Tensions in Practice: A package can be worth more than its parts¶
Two items with complementary value
Imagine allocating items X and Y using fixed declared bids. A would declare 13 for the pair and 0 for either item alone; B declares 7 for X; C declares 5 for Y. Compare a format admitting only those singleton bids with one also admitting A’s all-or-nothing package. The winner selection maximizes admitted bid totals without assigning an item twice. This isolates bid expressiveness; it does not model how bidders choose bids or what winners pay.
Clear items independently
Keep the admitted bid language and each item’s allocation simple.
Express complementary value
Allow a bidder to request a package whose value is not the sum of singleton values.
Why these aims pull against each other
Independent item clearing cannot act on a package bid it does not admit. Admitting packages makes overlapping bundles compete jointly and increases the general clearing problem.
Choose an arrangement to see what changes and what remains difficult.
Compare the same rows across alternatives. Cells state explicit toy quantities, membership or permissions; colors do not supply additional meaning.
What this choice protects
What it costs
When it fits
Compare the arrangements
Singleton bids
Only bids for one item are admitted. B receives X for a declared 7 and C receives Y for a declared 5; their bid sum is 12. A’s package is outside this format.
| Admitted bid | Bid sum | Selected | |
|---|---|---|---|
| B and C | X + Y | 12 | Yes |
| A package | Not admitted | 13 | No |
- What it protects
- Each item can be cleared independently without searching overlapping packages.
- What it costs
- The declared package value of 13 cannot compete; the selected bid sum is 12 in this example.
- When it fits
- Plausible where item values are sufficiently separable or simple clearing is worth restricting bid expressiveness.
Illustration note: This is an editorial, deliberately bounded illustration. Its stated rules and any numbers are invented, not observations, recommended settings, or predictions.
Package bids
Admit A’s all-or-nothing bid in addition to the singleton bids. The two disjoint singleton bids total 12; the package totals 13, so A receives both items.
| Admitted bid | Bid sum | Selected | |
|---|---|---|---|
| B and C | X + Y | 12 | No |
| A package | X with Y | 13 | Yes |
- What it protects
- The allocation can recognize the declared complementarity and select the larger admitted total.
- What it costs
- The format must represent packages and compare overlapping combinations. This tiny example hides the computational and strategic difficulties of larger instances.
- When it fits
- Plausible when complementarities matter and the admissible bundle set can be cleared with a suitable, explicit method.
Illustration note: This is an editorial, deliberately bounded illustration. Its stated rules and any numbers are invented, not observations, recommended settings, or predictions.
What this illustration does—and does not—establish
The source establishes the structural tension; the concrete alternatives and their conditional costs are editorial synthesis. No arrangement is a universal recommendation.
- All numbers and declarations are invented and held fixed. No equilibrium bidding, truthfulness, prices, revenue or real welfare is inferred.
- The two displayed allocation plans suffice for these bids; arbitrary package auctions require a larger feasibility search.
- A larger admitted bid total is the stated objective here, not proof of a universally preferable allocation.
Source entries
Auction Theory
Auction theory Combinatorial Valuation vs Clearing Tractability supplies the local tension. The setting, alternative arrangements, and stipulated consequences are editorial applications.
Combinatorial Valuation vs Clearing Tractability
But combinatorial clearing is NP-hard in general; real deployments have to approximate, restrict bid structure, or iterate via clock-proxy formats — each with its own strategic artifacts.