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We consider two-player games played on graphs equipped with costs on edges and introduce two winning conditions, cost-parity and cost-Streett, which require bounds on the cost between requests and their responses. Both conditions generalize the corresponding classical ω-regular conditions as well as the corresponding finitary conditions. For cost-parity games we show that the first player has positional winning strategies and that determining the winner lies in NP ∩ coNP. For cost-Streett gamesdoi:10.4230/lipics.fsttcs.2012.124 dblp:conf/fsttcs/FijalkowZ12 fatcat:4nnmkdmxhzdo7cpwfsralntzey