**Social Welfare and the Emergence of Negotiations**

Francisco Candel-Sánchez *University of Murcia Spain* 

#### **1. Introduction**

26 Will-be-set-by-IN-TECH

202 Social Welfare

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The presence of long-lasting conflicts in situations where reaching an agreement would benefit all parties is a pervasive phenomenon. This paradoxical fact, pointed out by Hicks (1932) in the context of strikes, has especially serious consequences when nations are engaged in war. One of the most common sources of conflict is bargaining over a territory, present in both secessionist movements as in international disputes1. This chapter aims to provide a rationale to the "Hicks paradox" in a context where two countries are engaged in open conflict for a territory. Although termination of the conflict (i.e. reaching a negotiated outcome) improves the welfare of both parties, it is shown that when bargainers act rationally, the probability that the conflict persists over time is positive.

There are two main ways in which the inefficiencies associated to delay in bargaining are explained from a rational choice perspective: asymmetric information and dynamic commitment problems2. The former invokes the existence of some sort of incomplete information in the bargaining environment. For instance, states might prefer fighting instead of achieving a peaceful settlement when there is uncertainty about the other's cost of fighting. The latter resorts to dynamic commitment problems whereby the players may face incentives to renege on agreements. Our contribution belongs to the first category.

We propose a game-theoretic approach to understand the relationship between conflict, social welfare and the likelihood that a negotiation process emerges. Although the problem we tackle here is applicable to several bargaining contexts, for expositional purposes we conduct our analysis through the example of an international dispute over territory3. In contrast with the standard view of conflict as a bargaining tool (where the threat of war enhances the bargaining power of the parties), we consider conflict as the status quo, and then focus on the strategic elements that influence conflict termination. Our contribution is then very related to the work of Wittman (1979), who envisions the end of war as a rational process.

The key element in our approach is that an inefficient outcome may arise as a consequence of the parties' ability to make public statements. When engaged in conflict, strong public pronouncements against the other's territorial claims undermine the possibilities of reaching a mutually beneficial agreement. Why would then the parties make such statements? Our

<sup>1</sup> For a survey of the theoretical work on bargaining and war, see Powell (2002).

<sup>2</sup> See Fearon (1995).

<sup>3</sup> Other contexts where our analysis would apply are: bargaining between a buyer and a seller, firm-union negotiations, settlements out of court in legal disputes, etc.

model, statements are costless6, but they entail consequences that affect the players' welfare. The players do not send revealing signals, although the strategic process that precedes negotiation is characterized by imperfect information. A threshold whose value is uncertain is introduced to account for all pieces of information that condition the likelihood of a pacific settlement and are not perfectly known to both parties. In this informational environment, we

Social Welfare and the Emergence of Negotiations 205

The chapter is organized as follows. Section 2 presents a dynamic model that incorporates the main elements of our analysis. Section 3 characterizes the generalized Nash bargaining solution assumed to prevail if negotiations are held. Section 4 is devoted to compute the statements made by both countries in a symmetric Bayesian Nash equilibrium. Section 5 investigates the welfare consequences of the equilibrium play of the countries, paying special attention to the relationship between the conflict costs and the likelihood of a negotiation

We consider a dynamic infinite horizon model in which two countries7, *L* and *R*, are in conflict over a piece of land of size 1. At any given period, the conflict can be resolved through negotiation, or it can persist at least one more period. Formally, the players face a problem consisting on how to reach an agreement about a division of the interval [0, 1]. This situation

At the beginning of each period, countries have the ability to make public statements about their respective territory claims. After the statements are made, two possibilities emerge: Either a negotiation process takes place, or the conflict continues. If the positions publicly announced are "too far" from each other, then no negotiation process starts, and each party *i* = *L*, *H* faces conflict costs *Hi*. This situation might occur, for instance, if pre bargaining

