Parallel versus Sequential Update and the Evolution of Cooperation with the Assistance of Emotional Strategies

Our study contributes to the debate on the evolution of cooperation in the single-shot Prisoner’s Dilemma (PD) played on networks. We construct a model in which individuals are connected with positive and negative ties. Some agents play sign-dependent strategies that use the sign of the relation as a shorthand for determining appropriate action toward the opponent. In the context of our model in which network topology, agent strategic types and relational signs coevolve, the presence of sign-dependent strategies catalyzes the evolution of cooperation. We highlight how the success of cooperation depends on a crucial aspect of implementation: whether we apply parallel or sequential strategy update. Parallel updating, with averaging of payoffs across interactions in the social neighborhood, supports cooperation in a much wider set of parameter values than sequential updating. Our results cast doubts about the realism and generalizability of models that claim to explain the evolution of cooperation but implicitly assume parallel updating.

💡 Research Summary

The paper investigates how cooperation can emerge and persist in societies where individuals are linked by both positive and negative relationships. Building on earlier work that typically assumes only positive ties, the authors introduce a signed network model in which each edge carries an emotional sign (positive or negative). Agents adopt one of three strategies: Unconditional Defection (UD), Unconditional Cooperation (UC), or a Conditional (COND) strategy that cooperates with positively‑signed partners and defects with negatively‑signed ones. The Conditional strategy acts as a low‑cost emotional cue, allowing agents to use the affective content of a tie rather than a full memory of past interactions.

The model allows three co‑evolving layers: (i) relational signs, (ii) network topology, and (iii) agents’ strategies. After each Prisoner’s Dilemma (PD) encounter, the sign of the link updates automatically: mutual cooperation makes the link positive, mutual defection makes it negative, and mixed outcomes generate a probabilistic sign change (with a higher probability of turning a positive link negative than vice‑versa). In addition, a rewiring mechanism captures social exclusion: with a small probability (Prew = 0.01) a frustrated agent may cut a tense link and form a new connection, preferably with a friend‑of‑a‑friend, mimicking transitive closure observed in real social networks.

The central methodological contribution is a systematic comparison of two strategy‑updating protocols. In the sequential (asynchronous) rule, at each time step a random connected pair plays the PD; the lower‑payoff player may adopt the partner’s strategy with probability Padopt. Thus only the dyadic payoff matters. In the parallel (synchronous) rule, all agents play the PD with all neighbors simultaneously, compute an average payoff, and then compare it with the averages of their neighbors. If any neighbor has a higher average, the focal agent randomly selects one of those better‑performing neighbors and adopts its strategy with probability Padopt. This synchronous update uses information from the whole neighbourhood and updates all agents at once.

Extensive simulations reveal a stark contrast between the two rules. Under parallel updating, the presence of COND agents dramatically boosts the spread of cooperation across a wide range of parameters. Cooperation thrives especially when the rewiring probability is moderate and the strategy‑adoption probability is low, allowing clusters of positive ties and conditional cooperators to form and stabilize. The emotional cue provided by the sign‑dependent strategy serves as a catalyst: it protects cooperative clusters from invasion by defectors and leverages the dynamic rewiring to prune hostile links.

Conversely, sequential updating leads to rapid domination by unconditional defectors. Because only the immediate payoff of the selected pair influences strategy change, cooperative ties have little chance to accumulate, and the conditional strategy cannot exploit its emotional information effectively. The system quickly converges to an all‑defect state regardless of rewiring or initial strategy composition.

The authors conclude that (1) emotional (sign‑based) conditional strategies can catalyze cooperation when the network is dynamic; (2) the co‑evolution of relational signs and topology is essential—without rewiring or sign change cooperation collapses; and most importantly (3) the choice of update timing is a decisive factor. Models that implicitly assume synchronous updating may overstate the robustness of cooperation, whereas asynchronous (sequential) dynamics paint a more pessimistic picture. The paper thus calls for explicit consideration of update mechanisms in future evolutionary‑game studies of social systems and highlights the interplay between emotions, network restructuring, and strategic adaptation as a fertile ground for understanding cooperative behavior.