# Model of Multi-Branch Trees for Efficient Resource Allocation
## 4 Mathematical Features
### Incentive Compatibility  
Our method is incentive-compatible because no director has an incentive to make fallacious assessments. Let us consider when a director assesses a project lower than its true value. This underestimation has a positive effect if an ancestor of the director (an evaluator who is not a director) chooses the project because the allocation to the other projects of the department can increase. However, if no ancestor of the director selects the project, the allocation decreases. As well, let us consider when a director assesses a project to be greater than its true value. This overestimation has a positive effect if no ancestor of the director selects the project, while allocation to the other projects decreases if an ancestor of the director does choose the project. Regardless of either underestimation or overestimation, it is possible to lose one's benefit compared with the case of true assessment. Therefore, a max-min strategist has no merit to make fallacious assessments. The following theorem supports our discussion rigorously.  
**Theorem 2**: _If a director adopts a max-min strategy, one has no incentive to make fallacious assessments._  
Let \(x_{i}\) be the assessment value of a project \(i\) in all projects, \(f\) be a project that a director wants to assess fallaciously, \(D\) be the department \(f\) belongs to, \(D_{-f}\) be the set of projects of \(D\) except for \(f\), \(q\) be the fallacious value of \(f\) that the director wants to assign instead of \(x_{f}\), and \(S\) be the total of the relative weights of all projects except for \(f\) (\(S=\sum_{i}x_{i}-x_{f}\)). Let \(T=\sum_{D_{-f}}x_{i}\). We set it so that the total of the budget goes to \(1\) without loss of generality.  
Case 1: The total number of evaluators assessing \(f\) is one (only a director). In this case, the relative weight of \(f\) only is changed to \(q\) from \(x_{f}\). Therefore, the allocation of \(i\), which is not \(f\), is changed to \(\frac{x_{i}}{5+q}\) from \(\frac{x_{i}}{5+x_{f}}\), and the allocation of \(f\) is changed to \(\frac{q}{5+q}\) from \(\frac{x_{f}}{5+x_{f}}\). Therefore, the total allocation of the department is changed to \(\frac{T+q}{5+q}\) from \(\frac{T+x_{f}}{5+x_{f}}\). The difference, \(\Delta_{1}\), is calculated as\[\Delta_{1}=\frac{(T-S)(x_{f}-q)}{(S+q)(S+x_{f})}. \tag{9}\]  
\(S>T\) is satisfied, and thus, the allocation of the department decreases if \(x_{f}>q\).  
Case 2: The total number of evaluators assessing \(f\) is more than one. In this case, no evaluators except for the director change their own assessments, and thus, the relative weights of none of the projects are changed expect for the elements of \(D_{-f}\). The relative weight of any \(x_{i}\) in \(D_{-f}\) goes to \(\frac{x_{f}}{q}\) times of its original value. Therefore, the total allocation of the department is changed to \(\frac{T\frac{x_{f}}{q}+x_{f}}{S+x_{f}+(\frac{x_{f}}{q}-1)T}\) from \(\frac{T+x_{f}}{S+x_{f}}\). The difference, \(\Delta_{2}\), is calculated as  
\[\Delta_{2}=\frac{(S-T)(x_{f}-q)T}{\big{(}S+x_{f}+(\frac{x_{f}}{q}-1)T\big{)}(S +x_{f})q}. \tag{10}\]  
\(S>T\) is satisfied, and thus, the allocation of the department decreases if \(x_{f}<q\).  
Considering these two cases, if a director makes a fallacious assessment, the total allocation of the department can decrease regardless of projects chosen by the ancestors. Moreover, if the ancestors choose suitable projects, the total allocation of the department can decrease regardless of the value of the fallacious assessment (the value is both greater and smaller than its true value).  
Note that we will extend the discussion above to analyze cases including that in which the ancestors adopt the max-min strategy, or in which the evaluators adopt the expected-utility principle.