Profit Calculation

Based on the prices that the pricebots choose, they will earn a certain amount of profit from buyers. Buyers will choose to buy from a pricebot according to the price at which the pricebot is selling the item. There are two kinds of buyers: ``any buyers'' and ``min buyers''. The any buyers will buy from any pricebot regardless of the price at which the pricebot is selling the item, while the min buyers will only buy from the pricebot(s) selling the item at the minimum price. This means that being the pricebot the min buyers buy from is often desirable, particularly if a pricebot is the only pricebot the min buyers are buying from, because on top of the profits that all the pricebots make from the any buyers, the pricebot selling at the minimum price will also get profits from the min buyers. The equation to determine the profits for some pricebot's price is split into two parts. First we determine the profits for that pricebot using that price when selling to an any buyer. Then we determine the profits for that pricebot using the same price when selling to a min buyer. Finally we sum together both profits to get the final profit for a pricebot selling at that price. The profit from selling to any buyers using price p is calculated with the equation,

$$\pi_{A} = (p - c) \cdot \frac{w_{A} \cdot b}{s}$$

where w_A is the percentage of the buyer population that is any buyers, c is the cost of the item, b is the number of buyers, and s is the total number of pricebots.

If p happens to be the minimum price at which an item is being sold in the round, then there will also be some profit from min buyers. This profit is given by the equation,

$$\pi_{B} = (p - c) \cdot \frac{w_{B} \cdot b}{s_B + 1}$$

where w_B is the percentage of the buyer population that is min buyers and s_B is the number of pricebots using price p.

The final profit, $\pi$, for p is given by,

$$\pi = \pi_A + \pi_B$$