Cellular automata (CA) is a class of simulation model in which discrete dynamic systems are simulated using cells and the cell transition behaviour is completely specified in terms of local relations (Toffoli and Margolus 1987). In CA, time advances in discrete steps. At each step a cell derives its next state from that of its close neighbours according to a set of local and uniform transition rules. The idea of cellular automata is closely associated with that of microscopic simulation in which the behaviour at a local scale gives rise to an emerging global organisation. Processes of positive feedback gradually confine the state transition of a system and the system is said to self-organise -- into stability, oscillation or chaos. In the application to urban economic systems, CA represents a modelling approach quite different from top-down and macroscopic approaches. The simulation reflects a new way of looking at the manner in which global organisational form emerges from an uncoordinated local decision-making process (Batty, 1995). The CA tradition had its origin in computational ecological experiments, the most famous of which is the Game of Life (Gardner 1970). Driven by a simple set of rules for turning a cell "alive" and "dead", the Game of Life was able to display complex behaviour. Since then, Artificial Life has grown into "a field of study devoted to understanding life by attempting to abstract the fundamental dynamics in other physical media -- such as computers -- making them accessible to experimental manipulation and testing (Langton et al. 1992, xiv). Urban systems are only one of many diverse domains in which the methods have been applied. CA's emphasis on atomistic behaviour; local information; spatial configuration; global-local interactions; and path dependency; makes it a powerful medium for exploring land market processes. (Examples of CA in urban economic, urban geography and planning include: Page 1999; Portugali and Benenson 1995; Webster and Wu 1999a, 1999b, 2001; Wu and Webster 1998, 2000; and White and Engelen 1993.)
The simulation presented below is a very simple conventional CA model in which there are 2 states (values 1 and 3 represent the same state an are necessary for the algorithm):
1(1,1,.,)->3; 2(1,1,.,)->3; 1(.,.,.,)->2; 2(.,.,.,)->2; 3(.,.,.,)->1
Reading across: if a cell (household) that is voluntarily restraining at the social optimum has at least two neighbours doing likewise, then maintain social optimum. If a cell producing at private optimum has at least two neighbours that are voluntarily restraining, then convert to social optimum. If a cell producing at social optimum does not possess at least two neighbours doing likewise, then convert it back to private optimum. Cells producing at private optimum and not having at least two voluntarily restraining neighbours continue to produce at private optimum. The fifth rule is purely algorithmic. The simulation starts with a random seeding of households switched to state 1 -- modelling some random start point in a neighbourhood's evolution at which a minority of households offer to make mutually beneficial agreements with neighbours. The CA proceeds with a 3x3 cellular neighbourhood.
It should be clear that the simulation is a discrete representation of the Coasian model in Figure 1 in which q1 and q2 are represented by discrete states (0,1). Elsewhere I have published continuous versions, which solve for optimal quantities of output and externalities by embedding simultaneous equations in the cell transition rules (Webster and Wu 2001). As well as making the algorithm more complex, such an approach also raises the number of parameters in the model, increasing problems of interpretation. The discrete CA model's relationship to the property rights propositions 1-3 discussed in Section II above should be briefly explained. Proposition 1 says that changes in the value of shared, or public domain goods, lead to demands for property rights reassignment. A neighbourhood of three households (Figure 1) each consuming/producing at q2 has less value than a neighbourhood of three similar households consuming/producing at q1. The difference in value is 3E. Proposition 1 says that the households in the lower value neighbourhood will seek to reassign property rights. This, any of them might do, by offering to restrain individual consumption/production. Such an offer is an attempt to create an informal contract or agreement (or institution, loosely defined) that effectively secures an individual's property right over his/her nuisance risk by imposing an obligation (backed up by threat) on neighbours. The success in bi-lateral institution-building depends, in the model, on the presence of neighbours willing to follow suit.