FairGamer: A Fairness-Centric Game Framework

• FairGamer is a fairness-centered framework of game models that formally incorporates fairness into objectives, admissible strategies, and implementation loops.
• It employs diverse fairness objectives—including balanced frequency, bounded discrepancies, and strong transition fairness—to ensure equitable long-run outcomes.
• The framework spans algorithmic analyses from CoNP-complete decision processes and fairness-aware learning methods to verifiable, blockchain-based implementations.

FairGamer is most naturally understood as a fairness-centered family of game models and game-mediated systems in which fairness is part of the formal objective, the admissible strategy space, or the deployment loop itself. In the literature closest to the term, this includes quantitative fairness games on finite colored graphs, fair quantitative games with mean-payoff or energy objectives, fair $\omega$-regular games with fairness constraints on both players, fairness-aware multi-agent reinforcement learning, fair bandit allocation, game-based auditing, and verifiable randomness mechanisms for blockchain games (Bianco et al., 2010). This breadth suggests that FairGamer is not a single canonical formalism but a research program organized around one question: how should game dynamics, incentives, and long-run outcomes be constrained so that fairness is enforceable, analyzable, and, in some settings, externally auditable.

1. Formal scope and game models

A foundational lineage for FairGamer is the finite-state, turn-based graph game. In "Quantitative Fairness Games" (Bianco et al., 2010), a $k$-colored arena is defined as

$A=(V_0,V_1,ini,E),$

where $V_0$ and $V_1$ partition a finite node set $V$, $ini\in V$ is the initial node, $E\subseteq V\times[k]\times V$ is a set of colored edges, and every node has at least one outgoing edge. Plays are infinite paths, strategies may be history-dependent or memoryless, and objectives are subsets of color sequences in $[k]^\omega$. This yields a finite-state, turn-based, edge-colored, deadlock-free model.

A second lineage replaces colors by numeric weights. "Fair Quantitative Games" (Anand et al., 28 Jan 2025) studies two-player weighted arenas

$G=(Q,E,w),$

where $k$0 is partitioned between the players, $k$1, and $k$2. Here the underlying objective is quantitative—mean-payoff or energy—while fairness is imposed as an additional constraint on one player’s moves.

A third lineage makes fairness part of the arena itself through a designated set of fair edges. In "Fair $k$3-Regular Games" (Hausmann et al., 2023), a fair game arena is

$k$4

with $k$5 the set of fair moves. A player is fair if they choose an edge in $k$6 infinitely often whenever its source vertex is visited infinitely often. This explicitly models the prohibition against postponing an enabled fair response forever.

Taken together, these models suggest that FairGamer is best viewed as a graph-game framework in which fairness can be encoded at several levels: edge colors and their frequencies, edge obligations, weighted objectives, or combinations of these. The common structural assumptions are finite arenas, infinite plays, adversarial or mixed-motive interaction, and fairness conditions that constrain asymptotic behavior rather than single moves.

2. Fairness objectives and semantic variants

The quantitative core of FairGamer appears in three frequency-based objectives. For an infinite path $k$7, the frequency of color $k$8 is

$k$9

when the limit exists. A path is balanced if every color has frequency $A=(V_0,V_1,ini,E),$0, and is frequency-$A=(V_0,V_1,ini,E),$1 if its color frequency vector equals a prescribed $A=(V_0,V_1,ini,E),$2 with $A=(V_0,V_1,ini,E),$3. A stronger notion is the bounded difference property, requiring a constant $A=(V_0,V_1,ini,E),$4 such that for all colors $A=(V_0,V_1,ini,E),$5 and all prefixes,

$A=(V_0,V_1,ini,E),$6

Balanced fairness enforces equal long-run shares; bounded fairness enforces prefix-wise discrepancy control; frequency-$A=(V_0,V_1,ini,E),$7 fairness enforces weighted long-run shares (Bianco et al., 2010).

These objectives are quantitative refinements of classical qualitative fairness. The classical condition merely requires that every color occur infinitely often. Quantitative fairness asks not only whether all colors recur, but how often and how evenly. In scheduling language, the distinction is between “nobody is starved forever” and “service shares are controlled” (Bianco et al., 2010).

A different fairness semantics is strong transition fairness. In fair quantitative games, fairness is specified by a set of designated edges $A=(V_0,V_1,ini,E),$8, all owned by a single player, and requires that the player take fair edges infinitely often if their source nodes are visited infinitely often (Anand et al., 28 Jan 2025). This is not a frequency condition over all events; it is an obligation attached to a chosen subset of transitions.

