Nullified-Game Consistency Axiom

• Nullified-Game Consistency Axiom is a principle in cooperative TU-games that ensures invariant allocations by fixing certain players' payoffs.
• It employs specific nullification operations (HM, Funaki, Moulin, and projection methods) to adjust coalition values consistently.
• The axiom uniquely characterizes allocation rules such as the Shapley, CIS, ENSC, and proportional division values with efficiency and fairness.

The nullified-game consistency axiom is a principle used in the axiomatic characterization of value allocation rules for cooperative games with transferable utility (TU-games). It formalizes the requirement that, when the payoffs of a fixed subset of players are set to their original values and those players are rendered strategically null, the allocation assigned to the remaining players must not change, provided an appropriate adjustment of the underlying cooperative game to account for the removed payoffs. This property, in conjunction with complementary fairness and efficiency axioms, uniquely specifies prominent allocation rules such as the Shapley value, CIS value, ENSC value, and the proportional division value within their respective domains and structural classes.

1. Formal Definition and Nullification Operations

Given a finite player set $N$ with $|N| = n \geq 3$ and a TU-game $v: 2^N \to \mathbb{R}$ with $v(\emptyset) = 0$, an allocation rule $\varphi: \mathcal{V} \to \mathbb{R}^N$ assigns a value $\varphi_i(v)$ to each player $i$. For any subset $S \subseteq N$, the $S$-nullified game $v|_S$ is defined by

$$(v|_S)(T) = v(T \cap S), \qquad \forall T \subseteq N,$$

so that players in $N \setminus S$ have become null (i.e., contribute nothing to any coalition). This operation is extended to residual games that fix the payoffs of nullified players and adjust coalition worths for the remaining (active) players according to specific rules.

Three standard residual game operators arise in the context of efficient, linear, and symmetric (ELS) values:

(a) HM-nullified game (Hart–Maschler):

$$R^{\mathrm{HM},S}(x, v)(T) = \begin{cases} v(T \cup (N \setminus S)) - \sum_{j \in N \setminus S} \varphi_j(v|_{T \cup (N \setminus S)}) & \text{if } T \cap S \neq \emptyset, \ 0 & \text{otherwise}. \end{cases}$$

(b) Funaki-nullified game (F):

$$R^{\mathrm{F},S}(x, v)(T) = \begin{cases} v(N) - \sum_{j \in N \setminus S} \varphi_j(v|_{\{j\}}) & \text{if } S \subseteq T, \ v(T \cap S) & \text{otherwise}. \end{cases}$$

(c) Moulin-nullified game (M):

$$R^{\mathrm{M},S}(x, v)(T) = \begin{cases} v(T \cup (N \setminus S)) - \sum_{j \in N \setminus S} \varphi_j(v^*|_{\{j\}}) & \text{if } T \cap S \neq \emptyset, \ 0 & \text{otherwise}, \end{cases}$$

where $v^*(T) = v(N) - v(N \setminus T)$ is the dual game.

More generally, for the class of games where division is proportional to stand-alone worths, the projection nullified-game consistency axiom (PNC) uses

$$R^S(x, v)(T) = \begin{cases} v(N) - \sum_{k \in N \setminus S} x_k & \text{if } S \subseteq T, \ v(T \cap S) & \text{otherwise}. \end{cases}$$

defined for $v$ in the set $$\widehat{\mathcal{V}} := \{v \in \mathcal{V} : v(\{i\}) \geq 0 \; \forall i \text{ or } v(\{i\}) \leq 0 \; \forall i\}$$.

2. Statement of the Nullified-Game Consistency Axioms

Let $I$ be one of $\{\mathrm{HM}, \mathrm{F}, \mathrm{M}\}$ for ELS values, or $P$ for the proportional rule. The $I$-nullified-game consistency axiom (I-NGC) posts that for all $v$ in the relevant game class, all $S \subseteq N$ with suitable nontriviality (\emph{e.g.}, $2 \leq |S| < n$ and some $j \in S$ with $v(\{j\}) \neq 0$), and each $i \in S$: $$\varphi_i(v) = \varphi_i\big(R^{I,S}(\varphi(v), v)\big).$$

For the proportional rule (PNC), the formal statement is: $$\forall v \in \widehat{\mathcal{V}},\; \forall S \subseteq N \text{ with } \exists j \in S: v(\{j\}) \neq 0,\; \forall i \in S:\quad \varphi_i(v) = \varphi_i(R^S(\varphi(v), v)).$$

This expresses that once the payoffs of nullified players have been fixed, the allocation to the remaining players determined by the rule in the residual, nullified game must coincide with their original allocations.

