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Incertezza e Sistemi Dinamici in Rete

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.

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Incertezza e Sistemi Dinamici in Rete

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  1. Incertezza e Sistemi Dinamici in Rete ErolGelenbe 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

  2. 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 ExchangeCognitive Network: A Computer-Communication Network where Resource Allocation including Routing is Achieved by Adaptive Procedures that Optimise QoS, Profit or Other Criteria

  3. Vision: The World’s Economy is a “Chain Reaction” of Electronic Economic TransactionsUsers and Services, Buyers and Sellers, are Agents with Interchangeable RolesComputer Networks are the Infrastructure of the World EconomyNetworks are Becoming Autonomic and Cognitive

  4. 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.

  5. AuctionsCognitive Networks

  6. Research carried out in the EU FP6 CASCADAS Project with Telecom Italia, Univ. of Modena, Fokus-Fraunhofer and other partnersand the UK ALADDIN Project withSouthampton University, Imperial College, Oxford, BristolUK 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

  7. 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

  8. 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?

  9. 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

  10. 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 All random variables are exponentially distributed

  11. 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]

  12. 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.

  13. 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

  14. 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 for 1  l  v, which was not owned by the SB.

  15. 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 n, K+1(Ski=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 , Ski=1 Xn i), or more specifically Xn, K+1= g n, K+1(Un - Ski=1 Xn i)

  16. Analytical Results Bids arrive to an auction according to a Poisson process; 1/ l is the average time between successive bids 1/d 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 = l/(l+d) E[ Income per unit time ] = (1- E[rV] ) lr(l+d)/(lr+ld+rd) The results generalize nicely to iid bid sizes, and to models in which a Markov renewal structure can be exploited

  17. 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 = l/(l+d) E[Sale price| V=v ]=[1-rv]/[1-r] Svl=1 Xl E[Income per unit time]= E[X](1-E[rV])lr(l+d)/(lr+ld+rd) 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]+d/l}

  18. Auctions with a minimum sale price s: E[Sale price| V=v ]=E[X] r {ls [1-rv-s]/d + s} where r = ls/(ls+ d) When the un-successful bidder re-bids with probability p and new bidders arive at rate g: l = g + l p [F – 1]/ F = g [1-p+p/ F]

  19. Income per unit time vs rate d 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

  20. Income per unit time vs rate d 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

  21. Comparison of the effect of a uniformly distributed and Poisson distributed value of the good on the Income per Unit Time

  22. Income per unit time vs rate d 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

  23. Income per unit time vs the rate at which bids arrive for d=0.1, and r=1 with the probability that an unsuccessful bidder will try again p=0.7

  24. Smart Price Formation EFFECTIVE SALE PRICE S = E[V] / e BIDDERS VALUE THE GOOD AT V BID AT RATE l

  25. Smart Price Formation by Watching the Market Sv = E[ Sale price| V=v ] < v Sv =[1-rv]/[1-r], where r = l/(l+d) <1 S=E[Sale price] = (1-E[rV])/[1-r] < E[V] v = eSv, or S = E[V]/e, e>1 If buyers are “stingy”, then e will be large

  26. 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+d/l) Smart Price Formation

  27. 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)

  28. Networked Auctions g = (g1, … , gN) 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 b i>0; the departure rate of bidders from auction i is: bi(y,x) = (y-1) bI yI(x), if x>0 bi (y,0) = y yI(0) The rate at which bids are accepted at auction I is di All rates refer to the inverses of average values of exponentially distributed iid random variables

  29. System Equations for the Numbers of Bidders

  30. Analytical Solution for Active Bidders

  31. Active Bidders’ Model Given the Value of the Good

  32. 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;

  33. 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

  34. 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.

  35. Sensible Bidding Policies 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 R_i= D_i+G_i : effective time for reaching the seller D_i: seller’s decision time G_i: CPN goal SRP is evaluated as: SRP2 is evaluated as:

  36. Implementationvia Autonomic AgentsEU FP6 Cascadas

  37. 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 seller • 2 sellers • 3 sellers • 4 sellers • Metrics of interest: • Average seller income p/sec; • Average % of unsuccessful bids; • Average % of unsuccessful bids p/sec; • Average bids p/sec;

  38. Average Income $/sec

  39. Average Unsuccessful Bids

  40. Average Bids per/sec

  41. Average unsuccessful bids per/sec

  42. Our Conclusions The World’s Economy is a “Chain Reaction” of Electronic TransactionsUsers and Services, Buyers and Sellers, are Agents with Interchangeable RolesComputer Networks are the Infrastructure of the World EconomyNetworks are Becoming Autonomic and CognitiveAn Inportant of Networks is to Support Economic Activity

  43. Our Conclusions The World’s Economy is a “Chain Reaction” of Electronic TransactionsComputer Networks are the Infrastructure of the World EconomyPrices are Formed by the “value” of goods, the level of economic activity, and by feedback between individual and collective behaviourNetwork QoS impacts the Outcome of Economic Activity, Values and PricesThe World Economy will be increasingly dominated by ITC Constructs and Mathematics

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