Harsanyi Set: Game Theory & Applications

• The Harsanyi set is a key concept in game theory and cooperative games, defining the collection of efficient equilibrium allocations and belief hierarchies.
• It underpins homotopy methods for tracing Nash equilibria and generalizes dividend sharing rules, including frameworks that recover the Shapley value.
• The concept extends to epistemic logic, social choice, and machine learning, offering flexible approaches to equilibrium selection and feature attribution.

The Harsanyi set is a foundational construct in game theory and cooperative game theory, with critical implications for equilibrium selection, belief hierarchies, efficient allocation of payoffs, and model interpretability. It arises in contexts ranging from the homotopy-based tracing of Nash equilibria in strategic games, the Möbius-transform-driven sharing of dividends in cooperative settings, and the depiction of agents’ beliefs in epistemic logic, to principled frameworks for feature attribution in machine learning. The following sections detail its mathematical underpinnings, conceptual extensions, complexity properties, distributional interpretations, applications in economics and XAI, and interplay with belief and information hierarchies.

1. Mathematical Foundations: Tracing Equilibria and Dividend Allocation

The Harsanyi set was originally formulated in the context of equilibrium selection via homotopy methods and as a generalization in the theory of cooperative game allocations. In strategic games, the Harsanyi–Selten equilibrium selection process traces a continuous path of equilibria from a simple game $G_0$—where each player has a dominant strategy and a unique equilibrium—to the game of interest $G$, which may possess multiple Nash equilibria. The traced path is defined by a linear homotopy: $G_t = (1-t)\,G_0 + t\,G, \quad t \in [0,1]$ At $t=0$, the game is $G_0$ with a known unique equilibrium, while at $t=1$ the game is $G$. The limit of this path isolates a canonical equilibrium—this collection, or in generic cases a singleton, forms the Harsanyi set for equilibrium selection (Goldberg et al., 2010).

In cooperative games, the Harsanyi set formalizes the possible redistributions of coalition dividends, based on the Möbius transform (Harsanyi dividends) of the game’s value function $v$: $$\phi_v(A) = \sum_{B \subseteq A} (-1)^{|A|-|B|} v(B)$$ Allocations are derived by applying sharing weights $\lambda$, so that for each player $j$,

$$\phi_v(j) = \sum_{A \ni j} \lambda_j(A)\, \phi_v(A)$$

When $\lambda_j(A) = 1/|A|$, the allocation recovers the Shapley value, but the structure is strictly more general (Idrissi et al., 16 Jun 2025). Thus, the Harsanyi set comprises all efficient allocations of dividends across coalitions according to admissible sharing rules.

2. Homotopy Methods and Complexity Implications

Implementing tracing procedures for equilibrium selection, as formalized by the Harsanyi–Selten process or path-following algorithms (including the Lemke–Howson algorithm), is computationally demanding. The problem of computing the prescribed equilibrium is PSPACE-complete. Specifically, tracking the homotopy path from the unique equilibrium of $G_0$ to the selected equilibrium of $G$—even when $G_0$ admits dominant strategies and the process conceptually singles out an equilibrium—is as hard as any problem solvable in polynomial space. There exists strong evidence that no polynomial-time procedure is possible under standard complexity-theoretic assumptions (Goldberg et al., 2010). In bimatrix games, this intractability extends to any equilibrium computable via Lemke–Howson, and shortcut attempts violate the inherent complexity encapsulated by the {\sc Oeotl} problem. The consequence is that conceptual refinement of equilibria through tracing does not resolve computational bottlenecks in equilibrium selection.

3. Dividend Sharing, k-additive Extensions, and the Selectope

The Harsanyi set is pivotal in the theory of cooperative games, particularly as the selectope generated by sharing the Möbius-transformed dividends over coalitions up to size $k$. Given a $k$-additive game—where dividends for coalitions $|S| > k$ vanish—the k-additive core is always nonempty for $k \geq 2$ and preserves coalitional rationality (Grabisch et al., 2011). Classical imputations (payoff distributions to individuals) are derived from the $k$-order imputations using sharing rules $q$: $$x_i = m^{(\varphi)}(i) + \sum_{\substack{S \subseteq N\|S| \geq 2,\, i \in S}} q(S,i) m^{(\varphi)}(S)$$ The set of all imputations thus obtainable,

$$S(v) = \{x^q(v): q \in Q(N)\} = \operatorname{conv}\{x^\alpha(v): \alpha \in A(N)\}$$

is the selectope or Harsanyi set. Under suitable choices of $q$ such that $q(S,i)>0$ for all $i\in S$, every preimputation (sum equal to $v(N)$) can be generated (Grabisch et al., 2011). The Harsanyi set therefore characterizes all possible imputations resulting from coalition dividend sharing under admissible allocation rules.

