[Paper Review] Characterizations of the position value for hypergraph communication situations

Erfang Shan, Zhang Guang|arXiv (Cornell University)|Jun 1, 2016

Game Theory and Voting Systems13 references3 citations

TL;DR

This paper provides the first axiomatic and non-axiomatic characterizations of the position value in hypergraph communication situations, introducing the uniform hyperlink game and k-augmented uniform hyperlink game whose Shapley values yield the position value. It establishes component efficiency and a new property, partial balanced conference contributions, as sufficient axioms for uniqueness.

ABSTRACT

The Mayser value (Myerson (1977)), the position value (Meessen (1988)) and the average tree solution (Herings et al.) are three most important allocation rules for (graph or hypergraph) communication situations. In 2005, an axiomatic characterization of the position value for arbitrary (graph) communication situations was given by Slikker (2005). However, an axiomatic characterization of the position value for arbitrary hypergraph communication situations has not yet been found and remains an open problem. In our manuscript, we first give two non-axiomatic characterizations of the position value for hypergraph communication situations by introducing the uniform hyperlink game and the $k$-augment uniform hyperlink game. Based on the non-axiomatic characterizations, we provide an axiomatic characterization of the position value for arbitrary hypergraph communication situations by employing component efficiency and partial balanced conference contributions.

Motivation & Objective

• To resolve the open problem of axiomatic characterization of the position value for arbitrary hypergraph communication situations.
• To extend non-axiomatic characterizations of the position value—previously known only for graph communication situations—into the hypergraph setting.
• To introduce and formalize a new property, partial balanced conference contributions, as a key component in the axiomatic framework.
• To demonstrate that the position value is uniquely determined by component efficiency and partial balanced conference contributions in hypergraph settings.

Proposed method

• Introduce the uniform hyperlink game (U(H), w) derived from a hypergraph communication situation (N, v, H), where the characteristic function w(S) aggregates the worth of components in the induced hypergraph on S.
• Define the k-augmented uniform hyperlink game as a refinement of the uniform hyperlink game to strengthen the characterization under specific structural conditions.
• Establish that the Shapley value of the uniform hyperlink game equals the position value for the original hypergraph communication situation.
• Propose the new property, partial balanced conference contributions, which generalizes Slikker’s (2005) balanced link contributions to hypergraphs by analyzing payoff differences when a player breaks a hyperlink.
• Use induction on the number of hyperlinks |H| to prove uniqueness of the position value under component efficiency and partial balanced conference contributions.
• Compare the proposed uniform hyperlink game with the hyperlink agent form (HAF) from Casajus (2007), showing equivalence in payoff but greater conceptual and computational simplicity of the new game.

Experimental results

Research questions

• RQ1 Can the position value for arbitrary hypergraph communication situations be characterized non-axiomatically using modified games?
• RQ2 Is there a new axiomatic foundation for the position value in hypergraph settings that generalizes existing results from graph-based communication situations?
• RQ3 How does the concept of balanced contributions need to be adapted to hypergraphs, where links are hyperedges involving multiple players?
• RQ4 What is the relationship between the uniform hyperlink game and the hyperlink agent form (HAF) in terms of payoff equivalence and structural simplicity?
• RQ5 Does the position value remain uniquely determined by component efficiency and a generalized form of balanced contributions in hypergraph contexts?

Key findings

• The position value for any hypergraph communication situation is equal to the Shapley value of the uniform hyperlink game (U(H), w).
• The position value is also equal to the Shapley value of the k-augmented uniform hyperlink game, providing an alternative non-axiomatic characterization.
• The position value satisfies component efficiency, meaning the total payoff to players in each connected component equals the worth of that component.
• The position value satisfies partial balanced conference contributions, a new axiom that captures the symmetry of payoff changes when a player breaks a hyperlink.
• The position value is the unique allocation rule satisfying both component efficiency and partial balanced conference contributions, as proven by induction on |H|.
• The uniform hyperlink game yields the same Shapley payoff as the Myerson value of the hyperlink agent form (HAF), but is structurally simpler and more directly applicable to characterization.