Evolutionary Dynamics

Evolution of the pricebot population is based on the total profits each pricebot type makes per generation. First the average profits per pricebot for each pricebot type per generation are calculated. The utility each pricebot type receives, util_i is then calculated as the difference between the average profits per pricebot for each pricebot type, and the average profits per pricebot for all pricebot types. These utilities are normalized. The percentage of pricebots for somepricebot type i at time t is given by f_i^t. To calculate the percentage of pricebots of type i at time t+1 we use the following,

$$f_i^{t+1} = f_i^{t} \cdot (1 + 0.05 \cdot util_i)$$

The 0.05 is being used to take some proportion of the utility rather than the entire utility.

In order to update f_i^t with f_i^t+1 we use,

$$f_i^{t} = \frac{f_i^{t+1}}{f^{t+1}}$$

where f^t+1 is the normalized sum of the f_i^t+1 for all i where i is a pricebot type.

The new number of expected pricebots for each pricebot type i at time t+1 is N_i^t₊1 and is given by,

$$N_i^{t+1} = w_{A} \cdot \frac{f_i^{t+1}}{f^{t+1}}$$

If there are extra pricebots due to the percentages not adding up completely, the extra pricebots are assigned in order to the game-theoretic, myopically-optimal, derivative-following, and q-learning pricebots until there are no extra pricebots left. Ideally we should choose randomly among the pricebot types to distribute extra pricebots, but that has not yet been implemented.