Price of Stability and Introduction to Mechanism Design

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1 Algorihmic Game Theory Summer 2017, Week 5 ETH Zürich Price of Sabiliy and Inroducion o Mechanim Deign Paolo Penna Thi i he lecure where we ar deigning yem which involve elfih player. Roughly peaking, we conider wo approache: Sugge he player he be poible equilibrium, and meaure how well hi approach perform (Price of Sabiliy); Provide compenaion o he player o induce hem o o behave in a deirable way (Mechanim Deign). 1 Price of Sabiliy Example 1 (neceiy of paymen rule). In hi graph, each edge can be buil a a co of 1. Each player i wan o go from i o i, and can pay a par of he co for building he neceary edge An edge exi if i co i covered. If a player canno connec o i arge node i hen he/he ha a huge co. The raegy of a player i i o pecify, for each edge e, how much he/he conribue (he co for he player i hi/her oal conribuion). Thi game ha no pure Nah equilibria (Exercie!). The above example ugge ha we need rule on how he co of an edge i hared among he player ha ue i. A naural cheme i o divide he co of an edge equally among hoe ha ue i. Thi lead o o-called fair co haring game, which are congeion game wih delay d r (x) = c r /x where c r 0 i he co for building reource r. We have n player and m reource. Player i elec ome reource, i.e., hi raegy e i S i [1,..., m]. The co c r i aigned in equal hare o he player allocaing r (if any). Tha i, for raegy profile denoe by n r () he number of player uing reource r. Then he co c i () of player i Verion : Ocober 23, 2017 Page 1 of 10

2 i c i () = r i d r (n r (S)) = r i c i /n r (). A in he previou lecure, he ocial co of a ae i he um of all player co co() = i c i (). Social co = Co of ued reource co() = c i () = c r. (1) i r: n r() 1 Here i he imple proof: co() = i c i () = i d r (n r ()) = r i n r () c r /n r () = c r. r: n r() 1 r: n r() 1 The price of anarchy for pure Nah equilibria can be a big a he number of player n, even in a ymmeric game. Example 2 (Lower bound on price of anarchy). For ɛ > 0, conider he following nework co haring game, in which edge label indicae he co c r of hi reource: 1 + ɛ n I i a pure Nah equilibrium if all player ue he boom edge, wherea he ocial opimum would be ha all uer ue he op edge. Alhough hi i a very ylized example, here are indeed example of uch bad equilibria occurring in realiy. A prime example are mediocre echnologie, which win again beer one ju becaue hey are in he marke early and ge heir hare. A a concree example conider ocial nework or meaging ervice. Even if you were o deign and code-up a revoluionary new ocial nework or meaging ervice, would people acually wich over from, ay, Facebook or Whaapp, if all heir friend are uing i? 1.1 Be Equilibria and Price of Sabiliy Le u fir urn o he price of abiliy, which compare he be equilibrium wih he ocial opimum. We can ee hi hi concep in wo way: Try o reduce he negaive effec of elfih behavior by uggeing a good equilibrium o he player (raher han le hem compue an equilibrium by hemelve); Underand if he inefficiency lo i ju a queion of equilibrium elecion, or i i inrinic in he fac ha player will alway play ome equilibrium. Verion : Ocober 23, 2017 Page 2 of 10

3 Definiion 3 (Price of Sabiliy). For a co-minimizaion game, he price of abiliy for Eq i defined a P os Eq = min p Eq co(p) min S co(), where Eq i a e of probabiliy diribuion over he e of ae S, and co(p) i he expeced ocial co of p. a a Recall definiion from previou lecure: co(p) := E p [co()] = S p()co(). Exercie 1. Prove ha in a ymmeric co haring game every ocial opimum i a pure Nah equilibrium. Therefore, he price of abiliy for pure Nah equilibria i 1. In aymmeric game hi i no longer rue a he ocial opimum i no necearily a pure Nah equilibrium. Example 4 (Lower bound on price of abiliy). Conider he following game wih n player. Each player i ha ource node i and deinaion node n n 1 + ɛ v A player ha wo poible raegie: Eiher ake he direc edge or ake he deour via v. The ocial opimum le all player chooe he indirec pah, which lead o a ocial co of 1 + ɛ. Thi, however, i no a Nah equilibrium. Player n, who currenly face a co of (1 + ɛ)/n, would op ou and ake he direc edge, which would give him co 1/n. Therefore, he only pure Nah equilibrium le all player chooe heir direc edge, yielding a ocial co of H n, where H n harmonic number. We have H n = Θ(log n). = n i=1 1 i = n i he n-h Theorem 5. The price of abiliy for pure Nah equilibria in fair co haring game i a mo H n. Proof. Le u fir derive upper and lower bound on Roenhal poenial funcion ha apply o any ae, wheher a equilibrium or no. Verion : Ocober 23, 2017 Page 3 of 10

