Alignment Games: Zero-Sum Set Selection

• Alignment games are a class of two-player, zero-sum search/intervention games where payoffs are determined by the misalignment between selected and hidden sets.
• They operationalize trade-offs between commission errors (unnecessary actions) and omission errors (missed targets) using cost and penalty functions in various settings.
• The framework includes continuous geometric models and discrete ratio-based equilibria, with special cases such as Matching Pennies emerging naturally.

Searching arXiv for the primary paper and closely related uses of the term "alignment games". Search results reviewed. Proceeding to synthesize the encyclopedia entry grounded in the provided paper and adjacent papers. Alignment games are a class of two-player, zero-sum search/intervention games in which the central issue is not merely whether a Searcher finds a Hider, but whether the Searcher’s action is well aligned with an underlying hidden state. In the formulation introduced in "Alignment Games" (Gerum et al., 2 Sep 2025), both players choose subsets of a ground set, and payoff is determined by their misalignment, measured through the cost of unnecessary action and the penalty for missed targets. The framework is motivated by operational problems in medical diagnostics, economic sanctions, and resource allocation, and is broad enough that classical models such as Matching Pennies emerge as special cases (Gerum et al., 2 Sep 2025).

1. Core model and interpretation

The core model specifies a ground set $Q$, a cost function $C:2^Q \to \mathbb{R}$, and a penalty function $\Pi:2^Q \to \mathbb{R}$. The Hider chooses a set $H \in \mathcal H \subseteq 2^Q$, and the Searcher chooses a set $S \in \mathcal S \subseteq 2^Q$. The payoff to the Hider is

$P(H,S)=C(S\setminus H)+\Pi(H\setminus S).$

Because the game is zero-sum, the Searcher minimizes this payoff and the Hider maximizes it (Gerum et al., 2 Sep 2025).

The decomposition of the payoff is the defining feature of the framework. The term $S\setminus H$ corresponds to places searched where nothing is there, interpreted as commission error or unnecessary action, and is charged through $C$. The term $H\setminus S$ corresponds to places that should have been searched but were missed, interpreted as omission error or missed target, and is charged through $\Pi$. Alignment games therefore make explicit the trade-off between acting too much in the wrong places and failing to act in the right places (Gerum et al., 2 Sep 2025).

When costs and penalties are equal, $C:2^Q \to \mathbb{R}$0, the payoff simplifies to symmetric-difference loss: $C:2^Q \to \mathbb{R}$1 In that regime, misalignment itself is the quantity being minimized or maximized. This establishes the paper’s central conceptual point: alignment is represented as set-theoretic agreement between hidden and chosen sets, and strategic behavior is organized around the geometry or combinatorics of that agreement (Gerum et al., 2 Sep 2025).

2. Continuous geometric forms

The paper develops continuous alignment games first on the unit circle $C:2^Q \to \mathbb{R}$2 with endpoints identified. The Hider chooses an arc of length $C:2^Q \to \mathbb{R}$3, the Searcher an arc of length $C:2^Q \to \mathbb{R}$4, and costs are proportional to arc length: $C:2^Q \to \mathbb{R}$5 If both starting points are chosen uniformly at random, the expected payoff is

$C:2^Q \to \mathbb{R}$6

This is the expected cost of the Searcher’s wasted arc plus the expected penalty for missed Hider territory (Gerum et al., 2 Sep 2025).

When both players choose arc lengths strategically, the equilibrium has a closed form: $C:2^Q \to \mathbb{R}$7 with value

$C:2^Q \to \mathbb{R}$8

The paper characterizes this as a ratio-governed solution: the Hider chooses the longer arc when omission penalties dominate, and the Searcher chooses the longer arc when commission costs dominate. This ratio structure is one of the article’s recurring themes and later reappears in discrete product-form equilibria (Gerum et al., 2 Sep 2025).

The circle game also exhibits sharp threshold behavior. If the Hider’s arc length $C:2^Q \to \mathbb{R}$9 is fixed and the Searcher chooses $\Pi:2^Q \to \mathbb{R}$0, then

$\Pi:2^Q \to \mathbb{R}$1

Hence, if $\Pi:2^Q \to \mathbb{R}$2, the Searcher chooses $\Pi:2^Q \to \mathbb{R}$3, while if $\Pi:2^Q \to \mathbb{R}$4, the Searcher chooses $\Pi:2^Q \to \mathbb{R}$5. Dually, if $\Pi:2^Q \to \mathbb{R}$6, the Hider chooses $\Pi:2^Q \to \mathbb{R}$7, and if $\Pi:2^Q \to \mathbb{R}$8, the Hider chooses $\Pi:2^Q \to \mathbb{R}$9. These are explicit threshold rules governed by the cost-to-penalty balance (Gerum et al., 2 Sep 2025).

