Algorithmic Game Theory Bercea Multicast and Network Formation - PowerPoint PPT Presentation

AGT- Network Formation Games Ioana Oriana Algorithmic Game Theory Bercea Multicast and Network Formation Games Introduction Potential Functions and Games Ioana Oriana Bercea Global Connection Game Department of Computer Science The Model University of Maryland Price of Stability College Park The Non- Cooperative Multicast December 1, 2010 Game The Model Price of Anarchy A new model Related Games Our Game Ioana Oriana Bercea AGT- Network Formation Games

AGT- Network Formation Games Motivation Ioana Oriana Bercea The overarching goal of Network Formation Games is to Introduction analyze the way (efficient) networks form under the existence Potential of selfish agents, excluding a central authority. Functions and Games Examples include: Global Connection Game The Internet, social networks in general The Model Price of Network design(routing etc) Stability The Non- Operations Research(facility location games) Cooperative Multicast Game The Model Price of Anarchy A new model Related Games Our Game Ioana Oriana Bercea AGT- Network Formation Games

AGT- Network Formation Games Ioana Oriana By selfish behavior, we basically mean: Bercea players want to minimize the expenses they incur for Introduction building the network Potential Functions and they also seek to obtain a high quality of service from the Games network Global Connection Game Informally, the activity of agents reduces to the agents The Model choosing a particular set of edges(generally, paths) according to Price of Stability such selfish behavior. The Non- Cooperative Multicast Game The Model Price of Anarchy A new model Related Games Our Game Ioana Oriana Bercea AGT- Network Formation Games

AGT- Network Formation Games Ioana Oriana Bercea We are interested in the quality( social cost ) of the network. Introduction does the game have a Nash equilibrium ? Potential if it does, how much worse it is than the optimum ? Functions and Games Global Connection Game The Model Price of Stability The Non- Cooperative Multicast Game The Model Price of Anarchy A new model Related Games Our Game Ioana Oriana Bercea AGT- Network Formation Games

AGT- Network Formation Games Ioana Oriana Bercea We are interested in the quality( social cost ) of the network. Introduction does the game have a Nash equilibrium ? Potential if it does, how much worse it is than the optimum ? Functions and Games Global Connection Game The Model Price of Main tool: Potential functions! Stability The Non- Cooperative Multicast Game The Model Price of Anarchy A new model Related Games Our Game Ioana Oriana Bercea AGT- Network Formation Games

AGT- Network Formation Games Interesting variations: Ioana Oriana Bercea What exactly are the costs inccured by the agents? Introduction the cost for using/building an edge congestion? Potential Functions and latency? Games Global Connection Game The Model Price of Stability The Non- Cooperative Multicast Game The Model Price of Anarchy A new model Related Games Our Game Ioana Oriana Bercea AGT- Network Formation Games

AGT- Network Formation Games Interesting variations: Ioana Oriana Bercea What exactly are the costs inccured by the agents? Introduction the cost for using/building an edge congestion? Potential Functions and latency? Games Global Connection Game How are we going to make the agents pay? The Model Price of Stability we can set up certain cost sharing mechanisms that The Non- decide the way agents pay for their strategies Cooperative Multicast Game The Model Price of Anarchy A new model Related Games Our Game Ioana Oriana Bercea AGT- Network Formation Games

AGT- Network Definition Formation Games For any finite game, an exact potential function is a function Ioana Oriana Bercea that maps evert strategy vector S to some real value and satisfies the following condition: Introduction Potential Functions and Games Global Connection Game The Model Price of Stability The Non- Cooperative Multicast Game The Model Price of Anarchy A new model Related Games Our Game Ioana Oriana Bercea AGT- Network Formation Games

AGT- Network Definition Formation Games For any finite game, an exact potential function is a function Ioana Oriana Bercea that maps evert strategy vector S to some real value and satisfies the following condition: Introduction Potential Functions and Games if S = ( S 1 , S 2 , ..., S k ) , S i ′ � = S i is an alternate strategy for Global some player i , and S ′ = ( S − i , S i ′ ),then Connection Game The Model Price of Stability Φ( S ) − Φ( S ′ ) = u i ( S ′ ) − u i ( S ) The Non- Cooperative Multicast A game that possesses an exact potential function is called an Game The Model exact potential game . Price of Anarchy A new model Related Games Our Game Ioana Oriana Bercea AGT- Network Formation Games

