Figure 2 shows a city of 122,500 households in three stages of neighbourhood evolution. The bright areas are stable neighbourhoods comprising households who have agreed to voluntarily curtail production/consumption activity to a mutually agreed social optimum. Areas with intermediate brightness are unstable neighbourhoods in which households fluctuate between private and social optimum levels of activities as they make and break agreements. Dark areas are inefficient neighbourhoods in the sense that all households maximise individual welfare but in so doing are individually and collectively worse off (each forgoes a welfare gain equal to the area E in Figure 1).
The neighbourhoods in Figure 2 have emerged purely as a result of households making bilateral agreements with neighbours. This means of course that the resultant morphology is dependent on the initial spatial distribution of households offering to make good neighbour contracts (the top image in Figure 2). Had the starting distribution been different, a very different city might have emerged. This point is illustrated in Table 1, which shows the frequency distribution of neighbourhood types emerging in 100 simulations. The simulated city in Table 1 has 2,500 households (50x50) and each simulation is randomly seeded with 375 (15%) households that make an opening bid to make a good neighbour contract.
Reading the bottom row of Table 1: from a starting point of 15 percent socially efficient households, the city has a 0.33 chance of equilibrating to a state in which 91-100 percent of households are in stable efficient neighbourhoods (bottom right cell). The greater probability, however is that only 0-10 percent of households will form such neighbourhoods (0.46). Reading the top row, which is also bi-modal, there is a 0.33 chance of the city stabilising with 0-10 percent of households in neighbourhoods where consumption/production is individually optimal but jointly sub-optimal (inefficient neighbourhoods). The most likely percentage of households in inefficient neighbourhoods at equilibrium, however, is 61-80 percent (0.43). Reading the middle row, there is a roughly equal chance of 0-10 percent and 21-30 percent of households forming unstable alliances as they pursue mutually beneficial contracts.
FIGURE 2: Emergence of neighourhoods in a city of 122,500 households
TABLE 1: Probability of Inefficient, Unstable and Efficient neighbourhood formation
FIGURE 3: Random allocation of neighbourly contact offers and two contrasting equilibrium states (bright neighbourhoods are efficient, dark are inefficient and chequered are unstable)