Subset Games: Models, Strategies, and Applications

• Subset games are a family of models where subsets serve as the primary objects for defining game states, strategies, and payoffs across various frameworks.
• They encompass cooperative, impartial, online, and search formulations, each with distinct methods such as subset team games and Rényi–Ulam search.
• Advanced algebraic and combinatorial techniques, including Möbius transforms and symmetry reductions, are crucial for analyzing and solving these games.

Subset games are studied in several distinct research traditions in which subsets are the primary objects of play, evaluation, or representation. In the cited literature, the term covers cooperative games whose payoff is defined for every assessing subset and coalition outcome, games in partition function form defined on embedded subsets, impartial games played on subset lattices or simplicial complexes, online graph problems whose solutions are subsets of vertices, and subset-membership search games with restricted lies (0907.2376, Grabisch, 2010, Khandhawit et al., 2011, Fuchs et al., 2022, Beluhov, 2016). This suggests that “subset games” is not a single canonical formalism, but a family of models organized around subset-indexed states, moves, or utilities.

1. Formal scope and recurring domains

Several non-equivalent objects are called subset games in the literature. What they share is that subsets are not merely auxiliary notation; they are the basic carriers of value, legality, or strategy.

Family	Primitive subset object	Representative formalism
Subset team games	Assessing subsets and coalitions	$u_A(S)=u_A(V(S))$
Partition function form	Embedded subsets	$(S,\pi)$ with $S\in\pi$
Subset take-away	Faces of a simplicial complex	remove $U(\sigma)$
Rényi–Ulam search	Query subsets of $[n]$	“Is the number in $Q\subseteq[n]$?”
Online vertex subset games	Solution subsets of vertices	irrevocable inclusion in $S\subseteq V$

A subset team game consists of a team of players $T$, a set $X$ of possible outcomes, a consequence function $V:2^T\to X$, and a payoff or utility function $(S,\pi)$0 defined for each subset $(S,\pi)$1; the induced subset utility is $(S,\pi)$2 (0907.2376). Games in partition function form instead use embedded subsets, defined as pairs $(S,\pi)$3 where $(S,\pi)$4 is a partition of $(S,\pi)$5 and $(S,\pi)$6, with $(S,\pi)$7 (Grabisch, 2010). Subset take-away begins with a simplicial complex $(S,\pi)$8 and allows a move on a face $(S,\pi)$9, removing the up-set

$S\in\pi$0

while Graph Chomp is the special case in which $S\in\pi$1 has only vertices and edges (Khandhawit et al., 2011).

In Rényi–Ulam games, the Seeker asks whether a hidden number belongs to a chosen subset $S\in\pi$2, and the Obscurer answers truthfully or falsely subject to a restriction $S\in\pi$3 on the truth/lie sequence (Beluhov, 2016). In online graph games, the solution itself is a subset of vertices, but the subset must be built online: a deterministic algorithm receives vertices under a neighborhood-reveal model, must make irrevocable decisions, and is equipped with an unlabeled map, an isomorphic copy of the full graph with vertex labels removed (Fuchs et al., 2022).

A common misconception is to treat these models as instances of a single theory. The papers instead show several independent subset-based formalisms, each with its own state space, equilibrium notion, and computational machinery.

2. Subset-valued utilities in cooperative and hedonic games

The most explicit utility-theoretic formalization is the subset team game. For disjoint coalitions $S\in\pi$4, the total marginal contribution of $S\in\pi$5 to $S\in\pi$6 is

$S\in\pi$7

the competitive contribution is

$S\in\pi$8

and the altruistic contribution is

$S\in\pi$9

with

$U(\sigma)$0

Sensibility requires $U(\sigma)$1 for all disjoint $U(\sigma)$2, cohesiveness requires $U(\sigma)$3 for all disjoint $U(\sigma)$4, a fully-cooperative game is one in which all coalitions are cohesive, and the ST-core of a consequence function $U(\sigma)$5 is the set of utility functions that create a fully-cooperative game (0907.2376).

The same framework introduces a cooperation space in which the pair $U(\sigma)$6 locates a subset’s behavior. Quadrant I, defined by $U(\sigma)$7 and $U(\sigma)$8, is both sensible and fully-cooperative; Quadrant II is sensible but non-cohesive; Quadrants III and IV are non-sensible. This decomposition is used to reinterpret the prisoner’s dilemma. In the paper’s two-player specification, each player’s altruistic contribution is $U(\sigma)$9, each player’s selfish contribution is $[n]$0, and the point $[n]$1 lies in Quadrant I, so cooperation is both sensible and cohesive (0907.2376).

