Prcng and Resource Allocaton Game Theoretc Models Zhy Huang Changbn Lu Q Zhang Computer and Informaton Scence December 8, 2009 Z. Huang, C. Lu, and Q. Zhang (CIS) Game Theoretc Models December 8, 2009 1 / 20
Motvaton The framework wth prces gnores a number of ssues that are mportant for analyss of resource allocaton n large-scale communcaton networks: 1 Centralzed sgnals may be mpractcal or mpossble 2 Prces are often set by multple servce provders wth the objectve of maxmzng revenue We nvestgate the mplcatons of proft maxmzng prcng by multple decentralzed servce provders. The model s of practcal mportance for a number of settngs: 1 Transportaton and communcaton networks 2 Markets n whch there are snob effects Z. Huang, C. Lu, and Q. Zhang (CIS) Game Theoretc Models December 8, 2009 2 / 20
Example l 1 (x) = x 2 /3 1 unt of traffc l 2 (x) = (2/3)x 1 The effcent allocaton that mnmzes the total delay cost l (x )x s x opt 1 = 2/3 and x opt 2 = 1/3 2 The equlbrum allocaton that equates delay on the two paths s x eq 1.73 and x eq 2.27 The equlbrum of traffc assgnment wthout prces can be neffcent. Z. Huang, C. Lu, and Q. Zhang (CIS) Game Theoretc Models December 8, 2009 3 / 20
Example (Cont d) l 1 (x) = x 2 /3 1 unt of traffc l 2 (x) = (2/3)x 1 Monopolst wll set prces p m 1 = (2/3)3 and p m 2 = (2/32 ). The resultng traffc n equlbrum wll be x m 1 = 2/3 and x m 2 = 1/3 2 Duopoly stuaton results n p1 d.61 and pd 2.44. The resultng traffc n equlbrum wll be x1 d.58 and x 2 d.42 Increasng competton can ncrease neffcency Z. Huang, C. Lu, and Q. Zhang (CIS) Game Theoretc Models December 8, 2009 4 / 20
Intuton for the Ineffcency of Duopoly The neffcency s related to a new source of monopoly power for each duopolst, whch they explot by dstortng the pattern of traffc: 1 Provder 1 charges hgher prce 2 Some traffc s pushed from route 1 to route 2 3 The congeston on route 2 s rased 4 The remanng traffc on route 1 become more locked-n Z. Huang, C. Lu, and Q. Zhang (CIS) Game Theoretc Models December 8, 2009 5 / 20
Model We are nterested n the problem of routng d unts of flow across I lnks. 1 I = {1,..., I }, set of lnks 2 x = [x 1,..., x I ], where x j denotes total flow on lnk j 3 l j (x j ), a convex, non-decreasng, and contnuously dfferentable flow-dependent latency functon for each lnk j n the network, l j (0) = 0 for all j 4 p j, prce per unt flow of lnk j 5 The cost per unt of traffc s the sum of prce and latency l j + p j 6 We assume that ths s the aggregate flow of many small users, who have a homogeneous reservaton utlty R and decde not to send ther flow f the effectve costs exceeds the reservaton utlty Z. Huang, C. Lu, and Q. Zhang (CIS) Game Theoretc Models December 8, 2009 6 / 20
Wardrop Equlbrum We adopt the Wardrop s prncple n characterzng the flow dstrbuton on the network. For a gven prce vector p 0, a vector x eq R I + s a Wardrop equlbrum f l (x eq ) + p = mn j {l j (x eq j l (x eq ) + p j }, wth x eq > 0, ) + p R, wth x eq > 0, I x eq d, wth I x eq = d f mn j {l j (x eq j ) + p j } < R. We denote the set of equlbrums at a gven prce vector p by W (p). Z. Huang, C. Lu, and Q. Zhang (CIS) Game Theoretc Models December 8, 2009 7 / 20
