Network Formation Games Ofir Geri

Network Formation GamesOfir Geri

Price of Anarchy SeminarSupervised by Prof. Michal Feldman

Tel-Aviv University

2/4/2014

Introduction

• We discuss games in which agents wish to construct a network

• Each agent has different terminals they wish to connect

• The cost of the network is shared between the agents

• Each agent may contribute to any link

Outline

• General cost-sharing connection games• Fair cost-sharing connection games• Capacitated symmetric cost-sharing

connection games

General Cost-SharingConnection Games

E. Anshelevich, A. Dasgupta, É. Tardos, and T. Wexler,“Near-Optimal Network Design with Selfish Agents”

The Connection Game

• There are players• is an undirected graph• Each player must connect a set of terminals in • An edge has a cost • Each player offers payments • The graph of bought edges is where

Basic Properties of Nash Equilibria

• The bought edges form a graph that is a forest• A player only contributes to the edges they

use• Each edge is either fully paid for or not paid at

all

A Game Without Nash Equilibrium

A Nash Equilibrium May RequireCost-Sharing

• NE: Player 2 pays 5 foredge , player 1 pays forthe rest

• Any NE must buy edge • Player 2 only contributes

to • A non-fractional NE

doesn’t exist

The Price of Anarchy

• Lower bound:In this example,

• The same holds when theedges are directed and thecosts are shared in a fairmanner


• Theorem: In every connection game with players,

• Proof: Let be the worst Nash equilibrium• The cost of each player is at most – If a player pays more than , they can buy instead

Single Source Games

• All players share a common source , and player has only one other terminal,

• In this class of games, the price of stability is 1

Single Source Games: Price of Stability

• Denote by the minimum cost Steiner tree that connects all terminal nodes

• Consider as the root• Let be the sub-tree that

is disconnected from ifedge is removed

𝑇 𝑒


• We show a Nash equilibrium that buys – We only need to define the payments

Algorithm 11. For all players and edges , set 2. For all edges in in reverse BFS order:

1. For players so that (until is paid for):1. If is a cut in , set 2. Define: 3. Define to be the cost of the lowest cost path from

to in under 4. Define 5. Define 6. Set


Claim: The algorithm yields a Nash equilibriumProof:• At any stage, emulates the cost of the lowest

cost path that does use an edge • We set the payment for to be at most the cost

of the alternative path to • A player cannot reduce their cost by deviating


• We only need to prove that the algorithm fully pays for

• For each edge , the players with terminals in must pay for – Otherwise, each player has a path that explains

why they can’t contribute more to – We show that if is not fully paid for, these paths

allow us to find a solution that is better than


• At some stage, denote by the alternate path that costs for player – Choose the one that includes as many ancestors of

in as possible• Lemma: is composed of three sub-paths– The first contains only edges from – The second contains only edges from – The third contains only edges from


Proof:• Once reaches a node in , it will use nodes from

since their cost is 0• Suppose starts with a path that contains edges

of , and continues with a path that contains edges of , leading to node in

• Let be the common ancestor of and


(Figure taken from Anshelevich et al., “Near-Optimal Network Design with Selfish Agents”)


• We prove that

• is strictly below – Otherwise, is contained in – We get that – Since , – is not the best deviation


• Consider an iteration during which player contributed to an edge in

• The total payment of is bounded by the cost of any path, including

• After reaches , the rest of the path to costs


• We proved that • If we replace with in – The cost can only decrease– contains a higher ancestor of than ()– Hence, a contradiction!


• We are ready to prove that the algorithm pays fully for

• Suppose that for an edge , • Recall that is the alternative path for • The highest ancestor of in that is also in will

be denoted (’s deviation point)• Let be the set of the highest deviation points,

such that every has an ancestor in


• Let be the sub-tree rooted at • Assume all players with

deviated to • Payments are not

increased• All edges in every

are still paid for

(Figure taken from Anshelevich et al.,“Near-Optimal Network Design with Selfish Agents”)


• Each path connects to – pays fully for the edges that are not in

• All terminals are connected to after the deviation

• The total cost of all players is less than , but is optimal– Hence, a contradiction!