The conflict cost is a parameter that influences the countries' welfare and hence affects their behavior. It reflects the resources lost in the conflict (guns, lives, etc.), or it can even be interpreted as the cost of a delay in reaching a mutually beneficial agreement. Unlike Sanchez-Pagés (2009) and Cramton and Tracy (1992), we consider that countries cannot choose

The countries' utilities after an agreement is reached are linear and given by *uL*(*y*) = *θLy* and *uR*(*y*) = *θ<sup>R</sup>* (1 − *y*), where *y* ∈ [0, 1] denotes the piece of land that goes to country *L* if the territorial status quo is modified through negotiation, and *θ<sup>i</sup>* denotes country's *i* = *L*, *R* valuation of the territory gained. It is implicitly assumed that there is no initial owner of the land. Therefore, the status quo refers to a situation in which the share of territory possessed

We analyze the strategic incentives faced by the conflicting parties before the possibility of negotiations arises. In particular, we consider that countries use the tactical approach of making public statements on *y* with the aim of creating a bargaining position. Let *xi* ∈ [0, 1] be the public statement made by country *i* = *L*, *R*, where *xL* is the territory claim made by *L*

<sup>6</sup> Croson et. al (2003) show through experiments that cheap talk in bargaining games have real effects,

<sup>7</sup> We use the terms "countries", "players" or "parties" indistinticly throughout the chapter.

statements made by country *L* are regarded as unacceptable by country *R*.

the intensity of the dispute (and hence, cannot affect the conflict costs).

are able to characterize a unique Bayesian Nash equilibrium.

process to emerge. Finally, Section 6 concludes.

is repeated every period until an agreement is reached.

and 1 − *xR* represents the territory claim of country *R*.

both in the short and in the long run.

**2. Model**

by each country is zero.

view is that countries use public claims as a rational strategy to better their position in a future negotiation process.

The declaration of a position as an irreversible commitment4 is used as a tactical approach to create a more advantageous bargaining environment. The politician who makes the claim is implicitly committed with the public to fulfill the terms of the pronouncement. Such statements are effective as they have the power to bind oneself. As Schelling (1956) argued:

*When national representatives go to international negotiations knowing that there is a wide range of potential agreement within which the outcome will depend on bargaining, they seem often to create a bargaining position by public statements, statements calculated to arouse a public opinion that permit no concessions to be made*. *If a binding public opinion can be cultivated, and made evident to the other side, the initial position can thereby be made visibly "final."* p. 287.

We construct a formal model in which two agents bargain about how to divide the [0, 1] interval (territory). There are two possibilities: either a negotiation process emerges and the interval is divided according to a certain division rule, or the negotiation process does not take place, and the countries must face conflict one more period. In the former case, the territory is divided according to the generalized Nash bargaining solution. If pre-bargaining public statements are "too far" from each other, negotiations don't start. Statements are represented by numbers belonging to [0, 1]. We define a threshold T, whose value is uncertain to the parties, to account for the distance about claims. If such a distance is above T, then neither party is willing to negotiate. Uncertainty about T reflects the parties' lack of information about each other's willingness to make concessions.

Countries are engaged in a game in which extreme claims lower the probability of a negotiated outcome, but they also better their bargaining position in case a pacific settlement is to be reached. The players must then calculate rationally their strategies in the absence of information about the realization of the threshold T. We are posing a situation in which there is a risk-return trade off between aggressive claims and diplomatic efforts (concessions) that increase the likelihood of a negotiated outcome. Faced with this trade-off, the players' optimal strategy includes making statements that involve in some cases persistence of the conflict.

In order to achieve this result, we first characterize the Bayesian Nash equilibrium vector of public statements. We find that there exists a set of parameters under which the conflict continues with positive probability at equilibrium. Remarkably, this probability turns out to be a non monotone function of the conflict costs. For low enough costs, the probability of conflict is decreasing, then it reaches a minimum at a certain point and when costs go beyond that point, it increases and converges asymptotically to a constant value. From this pattern we can extract some conclusions. First, lowering the conflict costs may increase the probability of conflict if such costs are low enough. Second, when conflict costs are sufficiently high, variations on them have a negligible effect on the likelihood that a negotiation process emerges.

