The problem of resource allocation has recently caught the public imagination, where the resource owner has to decide the allocation of the item among a group of self-interested agents. Since the valuation differs from agents, it is a natural objective for the owner to pursue the efficiency of the allocation, i.e., allocating the item to the agent with the highest valuation. In many scenarios, the owner does not really aim at making profits but hopes the wealth maintained among the agents. For example, the government wants to build a library in a community that values it most; a charity distributes a donation to the recipient who needs it most; a hospital allocates doctors to rural areas where doctors are highly demanded.
To find the agent with the highest valuation, one common alternative is to hold an auction  under some protocols such as the well-known Vickrey-Clarke-Groves (VCG) mechanism [17, 2, 4]. However, the payments under VCG will all be delivered to the auctioneer, which againsts our non-profit purpose. To maintain as much wealth as possible among the participants, many redistribution mechanisms based on VCG have been proposed [1, 8]. They redistribute the revenue generated by VCG back to all participants. However, these mechanisms can only be applied in static settings, where the resource owner can only allocate the item to the person whom she can directly contact with (her neighbours).
Accordingly, another issue emerges: how can the owner enroll more participants in the resource allocation problem in order to achieve a more efficient allocation? Advertising is a widely used method to disseminate information to attract more people. However, it should be paid in advance without a guarantee that there will be more participants or a more efficient allocation. Moreover, it is irrational for a resource owner who no longer cares about profit to pay something for the allocation. Therefore, in this paper, we consider a cost-free promotion by incentivizing participants to invite their neighbours in social networks , which is an enormous challenge as no one would be willing to invite more competitors without a profit.
Therefore, in this paper, we propose a novel network-based redistribution mechanism to tackle this challenge, where the reward redistributed to each agent is a monotone increasing function to the number of participants she invites. Although the agents are not paid in advance, they are still incentivized not only to report their valuation truthfully but also to diffuse the information to all their neighbours without sacrificing the non-deficit guarantee, which is one of the key features of our mechanism. Eventually, more agents will be informed about the resource allocation and a more efficient allocation will be achieved. Moreover, it also satisfies the desirable properties of traditional redistribution mechanisms such as individual rationality and asymptotically budget-balance.
Some interesting work related to information diffusion via networks has been studied recently. DBLP:conf/aaai/LiHZZ17 DBLP:conf/aaai/LiHZZ17 proposed an auction mechanism where the seller sells one item in a social network with the help of that participants are inviting each other to attract more participants. With this inspiration, soon afterwards Zhao:2018:SMI:3237383.3237400 Zhao:2018:SMI:3237383.3237400 generalized the mechanism for multiple homogeneous items in the same setting. Their attention is on how to maximize the seller’s revenue, which is different from ours. We aim for a more efficient allocation without profits. We refer to the idea of their work and design our redistribution mechanism to achieve the goal.
There exists a rich literature on redistribution mechanisms for resource allocation problems [10, 15, 5, 9]. Furthermore, redistribution mechanisms have also been extended to the setting of public project problems [7, 16, 6, 10]. However, none of the work took the natural connections between participants into account. More to the point, our mechanism promises desirable properties when participants are connected via their private links which cannot be achieved by the existing mechanisms.
We claim our three main contributions here. First, to the best of our knowledge, we are the very first to study the redistribution mechanism design problem in social networks. Second, we show the limitations of the classical Cavallo mechanism if it is directly extended in social networks. Third, we propose a novel network-based redistribution mechanism which improves the efficiency of the allocation without sacrificing all the desirable properties.
The structure of the paper is organized as follows. We first describe the background and basic definitions of the problem. Next, we extend the Cavallo mechanism in social networks and discuss its limitations. After that, we propose our network-based redistribution mechanism for the tree structures and show its outstanding properties. Finally, we generalize our mechanism in graphs and discuss future work.
We consider a setting where an owner wants to allocate an item in a social network , where each agent is a potential bidder with a private valuation for the item. Each agent has a private neighbour set . If there exists a directed edge from agent to agent , we say is ’s neighbour, denoted by . Let be the depth of agent , which is the length of the shortest path from the owner to . We say agent is agent ’s child neighbour if and , denoted by . The objective of the owner is to allocate the item to the agent with the highest valuation to the best of her ability and maintain as much wealth as possible among the agents. That is, she is aiming to minimize the surplus of the payment transfers in the mechanism.
