We now turn our attention to formulating a simple CRP which, although it is not a model of a practical, real-world problem, it is representative of the core attributes of a CRP.
2.6. 1 Purpose
The TSP is the archetypal VRSPj usefully understanding any VRSP requires a good understand ing of the sequencing problem inherent to the TSP. Moreover, the TSP invariably appears as a subproblem as part of a solution method for a VRSP.
The RM-CRP considers the components that describe a CRP. The class of CRPs is at least as large as the class of VRSPs and we currently have no analysis of, nor computational experience with, a CRP. We wish to determine a core representative CRP that forms the nucleus of the CRP in a similar role to that which the TSP serves for the VRSP. In this way we wish to be able to extrapolate useful conclusions about the class of CRPs from analysis of, and computational experience with, the core problem. Hence we apply the Principle of Occam's Razor.4
2.6.2 Fundamental Components
Having determined to design a competition variation of the TSP, or rather the TSSP since one player need not necessarily visit all locations, what are the absolutely essential problem attributes and relationships?
4The principle attributed to the English philosopher William of Occam that the fewest possible as sumptions are to be made in explaining something.
2.6. Core Problem: Competitive Prize Collection Problem 39
Addresses. Since we have a competition, we must have some customers or prizes over which to compete, each with some commensurate reward or penalty value. The TSSP is a combinatorial optimization problem and hence there is little difference between a graph or network structure and a Euclidean problem since both are converted to a inter-customer distance matrix. However, in a dynamic problem there is a considerable difference and the Euclidean problem is much less assumptive although the graph embedding is more constrained. Either possibility could be consid ered. We choose the Euclidean problem due to its similarity with pursuit-evasion games and the diminishing of the routing aspect. The graph embedding is considered further in Section 1 1.2.2.2.
Vehicles. One uncapacitated vehicle per decision maker.
Decision Makers. It turns out that considering two players is sufficiently complicated with out having to analyse the dynamic collusion possibilities of three or more players. Indeed, any attempt at analysing three or more players would require some understanding of the correspond ing two player analysis. Also, we choose to start with a noncooperative game, i.e., one in which any type of collusion, such as correlated strategies and side payments, is forbidden. A public objective implies that the players either must cooperate or can make assumptions regarding the opponent's behaviour. However, any analysis is then dependent upon the exact nature of the pub lic objective. Hence we discard public objectives and each player need not make any assumption about the private objective of the opponent.
Objective. The simplest objective is to claim as many prizes as possible. This has the im
portant property that it is memoryless-whatever the current state of play, the objective is the same.
Relationships. Nil, i.e., no restrictions, all prizes are contestable.
2 . 6 . 3 Definition
We are now in a position to precisely define a simple CRP for further computational study. Definition 2.6.1 (Competitive Prize Collection Problem (CPCP»
Let V
=
{I, . . . , n} be a set of "contestable" prizes. Associated with each prize i E V is a location (Xi, Yi) in the Euclidean plane and a value Vi>
O. For the Competitive Prize Collection Problem (CPCP)
, two independent players (A and B), with given initial locations(XA , YA) and (XB, YB) respectively, move continuously at the same constant speed in the Euclidean plane. Each player's private objective is to individually collect as much in total prize value as possible up until the overall deadline, >., expires. The value of each prize is only awarded to the first player who visits; a prize location which is visited simultaneously by both players has its associated value shared equally. At all times, each player has perfect observation of the state of the game position, i.e., where the players are currently located and which prizes remain unclaimed.
By considering the simplicity of each component of a CRP, we can see that the CPCP is indeed a core CRP. Evidently, the classification of the CPCP is 2/ /2/�2, own/V.
Axelrod [7] makes a number of important assumptions regarding the players in the Prisoner's
Dilemma (see Section 2.4.5) which we also apply to the CPCP.
• "There is no mechanism available to the players to make enforceable threats or commit
ments. Since the players cannot commit themselves to a particular strategy, each must take into account all possible strategies that might be used by the other player. Moreover the players have all possible strategies available to themselves."
• "There is no way to be sure what the other player will do on a given move. This eliminates
the possibility of reliable reputations such as might be based on watching the other player interact with third parties. Thus the only information available to the players about each other is the history of their interaction so far."
• "There is no way to eliminate the other player or run away from the interaction."
• "There is no way for the players to change the payoffs."
• "The players can communicate with each other only through the sequence of their own
behaviour."
• "There is no need to assume that the players are rational. They need not try to maximize
their rewards. Their strategies may simply reflect standard operating procedures, rules of thumb, instincts, habits, or imitation."
When A = 00 the CPCP is a constant sum game (see Section 2.4) since there is always time
to claim one more prize. However, when A < 00 the CPCP is a noncooperative general-sum game
since it is possible that the overall deadline expires before all prizes are claimed. In the latter case, the players may exhibit qualities of cooperation since both players could be better off by not entering into time consuming conflict. Brandenburger and Nalebuff [27] describe the paradigm of coopetition ( "COOPeration" plus "compETITION" ) in which cooperation between players can sometimes work out to both players' advantage.
2.6.3.1 Competing Salesmen Problem
Fekete and Schmitt [66] study the Competing Salesmen Problem (CSP), also a two-player competitive version of the TSP. The players take turns, moving one edge at a time within a graph, with customers located at the vertices of the graph. Both players always know the position of their opponent and the winner is the player who reaches a majority of the customers. A geometric variant of the CSP is also proposed, in which the customers are located in the Euclidean plane and players move with the same maximal speed. This geometric CSP appears to be equivalent to the CPCP with equal prize values but with the objective of gaining more prizes than the opponent and alternative moves.
Fekete and Schmitt focus on establishing whether a player can avoid a loss or force a draw in the graph-based CSP. They draw on the theory of similar combinatorical games, mainly consid ering the special case where the graph is a tree. This thesis, however, focuses on the design and