# Interactive Guide to Cooperative Auctions Three roommates need to divide bedrooms. One room is bigger, one has a balcony, and one is next to the kitchen. They can argue about it, draw straws, or let whoever moved in first claim the best room. None of these options are satisfying. Someone always feels shortchanged. An auction solves this problem elegantly. Each person bids for the rooms they want, the highest bid for each room wins, and the proceeds are shared equally. The person who gets the best room pays more rent, and the person who gets the worst room pays less. Nobody has to argue about what's fair, since the bids determine it exactly. This guide walks through how these **cooperative auctions** work, starting from the basics and building up to multi-item auctions, settlement mechanics, and less-obvious applications like group decision-making. Each auction has an interactive simulation you can experiment with. ## Regular Auctions Before we can see how cooperative auctions work, we need to cover how auctions work in general. We'll start with the simplest possible case: one item, two bidders, and no shared ownership. Pat is selling her bike and Nina and Omar both want it. They agree to use an ascending auction to sell the bike. In ascending auctions, the price rises over multiple rounds until only one bidder is left. This format is economically efficient (the item goes to the person who values it most) and easy to interpret. The price starts at zero and each round it increases by a fixed bid increment. The top of the simulator defines the things available for auction, the bidders who are participating, and the amount that each bidder would pay to have each item. Here, Nina is willing to pay up to $150 for the bike, and Omar is willing to pay up to $100. The progression of the auction is straightforward. Nina and Omar both continue bidding for the bike until the price gets to $100. At this point, Omar would be bidding for the bike at a price of $110, which is above his value for the bike, so he drops out. The auction stops since there's no more bidding activity. Notice how Nina wins the bike, but the price is based on Omar's value. This is a second-price mechanism, and it's an important feature of the auction. It means that Nina is not penalized for having a higher value for the bike. She only pays what the next-highest bidder would have paid for it. To see why second-price mechanisms are useful, we can consider a first-price auction format where Nina and Omar write down their prices, give them to Pat, and Pat accepts the higher one. If Nina writes down $150, she will have to pay the maximum amount she wanted to pay for the bike. She would prefer to lower her bid, ideally just above Omar's bid, to avoid paying more than necessary. Thus Nina is incentivized to predict Omar's value, which might be difficult to do. If Omar's value is higher than expected, she could risk losing the bike at a price where she would have still wanted it. Second-price mechanisms save bidders this wasted effort. If Nina only pays Omar's value, then Nina never has to worry about predicting his value. She only has to worry about what she would pay, knowing that if she wins, her payment is only going to be as high as necessary to beat Omar. The second-price mechanism means bidders' optimal strategy is to bid based on their true values. As long as bidders don't have valuations for items that depend on whether they win other items, ascending auctions preserve this optimality of truthful bidding. Note that in a real auction, Nina's and Omar's values are private. They only find out how the other person values the bike as the price rises through the rounds. We can see their values in the simulator, but this information is not normally visible. When bids are tied, the simulator will pick the bidder with the alphabetically earlier name as the high bidder. This keeps the exact results predictable for our analysis, but in real auctions this is randomized. Try swapping Nina and Omar's values. You'll observe that Omar now wins the bike, but at a price of $110 instead of $100. This is because the simulation always picks Nina as the high bidder in the first round. Omar and Nina then trade places as the high bidder each round after that. In round 10, Omar is the high bidder at a price of $90. Since Nina's value is $100, she is still willing to bid for the bike, and she does, becoming the high bidder in round 11 at a value of $100. Omar then bids again at a price of $110, and Nina drops out. The size of the bid increment thus determines how much error there is in the final prices. If you change the bid increment to $5, you'll see that Omar now wins the bike for $105. If you change it to $1, Omar wins for $101. This asymmetry between Nina and Omar is only because the auction simulator is deterministic. In real auctions, each would have an equal opportunity to become the high bidder in the first round. ## Cooperative Auctions In cooperative auctions, there isn't a separation between buyers (Nina, Omar) and sellers (Pat). Instead, each member of the community has an equal right to the auction proceeds, while also participating in the auction itself. Alex and Ben are housemates, and they need to decide who is going to have the bigger room, and at what price. They both have an equal claim to it, and they use a cooperative auction to determine who gets it and what the payment is. They each bid according to what they'd pay per month to have the bigger