| dc.description.abstract | The primary objective of this thesis was to develop new solution concepts for two?person multicriteria games, emphasizing the security aspect of the players’ strategies, and to demonstrate their application to two?target or combat differential games. Because of the multicriteria nature of the game, the solution concepts are those derived from efficiency or Pareto optimality.
In two?person zero?sum scalar?criterion games satisfying saddle?point equilibrium, it was not necessary to consider the security aspects separately, as the saddle?point strategies themselves satisfy the requirements of security. But in non?zero?sum scalar?criterion games, the equilibrium strategies and the security strategies (defined as minimax or maximin strategies) are, in general, not the same. A similar dichotomy is noticed in multicriteria games. In the literature, research on multicriteria games has mainly restricted itself to the analysis of equilibrium strategies. Not many results are available on strategies based on security notions. In games where the opponent can win by violating certain threshold levels in the performance criteria, the security offered by a strategy to a player against the opponent’s worst?case behaviour, in terms of each of the player’s criteria, becomes extremely important. This was the aspect dealt with in this thesis.
Following conventional practice, we first investigated two?person zero?sum multicriteria matrix games and proposed various solution concepts and obtained their existence results. The concept of Pareto saddle points was defined and shown to possess all the good properties of saddle points in zero?sum scalar?criterion matrix games. But Pareto saddle points seldom exist in multicriteria games. The class of matrices for which Pareto saddle points exist was identified through certain equivalence conditions. Equilibrium points always exist in these games (as shown in Shapley (1959) and Corley (1985)), but they do not possess many of the desirable properties of Pareto saddle points (such as ordered interchangeability and optimal security levels). Next, the Pareto Optimal Response Strategies (PORS) were defined and their relationship with the equilibrium strategies was established.
A new solution concept, in the form of Pareto Optimal Security Strategies (POSS), was proposed. Some necessary and sufficient conditions were proved separately, followed by a condition that is both necessary and sufficient, all based on scalarization techniques. These results are useful in proving the existence of POSS and in determining them. The main drawback is that the actual implementation of this procedure requires a search over a compact and convex set of scalarization vectors. An important result proved here was that only a finite number of scalarizations is sufficient to obtain all the POSS. The problem of obtaining these scalarization vectors was not addressed here; this is an important topic for further research, which would enhance the implementability and utility of the method. Another topic of research may involve the development of methods for determining POSS that do not require scalarization techniques. One possibility is to use the sets of saddle?point solutions of the individual matrix games A(k), k = 1,…,n, and construct the POSS sets from them.
The concept of security was extended to multicriteria matrix games with an additional structure of qualitative outcomes. Different regions of the criterion space were identified with certain qualitative outcomes. The result of the multicriteria matrix game was then expressed as the outcome associated with the region containing the payoff vector. Under certain restricted conditions, solutions were proposed with players adopting pure or mixed strategies. A hypothetical combat game model was used to illustrate the solution concepts. Though the example used here was oversimplified, it could serve as a model for aerial combat games between fighter aircraft, in which each player's control can be discretized over time, as in Kelley and Lefton (1978), and a matrix game could be formulated similar to Neuman (1987). This approach also permits easy incorporation of mixed strategies. The major effort in future research in this direction should focus on two aspects: formulation of the matrix and development of solution concepts more general than the one presented here. Another promising approach deserving investigation is the approachability–excludability theory of Blackwell (1956).
Next, non?zero?sum continuous?kernel games were considered and equilibrium, Pareto optimal response, and security strategies were defined. Their existence under suitable conditions was also established. Here, the zero?sum case was treated as a special case of the non?zero?sum situation. Necessary and sufficient conditions were obtained separately to determine solutions through scalarization techniques. But it was found that a finite number of scalarization vectors is, in general, not sufficient to obtain all the POSS. Therefore, there is scope for further research to develop methods that do not have this drawback.
The above solution concepts developed for static games were then extended to a class of dynamic games, namely two?person zero?sum multicriteria differential games. Initially, solution concepts such as equilibrium, PORS, POSS, and Pareto saddle points were defined for a fixed?duration game, and methods to determine them through scalarization techniques were presented. The utility of the strategies arising from these solution concepts under various conditions of play was also discussed. Subsequently, a two?target game was modelled as a two?person zero?sum bicriteria differential game. It was shown that, of all the solution concepts presented above, POSS is the best as far as winning strategies are concerned. An examination of the solution concepts developed by others for combat games shows that these mainly reduce to the equilibrium concept and consequently suffer from certain drawbacks. It was also shown that the scalarization technique to obtain POSS works only if the set of security?level vectors satisfies a certain directional convexity property. The need to develop alternative methods that do not require this property was also discussed.
It is possible to extend the solution concepts presented here for fixed?duration multicriteria games to games in which the termination conditions are specified more generally in terms of a target set. These games are similar to pursuit–evasion games and will have both a qualitative and a quantitative aspect. If the state lies within the capture region of the pursuer, the players may minimaximise a vector payoff with the termination of the game assured in the pursuer’s favour. This could be an area of further research in multicriteria differential games. This can be extended to multicriteria games with two targets. It would be different from the bicriteria formulation presented here, in which the qualitative concepts of avoidance and termination are quantified to obtain the two criteria. However, a general multicriteria game could have its criteria independent of these aspects.
The security notion has been shown to be very important in multicriteria games. Modelling of two?target games as bicriteria game problems naturally leads to solution concepts based on the security aspect. To show that the security notion has special significance in two?target games in its own right (and not merely because the two?target games have been modelled as multicriteria games), the bicriteria formulation for these games was abandoned and a qualitative analysis was carried out from the standpoint of security. Secured outcome strategies and regions were defined and the effect of preference ordering of outcomes by the players on them was discussed. Conditions under which the secured mutual?kill and draw regions of players match were also found. This aspect has been overlooked by others in analysing two?target games. Finally, a construction procedure to obtain secured?outcome regions through the qualitative solution of two pursuit–evasion games was also presented for a fairly large class of problems. Further research could be pursued in this direction to obtain more general construction procedures, such as when the barriers leak.
Finally, the bicriteria formulation for two?target games was revisited and an alternative method to the scalarization technique was proposed to obtain the various secured?outcome strategies. Two strategy?optimization problems were formulated for each player. The existence or non?existence of solutions to these problems identifies the secured?outcome region to which an initial state belongs and also the secured?outcome strategies to be adopted by a player. Conditions under which this procedure breaks down were also discussed. Further research on this aspect is possible. The major task of developing computational procedures for the strategy?optimization problem formulated here is a challenging area of further work.
In summary, the main contributions of this thesis are two?fold. The first is the development of a reasonably complete theory for two?person multicriteria static games of the matrix and continuous?kernel type from the point of view of security. The second is to present the bicriteria game formulation as a natural conceptual framework for analysing two?target differential games. The thesis offers many interesting and useful solution concepts and insights into these games. It is hoped that these contributions will be applicable to the more complex but practical problem of aerial combat. | |
