Selfish Caching in Distributed Systems: A Game-Theoretic Analysis By Byung-Gon Chun et al. UC Berkeley PODC’04

CSE658 wireless Seminar2 Outline Motivation Game Theory Problem Formulation Theoretical Results Simulation Results Conclusions

CSE658 wireless Seminar3 Motivation Traditional networks assume nodes are cooperative and follow certain protocol In reality, nodes may behave selfishly Seek to maximize their own benefit Many applications: P2P, ad hoc network Selfish Caching is to: Find appropriate object placement strategies for distributed replication groups of selfish nodes

Game Theory Definition: A formal way to analyze interaction among a group of rational players who behave strategically. Game a set of players: {1, 2, …, n} player i has a set of alternative strategies: S_i each strategy is called a pure strategy mixed strategy: choosing all strategies according to prob. dist. Strategy profile s= (s_1, s_2, …, s_n), where s_i є S_i Payoff function h_i(s): the payoff to player i when the players choose strategy profile s All above are common knowledge to each player

CSE658 wireless Seminar5 Game Theory Continued Nash equilibrium (NE) : strategy profile in which no player can be strictly better off by unilateral deviation Why NE? It is a stable, self-enforcing solution Theorem (Nash, 1951) – “For a game with finite players and each player having a set of finite strategies, there always exists an equilibrium in (mixed) strategy.” In most cases, NE does not achieve social optimum due to selfishness of each player

CSE658 wireless Seminar6 How to quantify the selfishness? Price of anarchy (Papadimitriou et al, 1999) loss of efficiency ratio of the social cost of the worst possible Nash equilibrium to the cost of social optimum Optimistic Price of anarchy Smallest ratio of NE and social optimum

Caching and Selfish Caching

CSE658 wireless Seminar8 Traditional Caching Problem Given: n servers; m objects distance matrix D: d_{ij} is the distance from server i to server j demand matrix W: w_{ij} is the demand of server i to object j Placement cost matrix P: α_{ij} is the placement cost of server i for object j No memory constraint Goal: Place objects into servers s.t. the total access cost + placement cost is minimized It is uncapacitated multi-facility location problem – NP hard

CSE658 wireless Seminar9 Selfish caching problem Each node tries to minimize its own access + placement cost Model selfish caching problem as a non-cooperative game with n players whose strategies are sets of objects to cache Each node (player) chooses a pure strategy that minimize its own cost

CSE658 wireless Seminar10 Basic Selfish Caching Cost Model Given: s_i: strategy chosen by server i (set of objects i places) Strategy profile: s = (s_1, s_2, …, s_n) the cost of server i: For each object, at least one server will choose to cache

CSE658 wireless Seminar11 Separability of Uncapacitated Version We can look at individual object placement separately due to no memory constraint Nash equilibrium of the multi-object caching game is the combination of the Nash Equilibria of single object caching game

CSE658 wireless Seminar12 Social Optimal Caching

CSE658 wireless Seminar13 Social Optimal Caching Social Optimal Cost: C(So) = minS

CSE658 wireless Seminar14 Single Object Selfish Caching Game Strategy: s i = 1, when i replicates ; 0, otherwise strategy profile: s = (s1, s2, …, sn) Cost of server i to access object j Social cost of s: ∑ ci(s)

CSE658 wireless Seminar15 Major Questions Does a pure strategy Nash equilibrium exist in single object selfish caching game? If so, what is the price of anarchy in such NE? How does price of anarchy vary under different demand distribution, underlying topology and placement cost? Can Nash equilibrium achieve social optimum?

CSE658 wireless Seminar16 Major Results Pure strategy NE exists in single-object caching game Price of anarchy can be bad: O(n) With payment as incentive for nodes to replicate, there exists pure strategy NE achieving social optimum

CSE658 wireless Seminar17 How to find the pure strategy NE in single object caching game? Assume placement cost the same for all servers: α ; server i’s demand is w_i Algorithm: 1. Remove “zero demand” servers from the server set N 2. Pick a server y s.t. α/w_y is the minimum 1) y places the object 2) Remove y and all z where w_z*d_{zy} ≤ α 3. Continue 2 until N is empty

CSE658 wireless Seminar18 Continued… Claim: The resulting configuration X is a pure strategy NE All such y have no incentive to remove its object All such z have no incentive to replicate No player has incentive to deviate, its NE.

