agt 2012 samos 1 braess’s paradox a. kaporis dept. of information & commun. systems eng.,...

AGT 2012 SAMOS 1

Braess’s Paradox

A. KaporisDept. of Information & Commun. Systems Eng., U.Aegean, Samos, Greece

&Research Academic Comp. Tech. Inst.,U. Patras Campus, Greece

Joint work with:

D. FotakisNational Technical University of Athens, School of Electrical and Computer Engineering

P. SpirakisDept. of Computer Eng. and Informatics, U. Patras, Greece

&Research Academic Comp. Tech. Inst.,U. Patras Campus, Greece

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Overview

Selfish routing on a network:

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Overview

Selfish routing on a network: an infinite number

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Overview

Selfish routing on a network: an infinite number of infinitesimally small users,

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Overview

Selfish routing on a network: an infinite number of infinitesimally small users, each wanting to minimize her travel time along the path she routes.

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Overview


Selfishness:

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Overview


Selfishness yields a routing that no user wants to reroute

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Overview


Selfishness yields a routing that no user wants to reroute (all used paths have the minimum latency)

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Overview


Selfishness yields a routing that no user wants to reroute (all used paths have the minimum latency), aka Wardrop equilibrium.

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Overview



But, such a selfish equilibrium may increase arbitrarily society’s cost (the sum all user’s travel times).

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Overview



But, such a selfish equilibrium may increase arbitrarily society’s cost (the sum all user’s travel times).

Central question:

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Overview



But, such a selfish equilibrium may increase arbitrarily society’s cost (the sum of all user’s travel times).

Central question:

Is there a cheap (wrt network designer)

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Overview




Central question:

Is there a cheap (wrt network designer) & fair (wrt users)

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Overview




Central question:

Is there a cheap (wrt network designer) & fair (wrt users) way to decrease society’s cost

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Overview




Central question:

Is there a cheap (wrt network designer) & fair (wrt users) way to

decrease society’s cost, while all users still route free & selfish?

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Overview

Yes!

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Overview

Yes! Exploit Braess’s Paradox

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Overview


That is,

No matter how new & luxurious is your network,

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Overview


That is,

No matter how new & luxurious is your network, you can make all users

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Overview


That is,

No matter how new & luxurious is your network, you can make all users to travel faster,

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Overview


That is,

No matter how new & luxurious is your network, you can make all users to travel faster, by (cruelly) destroying a part of it.

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A network toy example

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A flow dependent latency function per edge

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NE flow= red Zoro’s-subgraph

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NE path latency = 1+0+1= 2

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OPT flow= blue Ο-subgraph

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OPT path latency = ½+1= 3/2 < 2 = NE path latency

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A lazy designer just deletes edge uv & achieves optimum routing …

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A lazy designer just deletes edge uv & achieves optimum routing … so it is cheap (wrt the designer)!

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No user is sacrificed through slower paths …

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No user is sacrificed through slower paths … so it is fair (wrt users)!

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Opposed to Stackelberg strategies, for example:

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A leader has to control selfish routing, on a pair of links as:

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Leader is highly motivated by the optimum routing:

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But, users are reluctant to Leader’s view… since all are mad about the speedy link

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So, Leader sacrifices ½ of flow through the slower link (up)

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So, Leader sacrifices ½ of flow through the slower link (up)

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How about Taxes?

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How about Taxes?

• In a large class of networks—including all networks with linear latency functions—marginal cost taxes do not improve the cost of the Nash equilibrium [Cole, Dodis, Roughgarden].

• Disutility per user = path latency + taxes paid, is increased. • It is more expensive & complex (wrt a designer) to build (& operate) tax-stations

per road, than closing (once & for all) some roads.

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How about Taxes?




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How about Taxes?




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But, is Braess’s paradox happening only to a designer’s summer-night dream?

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• It might be just a “creature” of mathematical imagination, restricted to “breath” only in a optimization laboratory.

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No!

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No!

It appears almost always in a very broad class of random graphs [Valiant, Roughgarden, EC ‘06]

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No!

It appears almost always in a very broad class of random graphs [Valiant,

Roughgarden, EC ‘06]

It has been long observed in many large cities, such as NY [Kolata, New York Times ‘90].

“It is just as likely to occur as not” [Steinberg, ‘83].

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Formally, a routing instance G consists of:

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• a directed graph G with source node s and destination t.

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• a flow r>0 of infinite number of infinitesimally small users that wish to route through paths of G from node s to t.

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• a flow r>0 of infinite number of infinitesimally small users that wishes to route through paths of G from node s to t.

• edges endowed with flow-dependent latency functions (continuous, increasing, etc).

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G is paradox ridden:

if on a subgraph HÍ G, the selfish routing coincides to the optimum routing.

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G is paradox ridden:

if on a subgraph HÍ G, the selfish routing coincides to the optimum routing.

