Facility location game

Transcription

1 Mechaism Desig Facility locatio game Recall facility locatio game What are we desigig exactly? Pigzhog Tag Mechaism Desig, Slide

2 Mechaism Desig Facility locatio game We kow the set of types of each player (private locatio) We kow the set of all possible outcomes (locatios of the library) We kow the utility fuctio of each player We eed to desig the set of actios for each player We eed to desig a mappig from a actio profile to a outcome Pigzhog Tag Mechaism Desig, Slide 2

3 Mechaism Desig Mechaism desig I short To complete some basic compoets of a Bayesia game: private ifo give utility fuctio give rules of the game to be desiged Referece book 3, Chapter 0 Eric Maski, Mechaism desig, Nobel Prize 2007 Pigzhog Tag Mechaism Desig, Slide 3

4 Mechaism Desig Eviromet give Exted the ative social choice settig to oe where agets ca t be relied upo to disclose their prefereces hoestly. Start with a Bayesia game, but o actios. Defiitio (Eviromet) A eviromet is a tuple (N,O,,p,u), where N is a fiite set of agets; O is a set of outcomes; = is a set of possible joit type vectors; p is a (commo prior) probability distributio o ; ad u =(u,...,u ),whereu i : O 7! R is the utility fuctio for each player i. Pigzhog Tag Mechaism Desig, Slide 4

5 Mechaism Desig Mechaism Desig Defiitio (Mechaism) A mechaism (for a eviromet (N,O,, p,u)) is a pair(a, g), where A = A A,whereA i is the set of actios available to aget i 2 N; ad g : A 7! (O) maps each actio profile to a distributio over outcomes. (Determiistic mechaisms: A 7! O) Thus, the desiger gets to specify the actio sets for the agets the mappig to outcomes, over which agets have utility ca t chage outcomes; agets prefereces or type spaces Pigzhog Tag Mechaism Desig, Slide 5

6 Mechaism Desig Desig objective The problem is to pick a mechaism that will cause ratioal agets to behave i a desired way, specificallymaximizigthe mechaism desiger s ow objective fuctio, give that each aget holds private iformatio, ukow to the desiger agets play BNE desiger hopes that i BNE, his/her objective is reached Various equivalet ways of lookig at this settig choose the Bayesia game out of a set of possible Bayesia games that maximizes some performace measure desig a game that implemets a particular social choice fuctio C( ) i equilibrium, give that the desiger o loger kows agets prefereces ad the agets might lie Pigzhog Tag Mechaism Desig, Slide 6

7 Mechaism Desig Implemetatio i Domiat Strategies Defiitio (Implemetatio i domiat strategies) Give a eviromet (N,O,, p,u), a mechaism(a, g) is a implemetatio i domiat strategies of a social choice fuctio C :! O if the iduced game has a domiat strategy equilibrium s ( ), ad for ay type profile, wehave g(s ( )) = C( ). E.g., secod-price auctio implemets social-welfare maximizatio i domiat strategies. Pigzhog Tag Mechaism Desig, Slide 7

8 Mechaism Desig Implemetatio i Bayes-Nash equilibrium Defiitio (Bayes Nash implemetatio) Give a Bayesia game settig (N,O,, p,u), a mechaism(a, g) is a implemetatio i Bayes Nash equilibrium of a social choice fuctio C :! O if there exists a Bayes Nash equilibrium s ( ) of the iduced game (N, A,,p,u) such that for ay type profile 2, wehavethatg(s ( )) = C( ). E.g., For symmetric type distributios, first-price auctio implemets social-welfare maximizatio i BNE. Pigzhog Tag Mechaism Desig, Slide 8

9 Mechaism Desig Bayes-Nash Implemetatio Commets Bayes-Nash Equilibrium Problems: there could be more tha oe equilibrium which oe should I expect agets to play? agets could miscoordiate ad play oe of the equilibria Possible Fix: Symmetric Bayes-Nash implemetatio Thm: symmetric Bayesia game has symmetric Bayes-Nash Agai, first-price auctio Pigzhog Tag Mechaism Desig, Slide 9

