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Credit to slides by Carmine Ventre (part of cogestion game slide)

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##### Credit to slides by Carmine Ventre (part of cogestion game slide)

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**Congestion Games (Review and more definitions), Potential**Games, Network Congestion games, Total Search Problems, PPAD, PLS completeness, easy congestion problems, hard congestion problems Credit to slides by Carmine Ventre (part of cogestion game slide) Added/modified by Michal Feldman and Amos Fiat**(General) Congestion Games**• resources • roads 1,2,3,4 • players • driver A, driver B • strategies: which roads I use for reach my destination? • A wants to go in Salerno • e.g. SA={{1,2},{3,4}} • B wants to go in Napoli • e.g. SB={{1,4},{2,3}} • what about the payoffs? A B Roma Milano road 1 road 4 road 2 road 3 Napoli Salerno**Payoffs in (G)CG: an example**A SIRF (Small Index Road First) B • A choose path 1,2 • B choose path 1,4 • uA = - (c1(2) + c2(1)) = - 4 • uB = - (c1(2) + c4(1)) = - 5 Roma Milano road 1 road 4 road 2 road 3 Napoli Salerno B Costs for the roads {1,4} {2,3} c1(1)=2 c1(2)= 3 c2(1)=1 c2(2)= 4 c3(1)=4 c3(2)= 6 c4(1)=2 c4(2)= 5 {1,2} A {3,4}**Congestion games: special cases**• Symmetric CG • Si are all the same and payoffs are identical symmetric function of n-1 variables • Network CG • Each player has a starting and terminal node and the strategies are the paths in the network**Rosenthal’s result**• The class of GCG is “nice” • A pure Nash equilibrium exists • Introduces the potential function:**Potential functions**Several kind of potential functions: • Ordinal potential function • Weighted potential function • (Exact) potential function • Generalized ordinal potential function**Ordinal potential function**• A function P (from S to R) is an OPF for a game G if for every player i • ui(s-i, x) - ui(s-i, z) > 0 iff P(s-i, x) - P(s-i, z) > 0 • for every x, z in Si and for every s-i BoS = P1 = u1(B,B) – u1(S,B) > 0 implies that P1(B,B) – P1(S,B) > 0 u1(B,S) – u1(S,S) < 0 implies that P1(B,S) – P1(S,S) < 0 P1(B,B) – P1(S,B) > 0 implies that u1(B,B) – u1(S,B) > 0 and that u2(B,B) – u2(S,B) > 0 P1 is an ordinal potential function for BoS game**A**B (3,1) (4,0) 11 9 A P3 = G’ = (2,4) (1,0) 8 0 B u1(A,A) – u1(B,A) = 3 - 2 = 1/3 (P3(A,A) – P3(B,A)) P3 is a (1/3,1/2)-potential function for the game G’ u1(A,B) – u1(B,B) = 4 - 1 = 1/3 (P3(A,B) – P3(B,B)) u2(B,A) – u2(B,B) = 4 - 0 = 1/2 (P3(B,A) – P3(B,B)) u2(A,A) – u2(A,B) = 1 - 0 = 1/2 (P3(A,A) – P3(A,B)) Weighted potential function • A function P (from S to R) is a w-PF for a game G if for every player i • ui(s-i, x) - ui(s-i, z) = wi (P(s-i, x) - P(s-i, z)) • for every x, z in Si and for every s-i C D C P2 = PD = D u1(C,C) – u1(D,C) = 1 = 2 (P2(C,C) – P2(D,C)) u1(C,D) – u1(D,D) = 3 = 2 (P2(C,D) – P2(D,D)) P2 is a (2,2)-potential function for PD game u2(D,C) – u2(D,D) = 3 = 2 (P2(D,C) – P2(D,D)) u2(C,C) – u2(C,D) = 1 = 2 (P2(C,C) – P2(C,D))**Weighted potential function**m identical machines, n jobs, and {wi}i=1..n the weights of the jobs Jobs seek to minimize the makespan on their machine, Take If job i moves from machine j to j’, then the difference in load is and the difference in the potential function is**Weighted potential function (example)**m identical machines, {wi}i=1..n job weights If job i moves from machine j to j’, then the difference in load is**Ordinal potential function (example)**m machines, {wij}i=1..n,j=1..m job weights wij – weight of job i on machine j If job i moves from machine j to j’, then the difference in load is**(Exact) potential function**A function P (from S