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We focus on real world routing instances, with latencies as:

F. Kelly, “The Mathematics of traffic in networks”. The Princeton Companion to Mathematics, Princeton University Press.

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We focus on real world routing instances, with latencies as:

F. Kelly, “The Mathematics of traffic in networks”. The Princeton Companion to Mathematics, Princeton University Press.

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Designer’s problems. Given an instance G:

• Paradox-Ridden Recognition (ParRid): decide if G is paradox-ridden.

• Best Subnetwork Equilibrium Latency (BSubEL): find the best subnetwork HB of G and its equilibrium latency L(HB).

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• Minimum Latency Modification (MinLatMod): with each edge e endowed

with polynomial of degree d latency on flow x:

modify the latency:

So that:

(i) the Euclidian distance of the coefficient’s vectors is minimum

&

(ii) the induced common latency gives the cost of optimum flow on G

d

iie

iiee axaxl

0,, 0,)(

d

i

iei

iee axaxl0

,

~

,

~~

0,)(

AGT 2012 SAMOS 59


• Minimum Latency Modification (MinLatMod): with each edge e endowed

with polynomial of degree d latency on flow x:

modify the latency:

So that:

(i) the Euclidian distance of the coefficient’s vectors is minimum

&

(ii) the induced common latency gives the cost of optimum flow on G

d

iie

iiee axaxl

0,, 0,)(

d

i

iei

iee axaxl0

,

~

,

~~

0,)(

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Previous work on Braess’s paradox

• First observed by D. Braess in ’68 and inspired a vast amount of papers.

• A nice history of it in: [Roughgarden, Selfish Routing and the Price of Anarchy. MIT, ’05].

• Recent work shows that the paradox is quite likely to occur [Valiant, Roughgarden,

EC ‘06].

• For general latencies ParRid is NP-hard [Roughgarden, Selfish Routing and the Price of

Anarchy. MIT, ’05]. Also, it is hard even to approximate BSubEL beyond a critical value (4/3 for linear latencies & n/2 for polynomial ones) [Roughgarden,

Selfish Routing and the Price of Anarchy. MIT, ’05].

• MinLatMod: of the most important problems [Magnanti, Wong: Network design and

transportation planning: Models and algorithms, Transp. Sci. '84].

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Our results

• Recognizing paradox-ridden instances (ParRid): . This problem is NP-complete for arbitrary linear latencies [29]. We show that it

becomes polynomially solvable for the important case of strictly increasing linear latencies.

Furthermore,

• We reduce the problem ParRid with arbitrary linear latencies to the problem of generating all optimal basic feasible solutions of a Linear Program that describes the optimal traffic allocations to the constant latency edges.

AGT 2012 SAMOS 62

Our results

• Recognizing paradox-ridden instances (ParRid): . This problem is NP-complete for arbitrary linear latencies. We show that it becomes

polynomially solvable for the important case of strictly increasing linear latencies.

Furthermore,


AGT 2012 SAMOS 63

Our results

• Recognizing paradox-ridden instances (ParRid): . This problem is NP-complete for arbitrary linear latencies. We show that it becomes

polynomially solvable for the important case of strictly increasing linear latencies.

Furthermore,


AGT 2012 SAMOS 64

Our results

• Best Subnetwork Equilibrium Latency (BSubEL):

For linear latencies, it is hard even to approximate BSubEL beyond the ratio 4/3.

For instances with polynomially many paths, each of polylogarithmic length, and arbitrary linear latencies, we present a subexponential-time approximation scheme.

That is,

For any ε> 0, the algorithm computes a subnetwork with an ε –NE. The common latency is at most an additive term of ε/2 from the optimum common latency. The running time is exponential in poly(logm)/ ε2, m is the number of edges.

Also,

• For instances with strictly increasing linear latencies that are not paradox ridden, we show that there is an instance-dependent δ > 0, such that the equilibrium latency is within a factor of 4/3-δ from the equilibrium latency on the best subnetwork.

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Our results




That is,


Also,


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Our results




That is,


Also,


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Our results




That is,


Also,


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Our results

• Minimum Latency Modification (MinLatMod):

If the instance is not paradox-ridden however, it is not possible to turn the optimal flow into a Nash flow by just removing edges.

Enforcing the optimal flow is possible, if in addition to removing edges, the administrator can modify the latency functions.

We present a polynomial-time algorithm for the problem of minimally modifying the latency functions of the edges used by the optimal flow so that the optimal flow is enforced as a Nash flow on the subnetwork used by the optimal flow with the modified latencies.

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Our results





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Our results





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Our results





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Best Subnetwork Equilibrium Latency (BSubEL): the idea

• For instances with polynomially many paths, each of polylogarithmic length, and arbitrary linear latencies, we present a subexponential-time approximation scheme for the subgraph of minimum common latency.

• We have to search amongst exponentially many subgraphs to find the best HB one of the minimum latency L(HB).

