained after the i-th bid, while Ai is the state
entered after the i-th bid is accepted – the 0-state is a
“rest state” after a particular auction is complete
The random process representing the current state of,
or value attained by, the auction is Xt ; alternately Xt
is the index of the bid and f(Xt) may be the monetary
value
0<t1< … < tn < … , are instants at which the auction
starts, with auction end times at tn+En< tn+1 when the
buyer accepts the offer, and rest times Rn=tn+1- [tn+En]
Increments offered by successive bidders within any one auction are
random variables X1, … , Xi , … that may depend on the auction
number n .. Xni etc.
The time between the arrival of successive bids are r.v. {Tni }
After bid “ni” is received, the seller will wait some “think” or decision
time Dni , after which it will accept the bid if Dni < Tn,i+1, or
consider the next bid if Dni > Tn,i+1 unless a new bid arrives
If there is reneging or balking, the most recent bid may be revoked
after some time Bni if Bni < Dni. If the bid is revoked by the highest
bidder, then the next highest un-revoked bid becomes the valid
bid, and may also be reneged, etc.
The Mathematical Model
• Let ti : i=1,2… be the sequence of instants when
the auctions start, and t1=0
• We model the system as a continuous time
Markov chain {X(t): t  0} with state space 0,
O(l), A(O,l), R(l), and A(R,l) where 1  l  v
• X(ti+t)=0 denotes the case when the auction has
restarted for the i-th round and is yet to receive
any bids after time t  0.
• ti+1 = inf{t : t  ti , X(ti+1)=0 } defines the auction
restart instants
The Mathematical Model
• X(ti+t)=R(l) if at time ti+t during the i-th auction (ti+t <
ti+1), the price has attained valuation l for 1  l  v,
where the l-th bid was placed by the SB (disregarding
the previous l-1 bids).
• X(ti+t)=A(R,l) if at time ti+t for t  0 during the i-th
auction, the seller has accepted the bid placed by the SB
at price l, where 1  l  v.
• X(ti+t)=O(l) if at time ti+t during the i-th auction (ti+t <
ti+1), the price has attained valuation l for 1  l  v,
where the l-th bid was not placed by the SB.
• X(ti+t)=A(O,l) if at time ti+t for t  0 during the i-th
auction, the seller has accepted the highest bid at price l
The price attained during the n-th auction will then be the r.v.
Qn = SN(n)i=1 Xni
Furthermore bids may be a function of the value attained by the
good during the preceding bid, e.g.
X
n, K+1=
g
k
n, K+1(S i=1
Xn i)
or a function of the value Vn of the good as well, e.g.
Xn, K+1= g
n, K+1(Un
Xn, K+1= g
, Ski=1 Xn i), or more specifically
n, K+1(Un
- Ski=1 Xn i)
Analytical Results
- Bids arrive to an auction according to a Poisson process; 1/  is the average time between
successive bids
- 1/ is the average time that the seller waits before accepting a bid (possible decision
variable) – the corresponding time is an exponentially distributed r.v.
- 1/r is the average rest period after the end of an auction and before the next auction
restarts. Without loss of generality r=1; this time can have a general distribution
- Assume there is no balking or reneging
- The value of the good is fixed to a given r.v. V with arbitrary distribution function, identical
at each successive auction
- Then after analysis
E[ Sale price | V=v ] = [1-rv]/[1-r] < v, r = /(+)
E[ Income per unit time ] = (1- E[rV] ) r(+)/(r++r)
The results generalize nicely to iid bid sizes, and to models in which a Markov renewal
structure can be exploited
The results generalize to iid bid sizes, and to other
models in which a Markov renewal structure can be
exploited
E[Sale price| V=v ]=E[X] [1-rv]/[1-r] < vE[X],
r = /(+)
E[Sale price| V=v ]=[1-rv]/[1-r] Svl=1 Xl
E[Income per unit time]= E[X](1-E[rV])r(+)/(r++r)
For the Vickrey auction where the good is sold to the
highest bidder at the second highest price:
E[Sale price| V=v ]=E[X] r{[1-rv-1]/[1-r]+/}
Auctions with a minimum sale price s:
E[Sale price| V=v ]=E[X] r {s [1-rv-s]/ + s}
where
r = s/(s+ )
When the un-successful bidder re-bids with probability p
and new bidders arive at rate g:
 = g +  p [F – 1]/ F = g [1-p+p/ F]
Income per unit time vs rate  at which decisions are made for a high 8
down to low 2 (bottom) rate at which bids arrive. The value of the good
Is uniformly distributed between 80 and 100 units
Income per unit time vs rate  at which decisions are made for an
English auction with unit increments: comparison of the effect of the arrival rate of bids.
