Average Prices

Because the GT chooses a random low price occasionally, it is this lower price that the MY will generally be undercutting. From Figure 4b we see that initially the average generation prices for MY pricebots start out fairly high, at about 0.95, but then decrease to about 0.75 after around 85 generations. The MY prices eventually stabilize around 0.75$\pm$.1. This decrease in price from about 0.95 to about 0.75 is due to the sequential update mentioned previously. Due to the sequential update, when one MY pricebot updates its price, all pricebots calculate profits. All pricebots except the MY pricebot that updated its price are using previous prices. Some of these prices may have been calculated as many as 98 times ago (98 as there are 100 pricebots, so a pricebot could do 99 calculations before getting a chance to update its price). So some MY pricebot that had the min price during its update, will still be using this price, but it will no longer be the min price when another MY pricebot updates its price. To have a low price that is not the min price gives very low profit. These low profits will accumulate during the rounds and will cause the number of MY pricebots to decrease. The number of MY pricebots decreasing is responsible for causing the drop in average price from about 0.95 to about 0.75. Why is this? First we have to think about what the MY prices are like if there is no change in the number of MY pricebots. The more MY pricebots there are, the more ones there will be as prices. We can think about why this is if we consider the following: if there are 100 MY pricebots, and the first MY pricebot to update its price chooses the absolute minimum price possible as its price, then the next 99 MY pricebots will each in turn choose 1 as their new price. They will not choose the minimum price because they would then have to share profits with the very first MY pricebot that updated. It doesn't make sense to choose a price below the maximum price if no minimum price can be had, and since the maximum price is 1, then these 99 MY pricebots will choose 1 as their new price. Between the prices of 0.5 and 1 there are 100 possible prices to choose if we go by 0.005 increments. So there are always 100 possible prices to loop through that are less than one, once the MY pricebots start their undercutting behavior. This number of prices to loop through will not be affected by the number of MY pricebots. However, the number of ones chosen as prices will be affected by the number of MY pricebots. If there were 50 MY pricebots instead of 100, then we would see at most 49 ones instead of 99 before the MY pricebots would start undercutting each other again. Because of this, decreasing the number of MY pricebots decreases the average price because the number of prices of 1 being added into the average MY price is decreased.

The few MY pricebots there are will have either 1 (the maximum price to receive profit at) or close to 0.5 (due to undercutting the lowest GT price which is near the lowest reasonable price possible of 0.505; alternatively the MY could be undercutting another MY which had undercut the low GT price) as their prices. Having around half the prices being 1 and the other half being close to 0.5 gives us an average around 0.75. There are exceptions to this, such as the lowest GT price being higher than 0.51 for instance, and these are the instances that explain why the MY average price is not perfectly 0.75. As well, at the beginning of each generation, the MY pricebots choose their prices randomly. These random prices should also average out to something close to 0.75. So the average of 0.75 and 0.75 is 0.75. This explains why the MY chooses prices around 0.75. Why is it that the MY and GT pricebots eventually stabilize with about 97 GT and 3 my? What happens is that with 3 my, there will be prices of 1, 0.510, and 0.505 often: with these prices, the 1 will give a profit of 25 always (from $((1 - .5) \cdot 5000)/100)$) and the 0.510 will give a profit of 50 (from $(.510 - .5) \cdot 5000)$) when it is the min price, and then 0.505 will give a profit of 25 (from $(.505-.5) \cdot 5000)$) when it is the min price. So we'll have the .510 making enough profit from the 50 it makes to share the next time around when it is not the min price, so it will end up with a profit of 25 or more and will be above the GT profits. It is probable that more MY pricebots cannot be sustained above 3, so that even when the MY can increase their number of pricebots by 1, they end up dropping back down to 2 or 3 or maybe 4 because they cannot maintain the profits needed for a higher number of MY pricebots when they actually have that higher number of MY pricebots.

Figure 4a. Showing the individual pricebot prices for 5 MY pricebots. Figure 4b. Average price for 50 GT versus 50 MY initially