Fairness in Recurrent Auctions With Competing Markets and Supply Fluctuations (original) (raw)
FAIRNESS IN RECURRENT AUCTIONS WITH COMPETING MARKETS AND SUPPLY FLUCTUATIONS
Javier Murillo, Beatriz López, Víctor Muñoz, Dídac Busquets
The final published version of the paper can be found at:
http://onlinelibrary.wiley.com/doi/10.1111/j.1467-8640.2012.00408.x/abstract
FAIRNESS IN RECURRENT AUCTIONS WITH COMPETING MARKETS AND SUPPLY FLUCTUATIONS
JAVIER MURILLO, BEATRIZ LÓPEZ, VíCTOR MUÑOZ, AND DÍDAC BUSQUETS
Institut d’Informàtica i Aplicacions, Universitat de Girona, Girona, Spain
Abstract
Auctions have been used to deal with resource allocation in multiagent environments, especially in serviceoriented electronic markets. In this type of market, resources are perishable and auctions are repeated over time with the same or a very similar set of agents. In this scenario it is advisable to use recurrent auctions: a sequence of auctions of any kind where the result of one auction may influence the following one. Some problems do appear in these situations, as for instance, the bidder drop problem, the asymmetric balance of negotiation power or resource waste, which could cause the market to collapse. Fair mechanisms can be useful to minimize the effects of these problems. With this aim, we have analyzed four previous fair mechanisms under dynamic scenarios and we have proposed a new one that takes into account changes in the supply as well as the presence of alternative marketplaces. We experimentally show how the new mechanism presents a higher average performance under all simulated conditions, resulting in a higher profit for the auctioneer than with the previous ones, and in most cases avoiding the waste of resources.
Key words: fairness, fair auction, recurrent auctions, multi-unit auctions.
1. INTRODUCTION
Auctions have been widely used in electronic markets because they are a mechanism that buyers and sellers find easy to understand, and also because it eliminates the necessity to set an exact price for products (McAfee and McMillan 1987). This is the case of serviceoriented marketplaces (Lee and Szymanski 2005a) where the seller offers electronic services or resources over the Internet. In an auction, the resource supply and the existing demand determine the prices of products at every moment and, therefore, auctions seem appropriate for these markets.
Our work concerns the use of auctions in domains such as network bandwidth allocations, taking into account the recurrent nature of the market. In this scenario, the owner of the network sells units of bandwidth that agents need. In each run, the owner of the network assigns the units of bandwidth of each link to the agents by means of an auction. The latter send bids to buy a unit of bandwidth and the network owner clears the auction by determining the best assignment according to its objectives.
Agents compete for limited resources and it is assumed that the agents may have different wealth. In this environment, when the allocation of resources is repeated with the same or similar participants, the midterm tendency is for the richer agents to win the resources. Thus, the traditional auctions (TA) could produce the inevitable starvation of certain buyers (Lee and Szymanski 2005b), that may choose to leave the market.
Thus, when poor agents leave, this situation leads the richer bidders to have the power to set the prices of the resources, provoking a fall in prices that can collapse the market in the midterm. Some authors have equipped the traditional mechanisms with reservation prices, 1{ }^{1} which consist in the auctioneer setting a minimum price for the resources and all bids below the reservation price cannot be winners. However this adds a new problem: resource waste. With reservation prices, many resources are not sold, and the auctioneer obtains fewer revenue from the sale of resources.
- Address correspondence to Beatriz López, Institut d’Informàtica i Aplicacions, Universitat de Girona, Campus Montilivi s/n 17071 Girona, Spain; e-mail: beatriz.lopez@udg.edu
1{ }^{1} When we use the term “reservation price” we refer to the minimum price at which the auctioneer is willing to sell a resource. We assume that this value is private to the auctioneer and bidders do not know it. ↩︎
To solve these problems, fairness has been introduced in auctions regarding two main facets: its scope and its use. First, the application of fairness in recurrent auctions can be considered from either a local or a global point of view. A local point of view means treating each auction separately and trying to find a fair solution to each one. With a global point of view we can find that, overall, the solutions of a set of auctions are fair. In this paper, we focus on the global point of view. Second, fairness can be analyzed in light of the agent who is fair. From the point of view of the bidder agent, fairness means that the objective of the agent is not only to maximize benefits in the short term, but also to consider fairness at the time of making decisions. Fairness can benefit the whole society and consequently, the agent may also obtain greater individual benefits. In this case the solution to the allocation of resources is given by the interaction, negotiation, and coalition between agents, etc. (De Jong, Tuyls, and Verbeeck 2008). It is also possible to consider fairness from the point of view of the seller agent. In this case the behavior of bidders could be totally utilitarian and it is the seller agent who imposes fairness to distribute the resources (Lemaître et al. 2003; Muñoz et al. 2007). In this paper we focus on this second case where agents behave selfishly and try to maximize their own profits. Thus, we say that a mechanism is fair when all bidders have a similar probability of winning regardless of their wealth and the auctioneer is using fairness with the aim of increasing its revenue.
Related to this later target, Lee and Szymanski (2005) proposed a method based on agent classification and reservation prices. Particularly, three categories of agents were defined: definite winners, definite losers, and the winner or loser ones. This classification varies from auction to auction and depends on the the prices offered by the bidders and their ability to win according to the current demand. Based on an agent dropout probability model maintained by the auctioneer, resources are assigned to winner agents and winner or loser agents, making a trade off between avoiding resource waste and keeping agents interested in the auction without lowering prices. However, to avoid an excessive decrease in prices, the auctioneer sets a reservation price in each run. The reservation prices change according to an ad hoc mechanism. After Lee and Szymanski’s work, we proposed different methods in our study (Murillo et al. 2008), one of which was also based on reservation prices. In that case, a reinforcement learning method was used to determine the reservation price from run to run, based on the previous reservation prices and the price paid by the bidder. Thus, reservation prices are set one per agent instead of a general value for all the bidders as in Lee and Szymanski’s work.
Both approaches have been tested in static scenarios to demonstrate that fairness is able to reduce the problems of traditional recurrent auctions and avoid market collapse. However, the problems analyzed in recurrent auctions get worse in dynamic scenarios when agents can move among different competing marketplaces and the supply fluctuates. On one hand, if bidders are aware of the opportunity to deal with several firms, they can model the auctioneer’s behavior and leave the market if they know that there are other places with more chances for them. On the other hand, as in real world scenarios, the supply offered by the auctioneer may change, for example, due to technical problems or network failures. Thus, the competition for the resources increases, and richer agents have more opportunities to set the prices.
In this paper the three previous fair mechanisms proposed by the authors (Murillo et al. 2008) are revised and an additional one is proposed. All of the authors’ fair mechanisms, as well as previous works on fair recurrent auctions (Lee and Szymanski 2005a) are then analyzed in dynamic scenarios with variations on resource supply as well as the presence of alternative marketplaces, and compared with unfair mechanisms. From our knowledge, no previous work has performed this analysis of fair mechanisms on such dynamic environments. We empirically investigate the causes of the collapse of the mechanisms (fair and unfair)
under such circumstances, and we show how the new fair mechanism proposed in this paper performs the best.
2. BACKGROUND
This section defines the type of auctions used in this work and the problems that appear when auctions are recurrent.
2.1. Issues in Recurrent Multi-Unit Auctions (RMUA)
Auctions have been widely used as one of the most popular market mechanisms to match supply with demand. They achieve this goal by allowing buyers and sellers to agree on a price of a resource following a set of well-defined rules and procedures (Cramton, Shoham, and Steinberg 2006). In this paper we use the recurrent multi-unit, single-item, sealed-bid auction. Recurrent auctions are a sequence of auctions (of any kind) that are solved individually, but the outcomes of each auction could be influenced by previous auctions (Payne et al. 2006; Lee 2007). In a multi-unit auction the auctioneer offers several (identical) units of the same resource or item. If there is only one item being offered by the auctioneer (regardless of the number of units), the auction is called a single-item auction. For instance, in a network domain, an Internet service provider could auction bandwidth for his clients. A unit may be a communication packet and the total number of units being offered would depend on the bandwidth of the communication channel. This example is multi-unit because the auctioneer offers several units of bandwidth and is single-item because there is only one item auctioned: the bandwidth. Finally, regarding the bidding procedure, a sealed-bid auction is one where the auctioneer is the only agent that knows the bids sent by the bidders (i.e., one bidder does not know what other bidders bid).
In an auction process we can distinguish the following three main components:
- Bidding policies determine how agents decide which items to bid for and the price they are willing to pay for them.
- A market clearing or winner determination algorithm defines how the auctioneer selects the winning bids (i.e., how the items-and their units-are allocated to the agents).
- A pricing mechanism determines how the auctioneer decides the price to be paid by the winners.
Our research is concerned mainly with the second component. The market clearing or winner determination algorithm poses an optimization problem for auctioneers, which try to maximize their revenue (Sandholm 2002). With regard to the bidding policy, we assume that agents have a behavior that tries to maximize their profit using an adaptation strategy (Lee and Szymanski 2005b). This strategy consists in varying the bid price depending on whether they have won or lost in the previous auction, thus trying to get as close as possible to the optimal bid price with which they can obtain the resource. In each auction each bidder sends a single bid for one unit of the resource auctioned. Finally, in terms of pricing, each bidder pays the price it bid (first-price auction).
The winner determination algorithm strategy is conditioned by the kind of resource being auctioned, which can be either static or time sensitive. Static resources do not change their properties during a negotiation process (Chevaleyre et al. 2005), while time-sensitive resources are consumable or perishable (Lee and Szymanski 2005b). A resource is consumable if it is depleted by constant use (e.g., fuel is a consumable resource). A resource is
perishable if it vanishes or loses its value when held over an extended period of time (e.g., network bandwidth is a perishable resource because the bandwidth not used is not accumulable for the future). In this paper, we focus on perishable resources, which introduce some problems in recurrent auctions, as we describe in the next subsection.
