War and Negotiation in Two Dimensions Jeffrey Alan Carnegie
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- The Constant-Sum Assumption in War
- Figure 1: One Dimensional Visualization of War
- are expected utility from war for each A and B, respectively.
- The 2D Diagram of Rational War
- Figure 2: Two Dimensional Visualization of War Model
- Table I: Example of Mutual Gains: Differing Prize Values
- Caption: Each player’s value of the three prizes is shown as a portion of total utility.
- Figure 3: The 2D Diagram for the Mutual Gains Example
- Figure 4: 2D Diagram with Indivisible Prizes
- Figure 5: 2D Diagram with Inflated Expectations
- Figure 6: 2D Diagram with First Strike Advantage
- Figure 7: 2D Diagram with Selectorate Theory
- Figure 8: 2D Diagram with Relative Gains
- Figure 9: 2D Diagram with Negotiation Sequences
- Figure 10: Conditional Costs of War, Unimodal Duration Distribution
- Figure 11: Conditional Costs of War, Bimodal Duration Distribution
- Figure 12: 2D Diagram with Negotiation and Sunk Costs
| War and Negotiation in Two Dimensions
Jeffrey Alan Carnegie Department of Politics, New York University Word Count: 7353 Thanks to Muhammet Bas, Jeff Frieden, and participants in the Spring 2013 Graduate IR Seminar at Harvard for helpful comments. Abstract War has been visualized as a single-dimension constant-sum game. However, an additional dimension captures the possibility of mutual gains and a more complex negotiation process. I present a new tool for visualizing rational conflict. I use the tool to show how mutual gains may make peace easier or more difficult to achieve. This method intuitively depicts utility for each player with two relevant sets: the settlements that are possible to achieve under a legitimate division of prizes, and the agreements that would be preferable to war for both players. These sets may intersect, revealing a set of war-ending settlements. Further, this tool better illustrates the negotiation process. Offers can be shown as points, with the order of offers illustrated as lines between them. This technique can readily accommodate multiple theories of conflict simultaneously. I show examples of information asymmetry, first strike advantage, indivisible goods, commitment problems, the democratic peace, and sunk costs. The tool is useful for teaching rational explanations of war by explicitly describing sets of potential agreements. It also reveals that war may be caused by a failure of the negotiation process, even when a mutually beneficial settlement exists, because neither player wants to offer too much to the other player. While most of the rationalist explanations focus on limitations to the possible and acceptable sets, existing explanations have proven difficult for various reasons. I suggest more work should focus on the negotiation process as a neglected area of study. “You plume yourself on inferring the existence of a Straight Line; but I can see Straight Lines, and infer the existence of Angles, Triangles, Squares, Pentagons, Hexagons, and even Circles. Why waste more words?'” Flatland: A romance of many dimensions by Edwin A. Abbott Introduction One of the largest puzzles in political science is why costly wars occur. Just as with lawsuits, if an agreement can be reached by a settlement, avoiding the costs, then it is better for both parties. James Fearon summarized these rational models, those driven by utility maximization, of conflict in his 1995 paper, ``Rationalist Explanations for War.'' He used a one-dimensional diagram to intuitively demonstrate settlements that are mutually beneficial and those that are not. He explained informational asymmetries and commitment problems as two reasons for war. The diagram Fearon used for these explanations defined the bargaining range as the numerical range between the expected outcome of war minus costs for each player. His diagram represented a single divisible prize, an objective of the war that would go entirely to the winner. The bargaining range represented all possible divisions of the prize that both players would prefer to war. By using a diagram, Fearon offered an intuitive way to visualize a mathematical model of war. It proved a valuable tool, both for researching and teaching rationalist explanations of war. However, the one-dimensional diagram implicitly assumed that possible settlements are fully represented by the division of a single prize. It implicitly assumed that conflict was a constant-sum game, with no chance of mutual gains from multiple prizes. In this paper, I build on Fearon's diagram by relaxing the assumption of a constant-sum game. I suggest a better way to visualize war that covers existing explanations and expands on them, by simply adding an additional dimension. My method is to use a single dimension for the utility of each player, and to explicitly identify two sets: the set of possible settlements based on how prizes can actually be divided, and the set of acceptable settlements that all players prefer to war. The method I present here is easy to draw, intuitive to understand, and can incorporate any rationalist explanation using established game theory. In the next section, I discuss the limitations of the one-dimensional depiction and the constant-sum assumption. I then discuss the two dimensional diagram in detail, focusing on the effects of mutual gains. I further categorize rational theories of war into three groups: (1) limitations reduce the set of possible settlements; (2) high expected utilities from war reduce the area of the mutually acceptable settlements; or (3) a failure of negotiation leads to war despite the existence of possible acceptable settlements. Any or all of these explanations may occur simultaneously. Finally, I conclude with a discussion of insights generated by the new method. The Constant-Sum Assumption in War Prior to rational theories, realists focused on bandwagoning, balancing, and polarity as explanations for war. Liberals offered conflicts between different cultures and forms of government as explanations. These explanations offered reasons for changes in utility, but did not explain why negotiated agreements may fail. In the single divisible prize formulation, which I will call the 1D Diagram, these explanations may move the area of possible agreements toward one end or the other. These theories did not completely remove this bargaining range, however. The rationalist explanations addressed this point specifically, by explaining when this window of war-ending agreements closed. Figure 1: One Dimensional Visualization of War 
