Let \(\char{cmmi10}{0x54}\) : \(\char{cmmi10}{0x58}\) → \(\char{cmmi10}{0x58}\) be a function from a countably infinite set \(\char{cmmi10}{0x58}\) to itself. We consider the following problem: can we put a structure on \(\char{cmmi10}{0x58}\) with respect to which \(\char{cmmi10}{0x54}\) has some meaning? In this thesis, the following questions are addressed: when can we endow \(\char{cmmi10}{0x58}\) with a topology such that \(\char{cmmi10}{0x58}\) is homeomorphic to the rationals \(\char{msbm10}{0x51}\) and with respect to which \(\char{cmmi10}{0x54}\) is continuous? We characterize such functions on the rational world. The other question is: can we put an order on \(\char{cmmi10}{0x58}\) with respect to which \(\char{cmmi10}{0x58}\) is order-isomorphic to the rationals \(\char{msbm10}{0x51}\), naturals \(\char{msbm10}{0x4e}\) or integers \(\char{msbm10}{0x5a}\) with their usual orders and with respect to which \(\char{cmmi10}{0x54}\) is order-preserving (or order-reversing)? We give characterization of such bijections, injections and surjections on the rational world and of arbitrary maps on the naturals and integers in terms of the orbit structure of the map concerned.

This thesis, which is divided into five papers, explores biased decision making in naturalistic environments and its implications for the efficiency of financial markets and forecasts of competitive event outcomes. Betting markets offer a valuable real world decision making context, allowing analysis that is not possible using regular financial market data. The first paper surveys studies that have employed betting markets to investigate biased decision making and discusses why the extent of these biases is significantly less than in the laboratory. The second paper addresses unresolved issues relating to noise trading and herding in financial markets, by showing that noise trading is associated with increased market efficiency, that the extent of herding differs depending on the direction and timing of changes in market prices, and that this results in an economically significant inefficiency. The findings of this paper have important policy implications for wider financial markets: regulatory measures to protect investors from the destabilizing effects of noise appear to be self-defeating and herding is particularly prevalent when uninformed traders perceive that informed traders are participating in the market. The third and fourth papers address the favourite-longshot bias (FLB), where market prices under-/over-estimate high/low probability outcomes. These papers demonstrate that previous explanations of the bias are inconsistent with evidence of trading in UK betting markets by developing and testing the predictions of models that explain the bias in terms of competition between market makers and the demand preferences of bettors. Moreover, it is definitively shown that, when no market maker is involved, the bias is due to cognitive errors of traders rather than their preference for risk, because only prospect theory, and not risk-love, can explain a reduced FLB in events with strong favourites. The final chapter explores methodological concerns relating to estimates of forecast accuracy in models of discrete choice, and arrives at a much more rigorous understanding of the value of these estimates.