“In 1989, the Serbs commemorated their defeat at the hands of the Turks in the Battle of Blackbird Field, in 1389, and it formed the starting point for the Balkans wars of the 1990s.”
— Baumeister and Hastings (1997)

“You know the Spanish crusade against Muslims and the expulsions from Al Andalus [in 1492] are not so long ago.”
— Al Qaeda spokesman
(after claiming responsibility for the 2004 Madrid bombing)

It seems unlikely that remote events from over 500 years ago could have contemporary relevance to the groups above. Strange as it may seem, memory may matter - sometimes in a way that induces profoundly disturbing responses.
The “memories” that induced such horrific behavior — a self destructive war in one case and terrorism in another — were obviously not direct. Instead, they were shaped by oral or written accounts of history that were passed on through the generations. Our work explores a mechanism for memory creation of this kind, which social scientists in various fields refer to as social memory. According to anthropologist Carole L. Crumley, social memory is defined as follows:
“Social memory is the means by which information is transmitted among individuals and groups and from one generation to another. Not necessarily aware that they are doing so, individuals pass on their behaviors and attitudes to others in various contexts but especially through emotional and practical ties and in relationships among generations [...]”
Crumley’s definition is obviously neutral about how social memory is to be used. Societies do not always use their “practical ties and relationships among the generations” to create war, genocide, and terrorism. In some cases, however, it does appear that way. Historian William McNeill has this to say about the aftermath of World War I:
“The suddenness with which the tide of victory reversed directions after June 1918 gave Germans little time to adjust to defeat [...] The German army was still on French soil and its leaders could claim with enough plausibility to be convincing to those who wished to believe it that German soldiers had never been defeated in battle but lost the war because they had been betrayed by Social Democrats and other social revolutionaries in the rear [...] The Nazi movement founded itself on this myth [...]”
Our work attempts to develop these ideas formally. We investigate how destructive behavior can arise from social memory. In particular, we focus on how incorrect social memory can arise in equilibrium, even in the face of contrary physical evidence, and how it is critical in creating and perpetuating destructive conflict on a large scale.
Clearly, not all conflicts are universally destructive. Some yield clear winners and losers. Moreover, traditional equilibrium theories can account for at least some “moderately bad” outcomes without social memory having any special role. For instance, conflicts yielding long run payoffs above each player’s stage game minmax can be sustained in a standard repeated game if the players are patient enough. Alternatively, conflict can arise in a standard signalling model as a rational response to a partially revealing strategy.
Our interest in this work, however, lies in conflicts with no winners and losers; conflicts that are, in fact, so destructive that existing theories have trouble explaining them. In standard repeated games, for instance, the stage game minmax payoff builds in a modicum of self protection. A player can do no worse than to choose a stage game best response against anything that other players can conceivably do to him. Consider, for instance, the Battle of Verdun in 1916 in which hundreds of thousands of French and German soldiers died and where the final position of these armies at the end of the battle was no more than a few kilometers away from what it was at the beginning. It seems unlikely that such an outcome was above these countries’ minmax, and so a simple repeated game explanation would then require that both countries would be well compensated in the long run for carnage suffered in battle. Similarly, a drawback of signalling games is that they require agents to assign large likelihood to objective states of the world in which the conflict is desirable from at least one country’s point of view. Yet in some conflicts, World War I again being a case in point, decision makers are often aware at the outset that the hostilities are undesirable.1 
The goal of our work is not to discredit these models but rather to expand them in a way that can accommodate a formal definition and role for social memory in conflicts. We therefore posit a model that abstracts away from scenarios in which there are positive incentives for conflict, for instance because one social stratum stands to gain from what is an overall destructive conflict with another society. Consider a repeated game faced by two nations each period in which all out conflict is catastrophically bad for all involved in a way that everyone understands. Each nation in this conflict is a “dynastic” placeholder for individual decision makers who are finitely lived but care about what happens to future generations who inhabit the same nation-dynasty. Families, tribes, and ethnic groups are all ongoing entities with similar features.