CSE658 wireless Seminar19 Price of Anarchy (PoA) of Single Object Game PoA = C(WNE)/C(SO)

CSE658 wireless Seminar20 Inefficiency of NE

CSE658 wireless Seminar21 Payment Game Each player i offers a payment to another player j as incentive for j to replicate object Strategy for player i: (v_i, b_i, t_i) v_i: the player to whom i makes a bid b_i: the value of the i’s bid t_i: payment threshold beyond which i will replicate Total amount of bids received by i: R_i i replicates iff R_i >= t_i I_i = 1 if i replicates object, I_i = 0 if not

CSE658 wireless Seminar22 Payment Game Continued… If i makes bid b_i to j If j replicates the object, then i must pay j amount of b_i If j does not replicate, i does not pay j The outcome of the game is: {(I_i, v_i, b_i, R_i)} Cost Model

CSE658 wireless Seminar23 Results of Payment Game Theorem In the payment game, there is always Nash equilibrium that implements the social optimum configuration

Simulation

CSE658 wireless Seminar25 Proof Observation If node i replicates the object, j is the nearest node to i that replicates the object, and d ij < α, then i should have a threshold at least (α – d ij ) Consider the social optimal configuration Φ. i is one of the cache nodes, Qi the set of nodes that access object from i (not including i). For each j in Qi, δj is extra cost needed if not accessing j Claim: If node i replicates the object, j is the nearest node to i that replicates the object, if i is

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CSE658 wireless Seminar27 Varying Placement Cost

Effects of Payment and Different Underlying Topology

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CSE658 wireless Seminar31 Conclusions Introduce a non-cooperative game model for selfish caching Pure strategy NE exists and the price of anarchy can be O(n) With payments, price of anarchy can be improved as one

Distributed Selfish Replication By N. Laoutaris etc IEEE TPDS 2005

CSE658 wireless Seminar33 Network Model Object sets O={o_1, o_2…,o_N}, n players Accessing object from a node’s local cache costs t_l, from a remote node’s cache t_r, and from origin server t_s, with t_l <= t_r <= t_s Each node has memory capacity: C_i Goal is to maximize the excess gain (access cost reduced due to caching)

CSE658 wireless Seminar34 Game Model Node i’s strategy P_i: the set of objects replicated at i Global object placement P = {P_1, P_2,…, P_n}; P_-i = P_1 U … U P_{i-1} U P_{i+1} U… U P_n The gain for i under P is

CSE658 wireless Seminar35 There exists a NE Step 1 (Greedy Local): each node caches the top C_i popular object according to its own access demand Step 2: in the order of their node id, each node adjusts its placement (move in/out objects) according to the current placement of all other nodes. What obtained is a NE.

Sketch of proof (why its NE): Only need to show in step 2 above, for any node i, the changes in the cache placement by the nodes following i’s turn do not affect the optimality of i’s placement (i.e. the top C_i popular data do not change). Lemma 1: In step 2, for any node, only object with more than 1 copy in the network can be removed Otherwise, the object’s gain won’t be reduced and thus won’t be removed out Lemma 2: In sep 2, only object with zero copy in the network can be moved in By way of contradiction Each removed object does not affect i’s cache placement optimality since there are at one more copy in the network Each moved-in object does not affect i’s cache placement optimality since its gain must be lower than that of any objects currently in i’s cache. So each node has no incentive to change its cache placement.

CSE658 wireless Seminar37 Our approach Network Model (two major difference compared with existing work): Consider both memory capacity of each node and the number of hops between nodes; consider both caching cost and access cost Network nodes are the source nodes of all the objects

CSE658 wireless Seminar38 Game model becomes more complicated!! Thus a pure strategy NE is in question, even omitting the caching cost!