Caption: p is the expected chance of A winning the war, d=0.08 is the proportional cost of war, EA and EB are expected utility from war for each A and B, respectively. The 1D Diagram of war uses one dimension, with some prize split between two players. I show an example in Figure 1, using the proportional costs formulation from Powell (2006). At one extreme, player A gets the entire prize, and at the other extreme, player B gets the entire prize. Assuming both opponents have the same beliefs about the likelihood that A is victorious, p, and each has some expected proportional cost to war, d, there is a range of outcomes that are preferable to war for both opponents. The bargaining range, which is the set of war-ending outcomes, is clearly shown and readily calculated. The 1D Diagram depicts conflict on a single-scale, which may be taken as the division of a single prize, such as territory. However, it is better understood as utility for one player. By illustrating a two-player war in one dimension it forces the utility gain of one player to be proportionally equal to the utility loss of the other. This is the constant-sum assumption. The assumption implies that there is no way to leverage multiple gains in settlements. If the bargaining range entirely disappears, then war must occur under the 1D Diagram, even when there may be a division of prizes acceptable to both player because each player values different things. In many ways, negotiation is the art of understanding different values and using multiple gains. Abstractly, there are two relevant sets to consider for utility-driven explanations of war. First, the possible set contains all possible ways of dividing the prizes. Some divisions simply are not possible, such as all players getting their maximum (or all getting their minimum) utilities. Multiple prizes or concave utilities make it possible to find mutual gains as well as mutual losses. Second, the acceptable set contains all outcomes that are superior to each player's expected utility from war. If any player prefers to go to war, based on his possibly wrong estimate of the outcome of war, then the settlement would be rejected. The intersection of the possible set and acceptable set is the bargaining range, the set of settlement offers that would end or prevent the war. The 1D Diagram artificially limits the possible set to only constant-sum divisions of the prizes. Using this tool, the entire axis is the possible set, which takes focus away from limitations of this set as reasons for war. As a result, the bargaining range is the same as the acceptable set. However, both sets can change in size and shape, and so either can affect the bargaining range. By not illustrating some possible agreements, the 1D Diagram may give a false impression that negotiations are futile. For instance both players may have mutual optimism that would prevent a zero-sum settlement, but they can find an acceptable agreement if they take advantage of mutual gains. The constant-sum assumption implies that mutual gains are impossible. Brams and Taylor (2000) show an easy way to achieve a division that maximizes mutual gains in divorce settlements. If there exist multiple prizes in a conflict, that is multiple items over which the players bargain, it is almost certain that each player has different priorities. This makes mutual gains possible. Unfortunately, it also makes mutual losses possible, and this is a threat to bargaining, especially early on when neither player has a clear understanding of the other player's desires. Mutual gains may seem a trivial addition to the understanding of conflict, but there are scenarios where possible war-ending agreements only exist because of mutual gains. In these cases, war cannot be avoided in a constant-sum game, but it can be with mutual gains. Negotiation, like war, is an ongoing process rather than a single event. When a crisis occurs, there is an initial period of negotiation. If that does not settle the crisis, war begins. The players then simultaneously fight and negotiate. In negotiating, they make proposals to each other, which can be visualized as a walk along the graph. A player taking a hard bargaining approach takes small steps, which lengthens the time to reach an acceptable settlement. Uncertainty about an opponent's war costs or capabilities or discount rate may cause further difficulties in reaching the intersection of the two sets. As the negotiations proceed, the war also proceeds, further altering each player’s expectations and costs. I can mark each offer in a negotiation as a point in the 1D Diagram, and connect them with lines to show the sequence. However, lines will overlap in one dimension so that the exact sequence will not be seen if it is not monotonic. A more flexible depiction can graph utility functions that they 1D Diagram cannot. For example, player A may not value contested land linearly if it is of strategic importance, perhaps a protective bay that loses naval safety if shared, making it effectively indivisible. To illustrate indivisibility, the 1D Diagram must show utility rather than the prize division, and then inflate expected payoffs to war, which confuses the explanation with mutual optimism. A more obscure example is when player B is interested in lands that may yield natural resources, but finds that other