3.1.3. Farmers' Expected Returns

Based on the above basic assumptions, the farmers' expected returns were related to government regulation in their behavioral decision regarding the use of VAs. To regulate the use of VAs, government regulators make spot checks to monitor pig farmers' VA use, and punish the improper use of VAs in accordance with laws and regulations. Government regulation and punishment of pig farmers for improper VA use have an impact on their use of VAs. Therefore, the farmers' expected returns can be described as follows.

Farmers' expected return from proper VA use is:

$$\mathcal{W}\_1 = \mathcal{G} \tag{1}$$

Farmers' expected return from improper VA use is:

$$W\_2 = (1 - q) \times (\Delta G + G) + q \times (\Delta G + G - C\_1 - C\_2) \tag{2}$$

where Δ*G* = *θ* × *G*, where *G* is the farmers' return from proper VA use; Δ*G* is the farmers' extra return from improper VA use; *C*<sup>1</sup> are the financial penalties imposed by government regulators on farmers for improper VA use; *C*<sup>2</sup> are the social costs of discovered improper VA use for farmers including pressure from public opinion and moral pressure, etc.; *θ* is the ratio of farmers' increased return from improper VA use to that from proper use; *q* is the probability of the farmers' improper VA use to be checked by government regulators.

#### 3.1.4. Behavior Probability Model

As pig farmers' VA use is affected by multiple factors, the choice probability for VA use varies among farmers. Sun et al. [23] developed a mathematical model of behavior probabilities to assess the probability of choosing a certain behavior under the general reward expectation on that behavior. For individual pig farmers, behavior probability is a description of behavioral uncertainty, that is, the probability of a farmer choosing a certain use of VAs in the "behavior set". Correspondingly, for the pig farmer group, behavior probability is the proportion of individual farmers who choose a certain use of VAs in the group. If all individuals in the group have the same return expectation on each use of VAs, they will all choose the same use of VAs, and there is no need to discuss behavior probability. However, in fact, there is a big difference in farmers' return expectation on each use of VAs. The differences in cognitive capacity and bias regarding VA use specification, hazards of VA residues, and relevant laws and regulations among each individual actor in the group lead to different probabilities for each farmer in choosing the use of VAs. Based on the literature [23] and the knowledge of farmers, a behavior probability model was developed in this study to simulate the farmers' VA use during pig farming under different return expectations.

*IJERPH* **2018**, *15*, 1126

According to the assumptions, the farmers' VA use was simplified into two categories: either proper use *a*<sup>+</sup> or improper use *a*−. The behavior set was *A* = {*a*+, *a*−}. The following behavior probability model was developed:

$$\begin{cases} p\_i(a\_+) = \frac{\varepsilon \{\varrho\_0 + (\varrho\_{i1} + \varrho\_{i2} + \varrho\_{i3})w\_i(a\_+) - (\varrho\_{i4} + \varrho\_{i5} + \varrho\_{i6})w\_i(a\_-)\}}{1 + \varepsilon \{\varrho\_{i0} + (\varrho\_{i1} + \varrho\_{i2} + \varrho\_{i3})w\_i(a\_+) - (\varrho\_{i4} + \varrho\_{i5} + \varrho\_{i6})w\_i(a\_-)\}} \\ p\_i(a\_-) = 1 - p\_i(a\_+) = 1 - \frac{\varepsilon \{\varrho\_{i0} + (\varrho\_{i1} + \varrho\_{i2} + \varrho\_{i3})w\_i(a\_+) - (\varrho\_{i4} + \varrho\_{i5} + \varrho\_{i6})w\_i(a\_-)\}}{1 + \varepsilon \{\varrho\_{i0} + (\varrho\_{i1} + \varrho\_{i2} + \varrho\_{i3})w\_i(a\_+) - (\varrho\_{i4} + \varrho\_{i5} + \varrho\_{i6})w\_i(a\_-)\}} \\ \quad \quad \frac{1}{1 + \varepsilon \{\varrho\_{i0} + (\varrho\_{i1} + \varrho\_{i2} + \varrho\_{i3})w\_i(a\_+) - (\varrho\_{i4} + \varrho\_{i5} + \varrho\_{i6})w\_i(a\_-)\}}{p\_i(a\_-)} \end{cases} \tag{3}$$

where *ϕij* is the regression coefficient, *i* ∈ [1,2, ... ,N], *j* ∈ [1,2, ... ,6], and *ϕij* > 0. It should be noted that when *j* = 0, *ϕi*<sup>0</sup> ∈ (−∞,+∞). When *ϕi*<sup>0</sup> determines that the expected returns from the two different behavioral choices, i.e., proper and improper VA use, are both 0, that is, *wi* (*a*+) = *wi* (*a*−) = 0, the *i-*th actor's behavior occurs without a driving force. The behavior probability in this case is called spontaneous probability. In fact, the farmer's choice regarding VA use is influenced by their judgment of the expected return. Based on the behavior probability model, the probabilities of proper use *a*<sup>+</sup> and improper use *a*−, *pi* (*a*+) and *pi* (*a*−), were simulated under the influence of farmers' knowledge and return expectations. It is assumed that when *pi* (*a*+) ≥ *pi* (*a*−), the *i-*th actor chooses proper use; otherwise, they choose improper use. The group behavior probability was obtained by the observation of a total of *N* actors.