Initially, without any third-party platforms, the owner can only allocate the item to her neighbours since all the other agents cannot be reached directly. To attract more potential bidders, a feasible approach is to ask the agents to invite their neighbours to join the allocation. However, there is no reason for these bidders to invite more competitors. Thus, how to design the incentives for the agents to propagate the information without sacrificing desirable properties is the greatest challenge we need to overcome.
In this paper, we propose a novel network-based redistribution mechanism, where all the agents are willing to not only report their private valuation for the item but also invite all their neighbours to the mechanism voluntarily.
We start by defining some notations in the mechanism:
[leftmargin= 20 pt]
Let be the type of agent , which is ’s true private information.
Let be the type profile of all the agents, where is the type profile for agents except .
Let be the type space for agent and be the type profile space for all the agents.
Let be the reported type of agent , where is the valuation she reported and is the neighbours she has invited. Let , if agent is not invited.
Let be the graph constructed by the reported type profile .
Note that the reported type is not definitely the same as the true type. Therefore, we can easily observe that for each agent ,
A redistribution mechanism in the social network is defined by an allocation policy and a payment policy , where and .
Given a reported type profile , the payment policy represents the money paid by each agent and the allocation policy represents the item allocation result. Intuitively, we have
In our setting, we assume that there is no cost for an agent to spread the information to her neighbours. Therefore, given a reported type profile of all the agents, the utility of agent with type and reported type is defined as:
Therefore, the surplus of the payment transfer of the mechanism is defined as:
We say a reported type profile is feasible if for each agent with , there must exist at least one path from the owner to in . We say an allocation is feasible if at most one agent with is allocated the item, i.e., .
In the following discussion, we only focus on feasible type profiles and feasible allocations since infeasible cases will not happen in real-world application. Let be the set of all feasible reported type profiles.
Given a feasible reported type profile and a feasible allocation , the social welfare of allocation is defined by .
That is, the social welfare of an allocation is the sum of the valuations of all agents who win the item for this allocation. The higher the social welfare is, the more efficient the allocation is.
A redistribution mechanism is individual rational (IR) if for all , all and all , we have , where .
This is a general extension of the traditional definition of individual rationality. That is, all the agents participated in the mechanism will not have negative utilities as long as she truthfully reports her private valuation. Note that the definition does not require the agents to invite all their neighbours, which loosens the restriction of reporting true type.
A redistribution mechanism is incentive compatible (IC) if for all , all and all , we have , where and . is the corresponding reported type profile when changes her reported type such that the reported type of any agent is and the others are the same as those in .
For the traditional definition of incentive compatibility, all the buyers’ dominant strategy is to truthfully report their private valuation of the item. Here, we put forward a stricter extended definition of IC for the network setting, where all the agents are incentivized not only to report valuation truthfully but also to invite all their neighbours.
A redistribution mechanism is non-deficit (ND) if for all , all and all , we have .
That is, the surplus of the payment transfer is non-negative, which is reasonable because the owner or other outside parties has to pay for the deficit otherwise.
A redistribution mechanism is asymptotically budget-balanced (ABB) if for all , all and all , we have .
This is to say when the number of the participants goes to infinity, almost all the money received by the owner will be redistributed back to the participants.
Cavallo Mechanism in Social Networks
Considering the constraint of generalized IR, extended IC, ND and ABB, seemingly some traditional redistribution mechanisms can be easily applied to the new setting in social networks. Therefore, in this section, we first review the classical Cavallo mechanism  and show that it may lead to a deficit and disincentivize agents to diffuse the information.
The Cavallo mechanism modifies the VCG framework and redistributes the transfer payments back among the agents while keeping the specified desirable properties of VCG. The mechanism for a single item is outlined below:
Intuitively, the Cavallo mechanism can be viewed as two stages: the auction stage and the redistribution stage. In the first auction stage, the item will be allocated to the highest bidder and she pays the loss of other players because of her participation to the owner as defined in the traditional VCG mechanism. Then in the second redistribution stage, the owner redistributes the money received to all the agents in the mechanism. The money redistributed to agent is calculated by , where is the surplus lower-bound in VCG among the same agents over all possible reported valuation of . Specially, in the single-item setting, the payment in the first stage of the highest bidder is the second highest reported valuation. Let be the highest bidder among all the agents . The redistributed money is for the highest bidder and the second highest bidder, and for the others. Consequently, all the agents share the surplus and the rewards are independent of their reported valuation.