Subset team games also recover standard NTU and TU models as special cases. If $[n]$2, every NTU game with individual utilities $[n]$3 is an ST game. If $[n]$4 for all disjoint $[n]$5, then all subsets value every outcome equally and the game reduces to a TU game. The paper further isolates additive, co-additive, and bi-additive cases. For example, in additive games, $[n]$6 and the game is fully-cooperative if and only if $[n]$7 for all players $[n]$8 (0907.2376).

A related but distinct subset-based valuation appears in hedonic games. A preference profile is subset-additive if, for each player $[n]$9, there exists $Q\subseteq[n]$0 such that

$Q\subseteq[n]$1

for all coalitions $Q\subseteq[n]$2. The paper proves that any hedonic game is also a subset-additive hedonic game by defining

$Q\subseteq[n]$3

Subset-neutral hedonic games impose a common set-function $Q\subseteq[n]$4 shared by all players, and neutrally anonymous hedonic games further restrict preferences to coalition size through a common function $Q\subseteq[n]$5 (Suksompong, 2018).

Neutrality yields an exact potential:

$Q\subseteq[n]$6

A partition maximizing $Q\subseteq[n]$7 is Nash stable and therefore individually stable in subset-neutral hedonic games. For games satisfying the common ranking property, the paper gives an explicit algorithm: initialize $Q\subseteq[n]$8; repeatedly choose a coalition $Q\subseteq[n]$9 that ranks highest, breaking ties by largest size; remove $S\subseteq V$0 from $S\subseteq V$1; and output the resulting partition. In neutrally anonymous games this produces a partition that is core stable, individually stable, and contractually individually stable (Suksompong, 2018).

3. Embedded subsets, externalities, and subset strategy systems

In partition function form, the central objects are embedded subsets rather than ordinary coalitions. The set

$S\subseteq V$2

is endowed with the order

$S\subseteq V$3

where $S\subseteq V$4 on $S\subseteq V$5 is the refinement order. After adjoining a bottom element $S\subseteq V$6, the resulting structure is a lattice with top element $S\subseteq V$7, rank function $S\subseteq V$8 when $S\subseteq V$9 is a $T$0-partition, and cardinality

$T$1

where $T$2 is the Stirling number of the second kind (Grabisch, 2010).

The lattice admits explicit meet and join operations. For $T$3,

$T$4

and the join is obtained by taking $T$5 in the partition lattice, locating the blocks containing $T$6 and $T$7, merging them, and returning the resulting embedded subset. The lattice is not distributive, not atomistic, not geometric, and not modular, but it is upper semimodular (Grabisch, 2010).

This structure restores Möbius and zeta calculus for games with externalities. For a game $T$8 with $T$9, the Möbius transform is

$X$0

with inversion

$X$1

The unanimity basis $X$2 defined by $X$3 if $X$4 and $X$5 otherwise yields the decomposition $X$6. The paper also shows that, for $X$7, additive games do not exist on $X$8 except for $X$9 (Grabisch, 2010).

A different subset-strategy formalism appears in resource buying games. Here $V:2^T\to X$0, $V:2^T\to X$1, and each player $V:2^T\to X$2 chooses a feasible subset from a family $V:2^T\to X$3, together with a payment vector $V:2^T\to X$4. A resource $V:2^T\to X$5 is purchased if $V:2^T\to X$6, where $V:2^T\to X$7 is the load on $V:2^T\to X$8. For marginally non-decreasing cost functions, every resource buying game admits a pure Nash equilibrium, and a socially optimal profile can be stabilized by marginal-cost pricing. For marginally non-increasing cost functions, unweighted matroid resource buying games admit a pure Nash equilibrium and a polynomial-time algorithm, while matroids are exactly the right structure: for every non-matroid anti-chain $V:2^T\to X$9, there exists a two-player weighted resource buying game with marginally non-increasing costs and no pure Nash equilibrium (Harks et al., 2012).

This combination of embedded-subset lattices and subset strategy systems shows two different ways of treating externalities. One refines the state space of coalitions; the other constrains feasible actions to subsets of a common ground set.

4. Search, revelation, and online subset selection

In Rényi–Ulam games, subsets appear as adaptive queries. The Obscurer chooses a hidden number from $(S,\pi)$00, and the Seeker repeatedly asks whether the number lies in a subset $(S,\pi)$01. The answers form a binary truth/lie string in a prefix-closed, extensible language $(S,\pi)$02 determined by a restriction $(S,\pi)$03. The paper formalizes $(S,\pi)$04 by a restriction graph $(S,\pi)$05, a finite automaton-like object with labeled $(S,\pi)$06- and $(S,\pi)$07-arcs. A position assigns a nonnegative integer to each vertex of $(S,\pi)$08, and a strategy tree is a binary tree of such positions whose leaves have weight $(S,\pi)$09 or $(S,\pi)$10 (Beluhov, 2016).

