**5.3 Case 3 (***n* → ∞**)**

10 Will-be-set-by-IN-TECH

negotiation process to emerge and the social welfare are dependent on the conflict costs when

In this scenario, we have *dn* <sup>=</sup> *<sup>d</sup>*<sup>2</sup> <sup>=</sup> <sup>−</sup>(<sup>1</sup> <sup>+</sup> *<sup>δ</sup>*)*<sup>H</sup>* <sup>+</sup> *<sup>δ</sup>*2*W*. A conflict in period *<sup>t</sup>* involves costs *<sup>H</sup>*, but the emergence of a negotiation process opens the possibility of ending up with an outcome which is even worse. This outcome occurs if the parties fail to reach an agreement. Then, the payoff to each country is *d*<sup>2</sup> < *d*1. Comparing it with the former case, we should expect that now each country claims for more than one half of the territory. Therefore, the probability of

*<sup>W</sup>* <sup>=</sup> (*d*<sup>1</sup> <sup>−</sup> *<sup>d</sup>*<sup>2</sup> <sup>−</sup> <sup>1</sup>) *<sup>d</sup>*<sup>1</sup> <sup>+</sup> <sup>1</sup>

If we substitute *<sup>d</sup>*<sup>1</sup> <sup>=</sup> <sup>−</sup>*<sup>H</sup>* <sup>+</sup> *<sup>δ</sup><sup>W</sup>* and *<sup>d</sup>*<sup>2</sup> <sup>=</sup> <sup>−</sup>(<sup>1</sup> <sup>+</sup> *<sup>δ</sup>*)*<sup>H</sup>* <sup>+</sup> *<sup>δ</sup>*2*<sup>W</sup>* and *<sup>δ</sup>* <sup>=</sup> 0.5 into Eq. (15) above,

*<sup>W</sup>* <sup>=</sup> (0.25*<sup>W</sup>* <sup>+</sup> 0.5*H*) (−*<sup>H</sup>* <sup>+</sup> 0.5*W*) <sup>+</sup> <sup>1</sup>

There are two values of *W* that satisfy Eq. (16), but only one of them satisfies our assumption

The stationary value for the social welfare decreases in response to an increase in the conflict costs. The rationale behind this effect is as follows: If *H* increases, the disagreement utility becomes lower. This makes a negotiation process be less attractive for the countries, as long as the "bad outcome" of conflict today is not so bad compared to the disagreement outcome (two periods of conflict). As a consequence, the countries are willing to make more aggressive statements. This behavior decreases the probability of negotiations to emerge. Therefore, the

1 + 3*H* − 0.5*W*

<sup>1</sup> <sup>2</sup> , 1 

conflict instead of initiating a negotiation process is positive. This result is in accordance with

1 − *d*<sup>2</sup> − *d*<sup>1</sup>

2

<sup>5</sup>*<sup>H</sup>* <sup>+</sup> <sup>8</sup>*H*<sup>2</sup> <sup>−</sup> <sup>3</sup>

2 <sup>1</sup> <sup>+</sup> 2.5*<sup>H</sup>* <sup>−</sup> 0.75*<sup>W</sup>* (16)

> 4 1 2 − 1

<sup>1</sup> <sup>+</sup> 2.5*<sup>H</sup>* <sup>−</sup> 0.75*<sup>W</sup>* . (18)

<sup>8</sup> the probability that countries continue with the

provided that *H* > 0, *W* > 0 and

<sup>2</sup> implies *p*<sup>∗</sup> = 2*x*<sup>∗</sup> − 1 > 0. An important

. (15)

. (17)

<sup>2</sup> . For this purpose, we

From Eq. (14) we write the following expression for the social welfare:

*W* = 1. 428 6*H* − 0.571 43

Next we check that the equilibrium statements are higher than <sup>1</sup>

substitute the value for the social welfare in Eq. (16) into Eq. (9) to obtain:

*<sup>x</sup>*<sup>∗</sup> <sup>=</sup> <sup>1</sup> 2

<sup>8</sup> . The fact that *<sup>x</sup>*<sup>∗</sup> <sup>&</sup>gt; <sup>1</sup>

 − 3 4 , 1 8 .

