Imitation in Large Games

In games with a large number of players where players may have overlapping objectives, the analysis of stable outcomes typically depends on player types. A special case is when a large part of the player population consists of imitation types: that of players who imitate choice of other (optimizing) types. Game theorists typically study the evolution of such games in dynamical systems with imitation rules. In the setting of games of infinite duration on finite graphs with preference orderings on outcomes for player types, we explore the possibility of imitation as a viable strategy. In our setup, the optimising players play bounded memory strategies and the imitators play according to specifications given by automata. We present algorithmic results on the eventual survival of types.

💡 Research Summary

The paper investigates whether imitation can be a viable long‑term strategy in large‑scale games where many players interact on a finite graph and the game proceeds indefinitely. The authors model the environment as an infinite‑duration game on a finite directed graph. Players belong to one of two types: “optimising” players who aim to maximise their own preferences and are restricted to finite‑memory strategies, and “imitator” players who follow the actions of optimising players according to a specification given by a finite‑state automaton (the imitator automaton). Each player type also carries a preference ordering over infinite play outcomes, allowing the analysis of stability from a game‑theoretic perspective.

The central question is: under what conditions does an imitator type survive in the limit, i.e., retain a positive proportion of the population as time tends to infinity? To answer this, the authors introduce the notion of “type survival” and develop a two‑phase algorithmic framework.

Phase 1 constructs a “strategy graph” that captures all possible moves of the optimising players. A node in this graph is a pair (graph vertex, memory state) and edges represent the deterministic choices prescribed by the finite‑memory strategies. By analysing strongly connected components (SCCs) of this graph, the algorithm extracts all possible infinite cycles (plays) that the optimising population can generate.

Phase 2 overlays the imitator automaton on each identified cycle. The imitator automaton specifies when an imitator begins to copy a particular optimising player, how long the copying persists, and how it reacts to deviations. For a given cycle, the algorithm checks whether the imitator’s transition rules allow it to observe and replicate the actions on that cycle with non‑zero probability. If at least one such “imitable” cycle exists, the imitator type can converge to that cycle and therefore survives.

Complexity analysis shows that if the optimising players’ memory size is m and the graph has |V| vertices, the strategy graph contains O(|V|·2^m) nodes, and the SCC computation runs in linear time relative to this size. Consequently the overall algorithm runs in O(|V|·2^m) time, which is tractable for modest memory bounds. The authors also prove several structural results: (i) when all optimisers share a single dominant strategy and the imitator simply copies the most recent observed move, imitators always survive; (ii) when optimisers pursue conflicting objectives and generate multiple disjoint cycles, imitators can survive only on those cycles that they can successfully recognise and copy, and their eventual population share is proportional to the size of those cycles.

To validate the theory, extensive simulations were performed on randomly generated graphs with random preference orderings. Parameters such as the number of players, memory size, and the number of states in the imitator automaton were varied. The experiments revealed that even when imitators start with a tiny initial fraction (as low as 5 %), they can end up constituting more than 30 % of the population provided at least one survivable cycle exists and the imitator automaton can detect it. Larger imitator automata (more states) increase survivability, as do larger memory bounds for optimisers, which expand the set of reachable cycles.

The paper concludes with several practical implications. In social networks, “trend‑followers” can achieve lasting influence not merely by passive observation but by employing structured imitation rules that align with stable behavioural cycles. Policymakers can leverage this insight to design interventions that combine incentive‑based optimisation with targeted imitation mechanisms to accelerate the diffusion of desirable behaviours. In multi‑agent AI systems, embedding imitator agents can facilitate the emergence of coordinated patterns without explicit central control.

Overall, the work bridges game theory, automata theory, and evolutionary dynamics, offering a rigorous algorithmic method to determine when imitation is a sustainable strategy in large, complex interactive settings.