SCIENCE CHINA Information Sciences, Volume 64 , Issue 3 : 132206(2021) https://doi.org/10.1007/s11432-019-2825-y

## The greedy crowd and smart leaders: a hierarchical strategy selection game with learning protocol

• AcceptedFeb 5, 2020
• PublishedFeb 7, 2021
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### Acknowledgment

This work was supported by Tianjin Natural Science Foundation (Grant Nos. 20JCYBJC01060, 20JCQNJC01450) and National Natural Science Foundation of China (Grant No. 61973175).

### Supplement

Appendix

Proof of Theorem 4

Proof. We set $J_{1k}(s_k)~=~f^a_k(w_k)~-~C_1s$, $J_{2k}(s_k)~=~f^a_k(w_k)~-~C_2s_k$, and $s_1~=~[s_{11},~s_{12},\dots,~s_{1n_c}]~$ is the solutions of $J_{1k}$. So according to the Nash-equilibrium condition we have $$0 = \frac{\partial J_{1k}}{\partial s_{1k}}= \frac{f^a_k{}' (w_{1k})(1-v_{1k})K^a}{\sum_{m\in V^c}s_{1m}}-C_1. \tag{1}$$ Now we set $s_2~=~[s_{21},~s_{22},\dots,~s_{2n_c}]$ where $s_{2k}~=~C_1s_{1k}/C_2$. This makes $v_{1k}~=~v_{2k}$ and $w_{1k}~=~w_{2k}$ to be the same allocation of weights. $s_2$ also satisfies \begin{align} \frac{\partial J_{2k}}{\partial s_{2k}} = \frac{f^a_k{}' (w_{2k})(1-v_{2k})K^a}{\sum_{m\in V^c}s_{2m}}-C_2 &=\frac{C_2}{C_1} \frac{f^a_k{}' (w_{1k})(1-v_{1k})K^a}{\sum_{m\in V^c}s_{1m}}-C_2 = C_2 - C_2 = 0. \tag{2} \end{align} Namely $s_2$ is also the Nash-equilibrium of each function $J_{2k}(\cdot)$. Since $s_1$ and $s_2$ denote the same allocation, we know that all the Nash-equilibrium of $J_1$ is the Nash-equilibrium of $J_2$, and the inverse proposition holds the same. So the choice between $C_1$ and $C_2$ only affects the scale of the bid $s_k$ but not the final allocation of the auction.here

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