lands are not worth the cost to secure borders. Player B’s highest utility is at a 70% share of the prize, not a 100% share. The 1D Diagram has no ability to depict this. The exact method of prize division matters. Consider that a piece of contested land could be divided evenly by a north-south line, an east-west line, diagonal lines, irregularly, or even piecemeal like a checkerboard. Undoubtedly the latter would yield worse utility for both players than two contiguous blocks, and so is a ridiculous thought. However, the idea illustrates that not all prize divisions in war are created equal. The idea that one player's gain can only come from an equal loss to the other player is a non-trivial assumption. The 1D Diagram can only represent simple utility functions. The 1D Diagram can be expanded to additional players by adding more axes, but this is problematic. The constant-sum assumption limits the possibilities so that no sum can exceed one. The number of dimensions is one less than the number of players; the utility of the last player is such that the total utility sums to one. A three player game is a triangle. Some outcomes are censored because the two player utilities cannot sum to more than one. A four player game is similarly an irregular tilted tetrahedron, cut from the three corners of a cube at which the first three players get all of the prize. Additional players result in more complex pierces cut from hypercubes. The constant-sum assumption means more complex shapes as players are added. Diagrams of rational models of war are useful tools to visualize the numerical constraints on war. The 1D Diagram method is limited because it cannot capture mutual gains or depict complex utility functions over many prizes. Reality may not be so limited by the constant-sum assumption, and this provides more opportunities for war-ending agreements, as well as opportunities for failure. Depicting negotiations in one dimension fails to capture details of the process. It is also simpler to depict all player utilities, rather than calculate the utility of the final player in order to sum to one. For these reasons, I argue that the 1D Diagram imposes unnecessary limitations. The 2D Diagram of Rational War By relaxing the constant-sum assumption, war can take the form of a win-win or lose-lose game. However, this then requires an extra dimension to fully depict. The method I present here, which I will call the 2D Diagram, places each player's utility on a separate axis. While graphing utility requires some calculation from prize divisions, it more fully illustrates mutual gains, the possible and acceptable sets, and the negotiation sequences. While the 2D diagram adds an additional dimension, it depicts a game with multiple players as a cube or hypercube with no censored outcomes. Much simpler objects in multiple dimensions than tetrahedrons. A point can be described completely by a vector without having to calculate the utility of the last player. The 2D Diagram corrects many limitations of the 1D Diagram. An n-dimensional visualization, where n is the number of opponents, shows the underlying assumptions directly. Each dimension represents the utility of an outcome to an opponent. If there is a single infinitely divisible prize and two opponents value it linearly, then the possible set, N, is a line from (0,1) to (1,0). If however, the value of the prize is convex in each portion, such as the protective bay mentioned previously, then N is a convex line. If there are multiple prizes of different value to the opponents, then N includes points both above and below the diagonal line. The expected value of war to each opponent can be plotted as a line, even if the expectations are different. This defines the acceptable set, W, which all opponents weakly prefer to war. If N ∩ W ≠ ∅, then there exists an acceptable settlement that avoids war. By considering an international crisis in terms of these two sets, rather than as outcomes along a one-dimensional line, more scenarios can be readily visualized. Figure 2 shows an example of the 2D Diagram. The gray area is N, which is the possible settlements. The lined area is the set W, which is the acceptable payoffs that avoid war. The intersection is the acceptable possible settlements, those which end war. Figure 2: Two Dimensional Visualization of War Model 
Caption: The Possible Set is gray; the Acceptable Set is lined; the expected outcomes of war are dotted lines. Using this visualization, there are three situations that can prevent a peaceful resolution if conflict. One, situations reduce the possible settlements, N. Two, situations increase either player's expected payoff from war, reducing W. Three, situations result in a failure to find an existing Pareto optimal settlement. It may be that when negotiation fails to avert war, it is due to multiple causes rather than just one. Further, all of these reasons may exist empirically. Empirical studies which seek evidence for only one method will have difficulty because there are several reasons for war. Reiter (2009) shows that different wars exhibit behavior predicted by different models, revealing multiple extant causes. I will first show an example of mutual gains. Then I will discuss each type of situation before using the model to address concerns about the future. To illustrate mutual gains and how they may be important, consider two states in a crisis. Player A believes she can get an expected utility of 0.55 from war. Due to economic development, A has increased her military power and also changed her needs relative to the existing status quo (which she now values at 0.5). The states have strong informational asymmetry, so that player B has the expected utility of war at 0.55 as well. In a constant-sum game, there is no possible settlement that is acceptable to both states. War would proceed, until by Bayesian updating both states arrive at expected outcomes that overlap and allow a settlement to