Internet Routing Games

Transcription

1 CS498: UnderGraduate Project Internet Routing Games Submitted by: Pranay Borkar Roll no.: Mentored by: Prof. Sunil Simon Dept. of Computer Science and Engineering Indian Institute of Technology,Kanpur

2 Abstract Weakly acyclic games consist of classes of games in which certain global update dynamics guarantee to converge to Nash Equilibrium. We study in detail a subclass of weakly acyclic games - routing games introduced by Fabrikant, Papadimitriou(1) and Levin et al.(2) which explains the important aspects of routing on internet. In synchronous setting, we propose a scheduler that selects a set of players to deviate to their better response strategies such that we get a Nash Equilibrium in. We also design a distributed algorithm that follows the proposed scheduler using multiple token passing between the nodes. 1 Introduction In Game theory, convergence of a game to pure strategy Nash equilibrium is the most important objective. This objective is achieved by various simple and obvious dynamics like better response dynamics or best response dynamics. In these dynamics, at each step, a single player deviates from current strategy to new strategy that increases its utility. The convergence of games to Nash equilibrium under such dynamics is much studied in game theory. In the better or best response dynamics, weakly acyclic games is a class of games in which, for every strategy profile, there exist a improvement path to the Nash equilibrium. Potential games are weakly acyclic games in which the best response or better response dynamics guarantee to converge to a pure Nash equilibrium irrespective of the starting strategy profile. There is a lot of research on potential games. The class of weakly acyclic games is much broader than potential games. There are very few classes of games known that are weakly acyclic and not potential. One such famous class is of Rosenthal s congestion games in which player utilities are player specific. Our focus mainly is on a different kind of environment: routing on Internet. The Border Gateway Protocol(BGP) is protocol used to establish routes between different users on the internet. There have been advances in BGP obtained through game theoretic analyses. Fabrikant and Papadimitriou and, indepently Levin et al. stated that the BGP is a best response dynamics in routing games and the Pure Nash equilibrium in such games corresponds to stable routing in the network. In game theoretic framework, players are the source nodes that lie on a graphical network which aim to s their traffic to a unique destination on the Graph. Every node s neighbors is the strategy space of the player. 1

3 2 Model (This section contains definitions from (3)) In this class of games, the players are source nodes on a network G = (V, E). The players wish to s traffic to a unique destination node d. Each source node(player) i s strategy is to select an outgoing edge e i E(G) to forward the traffic or empty set φ(not forwarding the traffic). Therefore S i = {(u, v) φ (u, v) E and u = i} The strategy profiles s S in this game will be a directed subgraph (G s ) of the original graph G with each source node having a out-degree of at most one. For a given strategy profile s S induced route of a source node i, R s i is defined as the unique simple route to destination d in the subgraph if it exists or otherwise. Define the set of all possible paths from a source node i to the destination d as P i i.e. P i = s S Rs i. Each source node i has a routing policy with following two components:- ranking function π i that maps elements of P i to integers such that π i ( ) = 0 and π i (R) > 0 for R P i export policy that, for each neighboring node j V (G), specifies the set of routes R ij P i that a player i is willing to make available to j. To simplify notations, denote π i (R) < π i (Q) by R < i Q. A route R P i is permitted if each node on R is willing to export its sub-routes predecessor on R. Now, for a strategy profile s,the utility function is defined as :- π i (Ri s ) If Rs i is permitted and Rs i is not u i (s) = 0 If s i = φ 1 Otherwise We consider export all policy for the following discussions. Consider a general case of Internet Routing case in which every player deviates simultaneously to a better strategy(local update). For this case we show that even if there exist a Nash Equilibrium in the game, the simultaneous local update mechanism does not converge to Nash Equilibrium. Consider a game shown by following Graph. 2

