CS 684: Algorithmic Gme Theory Spring 217 Lecturer: Év Trdos Lecture 2: Jnury 27 Scrie: Alert Julius Liu 2.1 Logistics Scrie notes must e sumitted within 24 hours of the corresponding lecture for full credit. They re worth 4% of the course grde. See the first lecture s notes for dditionl exmples of dysfunctionl tournment incentives. Agin, Roughgrden s Twenty Lectures on Algorithmic Gme Theory is not required, ut the first third of the course will closely follow the second hlf of the ook. This lecture will follow Section 11. 2.2 Bress prdox: Recll from lst lecture Bress prdox: x 1 1 trvelers s t 1 x In this tomic prolem, i.e. with discrete trvelers, we hd four Nsh equiliri: one where ll 1 trvelers tke the -cost ridge etween nd ; two where only 99 do, nd the lst trveler tkes either the top or ottom pth; nd one where 98 tke the ridge nd one tkes ech of the top nd ottom pth. Wht if we insted consider non-tomic version of the prolem, where trvelers re infinitely smll? Let us normlize the flow of trvelers to 1, nd likewise the constnt edge costs to 1 s well: x 1 1 s t 1 x In this cse there is only one Nsh equilirium, where the entire flow of trvelers tkes the ridge. Proof: To see this, suppose tht: 2-1
2-2 Lecture 2: Jnury 27 A frction of the flow tkes the top pth. A frction tkes the ottom pth. The remining frction 1 tkes the ridge pth. The costs of the three pths re then 2 for the top pth. 2 for the ottom pth. 2 for the ridge pth. If nd re not oth nonzero, then the flow on the top or ottom pth will e incentivized to tke the ridge insted, s it is strictly cheper. If exctly one of nd re zero, the ottom or top pth (whichever one hs nonzero flow) will e strictly more expensive thn either of the other two pths nd so will e incentivized to switch. Therefore, the unique Nsh equilirium occurs when nd re oth zero. In generl, this non-tomic formultion is convenient s it llows us to use the powerful tools of clculus. Even if the popultion eing modeled is tomic in rel life, this non-tomic formultion my e good pproximtion if the popultion is lrge enough. 2.3 Routing gmes Our Bress prdox exmple is n instnce of routing gme. 2.3.1 The gme An instnce of such gme is defined y the following: A grph G consisting of set of vertices V nd set of directed edges E etween ordered pirs of vertices. For ech edge, cost c e (x) s function of the flow on tht edge. The cost functions re required to e ll nonnegtive (for nonnegtive flow, which we will require), continuous, nd monotoniclly nondecresing. While it cn e convenient to require tht they e strictly monotoniclly incresing, this will not e necessry for this lecture, nd is not true in our exmple ove. A set of pirs of source nd sink vertices s i, t i, with desired flow r i etween ech such pir of vertices. The pirs re required to e unique; tht is, for ny s i = s j we require t i t j, nd for ny t i = t j we require s i s j. Another wy of thinking out it is tht we collpse ll duplicte source-sink pirs into single one y summing their required flows r i + r j. 2.3.2 Solution A legl glol flow f in this gme consists of n ssignment of flow mgnitudes f P tht ll the required flows re stisfied, i.e. for ll i, to ech pths P such
Lecture 2: Jnury 27 2-3 r i = P :s i t i f P (2.1) Note tht the numer of possile pths P cn ecome exponentilly lrge in the numer of vertices V, even if only simple pths (i.e. not visiting ny vertex more thn once) re permitted! While this cn pose prolem for computtion, it is fine for our nlysis here. 2.3.3 Edge flows We will lso consider flow on individul edges e, denoted y f (e). This is simply the sum of the flow of ll pths tht go through tht edge: f (e) = P e f P (2.2) 2.3.4 Pth cost The cost c P (f) of given pth P given glol flow f is the sum of the cost of edges on tht pth: c P (f) = e P c e (f (e)) (2.3) 2.3.5 Nsh equilirium For this gme, Nsh equilirium is where no flow hs n incentive to switch from one pth to nother. More formlly, we re t Nsh equilirium if, for ll pths P : s i t i with nonzero flow f P >, nd ll possile lterntive pths Q : s i t i etween the sme source nd sink nodes, we hve c P (f) c Q (f) (2.4) The converse ( if nd only if ) holds s well tht is, if this condition is not met, there is lwys n incentive for some flow to switch. Proof: Suppose for glol flow f there re some P nd Q with f P > such tht Equivlently, for some δ >, c P (f) > c Q (f) (2.5) c P (f) c Q (f) = δ (2.6) This difference is the incentive to switch from P to Q. The question is whether we cn mintin n incentive for nonzero mount of flow switching.
