  The multipartite Greenberger-Horne-Zeilinger (GHZ) states are indispensable
elements for various quantum information processing tasks. Here we put forward
two deterministic proposals to dissipatively prepare tripartite GHZ states in a
neutral atom system. The first scheme can be considered as an extension of a
recent work [T. M. Wintermantel, Y. Wang, G. Lochead, \textit{et al}, {Phys.
Rev. Lett. \textbf{124}, 070503 (2020)}]. By virtue of the polychromatic
driving fields and the engineered spontaneous emission, a multipartite GHZ
state with odd numbers of atoms are generated with a high efficiency. This
scheme effectively overcomes the problem of dependence on the initial state but
sensitive to the decay of Rydberg state. In the second scenario, we exploit the
spontaneous emission of the Rydberg states as a resource, thence a steady
tripartite GHZ state with fidelity around $98\%$ can be obtained by
simultaneously integrating the switching driving of unconventional Rydberg
pumping and the Rydberg antiblockade effect.
