First, Second, Third, and All Pay Auctions- Strategies, and Revenues

This study presents the optimum bidding strategy for first, second, third and all pay auctions in this article. Assuming that all bidders are rational, meaning that they would all want to make the most profit and behave the same, and there exists an equilibrium strategy that all bidders use, compute a mathematical model for different types of auctions. First, believe that there exists an equilibrium strategy that the strategy is linear, if not then we would try non-linear equilibrium strategies, it could be any polynomial or even exponential. Solving it, there will be a bid function where bidders get the maximum profit while the probability of winning is high. Next step, calculate the expected revenue for the auctioneer using our results from the equilibrium strategy. They are the same due to the revenue equivalence theorem. It is discovered that under certain conditions, different auction formats yield the same expected revenue for the seller.