4. Epistemic Logic, Type Spaces, and Interactive Belief Hierarchies

In probabilistic epistemology, the Harsanyi set emerges from the construction of universal type spaces representing all coherent belief hierarchies over states of nature and types (Zhou, 2014, Bjorndahl et al., 2017). Each agent’s type encapsulates her beliefs about the state and about other agents’ types, recursively. The canonical model built via a background logical language $L$ yields the full set of admissible belief configurations—the Harsanyi set of interactive types—parameterized by the expressive power of $L$.

Formally, for rich enough $L$ (incorporating probability indices and modalities), one constructs maximally consistent sets (worlds), which are mapped to state and type spaces: $$X = \{d_0(\omega): \omega \in \Omega\}, \quad T_i = \{d_i(\omega): \omega \in \Omega\}$$ The universal type space, equivalent to the canonical model, contains all possible assignments consistent with the logic (Bjorndahl et al., 2017). When transitioning from probability frames to type spaces, this correspondence makes the Harsanyi set the essential object for epistemic modeling.

5. Applications in Social Choice, Discrete-Choice, and Distributional Welfare Analysis

In welfare economics, the classical Harsanyi set is the family of social welfare functions representable as weighted sums of individual utilities: $u = \kappa \int_I u_i d\mu(i)$ Harsanyi’s theorem states that, under a Pareto condition, utilitarian aggregation is the unique social welfare function (Echenique et al., 2 Nov 2024). In discrete-choice settings, utilitarian aggregation is recast at the level of stochastic choice rules: $$\rho_\pi(x,D) = \sum_i \lambda_i\, \rho_{\pi_i}(x,D), \quad \sum_i \lambda_i=1$$ This choice-analogue of the Harsanyi set sidesteps interpersonal utility comparisons and allows distributional welfare measures, such as median or quantile-based compensating variation (Echenique et al., 2 Nov 2024), thereby extending the Harsanyi set to encompass rankings sensitive to non-mean features of welfare change distributions.

6. Interpretability in Machine Learning: Feature Attribution via Cooperative Game Allocations

In post-hoc interpretability of machine learning models, the Harsanyi set provides a principled alternative to Shapley values for attribution tasks (Idrissi et al., 16 Jun 2025). For a value function $v$ over features $D$, dividends $\phi_v(A)$ are computed as above, and feature attributions are obtained by redistributing these dividends using an admissible weighting system $\lambda$: $$\phi_v(j) = \sum_{A \ni j} \lambda_j(A)\, \phi_v(A)$$ This framework supports flexibility, allowing practitioners to tailor attribution rules to model structure or interpretive goals. Shapley values correspond to the uniform weighting $1/|A|$; the full Harsanyi set realizes all such allocations that are efficient (sum to $v(D)−v(∅)$) and satisfy constrained sharing axioms. Integration into a three-step blueprint for interpretability—selecting the quantity of interest, value function, and allocation method—clarifies the formal distinction between what is being explained and how explanation is allocated (Idrissi et al., 16 Jun 2025).

7. Extensions, Nonstandard Interactions, and Constructions

Extensions of the Harsanyi dividend to OR-type interactions for neural network interpretability demonstrate that the set can be generalized to account for alternative interaction logics in high-dimensional representations (Chen et al., 6 Oct 2024). In image generation, feature components are disentangled via OR interactions: $$\Delta f_k = I_{\textrm{or}}(S_k) = - \sum_{S' \subseteq S_k} (-1)^{|S_k| - |S'|} u(N \setminus S')$$ Each component then controls a primitive regional pattern, and the set of all such components forms a "Harsanyi set" of regional primitives, generalizing the notion from cooperative games to deep learning feature analysis.

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In sum, the Harsanyi set is a unifying concept in game theory, cooperative allocations, epistemic logic, social choice theory, and machine learning model interpretation. Its structure encodes both mathematical elegance and practical flexibility, while its computational and axiomatic boundaries delineate the limits and possibilities for equilibrium selection, dividend allocation, and interpretability in research and applications.