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In Path the "Run" property field corresponds to the Monte Carlo iteration. In Path version 3.0 it only appears on the user interface for one scenario property, and that is the Transition Multipliers property. However if you look in the Path database (as you have done) then you’ll see that the full design for Path accepts a "Run" field for several other properties, with the Initial Conditions property (and its corresponding db table StratumInitialState) being one of these. This is intended for future use, where we eventually plan to allow users to optionally set several properties differently for each Monte Carlo iteration.
If you are setting properties directly through the database you should leave the Run property field blank for all properties except Transition Multipliers. For Transition Multipliers you have the option of leaving it blank or specifying values separately for each Iteration; if you leave it blank then Path will apply your property values to all iterations at runtime.
Hope this helps – Colin.