Our results demonstrate the possibility that participants, even in the face of imperfect but mounting evidence pointing to the truth, rely exclusively on intra−dynastic messages to gauge the best course of action. Weighing the various possible intentions and mistakes that may generate their predecessors’ behavior, we show that it is possible that the beliefs of the current participants may stray very far from what is actually taking place. In other words, social memory within and across societies can be systematically incorrect. Catastrophic and perpetual all out conflicts may then actually materialize in a variety of ways provided everyone making decisions today is sufficiently concerned about the future.
To illustrate the mechanism by which this occurs, consider a world in which there are two dynasties, France and Germany. These countries are engaged in a repeated “Game of Conflict”. In a Game of Conflict, there is at least one “destructive” action profile which takes both countries below their “individually rational” (minmax) payoff levels in the stage game. We refer to this profile as “all out war.” Notice that if both countries had perfect memory of the past then all out war could not occur with high frequency since, with perfect memory, long-run equilibrium payoffs must lie above the stage game minmax.
Consider an equilibrium in which all out war occurs with arbitrarily high frequency. Each war epoch starts at a predetermined date (but the logic easily extends to a war epoch brought about by some, possibly random, exogenous payoff irrelevant event like the assassination of an Archduke). A war epoch continues until some possibly distant point in the future at which time the slaughter will be over, and peace will prevail again. To rationalize this scenario each side believes that plunging into war is in the long run interest of their country. The question in the mind of both sides is: how good will peace be for them once it prevails? Critically, both sides must think that peacetime will be a lot worse for them if they fail to wage war today.
This is possible in the dynastic Game of Conflict because peaceful epochs come in three very different varieties: one that is good for both sides (“cooperation”), one that is bad for one side and good for the other (“domination” by one country over the other), and a third one in which the two sides are reversed. Suppose, then, that France did not wage war. Then the equilibrium entails that peace in that case simply consists of domination of France by Germany. This asymmetric peace continues for a short time until the next war epoch, at which point a long phase of all out war resumes. But why should France engage in such costly slaughter? After all, it does not have anything more to lose during peace times: these are already good for Germany and bad for France . Indeed, if France had full memory about the past history, it would have an incentive not to choose the war option since it would then see clearly that this option places its long run payoff below its stage game minmax.
This is where the possibility of systematically incorrect social memory becomes critical. We demonstrate how France, relying on its own historiography (passed down from one generation to the next) rather than on available evidence, suffers from the illusion that peace times are not asymmetrically biased against it. If such an illusion can be maintained for long enough in the face of mounting evidence against it, and if the future matters a lot for everyone, then our argument is complete and a world with very frequent destructive all out wars is compatible with rational equilibrium behavior.
To summarize, we show that there exist equilibria in the dynastic Game of Conflict with the following properties. (1) All out war occurs with arbitrarily high frequency. (2) Physical evidence is ignored−i.e., neither beliefs nor actions condition on evidence from the past.  (3) Social memory following counterfactual histories may be incorrect; in fact, there are certain possible histories after which social memory is maximally incorrect in the following sense. Using a standard hypothesis testing procedure, and given any level of confidence, the null hypothesis that a certain event took place is rejected by a statistician who observes the accumulated evidence. However, all participants’ beliefs assign probability 1 to the very same event.
It is worth noting that our construction does not depend on anyone’s failure to understand the consequences of war. Everyone in this model correctly anticipates that it will be horrific.2  Rather, what matters is the failure of future leaders to comprehend some of the various counterfactual options off path available to present leaders. Hence, moving forward, current leaders are led to the inescapable conclusion that war is necessary. This logic and its ugly consequences do not arise in standard repeated games with infinitely−lived players.
Crucially, messages can in principle convey more information than any imperfect physical evidence. This is because they are sent after the current action profile is observed. It turns out that this is suffcient to make viable equilibria in which physical evidence is ignored, but the messages convey the “wrong” information to future individuals. To underscore this point, we examine a variation of the dynastic model in which information is further degraded so that present actions of the rival country are not directly observed. All dynastic members observe the same imperfect evidence about the rival. We show that, in any pure strategy equilibrium, messages would be useless in this case, and so participants would be forced to confront the evidence. The perpetual state of frequent all out war sketched above cannot arise in this case.

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.

The Role of Social Memory in All-Out War

Recall McNeill’s discussion of the Nazi myth. The problem was not so much that the Germans did not know what happened. Rather, it was that they misunderstood why it happened. The catastrophe of the German defeat was interpreted by many, including Hitler, as a betrayal or a lack of will by civilians: “a deep distrust of civilian steadfastness, based on Hitler’s memories of 1918, governed Germany’s domestic policy during the first phases of World War II.”3 
But would Hitler have believed a message that exonerated civilians in the rear? In the model, the message deviating from the expected plays this role. This message indicates that the German civilian leadership chose the protective action D instead of the self-destructive action W. If the deviating message were sent, however, it would have been discounted completely by Hitler, who would have concluded instead that the message was sent in error, despite the fact that message is an equilibrium response to the self protective action D. In these circumstances, the German civilian leadership opts for W instead of D since the latter entails that a distrustful future German decision maker would unwittingly hurt Germany following the choice of D. In sum, the future German leader’s belief that his predecessor chose W on path is correct, but his understanding of why it was chosen is not.

Conclusion

Social memory is embodied in a society’s vicarious beliefs about the past. These beliefs are shaped by both intergenerational communication and the imperfect physical evidence from the past. To formalize it entails a detailed model of the intergenerational communication within dynastic societies.
We show that there exist equilibria in a canonical Game of Conflict in which “all out war” occurs with arbitrarily high frequency. In these equilibria physical evidence is ignored and, in fact, beliefs of one or both parties are maximally incorrect after certain events.
Significantly, these equilibria can occur despite the fact that there are no objective states of the world in which the conflict is desirable from anyone’s point of view. These outcomes could not be attained in a standard infinitely repeated game. Because messages can, in principle, convey more information than any imperfectly informative physical evidence, there are equilibria in which the current generation focuses only on the messages. Ironically, social memory can be maximally incorrect precisely because it relies on sources that can be more informative than hard evidence.
Two further issues bear mentioning here. The first concerns a critical robustness test for sequential equilibrium. The strategies used in this equilibrium remain sequentially rational even if we consider the trembles that generate the equilibrium beliefs when they are arbitrarily small, but before they have completely shrunk to zero. Intuitively, the robustness property we have just claimed tells us that the equilibrium on which we have focused throughout the paper survives the possibility that play does in fact stray off the equilibrium path with positive (albeit arbitrarily small) probability.
Second, we have focused our attention entirely on “bad” equilibria with frequent all out wars. The implication is that systematically wrong social memory is a bad thing. But there is a flip-side to this which highlights the possible “good” consequences of wrong social memory. Precisely because very bad payoffs can be sustained on path, these payoffs can be used as “punishments” off it.
Whether cooperation can in general be sustained more easily in the dynastic game is an open question at this point. The question of how the possibility of systematically inaccurate social memory might lead to the emergence of better equilibria is clearly both interesting and potentially important. We leave this issue for future research.

Luca Anderlini is Professor of Economics, Georgetown University.
Dino Gerardi is Associate Professor of Economics, Yale University.
Roger Lagunoff is Professor of Economics, Georgetown University.
This paper is an adaptation of “Social Memory and Evidence from the Past”, of the Yale Cowles Foundation Working Papers Series. The original paper may be read at http://cowles.econ.yale.edu/P/cd/d16a/d1601.pdf.