room. The cooperative auction is the same as a regular auction, but with an extra **equalization** step. Here, the equalization step takes the auction proceeds and redistributes them to Alex and Ben equally. Knowing that they'll get half of their payment back, each bids twice what they would be willing to pay the other person to have the room. Alex wins the space for a net payment of $25, whereas Ben is excluded from the space but gains $25. Alex obtains the space at a loss of $50 relative to if he hadn't won the space. In cooperative auctions, your bid is the wealth gap you're willing to accept, relative to not winning. If you win, you pay the final bid price and receive your share of redistribution, and your net position relative to non-winners equals your bid. This interpretation of bidder values holds true even as the number of bidders and the number of spaces increase. Try adding more bidders. Since these new bidders are treated as equal community members, they also obtain an equal share of the proceeds. Alex's relative loss of wealth versus the non-winning baseline remains $50 in all cases. Alex and Ben can layer their auction payments on top of their existing rent obligation to determine how much each needs to pay. If their rent is $2,000, an equal split would be $1,000 each. With the adjustments applied, their rent payments are $975 and $1,025. Notice how Ben is assigned no room. This is because the assignment of the smaller room is clear from the auction result. However, we can make this explicit by adding the smaller room as an option in the auction. Try adding it as an additional room. Neither person bids for it, because they have no value for it. But if you assign them a value of zero for the smaller room, you'll see that Ben bids for it only after the price for the bigger room has climbed past his value for it. ## Revaluation People's values for spaces can change over time, so it's best to keep possession time-bounded and hold another auction at the period end. When resources have ongoing value to the user, auctions shouldn't be one-time events, but an ongoing governance mechanism. For roommates, a good cadence could be annual auctions. For example, Ben might decide at some point that he actually values the bigger room more than he's being compensated for. Or Alex might decide that he's overpaying for the bigger room and would want to bid less for it next time. Just because the auction is held again does not mean that Alex and Ben are going to need to switch rooms. It's likely that Alex's value for the room is still higher than Ben's, especially considering how Alex will be willing to bid more for it by the perceived cost of moving, and Ben will bid less by his perceived cost of moving. The switching costs add some stickiness to the allocation, even when the auctions are recurring. ## Unequal Bidder Wealth You might be wondering if equal redistribution gives wealthier members an unfair advantage. Typically, in the status quo, someone just *has* the resource, often by inertia or social dynamics, and no one else gets anything. Compared to this baseline, the compensation that the rest of the community receives from auction settlement is a net gain. Better yet, equality of access is preserved. If people always use their compensation to bid for the resource in later auctions, then they will be able to access their equal share of the resource over time. Wealthier individuals might introduce money into the system at a higher rate, but every auction pushes balances to equalize. The only way access becomes unequal is if someone *chooses* to spend their distributions on something else, which means they preferred that outcome. Nobody is worse off than the no-auction baseline, and most people are better off. Cooperative auctions achieve a net increase in satisfaction because they allow people to give up what is less desired for what is more desired. ## Multi-item Auctions A year later, Alex and Ben take on a third housemate, Cam, and move into a larger place with three bedrooms that differ in size. One is large, one is medium-sized, and one is small. They hold an auction and each person expresses how much they'd pay for each room. Now that there's more than one desirable space, the method of choosing what to bid on starts to matter. In the simulator, bidders are modeled as wanting to get the best deal, defined as the largest difference between the bidders' value for a space and its current price. This difference is called the bidder's **surplus** for a space at a given price. Bidders do not always behave this way, and real ascending auctions have time during each round so bidders can change their values in response to new information or manually place their bids. This flexibility to shift bidding between rooms during the auction is why all the rooms are auctioned together. Separate auctions for each room wouldn't allow the bidders to properly respond to competition. In Round 4 of the auction, Cam switches from bidding on the large room to bidding on the medium room. In this round, bids on the large room are now at $40 (the high bid of $30 plus the bid increment of $10), and this is only $20 of surplus in Cam's eyes, since he sees the large room as worth $60. At the same time, he can still bid for the medium room for $0, and since Cam sees it as worth up to $30, it would provide Cam $30 of surplus. So Cam switches to bidding for the medium room, even though the large room hasn't yet reached Cam's value for