  Revision game is a very new model formulating the real-time situation where
players dynamically prepare and revise their actions in advance before a
deadline when payoffs are realized. It is at the cutting edge of dynamic game
theory and can be applied in many real-world scenarios, such as eBay auction,
stock market, election, online games, crowdsourcing, etc. In this work, we
novelly identify a class of strategies for revision games which are called
Limited Retaliation strategies. An limited retaliation strategy stipulates
that, (1) players first follow a recommended cooperative plan; (2) if anyone
deviates from the plan, the limited retaliation player retaliates by using the
defection action for a limited duration; (3) after the retaliation, the limited
retaliation player returns to the cooperative plan. A limited retaliation
strategy has three key features. It is cooperative, sustaining a high level of
social welfare. It is vengeful, deterring the opponent from betrayal by
threatening with a future retaliation. It is yet forgiving, since it resumes
cooperation after a proper retaliation. The cooperativeness and vengefulness
make it constitute cooperative subgame perfect equilibrium, while the
forgiveness makes it tolerate occasional mistakes. limited retaliation
strategies show significant advantages over Grim Trigger, which is currently
the only known strategy for revision games. Besides its contribution as a new
robust and welfare-optimizing equilibrium strategy, our results about limited
retaliation strategy can also be used to explain how easy cooperation can
happen, and why forgiveness emerges in real-world multi-agent interactions. In
addition, limited retaliation strategies are simple to derive and
computationally efficient, making it easy for algorithm design and
implementation in many multi-agent systems.
