Strategy Logic (SL) is a very expressive temporal logic for specifying and verifying properties of multi-agent systems: in SL, one can quantify over strategies, assign them to agents, and express LTL properties of the resulting plays. Such a powerful framework has two drawbacks: first, model checking SL has non-elementary complexity; second, the exact semantics of SL is rather intricate, and may not correspond to what is expected. In this paper, we focus on strategy dependences in SL, by tracking how existentially-quantified strategies in a formula may (or may not) depend on other strategies selected in the formula, revisiting the approach of [Mogavero et al., Reasoning about strategies: On the model-checking problem, 2014]. We explain why elementary dependences, as defined by Mogavero et al., do not exactly capture the intended concept of behavioral strategies. We address this discrepancy by introducing timeline dependences, and exhibit a large fragment of SL for which model checking can be performed in 2-EXPTIME under this new semantics. 1 Introduction Temporal logics. Since Pnueli's seminal paper [24] in 1977, temporal logics have been widely used in theoretical computer science, especially by the formal-verification community. Temporal logics provide powerful languages for expressing properties of reactive systems, and enjoy efficient algorithms for satisfiability and model checking [9]. Since the early 2000s, new temporal logics have appeared to address open and multi-agent systems. While classical temporal logics (e.g. CTL [8, 25] and LTL [24]) could only deal with one or all the behaviours of the whole system, ATL [2] expresses properties of (executions generated by) behaviours of individual components of the system. ATL has been extensively studied since then, both about its expressiveness and about its verification algorithms [2, 13, 16]. Strategic interactions in ATL. Strategies in ATL are handled in a very limited way, and there are no real strategic interactions in that logic (which, in return, enjoys a polynomial-time model-checking algorithm). Over the last 10 years, various extensions have been defined and studied in order to allow for more interactions [1, 7, 6, 18, 26]. Strategy Logic (SL for short) [7, 18] is such a powerful approach, in which strategies are first-class objects; formulas can quantify (universally and existentially) over strategies, store those strategies in variables, assign them to players, and express properties of the resulting plays. As a simple example, the existence of a winning strategy for Player A (with objective ϕ A) against any strategy of * Supported by ERC project EQualIS (308087).