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Recent Developments in the Theory of the Shapley Value · 1978-econ-15

Cooperative Game Theory · ICM 1978: Recent Developments in the Theory of the Shapley Value

MSC 91B · Mathematical economics

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Problem statement

Let (I,C)(I,\mathscr C) be a measurable space, where II is the set of players and C\mathscr C is a σ\sigma-field of coalitions. A game is a function v:C→Rv:\mathscr C\to\mathbb R such that v(∅)=0v(\varnothing)=0, and it is monotone if S⊇TS\supseteq T implies v(S)≥v(T)v(S)\geq v(T). An outcome is a finitely additive game.

For every nonatomic probability measure ν\nu on (I,C)(I,\mathscr C) and every quota q∈(0,1)q\in(0,1), define the nonatomic weighted-majority game

wq,ν(S)={0,ν(S)≤q,1,ν(S)>q,S∈C. w_{q,\nu}(S)= \begin{cases} 0,&\nu(S)\leq q,\\ 1,&\nu(S)>q, \end{cases} \qquad S\in\mathscr C.

Let QQ be the smallest set of games containing every such wq,νw_{q,\nu} and closed under pointwise addition, scalar multiplication, and multiplication.

For a measurable automorphism Θ:I→I\Theta:I\to I (a bijection such that both Θ\Theta and Θ−1\Theta^{-1} are measurable), define

(Θ∗v)(S)=v(Θ(S)),S∈C. (\Theta_*v)(S)=v(\Theta(S)),\qquad S\in\mathscr C.

A value on QQ is an operator φ:Q→{outcomes on (I,C)}\varphi:Q\to\{\text{outcomes on }(I,\mathscr C)\} satisfying all of the following:

  • Linearity: φ(av+bw)=aφ(v)+bφ(w)\varphi(av+bw)=a\varphi(v)+b\varphi(w) for all v,w∈Qv,w\in Q and a,b∈Ra,b\in\mathbb R.
  • Monotonicity: if v∈Qv\in Q is monotone, then φ(v)\varphi(v) is monotone.
  • Symmetry: φ(Θ∗v)=Θ∗φ(v)\varphi(\Theta_*v)=\Theta_*\varphi(v) for every measurable automorphism Θ\Theta and every v∈Qv\in Q.
  • Efficiency: (φv)(I)=v(I)(\varphi v)(I)=v(I) for every v∈Qv\in Q.

For a finite measurable partition Π\Pi of II, let F(Π)\mathscr F(\Pi) be the finite field generated by its atoms and let vΠv_\Pi be the finite game on those atoms defined by vΠ(A)=v(A)v_\Pi(A)=v(A) for A∈F(Π)A\in\mathscr F(\Pi). Write ψ\psi for the Shapley value of a finite game: if NN is its finite player set, then for i∈Ni\in N,

(ψv)({i})=1∣N∣!∑π(v(Piπ∪{i})−v(Piπ)), (\psi v)(\{i\}) =\frac1{|N|!}\sum_{\pi} \bigl(v(P_i^\pi\cup\{i\})-v(P_i^\pi)\bigr),

where the sum is over all permutations π\pi of NN and PiπP_i^\pi is the set of players preceding ii in π\pi; for a coalition A⊆NA\subseteq N, set (ψv)(A)=∑i∈A(ψv)({i})(\psi v)(A)=\sum_{i\in A}(\psi v)(\{i\}).

A value φ\varphi on QQ is called a partition value if, for every v∈Qv\in Q and every S∈CS\in\mathscr C, there exists a sequence of finite measurable partitions Π1,Π2,…\Pi_1,\Pi_2,\ldots such that each Πn+1\Pi_{n+1} refines Πn\Pi_n, S∈F(Π1)S\in\mathscr F(\Pi_1),

σ ⁣(⋃n≥1F(Πn))=C, \sigma\!\left(\bigcup_{n\geq1}\mathscr F(\Pi_n)\right)=\mathscr C,

and

lim⁡n→∞(ψvΠn)(S)=(φv)(S). \lim_{n\to\infty}(\psi v_{\Pi_n})(S)=(\varphi v)(S).

