Source-linked AI summary
Evolutionary games in the multiverse
Chaitanya S. Gokhale, Arne Traulsen
TL;DR
The paper examines multiplayer evolutionary games because many real-world interactions involve more than two individuals, while pairwise models may not capture their dynamics. It develops a general framework for games with multiple players and strategies, showing that multiplayer games can have substantially greater dynamical complexity, including multiple internal equilibria.
Problem
Many biologically and socially relevant interactions involve multiple individuals and strategies, but evolutionary game theory has focused extensively on pairwise interactions.
Method
The paper develops a general evolutionary-game framework using tensor notation for order-dependent payoffs and analyzes replicator dynamics across numbers of players and strategies.
Results
For d players and n strategies, the maximum number of isolated internal equilibria is (d-1)^(n-1), while two-player games have at most one.
Takeaways & Limitations
Multiplayer interactions can produce biodiversity states and dynamical patterns that pairwise models cannot capture, with applications including public goods and multiplayer stag hunts.
Abstract
from arXiv · showhide
Evolutionary game dynamics of two players with two strategies has been studied in great detail. These games have been used to model many biologically relevant scenarios, ranging from social dilemmas in mammals to microbial diversity. Some of these games may in fact take place between a number of individuals and not just between two. Here, we address one-shot games with multiple players. As long as we have only two strategies, many results from two player games can be generalized to multiple players. For games with multiple players and more than two strategies, we show that statements derived for pairwise interactions do no longer hold. For two player games with any number of strategies there can be at most one isolated internal equilibrium. For any number of players $\boldsymbol{d}$ with any number of strategies n, there can be at most (d-1)^(n-1) isolated internal equilibria. Multiplayer games show a great dynamical complexity that cannot be captured based on pairwise interactions. Our results hold for any game and can easily be applied for specific cases, e.g. public goods games or multiplayer stag hunts.
I. MODEL AND RESULTS
The paper extends evolutionary game dynamics from pairwise interactions to multiplayer and multistrategy settings, where payoff structure and equilibrium behavior become substantially richer. It derives bounds and conditions for internal equilibria, fixation, and stability in finite and infinite populations.
- Model: Multiplayer payoffs can depend on player order, requiring a d-index tensor with n^d entries rather than a simple payoff matrix.For randomly formed interaction groups, order-dependent payoffs can be averaged into the tabular representation.
- Two strategies: For two strategies, d-player replicator dynamics can have up to d−1 interior fixed points, with attainable stability patterns including multiplayer stag hunts.The maximum number of stable interior fixed points is d/2 for even d and (d−1)/2 for odd d.
- Two strategies: A payoff-difference sign change at every adjacent frequency is necessary for d−1 interior fixed points, while a single sign change is sufficient for one equilibrium in infinite populations.The necessary condition for the maximum is not sufficient, whereas one sign change limits selection-direction changes to at most one.
- Finite populations: The one-third law does not extend to higher-player games, because fixation above neutrality is not directly tied to internal equilibrium points.For weak selection, the relevant condition instead weights payoff entries toward low mutant frequencies, making initial invasion crucial.
- Multiple strategies: Random games usually have few internal equilibria: with four players and three strategies, the probability of observing at least two is approximately 24%.The analysis generates payoff structures with uniformly distributed entries and computes their equilibrium counts.
II. DISCUSSION
Multiplayer interactions are common in social and biological settings, and their intrinsic complexity motivates a general theory beyond pairwise games. The framework applies across player and strategy counts, including public goods, multiplayer stag hunts, and multiplayer snowdrift games.
- II. DISCUSSION: Multiplayer interactions occur in many-person settings, not merely as disconnected collections of pairwise games.The paper connects such interactions to cooperation, group hunting, and climate-related problems.
- II. DISCUSSION: The theory applies to games with any number of players and strategies.It is presented as a basis for analytical descriptions of situations considered very complex.
- II. DISCUSSION: The framework can be applied to public goods games, multiplayer stag hunts, and multiplayer snowdrift games.