Full Cooperation in Repeated Multi-Player Games on Hypergraphs

TL;DR

A novel framework for studying repeated multi-player interactions in structured populations -- repeated multi-player games on hypergraphs -- where multiple individuals within each hyperedge engage in a repeated game, and each player can simultaneously participate in many games is introduced.

Abstract

Nearly all living systems, especially humans, depend on collective cooperation for survival and prosperity. However, the mechanisms driving the evolution of cooperative behavior remain poorly understood, particularly in the context of simultaneous interactions involving multiple individuals, repeated encounters, and complex interaction structures. Here, we introduce a novel framework for studying repeated multi-player interactions in structured populations -- repeated multi-player games on hypergraphs -- where multiple individuals within each hyperedge engage in a repeated game, and each player can simultaneously participate in many games. We focus on public goods games, where individuals differ in their initial endowments, their allocation of endowments across games, and their productivity, which determines the impact of their contributions. Through Nash equilibrium analysis, we reveal the intricate interplay between full cooperation (all individuals contribute their entire endowments, maximizing collective benefits) and key factors such as initial endowments, productivity, contribution strategies, and interaction structure. Notably, while equal endowments are most effective in promoting full cooperation in homogeneous hypergraphs, they can hinder cooperation in heterogeneous hypergraphs, suggesting that equal endowments are not universally optimal. To address this, we propose two optimization strategies: one for policymakers to adjust endowment distributions and another for players to modify their contribution strategies. Both approaches successfully promote full cooperation across all studied hypergraphs. Our findings provide novel insights into the emergence of full cooperation, offering valuable guidance for both players and policymakers in fostering collective cooperation.

Figures (14)

