 Auctions are everywhere – on TV, in corporate boardrooms, all over the internet – and for just about everything, from trinkets and collectibles to natural resources and government contracts. Whether the stakes are silly or serious, auctions are an interesting combination of buying, selling, and gambling, all rolled into one.

In this lesson, students use probability, expected value, and polynomial functions to develop a profit-maximizing bidding strategy.

### Students will

• Calculate the probability of winning an auction with different bids distributed uniformly on an interval
• Write and graph functions that yield expected profit for various bids
• Find the vertex of a parabola in order to maximize expected profit from a quadratic function
• Develop and reason about higher-order polynomial expected value functions to maximize profit on an interval
• Make informal limit arguments about optimal biding strategy as the number of bidders in an auction increases

### Before you begin

The most important prerequisite for this lesson is an understanding of how to use values and probabilities to calculate an expected value. One of the expected value functions turns out to be quadratic, so students will have to find the vertex in order to maximize expected profits. If they can do this algebraically, that’s great, but they will also have the opportunity to find a maximum graphically. In fact, as they move on to higher-order polynomials, they will have to find local maxima graphically, unless you’re using this as a calculus lesson on maximization...which is a great idea.