The Contagion of Selfishness Under Frame Uncertainty: A Bateson Game Approach to Cooperation Collapse in Networked Populations

This deposit contains two components: a theory paper and an executed computational notebook implementing all simulations reported in the paper. The paper introduces frame uncertainty as the structural variable explaining why selfish behavior travels farther and more reliably through social networks than cooperation does. The theoretical precursor is the two-player Bateson Game, where a Sender observes the true interpretive frame while the Receiver remains uncertain, the Receiver's best response reverses across frames, clarification attempts are penalized, and no separating Perfect Bayesian Equilibrium exists. The present paper embeds that local mechanism in a graph-based population model to produce a Network Bateson Game, then proves four main results about how cooperation collapses under frame manipulation. The Network Bateson Game is defined on a finite connected graph where each node may act as a sender to some neighbors and as a receiver from others. The active sender share generates local frame uncertainty via the three Bateson axioms: frame-contingent best-response reversal, strict meta-communication penalty ruling out clarification on the equilibrium path, and sender leverage from misinterpretation. Each receiver maintains a posterior probability over a cooperative and an exploitative frame and updates it via a projected Rescorla-Wagner rule. The update has two terms: a conditional-cooperation prediction error proportional to the gap between observed neighborhood cooperation and an aspiration level, and a Bateson leverage term subtracting from the cooperative frame's credibility when active sender leverage is positive. Action selection follows a softmax rule mapping posterior frame credibility into a cooperation probability. The first main theorem, accelerated cooperation collapse, proves by induction that cooperation under Bateson dynamics is weakly below the no-manipulation benchmark at every date after activation and strictly below whenever sender leverage is positive for at least one receiver and neither update is simultaneously projected to the same boundary. The proof relies on componentwise comparison: since the Bateson update subtracts an extra nonnegative term, posteriors are always weakly lower in the Bateson process, softmax monotonicity then gives weakly lower cooperation probabilities, which in turn lower neighborhood averages and compound the effect at the next step. The second main theorem, contagion radius, derives an explicit upper bound on the number of rounds until a node at graph distance d from the active sender becomes frame-infected, defined as experiencing a posterior shift exceeding a threshold epsilon. Direct neighbors of the sender are infected within the ceiling of epsilon over alpha times delta-one rounds, where alpha is the learning rate and delta-one is the direct-exposure lower bound from the propagation assumption. Each subsequent degree of separation adds the ceiling of epsilon over alpha times delta-two rounds, where delta-two is the indirect-exposure lower bound. The total propagation time to distance d is therefore of order d divided by the learning rate. These bounds are then derived from the payoff matrix primitives of the certainty-trap specialization of the Bateson Game, yielding explicit formulas for the direct and indirect lower bounds in terms of the softmax parameters, the leverage coefficient, the aspiration level, and the expected degree of the graph. The third main theorem, tipping-point shift, uses a mean-field replicator model in which Bateson senders augment effective defector pressure by a leverage differential gamma. The no-manipulation benchmark collapses once the defector share exceeds a critical threshold. Under Bateson dynamics the collapse threshold shifts to that critical value divided by one plus gamma, which is strictly lower. A sender is therefore more dangerous than an ordinary defector because the sender not only contributes a defective action but also lowers the cooperative frame's credibility among neighboring receivers. A validity theorem for the mean-field approximation is proved using Chernoff concentration of degrees on Erdos-Renyi graphs with diverging mean degree, showing the network cooperation rate deviates from the deterministic mean-field trajectory by at most a constant divided by the square root of the expected degree, uniformly over any fixed horizon. The approximation fails on sparse, heterogeneous, or community-structured graphs where neighborhood self-averaging breaks down. A fourth main theorem, belief-behavior divergence, identifies the Bateson mechanism against alternative explanations. It proves that if aggregate cooperation is exactly equal across the Bateson and benchmark processes at some date, then the full latent posterior state must be equal as well. Conversely, when leverage has been active and beliefs have diverged, Bateson defectors are strictly more confident in their defection than benchmark defectors on matched behavioral slices, and the population belief variance under Bateson dynamics exceeds the benchmark variance whenever the variance of the belief gap exceeds twice the covariance between benchmark beliefs and the gap. A fifth theorem on irreversibility after source removal proves that once a connected subgraph has entered a certainty trap at belief zero, it remains there permanently under the boundary degree condition, even after all senders are deleted. The benchmark process by contrast can recover cooperation locally whenever sender removal raises some interior-belief node's neighborhood cooperation above its aspiration threshold. An asymmetry proposition formalizes the empirical downward ratchet: a single active sender can induce frame infection across the graph and force cooperation below the benchmark, but no single node whose only instrument is messaging can induce recovery inside a certainty-trapped component, because no separating equilibrium exists and the update mechanics already block escape. This distinguishes the Bateson mechanism from standard threshold or payoff-noise models. A proposed experimental design discriminates the mechanism using three treatments on small-world networks: a benchmark replicator treatment, a pure behavioral control with honest messages, and a Bateson treatment with frame-manipulating confederates removed at a fixed round. The identifying predictions are that the Bateson treatment should show lower cooperation after matching on behavioral slices, higher defector confidence, higher cross-sectional belief dispersion with possible bimodality, and persistent cooperation deficits after confederate removal while the benchmark recovers. The companion executed Jupyter notebook reproduces all numerical results and figures in 26 cells using NumPy, SciPy, NetworkX, and Matplotlib. Simulation 1 runs the benchmark and Bateson dynamics on an Erdos-Renyi graph with 500 nodes and confirms accelerated cooperation collapse. Simulation 2 maps the tipping-point shift across sender shares on Barabasi-Albert graphs, recovering the implied gamma from the observed threshold ratio and comparing it to the theoretical leverage parameter. Simulation 3 visualizes contagion propagation from a single hub sender on a Barabasi-Albert graph and plots average belief as a function of graph distance. Simulation 4 demonstrates the certainty trap as an absorbing state on a cluster of 20 nodes. A supplementary single-agent module verifies the local certainty-trap and ambiguity-trap convergence behavior from the original Bateson Game. A propagation-bound calibration cell computes the explicit direct and indirect lower bounds from payoff primitives, matching the numerical example in the paper. Simulation 5 implements the belief-behavior divergence comparison on matched behavioral slices across the Bateson and benchmark processes. Simulation 6 runs the irreversibility experiment, tracking aggregate cooperation and the mean belief of the largest certainty-trapped component after sender removal at round 300. Simulation 7 runs 1000 sessions of the proposed three-treatment small-world experiment, computing recovery indices, round-25 belief variance, and experimental power. A final summary cell exports all key statistics to JSON.