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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.

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    $\begingroup$ Please provide more context on where this method is found or used. Some references would be great too. $\endgroup$
    – luchonacho
    Commented Jul 17, 2017 at 12:52
  • $\begingroup$ If you are asking for solutions to differential equations, this question would be better in the mathematics section. If that is what it is, I could answer it, but not as clearly or succinctly as a professional mathematician. $\endgroup$ Commented Jul 18, 2017 at 4:33

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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.

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This is an old question but it may be helpful for the community to give a formal answer. Without context, it is hard to say so much. However, the seminal paper of Levhari and Mirman (1980) published in Bell Journal of Economics (now RAND Journal of Economics) is one of the references.

The idea simply consists of putting a specific form of value function.

For fisheries, suppose that there are $N$ symmetric harvesters and say that $c_{jt}$ is the harvest rate of an individual $j$ at time $t$ and $x$ is the stock of fish with a regeneration function $$x_{t+1}=f(x_t-\sum_{j=1}^{N}c_jt)$$ where $x_t-\sum_{j=1}^{N}c_{jt}$ is the escapement level. Put a simple concave regeneration function, say

$$x_{t+1}=(x_t-\sum_{j=1}^{N}c_{jt})^\alpha$$

where $\alpha$ is the regeneration rate of the stock. Say also that a harvester has the following utility with usual properties;

$$u(c)= \text{log} c$$

In a non-cooperative framework, a harvester maximizes the following program

$$V(x_t)=\text{log} c_{jt} + \beta V(x_t+1)$$

which is a standard Hamilton-Jacobi-Bellman equation (HJB hereafter). $\beta\in\left(0,1\right)$ is the discount factor. By inserting the equation for stock dynamics into HJB equation, we slightly reformulate it such as

$$V(x_t)=\text{log} c_{jt} + \beta V((x_t-\sum_{j=1}^{N}c_{jt})^\alpha)$$

Once we have all these elements. Now, it is time to present linear strategies for the harvest. This means that the harvest strategy of a player $j$ is linear in the resource stock $x_t$ (also valid for all other players since all players are symmetrical). We write

$$c_{jt}=\omega x_t$$

Now, it is time to guess the value function. In some papers, you can find the word "conjecture" or "Nash conjecture". Say that your value function takes the form

$$V(x)=a \text{log} x + b$$

Note that we do not know the constants $a$ and $b$. However, we can find it by inserting it in HJB equation.

$$a \text{log} x + b = \text{log}(\omega x) + \beta (a \alpha \text{log}((1- N \omega)x)+ b) $$

Once we have all these elements, it is easy to find $a$ and $b$ which are

$$a=\frac{1}{1- \beta \alpha}$$

and

$$b=\frac{\text{log}\omega+\beta \alpha a \text{log}(1- N \omega) }{1-\beta}$$

If you want to verify that the result holds, just put the value function in HJB and you will see that it works. If you want to find the value of $\omega$, you should derive the first-order condition and find the envelope condition, which are quite straightforward.

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