To Be a Truster or Not to Be: Evolutionary Dynamics of a Symmetric N-Player Trust Game in Well-Mixed and Networked Populations

TL;DR

This work introduces a symmetric N-player Trust Game (SNTG) where players alternate between investor and trustee roles and fitness is based on the average payoff across roles. It analyzes the evolutionary dynamics in both well-mixed and structured populations, highlighting that trust fails to evolve in well-mixed settings regardless of payoff nonlinearity, while network topology and payoff nonlinearity jointly shape outcomes in structured populations. The study provides analytical thresholds and demonstrates that nonlinear payoffs coupled with network structure lead to distinct dynamics on square lattices versus heterogeneous networks, with degree-based hub initialization offering a practical intervention to promote trust. Overall, the paper emphasizes that both payoff structure and network topology critically determine the emergence and maintenance of prosocial behavior in multi-agent systems, guiding interventions in real-world group dynamics.

Abstract

Trust and reciprocation of it form the foundation of economic, social and other interactions. While the Trust Game is widely used to study these concepts for interactions between two players, often alternating different roles (i.e., investor and trustee), its extensions to multi-player scenarios have been restricted to instances where players assume only one role. We propose a symmetric N-player Trust Game, in which players alternate between two roles, and the payoff of the player is defined as the average across their two roles and drives the evolutionary game dynamics. We find that prosocial strategies are harder to evolve with the present symmetric N-player Trust Game than with the Public Goods Game, which is well studied. In particular, trust fails to evolve regardless of payoff function nonlinearity in well-mixed populations in the case of the symmetric N-player trust game. In structured populations, nonlinear payoffs can have strong impacts on the evolution of trust. The same nonlinearity can yield substantially different outcomes, depending on the nature of the underlying network. Our results highlight the importance of considering both payoff structures and network topologies in understanding the emergence and maintenance of prosocial behaviours.

Figures (14)

• Figure 1: Game tree of the asymmetric 2-player binary TG, in which the role of each player is fixed. The payoffs of an investor are shown in green. Those of a trustee are shown in orange. We require $0<r<1$, where $r$ represents the relative productivity of the prosocial strategies. Adapted from Ref. Masuda:2012aa.
• Figure 2: Different definitions of the payoff in $N$-player games on networks. (a) The definition of the payoff for the present STNG. A focal player (the black circle) belongs to five groups (shown as the shaded area in each of the five copies of the local network). The focal player is assumed to earn payoffs from each group. The summed payoff drives evolutionary game dynamics, as is often the case for other $N$-player game dynamics on networks such as the PGG Santos:2008aaPerc:2013aa. (b) An alternative definition of the payoff in an $N$-player game on networks. With this definition, while the focal player belongs to the same five groups, one assumes that the focal player's payoff only originates from the one group centred around it. The network version of the original NTG uses this definition of the payoff Chica:2018wt.
• Figure 3: Evolutionary dynamics of the SNTG in the infinite well-mixed population, shown over the triangular faces of the 3-simplex $\triangle^3 = \{(y_{it},y_{iu},y_{nt},y_{nu}) : y_{it} +y_{iu} +y_{nt} +y_{nu}=1\}$ as the state space. Vertices IT, IU, NT, and NU correspond to homogeneous population states $y_{it}=1$, $y_{iu}=1$, $y_{nt}=1$, and $y_{nu}=1$, respectively. All trajectories converge to the line of stable equilibria on the NT-NU edge given by $y_{nt}+y_{nu}=1$ and $\frac{r}{r+1}< y_{nu}\le 1$. Therefore, investment (i.e. trust) does not evolve. Equilibria appear only on vertices and edges, but not in the interior of the faces. A filled circle represents a stable equilibrium (i.e., NU). Open circles represent unstable equilibria (i.e., IT, IU, and NT). On the NT-NU edge, the thick solid segment indicates stable equilibria; the hollow segment indicates unstable equilibria. Nonlinearity in the payoff function (i.e., $w\ne 1$) does not qualitatively change evolutionary outcomes compared to linearity (i.e., $w=1$). We set $N=5$, $N_I=3$, $r=0.8$, $\beta=10$, and $w\in\{0.6,1,1.6,2.5\}$.
• Figure 4: Properties of approximate equilibria of the SNTG in finite networks. (a) Square lattice. (b) Heterogeneous networks. In both (a) and (b), the first four columns show the fraction of each strategy as the function of $w$ and $r$. The fifth column shows the average payoff over all nodes, normalised by that of a population comprised entirely of $it$. We used $(w,r) \in \left\{ 0.2,0.4,0.6,0.8,1,1.2,1.4,1.6,1.8,2\right\} \otimes \left\{ 0.55, 0.6, 0.65, 0.7, 0.75, 0.8,0.85, 0.9, 0.95, 1\right\}$. We ran 50 simulations per parameter set and for 5000 generations each. For each $(w, r)$ pair, we obtained the approximate equilibrium fraction of each strategist as the average over the last 256 generations for each simulation and over the 50 simulations. We used $p \in \{ 1/5, 2/5, 3/5, 4/5 \}$.
• Figure 5: Analytical approximation to the threshold $r^*$ for the square lattice. One obtains $P_{it} = P_{nu}$ at $r=r^*$. The threshold $r^*$ is derived from a configuration composed of an $it$ cluster and an $nu$ cluster on the infinite square lattice shown in (a). Under this configuration, the strategy can only change on the border between the two clusters, as shown by the two square cells with white boundaries. An $it$ player may become $nu$ by imitating its $nu$ neighbour or vice versa, depending on whether $P_{it} < P_{nu}$ (corresponding to $r <r^*$) and $P_{it} > P_{nu}$ (corresponding to $r>r^*$), respectively.