it. Ben's values for the large and medium rooms are $110 and $80, and the same thing happens for Ben in round 5 when the price of the large room has reached $40. Ben now has the choice between bidding for it at $50, which gives him a surplus of $60, or bidding for the medium room at a price of $10, which gives him a surplus of $70. The smaller room is a better deal for him. Then in round 6, having been replaced by Ben as the high bidder for the medium room, Cam reevaluates his options. He can choose to bid for the large room at $50 and a surplus of $10, or bid for the medium room at a price of $20 and a surplus of $10 as well. Since these surpluses are equal, the tiebreaker favors the space with the higher value, so Cam bids for the large space, and replaces Alex as the high bidder for it. After that, Alex bids for the large room back, and Cam, no longer interested in the large room at a price of $70, bids for the medium room at a price of $20 and a surplus of $10. Then Ben reclaims the medium room in round 10, and Cam finally bids for the small room, seeing as none of the desirable spaces are cheap enough. The auction concludes after no new bids are placed in round 11. The final allocation is Alex taking the large room for $60, Ben taking the medium room for $30, and Cam taking the small room for $0. The proceeds are $90, and divided equally each person receives a $30 credit. Notice how Ben wins the medium room for $30, which is Cam's value for the room. But when Alex wins the large room, it's only for a value of $60, well below Ben's value of $110. This is because Ben was able to win the medium room for a reasonable price from his perspective, and he dropped out of bidding for the large room before it got as high as his value for it. If you change Cam's value for the medium room to be higher, like $40, you'll see that he bids for the medium room longer, preventing Ben from winning the medium room at the previous price, and leading to an additional round of bidding on the large room, which pushes its price from $60 to $80. As for the final payments, we can see the interpretation of the prices as the relative difference in wealth between winners and non-winners. With payments of $60 and $30, the proceeds are $90 and the equal share is $30. Cam, paying nothing in the auction, gains this payment. Alex pays $60 and gets back $30, for a net loss of $30 and a relative loss of $60 compared to Cam. Ben's $30 payment is exactly offset by the $30 distribution, and his net payment of $0 leaves him with exactly $30 less than Cam. The reason this interpretation of prices is useful is that it holds true regardless of the overall payment level. If you set Cam's value for the medium room to $0, then Ben is able to win the medium room for $0, and the total proceeds are only $60. With an equal share of $20 instead of $30, Alex's net payment rises from $30 to $40. Since Ben and Cam now receive only $20 of credit, Alex's payment still leaves him with exactly $60 less than Ben or Cam. The only thing that changed was the level of competition for the medium room, which changed the overall payment level, but did not affect the fact that the auction prices are the relative difference in wealth between winners and non-winners. This property remains true even as the auctions get big. Here's an example of a big auction with more complex dynamics. ## Competition and Prices In the Alex, Ben, and Cam example, the rooms followed a consistent sequence of desirability. The large room is worth the most, followed by the medium room, and then by the small room. But rooms can have differences other than size, and these differences can affect which room each person values most highly. To the extent that bidders don't compete for the same things, prices do not rise. For example, suppose that Cam actually wants the small room because it's the quietest. Now the only bidding activity is on the large room, and after Ben switches to the medium room, bidding stops. The price on the medium room did not need to exceed zero since there wasn't competition for it. And since it was cheaper, there was less competition for the large room as well. Alex pays $26.67 on net whereas Ben and Cam receive $13.33 on net. If Alex and Ben were the only ones interested in the large room, then why did Cam get paid? One answer is that Cam still has an equal claim to the large room. Another answer is that if we determine distributions based on each bidder's demand, then that creates a strong incentive to falsely over-report valuations. Cam doesn't need to pretend that he's interested in the large room because he knows he'll get a share of its value anyways. ## Equalization with Equal Allowances An alternate option to equal redistribution is to equalize up front by giving each member an equal allowance of internal points. The points are issued out of the community treasury on a consistent schedule and winning bids return these points to the treasury. This method is useful when the community doesn't want to use real money. However, points only retain value when auctions are recurring. Equal redistribution works for one-off auctions since settlement in real money directly resolves the unequal access. But without future auctions to spend points in, the value of points can collapse. For example, a workplace can use points to allocate desks, offices, and conference rooms, with auctions occurring every quarter or year for desks, and every day or week for conference room times. People