4 A B D C The Player in this Graph are source nodes A,B,C and D is the destination node. Consider export all policy for this game and the ranking function for individual nodes as ffollows: For player A: ABD > A AD For player B: For player C: BCAD > A CABD > A BD CAD The Nash Equilibriums in this Game are s 1 = (AB,BD,CA) and s 2 = (AD,BC,CA). Consider the current strategy profile is (AD,BD,CA). The possible simultaneous deviations by the players A and B are to strategies AB and BC respectively, which makes strategy profile to be (AB,BC,CA). Again the simultaneous deviations of A and B lead to strategy profile (AB,BD,CA). This is the profile where we started. Thus, this process won t converge and will continue forever. Now, define policy consistency for the ranking of paths by every individual player. Definition 2.1. Policy Consistency: Let i and j be two adjacent source nodes in G. The source node i is said to be policy consistent with j iff for every two routes Q, R P j such that i / Q, R, if R < j Q, then (i, j)r < i (i, j)q. If all the nodes are policy consistent with its neighbors and export all policy is followed, we say that policy consistency holds. Theorem 2.1. Consider synchronous setting for policy consistent Internet Routing Game. Consider a process of choosing a subset of players from a given strategy profile (s) to deviate to their better response strategy to get strategy profile (s ) such that, 3

5 any of the player connected to the destination in (s) is also connected to destination in the (s ), This process will converge to some Nash Equilibrium Proof. Consider the sum of utilities of all the players in a strategy profile as the progress function i.e. N φ(s) = u i (s) i=1 Now, consider we are in strategy profile s and we choose players that deviate as per our conditions to get strategy profile s. Choose any player i from the set of players. We have two cases: Case 1: i did not deviate its strategy in s. In the path from node to destination, Rs, i say no node deviated to any other strategy. Thus, Rs i = Ri s which implies u i (s ) = u i (s). On the other hand say that some node j on the Rs i deviated. Therefore for that player j, we have Rs j < j R j s. Also, i / R j s (or it will form a cycle and will disconnect j from destination), so by policy consistency, Rs i > i Rs i which implies u i (s ) > u i (s). Similarly, argument can be exted when more than one player on R ( s) i deviated which implies u i (s ) > u i (s). Thus, for this case u i (s ) u i (s). Case 2: i deviated its strategy in s. In this case say only player i deviated from its strategy to get path P i thus, Rs i < i P i. Say some other nodes on this path P i also deviated to get a path Rs i. Thus, by the argument given in case 1, Rs i > i Rs i which implies u i (s ) > u i (s). From case 1 and 2, utility function of any player either increases or remains same therefore, φ(s ) φ(s). φ(s ) will be equal to φ(s) when no player has deviated(nash equilibrium). Therefore, for other strategy profiles φ(s ) > φ(s). Also, as the game is finite and utilities of players have upperbound, φ(s) has a upperbound of argmax N s i=1 u i(s). Hence the process will converge to some Nash Equilibrium. 3 Scheduler The concept of scheduler was first introduced in (4). We define a scheduler function F : S A {A N}. For a strategy profile s only players i A will deviate to some of their better response strategies such that no player - connected to the destination in current strategy profile(s) - is disconnected from the destination in 4

6 the following strategy profile(s ). Theorem 3.1. For any strategy profile (s S) which is not a Nash Equilibrium, the set A is non empty. Proof. For strategy profiles in which every player is not connected to the destination, the set A will always have players not connected to the destination. In the case when every player is connected to destination, there will be at least 1 player(say i) who wants to deviate to some better response strategy as the strategy profile is not Nash Equilibrium. If player i wants to deviate to some other strategy such that its path to destination changes from Rs i to P then, i / P otherwise it will disconnect the player i. Say only player i deviates in strategy profile(s) to get strategy profile (s ). Now for any player j, if i / Rs j then, R j s = Rs, j thus j is still connected to the destination. For players j, if i Rs j then, j is connected to destination as the route from j to i is not changed and i is connected to the destination. Thus, for any strategy profile which is not a NE, the set A will have atleast 1 element. This implies that for any strategy profile, the scheduler will always be able to give a non empty set of players as output. 4 Distributed Algorithm following Scheduler In the network, we have N+1 nodes(n players and 1 destination) and N edges(1 strategy of each player). Assume that out of N, k players are connected to the destination. Consider G to be the subgraph of these k players and destination. Subgraph G has k+1 nodes and k edges(minimum this subgraph can have to be connected). If some of these k nodes are disconnected from the destination, it should be the case that the disconnected nodes form a cycle as the number of nodes and edges remain same. So, while designing a distributed algorithm we have to be sure that a cycle is never formed in the graph. 4.1 Initialization: First we start with a graph where every node is connected to the destination. To do so, run the following algorithm on every node i with connected as boolean variable: 5

