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Evolution of trust in a hierarchical population with punishing investors

Trust plays an essential role in the development of human society. According to the standard trust game, an investor decides whether to keep or transfer a certain portion of initial stake to a trustee. In the latter case, the stake is enhanced to signal the value of trust. The trustee then chooses how much to return to the investor. We here distinguish two types of investors and two types of trustees who can learn from each other. While a trustee can be trustworthy or untrustworthy, an investor could be normal or punishing one. The latter strategy punishes both untrustworthy trustees and normal investors who are reluctant to control misbehaving trustees. Importantly, we assume a hierarchical population where the portion of investors and trustees is fixed. By means of replicator equation approach, we study the $N$-player trust game and calculate the level of trust and trustworthiness. We find that the introduction of punishment can induce a stable coexistence state between punishing investors and trustworthy trustees. Furthermore, an intermediate fraction of investors can better promote the evolution of trust when the punishment intensity is low. For more intensive punishment, however, a higher fraction of investors can be more efficient to elevate the trust level. In addition, we reveal that appropriate increase of the punishment intensity can enlarge the attraction domain of the coexistence state.