Although the Cavallo mechanism is IR and ABB, we then show that it may run a deficit and agents may be not willing to diffuse the information in social networks.
The Cavallo mechanism runs a deficit.
We prove the proposition by showing a counter example in Figure 1. Since agent is the only one with positive valuation, the mechanism allocates the item to her. For agent and , their participation will not affect the result, thus they will pay nothing in the first auction stage. For agent , if she does not participate in the mechanism, no one else will gain the item, thus her payment is also zero. For agent , if she quits the mechanism, agent will not be involved in, so her payment is . Since there is only one positive-utility agent, all the agents will be redistributed nothing. Thus we have , which runs a deficit. ∎
The Cavallo mechanism disincentivizes the agents to diffuse the information.
By showing a counter example in Figure 1, we can easily prove the proposition. As agent is the highest bidder, she keeps the item and pays the second highest valuation . Then in the redistribution stage, all these agents will share the surplus. For agent and , , then . For agent , and , , then . However, if stops inviting and stops inviting , the allocation and the surplus will remain the same but the number of agents who share the surplus will decrease. Then and . Thus, the Cavallo mechanism disincentivizes the agents’ diffusion. ∎
Therefore, owing to the special constraint of social networks, extending the Cavallo mechanism simply is not feasible. In the following section, we will introduce our novel mechanism with all the desirable properties satisfied.
Redistribution Mechanism in Trees
To tackle the challenges on networks, we propose a network-based redistribution mechanism (NRM) which satisfies all the desirable properties mentioned before. In this section, we will first start with a special type network, tree structures, which provides a clearer presentation of the intuition behind. Later we will generalize our mechanism on common graphs.
Given a tree graph and a feasible reported type profile of all the agents, for each agent if there exists a simple path from the seller to through and , we say is ’s ancestor and is ’s descendant.
Some basic notations in the mechanism is defined as:
[leftmargin= 20 pt]
Let be the ancestor sequence of agent , where is an ancestor of agent and .
Let be the subtree of agent if is a tree consisting of and all its descendants in . Let be the number of agents in .
Let be the sibling set of agent , where all the agents in has the same parent as .
Let denote the highest reported valuation among all the agents in any set .
That is, for each agent , she cannot join in the mechanism if any agent in her ancestor sequence does not diffuse the information. Besides, without the invitation of agent , any agent in her subtree cannot receive the information.
Now we will propose our NRM in Trees. The detailed procedure is given in Algorithm 1.
Intuitively, although the network-based redistribution mechanism is centralized, it can be viewed as a sequential procedure. The item passes through the ancestor sequence of the highest bidder and each agent is required to pay the highest reported valuation without her participation for either passing or keeping the item. The item is allocated to the first agent whose valuation is higher than or equal to her required payment. The money each agent paid first compensates the last ancestor’s payment and the remaining part will be redistributed among her siblings and herself. The money redistributed to agent is the new required payment difference multiplied by the percentage of agents in ’s subtree over all the agents in the subtrees of the ancestor and its siblings considered, which is a monotone increasing function to the number of their descendants. The more their descendants are, the more they will be redistributed, which incentivizes the agents’ diffusion. The rest money which is not redistributed will be given to the owner as the surplus. Note that NRM is a centralized mechanism and all the operation process is run by the owner. Therefore, only the winner in NRM is the one who is required to pay the money to the owner while the ancestor sequence of the winner and their siblings will be redistributed rewards.
We take Figure 2 as an example. In Figure 2, the highest bidder is agent and the ancestor sequence is , which are colored in green. Those purple nodes are siblings of the ancestors, i.e., , and . Figure 2, 2and 2 shows a running process of NRM, where each subfigure represents the computational process for each step. For each subfigure, the nodes in grey are the ancestor and its siblings we focus on in this step, and the nodes in red circle are the highest bidder in the subtree of the ancestor or its siblings. The value on the left top is the required payment for the ancestor and the red arrows represents the payment transfer. In the first step in Figure 2, the required payment for agent is the highest reported valuation without her participation . Since she is the first one in the ancestor sequence, the payment can be directly used to redistribute among the siblings and herself. The total number of agents in the subtrees of , and is . For agent , without her participation the new required payment is also and the number of agents in her subtree is , thus the money redistributed to her is . For agent , the new required payment becomes if she quits the mechanism and , thus the money redistributed is . Similarly, we have . Then the surplus will be given to the owner. In the second step in Figure 2, we have . Since , the money first compensates agent ’s payment. Then the payment difference will be redistributed among , and . For each , and , the new required payment difference without their participation is also . Thus, we have and . In this step the surplus to owner is zero since . In the third step in Figure 2, is equal to . Thus the money will all be used to compensate agent ’s payment and remain nothing for redistribution. Thus we have . Since , the item will be allocated to agent . Till now, the NRM runs over. The winner is agent and the surplus is . Compared to the classical Cavallo mechanism, the owner allocates the item to agent with social welfare and only three agents , and have positive utilities while in NRM the social welfare is and agents have positive utilities. Therefore, our mechanism is more efficient and more agents have positive utilities.