to R) is an (exact) PF for a game G if it is a w-potential function for G with wi = 1 for every i C D C P4 = PD = D u1(C,C) – u1(D,C) = P4(C,C) – P4(D,C) u1(C,D) – u1(D,D) = P4(C,D) – P4(D,D) P4 is a potential function for PD game u2(D,C) – u2(D,D) = P4(D,C) – P4(D,D) u2(C,C) – u2(C,D) = P4(C,C) – P4(C,D)**Generalized ordinal potential function**A function P (from S to R) is an GOPF for a game G if for every player i ui(s-i, x) - ui(s-i, z) > 0 implies P(s-i, x) - P(s-i, z) > 0 for every x, z in Si and for every s-i in S-i A B A G’’ = P5 = B P5 is a generalized ordinal potential function for the game G’’ P5 is not an ordinal potential function for the game G’’ P5(A,B) – P5(A,A) > 0 implies that u1(A,B) – u1(A,A) > 0 but not that u2(A,B) – u2(A,A) > 0**Potential games**• A game that admits an OPF is called an ordinal potential game • A game that admits a weighted PF is called a weighted potential game • A game that admits an exact PF is called a potential game • In such games a Nash equilibrium is a local maximum for the potential**Equilibria in Potential Games**Thm (MS96) Let G be an ordinal potential game (P is an OPF). A strategy profile s in S is a pure equilibrium point for G iff for every player i it holds P(s) ≥ P(s-i, x) for every x in Si Therefore, if P has maximal value in S, then G has a pure Nash equilibrium. Corollary Every finite OP game has a pure Nash equilibrium.**An example**Nash equilibrium P4 maximal value C D C P4 = PD = D Thm (MS96) C D C PD(P4) = D**FIP: an important property**• A path in S is a sequence of states s.t. between every consecutive pair of states there is only one deviator • A path is an improvement path w.r.t. G if each deviator has a strict advantage ui(sk) > ui(sk-1) • G has the FIP if every improvement path is finite • Clearly if G has the FIP then G has at least one pure equilibrium • Every improvement path terminates in an equilibrium point**FIP: an important property (2)**Lemma Every finite OP game has the FIP. The converse is true? “No” • G’’ has the FIP ((B,A) is an equilibrium) • any OPF must satisfies the following impossible relations: • P(A,A) < P(B,A) < P(B,B) < P(A,B) = P(A,A) A B A G’’ = B Lemma Let G be a finite game. Then, G has the FIP iff G has a generalized ordinal potential function.**FIP: an important property (3)**Lemma Let G be a finite game. Then, G has the FIP iff G has a generalized ordinal potential function. Proof: Consider the graph of strict improvement moves. Give v the potential function value = the length of the longest path to a sink**Congestion vs (Exact) Potential Games**Thm Every congestion game is an (exact) potential game. Thm Every finite (exact) potential game is isomorphic to a congestion game.**Congestion vs (Exact) Potential Games**Thm Every finite (exact) potential game is isomorphic to a congestion game. • Potential game: • n players, k pure strategies each, potential P • Congestion game: • n players, k pure strategies each, 2kn resources • resource associated with {0,1}kn vector**Congestion vs (Exact) Potential Games**Thm Every finite (exact) potential game is isomorphic to a congestion game. • Congestion game: • player 1 ≤ i ≤n plays pure strategy 0 ≤ q ≤ k-1: • uses all resources rb where bit bik+q = 1 (2kn-1 res.) • For every strategy vector s (for both games) • b(s)where b(s)ik+q = 1 iff player i uses q in s • For n agents, the cost of rb(s) is P(s), for less agents the cost is 0 • For agent i, let b’ be such that b’jk+q= 0 if user j≠i uses strategy q, 1 otherwise. • Cost of resource rb’ with one user ui(s)-P(s)**Final project part 1**Improve the construction (use less resources) Why?