Algorithmic Game Theory - Heinz Nixdorf Institut · Complexity Symmetric Network Congestion Games Given a directed graph G = (V,E) with delay functions d e: {1,...,n} → Z, e ∈

Complexity

Congestion Games

Algorithmic Game Theory

Alexander Skopalik Algorithmic Game Theory 2013

Congestion Games

Complexity

Complexity of pure Nash equilibria


Congestion Games

Complexity

We investigate the complexity of finding Nash equilibria in different kinds ofcongestion games.

Our study is restricted to congestion games with non-decreasing delayfunctions.


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Symmetric Network Congestion Games

◮ Given a directed graph G = (V ,E) with delay functionsde : {1, . . . , n} → Z, e ∈ E .

◮ Player i wants to allocate a path of minimal delay between a source s anda target t.

1,2,9

4,5,6 1,2,3

1,9,9

7,8,9

s t

◮ In more general asymmetric network congestion games, different playersmight have different source-destination pairs.


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Symmetric Network Congestion Games

◮ It is known that there are instances of symmetric congestion games inwhich there are states such that every improvement sequence from thisstate to a Nash equilibrium has exponential length.

◮ Hence, applying improvement steps is not an efficient (i.e. polynomialtime) algorithm for computing Nash equilibria in these games.

◮ However, there is another algorithm which finds Nash equilibria inpolynomial time ...


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Complexity in Symmetric Network Congestion Games

Polytime algorithm via a reduction to min-cost flow:(Fabrikant, Papadimitriou, Talwar 2004)

◮ Each edge is replaced by n parallel edges of capacity 1 each.

◮ The ith copy of edge e has cost de(i), 1 ≤ i ≤ n.

1,2,3

7,8,9

1,9,9

4,5,6

1,2,9

s t

◮ Optimal solution minimizes Rosenthal’s potential function and, hence, is aNash equilibrium.


Congestion Games

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The relationship to local search

Rosenthal’s potential function allows us to interprete congestion games as localsearch problems:

Nash equilibria are local optima wrt potential function.

How difficult is it to compute local optima?


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The complexity class PLS

Definition (PLS (Polynomial Local Search))

PLS contains search problems with an objective function and a specifiedneighborhood relationship Γ. It is required that there is a poly-time algorithmthat, given any solution s,

◮ computes a solution in Γ(s) with better objective value, or

◮ certifies that s is a local optimum.


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Definition (PLS (Polynomial Local Search))

PLS contains search problems with an objective function and a specifiedneighborhood relationship Γ. It is required that there is a poly-time algorithmthat, given any solution s,

◮ computes a solution in Γ(s) with better objective value, or

◮ certifies that s is a local optimum.

Some examples for problems in PLS

◮ FLIP (circuit evaluation with Flip-neighborhood)

◮ TSP with 2-Opt-neighborbood

◮ Pos-NAE-kSat with Flip-neighborhood

◮ Max-Cut with Flip-neighborhood

◮ Congestion games wrt improvement steps


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Positive Not-All-Equal kSat (Pos-NAE-kSAT) – Definition

Input:

Formula over n binary variables x1, . . . , xn described by

◮ m clauses c1, . . . , cm each containing k positive literals, and

◮ m weights w1, . . . ,wm.


Congestion Games

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Positive Not-All-Equal kSat (Pos-NAE-kSAT) – Definition

Input:

Formula over n binary variables x1, . . . , xn described by

◮ m clauses c1, . . . , cm each containing k positive literals, and

◮ m weights w1, . . . ,wm.

◮ Clause ci is satisfied by an assignment A ∈ {0, 1}n if its literals are not allassigned the same value.

◮ The value of an assignment is the weighted sum of satisfied clauses.

◮ Assignments A and A′ are neighboring if they differ in exactly one position.

Task:Find a local optimum, i.e., an assignment without neighboring assignment ofhigher value.


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Positive Not-All-Equal 3Sat (Pos-NAE-3SAT) – Example

Example instance of Pos-NAE-3SAT:

c1 = x1x2x3; c2 = x1x2x4; c3 = x1x2x5; c4 = x3x4x5

w1 = 100; w2 = 110; w3 = 120; w4 = 100


Congestion Games

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Positive Not-All-Equal 3Sat (Pos-NAE-3SAT) – Example

Example instance of Pos-NAE-3SAT:

c1 = x1x2x3; c2 = x1x2x4; c3 = x1x2x5; c4 = x3x4x5

w1 = 100; w2 = 110; w3 = 120; w4 = 100

An example for a local optimum is the assignment

x1 = 0; x2 = 0; x3 = 1; x4 = 1; x5 = 1

of value 330. Each of the five neighboring assignments has a value of at most330.


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Max-Cut – Definition

Input:

A graph G = (V ,E) with edge weights w : E → N.

◮ A cut partitions V into two sets Left and Right.

◮ Two cuts are neighboring if one can obtain one from the other by movingonly one node from Left to Right or vice versa.

◮ The value of a cut is the weighted number of edges with one endpoint inLeft and one endpoint in Right.

Task:Find a local optimum, i.e., a cut without neighboring cut of higher value.


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Definition (PLS-reduction)

Given two PLS problems Π1 and Π2 find a mapping from the instances of Π1 tothe instances of Π2 such that

◮ the mapping can be computed in polynomial time,

◮ the local optima of Π1 are mapped to local optima of Π2, and

◮ given any local optimum of Π2, one can construct a local optimum of Π1

in polynomial time.


