The Ghost in the Model (and Other Effects of Floating Point Arithmetic)

Bistromathics … is … a revolutionary new way of understanding the behaviour of numbers. … Numbers written on restaurant bill pads within the confines of restaurants do not follow the same mathematical laws as numbers written on any other pieces of paper in any other part of the Universe. 1 Introduction This paper will explore the effects of errors in floating point arithmetic in two published agent-based models: the first a model of land use change (Polhill et al., 2001; Gotts et al., in press), the second a model of the stock market (LeBaron et al., 1999). The first example demonstrates how branching statements with floating point operands of comparison operators create a high degree of nonlinearity, leading in this case to the creation of 'ghost' agents — visible to some parts of the program but not to others. A potential solution to this problem is proposed. The second example shows how mathematical descriptions of models in the literature are insufficient to enable exact replication of work since mathematically equivalent implementations in terms of real number arithmetic are not equivalent in terms of floating point arithmetic. (Henceforth 'mathematics' and 'arithmetic' will refer to the use of real numbers by default.) 2 A model of land use change The following describes the set up of a model of land use change that has emergent effects arising from errors in floating point arithmetic. The model is based on a particular parameterisation, P, of FEARLUS (Framework for Evaluation and Assessment of Regional Land Use Scenarios), described in Polhill et al. (2001) and Gotts et al. (in press). In what follows, terms referring to objects in the model are capitalised, and italicised on first use. FEARLUS consists of a set of Land Parcels arranged on a 2D grid of squares, each of which is assigned a Land Use by Land Managers (the agents in the model) using a Land Use Selection Algorithm each Year (a cycle in the model). After Land Uses have been assigned, the Yield from each Land Parcel is calculated according to the Land Use selected, and this is used to derive an amount of Wealth accrued by the Land Manager owning the Land Parcel. This is followed by a process of exchange of Land Parcels between Land Managers. Those Land Managers with negative Wealth put up enough Land Parcels for sale to bring their Wealth to zero or more. Once …