The theory of games played on graphs provides the mathematical foundations to study numerous important problems in branches of mathematics, economics, computer science, biology, and other fields. One key application area in computer science is the formal verification of reactive systems. The system is modeled as a graph, in which vertices of the graph represent states of the system, edges represent transitions, and paths represent behavior of the system. The verification of the system in an arbitrary environment is then studied as a problem of game played on the graph, where the players represent the different interacting agents. Traditionally, these games have been studied either with Boolean objectives, or single quantitative objectives. However, for the problem of verification of systems that must behave correctly in resource-constrained environments (such as an embedded system) both Boolean and quantitative objectives are necessary: the Boolean objective for correctness specification and quantitative objective for resource-constraints. Thus we need to generalize the theory of graph games such that the objectives can express combinations of quantitative and Boolean objectives. In this project, we will focus on the following research objectives for the study of graph games with quantitative objectives:(1) develop the mathematical theory and algorithms for the new class of games on graphs obtained by combining quantitative and Boolean objectives;(2) develop practical techniques (such as compositional and abstraction techniques) that allow our algorithmic solutions be implemented efficiently to handle large game graphs;(3) explore new application areas to demonstrate the application of quantitative graph games in diverse disciplines; and(4) develop the theory of games on graphs with infinite state space and with quantitative objectives.since the theory of graph games is foundational in several disciplines, new algorithmic solutions are expected.