7 Algorithm 1: Initial algorithm at every node while connected==false do for j in neighbor(i) do if j.connected == true then currstrat = j; connected = true; 4.2 Algorithm We need to do updates such that no node gets disconnected from the destination in the strategy profile that follows Variables and methods At every node we maintain some variables as follows:- visited : boolean variable to check if the node is visited in current traversal. active : boolean variable to check if the node is currenty active. ancestors : list of all node that have path to destination going through this node. curr strat : it stores the current strategy of the player dev strat : it stores the strategy that player want to deviate to or, value of curr strat if it does not want to deviate. Also, the player has knowledge about its neighbors and state of the game. Assume procedures that can be called from any node as follows: 1. changeneighborvar() : It takes three arguments- node, variable and value. For a neighboring node node, it sets variable to value. 2. get() : It takes two arguments node, variable and returns the value of the variable of the node. 6

8 4.2.2 Token initialization Assume that all the selected players deviate their strategy at clock ticks t 1, t 2, t 3,... such that difference between t n and t n+1 is constant tdiff. Token initialization is done every time on the clock tick once the players have finished updating their strategies. For initializing token, we maintain a underlying cyclic graph structure and every-time select next node which wants to deviate. This can be viewed as Queue with every node going from front to back. Assume procedure next node(), that sets the active bit for the node that initializes the token propogation according to the above mentioned underlying graph. Algorithm 2: Algorithm at every node at clock ticks if time-prevtick = tdiff then if deviationpermitted then curr strat = dev strat; deviationpermitted = false; active = false; visited = false; next node(); Token passing Currently active node passes token to its current strategy as well as the strategy it wants to deviate to. Also the ancestor list of the nodes is updated accordingly. The variable visited is maintained to restrict the number of tokens visiting a particular node to 1. In case two different tokens visits a node, there may be a case that node gets disconnected from the destination. 7

9 Algorithm 3: Algorithm at every node when there is no clock tick if active then visited = true; if dev strat not in ancestors and get(dev strat,visited)== false then changeneighbor(dev strat,active,true); changeneighbor(dev strat,ancestor,ancestor+curr node); changeneighbor(curr strat,active,true); changeneighbor(curr strat,ancestor,ancestor+curr node); /* note here that the third argument will be list of ancestors of the current node plus the current node */ deviationpermitted = true; active = false; 4.3 Proof of Algorithm In this algorithm a cycle is never formed, as a node never deviates to a node that has a path or will have path in next strategy profile to destination that goes through current node. Thus a node connected to destination is never disconnected. Also at every clock tick atleast one node deviates to the better response thus the set A as defined in the section 3 while defining scheduler is never empty. Thus the proposed algorithm follows the scheduler defined. 5 Conclusion: We looked closely at various sub-classes of weakly acyclic games. Internet routing games was studied in detail. With some observation about the simultaneous deviations of the players, we proposed a scheduler that selects subset of players to deviate at a time such that we reach Nash Equilibrium. A distributed algorithm was then designed to follow the proposed scheduler with help of token passing. The algorithm uses a underlying cycle to initialize the token. This algorithm can be further improved so that it does not dep on this underlying graph for token initialization. 8

10 References [1] Alex Fabrikant and Christos H. Papadimitriou The complexity of game dynamics: BGP oscillations, sink equilibria, and beyond. In Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms (SODA 08). Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, [2] Hagay Levin, Michael Schapira, and Aviv Zohar Interdomain routing and games. In Proceedings of the fortieth annual ACM symposium on Theory of computing (STOC 08). ACM, New York, NY, USA, DOI: [3] Engelberg R., Schapira M. (2011) Weakly-Acyclic (Internet) Routing Games. In: Persiano G. (eds) Algorithmic Game Theory. SAGT Lecture Notes in Computer Science, vol Springer, Berlin, Heidelberg [4] S. Simon and K. R. Apt. Choosing products in social networks. In Proc. 8th International Workshop on Internet and Network Economics (WINE), volume 7695 of Lecture Notes in Computer Science, pages Springer,