occur. If all prizes are divisible, N is a series of points connected by lines as each prize is divided among the players. However, here I use a rounded leaf to generalize a multitude of prizes, as in Figure 2. If one or more prizes are indivisible, N can be quite irregular. For example, consider three contested prizes: agricultural land, a protective bay important to naval defense, and a mountainous area expected to contain useful minerals. State A has a manufacturing economy, so Player A puts more value on the mining land than the agricultural land. State B is still a primarily agricultural economy, so Player B places relatively more value on the agricultural land. The bay is protective to either state only if held exclusively, and so is effectively indivisible. However, A has the larger navy and values it relatively more. If each state were to divide these three prizes into utilities summing to one, it would look as in Table I. Table I: Example of Mutual Gains: Differing Prize Values
Caption: Each player’s value of the three prizes is shown as a portion of total utility. From this information, algebra will determine the possible settlements. If A has everything, the utilities are UA=1 and UB=0; the reverse if B gets all three prizes. The utility for the remaining six combinations of each player having two of the three prizes is readily found. Connecting these dots describes the possible settlements if each item can be divided with linear utility. This is shown in Figure 3. The upper left rhombus in light gray shoes settlements in which B has the protective bay. The lower right rhombus is the settlements when player A has the protective bay. There is some overlap, however, all of the war-ending settlements leave A with the bay. Brams and Taylor (1999) describe the ``adjusted winner'' method of fair division, which amounts to allotting each item to the person bayed on its relative value (A's value divided by B's value) until the desired settlement is reached. The method relies on giving each player her relatively most desire item to maximize mutual gain, which is why the ratio of value is important. The indivisible item means we must consider the utilities of the two cases in which each player retains control of that item. If A has the bay, he next prefers the mining land. He needs (0.15/0.50) = 0.3 of the mining land to get to a final utility of 0.55. That would leave (0.35/0.50) x 0.25 = 0.175 utility for B from the mining land and all of the agricultural land for a final value of 0.675. In fact B would settle for only (0.5/0.25) of the mining land. So it is acceptable for B to get between 20 and 70 percent of the mining land. If B has the bay, A requires all the mining land plus (0.5/0.10) = 0.5 of the agricultural land to get 0.55. That leaves 0.25 + (0.5/0.10) x 0.25 = 0.5 for B, which is not enough to avoid war. So all acceptable possible settlements leave A with the bay. Using the 2D Diagram, it is easy to identify the bargaining range in a game with one indivisible good and mutual optimism. In a constant-sum game, no settlement would end the war. Figure 3: The 2D Diagram for the Mutual Gains Example 
Caption: The Possible Set when A has the bay is dark gray; the Possible Set when B has the bay is light gray; the Acceptable Set is lined; the expected outcomes of war are dotted lines. In conflicts, many outcomes are not possible. For instance, the outcome of all players getting their maximum utility is unlikely; there would be no conflict in the first place. However, the 2D Diagram formulation of one axis for each player allows the full range of outcomes to be considered. The plausibility of each outcome can then be decided individually rather than assumed away. Further, the 1D Diagram is a subset of the 2D Diagram - a diagonal line running from all prizes for one player, and none for the other, to the opposite extreme. So the 2D Diagram takes nothing away from previous work, but rather adds to it. Researchers who find the constant-sum assumption reasonable can use either model equally well. Constrained Settlements War may result from reducing the set of possible settlements, and this includes most rationalist explanations. Indivisible prizes are one example (Fearon 1995 mentions these briefly). For instance, suppose that there are three prizes of equal value to both opponents (perhaps they are defensive locations that are not valuable if shared), but that they have a value of zero if divided. This is shown in Figure 4 with the gray regions reduced to ten points. The spacing is such that no point is Pareto superior to the war expectations. Figure 4: 2D Diagram with Indivisible Prizes 
Caption: The Possible Set is the square points; the Acceptable Set is lined; the expected outcomes of war are dotted lines. Fearon (1995) was more concerned with commitment problems than issue indivisibility. Powell (2006) argues that they are essentially the same thing. The result is similar. A commitment problem limits the possible settlements. If the problem is sufficiently severe, then no settlement is possible. The exact change to the set depends on the particular commitment problem. One type of commitment problem is time inconsistency, caused by time lags between parts of a settlement. Suppose states A and B argue over an indivisible bit of strategically important land. B proposes that A keep the land, but pay B for not attacking. However, once B is paid, they may still attack, resulting in the same expected payoff of war for A minus the payment. A prefers just to go to war right away. In the worst cases, no settlements are possible at all. Restrictions to settlements may also occur when there is a threat to the current executive. In particular, if the executive anticipates that negotiating a settlement will result in his removal (perhaps there is an ongoing civil war or institutional change), and he has some chance of retaining office if he wins the war, then war is almost certainly a better