Matroids are Immune to Braess Paradox

The famous Braess paradox describes the following phenomenon: It might happen that the improvement of resources, like building a new street within a congested network, may in fact lead to larger costs for the players in an equilibrium. In this paper we consider general nonatomic congestion games and give a characterization of the maximal combinatorial property of strategy spaces for which Braess paradox does not occur. In a nutshell, bases of matroids are exactly this maximal structure. We prove our characterization by two novel sensitivity results for convex separable optimization problems over polymatroid base polyhedra which may be of independent interest.Comment: 21 page

Path deviations outperform approximate stability in heterogeneous congestion games

We consider non-atomic network congestion games with heterogeneous players where the latencies of the paths are subject to some bounded deviations. This model encompasses several well-studied extensions of the classical Wardrop model which incorporate, for example, risk-aversion, altruism or travel time delays. Our main goal is to analyze the worst-case deterioration in social cost of a perturbed Nash flow (i.e., for the perturbed latencies) with respect to an original Nash flow. We show that for homogeneous players perturbed Nash flows coincide with approximate Nash flows and derive tight bounds on their inefficiency. In contrast, we show that for heterogeneous populations this equivalence does not hold. We derive tight bounds on the inefficiency of both perturbed and approximate Nash flows for arbitrary player sensitivity distributions. Intuitively, our results suggest that the negative impact of path deviations (e.g., caused by risk-averse behavior or latency perturbations) is less severe than approximate stability (e.g., caused by limited responsiveness or bounded rationality). We also obtain a tight bound on the inefficiency of perturbed Nash flows for matroid congestion games and homogeneous populations if the path deviations can be decomposed into edge deviations. In particular, this provides a tight bound on the Price of Risk-Aversion for matroid congestion games

Routing choices in intelligent transport systems

Road congestion is a phenomenon that can often be avoided; roads become popular, travel times increase, which could be mitigated with better coordination mechanisms. The choice of route, mode of transport, and departure time all play a crucial part in controlling congestion levels. Technology, such as navigation applications, have the ability to influence these decisions and play an essential role in congestion reduction. To predict vehicles' routing behaviours, we model the system as a game with rational players. Players choose a path between origin and destination nodes in a network. Each player seeks to minimise their own journey time, often leading to inefficient equilibria with poor social welfare. Traffic congestion motivates the results in this thesis. However, the results also hold true for many other applications where congestion occurs, e.g. power grid demand. Coordinating route selection to reduce congestion constitutes a social dilemma for vehicles. In sequential social dilemmas, players' strategies need to balance their vulnerability to exploitation from their opponents and to learn to cooperate to achieve maximal payouts. We address this trade-off between mathematical safety and cooperation of strategies in social dilemmas to motivate our proposed algorithm, a safe method of achieving cooperation in social dilemmas, including route choice games. Many vehicles use navigation applications to help plan their journeys, but these provide only partial information about the routes available to them. We find a class of networks for which route information distribution cannot harm the receiver's expected travel times. Additionally, we consider a game where players always follow the route chosen by an application or where vehicle route selection is controlled by a route planner, such as autonomous vehicles. We show that having multiple route planners controlling vehicle routing leads to inefficient equilibria. We calculate the Price of Anarchy (PoA) for polynomial function travel times and show that multiagent reinforcement learning algorithms suffer from the predicted Price of Anarchy when controlling vehicle routing. Finally, we equip congestion games with waiting times at junctions to model the properties of traffic lights at intersections. Here, we show that Braess' paradox can be avoided by implementing traffic light cycles and establish the PoA for realistic waiting times. By employing intelligent traffic lights that use myopic learning, such as multi-agent reinforcement learning, we prove a natural reward function guarantees convergence to equilibrium. Moreover, we highlight the impact of multi-agent reinforcement learning traffic lights on the fairness of journey times to vehicles

経路選択ゲームの均衡解における経路本数制限効果の検証

　近年では混雑集中による観光公害が社会問題として注目されている．それに対して，増加しつつあるスマートフォン等による経路探索サービスを用いて，公平でありながらも全体の混雑が緩和されるような経路推薦を各ユーザに行う取り組みが始まっている．そこで利用されるのは経路選択ゲームと呼ばれる交通・社会ネットワークを設計，分析するための数理モデルが利用されている．　経路選択ゲームは，グラフ上の頂点ペア間のフローを形成するポテンシャルゲームの一種である．ポテンシャルゲームには均衡と呼ばれる状態が存在し，経路選択ゲームの均衡とは全利用者の頂点ペア間に対して，どの経路を選択してもコストが同一になる状態のことである．経路選択ゲームの派生モデルとして，利用可能な経路集合がすべて一定の要素数以下からなるマトロイド制約付き経路選択ゲームがある．　経路選択ゲームは，均衡解を求解に長い計算時間を要する．求解時に利用する経路の本数に制限を設けることで計算時間を削減することが可能であるが，制限がない場合に比べて得られる均衡解に乖離が見られる．そこで，本研究では経路本数を制限して計算した場合に得られる均衡解を検証し，どの程度の乖離が発生するのかを実験的に評価する．　本研究では先行研究であるマトロイド制約付き経路選択ゲームを基として，資源制約付き経路選択ゲームを定式化した．資源制約付き経路選択ゲームは，経路選択ゲームおいて均衡解を求めるための定式化であるポテンシャル関数最小化問題に，利用可能な経路に関する上限を与える制約を追加したものである．　資源制約付き経路選択ゲームについて，格子グラフとランダムグラフそれぞれのインスタンスについて実験を行い，得られた解について一般の経路選択ゲームとの比較を行った．本研究で行った実験における解の比較では，ポテンシャル間数値，平均コスト共に有意な差は存在しないという結果を得た．また，レプリケータダイナミクスを用いた計算時間の比較実験では，通常の均衡解探索と上限数まで経路本数を削減した場合での均衡解探索の時間を比較し，削減した場合の方が高速に求解可能であることを確認した．さらに現実の交通網を模したネットワークでの実験を行い，得られる均衡解についての考察を行った．電気通信大学201