Algorihmic Game Theory, Summer 2017 Inroducion o Congeion Game Lecure 1 (5 page) Inrucor: Thoma Keelheim In hi lecure, we ge o know congeion game, which will be our running example for many concep in game heory. Before coming o he formal definiion, le u conider he following example. We are given he following direced graph; here are hree player, who each wan o reach heir repecive deinaion node from heir ar node. Edge label indicae he co each player incur if hi edge i ued by one, wo, or all hree player. So, if he edge label i a, b, c and he edge i ued by wo player, hen each player ha co b for hi edge. 1 1 2 2 3 3 Player 2 and 3 do no have any choice, bu player 1 ha. He can eiher ue he direc edge or go via 2 and 2. Tha i, we have he following wo ae. Sae A: Sae B: 1 1 1 1 2 2 2 2 3 3 3 3 ocial co: 4 + 2 + 2 = 8 ocial co: 3 + 3 + 3 = 9 We oberve ha Sae A ha a maller ocial co han Sae B. However, player 1 prefer Sae B becaue hi individual co i maller here. In conra o Sae A, Sae B i able; i i an equilibrium. We will inroduce a general model ha allow u o capure hee effec. We will ak queion uch a: Are here equilibria? How can hee equilibria be found? How much performance i lo due o elfihne? 1 Formal Definiion Definiion 1.1 (Congeion Game (Roenhal 1973)). A congeion game i a uple Γ = (N, R, (Σ i ) i N, (d r ) r R ) wih N = {1,..., n}, e of player
Algorihmic Game Theory, Summer 2017 Lecure 1 (page 2 of 5) R = {1,..., m}, e of reource Σ i 2 R, raegy pace of player i d r : {1,..., n} Z, delay funcion or reource r For any ae S = (S 1,..., S n ) Σ 1 Σ n, n r (S) = {i N r S i } : number of player wih r S i d r (n r (S)): delay of reource r δ i (S) = r S i d r (n r ): delay of player i The co of player i in ae S i c i (S) = δ i (S), ha i, player aim a minimizing heir delay. Our above example i a nework congeion game: There i a graph G = (V, E) wih dedicaed ource-ink pair ( 1, 1 ),..., ( n, n ) uch ha Σ i i he e of - pah. Definiion 1.2. A raegy S i i called a be repone for player i N again a profile of raegie S i := (S 1,..., S i 1, S i+1,..., S n ) if c i (S i, S i ) c i (S i, S i) for all S i Σ i. A ae S Σ 1 Σ n i called a pure Nah equilibrium if S i i a be repone again he oher raegie S i for every player i N. 2 Exience of Pure Nah Equilibria A our fir reul, we will how every congeion game ha a pure Nah equilibrium. We will alk abou improvemen ep. The pair of ae (S, S ) i an improvemen ep if here i ome player i N uch ha c i (S ) < c i (S) and S i = S i. Example 1.3. A equence of (be repone) improvemen ep: ar: afer fir improvemen (red player): 6, 6 6, 6 afer econd improvemen (blue player): 6, 6 afer hird improvemen (red player): 6, 6
Algorihmic Game Theory, Summer 2017 Lecure 1 (page 3 of 5) d r (k) d r (k) 1 2 3 4 5 6 1 2 3 4 5 6 Figure 1: Proof of Lemma 1.6: The conribuion of wo reource r and r o he poenial i he haded area. If a player change from r o r, hi delay change exacly a he poenial value (difference of red area). Theorem 1.4 (Roenhal 1973). For every congeion game, every equence of improvemen ep i finie. Thi reul immediaely implie Corollary 1.5. Every congeion game ha a lea one pure Nah equilibrium. Proof of Theorem 1.4. Roenhal analyi i baed on a poenial funcion argumen. every ae S, le Φ(S) = r R Thi funcion i called Roenhal poenial funcion. d r (k). Lemma 1.6. Le S be any ae. Suppoe we go from S o a ae S by an improvemen ep of player i. hen Φ(S ) Φ(S) = c i (S ) c i (S). Proof. The poenial Φ(S) can be calculaed by inering he player one afer he oher in any order, and umming he delay of he player a he poin of ime a heir inerion. Wihou lo of generaliy