10 Mechaism Desig Implemetatio Commets We ca require that the desired outcome arises i the oly equilibrium i every equilibrium (strog implemetatio) i at least oe equilibrium (weak implemetatio) Forms of implemetatio: Direct Implemetatio: agets each simultaeously sed a sigle message to the ceter Idirect Implemetatio: agets may sed a sequece of messages; i betwee, iformatio may be (partially) revealed about the messages that were set previously like extesive form Pigzhog Tag Mechaism Desig, Slide 0

11 Revelatio Priciple Impossibility Revelatio Priciple It turs out that ay social choice fuctio that ca be implemeted by some mechaism ca be implemeted by a truthful, direct mechaism! direct mechaism: A i = i truthful, direct mechaism: s i ( i)= i Hold for both DSE ad BNE Cosider a arbitrary (e.g., may be idirect) Pigzhog Tag Mechaism Desig With Urestricted Prefereces, Slide

12 Revelatio Priciple Impossibility Revelatio Priciple type θ type θ strategy s (θ ) M strategy s (θ ) Origial Mechaism outcome It turs out that ay social choice fuctio that ca be implemeted by some mechaism ca be implemeted by a truthful, direct mechaism! direct mechaism: A i = i truthful, direct mechaism: s i ( i)= i Hold for both DSE ad BNE Cosider a arbitrary (e.g., may be idirect) Pigzhog Tag Mechaism Desig With Urestricted Prefereces, Slide

13 Revelatio Priciple Impossibility Revelatio Priciple type θ type θ strategy s (θ ) M strategy s (θ ) Origial Mechaism outcome It turs out that ay social choice fuctio that ca be implemeted by some mechaism ca be implemeted by a truthful, direct mechaism! direct mechaism: A i = i truthful, direct mechaism: s i ( i)= i Hold for both DSE ad BNE Cosider a arbitrary (e.g., may be idirect) Recall that a mechaism defies a game, ad cosider a equilibrium s =(s,...,s ) Pigzhog Tag Mechaism Desig With Urestricted Prefereces, Slide

14 Revelatio Priciple Impossibility Revelatio Priciple type θ type θ strategy s (θ ) M strategy s (θ ) s (s (θ )) M s (s (θ )) Origial Mechaism ( outcome New Mechaism We ca costruct a ew direct mechaism, as show above This mechaism is truthful by exactly the same argumet that s was a equilibrium i the origial mechaism The agets do t have to lie, because the mechaism already lies for them. Pigzhog Tag Mechaism Desig With Urestricted Prefereces, Slide 2

15 Revelatio Priciple Impossibility Discussio of the Revelatio Priciple The set of equilibria is ot always the same i the origial mechaism ad revelatio mechaism of course, we ve show that the revelatio mechaism does have the origial equilibrium of iterest however, i the case of idirect mechaisms, eve if the idirect mechaism had a uique equilibrium, the revelatio mechaism ca also have ew, bad equilibria So what is the revelatio priciple good for? recogitio that truthfuless is ot a restrictive assumptio for aalysis purposes, we ca cosider oly truthful mechaisms, ad be assured that such a mechaism exists recogitio that idirect mechaisms ca t do (iheretly) better tha direct mechaisms Pigzhog Tag Mechaism Desig With Urestricted Prefereces, Slide 3

16 Revelatio Priciple Impossibility Impossibility Result Theorem (Gibbard-Satterthwaite) Cosider ay social choice fuctio C of N ad O. If: O 3 (there are at least three outcomes); 2 C is oto; that is, for every o 2 O there is a preferece profile > such that C(>) =o; ad 3 C is implemetable i domiat-strategy (= truthful i domiat-strategy = strategy-proof). the C is dictatorial. Pigzhog Tag Mechaism Desig With Urestricted Prefereces, Slide 4

17 Revelatio Priciple Impossibility What does this mea? We should be discouraged about the possibility of implemetig arbitrary social-choice fuctios i mechaisms. However, i practice we ca circumvet the Gibbard-Satterthwaite theorem i two ways: use a weaker form of implemetatio ote: the result oly holds for domiat strategy implemetatio, ot e.g., Bayes-Nash implemetatio relax the implicit assumptio that agets are allowed to hold arbitrary prefereces Pigzhog Tag Mechaism Desig With Urestricted Prefereces, Slide 5