Determine whether a partition value exists on QQ.

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The problem is unresolved. The algebra QQ contains finite linear combinations of products of arbitrarily many nonatomic weighted-majority games. The strongest verified partition-value theorem covers individual scalar bounded-variation games and products with at most two singular scalar factors, but no located result extends it to products of arbitrary finite length. Conversely, no counterexample to the existence of a partition value on all of QQ was found.

Robert J. Aumann, “Recent Developments in the Theory of the Shapley Value,” Proceedings of the International Congress of Mathematicians, Helsinki 1978, Academia Scientiarum Fennica, 1980, pp. 995–1003, direct bibliographic record. Section 5 explicitly poses the existence of a partition value on the algebra generated by all nonatomic weighted-majority games as an open question.

Abraham Neyman and Yair Tauman, “The Partition Value,” Mathematics of Operations Research 4(3) (1979), 236–264, doi:10.1287/moor.4.3.236. Theorem 5.52 constructs a partition value on the closed linear space generated by ASYMP\mathrm{ASYMP}, bv′NAbv'NA, the algebra AA generated by continuous scalar measure games, and the product spaces A∗bv′NAA* bv'NA, bv′NA∗bv′NAbv'NA*bv'NA, and A∗bv′NA∗bv′NAA*bv'NA*bv'NA. Since a weighted-majority game is a scalar bv′NAbv'NA game, this proves the desired conclusion for individual generators, their linear combinations, and products involving at most two discontinuous weighted-majority factors. It does not cover the unrestricted algebra QQ.

Abraham Neyman, “Weighted Majority Games Have Asymptotic Value,” Mathematics of Operations Research 13(4) (1988), 556–580, doi:10.1287/moor.13.4.556. This proves that every weighted-majority game has an asymptotic value, which is stronger than having a partition value for each single generator. It gives no common partition value for arbitrary products of generators.

Jean-François Mertens, “Values and Derivatives,” Mathematics of Operations Research 5(4) (1980), 523–552, doi:10.1287/moor.5.4.523. Mertens constructs values on much larger spaces containing nondifferentiable games with jumps. This establishes relevant axiomatic-value existence, but the construction is not by limits of Shapley values along refining finite partitions and therefore does not establish the partition-value property required here.

Abraham Neyman, “Values of Games with Infinitely Many Players,” in Robert J. Aumann and Sergiu Hart (eds.), Handbook of Game Theory with Economic Applications, vol. 3, Elsevier, 2002, pp. 2121–2167, doi:10.1016/S1574-0005(02)03019-9. Section 9.2 surveys partition values and records only the Neyman–Tauman theorem above, while separately describing values on the closed algebra generated by bv′NAbv'NA. It supplies no partition-value extension to that algebra.

Abraham Neyman and Rann Smorodinsky, “Asymptotic Values of Vector Measure Games,” Mathematics of Operations Research 29(4) (2004), 739–775, doi:10.1287/moor.1040.0118. This proves asymptotic-value results for certain nonsmooth vector-measure and multi-house games under additional atomic “sufficient variety” hypotheses, and explicitly notes that not all nonatomic two-house weighted-majority games have asymptotic values. These conditional asymptotic results neither cover the fully nonatomic algebra QQ nor provide the missing global partition value.

Thus the established literature handles each generator and substantial low-degree or specially structured subclasses, and axiomatic values exist on broader ambient spaces. What remains unproved is precisely the existence (or nonexistence) of one linear, monotone, symmetric, efficient value on all finite products in QQ that also has the stated partition-limit property.

References
  1. [1]ICM Conjectures · 1978-econ-15 link
  2. [2]ICM 1978: Recent Developments in the Theory of the Shapley Value link

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