2.2. Problems in Recurrent Multi-Unit Auctions
If the auctioneer can keep some of the units being auctioned (by doing so its revenue may be higher), the auction is said to have the free disposal condition. However, this free disposal condition has to be minimized when dealing with perishable resources, as it can produce what is known as the resource waste problem. Other problems related to recurrent auctions that should be dealt with in the market clearing mechanism are the asymmetric balance of negotiation power and the bidder drop problem. Next we describe these problems in detail.
2.2.1. Resource Waste Problem. When perishable resources are used, they cannot be
stored in warehouses for future sales; if the resources are not allocated, they lose their value or vanish completely. This is known as the resource waste problem in recurrent auctions, because if the auctioneer does not sell the resource in an auction, it cannot be sold in the forthcoming auction. On the other hand, the resource cannot be given for free, so a trade-off between the resource usage and the benefit of the auctioneer should be handled appropriately.
2.2.2. Asymmetric Balance of Negotiation Power. In most TA mechanisms, the bid
prices depend only on the customer’s willingness to pay for the traded goods. This means that only the intentions of customers, not those of the auctioneers, are reflected in the auction winning prices (Lee and Szymanski 2006). In the long run, the effect of this problem may cause the auction to collapse. For example, let us suppose that initially there are N bidders. A third of them are poor and bid 1€1 €, while the other two-thirds are richer and bid an amount over 5€5 €. After several rounds, the richer agents start lowering their bids down to 3€3 €, while the poor agents raise their bids to 2€2 €. In the end, the richer agents win with a bid close to that of the poor agents. In this case, the richer bidders have the power to set the price, not the auctioneer. In a recurrent auction, these bids can even go under the lower bids, if the poor agents have dropped out of the market. Note that this problem is different from bidder collusion (Sandholm 1999), although the effects are the same. In bidder collusion, the bidders form coalitions to force this situation, while the asymmetric problem is caused by the uneven wealth distribution of the agents.
2.2.3. Bidder Drop Problem. This problem occurs when bidder agents participating in many auctions always lose. These bidders could decide to leave the market because they are not making any profit. This has bad consequences for the auctioneer: the reduction in the number of bidders gradually decreases the price competition because the probability of winning increases for the remaining bidders. Bidders can decrease their bids without losing the chance to win, provoking the asymmetric balance of negotiation power and the consequent overall drop in bid price. To avoid collapse, some authors (Riley and Samuelson 1981) have introduced a reservation price in the auction. In this case the reservation price maintains the balance of the negotiation but produces resource waste. Note that the bidder drop problem could provoke the asymmetric balance of negotiation power but the asymmetrical balance can also appear in other circumstances without the bidder drop problem.
3. RELATED WORK
The fair auction mechanisms we will present are designed to be fair for a complete sequence of auctions. Although fairness has been studied for a long time in microeconomics, to our knowledge, there are very few previous studies that take into account fairness in a repeated auction process. The discriminatory price optimal recurring auction (DP-ORA) algorithm (Lee and Szymanski 2005a) is an exception. It is based on the supply and demand principle of microeconomics. The mechanism sets a reservation price b0b_{0} in each auction. This value is the maximum between the (2R/3)(2 R / 3) th highest bid value in the current auction and the auctioneer’s minimum desired benefit from the sale of a unit of the resource, where RR is the number of auctioned resources. Then, all bidders with a bid higher than b0b_{0} become winners. The remaining resources are shared between the loser agents following the Valuable Last Loser First Bidder Drop Control (VLLF-BDC) algorithm. This algorithm divides the surplus resources in two phases. In the first phase, the algorithm marks the bidders who lost in the last auction and increased their bid in the current one. Next, the algorithm allocates resources among the highest marked bids. If there are still resources to be allocated, then in the second phase the resources are allocated to the highest nonmarked bids. So, in some sense, the history analyzed by DP-ORA is as if we were considering a time window of length two: the authors analyze the history of the current and previous auctions to decide upon the current egalitarian distribution, and this egalitarian distribution is performed independently of the wealth of the agents. In the methods proposed in this paper, we will see how taking into account a longer history (time window), as well as the agents’ wealth, the mechanisms become more stable under dynamic circumstances.
In light of the agent being fair, (Lemaître et al. 2003), and DP-ORA too, consider bidders behaviors to be totally utilitarian and it is the seller agent who imposes fairness to distribute the resources with the aim of increasing its revenue, as we are also proposing. Particularly, in the earth observation satellite scenario presented in (Lemaître et al. 2003), the decision maker proposes a fair division amongst the agents proportional to their investment in the construction of the observation system. Their study includes a comparison between fairness constraints and efficiency constraints in the use of the satellite with three different models. The authors conclude that there is no best method. In this sense they are demonstrating that there is a compromise between optimality and fairness. From our experiments, we can see that these results can be true in the short term, but in the long run, fair methods also become efficient (they obtain as much revenue as unfair methods). These results lead us to say that being fair does not mean being quasi-rational, but rational in the long term (that is, individually, the agents improve their revenue).
Other studies analyze fairness from the point of view of the bidder agent, instead of the auctioneer’s point of view. That is, the bidders are aware of the process being fair. One of those processes is the envy-free approach. In an envy-free allocation, none of the agents would prefer to exchange its allocated goods with those allocated to another agent (Brams and Taylor 1996). However, this criteria alone is not sufficient (e.g., allocating no resources to any agent would be an envy-free allocation), because the efficiency (in the sense of utilitarian social welfare) is also important. Other studies (Chevaleyre et al. 2007) have shown how such efficient envy-free allocations can be attained through distributed negotiation among the agents. In such a setting, there is no central agent or authority (e.g., the auctioneer) to decide what the optimal allocation is, but the agents themselves perform a sequence of deals (exchanges of resources and payments) to find an efficient and envy-free allocation. The work presented by these authors, however, deals with an isolated allocation problem, and does not mention how distributed negotiation could be applied when the agents are faced with a sequence of allocation problems. In another study (Endriss et al. 2006) the concept
TABLE 1. Classification of Existing Literature on Fair Mechanisms.
Local view (single shot) | Global view (repeated game) | ||
---|---|---|---|
Static | Bidder (distributed) | Endriss et al. (2006) | De Jong et al. (2008) |
Chevaleyre et al. (2007) | Nawa et al. (2002) | ||
Seller (centralized) | Lemaître et al. (2003) | Lee and Szymanski (2005a) | |
Murillo et al. (2008) | |||
Dynamic | Bidder | - | De Jong et al. (2008) |
Seller | - | This work |
of egalitarian social welfare is defined to measure the degree of fairness in this envy-free scenario, as well as for a general purpose. The concept is based on the welfare measure of the agent that is the worst off. This measure is interesting from the perspective of a single-shot auction, but when dealing with a sequence of auctions, other measures that take into account the degree of satisfaction of agents in the long run, like the one we present in this paper, are also required.
Another interesting and related study is based on fairness and human behavior (De Jong et al. 2008). It presents three models for adding fairness to multiagent systems, in which the inequity aversion shown by humans is modeled (i.e., we tend to avoid situations where there are high inequalities among the members of society). Their work can be situated in the group of studies in which fairness is exhibited by the bidders (distributed settings) because there is no central agent coordinating the whole process. Specifically the authors present a model in which each agent is assigned a priority (for instance, according to its wealth), and the agents behave differently depending on the priorities of the other agents (e.g., a rich agent would be willing to give away more money to a poorer agent). These priorities are assigned a priori, and they do not change over time. The authors also address the repetitive aspect in their model, although it is mainly focused on adding punishment mechanisms (agents that do not behave fairly can be punished by the others) and on enabling the use of trust mechanisms to learn who to interact with. They mention that priority updates could be done, but they do not point out how could this actually be performed. In our work, priorities are dynamic, and they vary according to what has happened in the auction history.
Some studies aligned with these human studies are Nawa et al. (2002) and Paparistodemou et al. (2008). The first Nawa et al. (2002) analyzes the effect of risk attitudes in agents and fairness. The authors distinguish three kinds of agents: risk-averse, risk-neutral, and riskloving. All of the agents are trying to maximize their payoffs by using a Q-learning algorithm in a bargaining scenario. The most successful agents, that is, the ones that reach an agreement, are risk-averse agents that propose fair contracts. This confirms, as our results will show, that fair behavior is beneficial in the long term. On the other hand, Paparistodemou et al. (2008) study the effects of fairness and randomness in a repeated game. The authors distinguish two flavors of fairness: apparent and evident. The former is related to opportunities that are regarded as equal for all participants (like tossing dice), while the latter happens when one player takes the control of the game. The mechanisms we propose could be seen as a combination of these two types of fairness. The use of priorities and reservation prices makes the competition fair to all agents, regardless of their wealth (apparent fairness). On the other hand, the auctioneer is the sole agent in charge of applying the fair mechanisms (evident fairness).
Table 1 shows the classification of the previous studies in relation to the local/global point of view (whether fairness is exhibited by the bidder or the seller) and the type of
environment (static or dynamic). As far as we know, there has been no work dealing with dynamic environments in which the auctioneer is the agent being fair. Thus, the work we present in this paper would be a first attempt to handle such a scenario.
4. FAIR MECHANISMS FOR RECURRENT MULTI-UNIT AUCTIONS
To avoid, or somehow decrease, the problems of recurrent multi-unit auctions, the recurrent auction process should have some degree of fairness. We say that the result of a set of auctions is fair if all bidders are satisfied to the same degree. If the agents are satisfied, they remain in the auction, and the collapse of the market is avoided. But we must also take into account the maximization of the revenue of the auctioneer.