4 To obain an upper bound we can ue he definiion of he poenial funcion and he fac ha each reource i ued by no more han n player: Φ() = r n r() k=1 c r k = r: n r() 1 r: n r() 1 c r c r H n = co() H n. ( ) n r () A for he lower bound, each erm in he righ hand ide of (2) i a lea c r, hu Φ() c r = co(). r: n r() 1 Now le be a ae minimizing he poenial, and be a ae minimizing he ocial co. By definiion, Φ( ) Φ( ), hu implying co( ) Φ( ) Φ( ) co( ) H n. Noe ha mu be a pure Nah equilibrium (becaue i i a local minimum for he poenial). Thi prove ha here i a pure Nah equilibrium ha i only a facor of H n more coly han. We conclude ha in boh ymmeric and aymmeric co haring game our aumpion ha player play an equilibrium ha a raher mild impac. The poible ource of inefficiency i he fac ha player may no be able o elec a paricularly good equilibrium. Anoher ue of Poenial: bound he price of abiliy Theorem 6. In a co-minimizaion poenial game, if i a ae minimizing he poenial, hen co( ) P os min S co(). Proof. In poenial game every ae which i a local minimum for he poenial i a pure Nah equilibrium (in paricular, he global minimum ). 2 Inroducion o Mechanim Deign In hi ecion we inroduce mechanim wih money. The main idea i ha we provide compenaion o he player in order o induce hem o behave a we deire. (2) Verion : Ocober 23, 2017 Page 4 of 10

5 2.1 Example and Main Inuiion Problem (2 Link, 1 Packe) 1 2 We wan o elec he be (minimum co), bu he following complicaion arie: Each link i a player i and i i he ime (laency) ha hi link uffer if we ue i for ending our meage; We do no know i and we hu have o ak he player o repor o u i co; Player can chea and repor a co which i differen from he rue one. The reaon for a player i o chea i imple: if i link i eleced, hen he player ha a co i, while if i link i no eleced, here i no co for he player. For inance, if hee were he rue co: 3 5 he op player ha an incenive o declare a higher co (ay 6) o ha our algorihm elec he oher link. I i herefore naural o inroduce compenaion or paymen. Fir aemp: Pay he repored co Suppoe we ak each player wha i he co of i link, and we provide ha amoun of money a compenaion. Now he op link ill ha a reaon o chea: by reporing 4.99 he/he can ge a higher paymen (can peculae). Conider hi uiliy in he wo cae: Truh-elling (repor 3): receive 3 a paymen, and ha co 3 for being eleced; Uiliy = Paymen - Co = 3 3 = 0. Cheaing (repor 4.99): receive 4.99 a paymen, and ha co 3 for being eleced. Uiliy = Paymen - Co = > 0. Surpriingly here are paymen ha alway induce he player o ell he ruh. The idea come from he following ype of aucion for elling one iem o poenial buyer. 2nd Price Aucion (Vickrey) 1. Pick he highe offer (bid) 2. The winner pay he 2nd-highe offer Verion : Ocober 23, 2017 Page 5 of 10

6 In hi aucion, each buyer end hi/her offer in a ealed envelope, and he winner i deermined a above. Here buyer have a privae valuaion for he iem, and hey naurally ry o ge he iem for a lower price (uiliy = valuaion price). For example: Bid: 7, 10, 3 Winner: econd player Price: 7 Noe for inance, ha he econd buyer canno ge he iem for a beer price, e.g., by bidding 8; bidding 6 would make hi buyer o loe he aucion. Thi ranlae immediaely ino he following oluion for our problem: Second Aemp (Vickrey aucion): 1. Selec he be link (min repored co) 2. Pay hi link an amoun equal o he 2nd-be co 3 5 pick op link pay 5 One can argue on hi example ha neiher player can improve hi/her uiliy by cheaing. In fac hi i rue no maer which were he rue co of he wo link. We prove hi and more general reul below. Now we wan o build ome inuiion and generalize hi problem. Obervaion 7. In he example above, if we were paying he repored co, he maximum hi agen could peculae i 5 (he co of he oher link). To preven him/her from cheaing, we pay direcly hi amoun o him/her. Le u conider he ame problem, ju lighly more general. Problem (Shore Pah) Imagine we pay each eleced link an amoun equal o he maximum he/he could peculae if he paymen wa he repored co: pay = 10 1 = 9 pay = 10 3 = Verion : Ocober 23, 2017 Page 6 of 10

