In this paper, we develop a noncooperative game to model a non-life insurance market. Our first goal will be to analyze the effects of competition between insurers through different indicators: the solvency level, the market share, the underwriting results. Secondly, we will seek to further understand the genesis of insurance market cycles. Insurance market cycles have troubled actuaries and academics for decades: this game-theory focus will allow us to shed a different light on the subject.