Computer Generated Farmers: Information and Nash Equalibrium

What are we trying to do here?

Picture a hundred farmers in a simple economy, where two crops are farmed: Rice and Corn. In a similar manner as the fisherman’s dilemma, in which a day’s output for one fisherman far affects the yield of the other fisherman, the amount of corn produced affects how much rice is produced. Each crop has a separate price and cost curve. Each farmer has to make individual decision, given the current point on the curves, of what percentage of total yield to devote to corn and rice. Given 100 farmers, will a stable Nash Equilibrium be reached?

Why make a computer simulation of a simple game?

There are many games with very simple rules or agents result in complex aggregate results. These results are not necessarily very predicable–while there are only a number of desirable outcomes with 2x2 games, games which involve hundreds of agents playing many games in succession are unpredictable. Our simulation is simple to model a broad market; actors will not be given an extensive intelligence. By default, we expect a result consistent with a Nash Equilibrium. However, despite low intelligence and near perfect information (classical game theory), each farmer will indirectly be influencing the decisions of the other farmers, which could bring some interesting results.

Objective

We used computer gerated farmers to simulate a special economic system, in which each farmer had a limited number of friends who they could get information from to try to better their strategic outcome.

Setup

We will generate 100 digital farmers, each having a production capacity of 1, choice of 2 crops to make (corn and rice), and a profit function. We will call the amount of corn a farmer makes alpha. Since each farmer has a production capacity of 1, naturally the amount of rice this farmer makes will be 1-alpha. Each farmer will try to maximize their profit according to the price of crops in the market and the profit function. The profit function will take the form similar to p=a/x where p is profit, a is some constant, and x is the quantity of crops the farmer is producing. with no asymptote. This follows the rule that the more units of a product produced, the less the production cost (usually) is. The price function will also be determined by a P=b/X function, this time P standing for price of the crop, b is some constant, and X is the total number of the crop being produced. Therefore, the two functions will look like the following:

File:Function.png You have to be logged into the wiki to upload a image.Gribble 15:34, 12 Apr 2006 (EDT)

When the farmer makes a decision, he will treat the price as a constant, and maximize profit according to the profit function. This can be solved using some simple calculus.

Maximize <math>a1/x1+a2/x2</math> Subject to