7 which can be alo een a follow, Alernaive hore pah Shore pah Paymen (agen of co 1) Thi i perhap he main inuiion we ue in a general conrucion decribed in Secion Formal Seing We conider he following eing. A i a e of feaible alernaive (or oluion) Each player i ha privae rue co funcion i : A R which give hi/her co for every poible alernaive, i (a). Player i can mirepor hi/her rue co o ome oher co funcion c i : A R. If agen i receive a paymen p i and alernaive a i choen, hen hi/her uiliy i p i i (a). A mechanim i a combinaion of an algorihm elecing a oluion and a paymen rule aigning paymen o each agen. The mechanim ake in inpu he co repored by he player. A mechanim i a pair (A, P ) which on inpu he co c = (c 1,..., c n ) repored by he player, oupu A oluion A(c) A; A paymen P i (c) for each player i. The correponding uiliy for each agen i i u i (c i ) := P i (c) i (A(c)). Wha we wan i ha cheaing i never convenien for a player. Tha i, ruh-elling i a dominan raegy. Verion : Ocober 23, 2017 Page 7 of 10

8 A mechanim (A, P ) i ruhful if for all i, for all u i (c 1,..., c i 1, i, c i+1,..., c n i ) u i (c 1,..., c i 1, c i, c i+1,..., c n i ) (3) i }{{} rue co, for all c 1,..., c i 1, c i+1,..., c }{{ n, and for all c } i co repored by oher }{{} co repored by i. 2.3 VCG Mechanim We nex preen a very general conrucion of ruhful mechanim which will olve he hore-pah problem a a pecial cae. Inuiively, he mechanim doe he following: Compue he oluion minimizing he ocial co (um of player co) wih repec o he repored co (algorihm); Pay each agen hi/her marginal conribuion o hi oluion (paymen). Thi par ue he inuiion decribed a he end of Secion 2.1. Le u conider he ocial co of a oluion a wih repec o co c = (c 1,..., c n ) a co(a, c) := i c i (a) and le u denoe he opimum a op(c) := min co(a, c). a A A VCG mechanim i a pair (A, P ) uch ha A in an opimal algorihm: P i of he following form: co(a(c), c) = op(c) for all c; P i (c) = Q i (c i ) j i c j (A(c)) where Q i i an arbirary funcion independen of c i. Thi conrucion yield ruhful mechanim: Theorem 8 (Vickrey-Clarke-Grove). VCG mechanim are ruhful. Proof. Fix an agen i, and he co c i repored by he oher. Le := (c 1,..., c i 1, i, c i+1,..., c n ) be he vecor in which i i ruh-elling. Verion : Ocober 23, 2017 Page 8 of 10

9 Claim 9. The uiliy of i for i u i ( i ) = Q i (c i ) op( ). Proof. By definiion of he paymen ( ) u i ( i ) = P i ( ) i (A( )) =Q i (c i ) c j (A(c)) + i (A( )) Oberve ha for any oluion a A c j (a) + i (a) =c 1 (a) + + c i 1 (a) + i (a) + c i+1 (a) + + c n (a) Therefore j i Thi prove he claim. =co(a, ). Claim 10. The uiliy of i for c i j i u i ( i ) = Q i (c i ) co(a(, )) = Q i (c i ) op( ). u i (c i ) = Q i (c i ) co(a(c), ). Proof. The ame proof of he previou claim wih c i in place of i yield he deired reul. Now ruhfulne i immediae from he wo claim: ince by definiion of op op( ) co(a(c), ) we ge u i ( i ) = Q i (c i ) op( ) Q i (c i ) co(a(c), ) = u i (c i ). Exercie 2. Show ha he mechanim for hore-pah problem in Secion 2.1 i VCG mechanim (and hu ruhful no maer he co of he edge). Decribe he mechanim on general graph, auming ha he graph given i 2-conneced (removing one edge never diconnec he graph). Wha i he funcion Q i ()? Exercie 3. The above heorem i a weaker form of he original VCG heorem. Prove ha he following weaker condiion on he algorihm i ufficien. There exi a ube R A of oluion uch ha A i opimal wih repec o hi fixed ube of oluion. Tha i, for every c A(c) R; co(a(c), c) = min a R co(a, c). Verion : Ocober 23, 2017 Page 9 of 10

10 2.4 Longe Pah We conclude by decribing a imple problem which doe no have a ruhful mechanim, no maer how clever or complicaed we make he paymen. Longe Pah: Find he longe pah (inead of hore-one) eleced eleced Conider he op link, and any paymen P i for i. Truhfulne would require o deal wih boh hee cae: 1. (rue co = 3, fale co = 6): 2. (rue co = 6, fale co = 3): P i (3, 5) 0 P i (6, 5) 3 P i (6, 5) 6 P i (3, 5) 0 which canno be boh aified. So here i no ruhful mechanim (A, P ) for hi problem. Recommended Lieraure For he Price of Sabiliy ee Chaper and Chaper 19.3 in Algorihmic Game Theory, N. Nian e al., page , For ruhful mechanim wih money: Chaper 2 (2nd price aucion) and Chaper 7 (VCG) in Tim Roughgarden, Tweny Lecure on Algorihmic Game Theory, Cambridge Univeriy Pre, 2016 (boh for problem wih privae valuaion). Chaper 9 in Algorihmic Game Theory, N. Nian e al., page , Verion : Ocober 23, 2017 Page 10 of 10