player i i he la player ha we iner when calculaing Φ(S). Then he poenial accouned for player i correpond o he delay of player i in ae S. When going from S o S, he delay of i decreae by, and, hence, Φ decreae by a well (ee Figure 2 for an example. The lemma how ha Φ i a o-called exac poenial, i.e., if a ingle player decreae i laency by a value of > 0, hen Φ decreae by exacly he ame amoun. Furher oberve ha (i) he delay value are ineger o ha, for every improvemen ep, c i (S ) c i (S) 1, (ii) for every ae S, Φ(S) r R i=1 d r(i), (iii) for every ae S, Φ(S) r R i=1 d r(i). Conequenly, he number of improvemen i upper-bounded by 2 r R i=1 d r(i) and hence finie. For
Algorihmic Game Theory, Summer 2017 Lecure 1 (page 4 of 5) 3 Convergence Time of Improvemen Sep Roenhal heorem how ha any equence of improvemen ep i finie. However, i doe no give any guaranee how many improvemen ep are needed o reach a Nah equilibrium. A rivial upper bound on he lengh of any (finie) equence of improvemen ep i he overall number of ae, which i a mo 2 mn. However, hi i only a very poor guaranee and by no mean igh. We will how a ignificanly beer, namely polynomial, guaranee for ingleon congeion game. In hi ubcla of congeion game every player wan o allocae only a ingle reource a a ime from a ube of allowed reource. Formally: Definiion 1.7 (Singleon Game). A congeion game i called ingleon if, for every i N and every R Σ i, i hold ha R = 1. Alhough hi conrain on he raegy e i quie rericive, here are ill up o m n differen ae. Example 1.8 (Singleon Congeion Game). Conider a erver farm wih hree erver a, b, c (reource) and hree player 1,2,3 each of which wan o acce a ingle erver. 50 30 20 90 70 30 17 16 15 1 2 3 The colored arrow indicae a pure Nah equilibrium. Theorem 1.9. In a ingleon congeion game wih n player and m reource, all improvemen equence have lengh O(n 2 m). Proof idea: Replace original delay by bounded ineger value wihou changing he preference of he player. Upper bound on he maximum poenial wr new delay. Due o ineger value, decreae of poenial in an improvemen ep i a lea 1. Hence, lengh of every improvemen equence i bounded by maximum poenial. Proof. Sor he e of delay value V = {d r (k) r R, 1 k n} in increaing order. Define alernaive, new delay funcion: d r (k) := poiion of d r (k) in ored li. The new delay of a player i uing reource r in ae S i δ i (S) = d r (n r (S)).
Algorihmic Game Theory, Summer 2017 Lecure 1 (page 5 of 5) Obervaion 1.10. Le S and S be wo ae uch ha (S, S ) i an improvemen ep for ome player i wih repec o he original delay. Then (S, S ) i an improvemen ep for i wih repec o he new delay, a well. Furhermore, oberve ha d r (k) nm for all r R and k [n] becaue here are a mo nm elemen in V. Therefore, Roenhal poenial funcion wih repec o he new delay d r (k) can be upper-bounded a follow: Φ(S) = r R d r (k) r R n m n 2 m, where in he la ep we ue r R = n becaue every player ue exacly one reource. I hold ha Φ 1. Alo, Φ decreae by a lea 1 in every ep. Therefore, he lengh of every improvemen equence i upper-bounded by n 2 m. Example 1.11. The ored li of delay value in Example 1.8 i Hence, he old and new delay funcion are 15, 16, 17, 20, 50, 70, 90. d a () = (20, 50) da () = (4, 5, 6) d b () = (30, 70, 90) db () = (5, 7, 8) d c () = (15, 16, 17) dc () = () Recommended Lieraure D. Monderer, L. Shapley. Poenial Game. Game and Economic Behavior, 14:1124 1143, 1996. (Equivalence Congeion and Poenial Game) H. Ackermann, H. Röglin, B. Vöcking. On he impac of combinaorial rucure on congeion game. Journal of he ACM, 55(6), 2008. (Generalizaion of Theorem on Singleon Game)