The principal reduction is from arbitrary positions to weight-two positions. Theorem 1 states that the Seeker wins from all positions if and only if the Seeker wins from all positions of weight two; Theorem 2 states that the Seeker wins from all positions of the form $(S,\pi)$11 if and only if the Seeker wins from all unit-support weight-two positions. For regular restrictions with a finite restriction graph on $(S,\pi)$12 vertices, if the Seeker wins for $(S,\pi)$13, then the Seeker wins for all $(S,\pi)$14; moreover, if the Seeker wins for all $(S,\pi)$15, then the Seeker wins in $(S,\pi)$16 steps, with

$(S,\pi)$17

where $(S,\pi)$18 is the number of iterations of $(S,\pi)$19 until reaching $(S,\pi)$20 (Beluhov, 2016).

For restrictions characterized by forbidden substrings, the paper gives a complete classification for two forbidden substrings. It builds on the Czyzowicz–Lakshmanan–Pelc theorem for a single forbidden substring, which states that when $(S,\pi)$21, the Seeker wins if and only if $(S,\pi)$22. For $(S,\pi)$23, Theorem 15 classifies all Seeker-win pairs into three classes: all pairs expanded by a single-element Seeker-win set; $(S,\pi)$24 sporadic pairs; and six infinite families determined by

$(S,\pi)$25

together with its bitwise inversion (Beluhov, 2016).

Online vertex subset games place subsets under adversarial revelation. For a vertex-subset problem such as Vertex Cover, Independent Set, or Dominating Set, the adversary reveals the graph vertex-by-vertex under the neighborhood-reveal model, while the online algorithm must decide irrevocably whether the currently revealed vertex belongs to the solution subset $(S,\pi)$26. The algorithm is given an unlabeled map, an isomorphic copy of the graph with vertex labels removed, so it can recognize structures only up to automorphisms compatible with the reveal history (Fuchs et al., 2022).

The paper gives a general TQBF-to-online-game reduction scheme based on three kinds of gadgets: fake clause gadgets, dependency reveal gadgets, and ID gadgets. The goal is to extend a 3-SAT gadget reduction for the offline problem into a PSPACE reduction for the online game while preserving a one-to-one correspondence between satisfying assignments and feasible subset solutions. Using this framework, the online versions with a map of Vertex Cover, Independent Set, and Dominating Set are proved PSPACE-complete (Fuchs et al., 2022).

These two lines of work treat subsets differently. In Rényi–Ulam games, a subset is an information query. In online vertex subset games, it is the sought solution itself. In both cases, however, the adversarial structure is driven by partial revelation.

5. Impartial subset-removal, chomp, and saturation

Subset take-away is an impartial normal-play game on a simplicial complex $(S,\pi)$27. A move chooses a face $(S,\pi)$28 and replaces $(S,\pi)$29 by $(S,\pi)$30, where $(S,\pi)$31. Graph Chomp is the special case in which $(S,\pi)$32 contains only vertices and edges. The Sprague–Grundy function is

$(S,\pi)$33

A central symmetry theorem states that if $(S,\pi)$34 is an involution on $(S,\pi)$35 and the fixed-point set $(S,\pi)$36 is also a simplicial complex, then

$(S,\pi)$37

This reduction yields complete analyses for complete $(S,\pi)$38-partite graphs and for all bipartite graphs (Khandhawit et al., 2011).

For the complete graph $(S,\pi)$39, $(S,\pi)$40. More generally, if $(S,\pi)$41 is complete $(S,\pi)$42-partite and $(S,\pi)$43 is the number of odd $(S,\pi)$44, then

$(S,\pi)$45

For any bipartite graph $(S,\pi)$46 with $(S,\pi)$47 vertices and $(S,\pi)$48 edges,

$(S,\pi)$49

The same paper gives partial results for odd-cycle pseudotrees, including a $(S,\pi)$50 block formula for the family $(S,\pi)$51 and a classification of hairballs (Khandhawit et al., 2011).

On the Boolean lattice $(S,\pi)$52, Subset Takeaway becomes Chomp on a hypercube. A later computation-driven paper disproves two Gale–Neyman conjectures. In $(S,\pi)$53, the only winning first moves are $(S,\pi)$54-sets, so the top element $(S,\pi)$55 is not a winning first move. In the truncated lattices $(S,\pi)$56, the claim that the first player loses if and only if $(S,\pi)$57 divides $(S,\pi)$58 fails in both directions: $(S,\pi)$59 is a first-player loss although $(S,\pi)$60, and $(S,\pi)$61 is a first-player win although $(S,\pi)$62 (Brouwer et al., 2017).

