The Model

In this section, we develop a model of arbitrarily frequent and severe conflict. Two countries, France and Germany, face off in a stationary Game of Conflict, the details of which are laid out in the next subsection. Though the class of stage games to which our analysis applies is broad, for the sake of concreteness we work with the simple example below throughout the paper. We then posit the explicit demographic structure that defines each nation--dynasty. Social memory will be shown to be a well defined notion arising naturally from a standard (though perhaps underappreciated) aspect of sequential equilibrium.
A game of conflict. We work with an “augmented” Prisoners’ Dilemma, with each France and Germany having three potential courses of action: The first two are
the standard C, denoting “cooperate”, and D, denoting “defect” found in any Prisoners’ Dilemma, plus a third one, denoted W, representing “war.” When both countries choose W a state of “all-out war” ensues, in which both sides are engaged in destructive conflict with one another. Below is a diagram depicting this game, complete with expected payoffs for each country.

After a cursory examination, we can see that W is a dominated strategy and that if the two countries are to be engaged in this faceoff scenario indefinitely, then the optimal strategy for each country is D, to defect rather than to keep engaging in all out war. Our results are directly related to the possibility of a systematically incorrect social memory when the two countries face off repeatedly in a dynastic game. This is what we turn to next.
Dynastic game. Each France and Germany identifies a “dynasty,” which in our model describes a placeholder for successive generations of individual decision makers. At any given time, the dynasty is inhabited by one such decision maker who cares about his own payoff and those of future generations within his own dynasty. We hold that the decision makers may be replaced at the same time or at different times, however, each decision maker is assumed to have a finite lifespan. However, purely to simplify some of the details, we proceed with a version of the model in which all decision makers live exactly one period after which they are all simultaneously replaced.
In this instance, the individuals care both about the short-term and long-term payoffs for their respective countries. As such, we can interpret concern about the future payoffs for countries as altruism, since the individual decision makers won’t be alive to witness them. When a new decision maker steps into power, he gleans information about the past from two sources: imperfect, public evidence from past cycles, and a private message from his immediate predecessor. The decision-maker also is able to estimate the future footprint his current actions will leave, and he is also able to observe the actions of the other country. Before relinquishing his post, he gives his own private message to his successor in the same dynasty.
Equilibrium. Few focus on the sequential equilibria of the dynastic game, assuming that all decision makers are very concerned about the future of their respective countries. In other words, all participants act rationally given their beliefs. And the participants’ beliefs (on and off path) are subject to the “complete theory of mistakes” that is common to all decision makers.
The full details of how France and Germany can be caught in a sequential equilibrium in which all out war takes place “almost all the time” are intricate, but a basic outline can be sketched as follows. The two countries start off in a “phase” in which all out war takes place in most periods (the pair (W,W) is chosen) , but when it does not take place the two countries cooperate (the pair (C,C) is chosen). The structure of the sequential equilibrium also prescribes that if one country tries to alter this state of affairs (all out war in most periods, cooperation during peace periods), then a new “phase” will begin. The country which deviated from the initial phase will be punished in this new phase: while all out war will take place just as before during most periods, during the peace periods the deviating country will now cooperate while the other country defects (either the pair (C,D) or the pair (D,C) is chosen). The critical step in the construction is that the country that is being punished does not properly comprehend what is taking place. Upon starting the job, the decision maker in the punished country is told by his predecessor that their country is being punished, but simply does not believe this to be true. He believes that his predecessor’s account of the past is just a “mistake.” During his tenure as leader, the decision maker will then discover that his country is in fact being punished. However, his attempts to communicate this fact to future leaders are doomed to fail. Just as he misinterpreted the message he received, they would misinterpret his own.