There are two types of game-theoretic approach. The first of them attempts to explain the social phenomena: e.g. why the US and the (former) Soviet Union selected military confrontation, rather than peace. The second is to examine if the social improvement is achieved by letting members of a group to play games. As an example of the latter policy-oriented approach, we can refer to the introduction of rewards for the informers in bid rigging, thus producing "prisoners' dilemma" for the bidders. This paper follows the second approach. The Japanese government attempted to improve the Japanese academic performance, by changing the funding formula for the Japanese universities. This paper examines if this policy of letting members of a group playing a severer game works. In 2004, all the national universities in Japan were restructured as the "university corporations", rather similar to the "agencies" of British type. One of the main purposes of the restructuring is to introduce more competition into the higher education in Japan. The higher education is the foundation of the advanced technologies, and it was argued that the US supremacy of technology, which invents new products, stems from the superior higher education in the US. For the promotion of advanced technologies, the reform of higher education was initiated in 2004. It is well known that the higher education in Japan is not efficient compared with other advanced countries, due to the lack of competition. While the former national Japanese universities are no more national in the sense that teachers are no more public servants, the government basically provides operational funds in research and education. The reform consists in providing funds competitively, in the sense that the more excellent the universities become, the more funds they obtain. In order to examine the government policy, the contribution by Bowen  is utilized. Bowen  constructed a simple one-university behavior model, which maximizes prestige, not profit, under budget constraint. "Prestige" is objectively evaluated academic performance, and in this paper, it is regarded as the academic performance or academic level. In his model, prestige, or academic level, P, is a function of the quantity and quality of the teaching, T, and research, R, activities undertaken by the university, and it is assumed that P=TR. The unit cost of teaching is denoted by ct, while that of research is denoted by cr. According to Bowen , this university maximizes academic level under budget constraint, ctT+ crR=B, where B is assumed to be a function of P; B=B(P)=P; B is an increasing function of P. In this paper, the model is extended to two-university model, where B1=B1(P1,P2), and B2=B2(P1,P2). Here, Bi is the budget for the ith university provided by the government with B1+ B2=B: constant, and Bi is greater as Pi/(P1+P2) becomes greater. In this paper, assuming that the ith university maximizes Pi=TiRi subject to ctiTi+criRi=Bi(P1,P2) (i=1,2), we construct a Nash non-cooperative game, and examine if the severer assessment results in higher academic levels. Utilizing simulation approach with the specification, Bi(P1,P2)= B Pi/(P1+P2), first, it is shown that the government's policy does not work, so long as ct1= ct2 and cr1=cr2. Thus, when the cost structure between the two universities is identical, the solution in this Nash non-cooperative game is independent of the government policy parameter α. If the cost structure between the two universities is not identical, it is shown that the government can enhance the academic levels by the introduction of more competition. It is shown that the sum of the two academic levels increases as α increases. It must be noted, however, that when α is large, the solution of this Nash non-cooperative game is unstable, so that further increase of α results in the disappearance of one of the universities. Thus, simple introduction of severer competition cannot enhance the Japanese academic levels.