18 Quasiliear prefereces Quasiliear Mechaisms Properties Quasiliear prefereces restricted prefereces Defiitio (Quasiliear prefereces) Agets have quasiliear prefereces i a -player Bayesia game whe the set of outcomes is O = X R for a fiite set X, ad the utility of a aget i give joit type is give by u i (o, ) =u i (x, ) f i (p i ), where o =(x, p) is a elemet of O, u i : X 7! R is a arbitrary fuctio ad f i : R 7! R is a strictly icreasig. Questio: why is Quasiliear Pref. restricted? Pigzhog Tag Quasiliear Utility, Slide

19 Quasiliear prefereces Quasiliear Mechaisms Properties Quasiliear utility u i (o, ) =u i (x, ) f i (p i ) We split the mechaism ito a choice/allocatio rule ad a paymet rule: x 2 X is a discrete, o-moetary outcome p i 2 R is a moetary paymet (possibly egative) that aget i must make to the mechaism Implicatios: Pigzhog Tag Quasiliear Utility, Slide 2

20 Quasiliear prefereces Quasiliear Mechaisms Properties Quasiliear utility u i (o, ) =u i (x, ) f i (p i ) We split the mechaism ito a choice/allocatio rule ad a paymet rule: x 2 X is a discrete, o-moetary outcome p i 2 R is a moetary paymet (possibly egative) that aget i must make to the mechaism Implicatios: u i (x, ) is ot iflueced by the amout of moey a aget has agets do t care how much others are made to pay (though they ca care about how the choice a ects others.) Pigzhog Tag Quasiliear Utility, Slide 2

21 Quasiliear prefereces Quasiliear Mechaisms Properties Quasiliear utility u i (o, ) =u i (x, ) f i (p i ) We split the mechaism ito a choice/allocatio rule ad a paymet rule: x 2 X is a discrete, o-moetary outcome p i 2 R is a moetary paymet (possibly egative) that aget i must make to the mechaism Implicatios: u i (x, ) is ot iflueced by the amout of moey a aget has agets do t care how much others are made to pay (though they ca care about how the choice a ects others.) What is f i (p i )? Pigzhog Tag Quasiliear Utility, Slide 2

22 Quasiliear prefereces Quasiliear Mechaisms Properties Quasiliear Mechaism Defiitio (Quasiliear mechaism) A mechaism i the quasiliear settig (for a Bayesia game settig (N,O = X R,,p,u)) isatriple(a, x, p), where A = A A,whereA i is the set of actios available to aget i 2 N, x : A 7! (X) maps each actio profile to a distributio over choices, ad p : A 7! R maps each actio profile to a paymet for each aget. Pigzhog Tag Quasiliear Utility, Slide 3

23 Quasiliear prefereces Quasiliear Mechaisms Properties Further simplificatios: Direct Quasiliear Mechaism Defiitio (Direct quasiliear mechaism) A direct quasiliear mechaism (for a eviromet (N,O = X R,,p,u)) isapair(x, p). Itdefiesastadard mechaism i the quasiliear settig, where for each i, A i = i. Defiitio (Coditioal utility idepedece) A Bayesia game exhibits coditioal utility idepedece if for all agets i 2 N, for all outcomes o 2 O ad for all pairs of joit types ad 0 2 for which i = 0 i, it holds that u i(o, ) =u i (o, 0 ). I other words, u i oly depeds o i. Pigzhog Tag Quasiliear Utility, Slide 4

24 Quasiliear prefereces Quasiliear Mechaisms Properties Quasiliear Mechaisms with Coditioal Utility Idepedece Give coditioal utility idepedece, we ca write i s utility fuctio as u i (o, i ) it does ot deped o the other agets types A aget s valuatio for choice x 2 X: v i (x) =u i (x, i ) the maximum amout i would be willig to pay to get x Thik of v i as i s type from ow o Alterate defiitio of direct mechaism: ask agets i to declare v i (x) for each x 2 X Defie ˆv i as the valuatio that aget i declares to such a direct mechaism may be di eret from his true valuatio v i Also defie the tuples ˆv, ˆv i Pigzhog Tag Quasiliear Utility, Slide 5