The view of conflict termination presented here complements the usual approach to bargaining and wars, whereby investing resources in military weapons is used as a credible signal by countries wishing to convey certain private information to their opponents5. In our

<sup>4</sup> Muthoo (1996) shows how commitments which can be reversed at a cost affect the players' "share of the cake" at equilibrium.

<sup>5</sup> As in Kennan and Wilson (1993), delay can also be considered as a signal to convey private information credibly.

model, statements are costless6, but they entail consequences that affect the players' welfare. The players do not send revealing signals, although the strategic process that precedes negotiation is characterized by imperfect information. A threshold whose value is uncertain is introduced to account for all pieces of information that condition the likelihood of a pacific settlement and are not perfectly known to both parties. In this informational environment, we are able to characterize a unique Bayesian Nash equilibrium.

The chapter is organized as follows. Section 2 presents a dynamic model that incorporates the main elements of our analysis. Section 3 characterizes the generalized Nash bargaining solution assumed to prevail if negotiations are held. Section 4 is devoted to compute the statements made by both countries in a symmetric Bayesian Nash equilibrium. Section 5 investigates the welfare consequences of the equilibrium play of the countries, paying special attention to the relationship between the conflict costs and the likelihood of a negotiation process to emerge. Finally, Section 6 concludes.

## **2. Model**

2 Will-be-set-by-IN-TECH

view is that countries use public claims as a rational strategy to better their position in a future

The declaration of a position as an irreversible commitment4 is used as a tactical approach to create a more advantageous bargaining environment. The politician who makes the claim is implicitly committed with the public to fulfill the terms of the pronouncement. Such statements are effective as they have the power to bind oneself. As Schelling (1956) argued: *When national representatives go to international negotiations knowing that there is a wide range of potential agreement within which the outcome will depend on bargaining, they seem often to create a bargaining position by public statements, statements calculated to arouse a public opinion that permit no concessions to be made*. *If a binding public opinion can be cultivated, and made evident to the other*

We construct a formal model in which two agents bargain about how to divide the [0, 1] interval (territory). There are two possibilities: either a negotiation process emerges and the interval is divided according to a certain division rule, or the negotiation process does not take place, and the countries must face conflict one more period. In the former case, the territory is divided according to the generalized Nash bargaining solution. If pre-bargaining public statements are "too far" from each other, negotiations don't start. Statements are represented by numbers belonging to [0, 1]. We define a threshold T, whose value is uncertain to the parties, to account for the distance about claims. If such a distance is above T, then neither party is willing to negotiate. Uncertainty about T reflects the parties' lack of information

Countries are engaged in a game in which extreme claims lower the probability of a negotiated outcome, but they also better their bargaining position in case a pacific settlement is to be reached. The players must then calculate rationally their strategies in the absence of information about the realization of the threshold T. We are posing a situation in which there is a risk-return trade off between aggressive claims and diplomatic efforts (concessions) that increase the likelihood of a negotiated outcome. Faced with this trade-off, the players' optimal strategy includes making statements that involve in some cases persistence of the conflict. In order to achieve this result, we first characterize the Bayesian Nash equilibrium vector of public statements. We find that there exists a set of parameters under which the conflict continues with positive probability at equilibrium. Remarkably, this probability turns out to be a non monotone function of the conflict costs. For low enough costs, the probability of conflict is decreasing, then it reaches a minimum at a certain point and when costs go beyond that point, it increases and converges asymptotically to a constant value. From this pattern we can extract some conclusions. First, lowering the conflict costs may increase the probability of conflict if such costs are low enough. Second, when conflict costs are sufficiently high, variations on them have a negligible effect on the likelihood that a negotiation process

The view of conflict termination presented here complements the usual approach to bargaining and wars, whereby investing resources in military weapons is used as a credible signal by countries wishing to convey certain private information to their opponents5. In our

<sup>4</sup> Muthoo (1996) shows how commitments which can be reversed at a cost affect the players' "share of

<sup>5</sup> As in Kennan and Wilson (1993), delay can also be considered as a signal to convey private information

*side, the initial position can thereby be made visibly "final."* p. 287.

about each other's willingness to make concessions.

negotiation process.

emerges.

credibly.

the cake" at equilibrium.

