Iteratively Undominated Strategies

• Iteratively undominated strategies are defined as strategies that survive the repeated elimination of strictly dominated alternatives, offering a robust criterion for rational behavior.
• They are characterized through geometric proofs and algorithmic approaches, revealing strong connections with rationalizable strategies in both finite and infinite games.
• These strategies play a key role in mechanism design, auction theory, and dynamic games, enhancing stability and efficiency in strategic decision-making.

An iteratively undominated strategy is a central solution concept in game theory, arising from the process of iterated elimination of dominated strategies. It captures the notion of strategies that cannot be excluded via any sequence of strict dominance eliminations, and therefore remain as resilient rational choices after repeatedly pruning away inferior options. The concept plays a key role in equilibrium selection, algorithmic game solving, mechanism design, and beyond. This article provides a comprehensive, technical overview, traversing its formal definitions, algorithmic and geometric underpinnings, the relationship to rationalizability, computational considerations, extensions to infinite and imperfect-information games, and applications in modern economic theory and voting.

1. Formal Definition and Elimination Procedures

Let $G$ be a normal-form game with finite or possibly infinite strategy spaces $(A_i)_{i \in I}$ for players $i \in I$ and payoff functions $u_i$. A (pure or mixed) strategy $a_i \in A_i$ is said to be (strictly) dominated (relative to a reduction $R_{-i}$) if there exists another strategy $b_i \in A_i$ such that: $$u_i(a_i, a_{-i}) < u_i(b_i, a_{-i}) \quad \forall a_{-i} \in R_{-i}$$ The process of iterative elimination successively removes strictly dominated strategies, possibly transfinetely in infinite games. The set of strategies surviving all rounds constitutes the set of iteratively undominated strategies.

Two key elimination schemes arise, especially in infinite games (Crescenzi, 29 Jan 2025):

• Nested elimination: only non-dominated surviving strategies may be used as dominators in subsequent rounds,
• Universal elimination: any strategy (potentially already eliminated) can serve as a dominator.

The maximal reduction under iterated elimination is the set of strategies immune to the process—these are the iteratively undominated strategies. Formalizations allow successor and limit ordinals, thus encompassing both finite and well-founded infinite procedures (Apt et al., 2023).

2. Equivalence to Rationalizability and Geometric Characterization

In two-player finite games, the set of iteratively undominated strategies coincides exactly with the set of rationalizable strategies. Rationalizability requires that a strategy is a best response to some belief about the opponent's behavior. Bernheim and Pearce classically established this equivalence, now reinforced with new geometric proofs (Long, 25 May 2024):

• Each pure strategy corresponds to a linear function or hyperplane over the opponent belief simplex.
• A pure strategy is strictly dominated iff the union of halfspaces defined by alternative strategies covers the entire simplex.
• Via Carathéodory's theorem: any dominated strategy can be dominated by a mixed strategy with support size at most equal to the dimension of the belief simplex (e.g., number of opponent actions).
• This duality, along with Radon's theorem, tightly bounds the support size of necessary mixtures (i.e., any dominated strategy is dominated by a mixture over $\leq |B|$ alternatives, for opponent action set $B$).

These geometric insights anchor the operational meaning of iteratively undominated strategies in the structure of convex hulls and intersections of best-response regions, explaining why their identification is algorithmically tractable in many cases.

3. Order Independence, Uniqueness, and Infinite Games

In finite games, iterated elimination of strictly dominated strategies is order-independent: the maximal reduction does not depend on the sequence of eliminations (0910.5107, Patriche, 2013). This property often extends to infinite games with compact strategy spaces and appropriately continuous payoffs, provided certain boundedness or compactness conditions are met (Crescenzi, 29 Jan 2025). Specifically:

• Order independence may fail in infinite games unless each maximal reduction is bounded (i.e., every dominated strategy is dominated by an undominated dominator).
• In certain classes (e.g., quasisupermodular games, supermodular games, games with upper semicontinuous payoffs), the nested and universal procedures coincide, preserving order independence.

In strictly competitive or zero-sum games, the process is even more structured: in finite games with $n$ outcomes, a maximal iterated elimination of weakly dominated strategies (IEWDS) terminates in $n-1$ steps; in well-founded infinite games, the process can extend to transfinite ordinals but remains well-behaved under WD-admissibility conditions (Apt et al., 2023).

