The objective of the Ph.D. thesis is to establish a systematical framework on the study of some algorithmic game theoretical problems in graphs of fundamental importance such as rendezvous and searching games, congestion and load balancing games. We propose to study the rendezvous and searching games as a starting point, with the flexibility of tackling other fundamental games suggested by the candidate. We put particular emphasis on the generality and the structural properties of the problem of which the theoretical solution and the related analysis can bring new insights on the methodology of solving a class of generic problems. We also expect that the analysis conducted in the thesis will lead to a several distributed algorithms that can work in practical engineering scenarios, especially in distributed communication and networking paradigms.