• Can we approximate HB (best) with H* (almost best, that is, L(H*) ≤L(HB)+ε), which is “small” and, thus, “easy” to find?

How “small”?

• “small” means H* has only polylogarithmic many paths that receive flow.

• Because, this implies subexponentially many candidates for H*.

Central question:

Is it possible such a “David” H* to perform almost as good as a “Goliath” HB?

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How “small”?



Central question:


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How “small”?



Central question:


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How “small”?



Central question:


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How “small”?



Central question:


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How “small”?



Central question:


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How “small”?



Central question:


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How “small”?



Central question:

Does exist such a “small” H* that performs almost as good as a “big” HB?

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Is it possible such a “small” H* to perform almost as good as a “big” HB?

• If one has to bet, then she won’t put her penny on “David”…

But,

Our hopes come from a seemingly unrelated topic:

Once upon a time, there was a bimatrix game, with only 2 rivals.

But,

each rival was armed with a “Goliath” set of strategies.

So,

it was a headache to compute their best subsets of strategies to play.

Until,

a wise old man showed them that only a “David” subset of strategies can make them almost as happy as the best subset of strategies.

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• If one has to bet, then she won’t put her penny on a “small” subgraph…

But,



But,

each rival was armed with a “big” set of strategies.

So,


Until,

a wise old man showed them that only a “small” subset of strategies can make them almost as happy as the “big” best subset of strategies.

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But,



But,


So,


Until,

a wise old man showed them that only a “small” subset of strategies can make them almost as happy as the best subset of strategies.

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But,



But,


So,


Until,


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But,



But,


So,


Until,


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But,



But,


So,


Until,


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• If one has to bet, then she won’t put her penny on “small” subgraph…

But,



But,


So,


Until,

a wise old man showed them that only a “small” subset of strategies can make them almost as happy as the best subset of strategies.

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Is it possible such a “small” H* to perform almost as good as a “Goliath” HB?

• If one has to bet, then she won’t put her penny on “small” subgraph…

But,



But,


So,


Until,


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But,



But,


So,

it was a headache for them to compute their best subsets of strategies to play.

Until,


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But,



But,


So,


Until,


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But,



But,


So,


Until,

a wise old man showed them that only a “small” subset of strategies can make them almost as happy as the best “big” subset of their strategies.

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How each player finds small & almost optimal subsets of strategies? By the Probabilistic Method…

• Each rival selects sequentially & at random (according to an optimal but unknown distribution) a row (column) from her matrix.

• Strong tail bounds show that, with probability ≥ a positive constant, after polylogarithimic many random row (column) selections, her expected gain (by her random “David” subset) will be close to the optimal expected gain (by her “Goliath” subset).

• Thus, by the Probabilistic Method, there exists “small” subsets of rivals’s strategies that are close to the optimal expected gain.

• So, each rival has only to exhaustively search in subexponentially time for such “small” subsets.

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How each player finds small & almost optimal subsets of strategies? By the Probabilistic Method…

• Each rival selects sequentially & at random (according to an optimal but unknown distribution) a row (column) from her matrix.

• Strong tail bounds show that, with probability ≥ a positive constant, after polylogarithimic many random row (column) selections, her expected gain (by her random “David” subset) will be close to the optimal expected gain (by her “Goliath” subset).

• Thus, by the Probabilistic Method, there exists “small” subsets of rivals’s strategies that are close to the optimal expected gain.

• So, each rival has only to exhaustively search in subexponentially time for such “small” subsets.

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From a bimatrix game to an instance G (a graph!)

• A flow = infinite number of players (but, a bimatrix game has only 2 players) …

View the flow on paths as the mixed strategy of a single player (FLOW-player)!

• An instance G has no payoff matrices (but, a bimatrix has 2 such matrices)…

View the edge-path adjacency matrix as the payoff matrix of the FLOW-player

• There are congestion effects on G (but, a bimatrix has no congestion)

Show that congestion affects smoothly the random experiment.

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From a bimatrix game to an instance G (a graph!)







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From a bimatrix game to an instance G







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View the edge-path adjacency matrix as the payoff matrix of the FLOW-player.



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Show that congestion only affects smoothly the random experiment.

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m “short”paths from node s to t

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Let f the path-flow of the subgraph of the minimum common latency

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What if we search exhaustively for this precious f?

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Select k= Θ(logm/ε2) paths at random according to the unknown f

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These k= Θ(logm/ε2) paths with positive probability induce the minimum common latency (achieved by f) + ε

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These k= Θ(logm/ε2) paths with positive probability induce the minimum common latency (achieved by f) + ε

agt 2012 samos 1 braess’s paradox a. kaporis dept. of information & commun. systems eng.,...

Documents

overview selfish routing

infinite number slide

selfish equilibrium

small users

travel time

greece slide

minimum latency slide

wardrop equilibrium