In the literature we can find different definitions of fairness in the allocation of resources. Some authors define it as an equal distribution of resources, or as following some preestablished quotas (Lemaître et al. 2003). With this definition, each agent has a resource quota and a fair allocation is one that assigns the resources proportionally to these quotas at a higher degree. Other authors define fairness as all agents being satisfied equally or as equally as possible (Lee and Szymanski 2005b), which is more complete because the objectives of various heterogeneous agents may be different and may be satisfied by obtaining different resources. Also, if the satisfaction of the agent depends on the fulfillment of the quota, the second definition includes the first one. The motivation to use fairness in the second definition is to improve the system performance: for example, use fairness to help a service provider achieve more profit in a bandwidth communication network (Lee and Szymanski 2005a); or use fairness to help coordinate industrial discharges into a river, thereby getting better performance from a wastewater treatment plant (Muñoz et al. 2007). In these scenarios the auctioneer does not apply fairness in an altruistic way; the motivation for the auctioneer is that fairness achieves a medium-long term optimization of the revenue. This is our case.
In this section, we describe four fair mechanisms based on priorities and reservation prices. The mechanisms are presented from the simplest to the most complex. The first three (Priority Auction [PA], Customizable Reservation Price Auction [CRPA], and Customizable Reservation Price with Priorities [CRPAP]) are taken from our previous work (Murillo et al. 2008), although they were only tested in static environments. The fourth one (Dynamic Reservation Price Auction [DRPA]) is presented in this paper for the first time and it has been developed to overcome the limitations of the previous three.
4.1. Priority Auction (PA)
The motivation behind PA is to use priorities to prevent the starvation of poor bidders and their dropping out of the auction. The priority assigned to each bidder will increase the probability of poor bidders winning, and decrease the probability of the richest ones winning. This mechanism takes into account the history of each agent in previous auctions. Each agent is assigned a priority value depending on the number of won and lost auctions. The more lost auctions, the higher the priority. The priority values are updated after each auction, and they are used to clear the next auction. Because the history of the agents in a recurrent auction scenario is long, a time window could be used to calculate the priorities.
Formally, we define the priority of agent ii as follows
pi=∑j=N−T+1N(1−xi,j)Tp_{i}=\frac{\sum_{j=N-T+1}^{N}\left(1-x_{i, j}\right)}{T}
where xi,jx_{i, j} is the outcome of the jj th auction for agent ii ( 1 if the bid of the agent was a winner and 0 otherwise), NN is the total number of auctions, and TT is the length of the time window.
The clearing algorithm could then use these priorities in very different ways: transforming them into new constraints to be satisfied by the solution, letting them directly designate the set of winning agents, among others. We propose using the priorities to modify the value of the bids, and then select as winners the highest modified bids. More precisely, given a bid value viv_{i} of an agent with priority pip_{i}, a new bid valuation is computed as
vi′=f(vi,pi)v_{i}^{\prime}=f\left(v_{i}, p_{i}\right)
The priority is handled by the auctioneer, and this new value vi′v_{i}^{\prime} is the one used by the clearing algorithm to find an optimal solution. Note, however, that the winning bidders pay the original viv_{i} price. The function ff can be designed in many ways, and it allows the introduction of different fairness facets in the auction solution. Thus, the function should increase the chances of winning for a high priority agent, and decrease the chances of a low priority one. For example, we are currently using the product function as follows
vi′=f(vi,pi)=vi×piv_{i}^{\prime}=f\left(v_{i}, p_{i}\right)=v_{i} \times p_{i}
This priority-based mechanism is not strategy proof. That is, if the bidders are aware of how the winner determination algorithm works, they can manipulate it. However, we assume that bidders are honest in the sense that they change their bids only in response to the new prices, not to cheat and take advantage of the mechanism. This is a strong assumption that is valid for all the methods proposed in this paper. We are aware that we should address this issue and include mechanisms to prevent cheating behaviors. However, we will leave it for the next stage of our research.
The PA mechanism does not produce any resource waste as it always sells all the available units and, what is more, it reduces the effect of the bidder drop problem thanks to the use of priorities. However, this mechanism has a drawback: the revenue of the auctioneer can be reduced by selling resources at low prices.
4.2. Customizable Reservation Price Auction (CRPA)
The motivation for CRPA is to keep prices at an acceptable level. Some authors have suggested the use of a reservation price to accomplish this goal (Riley and Samuelson 1981), but in an environment where agents have different wealth it is not fair to have the same reservation price for all of them. CRPA proposes a different reservation price for each bidder. In this mechanism the reservation price of an agent ii is defined as the minimum price at which the auctioneer is willing to sell a good or service to the agent ii.
That means that the auctioneer does not accept any agent bid under its reservation price. The reservation price is initially the same for all the bidders, but it gradually varies as auctions are held. For each agent, if a bid price is higher than its reservation price, the reservation price for that agent is increased because the agent shows a higher willingness to buy the resource. Otherwise, if the reservation price is higher than the bid price, it decreases. A parameter γ∈[0,1]\gamma \in[0,1] is defined by the auctioneer to indicate the minimum percentage of increase and decrease of the reservation price. When bidder ii bids with a value higher than its reservation price, then its reservation price is incremented by half of the difference between the reservation price and the bid’s value, except if the difference is lower than γ×\gamma \times reservationPrice i{ }_{i}. In this case, the reservation price is incremented in γ×\gamma \times reservationPrice i{ }_{i} units. These reservation prices are private to the auctioneer. The procedure is shown in Algorithm 1.
Algorithm 1. updateReservationPrice(reservation Price \(_{i}\), bid \(_{i}\), \(\gamma)\)
minimum \(=\) reservation Price \(_{i} \cdot \gamma\)
difference \(=\) abs(bid \(_{i}-\) reservation Price \(_{i})\)
if bid \(_{i} \geq\) reservation Price \(_{i}\) then
reservation Price \(_{i}=\) reservation Price \(_{i}+\max (\) difference \(/ 2\), minimum \()\)
else
reservation Price \(_{i}=\) reservation Price \(_{i}-\max (\) difference \(/ 2\), minimum \()\)
end if
This mechanism is egalitarian because everybody can win or lose irrespective of its wealth. In addition, it prevents bidders with high wealth from reducing their price as low as possible to win, and it forces them to increase it to a minimum reservation price. Thus, it solves the problem of the asymmetric balance of negotiation power. However, the use of reservation prices produces resource waste as all the available resources are not always allocated. This mechanism learns the behavior of the agents and appreciates the effort being made to obtain the resources. Although agents do not know the reservation price that the auctioneer assigns to them, they can infer that they won by maintaining the bid value at an acceptable level within their capabilities. Of course, this requires the assumption that bidders are honest, as stated in the previous method.
4.3. Customizable Reservation Price with Priorities (CRPAP)
A way of avoiding the resource waste of the previous mechanism is to distribute the remaining resources among the nonwinning bidders. To do so, we propose a two-phase mechanism. In the first phase the resources are allocated following the same rules as in CRPA. In the second phase, if there are still nonallocated resources, the mechanism gives the resources to bidders with higher priority, calculated as in PA, without considering their bid prices. This second phase eliminates the resource waste problem and improves the level of fairness of the solutions. This mechanism is a combination of the CRPA and the PA mechanisms, because it uses the individual variable reservation price and the priority mechanism explained earlier.
4.4. Dynamic Reservation Price Auction (DRPA)
When the available resources are always the same, the previous fair mechanism (CRPAP) works correctly. However, if the supply varies, for example, due to technical malfunctions in a network communication scenario, the CRPAP mechanism could also collapse. That is, if there are fewer resources, and the bidders are the same, the number of loser agents increases. This leads to a decrease in the corresponding reservation prices at the same time that their priorities are increased, and then the auction may collapse. For example, Figure 1 shows a dynamic environment where there is a decrease of resources at time step 1000. When the supply is reestablished at time step 1100, the markets either collapse or decrease their performance (for more details see Section 5). This problem is due to priorities. In theory, priorities help poor agents, but when the supply varies, priorities can also favor medium and rich agents since a lot of poor bidders have disappeared from the market when the supply went down. When the supply recovers, medium and rich agents are not motivated to maintain
Figure 1. Average bidding price in dynamic markets: xx-axis: sequence of auctions; yy-axis; average price.
the prices at an acceptable level because they can win the resources in the fair distribution even with low priorities. In this situation, bidders have the negotiation power and the market collapses.
To avoid these problems we propose a mechanism called DRPA. It consists in combining the previous fair mechanisms (based on reservation prices and priorities) and adding a minimum priority value that avoids the collapse of the auctions when the supply varies.
In the first phase the resources are allocated following the same rules as in CRPA. In the second phase of the algorithm, the idea is to give the resources to bidders with higher priority, but only if their priority is higher than a predefined threshold (minimum Priority). This parameter prevents bidders with low priorities (that have won resources recently) and low bids from obtaining the resources, thereby avoiding the drop in prices. Algorithm 2 shows the pseudo-code of this mechanism.
Algorithm 2. DRPA (N: number of resources, B: bids)
Mark bids with a price higher than the reservation price of its sender
Sort the marked bids by price in descending order into \(\vec{P}\)
Select as winners the \(K\) first bids of \(\vec{P}\), where \(K=\min (N,|\vec{P}|)\)
if there are still available resources \((K<N)\) then
Mark bids of bidders with a priority higher than minimum Priority
Sort the marked bids by priority of its sender in descending order into \(\vec{D}\)
Select as winner the \(A\) first bids of \(\vec{D}\) where \(A=\min (N-K,|\vec{D}|)\)
end if
Update reservation prices according to the procedure of Algorithm 1
Update priorities according to Equation 1
5. EXPERIMENTATION
To evaluate the performance of the mechanisms presented earlier, we have experimented in four different scenarios
- First scenario: In this scenario the same framework used in other related work (Lee and Szymanski 2005b; Murillo et al. 2008) has been replicated. The supply is constant, and no other marketplaces are available.