3. Economic and Game-Theoretic Significance

Nullified-game consistency encodes a strong form of fixed-population consistency. Nullifying $N \setminus S$ means treating those players as if they permanently exit the bargaining process after their fixed payoffs are assigned, leaving a residual game among $S$ that is adjusted by subtracting committed payoffs from coalition values where relevant. The axiom ensures that the allocation mechanism is robust to such sequential reduction: the method of dividing the remaining surplus among $S$ is invariant under the prior removal of other players, provided those removals are conducted according to the prescribed nullification mechanics.

Economically, this implies that the method is consistent with iterative, coalitionwise bargaining in which the outcome is insensitive to the order or partitioning of side-deals, as long as nullified players are handled by fixing their payoffs and removing their influence from the residual game structure.

4. Interaction with Other Axioms and Characterization Results

Within each addressed value class, nullified-game consistency is paired with efficiency and a two-player fairness axiom to achieve uniqueness of the allocation rule:

Value Class / Rule	Nullified Consistency	Complementary Axioms	Characterized Value
ELS (Shapley, CIS, ENSC)	HM/F/M-NGC	Efficiency (E), Equal-Gain for Two (EG)	Shapley/CIS/ENSC
Proportional Value	PNC	Efficiency (E), Equal-Ratio for Two Players (ER)	Proportional Division Value

• Efficiency (E): $\sum_{i \in N} \varphi_i(v) = v(N)$.
• Equal-Gain for Two (EG): If all but $i, j$ are null, then $\varphi_i(v) - v(\{i\}) = \varphi_j(v) - v(\{j\})$.
• Equal-Ratio for Two (ER): $\varphi_i(v)v(\{j\}) = \varphi_j(v)v(\{i\})$ for all $i, j$.

The fundamental characterization theorems (Funaki et al., 7 Nov 2025, Funaki et al., 7 Nov 2025) establish that:

• For ELS values, each form of the nullified-game consistency axiom (HM-NGC, F-NGC, M-NGC), together with (E) and (EG), singles out the Shapley, CIS, or ENSC value, respectively.
• For the proportional division value, PNC together with (E) and (ER) is equivalent to the proportional rule.

The same works show that nullified-game consistency is independent of other usual invariance or decomposition axioms employed in classical characterizations, providing a robust alternative foundation for value selection.

5. Core Proof Strategies and Illustrative Example

The central idea in all uniqueness proofs is the reduction to two-player subgames, using the nullification operation to isolate player pairs, and then employing the two-player fairness/equality axiom to determine the difference in payoffs. Efficiency pins down the sum, ensuring uniqueness for all player allocations.

For example, in the Funaki-nullified case (F-NGC ⇒ CIS value), nullifying player 3 (at $x_3 = v(\{3\})$) in a three-player game with $v(\{1\})=1$, $v(\{2\})=2$, $v(\{3\})=0$, $v(\{1,2\})=5$, $v(\{1,3\})=1$, $v(\{2,3\})=2$, $v(N)=6$, the residual game for $S = \{1,2\}$ is $R^{F,\{1,2\}}(x,v)$ with $R(\{1\})=1$, $R(\{2\})=2$, $R(N)=6$. Applying the CIS formula,

$$CIS_1(w) = w(\{1\}) + \tfrac{1}{3}(w(N) - [w(\{1\}) + w(\{2\}) + w(\{3\})]) = 1 + \tfrac{1}{3}(6-3) = 2,$$

which is consistent with the original CIS allocation.

6. Independence, Comparisons, and Broader Role

The nullified-game consistency axiom is shown to be independent of traditional axioms such as homogeneity or additivity. Counterexamples demonstrate that value rules can satisfy efficiency and fairness criteria without meeting the consistency property, or vice versa, depending on their structure. This suggests the conceptual strength of nullified-game consistency as a substitute, not a derivative, of classic invariance and composition requirements.

In multi-axiom characterizations—for instance, those of the proportional value—nullified-game consistency can substitute more complex sets of axioms (homogeneity, normalization, decomposition), streamlining the theory of value rules and clarifying the roles of fixed-population and sequential consistency.

7. Summary of Impact and Applications

Nullified-game consistency axioms have direct implications for the design and justification of value allocation mechanisms in cooperative games, underpinning the robustness of leading solutions to player removal and side-deal adjustments. In the context of ELS values, these axioms delineate the boundary between the Shapley, CIS, and ENSC values, providing an axiomatic toolset for value selection in contexts such as cost-sharing, resource allocation, and fair division problems. Their role in characterizing the proportional division rule further indicates their generality and the centrality of fixed-population invariance as a guiding principle in value theory for TU-games (Funaki et al., 7 Nov 2025, Funaki et al., 7 Nov 2025).