How changes in synaptic connections underlie learning and memory is a central question in Neuroscience. Previous modeling efforts focused on biologically realistic learning rules and dynamics, but these rules usually produce bimodal weight distributions that are not aligned with experimentally observed lognormal distributions of synaptic weights. How biological learning rules preserve such unimodal distributions and at the same time retain information, i.e., learn, remains unknown. Here, we take a game-theoretical approach, asking whether learning is possible under the constraint of retaining a particular weight distribution of excitatory and inhibitory synapses. We assume a fixed set of weights and open a game of trading weights to achieve learning in a spiking neural network with asynchronous and irregular dynamics and a number of strong input stimuli. The aim of the game is to pattern-complete network responses to incomplete stimuli. We find that both functional swapping of weights between existing synapses, and structural swapping, in which entire connections can be destroyed or created de novo, can achieve robust pattern completion, but network stability can become compromised. To explore how a network learns multiple, overlapping stimuli without succumbing to instability, we expand to a multiplayer game in which each memory is a player and their strategies are the fraction of synapses allowed to change. We identify combinations of excitatory and inhibitory strategies that are positioned at equilibria so to increase network capacity without adding strong synapses. The resulting constraints of stable swapping form predictions for biological action in mechanistic rules.