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.

Implementation

100 digital farmers will be generated. Of importance to a farmer:

1)A fixed production capacity at 1 (noted as 100% on the cost function).

2)An alpha value, which is defined as the percentage of corn he wishes to produce.

3)(1-alpha) is the percentage of his yield that is rice.

4)Profit functions. Each farmer will try to maximize their profit according to the price of crops in the market and the profit function. The profit function for rice will take the form similar to P = A/X , where where p is profit, A is a constant, and X is the quantity of crops the farmer is producing, with no asymptote--the more units of a product produced, the less the production cost (usually) is. The price function of corn 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.

Figure 1: The Price and Cost Functions

The cost function is a simple direct relationship: as the amount of corn or rice produced increases, the cost decreases by slope -a or -c.

The maximation function

The economic actors in this experiment are programmed to maximize their returns and minimize the strategies of other actors. Given the information in figure 1, digital farmers will try to maximize x, the amount of crop to produce, given the following maximation function:

Figure 2: Maximation Function of Farmer

The Overall Flow of the Simulation

1) The program first initializes 100 farmers. Each farmer is given a different random alpha value to start with--therefore, each farmer will start with a different starting preference for how much corn to make.

2) The farmers are indexed, and using the fixed inverse price curves shown above, farmers one at a time will try a best response function (function maximize), until all farmers have had a chance to maximize or change strategies.

3) A frequency distribution graph will output the distribution of alpha levels at the end of each round of play.

4) There are 100 rounds of play. Therefore, as a default, we expect at some point for the distributions from round to round to stabilize. This would be a sign of Nash Equilibrium.

The Code of the Program

The programming was done in jEdit on the computers at the Tome Scientific building. The project consisted of a file which held the implementation functions and another file which defined the object FARMER. From a programming perspective, many functions which could have been publicly defined in the FARMER class were globalized and inserted into the implementation/main function file.

farmerGame.java

import java.io.*; public class farmerGame { private farmers[]; private int intervals[] = new int[10]; private float currentCornPrice; private float currentRicePrice; private float a; private float b; private float c; private float d; private float X; private float Y; private float totalAlpha; private float totalProfit;

public farmerGame(boolean i); { totalAlpha = 0; int i; farmers = new farmer[100]; if (i) { for (i=0; i<100; i++) { farmers[i].setAlpha((float)i/100); } } else { for (i=0; i<100; i++) { farmers[i].setAlpha(Math.random()); } } for (i=0; i<100; i++) { intervals[(int)(farmers[i].getAlpha() * 100)]++; } }

public void drawGraph() { int i; System.out.print("0-0.1 |"); for (i=0; i<intervals[0]; i++) { System.out.print("*"); } System.out.println(" (" + intervals[0] + ")"); System.out.print("0.1-0.2|"); for (i=0; i<intervals[1]; i++) { System.out.print("*"); } System.out.println(" (" + intervals[1] + ")"); System.out.print("0.2-0.3|"); for (i=0; i<intervals[2]; i++) { System.out.print("*"); } System.out.println(" (" + intervals[2] + ")"); System.out.print("0.3-0.4|"); for (i=0; i<intervals[3]; i++) { System.out.print("*"); } System.out.println(" (" + intervals[3] + ")"); System.out.print("0.4-0.5|"); for (i=0; i<intervals[4]; i++) { System.out.print("*"); } System.out.println(" (" + intervals[4] + ")"); System.out.print("0.5-0.6|"); for (i=0; i<intervals[5]; i++) { System.out.print("*"); } System.out.println(" (" + intervals[5] + ")"); System.out.print("0.6-0.7|"); for (i=0; i<intervals[6]; i++) { System.out.print("*"); } System.out.println(" (" + intervals[6] + ")"); System.out.print("0.7-0.8|"); for (i=0; i<intervals[7]; i++) { System.out.print("*"); } System.out.println(" (" + intervals[7] + ")"); System.out.print("0.8-0.9|"); for (i=0; i<intervals[8]; i++) { System.out.print("*"); } System.out.println(" (" + intervals[8] + ")"); System.out.print("0.9-1 |"); for (i=0; i<intervals[9]; i++) { System.out.print("*"); } System.out.println(" (" + intervals[9] + ")"); }

public void printStats() { totalAlpha=0; totalProfit=0; int i; for (i=0; i<100; i++) { totalalpha+=farmers[i].getAlpha(); totalprofit+=farmers[i].getProfit(); } System.out.println("Average alpha: " + (totalAlpha/100)); System.out.println("Average profit: " + (totalProfit/100)); System.out.println("Current price for Corn: " + currentCornPrice); System.out.println("Current price for Rice: " + currentRicePrice); }

public void changePrices() { currentCornPrice = (float)(X)/totalAlpha; currentRicePrice = (float)(Y)/totalAlpha; }

public void maximize(int i) { float temp = totalAlpha - farmers[i].getAlpha(); //check if 0 or 1 is better strategy. float t = (currentRicePrice - currentCornPrice + b - d + 2 * c) / (2 * a + 2 * c); if ((currentCornPrice - (-a * t + b)) * t + (currentRicePrice - (-c * (1-t) + d)) * (1-t) < (currentCornPrice - (-a + b))) { t = 1 } if ( farmer fake = new farmer()

farmers[i].setProfit(); totalAlpha = temp + farmers[i].getAlpha(); }

public static void main(String[] argv) throws IOException { farmerGame thisGame = new farmerGame(argv[0].equals("even")); thisGame.a = Float.parseFloat(argv[1]); thisGame.b = Float.parseFloat(argv[2]); thisGame.c = Float.parseFloat(argv[3]); thisGame.d = Float.parseFloat(argv[4]); thisGame.X = Float.parseFloat(argv[5]); thisGame.Y = Float.parseFloat(argv[6]); thisGame.changePrices(); } }

Farmer OBJECT