option. This situation effectively lowers the utility of all settlements for the threatened executive's state, and may eliminate them entirely if utility is based solely on reelection. Similarly, if the crisis is due to one state demanding a regime change, as liberal IR theory asserts, then settlements may also become unpalatable for the opposing state because settlements may include institutional changes that eliminate the opposing executive. Fearon (1995) suggests that side payments should resolve indivisibility issues, and these can be spread over time to settle commitment issues. Suppose both players are civil war opponents. They argue over who will be Prime Minister. Player A values the PM role at 0.3, with 3 Billion US Dollars worth 0.7. Players B values these at 0.2 and 0.8 respectively. Assuming equal power, if B takes the PM role and 1 billion USD, each player gets a utility of 0.4667. If they trade, each player gets utility 0.5333. The latter is a better compromise taking advantage of mutual gains. Combinations of indivisible prizes and monetary payments can be illustrated with the 2D Diagram. In fact, monetary payments can be shown as additional prizes in the 2D Diagram, or even as multiple prizes if payments are made on a time schedule to ensure that peace agreements are sustained. The argument for side payments is that there is always a divisible prize, namely money, which can be considered. Restrictions on possible settlements occur for two reasons. If both players value prizes equally, then the set is the diagonal line. But if mutual gains exist, the possible set is expanded. Nonlinearities in utility, indivisibility being the most extreme form, restrict the possible set. Commitment problems are essentially nonlinearities in the division of prizes. When one player believes the other player will not hold to an agreement, he assigns little utility to such a settlement. Because money is always an option, there should always be at least one divisible good, and so both indivisibility and commitment problems have met with some skepticism as singular explanations for war. The 2D Diagram, by avoiding the constant-sum assumption, illustrates these theories and also the criticisms against them. Figure 5: 2D Diagram with Inflated Expectations 
Caption: The Possible Set is gray; the Acceptable Set is lined; the expected outcomes of war are dotted lines. Acceptability A successful negotiated settlement must be better than (or equal to) the expected outcome of war for each player including expected costs of war. Increased expectations reduce the size of the acceptable set. Most explanations of war fall into this category. Fearon (1995) lists private information as one cause of war. This scenario, shown in Figure 5, results when each side has information about its own capabilities in war, but not about the other side. The lined area, W, has shrunk on both axes, leaving no intersection. Both sides over-estimate their likelihood of victory in war and so have high expectations. The situation occurs naturally due to the development of military technology in secret. Opponents are unaware of each other's advancements, and if both sides have not been in any war for some time, they have not exhibited their new technologies. Smith and Stam (2003) explain private information differently. Players may know their opponent's capabilities, but believe that different factors will be paramount in a war. They use the example of the Seven Weeks War between Austria and Prussia (among others). The Prussians believed the development of the needle gun was paramount. The Austrians believed unit cohesion was key, and that needle gun usage would undermine unit cohesion. It seems the Prussians were more correct, for they won. Essentially, both opponents knew the technology of the other, but they had different beliefs about how the technology would affect war. Both of these views constitute different prior beliefs about the outcome of war. As Smith and Stam explain, different prior beliefs converge as war proceeds by the process of Bayesian updating. If this is the only reason for war, Bayesian updating should make negotiated settlement possible as the war progresses. Their model can be used to predict the duration of war, but only if private information is the sole reason for war. Before war begins, both sides must base negotiations on only their prior beliefs. If there is a net first strike advantage from surprise, the attacker has a greater expectation of war if he attacks first. Each opponent must then consider his own expectation if he is the first attacker (one of them must be). This might be sufficient to create optimistic estimates. Each attacker must be considered separately, resulting in two Pareto superior regions, the first Pareto superior if A attacks first and the second Pareto superior if B attacks first. A negotiated settlement must be at the intersection of all three areas. If a settlement is Pareto superior to only one first attacker, the other will strike first. This is shown in Figure 6. The horizontally lined area is Pareto superior to the outcome of a first strike from A. A settlement in this region only prevents a first strike from A, but B will then strike first to gain the advantage. The vertically lined area is Pareto superior to the outcome of a first strike from B, inciting A to strike first. To avoid a first strike by either country, a settlement must be in the checkered region, which is not possible in this example. Figure 6: 2D Diagram with First Strike Advantage 
