Search for a unique Nash equilibrium in two public goods games: mixed integer programming technique

ABSTRACT

We provide the methods for searching for the unique Nash equilibrium using mixed integer programming techniques. We also simulate the model parameters using this technique and derive the numerical solution to show the characteristics of a key player providing both public goods as suggested by Bergstrom et al. (1986). We present two key results. First, when the number of players decreases, the appearance rate of the unique equilibrium accompanying a key player increases. Second, when the variance of the preference or price parameters decreases, the appearance rate of the unique equilibrium accompanying a key player increases.

KEYWORDS:

• mixed integer programming technique
• search for unique equilibrium
• simulation
• appearance rate

JEL CLASSIFICATIONS:

• H41
• D01
• C70
• C60

Acknowledgement

An earlier version of this article was presented at the Asian Meeting of the Econometric Society (Singapore) and the Operations Research Society of Japan (Osaka in Japan). We are grateful to reviewers and participants for their helpful comments. The research of the second and third authors was supported by Grants-in-Aid 20K01755 and 19K01711, respectively, from the Ministry of Education, Culture, Sport, Science and Technology of Japan.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Correction Statement

This article has been republished with minor changes. These changes do not impact the academic content of the article.

Notes

¹ This LCP is equivalent to (5) because the first condition of (8) is redundant. The reason why is as follows. As ${\xi }_i=0$ when $g_i^{\hspace{0.17em}}$> 0 from the third condition of (8), the fourth equation in (5) is reduced to $p_i{g}_i+q_i{h}_i+p_i{\alpha }_{i}G/{\beta }_i=w_i$, implying the first condition of (8). When $h_i^{\hspace{0.17em}}$ > 0, a similar argument holds.

Additional information

Funding

This work was supported by the Ministry of Education, Culture, Sport, Science and Technology of Japan [20K01755,19K01711].