Properties of NRM
In what follows, we show that our NRM satisfies all the desirable properties of IR, IC, ND and ABB in trees. Also, the allocation is more efficient than traditional Cavallo mechanism among the neighbours.
The network-based redistribution mechanism in trees is individually rational.
According to the algorithm of NRM, for each sibling of ancestors, they are not required to pay money and they will receive the money redistributed. Thus we have . For each ancestor of the winner, although they are required to pay money for either passing or keeping the item, their payment will be compensated by the next ancestor in the sequence. Thus they will be only redistributed the money from the mechanism, i.e., . For the winner of the item, her valuation must be greater than or equal to her required payment according to the allocation condition. Together with the redistributed money, her utility is . All the other agents pay nothing. Thus, agents’ utilities in NRM are non-negative and the mechanism is individually rational. ∎
The network-based redistribution mechanism in trees is incentive compatible.
As defined in the extended IC, all agents are required not only to report their truthful valuation but also to invite all their neighbours. Here we prove the theorem in two steps. First, fix whatever valuation for each agent, we prove that inviting all the neighbours is the dominant strategy. Next, fix whatever neighbours invited by each agent, we prove that reporting the truthful valuation is the dominant strategy. Thereby, for each agent, both reporting the truthful valuation and inviting all the neighbours is the dominant strategy.
In NRM, all the agents can be divided into four categories: the winner, winner’s ancestors, siblings of the ancestors and the others. Only the agents in the first three categories will gain non-zero utilities.
For the winner , her utility is . First, assume that her reported valuation is fixed and the neighbours she invited is . According to the allocation condition, no matter how many neighbours she invites, she will be still the winner since her valuation is at least equal to the required payment. The term remains the same. However, since is a monotone increasing function to the number of the descendants, will decrease if inviting fewer neighbours, which leads to a lower utility. Next, assume that her neighbours invited is fixed and her reported valuation is . If she is still the winner, her utility remains the same since it is not related to her reported valuation. If she becomes an ancestor of the new winner or the siblings of the ancestor, her utility will only consist the redistributed part , which is lower than the utility of being the winner.
For the winner’s ancestor , her utility is . First, assume that her reported valuation is fixed and the neighbours she invited is . According to the allocation condition, no matter how many neighbours she invites, she cannot be the winner since her valuation is lower than the required payment, i.e., . If she is still the ancestor or becomes a sibling, her utility will decrease after inviting fewer neighbours since the total amount of the money to be redistributed will not increase and the is a monotone increasing function to the number of the descendants. Next, assume that her neighbours invited is fixed and her reported valuation is . She has no chance to be the sibling. If she becomes the winner, her utility will be , which is lower than that of reporting truthfully. If she is still the ancestor, her utility will not change no matter what valuation she reports.
For the sibling of the ancestors , her utility is . First, assume that her reported valuation is fixed and the neighbours she invited is . She has no chance to be the winner or the ancestor according to the allocation condition. If she is still the sibling, her utility will decrease after misreporting since the total amount of the money to be redistributed will not increase and the is a monotone increasing function to the number of the descendants. Next, assume that her neighbours invited is fixed and her reported valuation is . She has no chance to be an ancestor. If she becomes the winner, her required payment will be the highest valuation without her participation, which is higher than her valuation. Thus her utility will decrease because . If she is still the sibling, her utility remains unchanged.
For any other agent , her utility is . First, assume that her reported valuation is fixed and the neighbours she invited is . The allocation will not change and she cannot become the winner, the ancestor or the sibling. So she will still gain nothing. Next, assume that her neighbours invited is fixed and her reported valuation is . The only possible way to gain something through misreporting valuation is to be the winner. However, the money she is required to pay must be higher than her valuation and the money redistributed to her must be zero. Thus her utility is negative.