Caption: The Possible Set is gray; the Acceptable Set if A strikes first is horizontally lined; the Acceptable Set if B strikes first is vertically lined the expected outcomes of war are dotted lines. Selectorate theory (Bueno de Mesquita et al 2003), rather than using linear utility functions for war, bases utility on electoral competition given the outcome of war. That model showed how democracies and autocracies may differ in approaching war. The size of the winning coalition, W, results in increased resources spent on a war (with a higher W being more democratic). In Figure 7, I have illustrated the possible beliefs about war based on W for the two players as a diamond of dashed lines. When both states are democracies (the vertically lined area) the expectations for war are at their lowest and a settlement is most likely. When both states are autocracies (the horizontally lined area) the expectations of war are at their highest level. I exaggerate this to illustrate the effects of the selectorate model, showing how regime type may eliminate all possible acceptable settlements. In actuality, the selectorate model would always have a Pareto superior settlement for any combination of W; war only occurs because N is a single point, an expected utility of a negotiated settlement. If the expected settlement is far from the expected outcome of war, then war occurs to correct the status quo imbalance. Because, the winning coalition size changes the expected results of a war, it is a restriction on the acceptable settlements. Figure 7: 2D Diagram with Selectorate Theory 
Caption: The Possible Set is gray; the Acceptable Set for two high W states (democracies) is horizontally lined; the Acceptable Set for two low W states (autocracies) is vertically lined the expected outcomes of war are dotted lines. The 2D Diagram can also show relative gains, which realist IR theory often suggests. I have previously assumed that each player has a fixed belief about the utility of war and will accept any settlement that is superior to this. But if players are concerned with relative gains, these straight lines become diagonal lines, moving toward (1,1) on the plot. For example, if each player wanted at least 75% of his opponent's utility, the borders for acceptable settlements would be (0,0) to (1,0.75) and (0,0) to (0.75,1). Combinations of relative and absolute gains are also possible. This model can readily illustrate how full or partially relative gains affect acceptable settlements. This is pictured in Figure 8. A demands to get at least a 48-52 utility split. B demands 44-56. The 2D Diagram can also show more complex borders for the set of acceptable settlements, if desired. Figure 8: 2D Diagram with Relative Gains 
Caption: The Possible Set is gray; the Acceptable Set is lined; the expected outcomes of war are dotted lines. If either opponent has a reduction in his expected costs of war, it moves his expected value of war up. This alone is insufficient to cause war (with zero cost, a settlement at the expected outcome of war is still possible), but it might exacerbate other situations. Costs may change for a variety of reasons. Technological innovation may reduce the costs. This has undoubtedly happened, allowing states to project power around the entire world rather than just within a regional area. Audience costs (Fearon 1994) represent social pressure on leaders to avoid war. Over the course of a war, audience costs may change, affecting negotiation. If one opponent increases his military forces in the war, it both increases his cost and increases his chance of winning. Unless the net benefit has diminishing returns, all states would always contribute all forces to war. Even if the size of military force increases infinitely, the probability of success goes to a maximum of one. This produces natural diminishing returns asymptotically. Last, some situations may provide utility only from war, regardless of the outcome. In such cases, any outcome from war is preferred to the same outcome from negotiation, despite the costs of war. These situations are not explained by typical rational explanations, but they can be easily modeled. For example, a military emperor (such as Alexander the Great or Attila the Hun) may seek war as a way to prove himself or build his own ego. He would rather make gains by war than take equivalent concessions. Similarly, if a leader is risk-seeking, he overvalues the uncertainty inherent in war. It may also be that the constituents of the leader demand war, possibly as a retaliation for some grievance. This could be due to cultural difference suggested by conflicts between civilizations (Huntington 1992) or by some constructivist theories. Most rationalist explanations for war have focused on the acceptable set. Mutual optimism, relative gains, polity type, cost changes, and risk-seeking behavior may all reduce this set. However, these explanations have been empirically difficult. Mutual optimism should result in short wars and negotiated concession changes after a defeat. Yet Reiter (2007) shows this is often not the case. Other explanations, like selectorate theory, always have a negotiated settlement and must assume there is no dynamic negotiation process to have war result. If players can negotiate freely as part of the game, these explanations fail. In the next subsection, I consider the last category of problems under the 2D Diagram, the negotiation process. Negotiation Failure Just as war is a process, not a single roll of the dice, negotiation is also a process. Any process may fail, and negotiations can be subject to many human failings. In the 2D Diagram it is easy to see that negotiation may fail, particularly if the region of acceptable possible settlements is already small due to some of the above issues. Players must identify the set of acceptable and possible settlements. Then they must try to understand their opponent's utility for different prizes. Even then, players may try to achieve their best result in this region through a back-and-forth negotiation process. Jumping right to a settlement might give away too much too soon, especially if an opponent's values are not well understood. Many books have been written on negotiation. In their classic book Getting to Yes (1981), Fischer and Ury explain principled negotiation techniques as a way to reach a mutually agreeable solution. These techniques include careful understanding of both your own and your opponent's BATNA (best alternative to negotiated agreement), i.e. war. In essence, this means carefully understanding the utilities of the alternatives as well as the value of the prizes for all parties. They note that hard negotiation practices, such as threatening to walk out after a take-it-or-leave-it offer, frequently result in inferior outcomes for both parties. Hard negotiation techniques may work if the offer is acceptable to the opponent, but it may also force the crisis into war if it is unacceptable, even though acceptable compromises exist. Figure 9: 2D Diagram with Negotiation Sequences 
