A review of cooperative rules and their associated algorithms for minimum-cost spanning tree problems

Abstract

Minimum-cost spanning tree problems are well-known problems in the operations research literature. Some agents, located at different geographical places, want a service provided by a common supplier. Agents will be served through costly connections. Some part of the literature has focused, mainly, in studying how to allocate the connection cost among the agents. We review the papers that have addressed the allocation problem using cooperative game theory. We also relate the rules defined through cooperative games with rules defined directly from the problem, either through algorithms for computing a minimal tree, either through a cone-wise decomposition.

Sharing cost of network among users with differentiated willingness to pay

“…Our 2 From our interviews with GRDF managers we understood that investments in network expansion are typically justi…ed by su¢ ciently high demand. 3 Apart Bergantiños and Martínez (2014), we are aware of only one other paper that considers consumers with di¤erentiated willingness to pay, namely Hougaard and Tvede (2020). This paper asks how to implement a welfare-maximizing network and allocate its costs when information on the agents' willingness to pay is private and it shows that is impossible to Nash-implement budget-balanced cost sharing rules, which is not directly related to our work.…”

“…A related economic literature studies the problem of sharing the cost of a minimal cost spanning tree (MCST) connecting di¤erent geographic locations to the source at the least cost (see review by Bergantiños and Vidal-Puga, 2021). Some cost-sharing rules are linked to the algorithms for constructing MCST proposed by operational research.…”

“…Asymmetry among consumers connected through the network which cost is to be shared relates us to Bergantiños and Martínez (2014). 3 In their model the agents have not only di¤erent willingness to pay for the good delivered through the tree-network (as in our model) but also produce di¤erent amounts of this good, with the main example being water transportation from the rainy to arid regions. The core of their problem is not necessarily full (conditions are provided).…”

Anonymity in sharing the revenues from broadcasting sports leagues

“…The broadcasting problem is an instance of resource allocation. Other well known instances to which the literature has paid attention are: airport problems (e.g., Littlechild and Owen, 1973), bankruptcy problems (e.g., O'Neill, 1982;Thomson, 2019a), telecommunications problems (e.g., van den Nouweland et al, 1996), minimum cost spanning tree problems (e.g., Bergantiños and Vidal-Puga, 2021), transport problems (e.g., Algaba et al 2019;Estañ et al 2021), inventory problems (e.g., Guardiola et al, 2021), liability problems with rooted tree networks (e.g. Oishi et al, 2023), knapsack problems (e.g., Arribillaga and Bergantiños 2023), pooling games (Schlicher et al 2020), m-attribute games (Özen et al, 2022), urban consolidation centers (Hezarkhani et al, 2019), and scheduling problems with delays (Gonçalves-Dosantos et al, 2020) .…”

Sharing Cost of Network Among Users with Differentiated Willingness to Pay

“…Either rule by Bergantiños and Martínez (2014) results in splitting a segment's cost equally among its users. This allocation is not in the 4 Our work is also related to the literature on sharing the cost of a speci…c tree, namely, minimal cost spanning tree, see recent review in Bergantiños and Vidal-Puga (2021).…”