8 .

<sup>&</sup>lt; 0 for all *<sup>H</sup>* <sup>≥</sup> <sup>1</sup>

*n* ≥ 2.

we obtain:

Notice<sup>12</sup> that *dW*

**5.2 Case 2 (***n* = 2**)**

conflict must be positive.

that *θ* + 2*H* > 2*δW*. This value is:

*dH n*=2

social welfare falls to a lower level.

<sup>1</sup> <sup>+</sup> <sup>2</sup>*<sup>H</sup>* <sup>&</sup>gt; *<sup>W</sup>* for all *<sup>H</sup>* <sup>≥</sup> <sup>1</sup>

From Eq. (18) it is straightforward that *x*<sup>∗</sup> ∈

conclusion is that if *<sup>n</sup>* <sup>=</sup> 2, and *<sup>H</sup>* <sup>≥</sup> <sup>1</sup>

<sup>12</sup> A solution for *<sup>W</sup>* exists whenever *<sup>H</sup>* <sup>∈</sup>/

In this third case, we have that lim*n*→<sup>∞</sup> <sup>−</sup> <sup>1</sup>−*δ<sup>n</sup>* 1−*δ <sup>H</sup>* <sup>+</sup> *<sup>δ</sup>nW* <sup>=</sup> <sup>−</sup> <sup>1</sup> <sup>1</sup>−*<sup>δ</sup> <sup>H</sup>*. We are dealing with the somehow extreme situation in which breaking down negotiations involves that conflict will last forever. We substitute *dn* <sup>=</sup> <sup>−</sup> <sup>1</sup> <sup>1</sup>−*<sup>δ</sup> <sup>H</sup>* into Eq. (14) to obtain:

$$W = \frac{0.25W^2 - H^2 + \frac{1}{2}}{1 + 3H - 0.5W}. \tag{20}$$

Solving Eq. (20) for *W* yields:

$$W = 2H - \frac{2}{3} \left[ 2 \left( \frac{3}{2} H + 3H^2 - \frac{1}{8} \right)^{\frac{1}{2}} - 1 \right]. \tag{21}$$

It is easy to see that *dW dH <sup>n</sup>*→<sup>∞</sup> <sup>&</sup>lt; 0 for all *<sup>H</sup>* <sup>≥</sup> <sup>1</sup> 12 <sup>√</sup><sup>15</sup> <sup>−</sup> <sup>1</sup> <sup>4</sup> . Moreover, for all *<sup>H</sup>* <sup>≥</sup> <sup>1</sup> <sup>8</sup> it holds that *dW dH <sup>n</sup>*→<sup>∞</sup> <sup>&</sup>lt; *dW dH n*=2 < 0. This suggests that the magnitude of the social welfare losses provoked by an increase in the conflict costs becomes higher as *n* grows large. Compared to the two former cases, notice that now the disagreement point does not depend on *W*. Therefore, the effect of an increase in the conflict costs can be easily computed as a variation of magnitude <sup>−</sup> <sup>1</sup> <sup>1</sup>−*<sup>δ</sup>* in the disagreement utility.