25 Quasiliear prefereces Quasiliear Mechaisms Properties Truthfuless Defiitio (Truthfuless) Aquasiliearmechaismistruthful if it is direct ad 8i8v i, aget i s equilibrium strategy is to adopt the strategy ˆv i = v i. Our defiitio before, adapted for the quasiliear settig Pigzhog Tag Quasiliear Utility, Slide 6

26 Quasiliear prefereces Quasiliear Mechaisms Properties E ciecy Defiitio (E ciecy) Aquasiliearmechaismisstrictly Pareto e ciet, orjust e ciet, if i equilibrium it selects a choice x such that 8v8x 0, X X v i (x) v i (x 0 ). i i A e ciet mechaism selects the choice that maximizes the sum of agets utilities, disregardig moetary paymets. How is this related to the Pareto e ciecy from GT? Pigzhog Tag Quasiliear Utility, Slide 7

27 Quasiliear prefereces Quasiliear Mechaisms Properties E ciecy Defiitio (E ciecy) Aquasiliearmechaismisstrictly Pareto e ciet, orjust e ciet, if i equilibrium it selects a choice x such that 8v8x 0, X X v i (x) v i (x 0 ). i i A e ciet mechaism selects the choice that maximizes the sum of agets utilities, disregardig moetary paymets. How is this related to the Pareto e ciecy from GT? if we iclude the mechaism as a aget, all Pareto-e ciet outcomes ivolve the same choice (ad di eret paymets) ay outcome ivolvig aother choice is Pareto-domiated: some agets could make a side-paymet to others such that all would prefer the swap Pigzhog Tag Quasiliear Utility, Slide 7

28 Quasiliear prefereces Quasiliear Mechaisms Properties E ciecy Defiitio (E ciecy) Aquasiliearmechaismisstrictly Pareto e ciet, orjust e ciet, if i equilibrium it selects a choice x such that 8v8x 0, X X v i (x) v i (x 0 ). i i Called ecoomic e ciecy to distiguish from other (e.g., computatioal) otios Also called social-welfare maximizatio Note: defied i terms of true (ot declared) valuatios. Pigzhog Tag Quasiliear Utility, Slide 7

29 Quasiliear prefereces Quasiliear Mechaisms Properties Budget Balace Defiitio (Budget balace) A quasiliear mechaism is budget balaced whe 8v, X i p i (s(v)) = 0, where s is the equilibrium strategy profile. regardless of the agets types, the mechaism collects ad disburses the same amout of moey from ad to the agets Pigzhog Tag Quasiliear Utility, Slide 8

30 Quasiliear prefereces Quasiliear Mechaisms Properties Budget Balace Defiitio (Budget balace) A quasiliear mechaism is budget balaced whe 8v, X i p i (s(v)) = 0, where s is the equilibrium strategy profile. regardless of the agets types, the mechaism collects ad disburses the same amout of moey from ad to the agets relaxed versio: weak budget balace: 8v, X i p i (s(v)) 0 the mechaism ever takes a loss, but it may make a profit Pigzhog Tag Quasiliear Utility, Slide 8

31 Quasiliear prefereces Quasiliear Mechaisms Properties Budget Balace Defiitio (Budget balace) A quasiliear mechaism is budget balaced whe 8v, X i p i (s(v)) = 0, where s is the equilibrium strategy profile. regardless of the agets types, the mechaism collects ad disburses the same amout of moey from ad to the agets Budget balace ca be required to hold ex ate: X E v p i (s(v)) = 0 i the mechaism must break eve or make a profit oly o expectatio Pigzhog Tag Quasiliear Utility, Slide 8

