Auction Strategies: First-Price vs. Second-Price Bidding

Setting the Stage: Auction Environments

Before we dissect the rules, it's crucial to understand the environment in which an auction takes place. The most important distinction is how bidders value the item.

To summarize the key ideas from the video:

• Private Value Auctions: Each bidder's valuation $v_i$ for the item is personal and independent of others' valuations. The value I place on a piece of art I want for my own enjoyment doesn't depend on how much you value it. This is the setting we will focus on for this lesson.
• Common Value Auctions: The item has a single true, objective value $V$ that is the same for everyone, but this value is unknown at the time of bidding. Each bidder has an estimate of this value. The oil well example from the video is classic: the amount of oil is the same no matter which company wins the drilling rights.

For the rest of this module, unless stated otherwise, we will assume a private value setting. This allows us to isolate the strategic choices related to the auction rules themselves.

First-Price Sealed-Bid (FPSB) Auction

This is perhaps the most intuitive auction format.

Rules

The rules are simple:

1. There are $n$ bidders, each with a private valuation $v_i$.
2. Each bidder $i$ simultaneously and secretly submits a bid, $b_i$.
3. The bidder with the highest bid wins the item.
4. The winner pays the price they bid.
5. A bidder's surplus (or payoff) is $v_i-b_i$ if they win, and 0 if they lose.

Strategic Implications

In a first-price auction, you face a fundamental trade-off.

As the video explains, bidding your true valuation $v_i$ is not a good idea. If you win by bidding $b_i=v_i$, your surplus is $v_i-v_i=0$. You gain nothing.

This creates a powerful incentive to shade your bid—that is, to bid some amount $b_i<v_i$.

• The lower you bid, the higher your potential surplus if you win.
• The higher you bid, the higher your probability of winning.

Your optimal bid depends on what you believe other bidders will do. How much you should shade your bid depends on how many bidders there are and what you believe about their valuations. This is a complex Bayesian game with no simple dominant strategy.

Second-Price Sealed-Bid (SPSB) Auction

This auction format, also known as a Vickrey auction, has a clever twist in its rules that dramatically changes the strategic landscape.

Rules

The bidding process is the same, but the payment rule is different:

1. There are $n$ bidders, each with a private valuation $v_i$.
2. Each bidder $i$ simultaneously and secretly submits a bid, $b_i$.
3. The bidder with the highest bid wins the item.
4. The winner pays the price of the second-highest bid.
5. A bidder's surplus is $v_i-b_{second}$ if they win, and 0 if they lose.

Let's look at a direct visual comparison of the two auction types.

Strategic Implications

The magic of the second-price auction is that it removes the trade-off that complicates the first-price auction. The optimal strategy is surprisingly simple and powerful.

In a second-price auction, bidding your true valuation $v_i$ is a dominant strategy.

This means that bidding $b_i=v_i$ is your best possible move, regardless of what any other bidder does. Let's build the intuition for this remarkable result:

Your bid only does one thing: it determines whether you win or lose. The price you pay is determined entirely by your opponents' bids. Let $b_{max\_other}$ be the highest bid among all other players.

1. Case 1: Your valuation is higher than the competition ($v_i>b_{max\_other}$)

   • If you bid your true value $b_i=v_i$, you win and pay $b_{max\_other}$. Your surplus is $v_i-b_{max\_other}$, which is positive. This is a good outcome.
   • Could you have done better? If you bid lower ($b_i^{\prime }<v_i$), you risk bidding below $b_{max\_other}$ and losing, getting a surplus of 0 instead of a positive one. If you bid higher ($b_i^{\prime }>v_i$), you still win and pay the same price $b_{max\_other}$. So, bidding differently doesn't help and might hurt.

2. Case 2: Your valuation is lower than the competition ($v_i<b_{max\_other}$)

   • If you bid your true value $b_i=v_i$, you lose and your surplus is 0. This is the correct outcome, because winning would mean paying $b_{max\_other}$, giving you a negative surplus of $v_i-b_{max\_other}$.
   • Could you have done better? If you bid lower, you still lose and get 0. If you bid higher ($b_i^{\prime }>v_i$) enough to win (i.e., $b_i^{\prime }>b_{max\_other}$), you are forced to pay a price higher than your valuation, resulting in a loss. Again, bidding differently doesn't help and might hurt.

In every possible scenario, you cannot improve your outcome by bidding anything other than your true value.

Comparing the Auctions

We've now seen that a seemingly small change in the payment rule leads to a profound difference in strategy. Let's summarize the comparison.

Here are the key points from that table and our discussion, presented in a slightly different format:

The fact that the SPSB auction induces truthful bidding is incredibly useful for auction designers who want to discover the true valuations of bidders.

Conclusion

In this lesson, we have taken our first steps into the world of auction theory. By comparing first-price and second-price sealed-bid auctions, we have uncovered a core principle in game theory and mechanism design: the rules of the game dictate the players' strategies.

Key Takeaways:

• In a First-Price Sealed-Bid (FPSB) auction, the winner pays their own bid, creating a strategic incentive to "shade" your bid below your true valuation.
• In a Second-Price Sealed-Bid (SPSB) auction, the winner pays the second-highest bid. This rule change makes bidding your true valuation a dominant strategy.
• The SPSB is considered "strategy-proof" because a bidder's best action is to be truthful, regardless of their beliefs about other players. The FPSB, in contrast, requires complex reasoning about your opponents.

We've established the intuition for why truthful bidding is dominant in a second-price auction. Given your interest in mathematical rigor, our next lesson will be dedicated to formalizing this critical result. We will walk through the complete proof to show, step-by-step, why any bid other than your true value is a weakly dominated strategy in an SPSB auction.