Fig 1.
Public goods game (PGG) on a hypergraph and four categories of higher-order update mechanisms.
a, In a PGG, each player is either a cooperator (pink) or a defector (blue). Every cooperator invest a cost c, while defectors do not. The total investment is multiplied by the synergy factor R to produce a benefit 3cR shared equally among all players, resulting in a cooperative dilemma. b, A nine-node hypergraph with four hyperedges (:
,
:
,
:
, and
:
). c, Players on the hypergraph participate in PGGs within the hyperedges. For example, the focal node (circled in red) belongs to two hyperedges,
and
, hence it participates two games and obtains the payoffs averaged across these two games. d- g, Four categories of two-stage updates: d, Group-and-individual-biased (HDB for higher-order death-birth, HIM for higher-order imitation); e, Group-biased (GMC for group-mutual comparison); f, Individual-biased (GIC for group-inner comparison); and g, Non-biased (HPC for higher-order pair-comparison).
Table 1.
Critical synergy factor for five higher-order update mechanisms.
Fig 2.
Numerical validations on homogeneous or heterogeneous hypergraphs.
The first row visualizes various hypergraphs: a, A homogeneous hypergraph with uniform hyperdegree and order; b, A hyperdegree-heterogeneous hypergraph with the same order but varying hyperdegrees following a power-law distribution; c, An order-heterogeneous hypergraph with the same hyperdegree but varying orders following a Poisson distribution; d, A hyperdegree-and-order-heterogeneous hypergraph with varying hyperdegrees and orders. The second row compares theoretical results (arrows) with simulation data (scatters) on these hypergraphs. Each dot represents the fixation probability times population size from independent simulations under weak selection
. The arrows indicate the theoretical results for critical synergy factors
listed in Table 1.
Fig 3.
Impact of varying hyperdegrees and orders.
a- b, Scatter plots of the critical synergy factor versus hyperdegree k ( a) and order g ( b) in homogeneous hypergraphs, where the hypergraphs have a size of 100, with a fixed order of 4 in a and a fixed hyperdegree of 4 in b. c- d, Scatter plots of the critical synergy factor versus hyperdegree heterogeneity in order-heterogeneous hypergraphs ( c) and order heterogeneity
in hyperdegree-heterogeneous hypergraphs ( d), where the hypergraphs have a size of 100, with an average order of 8 and an average hyperdegree of 8 in c and d. Here,
, where
is the second moment of the hyperdegree, and
, where
is the second moment of the order. The scatter points in all plots are based on numerical calculations according to Table 1, on hypergraphs with corresponding topological configurations. All hypergraphs are constructed using the configuration model based on the given hyperdegree and order sequences.
Fig 4.
Increasing overlap strength can foster cooperation.
a- c, Three homogeneous hypergraphs with large size N = 1000, hyperdegree k = 2, and order g = 4, differing only in the overlap strength between hyperedges, as defined in Eq (3). The overlap strengths are 1/4 in a (weak), 1/2 in b (moderate), and 5/8 in c (strong). d, Comparison of critical synergy factors for different overlap strengths under GMC, HDB, and HIM mechanisms. e- g, Another set of homogeneous hypergraphs with large size N, hyperdegree k = 3, and order g = 3, also differing in overlap strength: 1/3 in e (weak), 13/27 in f (moderate), and 5/9 in g (strong). h, Comparison of critical synergy factors for increasing overlap strengths under the same mechanisms as in d. Both comparison results ( d, h) show that stronger overlap decreases critical synergy factors across all the three mechanisms thereby promoting the emergence of cooperation.
Fig 5.
Intuitions from a viewpoint of defectors.
Each individual (node) plays as a cooperator (pink) or a defector (blue) and participates in public goods games (PGGs) within the groups they belong to. After a game round, both the groups and the individuals receive payoffs, based on which the focal individual (a defector) updates its strategy by selecting an individual for imitation or comparison through two-stage selection. For simplicity, the group payoff is assumed to be the average of its individuals’ payoffs and the individual payoff is the game outcome within a specific group. a, Group selection can be either group-biased or group-neutral. The focal player favors group due to its higher payoff under group-biased selection, whereas it randomly chooses among neighboring hyperedges under group-neutral selection. b, Individual selection can be individual-biased or individual-neutral. The focal player prefers the defector, as defectors generally outperform cooperators in individual payoffs within a group under individual-biased selection, while it randomly picks a member within the group under individual-neutral selection. c, Strategy updates occur based on imitation or comparison. Imitation involves copying a preferential individual’s strategy, whereas comparison tends to reinforce the original defector strategy due to the higher payoff itself.
Fig 6.
Higher-order updates tend to be more beneficial to cooperation than pairwise updates.
a, Higher-order updates involve two-stage selection: group selection followed by individual selection. b, Pairwise updates involve only individual selection. c, An empirical higher-order network of congress bills with N = 57 nodes and E = 108 edges (or hyperedges), where each node represents a congressperson and each hyperedge represents a bill composed of sponsors and co-sponsors. d, Comparisons of synergy factors between higher-order updates (HDB and HPC) and pairwise updates (DB and PC). e, Comparisons between the critical synergy factor for higher-order updates and
for pairwise updates. All seven-node subgraphs of the empirical network in c are extracted and sorted by
, with hyperedges formed by cliques. The results show that
in most of the seven-node hypergraphs, indicating that higher-order updates generally promote cooperation more effectively than pairwise updates.