- Second scenario: In the second scenario, we have done two experiments, adding variations in the supply of the resource to simulate dynamic environments.
- Third scenario: In the third scenario the resource supply is constant as in scenario 1 but we have changed the way bidders drop out: if a bidder decides to leave its provider, it joins the market of another, randomly selected auctioneer instead of disappearing from the system.
- Fourth scenario: The fourth scenario extends the third scenario. In this case bidders use trust and reputation information to choose their next provider.
In each scenario, the eight auctioneers are executing their auctions in parallel and each auctioneer has an initial population of 100 bidders. In the two former scenarios, when a bidder leaves its provider, it disappears from the simulation. On the other hand, in the last two scenarios, when a bidder leaves its provider it can join the market of any other provider. It is therefore likely that some providers will finish with more participants than those they started with. Hence, in these scenarios the different auctioneers (and thus, the different mechanisms) are competing against each other to attract more bidders.
To compare their performances, we have used the total revenue of the auctioneers during the recurrent auction, because their ultimate goal is to maximize their revenue. Total revenue is then analyzed regarding how fair the mechanisms are, because our hypothesis is that fairness improves revenue in the mid-long term. As secondary metrics, we have also used the percentage of resource waste and the number of bidders that remain in the auction at the end of the recurrent auction, to evaluate to what extent we are solving these problems of recurrent auctions, while improving revenue. Finally, in the last scenario we have also taken into account the number of recommendations that agents have done about each auctioneer, meaning that more recommendations is an indicator of the mechanism being beneficial for more bidders.
Thus, our scenarios evolve from static, simple environments, to more realistic ones. In the remainder of this section, we explain the experimental setup of each auction, a measure to compare the fairness of the mechanisms, and the different results obtained in each of the scenarios.
5.1. Experiment Setup
To test the proposed mechanisms, we have used and extended the experimentation framework provided by Lee and Szymanski (2005b) in which recurrent auctions are used to deal with e-service networking markets. This framework corresponds exactly to multi-unit single-item recurrent auctions (see Section 2.1) where the auctioneer offers several units of one resource and bidders bid for one unit of the resource at each auction. The recurrent auction is formed by 2000 multi-unit auctions.
There are eight auctioneers, each one with a different auction mechanism for the allocation of resources (four mechanisms are those explained in Section 4 and the others
are those explained later), offering 50 units of resources (i.e., perishable e-service units) in each auction (except in scenario 2 where the supply varies). Each auctioneer initially has 100 customers (bidders) assigned but this quantity can change over time due to the arrival or departure of bidders.
To compare the results of the proposed mechanisms, the following auction mechanisms have been considered
- DP-ORA: This is a fair mechanism (Lee and Szymanski 2005a) that sets a reservation price in each auction. Then, all bidders with a bid higher than this reservation price become the winners. 2{ }^{2} The remaining resources are shared between the loser agents according to the algorithm explained in Section 3. To our knowledge, this is the only recurrent auction mechanism in the literature that takes into account fairness. Our mechanisms should improve the results of the previous work.
- TA: In the TA mechanism the winners are the bidders with the highest bids. This is the simplest auction mechanism and it is used as a baseline approach to compare all the mechanisms. Our proposed mechanisms should work at least as well as this simple method.
- Cancelable Auction (CA): In a CA, if the resulting revenue of an auction does not meet the minimum requirements of the auctioneer, the entire auction is canceled. Thus, the cancelation of an auction wastes the entire stock of resources, making this the mechanism that is expected to waste the largest amount of resources. Our proposed mechanisms should reduce the resource waste of CA.
- Reservation Price Auction (RPA): In an RPA the auctioneer defines a reservation price (the same for all bidders) that indicates the minimum price the bidders should pay. Only bids higher than the auctioneer’s reservation price are considered during the winner selection. In an RPA, the reservation price restricts the number of winners and can waste part of the resources. This mechanism also defines a trade-off between revenue and resource waste but from a utilitarian point of view. We expect that the mechanisms proposed here will improve the results of RPA, by increasing revenue and decreasing resource waste through the use of fairness.
Regarding bidders, the initial bidding price is randomly selected from a uniform distribution over the range [ti/2,ti]\left[t_{i} / 2, t_{i}\right], where tit_{i} represents the upper bound on the willingness of customer ii to pay (or wealth). Based on the assumption that each bidder tries to maximize its expected profit and is honest, the following bidding policy has been considered. If a bidder has lost in the previous auction round, it increases its bidding price by a factor of α>1\alpha>1 to increase its probability of winning in the current round. The increase in the bidding price is limited by the upper bound of the bidder’s willingness to pay. If a bidder won in the last auction round, then with equal probability of 0.5 , it either decreases the bidding price by a factor β\beta or leaves it unchanged. The decrease attempts to maximize expected profit. The parameters α\alpha and β\beta are set in the experiments to 1.2 and 0.8 , respectively, because they are the same values used in Lee and Szymanski (2005a). In each auction bidders want to acquire 1 unit of the resource, therefore each bidder will send only one bid composed of one resource unit and a price between the 10%10 \% of its wealth and its wealth. To model bidder drop out, a tolerance of consecutive losses (TCL) value has been defined for each bidder. The TCL denotes the maximum number of consecutive losses that a bidder can tolerate before
- 2{ }^{2} Note that in the DP-ORA the reservation price is the maximum between the (2R/3)th highest bid, where R is the number of auctioned resources, and the auctioneer’s minimum desired revenue. Therefore it is not possible to have more bids with a price higher than the reservation price than items. ↩︎
dropping out of an auction. The TCL value of each customer is uniformly distributed over the range [2,10][2,10]. Thus, each bidder is modeled with a TCL value and a wealth. These values are private and unknown to the auctioneer.
Thus, each of the eight auctioneers of the simulation follows a different mechanism: there are five fair mechanisms (PA, CRPA, CRPAP, DRPA, DP-ORA) and three unfair mechanisms (TA, CA, RPA). From this point on, we will refer to the mechanisms by their abbreviations and we will add a superscript F for fair methods and N for unfair methods to make the reading easier. Because we claim that the use of fairness helps improving the performance of the mechanisms, we next define a quantitative measure of fairness, so that we can compare the different mechanisms taking into account the fair property.
5.2. Fairness Measure
The satisfaction of an agent is somewhat abstract and relative, as not all agents have the same needs and are not satisfied in the same way. For example, a bidder getting one resource out of four auctions may be satisfied with that, while another agent may need to win the resource in three out of four auctions to be satisfied. If both bidders won two resources out of four auctions, the first agent would be satisfied while the second would not. Thus, a metric should be defined according to the different preferences of the individual agents in the market. The aggregation of individual preferences can be modeled using the notion of social welfare (Chevaleyre et al. 2005). Welfare engineering addresses how to define the appropriate criteria and social mechanisms so that resource allocations converge to the optimal social criteria. There are different measures proposed including the satisfaction of the minimum needs of a large number of agents, fair division, leximin ordering, and envy freeness (see some of them in the related work section and a good summary of all of them in Chevaleyre et al. (2005)). However, the measures studied so far deal with a one-shot allocation process instead of a recurrent scenario.
We need to define a new measure that takes into account the sequence of winning and losing bids of an agent in a series of auctions. This measure must provide an aggregation of bidder satisfaction values of each auction round.
First, we propose a measure of the satisfaction degree of an agent in a single auction rr as
s(a,r)=xa,rs(a, r)=x_{a, r}
where, xa,rx_{a, r} is equal to 1 if the bid of agent aa won in auction r,0r, 0 otherwise (analogously to Equation 1). Recall that auctions are sealed bid and a bidder does not know the bids of the other bidders. We can therefore assume that there is no envy among the bidders. Next we define the satisfaction level of an agent aa in the auction sequence RR as the average of the satisfaction degree on each auction, as follows
s(a,R)=1∣P(R,a)∣×∑∀r∈P(R,a)ω(t(r),T)×s(a,r)s(a, R)=\frac{1}{|P(R, a)|} \times \sum_{\forall r \in P(R, a)} \omega(t(r), T) \times s(a, r)
where P(R,a)P(R, a) is the set of auctions belonging to RR where agent aa has participated, t(r)t(r) is the time when auction rr took place, ω(t(r),T)\omega(t(r), T) is a forgetting function to give less weight to older auctions and TT is the time window size.
Once we have a measure for each agent, the level of fairness of an auctioneer auc after a sequence of auctions can be defined as the inverse of the standard deviation of the average
Figure 2. Average bidding price in scenario 1 (Uniform wealth distribution, 1 simulation). For the sake of clarity, the results are split into two plots. Left-hand side: CAN,CRPAF,DP−ORAF,RPAN\mathrm{CA}^{N}, \mathrm{CRPA}^{F}, \mathrm{DP}-\mathrm{ORA}^{F}, \mathrm{RPA}^{N}. Right-hand side: TAN,PAF,CRPAPF,DRPAF\mathrm{TA}^{N}, \mathrm{PA}^{F}, \mathrm{CRPAP}^{F}, \mathrm{DRPA}^{F}.
satisfaction of agents belonging to the auctioneer, as follows
fairness(auc)=1/1∣A(Rauc)∣−1×∑∀a∈A(Rauc)(s(a,Rauc)−s(Rauc)‾)2\operatorname{fairness}(a u c)=1 / \sqrt{\frac{1}{\left|A\left(R_{a u c}\right)\right|-1} \times \sum_{\forall a \in A\left(R_{a u c}\right)}\left(s\left(a, R_{a u c}\right)-\overline{s\left(R_{a u c}\right)}\right)^{2}}
where RaucR_{a u c} is the set of auctions performed by auctioneer auc,A(Rauc)a u c, A\left(R_{a u c}\right) is the set of agents that have participated in any auction of the RaucR_{a u c} set, and s(Rauc)‾\overline{s\left(R_{a u c}\right)} is the average of the degree of satisfaction of the agents, defined as follows
s(Rauc)‾=∑∀a∈A(Rauc)s(a,Rauc)∣A(Rauc)∣\overline{s\left(R_{a u c}\right)}=\frac{\sum_{\forall a \in A\left(R_{a u c}\right)} s\left(a, R_{a u c}\right)}{\left|A\left(R_{a u c}\right)\right|}
With this measure, fairness(auc), we quantify how fair the different mechanisms are. A mechanism that treats all the bidders equally fair would have a high fairness(auc) value, independently of the amount of items being awarded by the auctioneer and the revenue obtained. Auctioneers look to maximize their revenue and the methods we propose, in particular, use fairness to achieve this goal. But their decision making does not take into account the evaluation measure defined in equation 6 . As said before, this measure is only used to compare the fair feature of the different mechanisms.