The equilibrium statements are now:

$$\mathbf{x}^\* = \frac{1}{2} \frac{1 + 4H}{1 + 3H - 0.5W}. \tag{22}$$

We have that *x*<sup>∗</sup> ∈ <sup>1</sup> <sup>2</sup> , 1 whenever *H* > −0.5*W*. By substituting *W* (Eq. (21)) in the latter inequality we find that it holds for values of *H* such that 6*H* > 2 <sup>3</sup> <sup>2</sup> *<sup>H</sup>* <sup>+</sup> <sup>3</sup>*H*<sup>2</sup> <sup>−</sup> <sup>1</sup> 8 1 2 − 1. This

an indeterminate number of periods (ranging from 1 to infinite). On the other hand, if the public statements are far enough from each other, then no negotiation process emerges and the conflict continues for at least one more period. As a difference with previous work, conflict is not considered as a bargaining instrument. Instead, conflict is the prevailing state, which can only be changed if the parties make diplomatic efforts (concessions, in the form of reasonable

Social Welfare and the Emergence of Negotiations 215

In this setting we conclude that, in general, the probability of conflict in period *t* is a non monotonic function of the conflict costs. For low costs it is decreasing, and it increases when costs are above a certain value. At this particular value, the probability of conflict reaches a minimum. A similar pattern is obtained with respect to the pre-bargaining equilibrium claims. The effect of conflict costs on the social welfare is more clear: it decreases monotonically as

These results suggest that the goals of maximizing the social welfare and minimizing the probability of conflict are not equivalent, since both are achieved at different levels of conflict costs. While the welfare is maximum when conflict is costless, the likelihood of a negotiation process to emerge achieves a maximum for positive conflict costs. Moreover, if such costs are very high, the probability of conflict achieves a constant value. This explains why in some instances a negotiation process does not emerge, even if it would be in the interest of the conflicting parties to initiate it. Although countries must bear certain costs every period the conflict lasts, they face even higher costs if negotiations emerge but no agreement is reached. The countries trade-off the conflict costs (today) against the expected outcome of negotiations, and therefore they make public statements that may involve the persistence of the conflict. The utility reached at the disagreement point when negotiations break down is a crucial element to determine the probability of such negotiations to emerge. In our model, the threat point is parameterized by "n", the number of periods the conflict will last if no agreement is reached. The analysis made in Section 5 reveals that the social welfare depends negatively on "n" for any given conflict cost. It is then desirable that the threat point of a negotiation process is not too harmful for the parties in conflict. This implies that an important issue in the design of the rules of a negotiation process should be to prevent and limit the consequences of breaking negotiations. Specifically, before the bargaining over territory takes place, it would be welfare enhancing that the parties could commit to restart negotiations at an early date in case no agreement were reached. In general, any measure conducive to mitigate the negative outcome represented by the threat point would improve the likelihood of achieving a pacific

If the countries could choose the intensity of the dispute, our model suggests that a moderate investment in resources devoted to the conflict proves optimal for a negotiation process to emerge. This conclusion is compatible with the approach that envisions conflict as part of a bargaining strategy, as in Sanchez-Pagés (2009). According to this view, the nations engaged in territorial disputes use limited confrontation as a way to convey information about their relative strength. The information revealed facilitates the emergence of a peaceful outcome. In our model, though, the line of explanation is a bit different. When conflict costs are positive, but not too high, the parties might find it beneficial to negotiate as long as the threat point in case of disagreement does not entail huge utility losses. However, if conflict costs become higher (for instance, if the countries engage in a nuclear arms race), then the threat point involves a final confrontation aimed to the absolute destruction of the enemy. We have shown that the territorial claims in this case tend to be more aggressive, thus lowering the probability

claims) to resolve the dispute through a pacific settlement.

the conflict costs increase.

settlement through negotiation.

of reaching a negotiated outcome.

inequality is satisfied by all *<sup>H</sup>* <sup>≥</sup> <sup>1</sup> 12 <sup>√</sup><sup>15</sup> <sup>−</sup> <sup>1</sup> <sup>4</sup> = 0.07 274 9. Hence, for conflict costs above this value, the probability of conflict is positive, i.e., *p*∗ > 0. Now we compute the equilibrium value for *x*∗ by substituting *W* in Eq. (21) into Eq. (22) to obtain:

$$\mathbf{x}^\* = \frac{1}{2} \frac{1 + 4H}{1 + 2H + \frac{1}{3} \left(2\left(\frac{3}{2}H + 3H^2 - \frac{1}{8}\right)^{\frac{1}{2}} - 1\right)}.\tag{23}$$

