Which sale rule would you choose, and how would you attack it?
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).
Every earlier topic in this section handed you a game and asked what the players do. A designer asks the opposite question: given what players do, which game should I build? You are about to be one. You run a market with one item for sale and a few bidders, each of whom knows privately what the item is worth to them.
A good market passes three tests, and each is a topic from this course. Do the bidders have a reason to lie? That is the incentive test of topics 14 and 26. Where does play end up? Real bidders do not solve equations; they adjust, and topic 28 showed where adjusting leads. Can the bidders agree not to compete? A group that can meet again (topic 27) and coordinate through a shared plan (topic 29) can, unless the design stops it.
The toy below lets you pick the rule, the reserve price (the lowest price at which the seller will sell) and the number of bidders, and then attack the market three ways. Press the first button to see where learners settle. Press the second to ask whether bidding your value is still a good idea when everyone else does it. Press the third to see what a bidding ring does to the seller.
What you find is that no rule wins every test. The second-price rule survives the second test, but it is the easiest to rig, and a reserve that is right for honest bidders can cost the seller money when the bidders are only learning. Market design is mostly a matter of choosing which failure you can live with.
The model. One item, n bidders, each with a value drawn independently and uniformly from the five numbers 0.2, 0.4, 0.6, 0.8, 1.0, and each bidding one of 0.0, 0.1, ..., 1.0. A bid below the reserve r is discarded and pays nothing. The highest remaining bid wins, and ties split the item evenly. The rules differ only in the price: first price charges the winner's bid; second price charges the winner the larger of r and the best rival bid; all pay charges every bidder whose bid clears the reserve, win or lose.
Test 1, honesty. In the second-price rule, bidding your value is a best reply to any rival bids (Vickrey, 1961): a higher bid can only win items at a price above their worth to you, and a lower one can only lose items that would have paid. The page checks it by trying every bid against every rival profile: the gain from lying is 0.000 with three bidders, with or without a reserve. In the first-price rule a bidder facing honest rivals gains 0.036 on average by shading, and in the all-pay rule 0.320, so honest bidding is not stable in either.
Test 2, learning. Give every bidder the multiplicative-weights rule of topic 28: after each round, raise the weight of every bid in proportion to the profit it would have earned, and report the average over 200 rounds. How far this is from a stable outcome is measured by the gain from deviating: how much a bidder could add, on average over its value, by changing its bid with the rivals' behaviour fixed. It is 0.003, 0.002 and 0.003 for the three rules, small enough to call these approximate equilibria. Yet the second-price learners do not become honest: they earn the seller 0.587, not the honest 0.600, because low bids that cost almost nothing stay in the mix. Honesty is a best reply, not the only one that learners find.
The reserve. A reserve trades revenue for efficiency, the share of the best possible value that reaches the bidder who wants it most. With honest second-price bidders and three bidders, the seller's revenue rises from 0.600 with no reserve to 0.653 with a reserve of 0.60, and efficiency falls from 100% to 97.1%. Higher reserves lose revenue again. For values spread uniformly over the whole interval from 0 to 1, Myerson (1981) showed that the best reserve is 1/2 whatever the number of bidders; on this five-point grid a search finds 0.60 for two and three bidders and 0.80 for four. With the same reserve of 0.60, learners in the second-price rule earn the seller only 0.577, against 0.587 with no reserve, so the advice depends on how the bidders behave.
Test 3, the ring. Suppose all n bidders agree beforehand to send only their best member to bid, and that member bids the reserve price. The item sells whenever the best value clears the reserve, and the price is the reserve, in any of the three rules. The seller's revenue is then (reserve) × (chance the best value is at least the reserve). For three bidders, the columns are the reserve, the seller's revenue from honest second-price bidders, their efficiency, and the seller's revenue from the ring:
| Reserve | Honest revenue | Efficiency | Ring revenue |
|---|---|---|---|
| 0.00 | 0.600 | 100.0% | 0.000 |
| 0.10 | 0.600 | 100.0% | 0.100 |
| 0.20 | 0.600 | 100.0% | 0.200 |
| 0.30 | 0.608 | 99.8% | 0.298 |
| 0.40 | 0.618 | 99.8% | 0.397 |
| 0.50 | 0.624 | 97.1% | 0.468 |
| 0.60 | 0.653 | 97.1% | 0.562 |
| 0.70 | 0.605 | 86.3% | 0.549 |
| 0.80 | 0.648 | 86.3% | 0.627 |
| 0.90 | 0.450 | 58.1% | 0.439 |
With no reserve the ring pays nothing at all, and with a reserve it pays that reserve and no more. So the ring costs the seller a great deal when the reserve is low and little when the reserve is high: the reserve price is the only part of this design that limits a ring. The ring is not an equilibrium of anything, since each member could do better by cheating on it; topic 27 is about what keeps such an agreement standing.
Choose the rule, the reserve and the number of bidders. Press 1 to let the bidders learn for 200 rounds, 2 to test whether honest bidding is a best reply, and 3 to see what a bidding ring does to the seller. Try each rule with and without a reserve.
Reserve prices Sellers who run auctions set a minimum acceptable price for the reason shown here: it is the lever that raises revenue when bidders are few. Myerson's 1981 paper (doi:10.1287/moor.6.1.58) derived the best reserve for the general case.
Bidding rings Agreements among bidders to hold prices down are treated as offences under the competition law of many countries. This page does not claim how common they are; it shows why a ring is hard to see in the bids alone, since all it does is bid the reserve.
Learning bidders Automated bidding programs adjust to what they see. The update used on this page is the multiplicative-weights rule surveyed by Arora, Hazan and Kale (doi:10.4086/toc.2012.v008a006).
The sources 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.
The model is a small, discrete one chosen so that every number on this page can be computed exactly and checked a second way. It is not a claim about any particular real market. No quantum link is claimed for this topic.
Check yourself
In the small market of topic 33, with three bidders, what does the second-price rule have that the first-price rule lacks?
In the second-price rule the price does not depend on your own bid, only on whether you win, so bidding your value cannot be beaten (Vickrey, 1961). The page's exact check finds a gain from lying of zero. A ring still works against it, and the learners on the page settle on lower bids that earn the seller less than honest bidding would.