Network creation games investigate complex networks from a game-theoretic point of view. Based on the original model by [PODC'03] many variants have been introduced. However, almost all versions have the drawback that edges are treated uniformly, i.e. every edge has the same cost and that this common parameter heavily influences the outcomes and the analysis of these games. We propose and analyze simple and natural parameter-free network creation games with non-uniform edge cost. Our models are inspired by social networks where the cost of forming a link is proportional to the popularity of the targeted node. Besides results on the complexity of computing a best response and on various properties of the sequential versions, we show that the most general version of our model has constant Price of Anarchy. To the best of our knowledge, this is the first proof of a constant Price of Anarchy for any network creation game.
Removing the parameter @math by restricting the agents to edge swaps was proposed and analyzed in @cite_6 @cite_20 . The obtained results are similar, e.g. the best known upper bound on the PoA is @math , there cannot exist a potential function @cite_17 and computing a best response is -hard. However, allowing only swaps leads to the unnatural effects that the number of edges cannot change and that the sequential version heavily depends on the initial network.
@cite_20 @cite_6 @cite_17
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A team consisting of an unknown number of mobile agents, starting from different nodes of an unknown network, have to meet at the same node. Agents move in synchronous rounds. Each agent has a different label. Up to f of the agents are Byzantine. We consider two levels of Byzantine behavior. A strongly Byzantine agent can choose an arbitrary port when it moves and it can convey arbitrary information to other agents, while a weakly Byzantine agent can do the same, except changing its label. What is the minimum number of good agents that guarantees deterministic gathering of all of them, with terminationq We solve exactly this Byzantine gathering problem in arbitrary networks for weakly Byzantine agents and give approximate solutions for strongly Byzantine agents, both when the size of the network is known and when it is unknown. It turns out that both the strength versus the weakness of Byzantine behavior and the knowledge of network size significantly impact the results. For weakly Byzantine agents, we show that any number of good agents permits solving the problem for networks of known size. If the size is unknown, then this minimum number is fp2. More precisely, we show a deterministic polynomial algorithm that gathers all good agents in an arbitrary network, provided that there are at least fp2 of them. We also provide a matching lower bound: we prove that if the number of good agents is at most fp1, then they are not able to gather deterministically with termination in some networks. For strongly Byzantine agents, we give a lower bound of fp1, even when the graph is known: we show that f good agents cannot gather deterministically in the presence of f Byzantine agents even in a ring of known size. On the positive side, we give deterministic gathering algorithms for at least 2fp1 good agents when the size of the network is known and for at least 4fp2 good agents when it is unknown. We resolve the worst-case price of anarchy (POA) of atomic splittable congestion games. Prior to this work, no tight bounds on the POA in such games were known, even for the simplest non-trivial special case of affine cost functions. We make two distinct contributions. On the upper-bound side, we define the framework of "local smoothness", which refines the standard smoothness framework for games with convex strategy sets. While standard smoothness arguments cannot establish tight bounds on the POA in atomic splittable congestion games, we prove that local smoothness arguments can. Further, we prove that every POA bound derived via local smoothness applies automatically to every correlated equilibrium of the game. Unlike standard smoothness arguments, bounds proved using local smoothness do not always apply to the coarse correlated equilibria of the game. Our second contribution is a very general lower bound: for every set L that satisfies mild technical conditions, the worst-case POA of pure Nash equilibria in atomic splittable congestion games with cost functions in L is exactly the smallest upper bound provable using local smoothness arguments. In particular, the worst-case POA of pure Nash equilibria, mixed Nash equilibria, and correlated equilibria coincide in such games. We study the problem of fair allocation for indivisible goods. We use the the maxmin share paradigm introduced by Budish as a measure for fairness. Procacciafirst (EC'14) were first to investigate this fundamental problem in the additive setting. In contrast to what real-world experiments suggest, they show that a maxmin guarantee (1- @math allocation) is not always possible even when the number of agents is limited to 3. While the existence of an approximation solution (e.g. a @math - @math allocation) is quite straightforward, improving the guarantee becomes subtler for larger constants. Procaccia provide a proof for existence of a @math - @math allocation and leave the question open for better guarantees. Our main contribution is an answer to the above question. We improve the result of ! to a @math factor in the additive setting. The main idea for our @math - @math allocation method is clustering the agents. To this end, we introduce three notions and techniques, namely reducibility, matching allocation, and cycle-envy-freeness, and prove the approximation guarantee of our algorithm via non-trivial applications of these techniques. Our analysis involves coloring and double counting arguments that might be of independent interest. One major shortcoming of the current studies on fair allocation is the additivity assumption on the valuations. We alleviate this by extending our results to the case of submodular, fractionally subadditive, and subadditive settings. More precisely, we give constant approximation guarantees for submodular and XOS agents, and a logarithmic approximation for the case of subadditive agents. Furthermore, we complement our results by providing close upper bounds for each class of valuation functions. Finally, we present algorithms to find such allocations for additive, submodular, and XOS settings in polynomial time. We study the size of memory of mobile agents that permits to solve deterministically the rendezvous problem, i.e., the task of meeting at some node, for two identical agents moving from node to node along the edges of an unknown anonymous connected graph. The rendezvous problem is unsolvable in the class of arbitrary connected graphs, as witnessed by the example of the cycle. Hence we restrict attention to rendezvous in trees, where rendezvous is feasible if and only if the initial positions of the agents are not symmetric. We prove that the minimum memory size guaranteeing rendezvous in all trees of size at most nis i¾?(logn) bits. The upper bound is provided by an algorithm for abstract state machines accomplishing rendezvous in all trees, and using O(logn) bits of memory in trees of size at most n. The lower bound is a consequence of the need to distinguish between up to ni¾? 1 links incident to a node. Thus, in the second part of the paper, we focus on the potential existence of pairs of finiteagents (i.e., finite automata) capable of accomplishing rendezvous in all bounded degreetrees. We show that, as opposed to what has been proved for the graph exploration problem, there are no finite agents capable of accomplishing rendezvous in all bounded degree trees.