I’m doing a Game Theory Project but unsure how to continue with the mathematics. My idea was bid pricing strategies in sealed first-price reverse auctions e.g. government contracts, raw material suppliers. So the government chooses the lowest bid (quality and other factors aside) and the bidders want to maximize their probability of winning while maintaining a high profit margin.
Suppose that company # 2 has information about company #1’s pricing distribution, as well as their cost. Call the bid prices p1 & p2. For example, say that company #1's bid pricing follows half a normal distribution with mean centered at their cost c1.
$\sqrt{2/pi}*(1/_1)e^{-(x-c_1)^2/2(_1)^2}$
and it has an area of 1 when calculated from 1 to infinity.
The standard deviation is unknown, so we’ll keep it at 1. So if company 1 is bids conservatively, they would bid with lower stdv. and if they were risky, vice versa. We know the optimal strategy is to bid to maximize the expected value which is probability (p2<p1)*(p2-c2)
That would make it an optimization of
$$(p_2-c_1)\int_{p_2}^\infty \sqrt{2/pi}*(1/_1)e^{-(x-c_1)^2/2(_1)^2}$$
Where I'm stuck is if instead of just one company having info on the other, they both knew of each other that they had a similar sort of distribution, and made like a simulataneous decision. Would the principle be the same? How would I go about calculating an equilibrium bidding price for both companies in that case?