 The distinction between game logics and logic games is not always sharp. Recent literature has seen a number of games whose design is connected to logic, yet they are not meant to analyze logical notions per se. Many of these games are played on graphs, as underlying game boards where players can perform different moves. Graph games are widely used in the computational literature for analyzing logical and algorithmic tasks ( Flum, Grädel &amp; Wilke 2007). However, in this section, we merely highlight one particular kind of graph game for its fruitful connections with logic which suggests a number of unusual issues.