Game Analysis of Investment in a Group with Stickiness

Abstract

This paper studies the investment in a group with stickiness. A group investment game is constructed. By analyzing the best response of each player, the equilibria are presented. Furthermore, the convergence region of each equilibrium is outlined, and the sensitivity analysis of the region to parameters is explored. The game with two players is given to illustrate the convergence region of each equilibrium. The explanation and illustration of the results are summarized. The findings indicate that a successful investment usually occurs in a group with “tolerance”, “open-sharing”, “efficiency”, “sensitivity”, and with large members.

Appendices

Appendix A: Proof of Theorem 2.2

Without any loss of generality, consider player i∈N. Then $\overline{x_{-i}^{*}}=1\geq \frac{d-1}{n-1}$. It follows from (12) that $\mathrm{BR}_{i}(\boldsymbol{x}^{*}_{-i})= \mathrm{BR}_{i\_\mathrm{B}}(\boldsymbol{x}^{*}_{-i})$ if $\overline{x_{-i}^{*}}< \frac{d}{n-1}$, and $\mathrm{BR}_{i}(\boldsymbol{x}^{*}_{-i})=\mathrm{BR}_{i\_\mathrm{A}}(\boldsymbol{x}^{*}_{-i})$ if $\overline{x_{-i}^{*}}\geq \frac{d}{n-1}$.

In the case $\overline{x_{-i}^{*}}< \frac{d}{n-1}$, we have $u_{i}(\mathrm{BR}_{i\_\mathrm{B1}}(\boldsymbol{x}^{*}_{-i}),\boldsymbol{x}^{*}_{-i})\leq 0$, $\mathrm{BR}_{i\_\mathrm{B2}}(\boldsymbol{x}^{*}_{-i})=1$ and $u_{i}(1,\boldsymbol{x}^{*}_{-i})=n\alpha k-1\geq 0$. Therefore, it follows from (10) that $\mathrm{BR}_{i\_\mathrm{B}}(\boldsymbol{x}^{*}_{-i})=\mathrm{BR}_{i\_\mathrm{B2}}(\boldsymbol{x}^{*}_{-i})=1$. Thus,

$$u_i\bigl(1,\boldsymbol{x}^*_{-i}\bigr)\geq u_i \bigl(x_i,\boldsymbol{x}^*_{-i}\bigr), \quad \forall x_i\in[0,1] \mbox{ and }\overline{x_{-i}^*}< \frac{d}{n-1}. $$

In the case $\overline{x_{-i}^{*}}\geq \frac{d}{n-1}$, we have $\mathrm{BR}_{i\_\mathrm{A}}(\boldsymbol{x}^{*}_{-i})=\min \{\frac{\alpha k-1}{2\beta}+\overline{\boldsymbol{x}^{*}_{-i}}, 1 \}=1$. Thus,

$$u_i \bigl(1,\boldsymbol{x}^*_{-i}\bigr)\geq u_i \bigl(x_i,\boldsymbol{x}^*_{-i}\bigr), \quad \forall x_i\in[0,1] \mbox{ and }\overline{x_{-i}^*}\geq \frac{d}{n-1}. $$

To sum up, $u_{i}(1,\boldsymbol{x}^{*}_{-i})\geq u_{i}(x_{i},\boldsymbol{x}^{*}_{-i}),\ \forall x_{i}\in[0,1]$.

In conclusion, the strategies $(x^{*}_{1},x^{*}_{2},\ldots,x^{*}_{n})$ are a Nash equilibrium, where for any i∈N, $x^{*}_{i}=1$.

Appendix B: Proof of Theorem 2.3

Without any loss of generality, consider player i∈N. Since d>1, then $\overline{x_{-i}^{*}}=0<\frac{d-1}{n-1}$. It follows from (12) that $\mathrm{BR}_{i}(\boldsymbol{x}^{*}_{-i})=\mathrm{BR}_{i\_\mathrm{C}}(\boldsymbol{x}^{*}_{-i})= \max \{\overline{\boldsymbol{x}_{-i}}-\frac{1}{2\beta}, 0 \}=0$. That is,

$$u_i \bigl(0,\boldsymbol{x}^*_{-i}\bigr)\geq u_i \bigl(x_i,\boldsymbol{x}^*_{-i}\bigr),\quad \forall x_i\in[0,1]. $$

In conclusion, the strategies $(x^{*}_{1},x^{*}_{2},\ldots,x^{*}_{n})$ are a Nash equilibrium, where for any i∈N, $x^{*}_{i}=0$.

Appendix C: Proof of Theorem 2.4

For any $\boldsymbol{x}\in S_{\mathbf{1}}^{(1)}$, we have $\overline{\boldsymbol{x}_{-i}}\geq \frac{d}{n-1}$. Let x ⁽⁰⁾=x, x ^(t) be given in Definition 2.1, and $K^{(t)}:= \{i\mid \mathrm{BR}_{i} (\boldsymbol{x}_{-i}^{(t-1)} )<1 \}$. If i∈K ⁽¹⁾, then $x_{i}^{(1)}= \overline{\boldsymbol{x}_{-i}^{(0)}}+\frac{\alpha k-1}{2\beta}\geq \frac{d}{n-1}+\frac{\alpha k-1}{2\beta}$. Thus, for any i∈N, we have

$$\overline{\boldsymbol{x}_{-i}^{(1)}}=\frac{1}{n-1}\sum_{j=1,j\neq i}^n x_{j}^{(1)}\geq \frac{d}{n-1}+\frac{|K^{(1)}\backslash\{i\}|}{n-1}\frac{\alpha k-1}{2\beta}, $$

where |⋅| measures the number of members of a set, K ⁽¹⁾∖{i} denotes the set K ⁽¹⁾ excluding the member i. Iteratively, we have

$$ \displaystyle \overline{ \boldsymbol{x}_{-i}^{(t)}}\geq \frac{d}{n-1}+\frac{\sum_{l=1}^t|K^{(l)}\backslash\{i\}|}{n-1} \frac{\alpha k-1}{2\beta}. $$

(17)

Therefore, when |K ^(t)|≥2, for any i∈N, the lower bound of $\overline{\boldsymbol{x}_{-i}^{(t)}}$ is increasing with respect to t, and the increasing step Δ satisfies $\Delta\geq \frac{\alpha k-1}{2(n-1)\beta}>0$. Since $\overline{\boldsymbol{x}_{-i}^{(t)}}\leq 1$ holds for all t>0, therefore, there exists a T ₁, when t>T ₁, we have |K ^(t)|<2. Otherwise, there exists an i such that $\overline{\boldsymbol{x}_{-i}^{(t)}}> 1$, which contradicts $\overline{\boldsymbol{x}_{-i}^{(t)}}\leq 1$. When t>T ₁ and |K ^(t)|=1, suppose K ^(t)={i}, it follows from (17) that the lower bound of $\overline{\boldsymbol{x}_{-i}^{(t)}}$ is unchanged, while, for any j∈N and j≠i, the lower bound of $\overline{\boldsymbol{x}_{-j}^{(t)}}$ is increasing. Thus, there exists a T ₂, when t>T ₂, |K ^(t)|=0, i.e., K ^(t)=ϕ, meaning that for any i∈N, $x_{i}^{(t)}=1$. Therefore, x∈D(1). Thus, $S_{\mathbf{1}}^{(1)}\subseteq D({\mathbf{1}})$.

Similarly, we have $S_{\mathbf{0}}^{(1)}\subseteq D({\mathbf{0}})$.