The Geometry of Escape: Antifragility, Convexity, and the Dissolution of the Bateson Game

Human interaction frequently unfolds under conditions of deepuncertainty about how one party interprets the behavior of the other.When individuals cannot reliably infer the interpretive frame throughwhich their actions are evaluated, even simple exchanges can generateself-reinforcing cycles of mistrust, strategic concealment, emotionalvolatility, and escalating conflict. The Bateson Game is a formalmodel of this phenomenon, capturing how frame-contingent best-responsereversals and costly meta-communication give rise to interpretivetraps and double bind dynamics. This paper develops a sequel to the Bateson Game by embedding frameuncertainty within a broader analysis of fragility, antifragility, andconvexity. We characterize fragility as a property of the geometry ofthe Receiver’s value function with respect to interpretiveuncertainty: under classical Bayesian evaluation with quadratic loss,this value function is strictly decreasing in the level of uncertainty.We show that this structure reproduces the core features of thedouble bind, in which attempts at clarification are themselvesinterpreted as suspicious or costly. We then introduce optionality actions that generate convex payoffstructures. When the Receiver can construct such actions, increases ininterpretive volatility become beneficial rather than harmful. In aminimal two–action, two–frame model, we construct a barbell strategywhose payoff is invariant across frames and demonstrate that itdominates classical responses under appropriate upside–downsideconditions. Extending the analysis to repeated interactions, we showthat the Receiver possesses an implicit exit option and derive theoptimal stopping rule for terminating the relationship. Convexityraises the continuation value and thereby reduces the likelihood ofexit. Taken together, these results demonstrate that convexity dissolvesthe Bateson Game. Fragility leads to interpretive traps, escalatingtension, and strategic paralysis, whereas convexity yields interpretiveresilience, learning, and adaptive escape. The geometry of payoffs,rather than the mere presence of uncertainty, determines whetherinteraction becomes pathological or self-correcting.