Approximate Nash Equilibria

• Definition:A strategy profile is a -approximate Nash equilibrium if no player can decrease their cost by factor of more than by deviating

• Intuitively, we want players not to profit much from deviating

Single Source Games

• Finding OPT is NP-Hard• Let be an -approximate minimum cost tree• We present a poly-time algorithm for finding a

(1+ε)-approximate Nash equilibrium , so that

Single Source Games

Algorithm 2• Define • Use Algorithm 1 to pay for all edges in , with

their cost decreased by – is not optimal, so the algorithm may fail to buy an

edge– In that case, construct a tree so that

and run Algorithm 1 iteratively

Single Source Games

• For each player and for each edge , the final payment is

Single Source Games

• Algorithm 1 can be run in polynomial time• Algorithm 1 may run at most times• The run-time of Algorithm 2 is polynomial in

and the network size

Single Source Games

• All edges are fully paid for• Suppose contains edges• Compared to the Nash equilibrium returned

by Algorithm 1, the payments increased in

General Connection Games

• The price of stability can be

• Every NE must buy a path that costs


General Connection Games

• We show that there is a 3-approximate Nash equilibrium that pays for

• Given a set of edges , a stable payment is a payment such that doesn’t have a profitable deviation, assuming the rest of is bought by the rest of the players– A Nash equilibrium consists of stable payments for

all players

Stable Payments

• Consider a payment scheme • Theorem: If every payment can be divided

into at most payments, such that each of them is a stable payment, then is an -approximate Nash equilibrium

Stable Payments

Proof:• Let be the best response of player to • Let be the sub-payments of • is still a possible deviation for • We get , hence

Connection Set: Definition

• From now on, a player either pays fully or pays nothing for each edge

• A set of edges is a connection set of if for every connected component in we have that either– Any player that has terminals in has all of its

terminals in , or– Player has a terminal in

Connection Sets

• Lemma: A connection set of player is a stable payment of with respect to

• Proof: Let be the best deviation of • is a set of edges so that connects all of ’s

terminals• If two terminals of another player are in

different components of , they are connected in

Connection Sets

• Thus, is a possible solution• is optimal,

• Therefore,

3-Approximate Nash Equilibrium

• Observe that the set of edges used only by is a connected set,

• We want each player to pay for 3 connection sets

• will be the first connection set, and from on, assume that all edges are used by at least two players


• Assume is a path • is the set of terminals at • For each terminal ,

define a path



• A payment that contains one edge from for every terminal of except the last terminal is a connection set



• A max-coupled-set is a set of edges such that every is contained in exactly the same paths , for



• Let be a max-coupled-setFor all components of except the two end components, any player that has a terminal in has all its terminals in – Suppose has a terminal in – If has a terminal before or after , the edges in

adjacent to can’t be in the same coupled-set


• A payment that contains at most one max-coupled-set from for every terminal of except the last terminal is a connection set– If a component does not contain a terminal of , it

is bordered by edges of the same max-coupled-set


• Finally, we wish to match at most two connection sets to each player

• We form a bipartite matching problem– is the set of all max-coupled-sets– is – There’s an edge between and if there is such

that – For players that don’t have a terminal in , form an

edge between the last terminal and if


• We use Hall’s Matching Theorem to assign a node to each max-coupled-set

• For , denote by the set of nodes that can be connected to

• We need to prove


• Sort the edges that are part of • If two edges belong to different max-coupled-

sets, there must be a path that contains only one of the edges– The player corresponding to must have a terminal

between the two edges• There must be a terminal from before the first

edge in


• We have shown that for all • Using the matching, each player can be

assigned (at most) two connecting sets• We get a 3-approximate Nash equilibrium• This is expanded by induction to the whole

tree

Fair Cost-SharingConnection Games

E. Anshelevich, A. Dasgupta, J. Kleinberg, É. Tardos,T. Wexler, and T. Roughgarden,

“The Price of Stability for Network Designwith Fair Cost Allocation”