Wardrop Equlbrum Monotoncty Wardrop Equlbrums satsfy ntutve monotoncty propertes: Proposton 1 For some p j < p j, let x W ( p j, p j ) and x W (p j, p j ), then x j x j and x x for all j. 2 For some Ĩ I, suppose that p j < p j for all j Ĩ and p j = p j for all j / Ĩ, and x W ( p) and x W (p), then j Ĩ x j j Ĩ x j. Z. Huang, C. Lu, and Q. Zhang (CIS) Game Theoretc Models December 8, 2009 8 / 20
Socal Optmum A flow vector x opt s a socal optmum f t s an optmal soluton for the socal problem max x 0, P I x d (R l (x ))x. I 1 When I x = d, the above socal problem s equvalent as to mnmze I l (x )x. When I x < d, we charge a penalty of R for each unt of undelvered traffc 2 The socal problem maxmzes the socal surplus,.e., the dfference between users wllngness to pay and total latency For a gven vector x 0, we defne the value of the objectve functon n the socal problem S(x) = I (R l (x ))x. Z. Huang, C. Lu, and Q. Zhang (CIS) Game Theoretc Models December 8, 2009 9 / 20
Socal Optmum and Prcng The socal optmum soluton x opt whch maxmzes I (R l (x ))x subject to x 0 and I x d shall satsfes that x opt, x opt j > 0, l (x opt ) + l (x opt )x opt = l j (x opt j ) + l j(x opt j )x opt j. So the prcng p = l opt (x )x opt + c for any constant c acheves socal optmum soluton. Z. Huang, C. Lu, and Q. Zhang (CIS) Game Theoretc Models December 8, 2009 10 / 20
Monopoly Prcng and Equlbrum The monopolst sets the prces to maxmze hs proft gven by Π(p, x) = I p x, x W (p). Ths s a two-stage dynamc prcng-congeston game: 1 The monopolst antcpates the demand of users, and sets the prces p 2 The users choose ther flow vectors x accordng to the Wardrop equlbrum gven the prces p Defnton (Monopoly Equlbrum) A par (p opt, x opt ) s a monopoly equlbrum f x opt W (p opt ) and Π(p opt, x opt ) Π(p, x), p 0, x W (p). Z. Huang, C. Lu, and Q. Zhang (CIS) Game Theoretc Models December 8, 2009 11 / 20
Wardrop Equlbrum Unqueness Proposton (Unqueness for Strctly Increasng Latences) Assume l s strctly ncreasng for all. For any prce vector p 0, the set of Wardrop Equlbrums, W (p), s a sngleton. Proposton (Weak Unqueness for General Case) For any prce vector p 0, for any Wardrop Equlbrums x, ˆx W (p), Π(p, x) = Π(p, ˆx). Z. Huang, C. Lu, and Q. Zhang (CIS) Game Theoretc Models December 8, 2009 12 / 20
Monopoly Equlbrum and Subgame-Perfect Equlbrum Defnton A par (p, x ) s a subgame-perfect equlbrum (SPE) of the prcng congeston game f x W (p ) and for all p 0, there exsts x W (p) such that Π(p, x ) Π(p, x). The defnton of the monopoly equlbrum s stronger than the defnton of subgame-perfect equlbrum. However, gven the weak unqueness property, a par (p eq, x eq ) s an monopoly equlbrum f and only f t s an subgame-perfect equlbrum of the prcng congeston game. Z. Huang, C. Lu, and Q. Zhang (CIS) Game Theoretc Models December 8, 2009 13 / 20
Monopoly Equlbrum and Socal Optmum Theorem (Acemoglu and Ozdaglar 06) The prce-settng by monopolsts acheves effcency. Proof Sketch. Suppose p s the prces by monopolsts. The correspondng x satsfes that l (x ) + p = mn j {l j (x j ) + p j } R, x > 0. If l (x ) + p < R for some x > 0, then the monopolst could rase the all prces by the same amount so that the set of equlbrums does not change but the revenue s ncreased. So l (x ) + p = R for all x > 0. So S(x) = I (R l (x ))x = I p x. concdes the monopolsts objectve functon. Z. Huang, C. Lu, and Q. Zhang (CIS) Game Theoretc Models December 8, 2009 14 / 20
