mae 298, lecture 12mae.engr.ucdavis.edu › dsouza › classes › ecs289-w11 › ...google adwords,...
TRANSCRIPT
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MAE 298, Lecture 12Feb 16, 2011
“Flows on spatial networks, Games onnetworks”
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Topics – Recap of last time
• Optimal allocation of facilities and transport networks:(Heuristic, qualitative – but analytic – approach)(Linear tradeoffs between geometric and network metrics)
– Michael Gastner (SFI) and Mark Newman (U Mich)
• Network flows on road networks(Details of demand, edge capacity, and feasible paths allextremely important!)
– I. User vs System Optimal– II. Braess’ Paradox / Nash Equilibrium– Michael Zhang (UC Davis)
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Nash equilibrium versus System optimalPrisoner’s Dilemma
Cooperate DefectCooperate 3, 3 0, 5
Defect 5, 0 1, 1
• Blue Cooperates/Red Cooperates — Blue gets payout “3”
• Blue Cooperates/Red Defects – Blue gets “0”
• Blue Defects/Red Defects – Blue gets “1”
• Blue Defects/Red Cooperates – Blue gets “5”
– Average expected payout for defect is “3”, for cooperate is“1.5”. Blue always chooses to Defect! Likewise Red alwayschooses Defect.
– Both defect and get “1” (Nash), even though each would geta higher payout of “3” if they cooperated (Pareto efficient).
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Braess Paradox
• Dietrich Braess, 1968 (And simpler example by Pigou, 1920)(Braess currently Prof of Math at Ruhr University Bochum, Germany)
• In a user-optimized network, when a new link is added, thechange in equilibrium flows might result in a higher cost,implying that users were better off without that link.
• Recall Zhang notation, qij is overall traffic demand from nodei to j. ta(νa) is travel cost along link a, which is a function oftotal flow that link νa.
• Equilibrium is when the cost on all feasible
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Getting from 1 to 4
Assume traffic demand q14 = 6. Originally 2 paths (a-c) and (b-d).
• ta(νa) = 10νa • tc(νc) = νc + 50• tb(νb) = νb + 50 • td(νd) = 10νd
=⇒ Eqm: ν = 3 on each link
C1 = C2 = 83
Add new link with te(νe) = νe + 10
Now three paths:
Path 3 (a - e - d), with νe = 0 initially, so C3 = 0 + 10 + 0 = 10
C3 < C2 and C1 so new equilibrium needed.
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• By inspection, shift one unit of flow form path 1 and from 2 respectively topath 3.
• Now all paths have flow f1 = f2 = f3 = 2.
• Link flow νa = 4, νb = 2, νc = 2, νd = 4, νe = 2.
ta = 40, tb = 52, tc = 52, td = 40, te = 12.
C1 = ta + tc = 92; C2 = tb + td = 92; C3 = ta + te + td = 92.
• 92 > 83 so just increased the travel cost!
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Braess paradox – Real-world examples
(from http://supernet.som.umass.edu/facts/braess.html)
• 42nd street closed in New York City. Instead of the predictedtraffic gridlock, traffic flow actually improved.
• A new road was constructed in Stuttgart, Germany, traffic flowworsened and only improved after the road was torn up.
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Braess paradox depends on parameter choices
• “Classic” 4-node Braess construction relies on details of q14and the link travel cost functions, ti.
• The example works because for small overall demand (q14),links a and d are cheap. The new link e allows a pathconnecting them.
• If instead demand large, e.g. q14 = 60, now links a and d arecostly! (ta = td = 600 while tb = tc = 110). The new path a-e-dwill always be more expensive so νe = 0. No traffic will flow onthat link. So Braess paradox does not arise for this choice ofparameters.
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How to avoid Braess?
• Back to Zhang presentation .... typically solve for optimal flowsnumerically using computers. Can test for a range of choicesof traffic demand and link costs.
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“The price of anarchy”
E. Koutsoupias, C. H. Papadimitriou“Worst-case equilibria,” STACS 99.
Cost of worst case Nash equilibrium / cost of system optimalsolution.
