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This thesis mainly focuses on the statistical modelling of a selection of games, namely, the minority game, the urn model and the Hawk-Dove game. Chapters 1 and 2 give a brief introduction and survey of the field. In Chapter 3, the key characteristics of the minority game are reproduced. In addition, the minority game is extended to include wealth distribution and leverage effect. By assuming that each player has initial wealth which rises and falls according to profit and loss, with the potential of borrowing and bankruptcy, we find that modelled wealth distribution may be power law distributed and leverage increases the instability of the system. In Chapter 4, to explore the effects of memory, we construct a model where agents with memories of different lengths compete for finite resources. Using analytical and numerical approaches, our research demonstrates that an instability exists at a critical memory length; and players with different memory lengths are able to compete with each other and achieve a state of co-existence. The analytical solution is found to be connected to the well-known urn model. Additionally, our findings reveal that the temperature is related to the agent's memory. Due to its general nature, this memory model could potentially be relevant for a variety of other game models. In Chapter 5, our main finding is extended to the Hawk-Dove game, by introducing the memory parameter to each agent playing the game. An assumption is made that agents try to maximise their profits by learning from past experiences, stored in their finite memories. We show that the analytical results obtained from these two games are in agreement with the results from our simulations. It is concluded that the instability occurs when agents' memory lengths reach the critical value. Finally, Chapter 6 provides some concluding remarks and outlines some potential future work.

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Integral equation theories are employed to study the solvation of large particles. The solutions studied consist of spherically symmetric solvent particles which interact through hard sphere and Lennard-Jones potentials, and hard sphere solutes. In particular, the Ornstein-Zernike (O-Z) equation is solved with the hypernetted-chain (HNC) closure, to obtain the pair correlation functions of mixtures at infinite dilution. The pair correlation functions in the O-Z equation and the H N C closure are expressed as power series in solute density to yield a pair of coupled equations which determine the derivatives with respect to solute density of the solvent-solvent pair correlation functions. The latter describe the perturbation of the solvent upon addition of a single solute particle. The derivatives are analysed to yield components that scale as the volume and surface area of the macroparticle, and which are then identified as changes in solvent structure due to the presence of a finite size particle and a flat surface respectively. From the pair correlation functions and their derivatives the excess internal energy, Helmholtz free energy, and entropy of solvation are calculated. The excess quantities are also separated into contributions from finite size and surface effects. Both components of the excess internal energy are negative at low densities, and become less negative for high density liquids. The magnitude and sign of the two contributions to the energy depend on physical conditions such as temperature and pressure. The excess entropy of solvation is negative for all systems studied, indicating that introduction of a macroparticle or a flat surface increases the spatial ordering of a bulk liquid. / Science, Faculty of / Chemistry, Department of / Graduate

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