## 2 Market Model
### Notations  
In our online DA market, we consider discrete time rounds, \(T=\{1,2,\dots\}\), indexed by \(t\). For simplicity, we assume the market is for a single commodity. The market has two types of agents, either sellers (\(S\)) or buyers (\(B\)), who arrive and depart dynamically over time. In each round, the agents trade multiple units of the commodity. The market is cleared at the end of every round to find new allocations. Each agent \(i\) has private information, called _type_, \(\theta_{i}=(v_{i},q_{i},a_{i},d_{i})\), where \(v_{i}\) is agent \(i\)'s valuation of a single unit of the good, \(q_{i}\) is the quantity of the goods that agent \(i\) wants to trade, \(a_{i}\) is the arrival time, and \(d_{i}\) denotes the departure time. The gap between the arrival time and departure time defines the agent's trading period \([a_{i},d_{i}]\), indexed by \(p\), during which the agent can modify the valuation of its unmatched bid. Moreover, agents can repeatedly participate in the market over several trading periods. We model our market as a wholesale spot market. In the market, seller \(i\) submits a bid of her goods at arrival time \(a_{i}\). At departure time \(d_{i}\), the salvage value of the goods evaporates because of its perishability unless it is traded successfully. Seller \(i\) must bring her goods to the market before her arrival. Therefore, seller \(i\) incurs a production cost in her trading period and considers it as valuation \(v_{i}\) of the goods at arrival time \(a_{i}\). Because of advance production and perishability, sellers face the distinct risk of failing to recoup the production cost in the trade. Buyers procure the goods to resell them in their retail markets. Arrival time \(a_{j}\) is the first time when buyer \(j\) values the item. For buyer \(j\), valuation \(v_{j}\) represents her assumed budget to procure the goods. In addition to trade surplus, buyer \(j\) gains profits by retailing the goods if she succeeds in procuring them before her departure time \(d_{j}\), which is the due time for a retail opportunity. Let \(\hat{\theta}^{t}\) denote the set of all the agents' types reported in round \(t\); a complete reported type profile is denoted as \(\hat{\theta}=(\hat{\theta}^{1},\,\hat{\theta}^{2},\,\dots,\hat{\theta}^{t},\, \dots)\); and \(\hat{\theta}^{\leq t}\) denotes the reported type profile restricted to the agents with a reported arrival of no later than round \(t\). In each trading period \(p\), agent \(i\) deals with a new trade and has a specific type \(\theta_{i}^{p}=(v_{i}^{p},\,q_{i}^{p},\,a_{i}^{p},\,d_{i}^{p})\). Report \(\hat{\theta}_{i}^{t}=(\hat{v}_{i}^{t},\,\hat{q}_{i}^{t},\,\hat{a}_{i}^{p},\, \hat{d}_{i}^{p})\) is a bid made by agent \(i\) in round \(t\) within trading period \(p\) (i.e., \(t\in[a_{i}^{p},\,d_{i}^{p}]\)). To be noted is that successful trades in previous rounds of period \(p\) reduce the current quantity of goods to \(q_{i}^{t}\leq q_{i}^{p}\). In the market, a seller's ask and a buyer's bid can be matched when they satisfy the following condition. **Definition 1** (Matching condition). : _Seller i's ask \(\hat{\theta}_{i}^{t}=(\hat{v}_{i}^{t},\,\hat{q}_{i}^{t},\,\hat{a}_{i}^{p},\, \hat{d}_{i}^{p})\) and buyer j's bid \(\hat{\theta}_{j}^{t}=(\hat{v}_{j}^{t},\,\hat{q}_{j}^{t},\,\hat{a}_{j}^{p},\, \hat{d}_{j}^{p})\) are matchable when_  
\[(\hat{v}_{i}^{t}\leq\hat{v}_{j}^{t})\wedge([\hat{a}_{i}^{p},\,\hat{d}_{i}^{p}] \cap[\hat{a}_{j}^{p},\,\hat{d}_{j}^{p}]\
eq\emptyset)\wedge(\hat{d}_{i}^{p}> \hat{d}_{j}^{p})\wedge(\hat{q}_{i}^{t}>0)\wedge(\hat{q}_{j}^{t}>0). \tag{1}\]  
The third term in the matching condition (1) (i.e., \(\hat{d}_{i}^{p}>\hat{d}_{j}^{p}\)) requires that seller's goods must not perish before the buyer's due date. Among matchable bids and asks, the market decides the final allocations based on its mechanism \(M\), which is composed of an _allocation policy_\(\pi\) and a _pricing policy_\(x\).