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4
The Price of Anarchy
Nash Equilibrium:
cost = 14+14 = 28
s t2x 12
5x50
(From Roughgarden Barbados Talk, http://theory.stanford.edu/ tim/)
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5
The Price of Anarchy
Nash Equilibrium: To Minimize Cost:
Price of anarchy = 28/24 = 7/6.�• if multiple equilibria exist, look at the worst one
s t2x 12
5x5
cost = 14+10 = 24cost = 14+14 = 28
s t2x 12
5x500
(From Roughgarden Barbados Talk, http://theory.stanford.edu/ tim/)
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Selfish routing and the POA on the Internet
T. Roughgarden and E. Tardos, How Bad is Selfish Routing?,FOCS ’00/JACM ’02
• Routing in the Internet is decentralized: Each router makes adecision, so path dynamically decided as packet passed on.
• Cost of an edge c(e), may be constant (infinite capacity) ordepend on the load.
• “Shortest path” routing (really lowest∑c(e) routing) typically
implemented.
• This is equivalent to “selfish routing” (each router chooses bestoption available to it).
• Resulting POA = 2!
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Braess and the POA for Internet traffic
Greg Valiant, Tim Roughgarden, Eva Tardos“Braess’s paradox in large random graphs”, Proceedings of the
7th ACM conference on Electronic commerce, 2006.
• Removing edges from a network with “selfish routing” candecrease the latency incurred by traffic in an equilibrium flow.
• With high probability, (as the number of vertices goes toinfinity), there is a traffic rate and a set of edges whose removalimproves the latency of traffic in an equilibrium flow by aconstant factor.
• Braess paradox found in random networks often (not just“classic” 4-node construction).
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Algorithmic game theory
• Since we know users act according to Nash, can we designalgorithms (mechanisms) that bring Nash and System Optimalas close together as possible?
• Typically we think of players who interact via a network, orwho’s connectivity is described by a network of interactions.
– Multiplayer games for users connected in a network orinteracting via a network.
– Designing algorithms with desirable Nash equilibrium.
– Computing equilibrium when agents connected in a network.
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focs 2001 20
mechanism design (or inverse game theory)
• agents have utilities – but these utilities are known only to them
• game designer prefers certain outcomes depending on players’ utilities
• designed game (mechanism) has designer’s goals as dominating strategies
(Papadimitriou, “Algorithms, Games, and the Internet” presented at STOC/ICALP 2001.)
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Some traditional games:
focs 2001 7
1,-1 -1,1
-1,1 1,-1
0,0 0,1
1,0 -1,-1
3,3 0,4
4,0 1,1
matching pennies prisoner’s dilemma
chicken auction
1 … n 1 . . n
u – x, 0
0, v – y
e.g.
(Papadimitriou, “Algorithms, Games, and the Internet” presented at STOC/ICALP 2001.)
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Mechanism design example:
focs 2001 25
e.g., Vickrey auction
• sealed-highest-bid auction encourages gaming and speculation
• Vickrey auction: Highest bidder wins, pays second-highest bid
Theorem: Vickrey auction is a truthful mechanism.
Theorem: It maximizes social benefit and auctioneer expected revenue.
(Papadimitriou, “Algorithms, Games, and the Internet” presented at STOC/ICALP 2001.)
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(Modified) Vickrey auctions in real life –Google AdWords, and Yahoo’s ad sales
• Bidding on a “keyword” so that your advertisement is displayedwhen a search user enters in this keyword
• You can safely bid the maximum price you think is fair, and ifyou win, you actually pay less!
• Mechanism design– Incentivizes users to bid what they think is fair(reveal their true utilities)– Keeps more people in the bidding– Does not necessarily maximize profits for seller
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Back to flows on spatial networks:
• David Aldous, “Spatial Transportation Networks with TransferCosts: Asymptotic Optimality of Hub and Spoke Models”
• Flows of material goods, self-organization:Helbing et al.
