DAGs with Ambiguous Temporal Ordering Between Nodes I am working on a project where I am attempting to estimate the causal impact of civil war peace agreements (treatment) on levels of violence (outcome). While developing the DAG, I came across an issue where some of the nodes cause each other (pictured below is a simplified version of the DAG designed to only include the nodes where this became a problem).

Peace agreements sometimes result in an agreement to authorize a third-party peacekeeping operation (PKO), but, in other cases, PKOs facilitate the establishment of a peace agreement. Sometimes, mediation between warring parties leads to the authorization of a PKO. In other cases, a PKO itself facilitates mediated talks between warring parties. There is no natural ordering to these phenomena. PKOs do not always precede mediation and mediation does not always precede PKOs (it depends on the context). Likewise, a similar relationship exists between peace agreements and PKOs.
One way of getting around this (I think, could be wrong) is to separate these three nodes into six nodes. Instead of PKO, Peace Agreement, and Mediation, I would have:

*

*Mediation t-1 (Mediation preceding PKOs)

*Mediation t (Mediation following PKOs)

*PKO t-1 (PKOs preceding mediation or peace agreements)

*PKO t (PKOs following mediation or peace agreements)

*Peace Agreement t-1 (Peace Agreement preceding PKOs)

*Peace Agreement t (Peace Agreement following PKOs)

Here is the DAG with the new lagged nodes incorporated:

I'm not sure that this actually solves the issue, however. After all, aren't PKO(t-1) and Mediation (t-1) & Peace Agreement (t-1) and PKO (t-1) mutually directed in the same way that their (t) counterparts are?
I was wondering if anyone could help with this issue.
Update on this question
Is another way to get around this is to assign an unobservable node to the dual-directed relationships? For example, consider the image below:

Take the PKO -> Mediation/Mediation -> PKO example. Mediation (t-1) can cause PKO (t) and PKO (t-1) can cause Mediation (t). Perhaps these can both be collapsed within the unobservable node given that the causal mechanism driving the variation in ordering is unknown?

If you have bi-directional arrows it's not a DAG (i.e. not a Directed Acyclic Graph), full stop. The 'acyclic' means there are no loops in the causal model.
This is because in a DAG (see, e.g., Hernán and Robins, 2020; Pearl, 2000, etc.):

*

*A uni-directed arrow means "the variable at the arrow's tail directly causes the variable at the arrow's head." A lack of an arrow between two variables means "no direct cause".

*Unidirected arrows also define temporal ordering, as in "the variable at the arrow's tail occurred before the variable at the arrow's head occurred."

That said, there are other causal graph formalisms, including ones involving digraphs (i.e. arrows which go both directions). One such formalism is Levins' "loop analysis" (Levins, 1974, Puccia & Levins, 1986, etc.) which uses signed digraphs (i.e. two variables may be connected by no arrow, a uni-directional arrow, or a bi-directional arrow, and arrowheads have signs) to model causal systems defined by every variable being a direct or indirect cause of every variable in the system at future times, and where the sign indicates "direct causal increase" (positively signed arrowheads) or "direct causal decrease" (negatively signed arrowheads). Also temporal ordering does not exist between variables, unlike in a DAG. Rather, this signed digraph formalism "looks head-on at the arrow of time" so that all times during which the causal system exists with the specified relationships are represented.



References
Hernán, M. A., & Robins, J. M. (2020). Causal Inference: What If. Chapman & Hall/CRC.
Levins, R. (1974). The Qualitative Analysis of Partially Specified Systems. Annals of the New York Academy of Sciences, 231, 123–138.
Pearl, J. (2000). Causality: Models, Reasoning, and Inference. Cambridge University Press.
Puccia, C. J., & Levins, R. (1986). Qualitative Modeling of Complex Systems: An Introduction to Loop Analysis and Time Averaging (C. J. Puccia & R. Levins, Eds.). Harvard University Press.