• Figure 1: Public goods games on hypergraphs. We investigate repeated public goods games in two settings: a single-group scenario (a) and a hypergraph framework with multiple groups (e.g., a two-group hypergraph in bcd). In each group, players engage in a repeated public goods game. Some players may participate in multiple games if they belong to multiple groups. In each round, player $i$ receives an initial endowment $e_i$ and decides what fraction of their endowment to contribute to each group they are part of. For example, player $i$ may allocate a fraction $x_{i1}$ to group 1. The contributed amount is then multiplied by the player's productivity $r_i$, and the total is evenly distributed among all participants in that group. The outcome of the public goods game is determined by three key factors: each player's initial endowment $e_i$, productivity $r_i$, and their contribution $x_{ih}$ to each group. a, The baseline model of a public goods game in a single-group scenario, where all players have identical endowments and productivity, contributing the maximum amount to the single group. b-d illustrate public goods games on hypergraphs with multiple groups, where five players participate in two distinct games, labeled $h_1$ and $h_2$. Player $i$ participates in all two games. b, Public goods games on hypergraphs with symmetric productivity and equal contributions, but unequal endowments $e_i\neq e_j$. c, Public goods games on hypergraphs with equal endowments and contributions, but asymmetric productivity $r_i\neq r_j$. d, Public goods games on hypergraphs with equal endowments and symmetric productivity, but unequal contributions $x_{i1}\neq x_{i2}$.
• Figure 2: Equal endowments foster cooperation under symmetric productivity, while unequal endowments are more effective when productivity is asymmetric. Homogeneous hypergraphs are categorized as either fully connected (a) or partially connected (d). Panels b, c, e, and f display the proportion of endowment combinations leading to the full cooperation, within the entire endowment space $E$ and for different continuation probability $\delta$. a, Fully connected hypergraphs with 3 to 6 nodes are color-coded blue, red, yellow, and green, respectively, with three players per game and total game counts of 1, 4, 10, and 20. d, Partially connected homogeneous hypergraphs with six nodes, pink and purple hyperedges have an average hyperdegree of 2, while dark green hyperedges have an average hyperdegree of 3. b and e show the proportion of endowment combinations leading to full cooperation within the endowment space $E$, with symmetric productivity ($r_i = 2$). c and f illustrate the proportion of endowment combinations leading to the full cooperation, with asymmetric productivity ($r_1 = 1.5$, $r_2 = 2$, $r_3 = 2.5$, and $r_4 = r_5 = r_6 = 2$). This setup ensures a consistent average productivity across hypergraphs with varying node counts. Black dashed lines indicate the theoretical optimum $e_i = \frac{1}{N}$, colored dashed lines represent optimized endowment allocations (see the Materials and Methods section), and dotted lines denote the continuation probability $\delta_{\text{equal}}^{*}$ required for equal endowments under asymmetric productivity. Overall, b and e confirm that equal endowments best promote cooperation under symmetric productivity, while c and f show that unequal endowments better promote cooperation in the presence of asymmetric productivity.
• Figure 3: Equal endowments most effectively promote cooperation under equal contributions to all games, while unequal endowments enhance cooperation when contributions vary from games. We consider a homogeneous partially connected hypergraph is used, where players 1, 2, and 3 participate in game $h_1$; players 2, 3, and 4 in $h_2$; players 4, 5, and 6 in $h_3$; and players 5, 6, and 1 in $h_4$. a, The impact of contribution proportion $p$ on the proportion of endowment combinations leading to full cooperation. Each player engages in two games, allocating a fraction of $p$ of their endowment to one game and $1-p$ to the other (e.g., player 1 contributes $x_{11} = p$ to $h_1$ and $x_{14} = 1-p$ to $h_4$). As $p$ increases from 0.5 to 1, the proportion of fully cooperative endowment combinations decreases. b, The proportion of fully cooperative endowment combinations is shown across a broad range of continuation probabilities $\delta$, with $p = 0.8$ from a. The red dashed line represents the optimized endowments for each $\delta$, while the dotted line indicates the $\delta_{\text{equal}}^{*}$ for equal endowments, demonstrating that unequal endowments better facilitate cooperation when players distribute their contributions unevenly across different games. c, Contribution deviations are further examined by shifting the contributions of players 1, 3, and 5 to the right and those of players 2, 4, and 6 to the left, revealing that increased deviations further reduce the proportion of cooperative outcomes. d, when $p = 0.8$, unequal endowments more effectively promote cooperation. Additionally, inset graphs near the optimized $\delta$ values validate the accuracy of the optimization algorithm.
• Figure 4: Unequal contributions across games enhance cooperation under asymmetric productivity or unequal endowments.a, A partially connected hypergraph used to identify optimal contribution strategies across the entire contribution space. b, The proportion of fully cooperative combinations under the symmetric productivity and equal endowment, e.g., $e_i = \frac{1}{6}$ and $r_i = 2$. The red dashed line represents $\delta$ for optimized contributions (see the Materials and Methods section), while the red dotted line indicates $\delta_{\text{c-equal}}^{*}$ for equal contributions. Their alignment indicates that equal contributions best promote cooperation. c, Introducing unequal endowments while maintaining symmetric productivity ($e_1=0.3$, $e_{2}=e_{3}=e_{4}=e_{5}=e_{6}=0.14$) creates a gap between optimized $\delta$ and equal contributions, suggesting that unequal contributions better facilitate full cooperation. Optimal contributions are highlighted in boxes: Player 1, participating in $h_1$ and $h_4$, receives a larger endowment, prompting other players in these hyperedges to contribute more, thereby enhancing cooperation. d, When endowments remain equal but productivity is asymmetric ($r_1=2.5$, $r_{2}=r_{3}=r_{4}=r_{5}=r_{6}=1.9$), a similar gap emerges, demonstrating that unequal contributions are more effective in fostering full cooperation. The optimal strategy in d reverses that of (c): higher productivity for Player 1 means other players in $h_1$ and $h_4$ should contribute less to promote full cooperation.
• Figure 5: Full cooperation on heterogeneous hypergraphs. We examine two heterogeneous hypergraphs: one with four players (a) and another with six players (b). a, Under a symmetric productivity payoff function, all players have a productivity of $r = 1.38$ and a continuation probability of $\delta = 0.9$. Players 2 and 3 distribute their endowments equally across both games. c, The endowment space for four players is visualized as a positive tetrahedron, where the equal endowment vector, $\boldsymbol{e} = \left\{\frac{1}{4}, \frac{1}{4}, \frac{1}{4}, \frac{1}{4}\right\}$, is marked by a red point at the center. The fully cooperative feasible region, shown in blue, reveals that allocating more endowments to players 2 and 3 enhances full cooperation. b, In a six-player heterogeneous hypergraph, players are allowed to adjust their contributions rather than adhering to equal contributions. d, As the contribution proportion $p$ varies from 0 to 1, the proportion of fully cooperative feasible combinations first increases and then decreases, as indicated by the brightness of the color bar. This suggests that equal contributions are no longer optimal. Instead, players with higher degrees should contribute more to hyperedges associated with lower-degree players to enhance cooperation. Overall, cooperation can be effectively promoted through two key mechanisms: allocating more endowments to players involved in a greater number of games and encouraging individuals to allocate their contributions toward hyperedges connected to lower-hyperdegree players.