Saving Game in MCST: Cooperative Value Game

• Saving game is a cooperative game reformulation that converts MCST cost problems into value games by measuring coalition savings relative to individual connection costs.
• It leverages key properties such as nonnegativity, monotonicity, and superadditivity to enable a relative-error Monte Carlo approximation of the Shapley value.
• The approach also provides effective null-player detection and supports a fully polynomial-time randomized scheme for computing Shapley values.

A saving game is a cooperative-game reformulation of a minimum-cost spanning tree (MCST) game that converts a cost game into a value game by measuring coalition savings relative to individual connection to the root. In the formulation introduced for MCST games, the saving game is the key device that makes relative-error (multiplicative) approximation of the Shapley value possible via Monte Carlo sampling, whereas exact computation in the original cost formulation is $\#P$-hard and prior Monte Carlo methods provide only additive-error approximations (Jimbo et al., 24 Mar 2026). In this setting, the saving game is not merely a change of notation: it yields a nonnegative, monotone, superadditive game with structural properties that support an FPRAS for Shapley-value approximation.

1. Construction from minimum-cost spanning tree games

Let $(N,c)$ be an MCST game on the complete graph $\langle [N_r],w\rangle$, where $r$ is the root and, for any coalition $S\subseteq N$,

$c(S)=\text{cost of an MST on } [S\cup\{r\}].$

The associated saving game $(N,v)$ is defined by

$$v(S)=\sum_{i\in S} c(\{i\}) - c(S), \qquad v(\emptyset)=0.$$

Since $c(\{i\})=w(r,i)$, this becomes

$$v(S)=\sum_{i\in S} w(r,i)-c(S),\qquad S\neq\emptyset.$$

The interpretation is explicit: $(N,c)$0 is the total amount of cost saved by coalition $(N,c)$1 compared with each player connecting separately to the root. The paper names this reformulation the MCST-saving game (Jimbo et al., 24 Mar 2026).

The motivation for this construction comes from a pathology of MCST cost games. In the original cost-game formulation, a player can have Shapley value $(N,c)$2 even when the player is not a null player in the cost-game sense. The paper gives the example with nodes $(N,c)$3 and edge weights

$(N,c)$4

where the MCST game has Shapley value $(N,c)$5, even though player $(N,c)$6 is not dummy. In the associated saving game, the Shapley value becomes $(N,c)$7. This change isolates the coalition surplus created by shared connectivity and removes an obstacle to multiplicative approximation.

2. Relation to the original MCST cost game

The saving game is linked to the original MCST game through an exact Shapley-value transformation. If $(N,c)$8 is the Shapley value of the cost game $(N,c)$9 and $\langle [N_r],w\rangle$0 is the Shapley value of the saving game $\langle [N_r],w\rangle$1, then

$\langle [N_r],w\rangle$2

Hence,

$\langle [N_r],w\rangle$3

This identity is central because approximating the saving-game Shapley value immediately yields an approximation of the original MCST cost allocation. The paper highlights the decomposition

$\langle [N_r],w\rangle$4

which makes the stand-alone connection cost the baseline and the saving-game Shapley value the coalition-generated discount (Jimbo et al., 24 Mar 2026).

The reformulation also clarifies the player-role correspondence. For the saving game, a player $\langle [N_r],w\rangle$5 is a null player iff $\langle [N_r],w\rangle$6. In the original MCST cost game, this corresponds to $\langle [N_r],w\rangle$7 being a dummy player: $\langle [N_r],w\rangle$8 The paper emphasizes that this exact null-player characterization holds cleanly for saving games, but not for MCST cost games directly.

A plausible implication is that the saving-game view separates individual baseline cost from cooperative surplus in a way that aligns the Shapley value more closely with null-player structure.

3. Structural properties of MCST-saving games

The approximation theory depends on three structural properties of the saving game.

First, it satisfies nonnegativity: $\langle [N_r],w\rangle$9 This follows because $r$0 is always a feasible spanning tree for $r$1, so

$r$2

Second, it satisfies superadditivity: for disjoint coalitions $r$3,

$r$4

Third, it satisfies monotonicity: if $r$5, then

$r$6

These properties imply that Shapley values are nonnegative. More importantly for approximation, the paper proves that in the simple $r$7-$r$8 case, any non-null player has a uniform positive lower bound on the Shapley value: $r$9 This lower bound is the bridge from additive concentration bounds to relative-error guarantees (Jimbo et al., 24 Mar 2026).

