When Random Outcomes Merge, Strategic Equilibrium Can Disappear

Optimal play after every history

A subgame-perfect equilibrium specifies strategies that remain optimal after every feasible history, including histories that equilibrium play never reaches. In a finite game with finitely many actions and perfect information, backward induction supplies such an equilibrium: solve the last decision, then work toward the beginning.

Continuous choices and random events complicate that reasoning. Even when payoffs are continuous, earlier players may have increasingly attractive choices without any best choice. Randomizing over actions need not solve the problem.

In Weak Atomic Continuity and Subgame Perfection, Enxian Chen, Wei He, Yeneng Sun and Hanping Xu identify a condition on Nature’s probability distributions that prevents this failure under their model’s other assumptions. Their paper is an arXiv preprint, version 1, posted on November 4, 2026. Its results are mathematical existence and counterexample theorems, rather than empirical findings.

Why nearby outcomes are not interchangeable

The central distinction concerns outcomes that become arbitrarily close but remain separately observable.

Players can choose different continuations after two distinct public outcomes, however similar those outcomes are. If the outcomes merge into one state at a limiting history, that distinction disappears. Continuation payoffs available before the merger may then become impossible to obtain.

The paper illustrates this with a two-player, five-stage game. Player 1 first chooses a cost parameter from zero and the reciprocals of positive integers. Player 2 decides whether to enter. After entry, Nature selects a publicly observed state, followed by further choices by both players.

For a positive initial choice x, Nature selects x or −x with equal probability. Backward induction makes entry optimal for player 2 and gives player 1 an expected payoff of 1 − x.

At x = 0, both random outcomes collapse to the same observed state. There is no hidden public label allowing players to distinguish two draws of zero. The continuation incentives change: under any strategies that are equilibria in every proper subgame, player 1 receives zero from choosing zero initially.

Player 1 therefore wants a positive x as small as possible. Payoffs approach one, but no choice attains it. Any randomized initial choice also yields less than one and can be improved upon. The authors show that this game has no subgame-perfect equilibrium, although every proper subgame has a pure-strategy equilibrium.

A modified version keeps Nature’s two outcomes separated by a positive offset. Then zero cost no longer merges the public events, and a pure-strategy equilibrium exists.

Tracking probability at individual outcomes

The proposed condition is weak atomic continuity, a concept originating in population genetics. An atom is an individual outcome with positive probability.

Ordinary weak convergence tracks expectations of bounded continuous functions. Weak atomic convergence additionally tracks the probability that two independent draws produce exactly the same outcome. For atoms with probabilities p, that coincidence probability is the sum of their squared probabilities.

Two distinct, equally likely outcomes have coincidence probability one-half. If they merge into one certain outcome, it becomes one. Ordinary weak convergence allows this merger; weak atomic convergence does not.

More generally, the condition requires each atom in a limiting distribution to be matched by nearby atoms whose locations and probability weights converge appropriately. Under the paper’s continuity assumptions, deterministic transitions, distributions on a fixed finite state space, and transitions that are atomless at every history satisfy the condition.

What the theorems establish—and what they do not

The authors prove existence of a measurable pure-strategy subgame-perfect equilibrium for both finite and infinite horizons when Nature’s transitions satisfy weak atomic continuity. Pure strategies prescribe an action rather than a lottery; equilibrium optimality is nevertheless checked against measurable randomized deviations.

The model has finitely many players, one mover at each stage, and full observation of previous moves. Strategic feasible-action sets are nonempty and compact and vary continuously in the specified sense. Payoffs are bounded and continuous. Nature’s state spaces need not be compact, and infinite-horizon payoffs need not be stationary or additively separable. The model includes neither simultaneous moves nor additional public randomization devices.

The proof’s key contribution concerns integrating sets of possible continuation payoffs over Nature’s next state. Weak atomic continuity preserves nonemptiness, compactness and upper hemicontinuity without requiring those sets to be convex.

The converse has a precise scope. For any weakly continuous transition on a compact metric parameter space that violates weak atomic continuity, the authors construct a two-player, five-stage game using that transition with no equilibrium, even in behavioral strategies. This does not mean every game violating the condition lacks equilibrium. It establishes necessity for a guarantee that holds uniformly over the remaining game primitives.