Accordingly, NRM is incentive compatible since all the agents have no incentive to either misreporting their valuation or inviting fewer neighbours. ∎
The network-based redistribution mechanism in trees runs no deficit.
According to the process of NRM, in each step, the ancestor pays the money required and shares the remaining part after compensation among . The total money redistributed is
Thus, the required payment can cover the compensation and the money redistributed. So NRM runs no deficit. ∎
The network-based redistribution mechanism in trees is asymptotically budget-balanced.
In each step, the money is redistributed among the ancestor and her siblings . Let the , where . The amount of the money redistributed is
As the number of participating agents goes to , the surplus for this step is
Thus in each step the surplus is asymptotically zero, so the NRM is asymptotically budget-balanced.
The network-based redistribution mechanism in trees is at least as efficient as Cavallo mechanism among neighbours.
According to the allocation condition, the winner is the agent whose reported valuation satisfies . Thus, NRM is at least as efficient as Cavallo mechanism among the owner’s neighbours. ∎
It seems that Theorem 5 is not that strong since there are no further guarantees for the efficiency except for the improvement compared to the Cavallo mechanism among the neighbours. However, even if we ignore redistribution and only consider weak budget balance (and IC, IR), efficiency approximation has not been found yet [13, 12]. Consider a simple line graph in Figure 3, where is the resource owner, has two neighbors and , and and ’s valuations are and and . If does not invite , wins the resource and pays zero, so will only invite if ’s reward is at least . If invites , to achieve efficiency, wins and her payment should be not more than for IR. To achieve no-deficit, should pay at least what receives. Eventually, their payments depend on their valuations, which seems to violate IC. That said, this simple setting is somewhat equivalent to the bilateral trading setting of one seller and one buyer studied by myerson1983efficient myerson1983efficient where behaves like the seller and is the buyer. If this is true, then their well-known impossibility theorem holds here (i.e. we cannot have efficiency, IC, IR and no-deficit at the same time). Even in bilateral trading settings, we haven’t seen good no-deficit examples to approximate efficiency. The well-known example is McAfee’s trade reduction for multiple buyers and multiple sellers, where efficiency is sacrificed to remove deficit, but the efficiency loss is diminished when the number of traders increases . However, it still does not guarantee a lower bound of efficiency in general. In the worst case when there is only one buyer and one seller, it has no efficiency guarantee.
Redistribution Mechanism in Graphs
In the previous section, we only studied the mechanism in tree structures. In real life, most social networks are common graphs. Hence in this section, we extend our NRM to more general cases without sacrificing all desirable properties.
Different from the tree cases, we extend the definitions and basic notations for graph settings.
Given a common graph and a feasible reported type profile of all the agents, for each agent if all the paths from to have to pass , we say is ’s ancestor and is ’s descendant.
[leftmargin= 20 pt]
Let be the ancestor sequence of agent .
Let be the subgraph of agent .
Let be the sibling set of agent .
That is, an ancestor for agent is a cut-point from the seller to . The subgraph of agent are those who cannot receive the information without ’s invitation. The siblings of an ancestor are the child neighbours of ancestor except herself.
Then the NRM can be simply extended in graphs by updating the definitions of the notations above in Algorithm 1.
Intuitively, the network-based redistribution mechanism in graphs is a generalization of that in trees. The sibling set who share the money with an ancestor are the child neighbours of the last ancestor. Seemingly, it is quite different from the ancestor’s brother neighbours with the same parent in tree cases. Actually, in tree structures, the ancestor herself is also one of the child neighbours of the last ancestor, which can be viewed as a special case of the common graphs.
Also an example is given to illustrate the mechanism. In Figure 4, the highest bidder is agent . According to the definition, the green nodes are the ancestors and purple nodes are their siblings, i.e., , and . In the running example in Figure 4 4, first for agent , her required payment will all be used to redistribute among , , and . The total number of agents in the subgraph of all , , and is . If quits the mechanism, and will be not able to receive the information and will be the new ancestor with payment . Thus we have . If quits the mechanism, only herself will be out of the network and will be the new ancestor with payment . Thus we have . Similarly, we have and . The surplus to the owner in this step is . In the same way, the required payment for agent is and the remaining money after compensation is . Then the money redistributed to each agent is and . The surplus is and will keep the item since . Till now, the mechanism runs over. The winner is agent , the social welfare is and the total surplus is .