can save up points to access a high value space when they need it, or spend more consistently on less contested spaces. Here's an example of a desk auction where the community uses "credits" that are issued on a recurring basis. ## The Exposure Problem Ascending auctions are optimal only if bidders don't have preferences that are interdependent between items. As an example of this interdependence, suppose that two coworkers want to sit next to each other, and so bid for adjacent desks. They might initially become high bidders for a pair of desks before facing competition. When outbid, they can switch to bidding for different desks, but this may only happen for one desk at a time. Thus they might only be able to release one of the desks, leaving them with a desk they don't want. In ascending auctions this phenomenon is called the **exposure problem**. One solution in ascending auctions is to combine some items together in bundles that are bid on as a unit. Then the people that want a pair of desks are able to bid for those bundles without the risk of the exposure problem. However, this system isn't perfect. Bidders may want different desks than the ones that are bundled. Or individual bidders may want only one of the desks in the bundle. The community has to decide what bundles best reflect the bidders' preferences. ## Decision Auctions Another application of cooperative auctions is delegating decisions. In decision auctions, the thing available for auction is the right to make a decision on behalf of the group. This decision-making right is like a scarce resource which the community collectively owns. For example, what movie to watch, or what restaurant to eat at, are decisions that can be resolved with cooperative auctions. Those who feel strongest about the outcome bid more for the decision right, and compensate everyone else for denying them the opportunity to have their choice. Compared to voting, decision auctions are fairer in that non-winners of the decision receive compensation for being denied their choice. On the flip side, voting always favors the majority's preferred outcome and leaves the minority without recourse. ## Nonzero Starting Prices In chore auctions, people compete to bid the lowest price to do a task on behalf of the community. Chore auctions set starting prices for items negative, so that winning the auction for the chore means receiving payment from the community to do it. As the auction progresses, payment climbs towards zero, representing a smaller and smaller reward, until there is only one bidder remaining. Setting starting prices to a positive value is also useful if a space has value when unallocated to a particular person. For example, a desk might be day-use by default, unless someone is willing to pay at least some minimum price to reserve it for themselves. Or, a common area can be reserved for events on Friday evenings, but only if a minimum payment is reached. The community can tune this minimum to determine how often the resource gets reserved instead of remaining common. ## Do it Yourself TinyLVT exists to make cooperative auctions easier to use. To run a cooperative auction with your group, see the [setup documentation](/docs). All usage below 50 MB of storage is free. Support for nonzero starting prices is coming soon. ## Terminology and Further Reading An ascending auction for a single item is called an **English auction**. An ascending auction for multiple items is called a **Simultaneous Ascending Auction (SAA)**. When a bidder values a set of items more than the sum of the values of the individual items, their valuation is called **superadditive** (or the items are **complements**). Ascending auctions do not handle superadditive valuations well. The auctions that do are combinatorial auctions, but since the space of solutions grows rapidly with the auction size, they quickly become computationally intractable. The combinatorial auction that preserves truthful bidding as a dominant strategy even when items are complements is the **Vickrey-Clarke-Groves (VCG) auction**. When bidder demand is not superadditive, SAAs approximate VCG auctions. Decision auctions are related to **quadratic voting**, which is a voting system where people can choose where to allocate their voting budget across the issues they care about. The cost of each additional unit of preference expression on an issue scales quadratically, which makes the marginal cost of expressing a strong preference higher than expressing several mild ones. This creates incentives for coalitions to trade votes across issues. Decision auctions instead have a linear cost, so participants do best by bidding on the issues they care about directly. Auction revenue **redistribution mechanisms** are studied for their incentive properties (Cavallo, Guo-Conitzer). The standard construction redistributes counterfactual revenue, which is strategyproof but redistributes zero when the number of bidders equals the number of items plus one. TinyLVT uses simple equal-split instead, which achieves full budget balance at all group sizes at the cost of a minor incentive to over-report: when there's a gap between the highest bidder's value and the second-highest bidder's value, a losing bidder can grow their rebate by bidding in this range, above their true value, as long as they stay below the winner's. The tradeoff favors full redistribution in small communities where budget balance matters more than perfect incentive compatibility.