**Computing NE in congestion games**• Sometimes easy (polytime) – symmetric network congestion games • Sometimes hard (PLS complete) – general congestion games, symmetric congestion games, general network congestion games**Symmetric network congestion games (easy)**• Graph G=(V,E), source vertex v, sink t • n players need to find a path from v to t. • Congestion on edge e is given by c(e,k) where k is the number of players using the edge (c ≥ 0) • Cost to player is sum of costs of edges player uses.**Symmetric network congestion games (easy) (s –strategy, v**– source, confusing) • Convert to min-cost flow • Replace edge e in G with n parallel links, of capacity 1, and with costs c(e,1), c(e,2), … c(e,n) • The cost of a min cost flow of value n, (strategy s), from v to t is equal to the potential function**Hardness results: Total Search Problems – TFNP**• A search problem S is a set of inputs IS* • Such that: • For every x IS there is an associated set of solutions • Recognizing if y is a solution can be done in polytime • A search problem is total if Sx is not empty, for all x • r-Nash (r player Nash, normal form) is total, needs accuracy parameter as input (true Nash can be irrational)**Total Search Problems – TFNP**• A search problem S is a set of inputs IS* • Such that: • For every x IS there is an associated set of solutions • Recognizing if y is a solution can be done in polytime • A search problem is total if Sx is not empty, for all x • r-Nash (r player Nash, normal form) is total, needs accuracy parameter as input (true Nash can be irrational)**Subclasses of TFNP – PPAD and PLS**• PPAD: Polynomial Parity Argument, Directed Version (or just that Papadimitriou has two p’s). • End of Line problem: • n input bit circuits P and S (predecessor and successor) • S(00…0) ≠ 00….0, P(00…0)=00…0 • Find the end of the line (S(x)=x) or a loop (P(S(x))=x. • PPAD – class of total search problems reducible to “end of line”. • In particular – includes Brower Fixed point (discrete version) • PPAD complete – natural notion. • Brower is complete, 2 Player Nash is also complete. • We saw how to use Brower to solve Nash, other direction also works (major recent result – we will return to it when I understand it).**Subclasses of TFNP – PPAD and PLS**• PLS: Polynomial Local Search • A local search problem P belongs to PLS if: • For every instance I, a polytime algorithm A computes an initial feasible solution • A polytime algorithm B that computes, for every instance I and every feasible solution S, the objective function value c(S) • A polytime algorithm C that determines if S is locally optimal for instance I, and if not gives a better solution in the neighborhood of I**PLS reductions**• PLS: Polynomial Local Search • A PLS reduction from Problem P to Problem Q: • A polytime function f that takes instances of P {IP} to instances of Q {IQ} • A polytime function g that maps pairs (SQ,IP) to solutions SP of IP. (SQ is a solution of f(IP)). • For all instances IP of P, if SQ is a local optimum of f(IP) then g(SQ,IP) is a local optimum of IP. • P is PLS-Complete – all problems in PLS are PLS reducible to P**PLS Complete problem – max cut (political party game)**• Undirected graph G, weights on edges • Find a partition of the vertices so that the sum of the weights of edges crossing the cut is maximized (minimize disagreements within the party, high weight – high disagreement)**NE in a congestion game is PLS complete (even a symmetric**congestion game) • For every edge e of weight w: two resources: reR and reL • Every player u has two pure strategies: • Use all resources (reL) for e = (u,v) • Use all resources (reR) for e = (u,v) • If re{L,R} is used by one player only, the cost is zero, if used by two, the cost is the weight of e