The Fair Connection Game

• We consider directed graphs• Each player chooses only which edges to use• We use Shapley (fair) cost-sharing:

Price of Anarchy and Price of Stability

• The fair connection game has an exact potential function:

• A pure Nash equilibrium always exists• (a tight bound)• (a tight bound)

Concave Cost Functions

• Each edge now has a non-decreasing concave cost function, – The cost per player decreases when more players

buy an edge• The function is still an exact potential function


• It holds that • Lemma: If there are such that for every

strategy profile ,

then


• For every strategy profile ,


• For every strategy profile ,

• Hence,

Games with Delays

• We can consider – is still concave if is concave

• Each player pays for • Latency tends to be convex

Games with Delays

• Claim:If for every , is concave and non-decreasing, is non-decreasing, and for every ,then – If is a polynomial of degree at most with non-

negative coefficients,

Games with Delays

• The potential is now • We have shown

Games with Delays

• From the assumption, • We get ,

hence

Game with Only Delays

• We assume the players share a single source• Finding a NE is possible using a reduction from

minimum cost flow– Replace each edge with parallel edges, with

capacity and cost – This graph emulates the potential function


• Let be the minimum cost Nash equilibrium, and be the minimum cost solution (OPT) for a game with twice as many players

• We show that • is the Nash equilibrium obtained by the

reduction to minimum cost flow


• Denote by the cost of the minimum cost path from to in the residual graph

• For each edge , consider the modified delay function


• Under the new costs :– The minimum cost of a path from to is at least – The minimum cost of a solution when the number

of players is doubled is at least • The difference in the cost between and :– For every and ,

– Globally, the difference is at most


• Therefore


• We have already seen cases where the price of stability is constant (the factor comes from the concave cost)

• Recall that for a class of function , we denote by the price of anarchy in non-atomic selfish routing (with delay functions from )

• In single source games with only delays, where ,

Weighted Players

• Player has a weight • If is the sum of weights of the players that use

edge , the cost of player will be • This is no longer a potential game

Weighted Players

• If each edge can be used by at most 2 players, there is a potential function

• For an edge that can be used by players , define

Weighted Players

• Define • is a potential function– Every game with two weighted players admits a

Nash equilibrium• Generally, the price of

stability can be – Consider players with

weights

Capacitated Cost-SharingConnection Games

M. Feldman and T. Ron,“Capacitated Network Design Games”

Model

• A variation of the fair cost-sharing connection game

• Each edge has a cost and capacity • A strategy profile is feasible if for every , • Not all games are feasible• Feasible games admit a pure Nash equilibrium

due to the potential function• We consider symmetric games


• In general networks, the price of anarchy can be arbitrary high

• In the following example, • The price of anarchy is

unbounded for everynetwork that embedsthis graph (Figure taken from Feldman and Ron,

“Capacitated Network Design Games”)

Graph-Theoretic Preliminaries

• Series composition

• Parallel composition

• Series-parallel network

s1

s2

t1

t2… …

s1

s2

t1

t2

…

…


• Lemma: Let be a feasible profile of agents in a series-parallel network. Given a profile for players, there exists a feasible path that uses only edges that are used in .


• We prove by induction– For a single edge, the lemma follows trivially–

Suppose in , agents use , and in , agents use . Either or .If , there is a suitable path in

–The path is obtained by concatenating two paths from


• We show that in series-parallel networks, • Let be a Nash equilibrium and let be the

optimal solution• The cost of each agent is at most – Otherwise they would deviate to using the lemma

• It follows that

The Price of Stability

• For incapacitated symmetric game, • The upper bound still holds• For capacitated games, this bound is tight

(Figure taken from Feldman and Ron, “Capacitated Network Design Games”)

Questions?

Documents

Network Formation Games Ofir Geri