2pa.md

Second Price Auctions

Overview

• Second price auctions (2PA) are a type of auction where the highest bidder pays the second highest bid
• In contrast to first price auctions (FPA), where the highest bidder pays her own bid

In this talk, going to go over

• Why 2PA work better than FPA
• Why you should care
• Some game theory behind all of this

Why should you care?

• You probably know already that all of our DSP data comes from auctions
  • A webpage has some space for an ad, they issue an RFP (request for proposals), multiple advertisers bid for that space, and the winner gets displayed
• What you might not know is: these auctions are all 2PA
• In fact, 2PA underlie most ads on the internet
• So if you want to understand how ads on the internet work, 2PA are a big part of that

Why are 2PA better than FPA?

• In a nutshell: 2PAs incentivize bidders to bid truthfully
  • Optimal strategy in 2PA is for everyone to bid what they actually value for the object - no matter what anyone else bids
• FPAs incentivize underbidding

Before seeing why 2PA is efficient, let's look at why FPA are not

Why do FPAs incentivize underbidding?

• Easy to see by looking at the payoffs:
  • When you win:
    • If bid = value, payoff = 0, just breaking even
      • For example, if you value a good at $10, and you pay $10 for it, on net, you've just broken even. You haven't made a net profit or loss.
    • If bid > value, payoff < 0
      • For example, if you value a good at $10 and you pay $12 for it, you've actually lost value
    • If bid < value, payoff > 0
      • For example, if you value a good at $10 and you pay $8 for it, you've gained a net profit of $2
  • When you lose: payoff = 0
    • And you can't improve that payoff by bidding non-truthfully:
    • In the graph, you can see that:
      • At B1, when you bid your true value, you lose and payoff = 0
      • If you decrease your bid below your value to B2, you still lose, and payoff = 0
      • If you increase your bid slightly to B3, above your value but below the price, you lose and payoff = 0
      • If you increase your bid all the way to B4, you're now the winner, but you pay more than the good is worth; so payoff < 0
  • So underbidding is always your best strategy

So why do 2PAs incentivize truthful bidding?

To prove this:¹

• Assume you bid truthfully. There are two possible cases: you win or you lose
  • In either case, deviating from truthful bid will hurt you
  • So, bidding truthfully must be your best strategy

Proof:

• Case 1: you win the auction by bidding honestly
  • At B1, you bid your value, and you pay the next highest bid, the price. Your payoff is B1 - price, which is positive
  • If you bid higher, at B2, then you still pay the same price. Your payoff is still B1 - price
    • Nothing has changed, except that you've added more risk: you might have had to pay more than your value
  • If you underbid slightly, at B3, then again nothing changes. You pay price, and your payoff is still B1 - price. You've just added the risk that someone would outbid you when you would have been willing to pay more.
  • If you drastically underbid, at B4, then you lose the auction. Your payoff is 0, and you feel regret because you would have been willing to pay more.
• Case 2: you lose the auction by bidding honestly
  • Like in the FPA case, at B1 when you bid your value, you lose the auction and your payoff is 0
  • If you bid lower, at B2, then you still lose and your payoff is 0. Also, you've added more risk: there are bids that beat B2 which you would have been willing to outbid
  • If you slightly increase your bid to B3, again you still lose and payoff = 0. You've added more risk, though: if you had won, your payoff would be negative.
  • If you drastically increase your bid to B4, you win the auction. But you would prefer to lose in this case, since your bid is higher than your value. Your payoff is negative.
  • So, no deviations from truthful bidding would have made you better off!
• So: any time you don't bid truthfully, you either hurt yourself or take on more risk

Why is bidding truthfully good?

OK so now the seller knows people's true values. So what? Why does this mean the 2PA is better?

You might think:

• Isn't the seller leaving money on the table with a 2PA? The winner pays less than what they were willing to bid
• Ex: bids = {11, 50, 88, 94}. The winner was willing to pay $94, but only paid $88, so $6 is left "on the table"
• But this reasoning requires us to think that people would bid the same way in a FPA and 2PA
  • But they don't, because what someone is willing to pay for an object is different from what they're willing to bid for that object, depending on the type of auction
  • So, in a sense, that $6 would not have been "on the table" in a FPA at all

So why use 2PA?

• Does this mean that 2PA earns the seller more money?
  • No -- 2PA and FPA are actually revenue equivalent.
• BUT: there are still reasons sellers might prefer 2PA
  • Because participants bid their true value, there is no strategy involved in 2PA. FPA requrie strategic underbidding. Therefore, 2PA is simpler
  • 2PA can provide useful information for the seller:
    • For example, say the seller wants to figure out how much something is worth. Since 2PA encourage truthful bidding, they are a good way to answer this question. FPA, not so much.

Footnotes

1: See this video

img1_FPA_losing_case.jpeg

img2_2PA_winning_case.jpeg

img3_2PA_losing_case.jpeg