The significance of this step is methodological. Prior Monte Carlo methods for MCST games gave additive-error approximations, but additive error can be weak when Shapley values differ greatly in magnitude across players. The saving-game reformulation avoids that weakness by placing the problem in a nonnegative value-game regime where multiplicative analysis becomes possible.

4. Null players, edge conditions, and lower bounds

A major theorem in the saving-game analysis is the null-player characterization. For an MCST-saving game $S\subseteq N$0 defined by $S\subseteq N$1, the following are equivalent for player $S\subseteq N$2:

1. $S\subseteq N$3;
2. $S\subseteq N$4 is a null player:

$S\subseteq N$5

1. for every $S\subseteq N$6,

$S\subseteq N$7

2. for every $S\subseteq N$8,

$S\subseteq N$9

Thus player $c(S)=\text{cost of an MST on } [S\cup\{r\}].$0 is null exactly when every edge $c(S)=\text{cost of an MST on } [S\cup\{r\}].$1 is at least as heavy as both root edges $c(S)=\text{cost of an MST on } [S\cup\{r\}].$2 and $c(S)=\text{cost of an MST on } [S\cup\{r\}].$3 (Jimbo et al., 24 Mar 2026).

The paper notes that this allows null players to be detected in $c(S)=\text{cost of an MST on } [S\cup\{r\}].$4 time per player, hence all null players can be eliminated in $c(S)=\text{cost of an MST on } [S\cup\{r\}].$5 time. In the simple $c(S)=\text{cost of an MST on } [S\cup\{r\}].$6-$c(S)=\text{cost of an MST on } [S\cup\{r\}].$7 case, the characterization sharpens further. For a player $c(S)=\text{cost of an MST on } [S\cup\{r\}].$8,

• $c(S)=\text{cost of an MST on } [S\cup\{r\}].$9 iff $(N,v)$0 is not null;
• equivalently, there exists $(N,v)$1 such that

$(N,v)$2

• and then

$(N,v)$3

This lower bound is what makes multiplicative approximation possible. A plausible implication is that the saving-game reformulation does not merely identify degenerate players; it also produces a quantitative floor for the contribution of every non-null player in the $(N,v)$4-$(N,v)$5 regime.

5. Monte Carlo approximation and the FPRAS

The algorithm samples a uniformly random permutation $(N,v)$6 of the players and computes each player’s marginal contribution in that order: $(N,v)$7 where $(N,v)$8 denotes the set of players appearing before $(N,v)$9 in $$v(S)=\sum_{i\in S} c(\{i\}) - c(S), \qquad v(\emptyset)=0.$$0. The Shapley value is the expectation of this marginal contribution: $$v(S)=\sum_{i\in S} c(\{i\}) - c(S), \qquad v(\emptyset)=0.$$1

Using $$v(S)=\sum_{i\in S} c(\{i\}) - c(S), \qquad v(\emptyset)=0.$$2 random permutations $$v(S)=\sum_{i\in S} c(\{i\}) - c(S), \qquad v(\emptyset)=0.$$3, the estimator is

$$v(S)=\sum_{i\in S} c(\{i\}) - c(S), \qquad v(\emptyset)=0.$$4

This estimator is unbiased: $$v(S)=\sum_{i\in S} c(\{i\}) - c(S), \qquad v(\emptyset)=0.$$5

For a fixed permutation, the algorithm incrementally updates MST costs for prefixes of the permutation. The paper shows that one can compute the needed sequence of coalition costs in

$$v(S)=\sum_{i\in S} c(\{i\}) - c(S), \qquad v(\emptyset)=0.$$6

time per sampled permutation, so total runtime is

$$v(S)=\sum_{i\in S} c(\{i\}) - c(S), \qquad v(\emptyset)=0.$$7

The key graph-theoretic observation is that the update from $$v(S)=\sum_{i\in S} c(\{i\}) - c(S), \qquad v(\emptyset)=0.$$8 to $$v(S)=\sum_{i\in S} c(\{i\}) - c(S), \qquad v(\emptyset)=0.$$9 can be handled on a planar auxiliary graph, allowing a linear-time MST computation on that subproblem (Jimbo et al., 24 Mar 2026).

