Offer of a reward does not always promote trust in spatial games

Abstract

Trust is one of the cornerstones of human society. One of the evolutionary pressure mechanisms that may have led to its emergence is the presence of incentives for trustworthy behavior. However, this type of reward has received relatively little attention in the context of spatial trust games, which are often used to build models in evolutionary game theory. To fill this gap, we introduce an inter-role reward mechanism in the spatial trust game, so that an investing trustor can choose to pay an extra cost to reward a trustworthy trustee. With extensive numerical simulations, we find that this type of reward does not always promote trust. Rather, while moderate rewards break the dominance of mistrust, thereby favoring investment, excessive rewards eventually stimulate a nonreturn strategy, ultimately suppressing the evolution of trust. Additionally, lower reward costs do not necessarily promote trust. Instead, more costly, but not excessive, rewards enhance the advantage of the original investment, consolidating the clusters of rewarders and improving trust. Our model thus provides evidence about the counterintuitive nature of the relationship between trust and rewards in a complex society.

Figures (9)

• Figure 1: Trust game model with rewards and spatial interaction structure. (a) The game tree for the trust game with a reward mechanism. The trustor first decides whether to invest (I), not invest (N), or reward (R). Then, the trustee decides whether to return (T) or not return (U), and a single round of the game ends after these two stages. (b) Individuals interact with their nearest neighbors, calculating the average payoff from interactions with multiple neighbors. (c) Individuals engage in strategy learning with their second nearest neighbors, ensuring that strategies are updated among individuals in the same role.
• Figure 2: Trust mainly emerges when returns promote investments and rewards offset the cost of reciprocation. The $(\beta, r)$ phase diagram for fixed $\gamma = 0.1$ shows that the system admits trust as a stable strategy when the return ratio $r$ is moderate, consistently with the conclusions of the classic two-strategy trust game. However, unlike what happens in the classic setting of the game, in which large return ratios suppress trust, our reward mechanism can support it for large enough rewards. The regions of different colors represent the stable strategy populations found for the corresponding parameters. Note that when the only strategy of the trustors is not to invest (N), the two strategies of the trustees are completely equivalent.
• Figure 3: A sufficiently high reward can promote trust even at high returns. (a) If $\beta=1$, the reward value is insufficient, as the two inequalities $\beta > 3r$ and $r > 1/3$ cannot be satisfied simultaneously, stopping trustees from adopting the reciprocation strategy T, and thus not promoting the emergence of trust as $r$ increases. (b) If $\beta=10$, the reward is sufficiently strong, satisfying $\beta > 3r$ for all values of $r$. Thus, even in high-$r$ regions, the mechanism drives trustees to reciprocate and trustors to choose investment or reward strategies, thereby promoting and stabilizing trust. (c) Except for removing strategy I, all other settings remain consistent with panel (a). Without strategy I suppressing strategy R, the reward strategy emerges at $r>0.3\overline{6}$. Subsequently, due to $\beta<3r$, the trust level decreases as r increases. (d) Except for excluding strategy I, all other settings remain consistent with panel (b). The reward strategy also emerges at $r > 0.3 \overline{6}$. Due to the reciprocal condition $\beta > 3r$, the population subsequently remains in the $\text{R}\&\text{T}$ phase regardless of how $r$ varies. In both panels, the reward cost for trustors is fixed at $\gamma=0.1$, and the color code is the same as in Fig. \ref{['fig:beta_r_phase']}.
• Figure 4: The sensitivity of I to U's invasion is crucial for maintaining I's survival, while the reward benefit of R is key to determining U's transformation into T. (a) When there is one U strategy among the trustee neighbors surrounding I, I's payoff is $\pi_I = 9r/4$. The payoffs for a single T or U are $3(1-r)$ or $3$, respectively (with a net payoff gap of $3r$). (b) When there is one U strategy among the trustee neighbors surrounding R, R's payoff is $\pi_I = (9r-4\gamma)/4$. The payoffs for a single T or U are $3(1-r)+\beta$ or 3, respectively (the magnitude of $\beta$ relative to $3f$ determines the conversion direction between T and U). (c) Regardless of the strategy distribution among the trustor's neighbors, the payoff for N is always 1, with a zero payoff difference between T and U among its neighbors.
• Figure 5: Greater rewards do not promote trust if the return ratios are not sufficiently large. For $\gamma=0.1$ and $r=0.6$, the adoption of reciprocation by the trustees reaches its maximum for $\beta=2$, decreasing afterward.