How much should you bid when you do not know what the others will pay?
Topics 11 to 15 asked who gains when several players choose. These nine go further: what changes when the moves come in turns (25), when players hold secrets (26), when they meet again (27), when they learn as they go (28), when a shared signal is allowed (29), when each chooses a route (30), when the question is how hard an equilibrium is to find (31), the one place where quantum physics changes a game's value (32), and then a market you design and attack yourself (33).
In an auction you know what the item is worth to you and not what it is worth to anyone else. That one fact turns bidding into a game of private information: each bidder has a secret type (their value), everyone knows how values are spread, and a strategy is a rule that turns a type into a bid. John Harsanyi showed in 1967 how to treat such a game as an ordinary one by adding a first move in which nature deals the types; the equilibrium of the result is a Bayesian Nash equilibrium.
Take the sealed-bid auction where the highest bid wins and the winner pays their own bid. If you bid your true value you win sometimes and earn nothing each time, so you should shade your bid below your value. How far? Shading more raises your profit when you win and lowers your chance of winning. With many rivals the chance of winning drops fast as you shade, so you shade less; with one rival you can shade a lot.
There is a quieter lesson. In the second-price auction the winner pays the second-highest bid, and bidding your true value is the best you can do whatever the others bid (William Vickrey, 1961). Yet on average the seller collects exactly the same in the two formats. That is the revenue equivalence theorem (Myerson, 1981): two very different-looking rules, one expected price.
Suppose n bidders have values drawn independently and uniformly from [0, 1] and each rival bids a fixed fraction c of their value. You, with value v, bid b. You win when every rival's value is below b/c, which has probability (b/c)n−1, so your expected profit is
u(b) = (v − b) · (b / c)n−1
Setting the derivative to zero gives b = v(n − 1)/n. For the rivals' rule to be the one you are best-responding to, c must equal that fraction, and it does: b(v) = v (n − 1)/n is a Bayesian Nash equilibrium.
Worked example, three bidders, your value 0.60. The rule says bid 0.40. The table checks it against nearby bids (a grid search over 2,000 bids finds the best at 0.40):
| Your bid | Chance of winning | Expected profit |
|---|---|---|
| 0.30 | 0.203 | 0.061 |
| 0.40 | 0.360 | 0.072 |
| 0.50 | 0.562 | 0.056 |
| 0.60 | 0.810 | 0.000 |
Revenue. The highest of n uniform values has mean n/(n + 1), and the winner pays the fraction (n − 1)/n of it, so the seller expects (n − 1)/(n + 1). In the second-price auction the seller gets the second-highest value, whose mean is the same. The expected revenue is:
| Bidders | Exact | Decimal |
|---|---|---|
| 2 | 1/3 | 0.333 |
| 3 | 1/2 | 0.500 |
| 5 | 2/3 | 0.667 |
| 10 | 9/11 | 0.818 |
Each added bidder raises the expected revenue, while switching between these two formats changes nothing. That is why auction designers worry first about attracting bidders.
Slide the number of bidders and your own value. The page finds the best bid by searching a grid of 1,000 bids and compares it with the formula, then runs 20,000 simulated auctions in each format to show the two revenues landing on the same number.
Selling the airwaves Governments sell radio spectrum by auction, in formats designed with this theory. In 2020 the Nobel memorial prize in economics went to Paul Milgrom and Robert Wilson for improvements to auction theory and for inventing new auction formats. Milgrom and Weber's model of bidding when values are correlated is Econometrica 50, 1089 (1982).
Selling attention Online advertising sells a slot in a fraction of a second to whoever bids most. Several large ad exchanges moved from second-price to first-price rules between 2017 and 2019, Google Ad Manager the last of the major ones, which changes what a sensible bidding program does: shade, as above, instead of bidding the value.
The sources Harsanyi, Management Science 14, 159 (1967), doi:10.1287/mnsc.14.3.159. Vickrey, Journal of Finance 16, 8 (1961), doi:10.1111/j.1540-6261.1961.tb02789.x. Myerson, Mathematics of Operations Research 6, 58 (1981), doi:10.1287/moor.6.1.58.
No quantum link is claimed for this topic.
Check yourself
Three bidders have private values uniform between 0 and 1. In a first-price sealed-bid auction, your value is 0.60. At the equilibrium of the game, what do you bid?
With n bidders the equilibrium bid is v(n − 1)/n. For n = 3 and v = 0.60 that is 0.60 × 2/3 = 0.40. Bidding your value wins but earns nothing; bidding lower raises your profit when you win and lowers your chance of winning.