I am trying to understand the intuition of the guess and verify method. Please could someone furnish me with a basic and very intuitive explanation of what we are trying to achieve and how this method helps us? I have been unable to find a summary elsewhere that explains the intuition in as jargon-free a way as possible.
As Dave Harris commented it depends a bit on the context. I know the guess and verify solution method mainly from solving value functions in differential resource games (more specifically the papers on fish wars), although I have also seen it used for value functions when there is no strategic interaction.
In such problems one is typically looking for a specific functional form that satisfies certain criteria, where the criteria are given by the problem. For a differential equation one of the criteria is that the functional form, when differentiated, returns the differential equation. For a value function, one of the criteria is that it satisfies the first order conditions and another that it satisfies Bellman's principal of optimality.
The specific functional form is often no more than an educated guess. In a paper on fish wars Fischer and Mirman try to identify a value function for a problem with an infinite time horizon, and they write that their guess is based on solving the problem in a finite 2,3 and 4 period setting and then inducting from that. They also write, however, that they cannot prove that this is the only functional form that satisfies the criteria.