32 Quasiliear prefereces Quasiliear Mechaisms Properties Idividual-Ratioality Defiitio (Ex iterim idividual ratioality) A mechaism is ex iterim idividual ratioal whe 8i8v i, E v i v i v i (x (s i (v i ),s i (v i ))) p i (s i (v i ),s i (v i )) 0, where s is the equilibrium strategy profile. o aget loses by participatig i the mechaism. ex iterim because it holds for every possible valuatio for aget i, but averages over the possible valuatios of the other agets. Pigzhog Tag Quasiliear Utility, Slide 9

33 Quasiliear prefereces Quasiliear Mechaisms Properties Idividual-Ratioality Defiitio (Ex iterim idividual ratioality) A mechaism is ex iterim idividual ratioal whe 8i8v i, E v i v i v i (x (s i (v i ),s i (v i ))) p i (s i (v i ),s i (v i )) 0, where s is the equilibrium strategy profile. o aget loses by participatig i the mechaism. ex iterim because it holds for every possible valuatio for aget i, but averages over the possible valuatios of the other agets. Defiitio (Ex post idividual ratioality) Amechaismisex post idividual ratioal whe 8i8v, v i (x (s(v))) p i (s(v)) 0, wheres is the equilibrium strategy profile. Pigzhog Tag Quasiliear Utility, Slide 9

34 Fially, a positive result! Recall that i the quasiliear utility settig, a mechaism ca be defied as a choice rule ad a paymet rule. The Groves mechaism is a mechaism that satisfies: truthfuless i domiat strategies e ciecy I geeral it s ot: budget balaced idividual-ratioal...though we ll see later that we ca recover these properties. Pigzhog Tag Groves Mechaism, Slide

35 The Groves Mechaism Defiitio (Groves mechaism) The Groves mechaism is a direct quasiliear mechaism (x, p), where X x (ˆv) = arg max ˆv i (x) x i X p i (ˆv) =h i (ˆv i ) ˆv j (x (ˆv)) j6=i Pigzhog Tag Groves Mechaism, Slide 2

36 The Groves Mechaism x (ˆv) = arg max x p i (ˆv) =h i (ˆv i ) X ˆv i (x) i X ˆv j (x (ˆv)) j6=i The choice rule should ot come as a surprise (why ot?) Pigzhog Tag Groves Mechaism, Slide 3

37 The Groves Mechaism x (ˆv) = arg max x p i (ˆv) =h i (ˆv i ) X ˆv i (x) i X ˆv j (x (ˆv)) j6=i The choice rule should ot come as a surprise (why ot?) Give the mechaism is both truthful ad e ciet: these properties etail the give choice rule. Pigzhog Tag Groves Mechaism, Slide 3

38 The Groves Mechaism x (ˆv) = arg max x p i (ˆv) =h i (ˆv i ) X ˆv i (x) i X ˆv j (x (ˆv)) j6=i The choice rule should ot come as a surprise (why ot?) Give the mechaism is both truthful ad e ciet: these properties etail the give choice rule. So what s goig o with the paymet rule? the aget i must pay some amout h i (ˆv i ) that does t deped o his ow declared valuatio the aget i is paid P j6=i ˆv j(x (ˆv)), the sum of the others valuatios for the chose outcome Pigzhog Tag Groves Mechaism, Slide 3

39 Groves Truthfuless Theorem Truth tellig is a domiat strategy uder the Groves mechaism. Cosider a situatio where every aget j other tha i follows some arbitrary strategy ˆv j.cosiderageti s problem of choosig the best strategy ˆv i.asa shorthad, we will write ˆv =(ˆv i, ˆv i ). The best strategy for i is oe that solves max ˆv i v i (x (ˆv)) p(ˆv) Substitutig i the paymet fuctio from the Groves mechaism, we have 0 max ˆv i v i (x (ˆv)) h i (ˆv i )+ X j6=i ˆv j (x (ˆv)) A Sice h i (ˆv i ) does ot deped o ˆv i,itissu 0 ciet to solve max ˆv i v i (x (ˆv)) + X j6=i ˆv j (x (ˆv)) A. Pigzhog Tag Groves Mechaism, Slide 4