The value of the good is uniformly distributed between 80 and 100 units
Comparison of the effect of a uniformly distributed and Poisson distributed
value of the good on the Income per Unit Time
Income per unit time vs rate  at which decisions are made for the
English and Vickrey auctions with unit increments. The arrival rate of bids is
4 (above) and 2 (bottom). The value of the good
Is uniformly distributed between 80 and 100 units
3.50E+00
3.00E+00
2.50E+00
Income
2.00E+00
1.50E+00
1.00E+00
5.00E-01
0.00E+00
0
0.5
1
1.5
2
2.5
-5.00E-01
Gamma
r=1 d=0.1 p=0.7
Income per unit time vs the rate at which bids arrive for =0.1, and r=1 with
the probability that an unsuccessful bidder will try again p=0.7
Smart Price Formation
EFFECTIVE
SALE PRICE
S = E[V] / e
BIDDERS
VALUE THE
GOOD
AT V
BID AT
RATE 
Smart Price Formation by Watching
the Market
Sv = E[ Sale price| V=v ] < v
Sv =[1-rv]/[1-r], where r = /(+) <1
S=E[Sale price] = (1-E[rV])/[1-r] < E[V]
v = e Sv, or S = E[V]/e, e1
If buyers are “stingy”, then e will be large
Smart Price Formation
If the bidders select value distribution such as
P(V=v)= av-1(1-a), v>0, a<1,
Then E[V]= 1/(1-a), E[rV]=r(1-a)/(1-ar),
S= 1/(1-ra) E[V] = S.e
e1: r = e(S - 1)/(eS - 1) < 1
Once e and either E[V] or S are known,
the bidding rate, or the decision rate, are set
via
r = 1/(1+/)
Networked Auctions
N Physically (Network) Interconnected Auctions for the same good (n(t),k(t))
• A client can only be at one of these auctions at a given instant of time
• n(t) = (n1(t), … , nN(t)) where ni(t) is the number of bidders at auction i
• k(t) = (k1(t), … , kN(t)) where ki(t) is the price currently attained at auction i
• ki(t) > 0 implies that ni(t)>0
• ni(t)=0 implies that ki(t)=0,
• While when ni(t)>0 then ki(t)>0
• The Mobile Bidder Model – a bidder at auction i who does not have an outstanding
(made but not yet accepted) bid may move from auction i to auction j with probability
P(i,j) or leave the auction system with probability P(i,N+1)
Networked Auctions
• g = (g 1, … , g N) where gi is the external arrival rate of bidders at auction i
• m i >0; the departure rate of bidders from auction i is:
mi(y,x) = (y-1) mi, if x>0
mi (y,0) = ymi
• The value of the good at auction i is the r.v. Vi with yI (x) = P[Vi > x] and yI (0) = 1
•  i >0; the departure rate of bidders from auction i is:
 i(y,x) = (y-1) I yI (x), if x>0
i (y,0) = y yI (0)
• The rate at which bids are accepted at auction I is i
•All rates refer to the inverses of average values of exponentially distributed iid
random variables
System Equations
for the Numbers of Bidders
Analytical Solution for Active Bidders
Active Bidders’ Model
Given the Value of the Good
Experimental Auction Scenario
Networked Auction Scenario:
• Roles: Seller, Bidder
• This Model assumes a centralised Auction Centre (AC) to which each bidder
goes before starting, and to which it returns after each unsuccessful bid
• An Auction is one Seller
• The AC advertises the Sellers to Bidders
• Sellers decide to accept a bid according to a Seller Decision Parameter
(SDP)
• In the executions, fixed at 0.1 acceptance/sec;
• Bidders contact the AC according to a Bidding Rate (BR)
• In the executions, fixed at 0.4 bids/sec;
• AC allows sellers and bidders to meet. Passive entity.
