Expected Profit in First-Price Sealed-Bid Auctions

Introduction

Hello! In our last lesson, we completed a rigorous proof showing that in a second-price auction, your optimal strategy is simply to bid your true private value. This is a powerful result because your best move doesn't depend on what you think others might bid.

However, the first-price sealed-bid (FPSB) auction is strategically much more complex. If you bid your true value $v_{i}$ and win, your profit is $v_{i} - v_{i} = 0$ . To gain any profit, you must bid less than your value, a practice known as "bid shading." But how much less? Bidding too low increases your potential profit but dramatically reduces your chance of winning. Bidding too high increases your chance of winning but squeezes your profit margin.

To solve this trade-off, a bidder needs to find the bid that maximizes their expected profit. Today's lesson is the crucial first step in that process. We will formulate the mathematical expression for a bidder's expected profit in a symmetric FPSB auction, under the assumption that bidders' private values are drawn from a uniform distribution. This sets the stage for our next lesson, where we will use this formula to derive the optimal bidding strategy itself.

Setting Up the First-Price Auction as a Bayesian Game

Let's formally define the environment. A first-price sealed-bid auction is a static game of incomplete information, also known as a Bayesian game.

Players: $n$ bidders, indexed by $i = 1, ..., n$ .
Types (Private Information): Each bidder $i$ has a private value $v_{i}$ for the item. This is their true willingness to pay. We'll assume these values are independently and identically drawn from a uniform distribution on an interval, for example, $U [0, 100]$ . This means any value between 0 and 100 is equally likely. Each bidder knows their own value but only knows the distribution from which others' values are drawn.
Actions: Each bidder $i$ submits a sealed bid, $b_{i}$ .
Payoffs (or Profit):
- If bidder $i$ wins (i.e., $b_{i}$ is the highest bid), their payoff is their surplus: $u_{i} = v_{i} - b_{i}$ .
- If bidder $i$ loses, their payoff is $u_{i} = 0$ .

Because a bidder doesn't know the others' values (and therefore their bids), they can't calculate their profit with certainty. Instead, they must aim to maximize their expected profit, which is the sum of possible outcomes weighted by their probabilities.

Formulating the Expected Profit

The expected profit for any action is always calculated as:

Expected Profit = (Profit if you win \times Probability of winning) + (Profit if you lose \times Probability of losing)

Since the profit from losing is zero, this simplifies to:

E [Profit] = (Profit if you win) \times (Probability of winning)

For a bidder $i$ with value $v_{i}$ who places a bid $b_{i}$ , the profit upon winning is simply $v_{i} - b_{i}$ . The challenging part is calculating the probability of winning.

A bidder wins if their bid $b_{i}$ is higher than all other $n - 1$ bids. To calculate this probability, we need to make an assumption about how others are bidding. In a symmetric Bayesian Nash Equilibrium (BNE), we assume all players use the same bidding strategy (or function), which we'll call $b^{*} (v)$ . This function takes a player's value $v$ and returns their optimal bid.

The following video will walk you through setting up the expected profit calculation in a two-player auction where values are drawn from $U [0, 100]$ . It makes a simplifying assumption that the symmetric strategy is linear ( $b^{*} (v) = a \cdot v$ ), which allows for a concrete calculation.

8. An Example for Bayesian Nash Equilibrium: First Price Auction (Game Theory Playlist 9)

Watch this segment from Selcuk Ozyurt's lecture series. It clearly explains how to construct the expected payoff (or profit) for a bidder who is contemplating what to bid, assuming their opponent follows a specific type of strategy.

Watch from the beginning to 9:43. 00:00 - 05:28: Pay close attention to how the game is defined: players, payoffs, and the assumption of private values from a uniform distribution (U[0, 100]). Note the assumption of a symmetric, linear bidding strategy, b_i = a \cdot v_i. 05:28 - 09:43: This is the core of our lesson. Follow the derivation of the 'expected payoff'. The key is to understand how the probability of winning is calculated based on the assumption about the opponent's strategy.

Deconstructing the Expected Profit Formula

Let's formalize the steps from the video. We are player $i$ with value $v_{i}$ . We are considering placing a bid $x$ . Our opponent is player $j$ .