40 Groves Truthfuless max ˆv i 0 v i (x (ˆv)) + X j6=i ˆv j (x (ˆv)) A. The oly way the declaratio ˆv i iflueces this maximizatio is through the choice of x. If possible, i would like to pick a declaratio ˆv i that will lead the mechaism to pick a x 2 X which solves 0 max x v i (x)+ X j6=i ˆv j (x) A. () Uder the Groves mechaism,! X x (ˆv) = arg max ˆv i (x) x i = arg max x 0 ˆv i (x)+ X j6=i ˆv j (x) A. The Groves mechaism will choose x i a way that solves the maximizatio problem i Equatio () whe i declares ˆv i = v i.becausethisargumetdoes ot deped i ay way o the declaratios of the other agets, truth-tellig is a domiat strategy for aget i. Pigzhog Tag Groves Mechaism, Slide 5

41 Proof ituitio exteralities are iteralized agets may be able to chage the outcome to aother oe that they prefer, by chagig their declaratio however, their utility does t just deped o the outcome it also depeds o their paymet sice they get paid the (reported) utility of all the other agets uder the chose allocatio, they ow have a iterest i maximizig everyoe s utility rather tha just their ow i geeral, DS truthful mechaisms have the property that a aget s paymet does t deped o the amout of his declaratio, but oly o the other agets declaratios the aget s declaratio is used oly to choose the outcome, ad to set other agets paymets Pigzhog Tag Groves Mechaism, Slide 6

42 Groves Uiqueess Theorem (Gree La ot) A e ciet social choice fuctio C : R X! X R ca be implemeted i domiat strategies for agets P with urestricted quasiliear utilities oly if p i (v) =h(v i ) j6=i v j(x (v)). it turs out that the same result also holds for the broader class of Bayes Nash icetive-compatible e ciet mechaisms. Pigzhog Tag Groves Mechaism, Slide 7

43 VCG mechaism Defiitio (Clarke tax) The Clarke tax sets the h i term i a Groves mechaism as h i (ˆv i )= X j6=i ˆv j (x (ˆv i )). Defiitio (Vickrey-Clarke-Groves (VCG) mechaism) The Vickrey-Clarke-Groves mechaism is a direct quasiliear mechaism (x, p), where x (ˆv) = arg max x p i (ˆv) = X j6=i X ˆv i (x) i ˆv j (x (ˆv i )) X ˆv j (x (ˆv)) j6=i Pigzhog Tag VCG, Slide

44 VCG discussio x (ˆv) = arg max x p i (ˆv) = X j6=i X ˆv i (x) i ˆv j (x (ˆv i )) X j6=i ˆv j (x (ˆv)) You get paid everyoe s utility uder the allocatio that is actually chose except your ow, but you get that directly as utility The you get charged everyoe s utility i the world where you do t participate Thus you pay your social cost Pigzhog Tag VCG, Slide 2

45 VCG discussio Questios: who pays 0? x (ˆv) = arg max x p i (ˆv) = X j6=i X ˆv i (x) i ˆv j (x (ˆv i )) X j6=i ˆv j (x (ˆv)) Pigzhog Tag VCG, Slide 3

46 VCG discussio x (ˆv) = arg max x p i (ˆv) = X j6=i X ˆv i (x) i ˆv j (x (ˆv i )) X j6=i ˆv j (x (ˆv)) Questios: who pays 0? agets who do t a ect the outcome Pigzhog Tag VCG, Slide 3

47 VCG discussio x (ˆv) = arg max x p i (ˆv) = X j6=i X ˆv i (x) i ˆv j (x (ˆv i )) X j6=i ˆv j (x (ˆv)) Questios: who pays 0? agets who do t a ect the outcome who pays more tha 0? Pigzhog Tag VCG, Slide 3

48 VCG discussio x (ˆv) = arg max x p i (ˆv) = X j6=i X ˆv i (x) i ˆv j (x (ˆv i )) X j6=i ˆv j (x (ˆv)) Questios: who pays 0? agets who do t a ect the outcome who pays more tha 0? (pivotal) agets who make thigs worse for others by existig Pigzhog Tag VCG, Slide 3