When both players are subject to fairness, the semantics becomes subtler. A play $A=(V_0,V_1,ini,E),$9 is $V_0$0-fair if every fair edge from a player-$V_0$1 vertex visited infinitely often is itself taken infinitely often. "Fair $V_0$2-Regular Games" (Hausmann et al., 2023) then defines a fair $V_0$3 game by two objectives: $V_0$4 governs mutually fair plays, $V_0$5 governs mutually unfair plays, and if exactly one player is fair, the fair player wins. This construction is essential because the naïve alternative—requiring both players simply to use fair strategies—leads to undetermined states (Hausmann et al., 2023).

A recurring misconception is that fairness is exhausted by equal treatment. The literature does not support that simplification. Frequency-$V_0$6 objectives encode fixed priorities rather than equality (Bianco et al., 2010); strong transition fairness constrains only selected edges (Anand et al., 28 Jan 2025); and in adaptive ML settings, fairness criteria such as statistical parity, equalized odds, and predictive value parity are explicitly described as conflicting in nature (Basu et al., 8 Aug 2025).

3. Solvability, reductions, and strategy complexity

The algorithmic profile of FairGamer is unusually sharp. For balanced, bounded, and frequency-$V_0$7 objectives, the decision problem “does player $V_0$8 have a winning strategy?” is CoNP-complete (Bianco et al., 2010). The upper bound follows from two structural facts: these goals are prefix independent and convex, and if player $V_0$9 can win, then player $V_1$0 has a memoryless winning strategy. The lower bound is by reduction from tautology checking for CNF formulas. In the one-player case, the picture changes: frequency-$V_1$1 on graphs is in PTIME via feasibility of a linear system, and the balanced and bounded one-player problems were already polynomial-time solvable (Bianco et al., 2010).

For fair $V_1$2-regular games, the parity case admits two complementary solution routes. First, fair parity/parity games are reduced in polynomial time to ordinary parity games by a gadget construction inspired by reductions for stochastic parity games. For fair parity/$V_1$3, the reduced game has at most $V_1$4 nodes and uses priorities in $V_1$5. For full fair parity/parity with parity maps $V_1$6 and $V_1$7, the reduction yields a parity game with $V_1$8 nodes and $V_1$9 priorities (Hausmann et al., 2023). Second, the same semantics can be compiled into a direct symbolic nested-fixpoint algorithm of alternation depth $V$0, giving a solver that avoids explicit gadget materialization (Hausmann et al., 2023).

Fair quantitative games provide a third algorithmic regime. When fairness is imposed on player $V$1, these games fall within previously studied $V$2-regular mean-payoff and energy games; when fairness is imposed on player $V$3, the paper presents this as a previously unstudied setting. For all variants of fair mean-payoff and fair energy under unknown initial credit, pseudo-polynomial algorithms compute the winning regions of both players, and the work also analyzes strategy complexity (Anand et al., 28 Jan 2025).

Strategy memory is asymmetric across the literature. In quantitative fairness games, player $V$4 can spoil with a memoryless strategy when spoiling is possible, but player $V$5 may require memory (Bianco et al., 2010). In fair parity games, positional strategies exist in the reduced ordinary parity game, but translating them back to the original fair game yields finite-memory strategies, with the proof giving an exponential upper bound on the size of winning strategies (Hausmann et al., 2023). FairGamer is therefore not, in general, a memoryless synthesis framework.

4. Learning-based FairGamer: MARL, bandits, and fair allocation

A learning-oriented interpretation of FairGamer replaces explicit winning conditions by fairness-aware optimization criteria. One influential direction uses Proportional Fairness, with the standard objective

$V$6

"Altruism and Fair Objective in Mixed-Motive Markov Games" (Xu et al., 9 Feb 2026) uses this objective to define fair altruistic utilities in log-payoff space, introduces Fair Markov Games, and derives fair Actor-Critic algorithms. In CleanUp, the proportional-fair method reaches roughly $V$7 apples by end of training, whereas the utilitarian method plateaus near $V$8; the proportional-fair solution also keeps the Gini coefficient low, while utilitarian optimization yields a highly unequal allocation with Gini near $V$9 (Xu et al., 9 Feb 2026). This suggests that, in some sequential social dilemmas, fairness is not merely a constraint on efficiency but can improve long-run collective performance.

