We consider mean field games in a competitive Markov decision process (MDP) framework as motivated by security maintenance and vaccination models. Each player has a continuum state, binary action, and a discounted cost function. The state stochastically deteriorates but can be reset to an ideal position. Under monotonicity conditions on the cost, we show a threshold structure of the policy and analyze ergodicity of the controlled state processes. We further prove uniqueness under positive externalities, and this is achieved by establishing some comparison theorems for regenerative processes. The extensions to long run average costs and a continuous time model with Poisson jumps will be discussed.
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