## 2 Market Model
### Strategic Bidding by Agents  
Agents are self-interested and their types are private information. At the beginning of a trading period, agent \(i\) submits a bid by making a claim about its type \(\hat{\theta}_{i}=(\hat{v}_{i},\hat{q}_{i},\hat{a}_{i},\hat{d}_{i})\
eq\theta_ {i}\) to the auctioneer. An agent's self-interest is exhibited in its willingness to misrepresent its type when this will improve the outcome of the auction in its favor. However, misrepresenting its type is not always beneficial or feasible for the agent.  
Reporting an earlier arrival time is infeasible for both sellers and buyers because the arrival time is the earliest timing at which they decide to sell or buy the goods in the market. Reporting a later arrival time or an earlier departure time can only reduce the chance of successful trades for the agents. For example, even though buyers like to take advantage of sellers' time discounting behavior in perishable goods markets, buyers' belated arrival is not beneficial for them because they might possibly miss seller's low-priced bids that could be matched if buyers bid earlier. Buyers can get better quotes by submitting a low-valued bid as early as possible. In other words, the buyers do not lose their chance of buying at a discounted price by biding low prices without misreporting their arrival time.  
For a seller, it is impossible to report a later departure time \(\hat{d}_{i}>d_{i}\) since the goods to be sold in the market perish by time \(d_{i}\). For a buyer, misreporting a later departure time \(\hat{d}_{j}>d_{j}\) may delay retailing the procured goods.  
As for quantity, it is impossible for a seller to report a larger quantity \(\hat{q}_{i}>q_{i}\) because the sold goods must be delivered immediately after trade in a spot market. Moreover, it is unreasonable for a buyer to report a larger quantity \(\hat{q}_{j}>q_{j}\) because excess orders may produce dead stocks for her.  
In addition, although a seller can misreport a smaller quantity \(\hat{q}_{i}<q_{i}\) with the intention of raising the market price, in that case, she needs to throw out some of her goods already produced for sale. If a buyer misrepresents a smaller quantity \(\hat{q}_{j}<q_{j}\) to lower the market price, she loses a chance of retailing more goods. Although these misreports might create larger profits for sellers and buyers, finding the optimal quantity for increasing their profits is not a straightforward task. Therefore, in this study, we assume that the agents do not misrepresent a quantity value in their type.  
On the other hand, we assume that an agent has incentives to misreport its valuation for increasing its profit because this is the most instinctive way for the agent to influence trades in a market. In a spot market for perishable goods, a seller may report a lower valuation \(\hat{v}_{i}<v_{i}\) when she desperately wants to sell the goods before they perish. In addition, a buyer may report a lower valuation \(\hat{v}_{j}<v_{j}\) to increase her profit when she can take advantage of the sellers' discounted bids.  
Consequently, in this study, we assume that agent \(i\) may report its limit price strategically (i.e., \(\hat{v}_{i}^{t}\) may not be equal to \(v_{i}\)) but reports truthful values of quantity \(q_{i}\), arrival time \(a_{i}\), and departure time \(\hat{d}_{i}\) in any round \(t\).