Olgopoly Prcng and Equlbrum 1 There s a set S of S servce provders 2 Each servce provder s S owns a dfference subset I s of the lnks 3 Servce provder s charges a prce p per unt on lnk I s 4 Gven the vector of prces of lnks owned by other servces provders p s, the payoff of servce provder s s Π s (p s, p s, x) = I s p x, x W (p s, p s ). We adopt the noton of Nash equlbrum and defne a vector (p eq, x eq ) 0 to be a Olgopoly Equlbrum f for all s S, x eq W (ps eq, p eq s ) and Π s (ps eq, p eq s, x eq ) Π s (p s, p eq s, x), p s 0, x W (p s, p eq s ). Z. Huang, C. Lu, and Q. Zhang (CIS) Game Theoretc Models December 8, 2009 15 / 20
Olgopoly Equlbrum and Subgame-Perfect Equlbrum Defnton A par (p, x ) s a subgame-perfect equlbrum of the prce competton game f x W (p ) and there exsts a functon x : R I + R I + such that x(p) W (p) for all p 0 and for all s S, Π s (p s, p s, x ) Π s (p s, p s, x(p s, p s)), p s 0. Smlar to the monopoly case, a par (p eq, x eq ) s an olgopoly equlbrum f and only f t s an subgame-perfect equlbrum of the prce competton game. Z. Huang, C. Lu, and Q. Zhang (CIS) Game Theoretc Models December 8, 2009 16 / 20
Effcency Metrc Gven a prce competton game wth latency functon {l } I, we defne the effcency metrc at some olgopoly equlbrum flow x eq as the rato of the socal surplus n x eq to the socal surplus n x opt : S(x eq ) S(x opt ). Ths effcency metrc concdes the noton of the prce of anarchy by [Koutsoupas and Papadmtrou 99]. Z. Huang, C. Lu, and Q. Zhang (CIS) Game Theoretc Models December 8, 2009 17 / 20
Tght Bound on the Effcency Metrc Theorem (Acemoglu and Ozdaglar 07) Consder a general parallel lnk network wth I 2 lnks and S servce provders, where provder s owns a set of lnks I s I. Then, for all prce competton games wth pure strategy equlbrum flow x eq, we have and the bound s tght. S(x eq ) S(x opt ) 5 6, Z. Huang, C. Lu, and Q. Zhang (CIS) Game Theoretc Models December 8, 2009 18 / 20
Upper Bound Example 1 A network wth I lnks, each s owned by a dfferent provder 2 The total flow s d = 1 3 The reservaton utlty s R = 1 4 The latency functons are gven by l 1 (x) = 0, l (x) = 3 (I 1)x, = 2,..., I. 2 The unque socal optmum for ths example s x opt = [1, 0,..., 0]. The olgopoly equlbrum s p eq = [1, 1 2,..., 1 2 ], x eq = [ 2 3, 1 3(I 1),..., 1 3(I 1) ]. Hence, the effcency metrc for ths example s 5 6. Z. Huang, C. Lu, and Q. Zhang (CIS) Game Theoretc Models December 8, 2009 19 / 20
Summary We nvestgate a model wth smple network structure, where there are a sngle source and a sngle destnaton and each route s a sngle edge between the source and the destnaton. Here s a few take-home ponts: 1 The equlbrum of traffc assgnment wthout prces can be neffcent 2 Increasng competton can ncrease neffcency 3 The extent of neffcency n the presence of olgopoly competton s bounded by 5 6 Z. Huang, C. Lu, and Q. Zhang (CIS) Game Theoretc Models December 8, 2009 20 / 20