Caption: The Possible Set is gray; the Acceptable Set is lined; the expected outcomes of war are dotted lines; settlement offers from A and B are a sequence of connected points. Negotiation can easily be visualized as a series of offers in the set of possible settlements, as seen in Figure 9. A state making an offer proposes a settlement. A hard bargaining player will make her next offer close to the previous one; a soft bargaining player makes offers relatively further apart. If the opponent's values for each prize are unknown, then part of the negotiation process is to learn those values. Until these values are established, it is possible to make an offer to an opponent that is mistakenly less than the previous offer. Ideally, offers should approach the border of the possible set in order to take advantage of mutual gains. However, there is a trade-off when choosing a strategy. A hard strategy may signal strength to an opponent, but it may also delay reaching an agreeable settlement, which raises the costs of an ongoing war. Further it makes it more likely that the final agreement is the opponent’s suggestion, which is always worse for the hard bargainer. A soft strategy will reach an agreement more quickly, but it may give up more than was necessary. Figure 10: Conditional Costs of War, Unimodal Duration Distribution 
Caption: Prior belief about the duration of war is the Gray curve; cost is linear in time; the expected cost conditional on having reached some point in time is the dashed line; the difference is the expected additional cost at any point in time. The conditional expectations of costs as war continues can be easily examined with simulations. Generally, as war approaches its finish, the expected costs asymptotes to the actual cost paid, as in Figure 10 with time on the x-axis. As sunk costs increase, the expected costs of further war then decrease. This has the effect of raising the expected utility of war relative to the possible settlement set. If the expected duration is a constant, the additional expected cost is also constant, but the paid cost is increasing. The two sets only move closer together if the distribution is strongly bimodal, as in Figure 11, and then only for a brief time. Figure 11: Conditional Costs of War, Bimodal Duration Distribution 
Caption: Prior belief about the duration of war is the Gray curve; cost is linear in time; the expected cost conditional on having reached some point in time is the dashed line; the difference is the expected additional cost at any point in time. As war continues, sunk costs decrease the set of possible acceptable settlements as shown in Figure 12. As players spend money on the war, they lose utility. The gray area moves toward (0,0). But as the war gets closer to its predicted end, the expected costs, conditional on the war having gone on for a time, decrease. But the total expected cost increases, because early ends have been eliminate as possibilities. So the lined area moves toward (0,0) as well, but at a slower rate. The intersection, the set of war-ending settlements, decreases in size with time. In this example, A's offers are close to a horizontal line, meaning that successive offers only cover sunk costs to B resulting in the same utility to B. In contrast, B's offers are nearly on the same place of the gray area each iteration, however this results in lost utility for both players between offers. Both players would benefit from a softer negotiating strategy to get a settlement more quickly. Negotiation failure may cause a war to go on longer than it should and result in a cost to both players. Figure 12: 2D Diagram with Negotiation and Sunk Costs 
Captions: Possible Set in gray decreases for both players; Acceptable Set in lined area grows larger; Bargaining Range gets smaller with time. Negotiation problems may arise due to cultural differences. Negotiators are human after all, and they can get offended or just have bad days. More experienced negotiators should be better at reaching a solution, yet this explanation is not terribly informative. A more interesting question is whether regime type systematically results in different negotiation strategies. Bailer (2012) argues that democracies are more likely to use soft negotiating strategies overall, but they are more likely to use hard strategies when under pressure from domestic interest groups. Little works has been done in this area. The classic bargaining models (Nash 1950; Rubinstein 1982) both predict immediate solutions to negotiation. These models are then unhelpful in considering how long negotiations will last, or whether hard or soft bargaining is preferable. Indeed, there must be some form of uncertainty, or any negotiation model will resolve instantly, as subsequent models have shown. Fey and Ramsay (2009) have used mechanism design to identify possible solutions to the negotiation problem, regardless of the specific game structure used. So far, they have identified that there should be a negotiated solution, but they also predict immediate resolution if any. Yet lengthy negotiations are the norm for international bargaining. In the end, negotiation problems are little explored, and researchers do not yet have a good understanding of how bargaining proceeds, how long it lasts, and why it