Managing competition between universities by increasing the quality of the educational process.

Sandru, Ovidiu Ilie ; Sandru, Ioana Maria Diana

1. INTRODUCTION

In the context of globalizing resources, including human, and of shifting to knowledge based economy, the concept of quality management in higher education institutions has gained a new dimension. The development of this concept has become a research object for researchers worldwide. The researches that we carried out have shown that without applying notions and methods related to exact sciences, efforts would be useless. The direction followed consists in searching for mathematical models which enable the transfer of theoretical knowledge into practical actions. In this respect we cite the paper (Sandru, 2008) where more classes of problems referring to education quality management are modeled by using the abstract notion of dynamic system (Kalman et al., 1969).

Due to the complex nature of the educational process, some subcategories of this process can be approached from various angles. For example, the issue of competition between two universities discussed in this paper can be solved by using game theory methods similar to the ones used by us in (Sandru & Sandru, 2009) while solving a marketing problem.

The competition between two universities to gain higher academic prestige is a serious issue for management decision factors. Such a competitional state can be controlled by mathematical means which enable managerial policies projections fit to reach the set goals. Further on, we aim to make a detailed presentation of how to express competition based management problems mathematically and to solve them effectively.

2. MATHEMATICAL MODEL OF THE COMPETITION STATE BETWEEN UNIVERSITIES

Changes in the attitude toward the educational process can generate if not states of conflict, then at least competition generated states between universities in the same field. For a simplified presentation we shall restrain the analysis of the interdependence relationships to two universities. Let us assume that in a certain geographic area there are two universities A and B. To attract some of the current or potential students of university B, university A adopts a set of strategies

[S.sup.A] = {[s.sup.A.sub.i]|i = 1,2,...,m},

where [s.sup.A.sub.i] represents a change in investments by [p.sub.i]% to sustain the quality in a certain field of activity "i", i = 1,2,...,m. In order to counteract the loss of students, university B adopts a set of strategies, as well,

[S.sup.B] = {[s.sup.B.sub.j]\j = 1,2,...,n},

where [s.sup.B.sub.j] represents the change in the investments to sustain quality by [q.sub.j]% for a certain field of activity "j", j = 1,2,...,n.

The efficiency of the measures undertaken by the two competing universities is measured by the number of students moving from B to A, or by the number of students changing their intention to pursue academic studies within university B in favor of a similar study programme within university A . Mathematically, we can express this by means of the function

f : [S.sup.A] x [S.sup.B] [right arrow] [??],

where f([s.sup.A.sub.i], [s.sup.B.sub.j]), represent the number of students moving from university B (or changing their option of studying at B) to university A, then, when university A adopts strategy [s.sup.A.sub.i] and university B adopts strategy [s.sup.B.sub.j], i = 1,2,...,m; j = 1,2,...,n.

A first conclusion we can draw at this point, is that the competition state between universities A and B can be abstractized through a game theory problem characterized by the triplet

([S.sup.A],[S.sup.B], f).

This observation will help us mathematically solve the conflict of interest between the two universities. We are now in the position to make a rigorous analysis of the ways to act and react of the two universities.

University A's management reasons this way: if we choose strategy [s.sup.A.sub.i] [member of] [S.sup.A], then university B's management chooses that strategy [s.sup.B.sub.j] [member of] [S.sup.B] which minimizes losses, namely

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

From this reason, university A's management chooses that strategy [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for which

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

University B's decision factors reason this way: if we choose strategy [s.sup.B.sub.j] [member of] [S.sup.B], then university A's management tries to adopt that strategy [s.sup.A.sub.i] [member of] [S.sup.A] which ensures the highest number of students, namely

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII];

therefore, we have to choose strategy [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for which

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Both competitors' reasoning regarding an optimal behaviour (from each one's point of view) stands for the principle "min-max". A simple reasoning shows that between the two values

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

there exists the relation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Indeed,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Due to this result, if

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

then, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] respectively, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], are optimal strategies for each of the two firms. From this reason the point ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]) is called equilibrium point of the game ([S.sup.A],[S.sup.B],f).

Unfortunately, not all games admit equilibrium points and in order to deal with such situations we have to undertake some preparations. Let us denote by Q the matrix of elements

[q.sub.ij] = f([s.sup.A.sub.i], [s.sup.B.sub.j]), i = 1,2,...,m, j = 1,2,...,n.

