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V. Simulation Results

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
FIGURE 2
FIGURE 2

FIGURE 2: Emergence of neighourhoods in a city of 122,500 households

TABLE 1: Probability of Inefficient, Unstable and Efficient neighbourhood formation

Percentage of households in three different kinds of neighbourhoods
0-10
11-20
21-30
31-40
41-50
51-60
61-70
71-80
81-90
91-100
Inefficient
33
0
0
5
8
11
22
21
0
0
Unstable
35
17
34
11
3
0
0
0
0
0
Efficient
46
10
5
3
1
2
0
0
0
33


Figure 3

FIGURE 3: Random allocation of neighbourly contact offers and two contrasting equilibrium states (bright neighbourhoods are efficient, dark are inefficient and chequered are unstable)


Figure 3 shows two examples of neighbourhood emergence at equilibrium. The distributions depicted in (b) and (c) have evolved from a random distribution of contract-offers such as that displayed in (a). Cities (b) and (c) are both at equilibrium but have very different proportions of households living in efficient, inefficient and unstable neighbourhoods. Table 1 suggests that there is a 0.46 chance of a neighbourhood pattern like (b) emerging and a 0.2 chance of a more efficient pattern like (c) emerging. Intriguingly, the chance that neighbourhood evolution stabilises with 100 percent households making mutually beneficial neighbour contracts is higher (0.33) than the chance of stabilising at some lesser percentage, such as the 50 percent in (c). The most likely neighbourhood pattern emerging from bi-lateral agreements is either one of two extremes: very few or very many households living in efficient neighbourhoods, with the very few outcome being the more probable by a small margin.

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