5.3. Results in the First Scenario: No Dynamics
In this scenario the number of resources in each auction is constant ( 50 units) and when a bidder ii loses more than TCLiT C L_{i} consecutive auctions, it drops out of the auction and disappears from the system. Figure 2 shows the average resource price at which each of the eight auctioneers has sold the resources during the 2000 auctions of one simulation. The xx-axis represents the temporal sequence of the auctions, and the yy-axis is the average price, which is the total benefit divided by the 50 units of resource, irrespective of whether all the resources have been sold or not. The wealth of bidders has been distributed following a
TABLE 2. Results for All Mechanisms with a Uniform Distribution of Wealth in Scenario 1 (No Dynamics). Averaged Results of 10 Simulations with the Standard Deviation in Brackets.
Mechanism | Total Revenue | % Resource Waste | Number of Bidders | |
---|---|---|---|---|
Fair | PAF\mathrm{PA}^{F} | 263226.26(53303.95)263226.26(53303.95) | 0.00%(0.00)0.00 \%(0.00) | 81.80(1.03)81.80(1.03) |
CRPAF\mathrm{CRPA}^{F} | 501216.90(38955.67)501216.90(38955.67) | 10.50%(2.03)10.50 \%(2.03) | 81.50(2.01)81.50(2.01) | |
CRPAPF\mathrm{CRPAP}^{F} | 495684.78(6798.01)495684.78(6798.01) | 0.00%(0.00)0.00 \%(0.00) | 91.10(1.10)91.10(1.10) | |
DRPAF\mathrm{DRPA}^{F} | 503950.25(21380.26)503950.25(21380.26) | 0.00%(0.00)0.00 \%(0.00) | 90.20(1.13)90.20(1.13) | |
DP−ORAF\mathrm{DP}-\mathrm{ORA}^{F} | 434270.21(25110.21)434270.21(25110.21) | 0.00%(0.00)0.00 \%(0.00) | 94.00(1.33)94.00(1.33) | |
Unfair | TAN\mathrm{TA}^{N} | 94585.65(6558.26)94585.65(6558.26) | 0.00%(0.00)0.00 \%(0.00) | 69.30(1.49)69.30(1.49) |
CAN\mathrm{CA}^{N} | 355414.14(39840.06)355414.14(39840.06) | 34.24%(7.15)34.24 \%(7.15) | 54.40(4.97)54.40(4.97) | |
RPAN\mathrm{RPA}^{N} | 336669.83(42836.59)336669.83(42836.59) | 38.26%(7.86)38.26 \%(7.86) | 50.50(6.43)50.50(6.43) |
uniform distribution in [2,10][2,10]. The results have been drawn in two plots for clearer readability. The numerical data is shown in Table 2, where the total revenue, the resource waste produced by each mechanism, and the number of bidders remaining with the auctioneer at the end of 10 simulations are also shown. The table values are the average of 10 simulations with the standard deviation in brackets.
In Figure 2 we can see how TAN\mathrm{TA}^{N} (right plot) collapses very quickly. Table 2 shows that this is due to the bidder drop problem (it has only 69.3 bidders at the end of the simulation). This drop of bidders means that the remaining bidders have the power to set the price and influence the consequent fall of prices. Thanks to the reservation prices, the RPA N{ }^{N} and CAN\mathrm{CA}^{N} mechanisms 3{ }^{3} can maintain the price at a higher level than TAN\mathrm{TA}^{N} as they are not affected by the asymmetric balance of negotiation power. However, the reservation prices produce a resource waste of 34.24%34.24 \% and 38.26%38.26 \% of the resources, respectively, and these mechanisms are also affected by the bidder drop problem ( 54.4 and 50.5 bidders, respectively, at the end of simulations). The mechanism CRPAF\mathrm{CRPA}^{F}, as well as RPAN\mathrm{RPA}^{N} (recall the difference between these two methods: CRPAF\mathrm{CRPA}^{F} has an adaptable reservation price for each bidder, while RPAN\mathrm{RPA}^{N} has a unique reservation price for all bidders) maintain the balance of negotiation power and the incentive for bidders to bid with higher prices, thereby achieving higher total revenue. CRPAF\mathrm{CRPA}^{F} also maintains a high number of bidders in the auction ( 81.5 bidders) due its fair behavior, but produces some resource waste (but less than CAN\mathrm{CA}^{N} and RPAN\mathrm{RPA}^{N} ). Other fair methods, such as DP−ORAF,CRPAPF\mathrm{DP}-\mathrm{ORA}^{F}, \mathrm{CRPAP}^{F}, and DRPAF\mathrm{DRPA}^{F}, show similar behaviors as those of CRPAF\mathrm{CRPA}^{F}, but without resource waste. They maintain the prices at an acceptable level and maintain a high level of bidders at the end of the simulations. Finally, PAF\mathrm{PA}^{F} is the only fair mechanism that is affected by the asymmetric balance of negotiation power, and this causes a price decrease that will collapse in the long term. Observe, however, that the asymmetric balance of negotiation power is due to the particularities of our experimental scenario: there are 50 units of resources and less than 100 bidders. Then, when some of the poorest bidders have dropped out of the auction, more than 50%50 \% of the bidders become winners; therefore the number of winning bidders is significantly higher than losers. Winning bidders are then lowering prices instead of raising them, making the average price fall at long-mid term.
Figure 3 shows the average and the standard deviation of total revenue earned in 2000 auctions for each auctioneer in 10 simulations, as well as the fairness level achieved by each
- 3{ }^{3} The CAN\mathrm{CA}^{N} mechanism is displayed as a gray area because whenever an auction is canceled, the prices fall down to zero, and so when plotting the results there are lots of lines reaching the xx axis. ↩︎
Figure 3. Total revenue and fairness obtained by all mechanisms in scenario 1 (average and standard deviation of 10 simulations).
Table 3. Results for Fair Mechanisms with a Exponential Distribution of Wealth in Scenario 1 (No Dynamics). Averaged Results of 10 Simulations with Standard Deviation in Brackets.
Mechanism | Total Revenue | % Resource Waste | Number of Bidders | |
---|---|---|---|---|
Fair | PAF\mathrm{PA}^{F} | 134113.39(17848.66)134113.39(17848.66) | 0.00%(0.00)0.00 \%(0.00) | 77.20(1.61)77.20(1.61) |
CRPAF\mathrm{CRPA}^{F} | 489161.87(28122.32)489161.87(28122.32) | 17.35%(2.78)17.35 \%(2.78) | 76.10(2.47)76.10(2.47) | |
CRPAPF\mathrm{CRPAP}^{F} | 511654.82(32149.30)511654.82(32149.30) | 0.00%(0.00)0.00 \%(0.00) | 90.80(0.92)90.80(0.92) | |
DRPAF\mathrm{DRPA}^{F} | 492633.98(30998.73)492633.98(30998.73) | 0.00%(0.00)0.00 \%(0.00) | 90.40(0.84)90.40(0.84) | |
DP−ORAF\mathrm{DP}-\mathrm{ORA}^{F} | 293116.79(18438.55)293116.79(18438.55) | 0.00%(0.00)0.00 \%(0.00) | 96.20(3.58)96.20(3.58) | |
Unfair | TAN\mathrm{TA}^{N} | 96529.49(6234.67)96529.49(6234.67) | 0.00%(0.00)0.00 \%(0.00) | 61.80(1.39)61.80(1.39) |
CAN\mathrm{CA}^{N} | 325410.58(3572.25)325410.58(3572.25) | 39.87%(0.60)39.87 \%(0.60) | 50.30(0.67)50.30(0.67) | |
RPAN\mathrm{RPA}^{N} | 228621.02(18923.73)228621.02(18923.73) | 58.23%(3.45)58.23 \%(3.45) | 33.90(2.84)33.90(2.84) |
mechanism. Fairness is expressed according to our satisfaction measure (see equation 6). In general, fair mechanisms have better performance than unfair ones. The four mechanisms with a higher revenue are also those with a higher fairness level: DRPA F,CRPAF,CRPAPF{ }^{F}, \mathrm{CRPA}^{F}, \mathrm{CRPAP}^{F}, and DP-ORA F{ }^{F}. Thus, fairness provides highest revenue in the long term.
Tables 3 and 4 show the results obtained by the mechanisms in scenario 1 when the wealth of agents follows an exponential distribution (this means few rich agents and many poor ones) and a Gaussian distribution (few rich, few poor, and many agents in the middle class), respectively. The results are similar to those obtained with the uniform distribution.
TABLE 4. Results for Fair Mechanisms with a Gaussian Distribution of Wealth in Scenario 1 (No Dynamics). Averaged Results of 10 Simulations with Standard Deviation in Brackets.
