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Distributed Constraint Optimization problems (DCOPs) have been studied as a fundamental model of multiagent 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. Similar classes of problems have been studied as Quantified (Distributed) Constraint Problems, where the variables of the CSP have existential/universal quantifiers. All constraints should be satisfied independently of the value taken by universal variables. In this paper, a Quantified Distributed Constraint Optimization problem (QDCOP) that extends the framework of DCOPs is presented. We apply existential/universal quantifiers to distinct uncooperative variables. A universally quantified variable is left unassigned by the optimization as the result has to hold when it takes any value from its domain, while an existentially quantified variable takes exactly one of its values for each context. We consider that the QDCOP applies the concept of game tree search to DCOP. If the original problem is a minimization problem, agents that own universally quantified 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 propose solution methods that are based on min-max/alpha-beta and ADOPT algorithms.

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