The equilibrium statement *x*∗ is above 0.75 for low values of *H*, is decreasing in *H* across the range (0.07 274 9, 0.540 57), and reaches a minimum value of 0.612 57 at *H* = 0.540 57. For *H* > 0.540 57, *x*∗ is strictly increasing and converges asymptotically to 0.633 97. The probability of conflict now varies between 0.587 44 (when *H* = 0.07275 and *x*∗ = 0.793 72) and 0.225 14 (when *H* = 0.540 57 and *x*∗ = 0.612 57). For *H* large enough, the probability of conflict is constant and equal to 0.267 94. The expected duration of the conflict ranges from 1. 423 9 periods (for *H* = 0.07275) to 0.290 56 periods (for *H* = 0.540 57). The expected conflict cost is then 0.103 59 if *H* = 0.07275, and 0.157 07 if *H* = 0.540 57.

In general, when analyzing the effects of variations in *H* on the probability of conflict, we are able to conclude that, in cases where *n* > 1: (i) *p*∗ achieves a maximum when the conflict costs are minimum13; (ii) *p*<sup>∗</sup> achieves a minimum for a certain value of *H*; (iii) *p*<sup>∗</sup> converges to a constant value as *H* → ∞. The pattern of variation of *p*<sup>∗</sup> is driven by the way how the equilibrium statement *x*∗ changes in response to *H*.

Therefore, tough public pronouncements are expected to be made when the conflict costs are minimum. From that point on, the claims moderate as *H* grows large. There is a certain value of *H* that yields the lowest equilibrium claim (i.e., the highest probability that negotiations emerge). When *H* is large enough, the claims tend to a constant value.

As the above analysis reveals, favoring the own position with aggressive claims can either be the product of extremely low or extremely high conflict costs. In the first case, the countries find it profitable to take the risk of aborting a negotiation process, as long as the benefits of a higher bargaining power in future negotiations outweigh the costs of facing conflict today. For the second case, it must be considered that failure to reach an agreement in negotiations involves a certain persistence of the conflict (during *n* ≥ 1 periods) and then high values of *H* lower considerably the disagreement utilities. Clearly, the worse outcome is represented by the disagreement point, the less attractive is a negotiation process. Then, strong public pronouncements that lower the probability of negotiations should not come as a surprise.

#### **6. Conclusions**

This chapter studies the relationship between social welfare, the cost of conflict and the emergence of a negotiation process. For this purpose, we develop a model in which two countries are bargaining over a fixed size territory. While no agreement is reached, the parties keep engaged in a conflict that entails costs in each period. The countries have the ability to make public statements representing claims over the territory. If the claims are not "too extreme", a negotiation process emerges that yields as an outcome the generalized Nash bargaining solution, where the bargaining power of each country is conditioned upon the previous claims. The disagreement point of this negotiation process is to face conflict during

<sup>13</sup> These results are in line with that of Wittman (1979), who finds that a reduction of hostilities may reduce the probability of a settlement taking place and thus prolong the war

12 Will-be-set-by-IN-TECH

value, the probability of conflict is positive, i.e., *p*∗ > 0. Now we compute the equilibrium

The equilibrium statement *x*∗ is above 0.75 for low values of *H*, is decreasing in *H* across the range (0.07 274 9, 0.540 57), and reaches a minimum value of 0.612 57 at *H* = 0.540 57. For *H* > 0.540 57, *x*∗ is strictly increasing and converges asymptotically to 0.633 97. The probability of conflict now varies between 0.587 44 (when *H* = 0.07275 and *x*∗ = 0.793 72) and 0.225 14 (when *H* = 0.540 57 and *x*∗ = 0.612 57). For *H* large enough, the probability of conflict is constant and equal to 0.267 94. The expected duration of the conflict ranges from 1. 423 9 periods (for *H* = 0.07275) to 0.290 56 periods (for *H* = 0.540 57). The expected conflict