2-4 Lecture 2: Jnury 27 Let ɛ e n mount of flow tht we propose to switch from P to Q. c P nd c Q re oth sums of edge costs, which re continuous; since sums nd differences of continuous functions re lso continuous, this mens tht c P c Q vries continuously with ɛ. By the definition of continuity, for some sufficiently smll ɛ, c P c Q remins positive, tht is, the incentive to switch, remins lrger thn zero. 2.4 Price of nrchy How d is it if everyone cts selfishly, compred to if centrl coordintor tries to minimize glol cost? 2.4.1 Glol cost To nswer this question, we must first define glol cost. We could define this s the totl of ll pth costs, weighted y the mount of flow tking tht pth: c (f) = P f P c P (f) (2.7) Or we could define this s the totl of ll edge costs, weighted y the mount of flow tking tht edge: c (f) = e f (e) c e (f (e)) (2.8) In fct, these two definitions re equivlent. Proof: Strting with the pth-sed definition of cost (Eqution 2.7), we sustitute in the definition of pth cost (Eqution 2.3): f P c e (f (e)) (2.9) P e P We cn move f P inside the inner summtion: f P c e (f (e)) (2.1) P e P Now we re summing over ll P, e pirs such tht e P. It is therefore equivlent to switch the nesting of the summtions: f P c e (f (e)) (2.11) e P e Since c e (f (e)) does not depend on P, we cn move it outside the inner summtion: c e (f (e)) f P (2.12) P e e Finlly, using the definition of edge flow (Eqution 2.2) we cn replce the inner summtion c e (f (e)) f (e) (2.13) mtching the edge-sed glol cost definition (Eqution 2.8) s desired. e
Lecture 2: Jnury 27 2-5 2.4.2 Defining price of nrchy Now we cn define the price of nrchy. Let f e the glol flow tht minimizes glol cost, nd f e the Nsh equilirium (the one with the gretest glol cost if it is not unique). The price of nrchy is defined s price of nrchy = c (f) c (f ) (2.14) 2.4.3 Exmple: Bress prdox In our Bress prdox scenrio: the socilly optiml solution is for hlf of the flow to tke the top pth nd hlf the ottom. Every unit of flow experiences cost of 1.5, so the glol cost is 1.5. Menwhile in the Nsh equilirium the entire flow tkes the ridge for cost of 2, so this exmple hs price of nrchy of 4/3. 2.4.4 Another exmple Here is nother exmple with just two vertices nd two edges. 1 1 s t x There is only one Nsh equilirium, nmely the entire flow tking the ottom edge otherwise, the ottom edge costs less thn the top edge, nd the top flow hs n incentive to switch to the ottom edge. The glol cost for the Nsh equilirium is 1. To find the optiml solution, we cn use clculus. Suppose x flow tkes the ottom pth, nd the rest tkes the top pth. The glol cost is then c (x) = 1 x }{{} + }{{} x 2 cost from top edge cost from ottom edge (2.15) Rememer tht the optimum of differentile function will occur t the ounds (not in this cse, since oth ounds hve glol cost of 1), or t locl extremum, where the derivtive is zero: dc = 1 + 2x (2.16) dx The minimum thus occurs t x = 1 2, just like our Bress prdox exmple. Plugging x ck into the glol cost (Eqution 2.15), we hve totl cost of 3 4, nd price of nrchy of 4 3 gin just like our Bress prdox exmple. However, unlike our Bress prdox exmple, in this cse the socil solution is in some sense unfir : the flow on the top pys unit cost of 1, while the flow on the ottom pys unit cost of 1 2.
2-6 Lecture 2: Jnury 27 A Appendix: Undirected grphs Note tht our definition ws for directed grph. It is possile to reduce gme on n undirected grph (where the cost is function of the totl flow in oth directions) to gme on directed grph vi the use of gdget: c ecomes u c v B Appendix: Multigrphs In our grph exmple ove (Section 2.4.4), we llow multiple edges etween the sme pir of nodes (with different costs). Agin, gme on such grph cn e reduced to gme on conventionl directed grph vi the use of gdget: c c 1 ecomes c u c 1 v