We consider a dynamic infinite horizon model in which two countries7, *L* and *R*, are in conflict over a piece of land of size 1. At any given period, the conflict can be resolved through negotiation, or it can persist at least one more period. Formally, the players face a problem consisting on how to reach an agreement about a division of the interval [0, 1]. This situation is repeated every period until an agreement is reached.

At the beginning of each period, countries have the ability to make public statements about their respective territory claims. After the statements are made, two possibilities emerge: Either a negotiation process takes place, or the conflict continues. If the positions publicly announced are "too far" from each other, then no negotiation process starts, and each party *i* = *L*, *H* faces conflict costs *Hi*. This situation might occur, for instance, if pre bargaining statements made by country *L* are regarded as unacceptable by country *R*.

The conflict cost is a parameter that influences the countries' welfare and hence affects their behavior. It reflects the resources lost in the conflict (guns, lives, etc.), or it can even be interpreted as the cost of a delay in reaching a mutually beneficial agreement. Unlike Sanchez-Pagés (2009) and Cramton and Tracy (1992), we consider that countries cannot choose the intensity of the dispute (and hence, cannot affect the conflict costs).

The countries' utilities after an agreement is reached are linear and given by *uL*(*y*) = *θLy* and *uR*(*y*) = *θ<sup>R</sup>* (1 − *y*), where *y* ∈ [0, 1] denotes the piece of land that goes to country *L* if the territorial status quo is modified through negotiation, and *θ<sup>i</sup>* denotes country's *i* = *L*, *R* valuation of the territory gained. It is implicitly assumed that there is no initial owner of the land. Therefore, the status quo refers to a situation in which the share of territory possessed by each country is zero.

We analyze the strategic incentives faced by the conflicting parties before the possibility of negotiations arises. In particular, we consider that countries use the tactical approach of making public statements on *y* with the aim of creating a bargaining position. Let *xi* ∈ [0, 1] be the public statement made by country *i* = *L*, *R*, where *xL* is the territory claim made by *L* and 1 − *xR* represents the territory claim of country *R*.

<sup>6</sup> Croson et. al (2003) show through experiments that cheap talk in bargaining games have real effects, both in the short and in the long run.

<sup>7</sup> We use the terms "countries", "players" or "parties" indistinticly throughout the chapter.

consequence of such a breakdown is to suffer conflict along *n* more periods, until the parties in conflict are again ready for dialogue. The number of periods of conflict, *n*, represents a

Social Welfare and the Emergence of Negotiations 207

The pair (*dLn*(*HL*, *WL*), *dRn*(*HR*, *WR*)) is the disagreement or threat point<sup>9</sup> in this bargaining

The share of territory that goes to country *<sup>L</sup>*, *<sup>y</sup>*(*αL*, *<sup>α</sup>R*), depends negatively on country *<sup>L</sup>*�

such effects increases as the discount factor *δ* approaches one. The reason is that what the bargainers obtain if they fail to reach an agreement is proportional to the conflict costs. Hence,

higher are the costs avoided (i.e., the higher is *Hi*) the lower share of land is obtained in the agreement. This effect is reversed when we consider the conflict costs of the opponent

similar interpretation can be given to the share of the interval [0, 1] that goes to country *R*,

We consider that the bargaining power of country *i* = *L*, *R*, *αi*, is given by the relative weight

assumption highlights the importance of pre-bargaining claims in further negotiation and is

Therefore, the Nash bargaining outcome as a function of the pre bargaining public statements

A quick inspection to Eq.(4) reveals that, for any given *xR*, *HR*, *HL* and *δ*, function *y* is increasing and concave in *xL*. By symmetry, the same occurs to 1 − *y* with respect to 1 − *xR*.