In general, when analyzing the effects of variations in *H* on the probability of conflict, we are able to conclude that, in cases where *n* > 1: (i) *p*∗ achieves a maximum when the conflict costs are minimum13; (ii) *p*<sup>∗</sup> achieves a minimum for a certain value of *H*; (iii) *p*<sup>∗</sup> converges to a constant value as *H* → ∞. The pattern of variation of *p*<sup>∗</sup> is driven by the way how the

Therefore, tough public pronouncements are expected to be made when the conflict costs are minimum. From that point on, the claims moderate as *H* grows large. There is a certain value of *H* that yields the lowest equilibrium claim (i.e., the highest probability that negotiations

As the above analysis reveals, favoring the own position with aggressive claims can either be the product of extremely low or extremely high conflict costs. In the first case, the countries find it profitable to take the risk of aborting a negotiation process, as long as the benefits of a higher bargaining power in future negotiations outweigh the costs of facing conflict today. For the second case, it must be considered that failure to reach an agreement in negotiations involves a certain persistence of the conflict (during *n* ≥ 1 periods) and then high values of *H* lower considerably the disagreement utilities. Clearly, the worse outcome is represented by the disagreement point, the less attractive is a negotiation process. Then, strong public pronouncements that lower the probability of negotiations should not come as a surprise.

This chapter studies the relationship between social welfare, the cost of conflict and the emergence of a negotiation process. For this purpose, we develop a model in which two countries are bargaining over a fixed size territory. While no agreement is reached, the parties keep engaged in a conflict that entails costs in each period. The countries have the ability to make public statements representing claims over the territory. If the claims are not "too extreme", a negotiation process emerges that yields as an outcome the generalized Nash bargaining solution, where the bargaining power of each country is conditioned upon the previous claims. The disagreement point of this negotiation process is to face conflict during

<sup>13</sup> These results are in line with that of Wittman (1979), who finds that a reduction of hostilities may reduce

the probability of a settlement taking place and thus prolong the war

1 + 4*H*

<sup>2</sup> *<sup>H</sup>* <sup>+</sup> <sup>3</sup>*H*<sup>2</sup> <sup>−</sup> <sup>1</sup>

8 1 2 − 1

<sup>4</sup> = 0.07 274 9. Hence, for conflict costs above this

. (23)

12

1 + 2*H* + <sup>1</sup>

value for *x*∗ by substituting *W* in Eq. (21) into Eq. (22) to obtain:

cost is then 0.103 59 if *H* = 0.07275, and 0.157 07 if *H* = 0.540 57.

emerge). When *H* is large enough, the claims tend to a constant value.

equilibrium statement *x*∗ changes in response to *H*.

**6. Conclusions**

*<sup>x</sup>*<sup>∗</sup> <sup>=</sup> <sup>1</sup> 2 <sup>√</sup><sup>15</sup> <sup>−</sup> <sup>1</sup>

3 2 <sup>3</sup>

inequality is satisfied by all *<sup>H</sup>* <sup>≥</sup> <sup>1</sup>

an indeterminate number of periods (ranging from 1 to infinite). On the other hand, if the public statements are far enough from each other, then no negotiation process emerges and the conflict continues for at least one more period. As a difference with previous work, conflict is not considered as a bargaining instrument. Instead, conflict is the prevailing state, which can only be changed if the parties make diplomatic efforts (concessions, in the form of reasonable claims) to resolve the dispute through a pacific settlement.

In this setting we conclude that, in general, the probability of conflict in period *t* is a non monotonic function of the conflict costs. For low costs it is decreasing, and it increases when costs are above a certain value. At this particular value, the probability of conflict reaches a minimum. A similar pattern is obtained with respect to the pre-bargaining equilibrium claims. The effect of conflict costs on the social welfare is more clear: it decreases monotonically as the conflict costs increase.

