I've been trying to work the following problem out but I can't quite seem to understand it, or the whole concept of first-place auctions. I don't understand how we get to the equilibrium.
The problem is the following: "Consider a first-price, sealed-bid auction in which a bidder’s valuation can take one of three values: 5, 7, and 10, occurring with probabilities 0.2, 0.5, and 0.3, respectively. There are two bidders, whose valuations are independently drawn by Nature. After each bidder learns her valuation, they simultaneously choose a bid that is required to be a positive integer. A bidder’s payoff is zero if she loses the auction and is her valuation minus her bid if she wins it. Determine whether it is a symmetric Bayes–Nash equilibrium for a bidder to bid 4 when her valuation is 5, 6 when her valuation is 7, and 9 when her valuation is 10."
In this case, we have discrete bids and discrete valuation, as well as different probabilities for each. Could someone help me out understand the problem?