Mechanism | Total Revenue | % Resource Waste | Number of Bidders | |
---|---|---|---|---|
Fair | PAF\mathrm{PA}^{F} | 468783.48(29296.04)468783.48(29296.04) | 0.00%(0.00)0.00 \%(0.00) | 86.90(1.96)86.90(1.96) |
CRPAF\mathrm{CRPA}^{F} | 482505.96(6383.20)482505.96(6383.20) | 8.78%(0.97)8.78 \%(0.97) | 82.70(0.82)82.70(0.82) | |
CRPAPF\mathrm{CRPAP}^{F} | 499247.06(4932.84)499247.06(4932.84) | 0.00%(0.00)0.00 \%(0.00) | 90.20(1.03)90.20(1.03) | |
DRPAF\mathrm{DRPA}^{F} | 502299.17(4909.11)502299.17(4909.11) | 0.00%(0.00)0.00 \%(0.00) | 90.40(0.97)90.40(0.97) | |
DP−ORAF\mathrm{DP}-\mathrm{ORA}^{F} | 485362.19(9632.47)485362.19(9632.47) | 0.00%(0.00)0.00 \%(0.00) | 95.00(0.94)95.00(0.94) | |
Unfair | TAN\mathrm{TA}^{N} | 143687.43(69252.21)143687.43(69252.21) | 0.00%(0.00)0.00 \%(0.00) | 77.80(2.29)77.80(2.29) |
CAN\mathrm{CA}^{N} | 354506.32(30767.49)354506.32(30767.49) | 33.64%(5.91)33.64 \%(5.91) | 57.40(5.25)57.40(5.25) | |
RPAN\mathrm{RPA}^{N} | 35082.90(3950.66)35082.90(3950.66) | 35.08%(3.95)35.08 \%(3.95) | 53.90(3.95)53.90(3.95) |
Figure 4. Average number of bidders in scenario 1 (Uniform wealth distribution, 10 simulations). At right the result of the 2000 auctions, at left the detail of the 100 first auctions.
Note, however, that with the exponential distribution of wealth, CRPAF\mathrm{CRPA}^{F} produces more resource waste, so its performance is slightly affected, as is that of DP-ORA F{ }^{F}.
Figure 4 shows the evolution of the average number of bidders in 10 simulations with each of the methods during 2000 auctions in the right plot and the detail of the 100 first auctions in the left plot. The number of bidders stabilizes during the first 20 auctions and from then on changes are much smaller and stable. The plots shows the difference between fair methods ( DRPAF,CRPAPF,CRPAF,PAF\mathrm{DRPA}^{F}, \mathrm{CRPAP}^{F}, \mathrm{CRPA}^{F}, \mathrm{PA}^{F}, and DP−ORAF\mathrm{DP}-\mathrm{ORA}^{F} ) with an average value higher than 80 bidders and unfair ones (TAN,CAN\left(\mathrm{TA}^{N}, \mathrm{CA}^{N}\right., and RPAN)\left.\mathrm{RPA}^{N}\right) with a value less than 70 . In this domain the stabilization of bidders causes a stabilization of prices (Figure 2) without changes to the supply. The bidders that remain in the system pressure the market at a constant rate, so revenue is also stabilized.
Finally, Figure 5 shows how bidders drop out of the different mechanisms, depending on their tolerance to losses (TCL). The first thing to note is that unfair mechanisms ( CAN\mathrm{CA}^{N}, TAN\mathrm{TA}^{N}, and RPAN\mathrm{RPA}^{N} ) are those with a higher dropout degree, as quantitative results of Table 2 corroborate. In general, and as it could be expected, the lowest the value of TCL, the highest the frequency of dropping out of an auction. Particularly, all fair mechanisms follow a skewed right distribution pattern. In unfair methods, however, the wealth of bidders has a
Figure 5. Average number of bidder drops in 10 simulations depending on the bidder’s tolerance (TCL).
huge importance, and poor bidders leave the auction even if they have a high TCL. For this reason the pattern is less skewed than for fair methods.
As a summary, the results of this scenario indicate that the mechanisms that incorporate fairness and reservation prices minimize the problems of recurrent multi-unit auctions. Fairness gives bidders incentives to stay in the auction, and reservation prices help maintain the equilibrium in the negotiation power. In addition, this combination maintains resource prices at a high level. However, this scenario is static: in a real environment there are other factors that can change the nature of the auctions. For this reason, we introduce some of these changes into the scenario.
5.4. Results in the Second Scenario: Variation of Supply
Reality is not static, and in a bandwidth communication domain it is feasible to expect malfunctions, technical problems with the network, and maintenance issues that decrease the available supply. To model this we have performed two experiments to vary the supply of the auctioneers. The first one consists of a sudden reduction of the supply. The initially available supply is set to 50 units, but it is reduced from 50 to 30 units at time step 1000, and then it is restored back to 50 units at time step 1100. From that time onwards, it does not change anymore. In the second experiment, the resource quantity intermittently oscillates between 40 and 60 units. The supply is initially set to 40 units, and it is gradually increased, reaching 60 units at time step 500 . Then the supply gradually decreases, arriving to 40 units again at time step 1000. This oscillation is repeated every 1000 auctions.
Figure 6 shows the average price in one simulation of the first experiment. We can see how prices rise for all methods during the time with scarce units of resource (less supply,
Figure 6. Average bidding price in scenario 2 with a temporal reduction of supply (Uniform distribution of wealth, 1 simulation). Left-hand side: CAN,CRPAF,DP−ORAF,RPAN\mathrm{CA}^{N}, \mathrm{CRPA}^{F}, \mathrm{DP}-\mathrm{ORA}^{F}, \mathrm{RPA}^{N}. Right-hand side: TAN,PAF,CRPAPF\mathrm{TA}^{N}, \mathrm{PA}^{F}, \mathrm{CRPAP}^{F}, DRPA F^{F}.
prices go up). During this period all methods suffer the drop of the poorest bidders, and as a consequence, some of the mechanisms are not able to return to their same or close steady-state when the supply is restored. Particularly, DP-ORA F,CRPAPF{ }^{F}, \mathrm{CRPAP}^{F}, and PAF\mathrm{PA}^{F} are fair methods that are not able to return to their steady state. We can find the explanation on the fair distribution of the resources. These mechanism focus on avoiding resource waste, and they are always trying to sell all the resources. That is, if we classify the bidders in three classes, rich, middle class, and poor agents, the bidders from the middle class put pressure on the upper class. If bidders from the high class relax, the bidders of the middle class get the resources, and the rich ones do not get the resources, consequently they maintain prices at an acceptable level. When bidders of the middle or upper class get the resources at low prices due to the fair distribution, then they have no incentive to increase their bids, and prices begin to decrease. In this circumstance, the bidders of the upper class have the power to fix the price, and obviously they try to decrease it to the minimum possible, and the market collapse. This does not happen with RPA N{ }^{N} and CRPAF\mathrm{CRPA}^{F} because they sacrifice resource waste for maintaining the prices at an acceptable level. Finally, DRPA F{ }^{F} does not collapse and it is trading off reservation prices and resource waste.
Table 5 shows how the performance has dropped for all mechanisms that have failed to keep the price after a reduction on the resource supply. The mechanisms that have obtained more revenue are CRPAF\mathrm{CRPA}^{F} and DRPAF\mathrm{DRPA}^{F}, but they have produced resource waste. On the other hand, the mechanisms that have not produced resource waste have collapsed. Therefore, in these situations it is necessary to produce some resource waste to maintain prices at acceptable levels.
Figure 7 shows the evolution of the average number of bidders in 10 simulations with each of the methods during 2000 auctions in this experiment. We can see how the number of bidders is drastically reduced at time step 1000, when supply fluctuation begins. If we correlate these results with those in Figure 6, we can see that the reduction of bidders reduces auctioneer revenue. So, we cannot guarantee any stabilization of prices when supply fluctuation appears, and we corroborate the importance of keeping agents interested in the auction for the stability of prices.
TABLE 5. Results for All Mechanisms with a Uniform Distribution of Wealth in Scenario 2 (Variation of Supply). Averaged Results of 10 Simulations with Standard Deviation in Brackets.
Mechanism | Total Revenue | % Resource Waste | Number of Bidders | |
---|---|---|---|---|
Fair | PA F^{F} | 208304.84(43389.46)208304.84(43389.46) | 0.00%(0.00)0.00 \%(0.00) | 60.20(1.47)60.20(1.47) |
CRPA F^{F} | 451891.36(19727.05)451891.36(19727.05) | 23.66%(1.00)23.66 \%(1.00) | 53.50(1.35)53.50(1.35) | |
CRPAP F^{F} | 316905.30(21666.03)316905.30(21666.03) | 0.00%(0.00)0.00 \%(0.00) | 56.40(0.52)56.40(0.52) | |
DRPA F^{F} | 451442.78(40175.99)451442.78(40175.99) | 18.97%(0.79)18.97 \%(0.79) | 56.20(1.23)56.20(1.23) | |
DP-ORA F^{F} | 285722.23(18388.61)285722.23(18388.61) | 0.00%(0.00)0.00 \%(0.00) | 58.40(1.26)58.40(1.26) | |
Non-Fair | TA N^{N} | 97984.91(5299.34)97984.91(5299.34) | 1.17%(1.04)1.17 \%(1.04) | 48.70(1.15)48.70(1.15) |
CA N^{N} | 228217.64(64154.85)228217.64(64154.85) | 60.02%(8.73)60.02 \%(8.73) | 15.00(15.81)15.00(15.81) | |
RPA N^{N} | 328682.72(29768.08)328682.72(29768.08) | 38.67%(5.23)38.67 \%(5.23) | 47.10(2.56)47.10(2.56) |
Figure 7. Average number of bidders in scenario 2 (uniform wealth distribution, 10 simulations).