These results suggest that the goals of maximizing the social welfare and minimizing the probability of conflict are not equivalent, since both are achieved at different levels of conflict costs. While the welfare is maximum when conflict is costless, the likelihood of a negotiation process to emerge achieves a maximum for positive conflict costs. Moreover, if such costs are very high, the probability of conflict achieves a constant value. This explains why in some instances a negotiation process does not emerge, even if it would be in the interest of the conflicting parties to initiate it. Although countries must bear certain costs every period the conflict lasts, they face even higher costs if negotiations emerge but no agreement is reached. The countries trade-off the conflict costs (today) against the expected outcome of negotiations, and therefore they make public statements that may involve the persistence of the conflict.

The utility reached at the disagreement point when negotiations break down is a crucial element to determine the probability of such negotiations to emerge. In our model, the threat point is parameterized by "n", the number of periods the conflict will last if no agreement is reached. The analysis made in Section 5 reveals that the social welfare depends negatively on "n" for any given conflict cost. It is then desirable that the threat point of a negotiation process is not too harmful for the parties in conflict. This implies that an important issue in the design of the rules of a negotiation process should be to prevent and limit the consequences of breaking negotiations. Specifically, before the bargaining over territory takes place, it would be welfare enhancing that the parties could commit to restart negotiations at an early date in case no agreement were reached. In general, any measure conducive to mitigate the negative outcome represented by the threat point would improve the likelihood of achieving a pacific settlement through negotiation.

If the countries could choose the intensity of the dispute, our model suggests that a moderate investment in resources devoted to the conflict proves optimal for a negotiation process to emerge. This conclusion is compatible with the approach that envisions conflict as part of a bargaining strategy, as in Sanchez-Pagés (2009). According to this view, the nations engaged in territorial disputes use limited confrontation as a way to convey information about their relative strength. The information revealed facilitates the emergence of a peaceful outcome. In our model, though, the line of explanation is a bit different. When conflict costs are positive, but not too high, the parties might find it beneficial to negotiate as long as the threat point in case of disagreement does not entail huge utility losses. However, if conflict costs become higher (for instance, if the countries engage in a nuclear arms race), then the threat point involves a final confrontation aimed to the absolute destruction of the enemy. We have shown that the territorial claims in this case tend to be more aggressive, thus lowering the probability of reaching a negotiated outcome.

Further research should focus on the public good nature of negotiation processes. Once the process is underway, no agent can be excluded from its benefits. Moreover, taking part in negotiations does not exclude other agents of being engaged in the process. Provision of the public good "negotiation" is clearly efficient, as long as the conflicting parties obtain a mutual benefit. However, contributing to this public good (i.e. making diplomatic efforts in the form of less demanding pre-bargaining claims) is a strategic decision plagued by the free rider problem. Interpreting the emergence of negotiations as a game of voluntary contributions, allow us to conclude that, in general, negotiations are under-provided. This view of the problem allows for a mechanism design solution to resolve international disputes over territory.

#### **7. Acknowledgements**

Financial support from the Spanish Ministry of Science and Innovation through MEC/FEDER grant ECO2010-21624 is gratefully acknowledged.

#### **8. References**


14 Will-be-set-by-IN-TECH

216 Social Welfare

Further research should focus on the public good nature of negotiation processes. Once the process is underway, no agent can be excluded from its benefits. Moreover, taking part in negotiations does not exclude other agents of being engaged in the process. Provision of the public good "negotiation" is clearly efficient, as long as the conflicting parties obtain a mutual benefit. However, contributing to this public good (i.e. making diplomatic efforts in the form of less demanding pre-bargaining claims) is a strategic decision plagued by the free rider problem. Interpreting the emergence of negotiations as a game of voluntary contributions, allow us to conclude that, in general, negotiations are under-provided. This view of the problem allows for a mechanism design solution to resolve international disputes

Financial support from the Spanish Ministry of Science and Innovation through MEC/FEDER

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