Matrix Q will be called the game matrix ([S.sup.A],[S.sup.B],f), while games of this kind (with SA and SB finite) will be named matricial games.

Observations: 1) As shown above, a matricial game is perfectly determined by its matrix.

2) The matricial game,

([S.sup.A], [S.sup.B],f) = ([S.sup.A], [S.sup.B], Q),

admits an equilibrium point if there exists an element [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] of matrix Q which simultaneously is: the smallest in its row; the biggest in its column.

This being said, if the problem studied by us does not admit equilibrium points we shall extend the finite game ([S.sup.A],[S.sup.B],f), by which this problem is being modeled, to the infinite game ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]), defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [sup.t][beta] is the transposed of the line matrix [beta].

Observations: 1) Within the abridged game ([S.sup.A], [S.sup.B], f) the strategies of the extended game ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]), namely

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

can be regarded either as random variables which can express a decision taking [s.sup.A.sub.i], (i = 1,2,...,m ) with probability [[alpha].sub.i] (in case of university A), respectively, a decision taking [s.sup.B.sub.j], ( j = 1,2,...,n) with probability [[beta].sub.j] (in case of university B), or as mixed strategies, where (the now called pure) strategy [s.sup.A.sub.i], ( i = 1,2,...,m) is applied [[alpha].sub.i] x 100% (within strategy [s.sup.A.sub.[alpha]), respectively, where the pure strategy [s.sup.B.sub.j], (j = 1,2,..., n) is applied [[beta].sub.j] x 100% (within strategy [s.sup.B.sub.[beta]]).

2) A mixed strategy becomes a pure strategy if in the representation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

vectors [alpha] = ([[alpha].sub.1],..., [[alpha].sub.m]), respectively, [beta] = ([[beta].sub.1],...,[[beta].sub.n]) have one single component different from 0 and equal with 1.

3) The following value

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

represents the average gain expected by A (the average loss expected by B) when strategies [s.sup.A.sub.[alpha]], [[S.sup.B.sub.[beta]] are chosen.

4) The main qualitative difference between the basic games ([S.sup.A], [S.sup.B], f) and the extended ones ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]) is given by a major result known as Neumann's theorem according to which any extended game admits an equilibrium point. Indeed, by using linear programming techniques, it can be shown that the function

[??]([s.sup.A.sub.[alpha]], [S.sup.B.sub.[beta]]) = [alpha][Q.sup.t][beta],

admits an equilibrium point for any matrix Q. For a thorough documentation in the field of game theory we recommend the papers (Neumann & Morgerstern, 2007; Owen, 1995).

In conclusion, if according to point 4) of the observation above, ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]) is the equilibrium point of function [??] within the extended game ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]); in other words, if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are optimal strategies of the game ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]), then, according to point 1) of the same observation, strategies [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] will be represented by the mixed optimal strategies of the unextended game ([S.sup.A],[S.sup.B], f).

3. CONCLUSION

The theoretical results presented in this paper help university managers undertake a thorough and rigorous analysis of different situations dealing with, analysis which enables a correct reasoning and an adoption of best strategies, also providing an effective set of mathematical instruments to be used.

4. REFERENCES

Kalman, R. E.; Falb, P. L. & Arbib, M. A. (1969). Topics in Mathematical System Theory, McGraw-Hill, ISBN 0754321069, New York

Neumann, J. & Morgenstern, O. (2007). Theory of Games and Economic Behavior, Princeton University Press, ISBN-13: 9780691130613, ISBN 0691130612, New Jersey

Owen, G. (1995). Game Theory, Academic Press Inc, ISBN 012-531151-6, San Diego

Sandru, I. M. D. (2008). The optimal design of the quality management concepts using mathematical modeling techniques, In: Proceedings of the 10 th WSEAS International Conference on Mathematical and Computational Methods in Science and Engineering, Iliescu, M. et al., pp 334-339, ISBN 978-960-474-019-2, Politehnica University, November 2008, WSEAS Press, Bucharest

Sandru, O. I. & Sandru, I. M. D. (2009). Regarding marketing problems as dynamic system theory problems, In: Proceedings of the WSEAS International Conference, Hashemi, S. & Vobach, C., pp 183-187, ISBN 978-960-474-073-4, University Houston, May 2009, WSEAS Press, Houston