49 VCG discussio x (ˆv) = arg max x p i (ˆv) = X j6=i X ˆv i (x) i ˆv j (x (ˆv i )) X j6=i ˆv j (x (ˆv)) Questios: who pays 0? agets who do t a ect the outcome who pays more tha 0? (pivotal) agets who make thigs worse for others by existig who gets paid? Pigzhog Tag VCG, Slide 3

50 VCG discussio x (ˆv) = arg max x p i (ˆv) = X j6=i X ˆv i (x) i ˆv j (x (ˆv i )) X j6=i ˆv j (x (ˆv)) Questios: who pays 0? agets who do t a ect the outcome who pays more tha 0? (pivotal) agets who make thigs worse for others by existig who gets paid? (pivotal) agets who make thigs better for others by existig Pigzhog Tag VCG, Slide 3

51 VCG properties x (ˆv) = arg max x p i (ˆv) = X j6=i X ˆv i (x) i ˆv j (x (ˆv i )) X j6=i ˆv j (x (ˆv)) Because oly pivotal agets have to pay, VCG is also called the pivot mechaism It s domiat-strategy truthful, because it s a Groves mechaism Pigzhog Tag VCG, Slide 4

52 Selfish routig example A F C E B D R - R - R What outcome will be selected by x? Pigzhog Tag VCG, Slide 5

53 Selfish routig example A F C E B D R - R - R What outcome will be selected by x? path ABEF. Pigzhog Tag VCG, Slide 5

54 Selfish routig example A F C E B D R - R - R What outcome will be selected by x? path ABEF. How much will AC have to pay? Pigzhog Tag VCG, Slide 5

55 Selfish routig example 3 A 2 R B C R - D 2 R 5 E F What outcome will be selected by x? path ABEF. How much will AC have to pay? The shortest path takig his declaratio ito accout has a legth of 5, ad imposes a cost of 5 o agets other tha him (because it does ot ivolve him). Likewise, the shortest path without AC s declaratio also has a legth of 5. Thus, his paymet p AC =( 5) ( 5) = 0. This is what we expect, sice AC is ot pivotal. Likewise, BD, CE, CF ad DF will all pay zero. Pigzhog Tag VCG, Slide 5

56 Selfish routig example A F C E B D R - R - R How much will AB pay? Pigzhog Tag VCG, Slide 6

57 Selfish routig example 3 A 2 R B C R - D 2 R 5 E F How much will AB pay? The shortest path takig AB s declaratio ito accout has a legth of 5, ad imposes a cost of 2 o other agets. The shortest path without AB is ACEF, which has a cost of 6. Thus p AB =( 6) ( 2) = 4. Pigzhog Tag VCG, Slide 6

58 Selfish routig example A F C E B D R - R - R How much will BE pay? Pigzhog Tag VCG, Slide 7

59 Selfish routig example A F C E B D R - R - R How much will BE pay? p BE =( 6) ( 4) = 2. Pigzhog Tag VCG, Slide 7

60 Selfish routig example A F C E B D R - R - R How much will BE pay? p BE =( 6) ( 4) = 2. How much will EF pay? Pigzhog Tag VCG, Slide 7

61 Selfish routig example A F C E B D R - R - R How much will BE pay? p BE =( 6) ( 4) = 2. How much will EF pay? p EF =( 7) ( 4) = 3. Pigzhog Tag VCG, Slide 7

62 Selfish routig example A F C E B D R - R - R How much will BE pay? p BE =( 6) ( 4) = 2. How much will EF pay? p EF =( 7) ( 4) = 3. EF ad BE have the same costs but are paid di eret amouts. Why? Pigzhog Tag VCG, Slide 7

63 Selfish routig example 3 A 2 R B C R - D 2 R 5 E F How much will BE pay? p BE =( 6) ( 4) = 2. How much will EF pay? p EF =( 7) ( 4) = 3. EF ad BE have the same costs but are paid di eret amouts. Why? EF has more market power: for the other agets, the situatio without EF is worse tha the situatio without BE. Pigzhog Tag VCG, Slide 7