AGT- Network Formation Games Ioana Oriana Theorem 19.11 Bercea Every potential game has at leat one pure Nash equilibrium, Introduction namely the strategy S that minimizes Φ( S ). Potential Functions and Games Proof Global Connection S is stable when no player can increase his/her utility by Game The Model choosing a different strategy, i.e. when Φ( S ) is at a local Price of Stability minimum. The Non- Cooperative Multicast Game The Model Price of Anarchy A new model Related Games Our Game Ioana Oriana Bercea AGT- Network Formation Games

AGT- Network Formation Games Theorem 19.12 Ioana Oriana Bercea In any finite potential game, best response dynamics always Introduction converge to a Nash equilibrium. Potential Functions and Games Proof Global Best response dynamics is the strategy which produces the Connection Game most favorable outcome for a player, taking other player’s The Model Price of strategies as given. Therefore, it simulates local search on Stability The Non- Φ.(improving moves for players decrease Φ) Cooperative Multicast Game The Model Price of Anarchy A new model Related Games Our Game Ioana Oriana Bercea AGT- Network Formation Games

AGT- Network Formation Games Ioana Oriana Bercea Theorem 19.13 Introduction Assume that, for any outcome S , Potential Functions and cost ( S ) Games ≤ Φ( S ) ≤ B · cost ( S ) A Global Connection Game for some constants A , B > 0. Then the price of stability is at The Model Price of most AB . Stability The Non- Cooperative Multicast Game The Model Price of Anarchy A new model Related Games Our Game Ioana Oriana Bercea AGT- Network Formation Games

Layout AGT- 1 Introduction Network Formation Games 2 Potential Functions and Games Ioana Oriana Bercea 3 Global Connection Game The Model Introduction Price of Stability Potential Functions and Games 4 The Non-Cooperative Multicast Game Global The Model Connection Game Price of Anarchy The Model Price of Stability 5 A new model The Non- Related Games Cooperative Multicast Our Game Game The Model Price of 6 Further ideas Anarchy A new model Related Games Our Game Ioana Oriana Bercea AGT- Network Formation Games

AGT- Network Formation Games Ioana Oriana The Model: Bercea a directed graph G = ( V , E ) Introduction Potential nonnegative edge costs c e for all edges e ∈ E . We will Functions and consider these costs to be fixed, though there are games in Games Global which that is not the case. Connection Game k players, each with a specified source node s i and sink The Model Price of node t i Stability The Non- Cooperative Multicast Game The Model Price of Anarchy A new model Related Games Our Game Ioana Oriana Bercea AGT- Network Formation Games

AGT- Network Formation Games Ioana Oriana Bercea Introduction Player i ’s goal is to build a network in which t i and s i are Potential connected, while minimizing construction costs. A strategy for Functions and Games player i is therefore a path from t i to s i . Global Connection Game The Model Price of Stability The Non- Cooperative Multicast Game The Model Price of Anarchy A new model Related Games Our Game Ioana Oriana Bercea AGT- Network Formation Games

AGT- Network Formation Games Ioana Oriana Apart from these parameters, we also need to set up the cost Bercea sharing mechanism, the way in which agents will contribute to Introduction building a network and, in particular, the set of edges in their Potential strategy. Functions and Games Global Connection A natural choice is the Shapley cost-sharing mechanism , Game also known as the fair mechanism, fair cost allocation, The Model Price of Stability egalitarian cost sharing etc. The Non- Cooperative Multicast Game The Model Price of Anarchy A new model Related Games Our Game Ioana Oriana Bercea AGT- Network Formation Games

AGT- Network Formation Games Ioana Oriana Bercea That just means that all the agents using a particular edge share its cost. Formally, if k e is the number of players whose Introduction path contains e , then e assigns a cost share of c e / k e to each of Potential Functions and them. Games Global Connection Game Also, the social objective for this game is simply the cost of the The Model constructed network.(sum of the cost played by all players) Price of Stability The Non- Cooperative Multicast Game The Model Price of Anarchy A new model Related Games Our Game Ioana Oriana Bercea AGT- Network Formation Games