The FPRAS proof uses bounded marginal contributions, Hoeffding’s inequality, and the positive lower bound for non-null players. In the simple $c(\{i\})=w(r,i)$0-$c(\{i\})=w(r,i)$1 case,

$c(\{i\})=w(r,i)$2

If

$c(\{i\})=w(r,i)$3

then for each player $c(\{i\})=w(r,i)$4,

$c(\{i\})=w(r,i)$5

If

$c(\{i\})=w(r,i)$6

then simultaneously for all players,

$c(\{i\})=w(r,i)$7

Because the required sample size is polynomial in $c(\{i\})=w(r,i)$8, $c(\{i\})=w(r,i)$9, and $$v(S)=\sum_{i\in S} w(r,i)-c(S),\qquad S\neq\emptyset.$$0, this is a fully polynomial-time randomized approximation scheme (FPRAS).

6. General nonnegative weights and transfer back to MCST games

For arbitrary nonnegative weights, the paper decomposes the game into a sum of simple $$v(S)=\sum_{i\in S} w(r,i)-c(S),\qquad S\neq\emptyset.$$1-$$v(S)=\sum_{i\in S} w(r,i)-c(S),\qquad S\neq\emptyset.$$2 games. Let the distinct positive weights be

$$v(S)=\sum_{i\in S} w(r,i)-c(S),\qquad S\neq\emptyset.$$3

and define thresholded $$v(S)=\sum_{i\in S} w(r,i)-c(S),\qquad S\neq\emptyset.$$4-$$v(S)=\sum_{i\in S} w(r,i)-c(S),\qquad S\neq\emptyset.$$5 weights

$$v(S)=\sum_{i\in S} w(r,i)-c(S),\qquad S\neq\emptyset.$$6

Then the MCST game decomposes as

$$v(S)=\sum_{i\in S} w(r,i)-c(S),\qquad S\neq\emptyset.$$7

and the saving game decomposes analogously: $$v(S)=\sum_{i\in S} w(r,i)-c(S),\qquad S\neq\emptyset.$$8 By additivity of the Shapley value,

$$v(S)=\sum_{i\in S} w(r,i)-c(S),\qquad S\neq\emptyset.$$9

This reduces the general case to repeated application of the simple-case estimator, with sample complexity increasing by a factor involving $(N,c)$00. If

$(N,c)$01

then each player’s relative error is at most $(N,c)$02 with probability at least $(N,c)$03; and if

$(N,c)$04

then this holds simultaneously for all players (Jimbo et al., 24 Mar 2026).

Once $(N,c)$05 is approximated for the saving game, the cost-game Shapley value is recovered by

$(N,c)$06

The paper explicitly states that the same approximation guarantees transfer to the original MCST game, with denominator $(N,c)$07 in the relative error bound: $(N,c)$08

The principal consequence is that the saving game becomes the mechanism that converts an intractable cost-allocation problem into a positive-valued cooperative game where multiplicative approximation is feasible.

7. Terminological scope and other uses of “saving game”

The term “saving game” is used in several unrelated literatures. In the cooperative-game-theoretic sense developed above, it denotes the value-game reformulation of an MCST cost game (Jimbo et al., 24 Mar 2026). In other papers, however, the phrase refers to different objects.

In health-economics game theory, “Saving Game” denotes a sequential producer–insurer game for life-saving medicines in which a monopoly patent-holder sets the drug price, an insurer sets a premium, and heterogeneous patients decide whether to insure and whether to purchase treatment (Coculescu et al., 19 Sep 2025). In energy systems, the expression appears in the context of a large-scale energy social game in a smart residential building, where occupants compete for points and incentives by reducing energy use (Das et al., 2019). In algorithmic graph theory, a closely related usage appears in the Firefighter setting, where the task is to save a designated critical set of vertices from a spreading fire; the paper formulates Saving A Critical Set (SACS) and proves that it is fixed-parameter tractable parameterized by the number of firefighters in the non-spreading model (Choudhari et al., 2017).

These usages are terminologically adjacent but conceptually distinct. The MCST-saving game is a cooperative value-game transformation; the producer–insurer model is a sequential market game; the energy social game is an incentive mechanism with human-in-the-loop control; and the firefighting formulation is a graph-theoretic saving problem. This suggests that “saving game” functions as a polysemous label whose meaning is determined by the surrounding field rather than by a single established cross-disciplinary definition.