Connectedness in weighted consensus division of graphical cakes between two agents
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
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2023
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| _version_ | 1866908557151043584 |
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| author | Hanke, Josef Pires, Ana Rita |
| author_facet | Hanke, Josef Pires, Ana Rita |
| contents | Austin's moving knife procedure was originally introduced to find a consensus division of an interval/circular cake between two agents, each of whom believes that they receive exactly half of the cake. We generalise this in two ways: we consider cakes modelled by graphs, and let the two agents have unequal, arbitrary entitlements. In this setting, we seek a weighted consensus division -- one where each agent believes they received exactly the share they are entitled to -- which also minimises the number of connected components that each agent receives. First, we review the weighted consensus division of a circular cake, which gives exactly one connected piece to each agent. Next, by judiciously mapping a circle to a graph, we produce a weighted consensus division of a star graph cake that gives at most two connected pieces to each agent -- and show that this bound on the number of connected pieces is tight. For a tree, each agent receives at most h+1 connected pieces, where h is the minimal height of the tree. For a connected graphical cake, each agent receives r+2 connected pieces, where r is the radius of the graph. Finally, for a graphical cake with s connected components, the division involves at most s+2r+4 connected pieces, where r is the maximum radius among all connected components. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_12260 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Connectedness in weighted consensus division of graphical cakes between two agents Hanke, Josef Pires, Ana Rita Combinatorics Computer Science and Game Theory Austin's moving knife procedure was originally introduced to find a consensus division of an interval/circular cake between two agents, each of whom believes that they receive exactly half of the cake. We generalise this in two ways: we consider cakes modelled by graphs, and let the two agents have unequal, arbitrary entitlements. In this setting, we seek a weighted consensus division -- one where each agent believes they received exactly the share they are entitled to -- which also minimises the number of connected components that each agent receives. First, we review the weighted consensus division of a circular cake, which gives exactly one connected piece to each agent. Next, by judiciously mapping a circle to a graph, we produce a weighted consensus division of a star graph cake that gives at most two connected pieces to each agent -- and show that this bound on the number of connected pieces is tight. For a tree, each agent receives at most h+1 connected pieces, where h is the minimal height of the tree. For a connected graphical cake, each agent receives r+2 connected pieces, where r is the radius of the graph. Finally, for a graphical cake with s connected components, the division involves at most s+2r+4 connected pieces, where r is the maximum radius among all connected components. |
| title | Connectedness in weighted consensus division of graphical cakes between two agents |
| topic | Combinatorics Computer Science and Game Theory |
| url | https://arxiv.org/abs/2312.12260 |