In the second experiment the units of resource offered by the auctioneer gradually vary oscillating between 40 and 60 units each 500 auctions. This oscillation seems feasible in a network communication domain. The decrease in supply from 50 to 40 could be caused by technical problems due to some temporarily unavailable links. On the other hand, the increase in supply can be caused by the temporary availability of some bandwidth that is usually not sold and reserved for backup purposes. The plots in Figure 8 show the ups and downs of prices as resources vary. CRPA F^{F} and DRPA F^{F} are the mechanisms that show better performance, as in periods of abundant resources they maintain the prices at a higher level than the other mechanisms. As already shown in Figure 6, the prices rise with low supply, but with CRPAP F^{F} it seems that the rises are becoming smaller every time and in the long
Figure 8. Average bidding price in scenario 2 with an oscillation of supply (Uniform wealth distribution, 1 simulation). Left-hand side: CAN,CRPAF,DP−ORAF,RPAN\mathrm{CA}^{N}, \mathrm{CRPA}^{F}, \mathrm{DP}-\mathrm{ORA}^{F}, \mathrm{RPA}^{N}. Right-hand side: TAN,PAF,CRPAPF,DRPAF\mathrm{TA}^{N}, \mathrm{PA}^{F}, \mathrm{CRPAP}^{F}, \mathrm{DRPA}^{F}.
term the auction collapses. This experiment shows how the dynamics in supply can cause problems in the auction mechanisms even with fair methods. The only fair mechanisms that maintain the stability in such situations are CRPAF\mathrm{CRPA}^{F} and DRPAF\mathrm{DRPA}^{F}.
From the results we can conclude that previous fair mechanisms with reservation prices can collapse in a scenario where the supply of the auctioneer is reduced over time. This occurs to the methods that distribute a quantity of resources fairly (DP-ORA F{ }^{F} and CRPAP F{ }^{F} ). On the other hand, DRPAF\mathrm{DRPA}^{F} avoids the collapse by introducing a minimum priority that prevents the fairly distributed resources from being allocated to middle- and upper-class agents.
5.5. Results in the Third Scenario: Arrival of New Customers
In environments where agents need resources, it is logical to think that if an agent leaves its supplier, its need for resources does not disappear and therefore the agent will try to find those resources elsewhere. Thus, in this scenario, when an agent leaves an auctioneer, it randomly joins the market of another auctioneer in search of the resources it needs. Thus, after each auction, some bidders might leave an auctioneer, while others could arrive. Each bidder is initially assigned to participate in a given market controlled by an auctioneer using its corresponding auction mechanism. However, when the bidder decides to randomly change of auctioneer, it leaves its current market and it joins the market of another auctioneer which uses another auction mechanism. Note that the bidders that change to a new auctioneer are usually poor because they are usually losers with their previous auctioneers. When a bidder changes auctioneer, its configuration values (wealth and TCL) remain the same, but the TCL counter is set to 0 , and the auctioneer of the new market sets the priority to the agent to default ( 0.5 ). Both measures, the TCL counter and priority, are dynamic and depend on the agent-auctioneer interactions; because the bidder moves to a new market, no previous interaction exists, and then the default values should be assigned.
Table 6 presents the averaged results after 10 simulations in this scenario. We expected a general increase in the revenue of all the auctioneers, because now bidders are not disappearing from the simulation, but moving from one auctioneer to another (and in the end, more bidders are kept in the simulation). The results show that, effectively, most of
TABLE 6. Results for Uniform Distribution of Wealth in Scenario 3 (Arrival of New Customers). Averaged Results of 10 Simulations with Standard Deviation in Brackets.
Mechanism | Total Revenue | Improv. (scen. 1) | % Resource Waste | Number of Bidders | |
---|---|---|---|---|---|
Fair | PA F^{F} | 448882.08(26294.13)448882.08(26294.13) | 71%71 \% | 0.00%(0.00)0.00 \%(0.00) | 109.70(4.32)109.70(4.32) |
CRPA F^{F} | 532526.97(37561.61)532526.97(37561.61) | 6%6 \% | 0.90%(0.16)0.90 \%(0.16) | 111.00(4.24)111.00(4.24) | |
CRPAP F^{F} | 526125.96(33562.50)526125.96(33562.50) | 6%6 \% | 0.00%(0.00)0.00 \%(0.00) | 113.70(4.42)113.70(4.42) | |
DRPA F^{F} | 547996.64(30969.23)547996.64(30969.23) | 9%9 \% | 0.00%(0.00)0.00 \%(0.00) | 114.30(3.26)114.30(3.26) | |
DP-ORA F^{F} | 508243.53(38068.57)508243.53(38068.57) | 17%17 \% | 0.00%(0.00)0.00 \%(0.00) | 117.20(4.83)117.20(4.83) | |
Non-Fair | TA N^{N} | 277304.95(16488.65)277304.95(16488.65) | 193%193 \% | 0.00%(0.00)0.00 \%(0.00) | 95.20(3.05)95.20(3.05) |
CA N^{N} | 353185.90(35803.38)353185.90(35803.38) | −1%-1 \% | 34.56%(6.43)34.56 \%(6.43) | 71.00(5.77)71.00(5.77) | |
RPA N^{N} | 328840.91(31884.12)328840.91(31884.12) | −2%-2 \% | 39.67%(5.78)39.67 \%(5.78) | 67.90(4.33)67.90(4.33) |
the mechanisms improve overall, some of them improve drastically, and only two, CAN\mathrm{CA}^{N} and RPA N^{N}, have slightly decreased their gains. The Improvement column in Table 6 shows the increase or decrease of revenue in proportion to scenario 1. TA N^{N} has been greatly benefited by the arrival of new bidders and has increased its profits an average of 193%193 \%. Despite this large increase, the TAN\mathrm{TA}^{N} mechanism is still the one with the worst revenue. As shown in the Number of Bidders column, fair mechanisms end up with more bidders than they had initially. The fair mechanism that has increased its revenue the most is the PAF\mathrm{PA}^{F} with a growth of 71%71 \%, followed by DP-ORA F{ }^{F} with 17%17 \% and DRPA F{ }^{F} with 9%9 \%. All of the differences between scenario 1 and scenario 3 for fair methods are statistically significant according to the t−t- test.
Summarizing all the results of this scenario, we can say that fair mechanisms ( PAF\mathrm{PA}^{F}, CRPAF,CRPAPF,DRPAF\mathrm{CRPA}^{F}, \mathrm{CRPAP}^{F}, \mathrm{DRPA}^{F}, and DP−ORAF\mathrm{DP}-\mathrm{ORA}^{F} ) are more able to retain newly arrived bidders. The bidders who go to other auctioneers because they did not find the resources they wanted, sometimes obtain the resources in a fair auction and, consequently, they remain with the new auctioneer and help to maximize the auctioneer’s revenue.
5.6. Results in the Fourth Scenario: Customers with Trust
Trust and reputation are increasingly used in electronic marketplaces. For example, in eBay auctions, users can view the reputation of sellers based on the opinions of users who have previously bought from them, and this information can be used to decide whether to trust these sellers or not. Reputation in this market is vital: buyers have to trust sellers because sellers could not send the products or the products might not have the expected quality. Having a good reputation is a key element for success in a competitive market; customers will move to the most reputable suppliers. The reputation of a supplier can be the factor that determines whether an agent decides to sign a contract with one supplier or another (Giardini, Tosto, and Conte 2008).
For these reasons, we added a trust model to the agents in the last scenario. As a result, the agents have different degrees of trust toward their suppliers. The objective is to approximate the agents’ behavior to that of a real environment. With this trust model, an agent leaves an auctioneer in which it has not enough trust. In addition, when it decides to leave its supplier, it joins the auctioneer with the highest reputation, instead of moving randomly as in the previous scenario.
The trust model of each bidder agent is composed of two components:
- The first component is direct trust. Each agent has direct trust in each provider it has interacted with. It is based on the direct experiences of the bidder. This trust is an estimator of the resources the bidder will obtain from the provider (Góme and Earle 2007).
In this scenario the TCL mechanism to model bidder dropout used in the previous scenarios is replaced by the index of desirable resources, IDR. Each bidder has a minimum percentage of resources that the agent needs (the IDR). This is a private value of each bidder, and it is uniformly randomly assigned from the interval [0.1,0.4][0.1,0.4]. For example, if an agent has an IDR value of 0.2 , it means that the agent wants to win at least two resources out of every 10 auctions. Then, the direct trust of this agent on its current provider is computed as the percentage of obtained resources over the number of auctions it has participated. If this percentage is lower than the IDR, then the bidder is not satisfied with its provider, and will leave it, in the case that the agent completely knows the provider (see knowledge explanation later). We assume that bidder dissatisfaction is the only motive to change an auctioneer. We do not consider other motivations such as better prices because we are dealing with sealed-bid auctions where bidders do not know the prices offered by the other bidders.
- The second component of trust is reputation (also called indirect trust), which is related to the experience of other agents. After every auction, each bidder randomly chooses another agent to ask about its auctioneer. If the asked agent is satisfied with its provider, it recommends its auctioneer to the other agent. Otherwise the asked agent does not recommend any auctioneer. The agent will use these recommendations when it decides to leave its current auctioneer and join a new one. The bidder agent chooses the auctioneer with the most recommendations. We assume that agents are honest and do not lie when they are giving recommendations.
Agents also have a parameter called minimumKnowledge (MK). This parameter indicates the minimum number of auctions needed to determine if an agent knows its auctioneer. When an agent has participated in more than MK auctions with its auctioneer, the agent can say that it knows enough about the auctioneer to leave if it is not achieving its objectives. In the same way, an agent would not recommend its auctioneer to another agent if it did not know enough about its auctioneer.
Table 7 shows the results obtained in this scenario. Examining the data we can see that the fair mechanisms obtain the highest number of recommendations and this means that they have increased the number of bidders with respect to scenario 3 (see column Improvement). All unfair mechanisms have decreased the number of bidders, while fair mechanisms have increased it. In conclusion, we can say that agents are more satisfied with fair mechanisms and this popularity means that auctioneers with them get more customers.