• Jamming and flow (phase transitions):Nishinari, Liu, Chayes, Zechina.
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Our modern infrastructure
Layered, interacting networks
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“Layered complex networks”
[ M. Kurant and P. Thiran, “Layered Complex Networks”, PhysRev Lett. 89, 2006.]
• Offer a simple formalism to think about two coexisting networktopologies.
• The physical topology.
• And the virtual (application) topology.
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Example 1:WWW and IP layer views of the Internet
• Each WWW link virtually connects two IP addresses.
• Those two IP nodes are typically far apart in the underlyingIP topology, so the virtual connection is realized as a multihoppath along IP routers.
• (Of course the IP network is then mapped onto the physicallayer of optical cables and routers.)
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Example 2:Transportation networks
Up until now separate studies of:
1. Physical topology (of roads)
2. Real-life traffic patterns
Want a comprehensive view analyzing them both together.
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The “multilayer” formalism
Consider two different networks:
• Gφ = (V φ, Eφ); the physical graph.
• Gλ = (V λ, Eλ); the logical/application-layer graph.
Assume both sets of nodes identical, V φ = V λ.
• The logical edges are made of a collection of physical edges.M(eλ) is a mapping of logical onto physical edges.
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The load on a node
• Load on node i, l(i), is the sum of the weights of all logicaledges whose paths traverse i.
• E.g., in a transportation network l(i) is the total amount oftraffic that flows through node i.
Here Gφ is physical connection (e.g., road, rail line), Gλ “logical”is the actual flow.
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Application
Study three transportation systems:
1. Mass transit system of Warsaw Poland (city).
2. Rail network of Switzerland (country).
3. Rail network of major trains in the EU (continent).
• Physical graph
• Logical graph: They can estimate the real load from thetimetables (some assumptions; decompose into units (onetrain, one bus, etc), independent of number of people).
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Physical versus logical
Physical
• exponentially decaying degree distributions.
• Diameter ∼√N
Logical
• Right skewed degree distributions
• Shorter shortest paths (e.g, number of transfers).
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Load
They can estimate the real load from the timetables (someassumptions; decompose into units (one train, one bus, etc),
independent of number of people).
Two typical load estimators:
1. The node degree of the physical network.
2. Betweenness of the physical network.
(Note, these estimators are the ones currently in use in almostall cases: 1) Resilience of networks to edge removal, 2)
Modeling cascading failures, etc.....)
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FindingsM. Kurant P. Thiran, “Layered Complex Networks”, Phys Rev Lett. 89, 2006.
• All three estimators 1) real load, 2) degree, 3) betweennessdiffer from one-another.
• Using the two-layer view can see the logical graphs may haveradically different properties than the physical graphs.
• May lead to reexamination of network robustness (previousstudies on Internet, power grid, etc, based on physical layer).
• M. Kurant, P. Thiran and P. Hagmann “Error and AttackTolerance of Layered Complex Networks”, Phys. Rev. E, Vol.76, 2007.Multilayer systems much more vulnerable to error and intentional attack than they seem
from single-layer perspective.
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Summary of spatial flows
• Optimal location of facilities to maximize access for all.
• Designing “optimal” spatial networks (collection/distributionnetworks – subways, power lines, road networks, airlinenetworks).
• Details of flows on actual networks make all the difference!
– Users act according to Nash
– Braess paradox (removing edges may improve a network’sperformance!)
– The “Price of Anarchy” (cost of worst Nash eqm / cost ofsystem optimal)
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Next topics
• Advanced network visualization, Weds Feb 23.Guest lecture: Isaac Liao and Chris Muelder
• Diffusion and cascades on networks– Physical diffusion (network Laplacian)– Cascades of “infection”– Applications to:• Cascades of ideas / adoption of new technologies in socialnetworks• Information cascades in blogs• Spread of innovation (Very interesting new developmentsusing a game-theoretic approach as opposed to adiffusion/epidemic approach).