The same paper develops a memoized, symmetry-aware algorithm for recursively defined functions on posets. It computes the number of linear extensions of $(S,\pi)$63 exactly for the first time, obtaining a $(S,\pi)$64-digit integer with

$(S,\pi)$65

and derives a closed product formula for $(S,\pi)$66 (Brouwer et al., 2017).

Another impartial subset-claiming model is the intersecting saturation game on

$(S,\pi)$67

Fast and Slow alternately claim unclaimed $(S,\pi)$68-sets while maintaining an intersecting family; the game ends when the claimed family is maximal intersecting. If $(S,\pi)$69 denotes the family of intersecting $(S,\pi)$70-uniform set families, then the game saturation numbers satisfy

$(S,\pi)$71

The key parameter is the covering number $(S,\pi)$72: for a maximal intersecting family with $(S,\pi)$73,

$(S,\pi)$74

Thus Fast attempts to force large cover number, while Slow attempts to force small cover number (Patkos et al., 2012).

A further convexity-based removing game is played on a graph $(S,\pi)$75 by selecting vertices until the convex hull of the remaining unselected vertices is too small. On grid graphs $(S,\pi)$76, geodetically convex sets are precisely axis-aligned rectangles. The paper studies the achievement game $(S,\pi)$77, called $(S,\pi)$78, and the avoidance game $(S,\pi)$79, called $(S,\pi)$80. For grids,

$(S,\pi)$81

and

$(S,\pi)$82

and for lattice graphs $(S,\pi)$83,

$(S,\pi)$84

The analysis relies on delayed gamegraphs, option-preserving maps, and case analysis diagrams (Benesh et al., 13 May 2025).

6. Structural methods, algebraic viewpoints, and open directions

Across these literatures, several technical methods recur. One is symmetry reduction. In subset take-away and Graph Chomp, involutions reduce a position to its fixed-point core when that core remains a simplicial complex (Khandhawit et al., 2011). In convex-hull removing games, option-preserving maps reduce grid games to matrix games, and delayed gamegraphs isolate parity effects (Benesh et al., 13 May 2025). In Boolean-lattice computations, canonicalization under $(S,\pi)$85 and memoization make exact Grundy and linear-extension calculations feasible up to $(S,\pi)$86 (Brouwer et al., 2017).

A second recurrent method is algebraic decomposition. In partition function form, the zeta and Möbius operators on the lattice of embedded subsets produce Harsanyi-like dividends and unanimity bases (Grabisch, 2010). In subset-neutral hedonic games, a common subset-value function yields the potential

$(S,\pi)$87

which aligns unilateral improvement with increases in $(S,\pi)$88 (Suksompong, 2018). In subset team games, the identity $(S,\pi)$89 separates coalition-wide gains into competitive and altruistic components (0907.2376).

A third method is automaton or gadget encoding. Restriction graphs for Rényi–Ulam games encode admissible truth/lie histories and enable reductions to weight-two positions and two-string games (Beluhov, 2016). TQBF gadget frameworks for online vertex subset games use fake clause, dependency reveal, and ID gadgets to enforce quantifier order and information asymmetry (Fuchs et al., 2022). These constructions show that subset games may be algorithmically easy on highly symmetric families and PSPACE-complete under adversarial revelation.

A more idiosyncratic algebraic viewpoint is proposed for board games by representing each legal move as a matrix in a proper subset

$(S,\pi)$90

with $(S,\pi)$91, or in the general two-player case

$(S,\pi)$92

The paper argues that these movement sets are neither rings nor groups because the additive identity and multiplicative identity are excluded and closure may fail (Hwang et al., 18 Feb 2025). This is a different use of “subset games” from the cooperative, impartial, and online models above, but it reinforces the same structural theme: the legally relevant objects form a constrained subset of a larger ambient space.

The open problems are correspondingly diverse. In subset team games, dynamic evolution, learning, and empirical calibration are identified as natural extensions (0907.2376). In Rényi–Ulam games, the full classification for $(S,\pi)$93 remains open, and three forbidden substrings already produce arithmetic phenomena such as the condition $(S,\pi)$94 for $(S,\pi)$95 (Beluhov, 2016). In intersecting saturation games, Conjecture 3.1 asks whether there exists $(S,\pi)$96 such that

$(S,\pi)$97

which would amount to forcing cover number linear in $(S,\pi)$98 (Patkos et al., 2012). In grid and lattice removing games, the odd-parity achievement nim-number in dimensions $(S,\pi)$99 is left unresolved (Benesh et al., 13 May 2025).

Taken together, these results show that subset games are best understood through the structures they place on subsets: coalition assessment, partition environment, closure under removal, admissible query histories, or online feasibility under reveal constraints. The unifying object is the subset itself, but the theory depends on which ambient structure—lattice, simplicial complex, partition lattice, automaton, or graph—governs its legal or strategic role.