# Model of Multi-Branch Trees for Efficient Resource Allocation
## 1 Introduction
Moreover, they tend to spend budgets wastefully at the end of a fiscal year. With this research background in mind, the objective of our study is to find a feasible allocation method that is better than the primitive method based on incrementalism [23; 24] even if there is no appropriate unified criterion. Here, we propose an allocation method that can distribute budgets efficiently with imperfect information, not perfect information, in cooperative games. To do so, we consider two points: minimizing assessment costs and satisfying allocational efficiency. Assessment costs here mean the costs of effort spent on assessments including the costs of designing a unified criterion that can be applied to all objects for allocation. Actually, designing such a criterion is almost impossible, and thus, we should consider a method that determines all allocation by integrating many assessments, the scope of which is local only. We developed a method in which multiple evaluators assess all objects for allocation in a manner like the division of labor, and allocation is calculated by integrating those assessments. Our method minimizes assessment costs. The second point, allocational efficiency, is also an important issue. Let us consider a department where two projects would like to get money. The director of the department may be one the best people to assign money to those projects. Therefore, the director's assessment on the relative weights between the two can be extremely important information when considering budget allocation. We believe that the popularity of the allocation method based on incrementalism and precedent that is widely adopted in Japan is empirical evidence that budgets being distributed by directors to their own projects in their department is one of the most effective allocation methods. Thus, are there any allocation methods that are more rational than a method that uses budget ceilings based on incrementalism? To consider this point, let us consider a small company that is working on budget allocation. The executive is the person in charge of the allocation, and there are several departments in the company. Each department is managed by a director and has several projects to which part of the budget needs to be assigned. If the directors precisely assess the ratesof allocation for the projects in their departments, is it possible to regard assessing one project only among all projects as an assessment of all projects in the departments? An executive cannot consider all projects in one's company; however, if the executive selects one project in each department, it can be considered that the executive assesses all projects due to supporting by the directors who assess the remaining projects. The executive may not understand each project in detail. In this case, however, the executive can request that the directors send reports on the selected projects. By doing this, the executive may know the selected projects as do the directors. The executive can assess the relative values between departments from their own viewpoint by comparing each selected project. Here, we propose a protocol of allocation based on the above discussion. In this protocol, a director's assessment is restricted to one's own projects in one's own department, and both an executive's and mid-level executives' assessments are also restricted to each representative project in each of the branches or departments that they manage. At the same time, we develop a calculation method to integrate such assessments. In our method, we use a multi-branch tree structure, where a set of leaf nodes represents either projects or policies and a set of non-leaf nodes represents either directors or executives. Our method may be balanced between minimizing assessment costs and allocational efficiency, and thus, we believe it to be novel. The rest of this paper is organized as follows. We describe our novel budget allocation method for both a tournament model and a generalized model in Section 2. Some examples are provided in Section 3.