Game-theoretic (GT)

The game-theoretic (GT) pricebots choose prices according to the following formula [1].

$$p = c + \frac{w_A(v-c)}{w_A + w_B(1-rand)^{s-1} \cdot s}$$

where p is the new price and rand is a random number between 0 and 1. This formula determines the prices that would occur at a Nash equilibrium, that is, it determines the prices for which all pricebots would be maximizing profits, and from which no pricebot wishes to deviate [Nash (1951) as referenced by [1]].

With c = 0.5, w_A = 0.5, w_B = 0.5, v = 1, s = 100, and rand equal to a random number between 0 and 1, most of the time p will be 1; occasionally p will be a lower price such as 0.56 or 0.7. Taking the average then, of all of the GT prices for one generation gives an average price of about 0.97. Note that it doesn't make that much of a difference whether GT pricebots update simultaneously or sequentially because they use an equation to calculate the next price and are not going to choose a price based on the price of what another pricebot has chosen, unlike with the myopically-optimal pricebots.

With GT, the more pricebots, the higher the average GT price. Presumably this is because it makes more sense to choose a higher price since there is little chance of being the only pricebot selling at the minimum price. If a pricebot can't be the only, or one of the few pricebot(s) selling at the minimum price, there is no reason to choose a low price since the pricebot will not receive any profit from the min buyers.