Static search games played over graphs and general metric spaces

We define a general game which forms a basis for modelling situations of static search and concealment over regions with spatial structure. The game involves two players, the searching player and the concealing player, and is played over a metric space. Each player simultaneously chooses to deploy at a point in the space; the searching player receiving a payoff of 1 if his opponent lies within a predetermined radius r of his position, the concealing player receiving a payoff of 1 otherwise. The concepts of dominance and equivalence of strategies are examined in the context of this game, before focusing on the more specific case of the game played over a graph. Methods are presented to simplify the analysis of such games, both by means of the iterated elimination of dominated strategies and through consideration of automorphisms of the graph. Lower and upper bounds on the value of the game are presented and optimal mixed strategies are calculated for games played over a particular family of graphs

Search-and-rescue rendezvous

We consider a new type of asymmetric rendezvous search problem in which player II needs to give player I a ‘gift’ which can be in the form of information or material. The gift can either be transfered upon meeting, as in traditional rendezvous, or it can be dropped off by player II at a location he passes, in the hope it will be found by player I. The gift might be a water bottle for a traveller lost in the desert; a supply cache for Captain Scott in the Antarctic; or important information (left as a gift). The common aim of the two players is to minimize the time taken for I to either meet II or find the gift. We find optimal agent paths and drop off times when the search region is a line, the initial distance between the players is known and one or both of the players can leave gifts. A novel and important technique introduced in this paper is the use of families of linear programs to solve this and previous rendezvous problems. Previously, the approach was to guess the answer and then prove it was optimal. Our work has applications to other forms of rendezvous on the line: we can solve the symmetric version (players must use the same strategy) with two gifts and we show that there are no asymmetric solutions to this two gifts problem. We also solve the GiftStart problem, where the gift or gifts must be dropped at the start of the game. Furthermore, we can solve the Minmax version of the game where the objective function is to minimize the maximum rendezvous time. This problem admits variations where players have 0, 1 or 2 gifts at disposal. In particular, we show that the classical Wait For Mommy strategy is optimal for this setting

Perspectives on the relationship between local interactions and global outcomes in spatially explicit models of systems of interacting individuals

Understanding the behaviour of systems of interacting individuals is a key aim of much research in the social sciences and beyond, and a wide variety of modelling paradigms have been employed in pursuit of this goal. Often, systems of interest are intrinsically spatial, involving interactions that occur on a local scale or according to some specific spatial structure. However, while it is recognised that spatial factors can have a significant impact on the global behaviours exhibited by such systems, in practice, models often neglect spatial structure or consider it only in a limited way, in order to simplify interpretation and analysis. In the particular case of individual-based models used in the social sciences, a lack of consistent mathematical foundations inevitably casts doubt on the validity of research conclusions. Similarly, in game theory, the lack of a unifying framework to encompass the full variety of spatial games presented in the literature restricts the development of general results and can prevent researchers from identifying important similarities between models. In this thesis, we address these issues by examining the relationship between local interactions and global outcomes in spatially explicit models of interacting individuals from two different conceptual perspectives. First, we define and analyse a family of spatially explicit, individual-based models, identifying and explaining fundamental connections between their local and global behaviours. Our approach represents a proof of concept, suggesting that similar methods could be effective in identifying such connections in a wider range of models. Secondly, we define a general model for spatial games of search and concealment, which unites many existing games into a single framework, and we present theoretical results on its optimal strategies. Our model represents an opportunity for the development of a more broadly applicable theory of spatial games, which could facilitate progress and highlight connections within the field

Two point one sided rendezvous

In a rendezvous search two or more teams called seekers try to minimize the time needed to find each other. In this paper, we consider two seekers in the plane. This is a one sided problem since Seeker 1 begins at a predetermined point O. Seeker 2 begins at one of a finite set of points xi with probability pi. We first discuss the general situation and then consider the specific case when Seeker 2 can begin from one of two points.Rendezvous Search Discrete search