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The construction industry is generally characterized by high emissions, making its transition to low-carbon practices essential for achieving a low-carbon economy. However, due to information asymmetry, there remains a gap in research regarding the strategic interactions and reward/punishment mechanisms between governments and firms throughout this transition. This paper addresses this gap by investigating probabilistic and static reward and punishment evolutionary games. The findings indicate that (1) Probabilistic rewards and penalties policies are more effective during the initial stages of the transition, whereas static mechanisms are more conducive to ensuring long-term stability. (2) The maximum values of rewards and penalties significantly influence the evolution of the low-carbon transition, with higher incentives enhancing motivation and more significant penalties imposing stricter constraints. (3) An increase in the cost of government involvement facilitates the low-carbon transition. (4) The benefits to both government and enterprises are critical in determining the application of static versus probabilistic rewards and penalties. The government may decide to cap probabilistic rewards and penalties by the magnitude of the benefits or adopt static rewards and penalties. This study offers theoretical support and a decision-making framework for developing effective low-carbon policies. [ABSTRACT FROM AUTHOR]