Pith Number

pith:GZT2Q5OF

pith:2026:GZT2Q5OF2IAAH7RQ4M2M657MBL

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A Tractable Class of Cooperative Games Defined by Directed Networks: Unanimity Decomposition and Shapley Value

David Ryz\'ak, Tom\'a\v{s} Kroupa

Directed network games decompose into unanimity games, giving closed-form Shapley values and a nonempty core.

arxiv:2605.18157 v1 · 2026-05-18 · cs.GT · econ.TH

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4 Citations open

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Claims

C1strongest claim

The resulting game has a convenient representation in terms of unanimity games. This representation enables closed-form polynomial-time formulas for the Shapley and Banzhaf values. We further establish that the game has a nonempty core and is totally balanced.

C2weakest assumption

The coalitional value is defined to combine an internal interaction term given by the induced subgraph game with an external component based on minimal incoming edges from outside the coalition.

C3one line summary

Introduces directed network cooperative games with unanimity decomposition that admit closed-form Shapley and Banzhaf values and possess a nonempty core.

References

14 extracted · 14 resolved · 1 Pith anchors

[1] Trust in Shapley: A Cooperative Quest for Global Trust in P2P Network

[2] International Foundation for Autonomous Agents and Multiagent Systems

[3] Olga N. Bondareva. Some applications of linear programming methods to the theory of cooperative games.Problemy Kybernetiki, 10:119–139, 1963. 13 1963

[4] Operations research games: A survey.TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, 9:139–199, February 2001 2001

[5] The degree measure as utility function over positions in graphs and digraphs.European Journal of Operational Research, 299(3):1033–1044, 2022 2022

Formal links

1 machine-checked theorem link

Receipt and verification

First computed	2026-05-20T00:05:48.658529Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

3667a875c5d20003fe30e334cf77ec0af99e9670a78c3cad3d78a427604e649f

Aliases

arxiv: 2605.18157 · arxiv_version: 2605.18157v1 · doi: 10.48550/arxiv.2605.18157 · pith_short_12: GZT2Q5OF2IAA · pith_short_16: GZT2Q5OF2IAAH7RQ · pith_short_8: GZT2Q5OF

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/GZT2Q5OF2IAAH7RQ4M2M657MBL \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: 3667a875c5d20003fe30e334cf77ec0af99e9670a78c3cad3d78a427604e649f

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "9d7f95b8e1560443dde4035807e8dfc1796ff00594528592bbd5d6ee15966ce2",
    "cross_cats_sorted": [
      "econ.TH"
    ],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "cs.GT",
    "submitted_at": "2026-05-18T10:02:55Z",
    "title_canon_sha256": "58328416ac728ef3dded7fe9b3e4efa9b8d14c77089e5429bc5af8bc9da1a100"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.18157",
    "kind": "arxiv",
    "version": 1
  }
}