Our Payoff if we Win: $v_{i} - x$
Assumption about Opponent's Strategy: We assume player $j$ is using the symmetric strategy $b_{j} = a \cdot v_{j}$ , where $v_{j} \sim U [0, 100]$ and $a$ is some constant we'll determine in the next lesson.
Condition for Winning: We win if our bid $x$ is higher than our opponent's bid $b_{j}$ .
$x > b_{j} ⟹ x > a \cdot v_{j}$
Probability of Winning: We need to find the probability that the condition above holds. Rearranging for the opponent's unknown value $v_{j}$ , we get:
$v_{j} < \frac{x}{a}$
Since player $j$ 's value $v_{j}$ is drawn from a uniform distribution $U [0, 100]$ , the probability that $v_{j}$ is less than some number $y$ is given by the cumulative distribution function (CDF), which for this distribution is $F (y) = y /100$ .
Therefore, the probability of winning is:
$Prob (Win) = Prob (v_{j} < \frac{x}{a}) = F (\frac{x}{a}) = \frac{x / a}{100} = \frac{x}{100 a}$
Putting it all together: We can now write the full expression for our expected profit, which we'll denote as $E [u_{i} (x; v_{i})]$ , for making a bid $x$ given our value is $v_{i}$ :
$E [u_{i} (x; v_{i})] = (v_{i} - x) \times \frac{x}{100 a}$
This equation captures the fundamental trade-off: as your bid $x$ increases, the term $(v_{i} - x)$ gets smaller (less profit), but the term $x / (100 a)$ gets larger (higher chance of winning).

Generalizing to N Bidders

The logic extends directly to a game with $n$ bidders. If player $i$ bids $x$ , they win only if their bid is higher than all other $n - 1$ bids.

x > b_{j} for all j \neq = i

Assuming all other players use the strategy $b_{j} = a \cdot v_{j}$ , this means:

v_{j} < \frac{x}{a} for all j \neq = i

Since each $v_{j}$ is drawn independently, the probability that all $n - 1$ of these conditions hold is the product of their individual probabilities:

Prob (Win) = (Prob (v_{j} < \frac{x}{a}))^{n - 1} = (\frac{x}{100 a})^{n - 1}

So, for an $n$ -player auction where $v_{i} \sim U [0, 100]$ , the expected profit for bidding $x$ is:

E [u_{i} (x; v_{i})] = (v_{i} - x) (\frac{x}{100 a})^{n - 1}

Often in academic models, the uniform distribution is normalized to $U [0, 1]$ . In that case, the CDF is simply $F (y) = y$ , and the expected profit becomes:

E [u_{i} (x; v_{i})] = (v_{i} - x) (\frac{x}{a})^{n - 1}

This generalized formula is what we will need to find the equilibrium bidding strategy.

Conclusion

Today we took the first, essential step toward solving the first-price auction. We moved from the simple world of dominant strategies in second-price auctions to the more complex domain of Bayesian games, where players must manage a trade-off between the probability of winning and the profit from a win.

Key Takeaways:

In a first-price auction, rational bidders "shade" their bids (bid below their true value) to secure a positive profit.
A bidder's goal is to choose a bid that maximizes their expected profit.
The expected profit is a function of the bidder's value ( $v_{i}$ ) and their chosen bid ( $b_{i}$ ): $E [Profit] = (v_{i} - b_{i}) \times Prob (Win)$ .
To calculate the probability of winning, we must make an assumption about how others will bid. We assume they follow a symmetric strategy, $b^{*} (v)$ .
For an $n$ -bidder auction with values from $U [0, 1]$ and an assumed linear strategy $b^{*} (v) = a v$ , a bidder with value $v_{i}$ choosing to bid $x$ has an expected profit of: $E [u_{i} (x; v_{i})] = (v_{i} - x) (x / a)^{n - 1}$ .

We have successfully formulated the objective function. In our next lesson, we will solve the optimization problem: what bid $x$ maximizes this function? The answer will give us the symmetric Bayesian Nash Equilibrium bidding strategy. You have a keen interest in mathematical physics, which often involves setting up and solving optimization problems (like minimizing action or energy); you'll find the process in our next lesson very similar.