How Albanian Private Universities Can Use Game Theory for Optimization of Scholarship Offers

Article excerpt


There are 46 private universities in Albania. We believe that the tuition fee and scholarship for high GPA students are two important components of the competition between private universities. This study is a first attempt in applying Game Theory for optimization of scholarship offers by Albanian private universities during academic years. There is a conflict between the utility functions of shareholders to maximize their profits and the utility functions of the students enrolled in private universities to minimize their expenses. The best solution of the conflict is a Nash equilibrium or Bayes-Nash-Harsanyi equilibrium, respectively. We prove the existence of these equilibria. However, there exist many equilibria, depending of private information. We interpret the Boyes-Nash-Harsanyi equilibrium solution as the set of possible rational concessions made by all players.

Keywords: private universities, scholarship, optimization, game theory, Nash equilibrium.

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1. Introduction

Actually, there are 46 private universities in Albania. Strengthening the position of private universities in the Republic of Albania in the actual conditions is a complicated problem involving several variables: profit, risk, individual goals and preferences, teaching and scientific levels, financial support, competitors, social and legal rules, corruption, etc. In this study, using Game Theory, Probability and Mathematical Statistics, we will try to find an optimal scholarship strategy in the private universities of Albania during the academic years. We apply Nash equilibrium, Bayes-Nash-Harsanyi equilibrium, and subjective equilibrium.

Main components of competition between private universities are:

1. Tuition fee

2. Scholarship for high GPA students, financial support for special cases

3. Degree of Bachelor of Science, Master Degrees, PhD. Degrees that are offered

4. Quality of teaching

5. Employment percentage of graduates

6. Quality of the library, availability of electronic data, computerization

7. Campuses and buildings

8. Research, international or natural scientific conferences, and publications

9. Honorary doctors and famous alumni

10. International cooperation.

The number of new students enrolled in a private university during different academic years is a random variable, depending on a plethora of economic, political, social, competitive, psychological and personal factors. In the following, we investigate the strategic game as a model of new students enrolled at Albanian private universities.

A game in strategic (or normal) form has three components: the set of all players which we assume to be finite {1, 2, ..., m}, the pure strategy space Sk for each player K, and the utility functions Uk = Uk(s) for each profile of strategies (situation) s = (s1, s2, Each player's objective is to maximize his own utility function, and this may involve "helping" or "hurting" the other players. For economists, the most familiar interpretation of strategies may be as choices of prices or output levels.

A mixed strategy is a probability distribution over pure strategies. We assume that each player's randomization over the space of pure strategies is independent of those of his (her) opponents. The utility functions to a profile of mixed strategies are the expectations (mathematical expectations) of the corresponding pure-strategy utility functions.

According to A. N. Kolmogorov, bargaining is a random process to settle disputes and reach mutually beneficial agreements. Typical situations of bargaining are characterized by two or more participants (agents) who have common interest in cooperating, but conflicting interests in the way of doing so. The outcomes of bargaining depend on agents' attitudes towards their bargaining items and their mathematical expectations from the realized bargaining. …