4. Algorithmic and Computational Aspects

The computational complexity of identifying iteratively undominated strategies depends sensitively on the definition of dominance (strict, dominance, weak dominance) and the game's payoff structure (0910.5107):

• For strict dominance in finite games, identifying and removing dominated strategies is in P or NL; the order of elimination does not affect the survivor set.
• For weak dominance, the process can be NP-complete due to sensitivity to elimination order and multiple possible survivor sets.
• Limiting the number of distinct payoff values or restricting the game class (e.g., constant-sum games) typically reduces computational complexity, sometimes yielding logspace algorithms.
• In imperfect-information extensive-form games, dominated actions (at individual information sets) can be identified and iteratively removed in polynomial time using sequence-form representations and linear programming (Ganzfried, 13 Apr 2025). Iterative action removal greatly compresses the game tree for subsequent equilibrium computations.

In parity games and other dynamic/evolutionary environments, the iteratively undominated strategies can be interpreted as fixed points of monotonic (e.g., BeLLMan-Ford-type) update operators (0806.2923, Viossat, 2011), inheriting both algorithmic and dynamic stability properties.

5. Extensions to Infinite, Stochastic, and Collective Environments

The iterated elimination paradigm generalizes beyond finite or deterministic strategy spaces:

• In infinite (e.g., one-player or continuous-strategy) games, nested and universal elimination procedures may diverge; unique maximal reductions exist only under boundedness or compactness (Crescenzi, 29 Jan 2025).
• In stochastic mechanisms and implementation theory, the iterative deletion of strictly dominated strategies (UD$^\infty$) enables the realization of non-dictatorial social choice functions in stochastic finite-action mechanisms (when $|Z|\geq3$ outcomes)—contrast with deterministic or non-iterated cases, where dictatorship is generally unavoidable (Xiong, 25 Sep 2025).
• In voting and collective choice, $(t,\alpha)$-undominated sets represent committees robust to domination by outside candidates, generalizing the classical Condorcet winning set to groupwise, fractional, and depth-$t$ protection, with constructive bounds and iterative algorithms for their formation (Nguyen et al., 27 Jun 2025).

These extensions demonstrate the flexibility of the iteratively undominated strategy concept as a robust criterion for rationalizability, implementability, and collective acceptability in diverse settings.

6. Applications in Mechanism Design, Auction Theory, and Beyond

Iteratively undominated strategies underpin a range of applications:

• Mechanism design: Robust implementation concepts employ iterated undomination (or the related "implementation in advised strategies" notion) to bridge computational tractability and economic optimality in settings where dominant strategy truthfulness is infeasible (Cai et al., 2019, Guo et al., 2012). Advising mechanisms guide agents toward iteratively undominated strategies via improvement advice functions.
• Auctions: In uniform price and second-price auctions, the equilibria supported on undominated or iteratively undominated strategies guarantee (worst-case) welfare bounds and collusion resistance. The iterated elimination approach refines equilibrium selection and rules out unstable or manipulative strategy profiles (Markakis et al., 2012, Mihara, 2011, Zhang, 2022).
• Dynamic/Evolutionary Games: Monotone evolutionary dynamics systematically eliminate dominated (and thus iteratively dominated) strategies, providing dynamical justification for refinement concepts (Viossat, 2011).
• Decision under Uncertainty: In the presence of non-expected utility or time-inconsistency preferences, the resolute choice framework selects iteratively undominated strategies by enforcing cooperation between decision "selves" in the decision tree, overcoming dynamic inconsistency (Jaffray, 2013).

Tables of principal algorithmic and structural results:

Context	Survivor Set	Complexity / Structure
Finite Normal-Form, Strict Dominance	Unique, order-indep.	P or NL (often)
Finite Normal-Form, Weak Dominance	May not be unique	NP-complete
Infinite (compact, u.s.c.)	Unique (if bounded)	Needs verification
Parity Games, Non-deterministic strategies	Unique, sub-exp. iter.	O(1.724<sup>n</sup>) steps
Stochastic Implementation, $|Z|\geq3$	Non-dictatorial possible (UD$^\infty$)	Construction needed

7. Theoretical and Practical Significance

Iteratively undominated strategies embody a foundational equilibrium refinement bridging classical solution concepts (dominance, rationalizability, Nash equilibrium) and modern algorithmic, stochastic, and collective-choice frameworks. They admit geometric and dynamic interpretations, robust algorithmic identification (in structured settings), and enable sharper welfare and implementation guarantees. As a result, they serve as both a technical tool and conceptual lens for understanding rational strategic behavior, efficient mechanism design, dynamic system evolution, and collective decision-making.

Their applicability and interpretative power continue to stimulate research at the intersection of mathematics, economics, computer science, and collective choice theory, with ongoing developments addressing infinite games, learning in extensive form, and robust equilibrium selection in mechanism design.