  Evolutionary prisoner's dilemma games are studied with players located on
square lattice and random regular graphs defining four neighbors for each one.
The players follow one of the three strategies: tit-for-tat, unconditional
cooperation, and defection. The simplified payoff matrix is characterized by
two parameters: the temptation $b$ to choose defection, and the cost $c$ of
inspection reducing the income of tit-for-tat. The strategy imitation from one
of the neighbors is controlled by pairwise comparison at a fixed level of
noise. Using Monte Carlo simulations and the extended versions of pair
approximation we have evaluated the $b-c$ phase diagrams indicating a rich
plethora of phase transitions between stationary coexistence, absorbing and
oscillatory states, including continuous and discontinuous phase transitions.
By reasonable costs the tit-for-tat strategy prevents extinction of cooperators
across the whole span of $b$ values determining the prisoner's dilemma game,
irrespective of the interaction graph structure. We also demonstrate that the
system can exhibit a repetitive succession of oscillatory and stationary states
upon changing a single payoff value, which highlights the remarkable
sensitivity of cyclical interactions on parameters that define the strength of
dominance.