5.7. Discussion
With the results from all of the scenarios we can generate the summary in Table 8. This table shows the obtained revenue, the level of fairness of each mechanism and to what degree the mechanisms are affected by the recurrent multi-unit auction problems described in Section 2.2. Regarding the bidder drop problem, the least affected mechanism is DP-ORA F{ }^{F}, but the mechanisms PAF,CRPAF,CRPAPF\mathrm{PA}^{F}, \mathrm{CRPA}^{F}, \mathrm{CRPAP}^{F}, and DRPAF\mathrm{DRPA}^{F} are a short distance below it. Regarding the resource waste problem, the mechanisms TAN,DP−ORAF,PAF\mathrm{TA}^{N}, \mathrm{DP}-\mathrm{ORA}^{F}, \mathrm{PA}^{F}, and CRPAPF\mathrm{CRPAP}^{F} are not affected by this problem, which slightly affects DRPAF\mathrm{DRPA}^{F}. The rest of the mechanisms
Table 7. Results for Uniform Distribution of Wealth in Scenario 4 (Customers with Trust). Averaged Results of 10 Simulations with Standard Deviation in Brackets.
Mechanism | Total Revenue | % Resource Waste | Number of Bidders | Improv. (scen. 3) | Number of Recommendations |
---|---|---|---|---|---|
PA F^{F} | 445684.32(43506.40)445684.32(43506.40) | 0.00%(0.00)0.00 \%(0.00) | 110.50(5.06)110.50(5.06) | 0.73%0.73 \% | 178005.50(2667.90)178005.50(2667.90) |
CRPA F^{F} | 529515.67(44930.74)529515.67(44930.74) | 0.82%(0.21)0.82 \%(0.21) | 115.20(3.08)115.20(3.08) | 3.78%3.78 \% | 185570.00(1296.40)185570.00(1296.40) |
CRPAP F^{F} | 536566.79(17068.58)536566.79(17068.58) | 0.00%(0.00)0.00 \%(0.00) | 122.10(7.52)122.10(7.52) | 7.39%7.39 \% | 187322.90(1052.01)187322.90(1052.01) |
DRPA F^{F} | 546594.38(19530.43)546594.38(19530.43) | 0.00%(0.00)0.00 \%(0.00) | 115.10(5.97)115.10(5.97) | 0.70%0.70 \% | 186343.20(1183.79)186343.20(1183.79) |
DP-ORA F^{F} | 496080.97(37991.97)496080.97(37991.97) | 0.00%(0.00)0.00 \%(0.00) | 128.60(3.13)128.60(3.13) | 9.73%9.73 \% | 197076.80(1556.35)197076.80(1556.35) |
TA N^{N} | 254016.99(20741.33)254016.99(20741.33) | 0.00%(0.00)0.00 \%(0.00) | 94.70(3.68)94.70(3.68) | −0.52%-0.52 \% | 160800.80(635.67)160800.80(635.67) |
CA N^{N} | 374335.30(25401.71)374335.30(25401.71) | 30.67%(4.54)30.67 \%(4.54) | 63.20(3.76)63.20(3.76) | −10.98%-10.98 \% | 117431.10(6393.71)117431.10(6393.71) |
RPA N^{N} | 323737.43(47015.53)323737.43(47015.53) | 40.67%(8.60)40.67 \%(8.60) | 50.60(8.64)50.60(8.64) | −25.48%-25.48 \% | 96442.90(13863.98)96442.90(13863.98) |
TABLE 8. Summary of Results.
Mechanism | Bidder Drop Problem | Resource Waste | Asym. Balance of Negotiation Power | Fairness | Revenue | |
---|---|---|---|---|---|---|
Fair | PA F^{F} | low | very low | high | high | low |
CRPA F^{F} | low | medium | very low | high | very high | |
CRPAP F^{F} | low | very low | medium | very high | medium | |
DRPA F^{F} | low | low | very low | very high | very high | |
DP-ORA F^{F} | very low | very low | medium | very high | medium | |
Unfair | TA N^{N} | very high | very low | very high | very low | very low |
CA N^{N} | high | very high | low | low | low | |
RPA N^{N} | high | very high | very low | very low | low |
are affected in a serious way. The asymmetric balance of negotiation power problem does not affect the mechanisms DRPA F,RPAN{ }^{F}, \mathrm{RPA}^{N}, and CRPA F{ }^{F}. DP-ORA F{ }^{F} and CRPAP F{ }^{F} have a medium level because they may be affected by this problem in dynamic situations. Thus, according to our hypothesis, obtaining the maximum revenue can be achieved by a fair mechanism, as DRPA F^{F} does. DRPA F^{F} and CRPA F^{F} achieve the highest revenue for the auctioneer and both mechanisms behave the best when supply varies and there is market competition. However, if we look at the results in more detail, CRPA F{ }^{F} is always behind the revenue obtained by DRPA F{ }^{F}, except for the case of varying supply where CRPA F{ }^{F} obtains just a bit more than CRPA F{ }^{F}. Moreover, CRPA F{ }^{F} causes higher resource waste, and it keeps fewer bidders interested than DRPA F{ }^{F} does. Finally, it is important to note that CRPA F{ }^{F} has shown certain instability regarding resource waste and number of bidders when the wealth distribution of bidders varies, while DRPA F{ }^{F} behaves similarly with all wealth distributions. Therefore, we can conclude that the new method proposed in this paper, DRPA F{ }^{F}, outperforms the previous ones.
Nevertheless, we should verify in a future whether the mechanisms behave differently when using other values for the simulation parameters (e.g., TCL, IDR, wealth, etc.), as well as for their internal parameters (e.g., α,β,γ\alpha, \beta, \gamma, etc.). The values of agent wealth, bidding price distribution, reservation prices modifications (α,β)(\alpha, \beta), and TLC have been taken from Lee and Szymanski (2005b) to make our work comparable to DP-ORA. On the other hand, the minimum priority, minimum knowledge, and the index of desirable resources (IDR)
have been empirically set in this work. Thus, some kind of sensibility analysis should be performed to assess the effect of each parameter on the behavior of the mechanisms.
6. CONCLUSION
In environments where resources are perishable and the allocation of resources is repeated over time with the same set or a very similar set of agents, recurrent auctions come up. A recurrent auction is a sequence of auctions where the result of one auction can influence the following ones. These kinds of auctions have particular problems, however, when the wealth of the agents is unevenly distributed and resources are perishable. The bidder drop problem appears when many bidder agents decide to leave the market (because they are participating in many auctions and they are always losing); this lowering of the demand also lowers the prices and, consequently, the auctioneer gets less profit and the auction could even collapse. The resource waste problem turns up when not all the resources are sold (which could be a solution to the previous problem) and they lose their value because perishable resources cannot be stored for future sales. Finally, the asymmetric balance of negotiation power occurs when the richest agents gain enough power to set the prices and they set it at the minimum price possible, causing the collapse of the market.
Fair mechanisms have already been proposed to deal with these problems. The experiments in a static environment show that the mechanisms that incorporates fairness and reservation prices minimize the problems of recurrent multi-unit auctions (bidder drop problem, resource waste, and asymmetric balance of negotiation power). Fairness gives bidders incentives to stay in the auction and reservation prices help to maintain the equilibrium in negotiation power. However, experiments in dynamic scenarios show that even fair mechanisms with reservation prices can collapse when the supply of the auctioneer is reduced during an elapse of time. This occurs to the methods that fairly distribute a quantity of resources (DP-ORA F{ }^{F} (Lee and Szymanski 2005b) and CRPAP F{ }^{F} (Murillo et al. 2008)). The DRPA F{ }^{F} mechanism proposed here avoids collapsing by introducing a minimum priority that prevents the fairly distributed resources from arriving to the middle-upper class agents. We can also say that in scenarios where customers can move from one provider to another, fair mechanisms are more able to retain the newly arrived customers. That is, when a customer leaves an auctioneer and joins another one, if the new provider uses a fair mechanism to allocate resources, the customer’s probability of achieving its goals is higher than if the auctioneer was unfair. On the other hand, when bidders use trust and reputation information to decide which auctioneers to join, fair auctioneers achieve greater popularity among bidders and, consequently, the number of arrivals is higher than for unfair mechanisms. In particular, we have experimentally tested how one of our proposed mechanisms, the dynamic reservation price auction (DRPAF)\left(\mathrm{DRPA}^{F}\right) presents the highest averaged performance of all the simulated conditions, including the variation of supply, and makes a profit for the auctioneer while avoiding, in most cases, the resource waste problem.
In future work, we will consider the improvement of our priority mechanism, because until now we have been using a very simple method, based on wins and loses. Thus, a learning method to take into account the bid values, or the number of items involved in the auctions won or lost, would be worthwhile. Additionally, functions to combine the priority with the bid value other than the product should be also studied. Finally, we should also study whether the fair mechanisms we have proposed are strategy-proof or not. One way of doing so is by analyzing the pricing mechanism we use. As a first instance, we have used a first-price auction, but other auction schemas, as Vickrey-Clark-Gloves (VCG) mechanism (Clarke 1971; Groves 1973) can be tested. The main advantage of the VCG mechanism is that
it has been proved to be strategy proof in nonrecurrent scenarios. However, in some works (Suyama and Yokoo 2004; Rothkopf 2007) such strategy-proof property is being questioned, and so we should deeply analyze the convenience of using such mechanism.
ACKNOWLEDGMENTS
This work has been done with the support of the Commissioner for Universities and Research of the Department of Innovation, Universities and Enterprises of the Generalitat of Catalonia, the European Social Funds and the Spanish MICINN project SuRoS (TIN200804547).
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