A useful summary of the main continuous and discrete settings is given below.

Setting	Equilibrium feature	Value
Unit circle, unequal $H \in \mathcal H \subseteq 2^Q$0	$H \in \mathcal H \subseteq 2^Q$1	$H \in \mathcal H \subseteq 2^Q$2
Unit interval, $H \in \mathcal H \subseteq 2^Q$3, both choose half-intervals	Equal mixing between $H \in \mathcal H \subseteq 2^Q$4 and $H \in \mathcal H \subseteq 2^Q$5	$H \in \mathcal H \subseteq 2^Q$6
Discrete, $H \in \mathcal H \subseteq 2^Q$7	$H \in \mathcal H \subseteq 2^Q$8	$H \in \mathcal H \subseteq 2^Q$9
One-element game $S \in \mathcal S \subseteq 2^Q$0	Threshold support governed by $S \in \mathcal S \subseteq 2^Q$1	$S \in \mathcal S \subseteq 2^Q$2 or $S \in \mathcal S \subseteq 2^Q$3 depending on regime

3. Unit-interval geometry and equal-cost structure

On the unit interval $S \in \mathcal S \subseteq 2^Q$4, the paper concentrates on the equal-cost case $S \in \mathcal S \subseteq 2^Q$5, so payoff is simply

$S \in \mathcal S \subseteq 2^Q$6

In this regime, the structure differs from the circle because boundary effects matter. When both players choose interval lengths strategically, a striking equilibrium is that each player randomizes equally between $S \in \mathcal S \subseteq 2^Q$7 and $S \in \mathcal S \subseteq 2^Q$8, and the value is

$S \in \mathcal S \subseteq 2^Q$9

The paper describes this as a geometric “split the interval in half” equilibrium (Gerum et al., 2 Sep 2025).

If the Hider’s interval length $P(H,S)=C(S\setminus H)+\Pi(H\setminus S).$0 is fixed, the Hider mixes equally between $P(H,S)=C(S\setminus H)+\Pi(H\setminus S).$1 and $P(H,S)=C(S\setminus H)+\Pi(H\setminus S).$2. The Searcher chooses the empty set if $P(H,S)=C(S\setminus H)+\Pi(H\setminus S).$3, and the whole interval $P(H,S)=C(S\setminus H)+\Pi(H\setminus S).$4 if $P(H,S)=C(S\setminus H)+\Pi(H\setminus S).$5, with value

$P(H,S)=C(S\setminus H)+\Pi(H\setminus S).$6

The dual statement holds when the Searcher’s length is fixed. This is another threshold phenomenon, but now induced by geometry and boundary effects rather than unequal commission and omission weights (Gerum et al., 2 Sep 2025).

When both players choose intervals of fixed length $P(H,S)=C(S\setminus H)+\Pi(H\setminus S).$7, the problem becomes a geometric optimization over shifts. Writing the left endpoints as $P(H,S)=C(S\setminus H)+\Pi(H\setminus S).$8 and $P(H,S)=C(S\setminus H)+\Pi(H\setminus S).$9, the payoff is

$S\setminus H$0

Letting $S\setminus H$1, the equilibrium is given by a uniform mixture over explicitly constructed grids of left endpoints: $S\setminus H$2 for the Hider, and

$S\setminus H$3

for the Searcher, with value

$S\setminus H$4

The geometric interpretation supplied in the paper is especially concise: the Hider’s intervals are arranged to minimally cover the interval, while the Searcher’s intervals are arranged to be maximally non-overlapping (Gerum et al., 2 Sep 2025).

As $S\setminus H$5, the interval game approaches the circle case because boundary effects vanish, and the value tends toward

$S\setminus H$6

which is the analogous circle value when the two lengths are equal. This establishes a direct connection between the bounded and periodic geometric models (Gerum et al., 2 Sep 2025).

4. Discrete alignment games and ratio-based equilibria

The discrete model takes $S\setminus H$7 with additive costs and penalties: $S\setminus H$8 and for $S\setminus H$9,

$C$0

The payoff becomes

$C$1

This is the discrete analogue of the continuous commission–omission decomposition (Gerum et al., 2 Sep 2025).

When both players can choose any subset of $C$2, so that $C$3, the paper gives a complete mixed-strategy solution. Defining

$C$4

the optimal Hider strategy is

$C$5

while the Searcher uses the complementary distribution $C$6. The value is

$C$7

The paper identifies this as its key ratio result: when costs and penalties differ, optimal play is governed by the elementwise cost-penalty ratios $C$8 (Gerum et al., 2 Sep 2025).

