Game theoretic approach for fertilizer application: looking for the propensity to cooperate

Abstract

This paper continues the research implemented in previous work of (Schreider et al. in Environ. Model. Assess. 15(4):223–238, 2010) where a game theoretic model for optimal fertilizer application in the Hopkins River catchment was formulated, implemented and solved for its optimal strategies. In that work, the authors considered farmers from this catchment as individual players whose objective is to maximize their objective functions which are constituted from two components: economic gain associated with the application of fertilizers which contain phosphorus to the soil and environmental harms associated with this application. The environmental losses are associated with the blue-green algae blooming of the coastal waterways due to phosphorus exported from upstream areas of the catchment. In the previous paper, all agents are considered as rational players and two types of equilibria were considered: fully non-cooperative Nash equilibrium and cooperative Pareto optimum solutions. Among the plethora of Pareto optima, the solution corresponding to the equally weighted individual objective functions were selected. In this paper, the cooperative game approach involving the formation of coalitions and modeling of characteristic value function will be applied and Shapley values for the players obtained. A significant contribution of this approach is the construction of a characteristic function which incorporates both the Nash and Pareto equilibria, showing that it is superadditive. It will be shown that this approach will allow each players to obtain payoffs which strictly dominate their payoffs obtained from their Nash equilibria.

Appendix

This appendix provides details for obtaining the optimal solutions of expressions (7) and (8).

For $u_{i}(\alpha_{i}^{r})$ the first derivative is:

(A.1)

to maximize $u_{i}(\alpha_{i}^{r})$ we solve $\frac{\partial u_{i}}{\partial\alpha_{i}^{r}}=0$, however the obtained solution must satisfy $A_{r1}\leq\alpha_{i}^{r}\leq A_{r1}$ for i=1,2,…,N and r=1,2,…,R. We outline the following steps to find the solution:

Step 1.

Assume $\alpha_{i}^{r}>\frac{L}{(1-q_{i}^{r})}$ hence $I(\alpha_{i}^{r}(1-q_{i}^{r})>L)=1$. Then from (A.1) we obtain:

If $\alpha_{i}^{r_{*}}>\frac{L}{(1-q_{i}^{r})}$ is hold, $\alpha_{i}^{r_{*}}$ is a candidate solution for maximizing $u_{i}(\alpha_{i}^{r})$ (the second derivative condition as shown in the main text is satisfied). Otherwise, $\alpha_{i}^{r_{*}}$ is not feasible and candidate solutions are $\frac{L}{(1-q_{i}^{r})}+\epsilon$ (where ϵ is a very small value close to 0), A _r1 if $A_{r1}>\frac{L}{(1-q_{i}^{r})}$ and A _r2 if $A_{r2}>\frac{L}{(1-q_{i}^{r})}$.

Step 2.

Assume $\alpha_{i}^{r}\leq\frac{L}{(1-q_{i}^{r})}$ hence $I(\alpha_{i}^{r}(1-q_{i}^{r})>L)=0$ and we obtain:

If $\alpha_{i}^{r_{**}}\leq\frac{L}{(1-q_{j}^{r})}$ is true, $\alpha_{i}^{r_{**}}$ is a candidate solution for maximizing $u_{i}(\alpha_{i}^{r})$ (the second derivative as shown in the text is satisfied). Otherwise $\alpha_{i}^{r_{**}}$ is not feasible and candidate solutions are $\frac{L}{(1-q_{i}^{r})}$, A _r1 if $A_{r1}\leq\frac{L}{(1-q_{i}^{r})}$ and A _r2 if $A_{r2}\leq\frac{L}{(1-q_{i}^{r})}$.

Step 3.

Evaluate candidate solutions obtained from Step 1 and Step 2 and see which one maximize the following term (which is the part of $u_{i}(\alpha_{i}^{r})$ that encompasses $\alpha_{i}^{r}$):

For Z(s), the first derivative is:

(A.2)

Since $\frac{\partial Z(s)}{\partial\alpha_{i}^{r}}$ does not contain any $\alpha_{j}^{r}$s (i≠j), Steps 1 to 3 can be applied similarly to obtain the solution for (A.2).