## 1. Introduction and Literature

## 2. Motivation and Examples

## 3. Ideal Reactive Equilibrium

- Deliberate deviation of order n, $n\ge 1$: Starting from any set of initial choices for all the players, a deliberate deviation of order n by a particular player at a particular information set is a departure by that player from his/her initial choice to another of his/her available choices (pure action, no randomization), and he/she supposes that n of his/her immediate followers can observe the deviation. That is, we erase the information sets (if there are any) for those n players’ choices that follow the particular player.After a deliberate deviation, any player may respond with his/her best response to any changes that have occurred. This includes the original player. After that, any other player may respond with his/her best response to all the changes that have occurred thus far, and so on, until everyone is using a best response and we are at a (possibly different) Nash equilibrium.
- Successful deliberate deviation: A deliberate deviation of any order is said to be successful if after the deviation and after all players are allowed to respond using best responses, the final outcome results in an increase in the payoff for the deviating player.If a deliberate deviation is successful, then the outcome is that we remain at the final Nash equilibrium after deviation and best responses. If it is not successful, then the original deviator retracts his/her deviation, and we are back at the original starting point.
- Thought process: A thought process is a sequence of outcomes starting with a deliberate deviation and then tracking all the best responses until we arrive at the resolution of that deviation. If the deviation is unsuccessful, then no movement occurs under the thought process; if it is successful, then the thought process tracks all the various stages of outcomes until we reach the final Nash equilibrium. Then, another deliberate deviation may be be undertaken by any player, and the thought process continues to track the outcomes of that deviation and all the best responses that follow, as before. This continues until either there are no more successful deviations available or until we observe that the thought process is cycling indefinitely through a subset of the terminal nodes.
- Ideal reactive equilibrium: A Nash equilibrium is an ideal reactive equilibrium if:
- all thought processes end with no change in the players’ choices,
- or if all thought processes cycle endlessly through the same subset of terminal nodes, then the original Nash equilibrium is IRE if either its outcome is part of the subset through which the thought process cycles or, in the case of the original Nash equilibrium being a mixed strategy Nash equilibrium, if the cycling utilizes all the actions with positive probability weights in the mixed strategy.

## 4. Properties and More Examples

## 5. Signalling Games

## 6. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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1. | |

2. | It is well understood that this is not a pure dichotomy. Generally, tremble-based refinements being normal form concepts mainly apply to static games, and forward induction refinements apply to dynamic games. However, there are exceptions. Extensive form trembling hand perfection though originating from the tremble philosophy exclusively applies to extensive form games. Iterated elimination of weakly-dominated strategies (which is not a true refinement, but rather a means to simplify a game), a completely normal form process, can at times coincide with forward induction solutions. Finally, the Kohlberg and Mertens stable sets [8] are also a normal form construct, but provide much insight into equilibria for incomplete information extensive form games. |

3. | Different refinements use different definitions of what is profitable. Differences are achieved by modifying the criteria of what constitutes the relevant set of best responses to types benefiting from choosing the out-of-equilibrium signal. |

4. | However, it is not the only way to eliminate incredible threats. |

5. | It is an incredible threat because if that point was actually reached, the incumbent would not fight, since his/her payoff is lower if he/she fights. |

6. | This is an example from Gibbons [12], p. 233. |

7. | “If possible” refers to the situation where both L and M were dominated by R, which would happen if we lower the payoff to Player 1 at L’ to, say, $\frac{3}{2}$. Now, it is not possible to assign both nodes a probability of zero. |

8. | |

9. | “Virtual observability” is a term coined by Camerer, Knez and Weber [15] in experiments where they tested the manipulated Nash thought process. |

10. | It is important to understand that a similar game tree with only the order of play reversed would be a completely different game that does not in any way depict our scenario, because the order of play there does not match the order of play given in the scenario. |

11. | There are many other authors that have experimental results where they use other games (such as weak link games or resource dilemma games) and find that order of play does matter, such as Camerer, Knez and Weber [15], Budescu, Suleiman and Rapoport [19], Guth, Huck and Rapoport [20], and Muller and Sadanand [21] |

12. | Thomspon [22] and Dalkey [23] did some pioneering work establishing some fundamental invariances for extensive form games, which for years have been held as very basic tenets of game theory. Among them was normal form invariance. The idea was that games with the same normal form should have the same equilibrium solutions. However, experimental evidence shows that normal form invariance breaks down. |

13. | Hammond defines sophisticated Nash equilibrium as an answer to this question, but he states that “the precise relationship between sophisticated and manipulated Nash equilibrium deserves further exploration,” and thus, at this point, it is unknown. |

14. | An interesting exception is the Gintis [29] local best response criterion, which agrees with ideal reactive equilibrium in its equilibrium selection for Selten’s horse. However, it differs from the ideal reactive equilibrium in many ways, including its use of cardinal payoffs rather than ordinal ones, as seen from its treatment of the beer and quiche game. |

15. | This is similar to what happened in the beer and quiche game. Starting with pooling at quiche, the strong switched to beer, and the receiver changed to no duel for beer. However, the receiver also switched his/her response to duel for quiche. Thus, it is important to also follow what changes happen to the receiver’s response to the original signal, as this induces further changes by the senders that were staying with the original signal. The changes that resulted in the beer and quiche example are that the other sender (weak) also joined the strong in messaging beer; much like in this example, ${t}_{3}$ also joined the others with the ${m}^{\prime}$ signal. However, for this example, it does not end there; a few more best responses occur before the thought process rests. |

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