Near-Optimal Network Design with Selfish Agents

1. Near-Optimal Network Design with Selfish Agents By Elliot Anshelevich, Anirban Dasgupta, Eva Tardos, Tom Wexler STOC’03 Presented by Mustafa Suleyman CIFTCI

2. Outline • Introduction • Model & Basic Results • Single Source Games • General Connection Games • NP Completeness • Conclusion • Questions/Homework

3. Introduction • Many networks, are developed, built, and maintained by a large number of agents all of whom act selfishly and have relatively limited goals. • The stable outcomes of the interactions of non-cooperative selfish agents correspond to Nash equilibrium. • Nash equilibria in network games can be much more expensive than the best centralized design. • price of anarchy : increase in cost caused by selfish behavior. • Nash Equilibrium: the best solution that selfish agents can agree upon, i.e. once the solution is agreed upon, the agents do not find it in their interest to deviate.

4. Introduction • Network Design Game: • Every agent has a specific connectivity requirement: each agent has a set of terminals and wants to build a network in which his terminals are connected. • Possible edges in the network have costs and each agent’s goal is to pay as little as possible. • Simple model of network creation. • Framework for understanding those networks that a central authority could persuade selfish agents to purchase and maintain. • selfish agents will share costs - cooperation

5. Introduction • Connection Game: • Given an undirected graph G=(V,E), non-negative edge costs • N players, each has a specific set of terminals • Players purchase edges to connect their terminals to the network • They can share costs • Each player wants to minimize their total payments • Each player wants its terminals to be connected • Centralized optimum network problem : generalized Steiner Tree problem • Deterministic Nash equilibrium of the network game and price of anarchy

6. Introduction • A two-parameter optimization problem • How close is the cost at Nash equilibrium to the optimum centralized cost? • How difficult is to find the equilibrium? • Assumption: length of connecting path is not important

7. Model & Basic Results • The Connection Game: • Let an undirected graph G = (V,E) be given, with each edge e having a nonnegative cost c(e). Each player i has a set of terminal nodes that he must connect. The terminals of different players do not have to be distinct. A strategy of a player is a payment function pi,where pi(e) is how much player i is offering to contribute to the cost of edge e. Any edge e such that ∑ipi(e) ≥ c(e) is considered bought, and Gpdenotes the graph of bought edges with the players offering payments p = (p1, . . . , pN). • Since each player must connect his terminals, all of the player’s terminals must be connected in Gp. However, each player tries to minimize his total payments, ∑eЄEpi(e) .

8. Model & Basic Results • The Connection Game: • A Nash equilibrium of the connection game is a payment function p such that, if players offer payments p, no player has an incentive to deviate from his payments. • This is equivalent to saying that if pjfor all j = i are fixed, then pi minimizes the payments of player i. • A (1+ε)-approximate Nash equilibrium is a function p such that no player i could decrease his payments by more than a factor of 1 + ε by deviating, i.e. by using a different payment function pi’.

9. Model & Basic Results • Properties of Nash Equilibrium • Suppose we have a Nash equilibrium p, and let Tibe the smallest tree in Gpconnecting all terminals of player i. Then : (1) Gp is a forest. (2) each player i only contributes to costs of edges on Ti (3) each edge is either paid for fully or not at all. • Nash Equilibrium may not exist • Price of anarchy may vary from 1 to N

10. Model & Basic Results

11. 1 t s N Model & Basic Results

12. Single Source Games • Definition 3.1 A single source game is a game in which all players share a common terminal s, and each player i has exactly one other terminal ti. • Theorem 3.2 In any single source game, there is a Nash equilibrium that purchases T∗, a minimum cost Steiner tree on all players’ terminal nodes.

13. Single Source Games • Proof : Given T∗, construct payment strategies p such that we are in Nash equilibrium. • T∗is rooted at s • Let Tebe the subtree of T∗disconnected from s when e is removed. • Determine payments to edges by considering edges in reverse breadth first search order. • Determine payments to the subtree Tebefore we considering edge e. • c’ : the cost that player i faces if he deviates in the final solution • Never allow i to contribute so much to e that his total payments exceed his cost of connecting tito s.

14. Single Source Games

15. Single Source Games • We ensure that player i’s contributions to edges are always less than his cost of connecting tito s. Therefore it is never in player i’s interest to deviate. Since this is true for all players, p is a Nash equilibrium. • Authors also prove that the algorithm succeeds in paying for T∗. • Therefore, optimistic price of anarchy is 1. • However, the algorithm requires a minimum cost Steiner tree as input, which is computationally infeasable.