A second line studies repeated intervention allocation. In "Improving Fairness in Adaptive Social Exergames via Shapley Bandits" (Gray et al., 2023), the central pathology is the Greedy Bandit Problem: a standard multi-armed bandit repeatedly favors players whose preferences yield the highest aggregate return, thereby underserving others over time. The proposed Shapley Bandit tracks each player’s cumulative Shapley contribution and treatment count, then minimizes the mismatch

$ini\in V$0

In a user study with $ini\in V$1, the Shapley condition achieved post-session motivation $ini\in V$2 with $ini\in V$3 versus control, and reversed the disparity–nonadherence correlation from $ini\in V$4 under the Greedy Bandit to $ini\in V$5, with the difference significant at $ini\in V$6 (Gray et al., 2023).

A third line integrates fairness directly into policy-gradient training. "Fairness Aware Reinforcement Learning via Proximal Policy Optimization" (Malfa et al., 6 Feb 2025) augments PPO with a fairness penalty derived from demographic parity, counterfactual fairness, and conditional statistical parity. The penalty has a retrospective part based on realized disparities in past rewards and a prospective part based on value estimates for future outcomes. In Allelopathic Harvest, fair-PPO yields fairer policies than classic PPO across all fairness metrics, but at a reward cost explicitly termed the Price of Fairness (Malfa et al., 6 Feb 2025).

At the online allocation level, "BanditQ: Fair Bandits with Guaranteed Rewards" (Sinha, 2023) imposes minimum reward accrual rates per protected arm through virtual queues and achieves regret and fairness-violation bounds of order $ini\in V$7 in the stochastic setting. This line is especially relevant when FairGamer is interpreted as a scheduler or allocator that must guarantee minimum long-run realized benefit rather than merely minimum selection probability (Sinha, 2023).

5. Auditing, public participation, and adaptive fairness loops

A broader sociotechnical reading treats FairGamer as a fairness-production mechanism rather than only a solved game. "Games for Fairness and Interpretability" (Chu et al., 2020) proposes mass-appeal games that are engaging, reveal model behavior, and produce data useful for improving fairness and interpretability. The paper distinguishes two main categories: Humans vs. AI, where humans generate inputs another human can interpret but the model cannot, and Break the Bot, where players seek minimal input changes that induce maximal output changes. The contribution is explicitly conceptual rather than algorithmic, but it positions gameplay as both a data pipeline and an accountability mechanism (Chu et al., 2020).

"The Fair Game: Auditing & Debiasing AI Algorithms Over Time" (Basu et al., 8 Aug 2025) shifts the focus from one-shot fairness checks to a continual auditor–debiaser loop around a deployed model. The framework has five conceptual pieces: an input dataset or evolving data stream, a base AI/ML model, model properties of interest, an auditor, and a debiasing or alignment module. It formalizes an evolving fairness property $ini\in V$8, an anytime-accurate PAC auditor satisfying

$ini\in V$9

and a long-run average audited bias objective

$E\subseteq V\times[k]\times V$0

The paper reports no empirical experiments or quantitative results for the framework itself, but its significance lies in treating fairness as a sequential control problem in a changing environment (Basu et al., 8 Aug 2025).

These works suggest a distinct but compatible FairGamer interpretation: fairness is not only a winning condition or regularizer, but also a dynamic governance loop in which human interaction, auditing, and policy adaptation co-evolve.

6. Verifiable implementation and trust models

A systems-level FairGamer must also specify how fairness claims are externally verified. "Building Random, Fair, and Verifiable Games on Blockchain" (Chen et al., 2023) addresses this through raffle smart-contract designs on Sui Network. The paper studies two randomness mechanisms: DRAND committee-based decentralized random beacons and single private-key-based verifiable random functions. It then develops several storage and input-management designs, including Object Tables, Delegate Object Creation, and Zero-Knowledge Proof or Merkle-root-based compression.

The trust model differs sharply across designs. DRAND-based settlement is slower but more decentralized: the contract verifies the beacon on-chain, winner selection follows public code, and anyone can often trigger settlement. Single-key VRF-style settlement is faster, but only the host can generate the needed signature, so availability and non-manipulation depend on that key holder. The paper is explicit that transaction fees for DRAND beacons and private-key-based VRFs are similar; Object Tables incur higher overall transaction fees; and the ZKP setup fee is cheap but becomes very expensive during the verification process (Chen et al., 2023).

This implementation literature clarifies an important point that also applies to the rest of FairGamer: fairness is not a single property. It may be quantitative fairness over infinite plays, strong transition fairness on designated edges, proportional fairness over long-run returns, fairness-aware allocation under adaptive learning, or public verifiability of random outcomes. The research record also shows that these choices carry different complexity classes, different memory requirements, different reward costs, and different trust assumptions. In that sense, FairGamer names not one solution but a technical landscape in which fairness is treated as a first-class game-theoretic and systems property rather than a post hoc evaluation score.