In this paper, we propose a Quantifie Distributed Constraint Optimization problem (QDCOP) that extends the framework of Distributed Constraint Optimization problems (DCOPs). DCOPs have been studied as a fundamental model of multi-agent cooperation. In traditional DCOPs, all agents cooperate to optimize the sum of their cost functions. However, in practical systems some agents may desire to select the value of their variables without cooperation. In special cases, such agents may take the values with the worst impact on the quality of the result reachable by the optimization process. We apply existential/universal quantifier to distinct uncooperative variables. A universally quantifie variable is left unassigned by the optimization as the result has to hold when it takes any value from its domain, while an existentially quantifie variable takes exactly one of its values for each context. Similar classes of problems have recently been studied as (Distributed) Quantifie Constraint Problems, where the variables of the CSP have quantifiers All constraints should be satisfie independently of the value taken by universal variables. We propose a QDCOP that applies the concept of game tree search to DCOP. If the original problem is a minimization problem, agents that own universally quantifie variables may intend to maximize the cost value in the worst case. Other agents normally intend to optimize the minimizing problems. Therefore, only the bounds, especially the upper bounds, of the optimal value are guaranteed. The purpose of the new class of problems is to compute such bounds, as well as to compute sub-optimal solutions. For the QDCOP, we also propose several methods that are based on min-max/alpha-beta and ADOPT algorithms.