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Kellog MEDS program has two concentrations. One is the MECS concentration (the one you reference), which to my understanding is a quant heavy strategy theory/micro theory/game theory PhD and is a more theoretical program with more of a quantitative and economic focus than some of the more OB oriented strategy programs. Still, it is arguably the top place in the world to go to if you are interested in game theory the more quant heavy econ oriented strategy stuff, and is a very well regarded program in the econ world (though I am not sure of the perception of the program in the strategy world). The second program is an OM concentration, and is like an other OM/OR program and probably would not fit your interests.
Columbia Business School has some excellent faculty in the management dept, though I have no idea how good the PhD program is.
NYU Stern's Strategy PhD is technically jointly offered by the econ and OB departments, but it is divided into two concentrations. One is the econ concentration, and is run by Stern's econ department and is more like Kellog's MECS program, while the other concentration is a management concentration that is run by the OB department. I don't know much about the quality of the program, but their placement in recent years has been excellent.
While I am an econ/finance person and not a strategy person, to my understanding there are two broad divisions within the field of strategy. There are schools and programs that take more of an OB/psychology/economic sociology approach and there are those that take a more quantitative/microeconomic/game theory approach. While all schools and programs study strategy, the different approaches to study fit different students.
As to the GRE V, if you are not a native english speaker and are required to take the Toefl, then they will probably not count the GRE V score for much if your Toefl scores are good. Even for native english speakers, as long as you score >500 or >550 on the GRE V, most schools will not care beyond that.
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