The equal-cost special case collapses this ratio dependence. If $C$9 for all $H\setminus S$0, then

$H\setminus S$1

so every subset is chosen with equal probability,

$H\setminus S$2

and the game value becomes

$H\setminus S$3

This sharp contrast between ratio-driven unequal-cost play and uniform equal-cost mixing is one of the paper’s most transparent structural distinctions (Gerum et al., 2 Sep 2025).

A plausible implication is that the discrete model separates two very different sources of strategic complexity. When $H\setminus S$4, complexity is encoded in local cost-penalty asymmetries. When $H\setminus S$5, the relevant structure shifts toward combinatorial constraints, support selection, and threshold effects.

5. Cardinality constraints, complement symmetry, and special cases

The paper next studies the equal-cost case under cardinality constraints. For the game

$H\setminus S$6

where the Hider must choose exactly $H\setminus S$7 elements and $H\setminus S$8, the value is

$H\setminus S$9

For $\Pi$0, the Hider mixes equally between a $\Pi$1-subset $\Pi$2 and its complement in $\Pi$3, while the Searcher uses a nested family of subsets concentrated on the top $\Pi$4 costs. The structural interpretation given in the paper is that only the top $\Pi$5 costly locations matter: play concentrates on the most important sites (Gerum et al., 2 Sep 2025).

A key lemma underlying these transfers is complement symmetry: $\Pi$6 This implies that if a strategy pair is optimal in one game, the complemented pair is optimal in the complement game, and the game value transforms as

$\Pi$7

in the dual setting. The lemma allows results to move between “Hider constrained” and “Searcher constrained” formulations (Gerum et al., 2 Sep 2025).

The smallest constrained game already recovers a classical zero-sum model. In

$\Pi$8

if the players choose different elements $\Pi$9, then

$C:2^Q \to \mathbb{R}$00

while if they choose the same element,

$C:2^Q \to \mathbb{R}$01

For $C:2^Q \to \mathbb{R}$02 and $C:2^Q \to \mathbb{R}$03, this is exactly Matching Pennies, so Matching Pennies appears as a special case of alignment games (Gerum et al., 2 Sep 2025).

For $C:2^Q \to \mathbb{R}$04, the equilibrium is threshold-based. The support is determined by

$C:2^Q \to \mathbb{R}$05

with one regime when $C:2^Q \to \mathbb{R}$06, in which both players mix over all $C:2^Q \to \mathbb{R}$07 elements using

$C:2^Q \to \mathbb{R}$08

and another regime when $C:2^Q \to \mathbb{R}$09, in which the optimal supports shrink to the top $C:2^Q \to \mathbb{R}$10 or so elements and the value changes to

$C:2^Q \to \mathbb{R}$11

The paper emphasizes that these support changes are discontinuous as parameters cross critical values (Gerum et al., 2 Sep 2025).

6. Conceptual significance and adjacent uses of the term

The original framework’s broader contribution is to introduce a general mathematical model for strategic intervention under uncertainty in which the agent must balance acting too broadly and paying unnecessary cost against acting too narrowly and incurring missed-target penalties. The paper identifies medical diagnosis, border/security screening, sanctions and regulatory enforcement, ecology and resource management, and inspection and patrolling problems as relevant domains, and situates the framework as an extension of geometric and search games in which “successful hit” logic is replaced by a misalignment-based objective (Gerum et al., 2 Sep 2025).

A common misconception is to read the term alignment games through the lens of LLM alignment. In the 2025 paper, the term denotes a zero-sum class of set-selection games with payoff

$C:2^Q \to \mathbb{R}$12

not a preference-learning or dialogue-alignment protocol. Later work uses the same phrase or closely related phrases in different ways. "Reference Games as a Testbed for the Alignment of Model Uncertainty and Clarification Requests" treats reference games as a controlled setting for testing whether models recognize uncertainty and ask for clarification (Ali et al., 12 Jan 2026). "Distributional Alignment Games for Answer-Level Fine-Tuning" formulates answer-level fine-tuning as a two-player game between a Policy and a Target distribution (Mohri et al., 29 Apr 2026). "Common-agency Games for Multi-Objective Test-Time Alignment" models alignment objectives as strategic principals that allocate token-level incentives to a shared LLM (Chen et al., 8 May 2026). This suggests a later terminological broadening rather than identity with the zero-sum search/intervention framework.

Within the original theory, the main conceptual takeaways are explicit. First, alignment is misalignment: payoffs depend on the symmetric difference $C:2^Q \to \mathbb{R}$13. Second, commission and omission errors are separated and weighted by $C:2^Q \to \mathbb{R}$14 and $C:2^Q \to \mathbb{R}$15. Third, unequal costs produce ratio-based equilibria, while equal costs create geometry and thresholds. Fourth, classical games emerge as special cases. These features together define alignment games as a distinct mathematical foundation for analyzing the tension between comprehensive coverage and precise targeting under uncertainty (Gerum et al., 2 Sep 2025).