• Model: Iterative English
• Initial low price increased by iterative bids;
• Initial price: 0;
• Bids increase the price by 1 unit;
Auction Dynamics
•Sellers advertise an auction at the AC and update the information;
•Bidders contact the AC (according to their BR) and retrieve the list of
auctions currently taking place
•Bidders query a local oracle, to decide the best auction to bid for, and
submit a bid
• Oracles determine the best auction through employment of a
decision policy.
•If the proposed price > current price, the seller considers the bid
• Sets an exponentially generated timeout based on its SDP
• Notifies the currently known competitors and updates the price at
the AC
•If another (good) bid arrives before expiration of the timeout, the
process is repeated.
•If the timeout expires, the auction terminates
• The current highest bidder is declared winner.
• The auction is deleted from the AC – but we may have recurrent
auctions with a series of goods being sold
Focus: Bidders’ choice for the seller
•Bidder gets the list of currently available auctions from the AC
•Employs a sensible decision policy to determine the best seller
to which its should submit a bid
•4 policies employed, and compared:
• Random: seller randomly chosen. Not a SRP;
• SRP: seller chosen through a SRP that considers the gain over the
selling price of the past auction;
• SRP2: seller chosen through the SRP modified to account the gain
over the price of the current auction;
• SP: seller chosen according to the gain over the current price. Not
a SRP.
Sensible Bidding Policies
R_i= D_i+G_i : effective time for reaching
the seller
D_i: seller’s decision time
G_i: CPN goal
i: The auction number
V_i: Item’s value
Sit : Most recently observed
selling price at auction I
Si Sita + Si(1-a): Historical
average of selling price
P_i: current bid
SRP is evaluated as:
pi =
(Vi - Si ) + /Ri
n
+
(V
S
)
 i i /Ri

i=1
SRP2 is evaluated as:

pi =
(Vi - Si ) + * (Vi - Pi ) + /Ri
n
+
+
(V
S
)
*
(V
P
)
/Ri
 i i
i
i
i=1
Implementation
via Autonomic
Agents
EU FP6 Cascadas
Experimental Setting
•Platform completely distributed among 10 CPN machines:
• 1 AC
• 5 bidders
• n sellers
•Bidders, sellers and AC are all agents
• Each machine hosting one agent
•Experiments ran for scenarios with:
•
•
•
•
1
2
3
4
seller
sellers
sellers
sellers
•Metrics of interest:
•
•
•
•
Average
Average
Average
Average
seller income p/sec;
% of unsuccessful bids;
% of unsuccessful bids p/sec;
bids p/sec;
Average Income $/sec
Income per second
Income p/sec
1.13
0.93
SRP
Random
0.73
SRP2
SP
0.53
0.33
1 Seller
2 Sellers
3 Sellers
No. of Sellers
4 Sellers
Average Unsuccessful Bids
% of Unsuccesful Bids
Unsuccessful Bids (%)
11
9
SRP
7
Random
SRP2
SP
5
3
1 Seller
2 Sellers
3 Sellers
No. of Sellers
4 Sellers
Average Bids per/sec
Avg. Bids/sec
1.6
1.4
Bids/sec Received
1.2
1
Bids [SRP]
Bids [Random]
0.8
Bids [SRP2]
Bids [SP]
0.6
0.4
0.2
0
1 Seller
2 Sellers
No. of Sellers
3 Sellers
4 Sellers
Average unsuccessful bids per/sec
Avg Unsuc. Bids/sec
0.18
0.16
0.14
0.12
Random
0.1
SRP
SP
0.08
SRP2
0.06
0.04
0.02
0
1 Seller
2 Sellers
3 Sellers
4 Sellers
Our Conclusions
The World’s Economy is a “Chain Reaction” of Electronic Transactions
Users and Services, Buyers and Sellers, are Agents with
Interchangeable Roles
Computer Networks are the Infrastructure of the World Economy
Networks are Becoming Autonomic and Cognitive
An Inportant of Networks is to Support Economic Activity
Our Conclusions
The World’s Economy is a “Chain Reaction” of Electronic Transactions
Computer Networks are the Infrastructure of the World Economy
Prices are Formed by the “value” of goods, the level of economic activity,
and by feedback between individual and collective behaviour
Network QoS impacts the Outcome of Economic Activity, Values and
Prices
The World Economy will be increasingly dominated by ITC Constructs
and Mathematics