# Efficient Strategies Algorithms for Resource Allocation Problems
## 3 Proposals of Strategies for Decision Making in an Interaction
### Definition and Payment Function or Utility Function  
Here, we introduce the new resource allocation strategy implemented, in order to respond to the problems formulated in the previous section. The development of the allocation strategy will consist of making a study starting by giving a mathematical definition of the strategy, then highlighting the utility functions or payment functions and finally establishing the rules to compare strategies.  
A strategy is a tuple \((c_{1},c_{2},\ldots,c_{n})\) such as \(c_{1}+c_{2}+\ldots+c_{n}\leq R\), with:  
* \(c_{i}\): the battlefield \(i\),
* \(n\): the number of battlefields,
* \(R\): the available resource.  
Let there be two strategies \(X=(x_{1},x_{2},x_{N})\) and \(Y=(y_{1},y_{2},\cdots,y_{N})\) with \(N\) the number of battlefields. Let us take \(U\) and \(V\) respectively the utility functions of strategies \(X\) and \(Y\), initially at 0. There are two possible cases: either the fields are homogeneous and their value will be 1, or they are heterogeneous and in this case a \(N\)-tuplet \(W=(w_{1},w_{2},\ldots,w_{N})\) will be introduced whose \(w_{i}\) corresponds to the value of the field \(i\). The usefulness of the strategies is obtained as follows:  
For \(i=1\ldots N\):  
* Homogeneous fields (the fields have the same values) If \(X_{i}<Y_{i}\), we increment \(V\) If \(X_{i}>Y_{i}\) we increment \(U\)
* \(n\): Heterogeneous fields (the values of the fields are different) If \(X_{i}<Y_{i}\) we add to \(V\) the value \(w_{i}\) If \(X_{i}>Y_{i}\) we add to \(U\) the value \(w_{i}\)  
In both cases (homogeneous and heterogeneous fields), if there is an equality between \(X_{i}\) and \(Y_{i}\), the values of the two utility functions remain unchanged. For the two cases previously seen, we considered that the resources are homogeneous. In the case of heterogeneous resources:  
* You must take a resource as a reference (resource whose value will be 1 if possible or the one with the smallest value).
* Each resource will have a value which will depend on the reference one.
* We create new strategies with the added values and we apply the above rules (case where the fields are homogeneous or heterogeneous).  
In order to better understand this last part (case of heterogeneous resources), let's see together a concrete example.  
In this example, we have ten (10) resources to allocate on three (3) battlefields. The resources are heterogeneous and their values can be found in Table 1.  
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline Resources & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\ \hline Values & 2 & 1 & 2 & 5 & 1 & 1 & 2 & 1 & 1 & 1 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Resources with their values.  
* We can take, as resource reference, resource \(n^{0}2\) in green color.
* Resource \(n^{0}1=2\times(resource\ n^{0}2)\) and so on for the others.
* Creation of new strategies with the added values of the resources allocated on the same field.  
Let us take two (2) strategies \((1,2,7)\) and \((2,2,6)\) as shown in Figure 1 and apply the transformation rule to get new strategies:  
1. the number of resources allocated to the different fields
2. the list of resources allocated to the fields
3. adding the values of resources on the same field and finally obtaining the new strategy.