Disclaimer. Don't rely on these old notes in lieu of reading the literature, but they can jog your memory. As a grad student long ago, my peers and I collaborated to write and exchange summaries of political science research. I posted them to a wiki-style website. "Wikisum" is now dead but archived here. I cannot vouch for these notes' accuracy, nor can I say who wrote them.
Wagner. 2000. Bargaining and war. American Journal of Political Science 44 (July): 469-484.
There are two ways to theorize about bargaining and war:
"Because decision theory implies that any uncertain expectations can be represented by a lottery, a costly lottery that is used to represent a war can be interpreted in two ways. One is that it represents a purely military contest fought until one or another state is incapable of further fighting, and thus a prewar choice between such a contest and a negotiated settlement is the result of a take-it-or-leave-it offer made by one state to another: if this offer is rejected then a fight to the finish will occur. The other possible interpretation is that bargaining continues to be possible after fighting begins, and the costly lottery of war merely represents the prewar expectations of states concerning the terms of the final settlement and the costs that will be suffered before it is reached."
Most models look at war as the result of a choice between bargaining and fighting (e.g. Fearon 1995, Gartzke 1999), the first of the two ways to connect bargaining and war. These models suffer from some common problems: "Thus formal models that focus on a prewar choice between war and a negotiated settlement either fail to distinguish between the outcome of fighting and the outcome of bargaining (Morrow 1985), or assume implicitly (Powell 1996) or explicitly (Fearon 1992,1995a) that bargaining ends when war begins. They are therefore at best either irrelevant to historical wars (since they ignore the possibility of negotiated settlements once war has begun) or contain no information (since they do not tell us what is the basis for the expectations represented by the costly lottery)."
So this is the puzzle Wagner intends to solve:
"Thus no one has developed a model encompassing both war initiation and war termination that would allow us to examine the relation between fighting and bargaining. Without one we cannot hope to understand why war occurs, how long it lasts, why it takes the form it does, or why it stops. I will present such a model in the next section."
Wagner addresses this by developing a model of war and bargaining.
Following Clausewitz, Wagner differentiates between "absolute war" (or "war in theory"--a war to destroy/disarm your opponent) and "real war" (the more limited wars that we usually see).
Bargaining leads to real war (as bargaining continues), which leads to absolute war (if bargaining fails). States can stop at any point.
The whole thing is driven by each side's expected payoff of absolute war. This payoff is determined by W (the value of winning), L (the value of losing), and p (the probability of winning or losing). The trouble is, we often don't know p. (He also brings in q, the probability that L will lose (be disarmed) by a certain date, and c, the cost for each period's fighting.)
Bargaining during war is a Rubinstein bargaining game. A makes an offer; if B rejects it, she makes a counteroffer; and so on. At each stage, each player's decision of whether to accept an offer depends on the variables above, and on the current expected outcome of the absolute war. If bargaining breaks down (i.e. the two sides can never agree), then absolute war continues until one side vanquishes the other. If, at some point, the two sides make a bargain, then the war ends (and we saw only a "real" war, not an "absolute" war).
The problem with Rubinstein: "However, none of this tells us anything about the probability of war, since if everything in this model is common knowledge the state that might be attacked can always prevent an attack by making the required concession and therefore the probability of war is zero. Thus to understand what this analysis implies about the probability of war we must first understand why a war might occur at all." (This was Fearon's (1995) complaint.)
So why does war occur in the first place?
To explain "real" war, we replace p, q, and c with each side's expectations about p, q, and c. "The answer Clausewitz gives is that military operations can be designed not to contribute to the enemy's defeat, but to influence instead his expectations of the future course of the war (Clausewitz 1976,92)." The two states have competing ideas about how a total war might end; limited fighting helps reveals information about which outcome if more likely: "A more accurate analogy might be to say that it is an experiment that allows states to test competing hypotheses about the outcome of a war fought to the finish."
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