Cooperation, Retaliation and Forgiveness in Revision Games

TL;DR

Limited retaliation strategies show significant advantages over Grim Trigger, which is currently the only known strategy for revision games, and can be used to explain how easy cooperation can happen, and why forgiveness emerges in real-world multi-agent interactions.

Abstract

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.

Figures (4)

• Figure 1: Example of MPC cooperative plan $x(t)$.
• Figure 2: Equilibrium payoffs of LR and GT in revision Prisoner's Dilemma with different error rates. In each subfigure, $50$ different revision games are simulated, with lengths varying from $T=0^+$ to $T=50$. The $y$-axis is the expected equilibrium payoff. Payoff in each simulated game is depicted by a colored dot, and all dots from $50$ revision games compose a curve.
• Figure 3: Equilibrium Payoffs in the Prisoner's Dilemma revision games with different error rates. GT strategy is the same in all subfigures while LR strategies have different retaliation fierceness $k$.
• Figure 4: $(a)$ and $(b)$ show two Limited Retaliations strategies with different MPC plans for revision Cournot game, and $(c)$ to $(h)$ show their equilibrium payoffs under different error rates.

Theorems & Definitions (20)

• Definition 1: NE
• Definition 2: SPE
• Definition 3: LR Strategy
• Definition 4: Cooperative Plan
• Lemma 1: One-Shot-Deviation Principle
• proof
• Definition 5: Deviation Gain
• Definition 6: Retaliation Loss
• Theorem 1: SPE Constraint
• Theorem 2