# On Multidimensional Congestion Games +
### Our Contribution and Significance  
The works of [PERSON] et al. (2018) and [PERSON] et al. (2019) aim at characterizing the impact of social ignorance in the most general case, that is, without imposing any particular structure on the social knowledge graphs defining the graphical game. Nevertheless, real-world-based knowledge relationships usually obey some regularities and present recurrent patterns: for instance, people tend to cluster themselves into well-structured collaborative groups (cliques) due to family memberships, mutual friendships, interest sharing, business partnerships. To this aim, we introduce and study _multidimensional (weighted) congestion games_, that is, (weighted) congestion games whose set of players are partitioned into \(d+1\) clusters \(C_{0},C_{1},\ldots,C_{d}\). Players in \(C_{0}\) have full information about all the other participants in the game, while players in \(C_{i}\), for any \(1\leq i\leq d\), have full information only about the members of \(C_{0}\cup C_{i}\) and are unaware of all the others. It is not difficult to see (and we provide a formal proof of this fact in the next section) that each multidimensional (weighted) congestion game is a graphical (weighted) congestion game whose social knowledge graph \(G\) is undirected and verifies \(\delta(G)=d\). In addition, \(G\) possesses the following, well-structured, topology: it is the union of \(d+1\) disjoint cliques (each corresponding to one of the \(d+1\) clusters in the multidimensional (weighted) congestion game) with the additional property that there exists an edge from all the nodes belonging to one of these cliques (the one corresponding to cluster \(C_{0}\)) to all the nodes in all the other cliques. We believe that the study of graphical games restricted to some prescribed social knowledge graphs like the ones we consider here, may be better suited to understand the impact of social ignorance in non-cooperative systems coming from practical and real-world applications. Moreover, the particular social knowledge relationships embedded in the definition of multidimensional (weighted) congestion games perfectly model the situation that generates when several independent games with full information are gathered together by some promoting parties so as to form a sort of \"global super-game\". The promoting parties become players with full information in the super-game, while each player in the composing sub-games maintains full information about all the other players in the same sub-game, acquires full information about all the promoting parties in the super game, but completely ignores the players in the other sub-games. Such a composing scheme resembles, in a sense, the general architecture of the Internet, viewed as a self-emerged network resulting from the aggregation of several autonomous systems (AS). Users in an AS have full information about anything happening within the AS, but, at the same time, they completely ignore the network global architecture and how it develops outside their own AS, except for the existence of high-level network routers. High-level network routers, instead, have full information about the entire network. Furthermore, multidimensional (weighted) congestion games are also useful to model games in which players belong to different types and the level of competition that each player experiences on each selected resource depends on her type and on the types of the other players using the resource. Consider, for instance, a machine which is able to perform \(d\) different types of activities in parallel and a set of tasks requiring the use of this machine. Tasks are of two types: simple and complex. Simple tasks take the machine busy on one particular activity only, while complex tasks require the completion of all the \(d\) activities. Hence, complex tasks compete with all the other tasks, while simple ones compete only with the tasks requiring the same machine (thus, also with complex tasks).