First-price sealed-bid auction

A first-price sealed-bid auction (FPSBA) is a common type of auction. It is also known as blind auction. In this type of auction, all bidders simultaneously submit sealed bids so that no bidder knows the bid of any other participant. The highest bidder pays the price that was submitted.

In a FPSBA, each bidder is characterized by their monetary valuation of the item for sale.

Suppose Alice is a bidder and her valuation is ${\displaystyle a}$. Then, if Alice is rational:

Alice would like to bid the smallest amount that can make her win the item, as long as this amount is less than ${\displaystyle a}$. For example, if there is another bidder Bob and he bids ${\displaystyle y}$ and ${\displaystyle y<a}$, then Alice would like to bid ${\displaystyle y+\varepsilon }$ (where ${\displaystyle \varepsilon }$ is the smallest amount that can be added, e.g. one cent).

Unfortunately, Alice does not know what the other bidders are going to bid. Moreover, she does not even know the valuations of the other bidders. Hence, strategically, we have a Bayesian game - a game in which agents do not know the payoffs of the other agents.

The interesting challenge in such a game is to find a Bayesian Nash equilibrium. However, this is not easy even when there are only two bidders. The situation is simpler when the valuations of the bidders are independent and identically distributed random variables, so that the valuations are all drawn from a known prior distribution.

Suppose there are two bidders, Alice and Bob, whose valuations ${\displaystyle a}$ and ${\displaystyle b}$ are drawn from a continuous uniform distribution over the interval [0,1]. Then, it is a Bayesian-Nash equilibrium when each bidder bids exactly half his/her value: Alice bids ${\displaystyle a/2}$ and Bob bids ${\displaystyle b/2}$.

PROOF: The proof takes the point-of-view of Alice. We assume that she knows that Bob bids ${\displaystyle f(b)=b/2}$, but she does not know ${\displaystyle b}$. We find the best response of Alice to Bob's strategy. Suppose Alice bids ${\displaystyle x}$. There are two cases: