Land Use with Contract

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This Demonstration explores an abstract evolving world in which plots of land each with two potential uses lie on a one-dimensional periodic lattice. The payoff each landowner receives on each iteration is a function of its choice of how to use its land coupled with the choices of its immediate neighbors. However, each landowner is permitted to contract without cost with a certain number of its neighbors to bind themselves to a pattern of land use. A group of five contiguous landowners can bind themselves, for example, to choose on each iteration the pattern of land use that maximizes their total payoff, given the land use choices of the neighboring noncontracting land users. They can agree to split their total payoff such that each would receive at least as much as if they did not cooperate.


You choose a random initial land use configuration. You likewise choose the maximum number of adjacent landowners that will participate in a contract. Some contracts will involve a smaller number of parties when the maximum number of parties fails to evenly divide the lattice size of 60. You also choose where in the lattice the "seams" in the contract will occur, that is, places where neighboring landowners will be parties to different contracts. You choose "scores" that each plot of land receives as a result of its configuration and those of the two adjacent landowners. These scores are constrained to be reflection symmetric. You also determine whether the payoffs each landowner receives are simply equal to those scores ("absolute" scoring weights) or whether they are normalized so that the payoff is equal to the proportion a particular score bears to the total scores ("relative" scoring weights).

The system responds with three pieces of information. The first is a tab view showing, for each configuration of neighbors, the behavior of the contracting parties that results in the highest payoffs. A different tab exists for contracts of different sizes ranging from the largest possible size down to a single landowner (without a contract). The second piece of information is an array plot showing the evolution of this world. Each row represents land use at a point in time, with later rows representing later points in time. Depending on your choice, each cell of the array shows the land use adopted by that cell (1 is red, 0 is violet) or the score each cell receives as a result of its land use coupled with the land use of its neighbors. Higher scores are evidenced by colors at the red end of the "rainbow" spectrum; lower scores are evidenced by colors at the violet end. You can also choose whether to display the "seams" in the contracts, which are represented as white vertical lines. The third piece of information is a logarithmic plot of the total scores. You determine whether the vertical plot range extends dynamically from the minimum to the maximum total possible given the scoring weights or whether to use a larger fixed plot range.


Contributed by: Seth J. Chandler (March 2011)
Open content licensed under CC BY-NC-SA



An interesting issue to pursue is the relationship between contract size and average payoffs. If, for example, average payoff rapidly approaches an asymptotic value as contract size increases, private land use arrangements such as covenants running with the land or private covenants may be sufficient to achieve sensible use of land without a need for government intervention such as zoning. If, however, average payoff only slowly approaches the global maximum that could theoretically be achieved were all landowners parties to a contract, government intervention would have greater warrant due to the significant costs of contracting with a large number of people.

Where the lattice size is evenly divisible by the number of parties to each contract, the system described in this Demonstration can likely be emulated by a cellular automaton with a smaller lattice and a high number of colors. There are similarities between the system described in this Demonstration and a block cellular automaton.

The Mathematica code underlying this Demonstration is written in a way such that it can readily be extended to situations in which the payoffs each landowner receives are a function of its behavior and a broader set of neighbors.

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