Since the network-based redistribution mechanism in graphs is a generalization of that in trees, we can easily obtain the following corollary.
The network-based redistribution mechanism in graphs runs no deficit and is IR, IC, ABB and at least as efficient as Cavallo mechanism among neighbours.
In this paper, we considered the redistribution mechanism design problem in social networks, where the owner wants to allocate one item and hopes the wealth maintained among the agents. The objective is to incentivize agents participated to invite all their neighbours to the mechanism so that the owner can make the allocation more efficient. The classical Cavallo mechanism performs well in the traditional setting; however, it may lead to a deficit and disincentivize the agents to diffuse the information. To overcome the challenge, we propose a novel network-based redistribution mechanism which incentivizes agents to invite all their neighbours. The mechanism works not only for the tree structures but also for the common graphs. Moreover, the mechanism satisfies all the desirable properties of individual rationality, incentive compatibility, asymptotically budget-balance and non-deficit. The allocation is also more efficient.
Our work has many interesting aspects for further investigation. We only consider the single-item situation in this paper, so it may be a challenge to extend the mechanism for multiple items . Since the wealth is redistributed among a particular group of agents in our mechanism, another valuable direction can be finding a way to redistribute more fairly. In addition, it is also worthwhile to generalize our mechanism for public project problems [2, 10].
-  (2006) Optimal decision-making with minimal waste: strategyproof redistribution of vcg payments. In Proceedings of the Fifth International Joint Conference on Autonomous Agents and Multiagent Systems, AAMAS ’06, New York, NY, USA, pp. 882–889. External Links: Cited by: Introduction, Cavallo Mechanism in Social Networks.
-  (1971) Multipart pricing of public goods. Public choice 11 (1), pp. 17–33. Cited by: Introduction, Conclusion.
-  (2010) Networks, crowds, and markets - reasoning about a highly connected world. Cambridge University Press. Cited by: Introduction.
-  (1973) Incentives in teams. Econometrica 41 (4), pp. 617–631. Cited by: Introduction.
Redistribution mechanisms for assignment of heterogeneous objects.
Journal of Artificial Intelligence Research41, pp. 131–154. Cited by: Introduction.
-  (2013) Undominated groves mechanisms. Journal of Artificial Intelligence Research 46, pp. 129–163. Cited by: Introduction.
-  (2011) Budget-balanced and nearly efficient randomized mechanisms: public goods and beyond. In International Workshop on Internet and Network Economics, pp. 158–169. Cited by: Introduction.
-  (2011) VCG redistribution with gross substitutes. See DBLP:conf/aaai/2011, External Links: Cited by: Introduction.
-  (2012) Worst-case optimal redistribution of vcg payments in heterogeneous-item auctions with unit demand. In Proceedings of the 11th International Conference on Autonomous Agents and Multiagent Systems-Volume 2, pp. 745–752. Cited by: Introduction.
-  (2019) An asymptotically optimal VCG redistribution mechanism for the public project problem. See DBLP:conf/ijcai/2019, pp. 315–321. External Links: Cited by: Introduction, Conclusion.
-  (2009) Auction theory. Academic press. Cited by: Introduction.
-  (2019) Diffusion and auction on graphs. See DBLP:conf/ijcai/2019, pp. 435–441. External Links: Cited by: Properties of NRM.
-  (2017) Mechanism design in social networks. In Proceedings of the Thirty-First AAAI Conference on Artificial Intelligence, pp. 586–592. Cited by: Properties of NRM.
-  (1992) A dominant strategy double auction. Journal of economic Theory 56 (2), pp. 434–450. Cited by: Properties of NRM.
-  (2009) Almost budget-balanced vcg mechanisms to assign multiple objects. Journal of Economic theory 144 (1), pp. 96–119. Cited by: Introduction.
-  (2012) Redistribution of vcg payments in public project problems. In International Workshop on Internet and Network Economics, pp. 323–336. Cited by: Introduction.
-  (1961) Counterspeculation, auctions, and competitive sealed tenders. Journal of Finance 16 (1), pp. 8–37. Cited by: Introduction.
-  (2018) Selling multiple items via social networks. In Proceedings of the 17th International Conference on Autonomous Agents and MultiAgent Systems, AAMAS ’18, Richland, SC, pp. 68–76. External Links: Cited by: Conclusion.