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First example for a PLS-reduction

Theorem:Pos-NAE-3Sat ≤PLS Pos-NAE-2Sat


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Proof:

For each 3-clause (x1, x2, x3) of weight w introduce the three 2-clauses(x1, x2), (x1, x3), (x2, x3) each of weight w/2.


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Proof:

For each 3-clause (x1, x2, x3) of weight w introduce the three 2-clauses(x1, x2), (x1, x3), (x2, x3) each of weight w/2. The value of an assignment in the

2SAT instance is identical to its value in the 3SAT instance as a 3-clause issatisfied if and only if exactly two of the three corresponding 2-clauses aresatisfied.

Hence, the local optima of both instances coincide, and the conditions of aPLS-reduction are fulfilled.


Congestion Games

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Second example for a PLS-reduction

Theorem:Pos-NAE-2Sat ≤PLS Max-Cut


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Second example for a PLS-reduction

Theorem:Pos-NAE-2Sat ≤PLS Max-Cut

Proof:

Each variable is represented by a node. Each clause and its weight isrepresented by a weighted edge. Multi-edges are merged to single edges byadding their weight.

Given an assignment for the variables, 0 is interpreted as Left and 1 isinterpreted as Right. This way, the local optima of both instances coincide, andthe conditions of a PLS-reduction are fulfilled.


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Definition (PLS-completeness)

A problem Π∗ in PLS is called PLS-complete if, for every problem Π in PLS, itholds Π ≤PLS Π∗.

Examples for PLS-complete problem:

◮ A master reduction shows that FLIP is PLS-complete.

◮ A PLS-reduction from FLIP to POS-NAE-3SAT shows that the latterproblem is PLS-complete as well.

◮ Hence, POS-NAE-2SAT and Max-Cut are PLS-complete, too.


Congestion Games

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Complexity of congestion games

Results from [Fabrikant, Papadimitriou, Talwar 2004]

network games general games

symmetric ∃ poly-time Algo PLS-complete

asymmetric PLS-complete PLS-complete

We present only one of these PLS-completeness proof, the one for general,asymmetric congestion games (the simplest one).


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PLS-hardness proof for general congestion games

We prove a PLS-reduction from Max-Cut to congestion games.


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We prove a PLS-reduction from Max-Cut to congestion games. First of all, we

observe that Max-Cut can be represented as a game:

Party Affiliation Game (Max-Cut)

Players correspond to nodes in a weighted graph G = (V ,E).

◮ Every player has 2 strategies: left or right.

◮ A state of the game yields a cut, i.e., a partition of V into left and rightnodes.

◮ Edge weights represent antisympathy.

◮ Players maximize the sum of weights of incident edges crossing the cut.

◮ Nash equilibria correspond to local optima of Max-Cut.


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Minimization Variant of the Party Affiliation Game

The strategies of a node are

◮ left: choose the left hand side of the cut

◮ right: choose the right hand side of the cut

The costs for these strategies are

◮ left: sum of the weights of the incident edges to the left

◮ right: sum of the weights of the incident edges to the right

Both games have the same transition graph as, for each player, minimizing theweights of incident edges on “her side”, is equivalent to maximizing the sum ofedges leading to the “other side”.


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Now the minimization variant can be described in terms of a congestion game.

Party Affiliation Congestion Game:

◮ Represent each edge e by two resources eleft , eright with delay functionsd(1) = 0 and d(2) = we .

◮ For each player the strategy Sleft contains resources eleft for all incidentedges; strategy Sright contains resources rright for all incident edges.


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Now the minimization variant can be described in terms of a congestion game.

Party Affiliation Congestion Game:

◮ Represent each edge e by two resources eleft , eright with delay functionsd(1) = 0 and d(2) = we .

◮ For each player the strategy Sleft contains resources eleft for all incidentedges; strategy Sright contains resources rright for all incident edges.

Players in this congestion game have exactly the same cost as players in theminimization variant of the party affiliation game.

Hence, the Nash equilibria of this congestion game coincide with local optimaof the Max-Cut instance, which yields a PLS-reduction from Max-Cut tocongestion games. (PLS completeness)


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Recommended Literature

◮ D. Monderer, L. Shapley. Potential Games. Games and EconomicBehavior, 14:1124–1143, 1996. (Equivalence Congestion and PotentialGames)

◮ M. Yannakakis. Computational complexity. Chapter 2 in E. Aarts and J.Lenstra (Eds), “Local Search in Combinatorial Optimization”, pages19–55, 1997. (Survey of the class PLS)

◮ A. Fabrikant, C. Papadimitriou, K. Talwar. The complexity of pure Nashequilibria. STOC 2004. (PLS-Completeness in Congestion Games)

◮ H. Ackermann, H. Roglin, B. Vocking. On the impact of combinatorialstructure on congestion games. Journal of the ACM, 55(6), 2008.(Matroid Games, Exponential Convergence, PLS-Completeness)


Congestion Games

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Algorithmic Game Theory - Heinz Nixdorf Institut · Complexity Symmetric Network Congestion Games Given a directed graph G = (V,E) with delay functions d e: {1,...,n} → Z, e ∈