is doe not resolve immediately. Conclusion I have presented a new method for visualizing international crises by using a multi-dimensional graph with one axis for the utility of each player. I defined a set of feasible divisions of war prizes, the possible set, allowing mutual gains through different values of prizes. I defined a set of settlements that are preferable to war for all players, the acceptable set, by considering each state's own expected value for war. The intersection of the two sets is the bargaining range, the possible acceptable negotiated settlements. I showed that the tool I propose here overcomes some limitations of the 1D Diagram method. In particular, the constant-sum assumption artificially limits settlements by ignoring mutual gains. The 2D Diagram method more explicitly depicts the possible and acceptable sets, allowing both to be considered, while the 1D Diagram ignores constraints to the possible set. The 2D Diagram better demonstrates complex utility functions, multiple players, the full bargaining range, and negotiation sequences. I categorized rational theories into three explanations of war. I explained restrictions to the possible and acceptable sets, but focused particularly on negotiation failures. These failure can lead to conflict even when there is a robust bargaining range. I analyzed extended wars by showing how the bargaining range shrinks as war plays out. I found that this intersection set only grows if the expected duration was strongly bimodal, such as when there were large informational asymmetries, and then only briefly. More typically, negotiation becomes more difficult as the war continues due to sunk costs. However, little work has been done that predicts the process of negotiation. Most models find immediate solutions to any bargaining problem. Such models may estimate final outcomes, but are uninformative about the negotiation process, duration, or bargaining strategies. More models of negotiation are needed to examine exactly how bargaining failures can occur. One negation failure that may lead to war is simply that players are unable to clearly identify the bargaining range and make an offer within the range to end the war. As I have shown, sunk costs can make the bargaining range a moving target. This is exacerbated if players are hard bargainers, unwilling to change too much between offers. In such cases, the first offer must be close to the bargaining range for subsequent offers to be acceptable. Another possibility is that one player makes an offer in the bargaining range, but the other player rejects it if she thinks she can get a better offer later on and is willing to bear the further costs of war. A last idea is that players are driven by ego or hubris to avoid appearing weak by giving in to any offer. In light of the selectorate model, a leader who might be punished by selectors for appearing weak in war would not yield. Further work is also needed to examine the role of regime type on negotiation, with both formal models and empirical evidence. Do democracies use different bargaining strategies, or come to different agreements, than autocracies? Mechanism design may also provide a useful way to examine negotiation formally, without being limited to assumptions about the negotiation procedure. But it must offer insight into the negotiation process, and not leave it all in a black box. Educational tools, such as these diagrams of rationalist theories of war, are important for discussing research as well as teaching. They add insight by questioning implied assumptions and allowing researchers to view problems from new angles. A comparison of the 1D Diagram and 2D Diagram directly leads to discussions about the plausibility of mutual gains and the negotiation process. I hope that the 2D Diagram can be used to generate even more ideas about the underlying reasons for conflict. A large body of research was generated after Fearon’s (1995) paper, but further research has led to troubled findings. Commitment problems and indivisibility should be solved by side payments. Mutual optimism should be fleeting once a war begins. In reality, as Reiter (2007) shows, players may act in ways inconsistent with rationalism altogether. Perhaps this more general visualization technique will help generate ideas that resolve these difficulties. References Bailer, Stefanie (2012) Strategy in the climate change negotiations: do democracies negotiate differently? Climate Policy, 12(5): 534-551. Brams, Steven J, and Alan D Taylor (1999) The Win-Win solution: guaranteeing fair shares to Everybody. New York: WW Norton and Company. Bueno de Mesquita, Bruce, Alastair Smith, Randolph M Siverson and James D Morrow (2003) The Logic of Political Survival. Cambridge, MA: MIT Press. Fearon, James D (1994) Domestic Political Audiences and the Escalation of International Disputes. The American Political Science Review 88(3): 577-592. Fearon, James D (1995) Rationalist Explanations for War. International Organization 49(3): 379-414. Fey, Mark, and Kristopher W Ramsay (2009) Mechanism design goes to war: peaceful outcomes with interdependent and correlated types. Review of Economic Design 13(3): 233-250. Fischer, Roger, and William Ury (1981) Getting to Yes: Negotiating Agreement Without Giving in. New York: Penguin Group, USA. Huntington, Samuel P (1993) The clash of civilizations? Foreign Affairs 72(3): 22-49. Nash, John (1950). The Bargaining Problem. Econometrica 18(2): 155–162.. JSTOR 1907266. Powell, Robert (2006) War as a Commitment Problem. International Organization 60(1): 169-203. Rubinstein, Ariel (1982) Perfect equilibrium in a bargaining model. Econometrica: Journal of the Econometric Society 50(1): 97-109. Reiter, Dan (2009) How wars end. Princeton, NJ: Princeton University Press. Smith, Alastair, and Allan C Stam (2004) Bargaining and the Nature of War. Journal of Conflict Resolution 48(6): 783-813. 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