CMU Noncooperative games 3: Price of anarchy. Teacher: Ariel Procaccia

Transcription

1 CMU Nncperative games 3: Price f anarchy Teacher: Ariel Prcaccia

2 Back t prisn The nly Nash equilibrium in Prisner s dilemma is bad; but hw bad is it? Objective functin: scial cst = sum f csts NE is six times wrse than the ptimum Cperate Defect Cperate -1,-1-9,0 Defect 0,-9-6,-6 2

3 Anarchy and stability Fix a class f games, an bjective functin, and an equilibrium cncept The price f anarchy (stability) is the wrst-case rati between the wrst (best) bjective functin value f an equilibrium f the game, and that f the ptimal slutin In this lecture: Objective functin = scial cst Equilibrium cncept = Nash equilibrium 3

4 Example: Cst sharing n players in weighted directed graph G Player i wants t get frm s i t t i ; strategy space is s i t i paths s s 2 Each edge e has cst c e Cst f edge is split between all players using edge Cst f player is sum f csts ver edges n path t 1 t 2 4

5 Example: Cst sharing With n players, the example n the right has an NE with scial cst n Optimal scial cst is 1 Price f anarchy n Prve that the price f anarchy is at mst n s n 1 t 5

6 Example: Cst sharing Think f the 1 edges as cars, and the k edge as mass transit Bad Nash equilibrium with cst n Gd Nash equilibrium with cst k Nw let s mdify the example s 1 s t s n k 6

7 Example: Cst sharing OPT= k + 1 Only equilibrium has cst k H(n) price f stability is at least Ω(lg n) s 1 s 2 s n k + 1 We will shw that the price f stability is Θ(lg n) k 1 k 2 k n t 7

8 Ptential games A game is an exact ptential game if there n exists a functin Φ: i=1 S i R such that n fr all i N, fr all s i=1 S i, and fr all s i S i, cst i s i, s i cst i s = Φ s i, s i Φ(s) Why des the existence f an exact ptential functin imply the existence f a pure Nash equilibrium? 8

9 Ptential games Therem: the cst sharing game is an exact ptential game Prf: Let n e s be the number f players using e under s Define the ptential functin n e (s) Φ s = c e k e k=1 If player changes paths, pays edge, gets c e n e s c e n e s +1 fr each new fr each ld edge, s Δcst i = ΔΦ 9

10 Ptential games Therem: The cst f stability f cst sharing games is O(lg n) Prf: It hlds that cst s Φ s H n cst(s) Take a strategy prfile s that minimizes Φ s is an NE cst s Φ s Φ OPT H n cst(opt) 10

11 Cst sharing summary In every cst sharing game NE s, cst s n cst(opt) NE s such that cst s H n cst(opt) There exist cst sharing games s.t. NE s such that cst s n cst(opt) NE s, cst s H n cst(opt) 11

12 Cngestin games Generalizatin f cst sharing games n players and m resurces Each player i chses a set f resurces (e.g., a path) frm cllectin S i f allwable sets f resurces (e.g., paths frm s i t t i ) Cst f resurce j is a functin f j (n j ) f the number n j f players using it Cst f player is the sum ver used resurces 12

13 Cngestin games Therem [Rsenthal 1973]: Every cngestin game is an exact ptential game Prf: The exact ptential functin is n j (s) Φ s = f j i i=1 Therem [Mnderer and Shapley 1996]: Every ptential game is ismrphic t a cngestin game j 13

14 Netwrk frmatin games Each player is a vertex v Strategy f v: set f undirected edges t build that tuch v Strategy prfile s induces undirected graph G(s) Cst f building any edge is α cst v s = αn v s + u d(u, v), where n v = #edges bught by v, d is shrtest path in #edges cst s = d u, v + α E u v 14

15 Example: Netwrk frmatin NE with α = 3 Subptimal Optimal 15

16 Example: Netwrk frmatin Lemma: If α 2 then any star is ptimal, and if α 2 then a cmplete graph is ptimal Prf: Suppse α 2, and cnsider any graph that is nt cmplete Adding an edge will decrease the sum f distances by at least 2, and csts nly α Suppse α 2 and the graph cntains a star, s the diameter is at mst 2; deleting a nn-star edge increases the sum f distances by at mst 2, and saves α 16

17 Example: Netwrk frmatin Pll: Fr which values f α is any star an NE, and any cmplete graph an NE 1. α 1, α 1 2. α 2, α 1 3. α 1, nne 4. α 2, nne 17

18 Example: Netwrk frmatin Therem: 1. If α 2 r α 1, PS = 1 2. Fr 1 < α < 2, PS 4/3 Prf: Part 1 is immediate frm the lemma and pll Fr 1 < α < 2, the star is an NE, while OPT is a cmplete graph Wrst case rati when α 1: 2n n 1 (n 1) n n 1 + n(n 1)/2 = 4n2 6n + 2 3n 2 3n <

19 Example: Netwrk creatin Therem [Fabrikant et al. 2003]: The price f anarcy f netwrk creatin games is O( α) Lemma: If s is a Nash equilibrium that induces a graph f diameter d, then cst(s) O d OPT 19

20 Prf f lemma OPT = Ω αα + n 2 Buying a cnnected graph csts at least n 1 α There are Ω n 2 distances Distance csts dn 2 fcus n edge csts There are at mst n 1 cut edges fcus n nncut edges 20

21 Prf f lemma Claim: Let e = (u, v) be a nncut edge, then the distance d(u, v) with e deleted 2d V e = set f ndes s.t. the shrtest path frm u uses e Figure shws shrtest path aviding e, e = (u, v ) is the edge n the path entering V e P u is the shrtest path frm u t u P u d P v d 1 as P v e is shrtest path frm u t v e v u V e P v P u vv uu ee 21

22 Prf f lemma Claim: There are O(nn/α) nncut edges paid fr by any vertex v Let e = (u, v) be an edge paid fr by v By previus claim, deleting e increases distances frm v by at mst 2d V e G is an equilibrium α 2d V e V e α/2d n vertices verall can t be mre than 2nn/α sets V e 22

23 Prf f lemma O(nn/α) nncut edges per vertex O(nn) ttal payment fr these per vertex O(n 2 d) verall 23

24 Prf f therem By lemma, it is enugh t shw that the diameter at a NE 2 α Suppse d u, v 2k fr sme k By adding the edge (u, v), u pays α and imprves distance t secnd half f the u v shrtest path by 2k 1 + 2k = k 2 If d u, v > 2 α, it is beneficial t add edge 24