16. Single Source Games • Theorem 3.6 Suppose we have a single source game and an α-approximate minimum cost Steiner tree T. Then for any ε > 0, there is a poly-time algorithm that returns a (1 + ε)-approximate Nash equilibrium on a Steiner tree T’,where c(T) ≤ c(T’). • Proof Scetch: The proof of Theorem 3.2 suggests such an algorithm that forms a cheaper tree whenever a Nash equilibrium cannot be found. To ensure polynomial-time convergence, we force the algorithm to make only substantial improvements.

17. General Connection Games • Each player can have more than one terminals. • They do not necessarily share the source terminal. • Price of anarchy can be as large as N. • Optimistic price of anarchy may be quite large as well. • Expensive Nash equilibrium for the multi-source case • How cheap α-approximate Nash equilibrium with small α can be?

18. General Connection Games Optimal centralized cost : 1 + 3 ε Optimistic price of anarchy : ~ N-2

19. General Connection Games • Theorem 4.1 For any optimal centralized solution T∗, there exists a 3-approximate Nash equilibrium such that the purchased edges are exactly T. • Definition 4.2 A connection set S of player i is a subset of edges of Tisuch that for each connected component C of the graph T∗\ S, we have that either • (1) any player that has terminals in C has all of his terminals in C, or • (2) player i has a terminal in C. Ti is the unique smallest subtree of T∗that connects all terminals of player i.

20. General Connection Games • Lemma 4.3 Let p be a payment function purchasing T∗that obeys the following properties. • (1) If p(e) > 0, then e is bought fully by a single player. • (2) Each player i only buys edges that lie in his tree Ti. If the set of edges that each player buys is a union of at most α connection sets, then p is an α-approximate Nash equilibrium.

21. General Connection Games • Theorem 4.1 contains a polynomial-time algorithm for generating a 3-approximate Nash equilibrium on T∗. • We can use the ideas from Theorem 3.6 to create an algorithm that given an α- approximate Steiner forest T, finds a (3 + ε)-approximate Nash equilibrium that pays for a Steiner forest T’ with c (T ‘ ) ≤ c (T). • However, this algorithm requires a polynomial time optimal Steiner tree finder as a subroutine. The algorithm of Theorem 4.1 generates at most 3 connection sets for each player i. We can check if each connection set is actually cheaper than the cheapest deviation of player i, which is found by the cheapest Steiner tree algorithm. If it is, then we have a (3 + ε)-approximate Nash equilibrium. Otherwise, we can replace this connection set with the cheapest deviation tree and run this algorithm over again. • If we use a 2-approximate Steiner forest T, and an optimal Steiner tree 1.55-approximation algorithm as our subroutine, then we get a (4.65 + ε)- approximate Nash equilibrium on T ‘ with c (T ‘ )≤ 2.OPT, in time polynomial in n and ε −1.

22. NP Completeness Xi = true if left path purchased, false otherwise Cj = (X1 V X2 V ~ X3 )

23. NP Completeness • If there is a satisfying formula, then one of the inner edges in clause gadget is connected. It leads to a Nash equilibrium when the players buy 2 more edges each to connect to the sources.

24. NP Completeness • If there is Nash equilibrium, then the only way is through the inner edges. It means at least one variable is true in each clause, therefore the formula is satisfiable.

25. Conclusions • Single Source games: • there is a Nash equilibrium, the cost of which is equal to the cost of the optimal network. • the optimistic price of anarchy is 1. • given an ε > 0 and an α-approximate solution to the optimal network, we show how to construct in polynomial time an (1+ε)-approximate Nash equilibrium (players only benefit by a factor of (1 + ε) in deviating) whose total cost is within a factor of α to the optimal network.

26. Conclusions • General Connection Games • There may not exist a deterministic Nash equilibrium. • When deterministic Nash equilibrium do exist, the costs of different equilibrium may differ by as much as a factor of N, the number of players, and even the optimistic price of anarchy may be nearly N. • there is always a 3-approximate equilibrium that pays for the optimal • We can construct in polynomial time a (4.65+ε)-approximate Nash equilibrium whose total cost is within a factor of 2 to the optimal network • Determining whether or not a Nash equilibrium exists is NP-complete when the number of players is O(n).

27. Questions?

28. Homework • Briefly explain Nash Equilibrium, price of anarchy and optimist price of anarchy. • What is the upper bound on price of anarchy? Give a brief informal proof. • There are two optimization problems in this paper. Formulate one using the outline presented in class.