This section is devoted to analyze the equilibrium values for the pre-bargaining claims made by the countries in conflict. These equilibrium values depend on parameters such as the conflict costs (*Hi*), the duration of the conflict if no agreement is reached after negotiation (*n*) and the discount factor (*δ*), and they also depend on the stationary value for the social welfare (*Wi*). In order to obtain closed solutions and simplify calculus, we compute the equilibrium

<sup>9</sup> Observe that the threat point varies as a function of parameters *δ*, *H* and *n*. We are only concerned on the effects of such changes on the Nash bargaining solution. To see how the influence of the threat point

on the bargaining outcome varies across different type of solutions, see Anbarci et. al (2002).

*Hi* can be seen as implicit gains (costs avoided) from reaching an agreement. The

*s* public statement with respect to the sum of both countries' statements. This

(<sup>1</sup> <sup>−</sup> *dRn*) + <sup>1</sup> <sup>−</sup> *xR*

*xL* + 1 − *xR*

*<sup>α</sup><sup>L</sup>* [(<sup>1</sup> <sup>−</sup> *<sup>y</sup>*) <sup>−</sup> *dRn*]

*<sup>y</sup>*(*αL*, *<sup>α</sup>R*) = *<sup>α</sup><sup>L</sup>* <sup>−</sup> *<sup>α</sup>LdRn* <sup>+</sup> *<sup>α</sup>RdLn*. (3)

*<sup>α</sup><sup>R</sup>* .

*s* conflict costs (*HR*). The magnitude of

*s* bargaining power (*αL*). A

*xL*+1−*xR* , and *<sup>α</sup><sup>R</sup>* <sup>=</sup> <sup>1</sup>−*xR*

*dLn*. (4)

*s*

*xL*+1−*xR* .

measure of the disagreement costs once the negotiations are underway.

*<sup>y</sup>*(*αL*, *<sup>α</sup>R*) = arg max {*y*} [*<sup>y</sup>* <sup>−</sup> *dLn*]

Solving the maximization problem stated above yields the following outcome:

problem. The *generalized Nash bargaining solution* is then given by:

country. Moreover, *<sup>y</sup>* also depends positively on country *<sup>L</sup>*�

central to our analysis. Specifically, we assume that: *α<sup>L</sup>* = *xL*

*xL* + 1 − *xR*

*<sup>y</sup>*(*xL*, *xR*) = *xL*

outcome under the assumption that the players are symmetric.

**4. Bayesian Nash equilibrium statements**

conflict costs (*HL*) but positively on country *R*�

<sup>1</sup>−*δ<sup>n</sup>* 1−*δ* 

<sup>1</sup> <sup>−</sup> *<sup>y</sup>*(*αL*, *<sup>α</sup>R*).

of country *i*

is given by

�

We define a threshold *T*, such that if *xL* − *xR* > *T*, negotiations don't take place and conflict continues. If *xL* − *xR* ≤ *T*, a bargaining process takes place whose final outcome is assumed to be given by the generalized Nash bargaining solution. In case negotiations don't start, the conflict is ongoing and each country *i* faces a cost *Hi* during the present period. At the beginning of the next period, countries can make again public statements thus opening the possibility of a negotiation in that period. The discount factor between periods is *δ* < 1.

We denote by *Wi* the expected utility of country *i* = *L*, *R* at the beginning of period *t*, and usually refer to it as the *social welfare*. As long as all periods are identical, and we consider an infinite time horizon, the game that starts in period *t* is identical to the game that starts in period *t* + 1. This allows us to truncate the infinite horizon game and restrict our analysis to the equilibrium strategies of any given period (say *t*). The equilibrium payoff of country *i* will also be *Wi* at the beginning of period *t* + 1. Hence, using the discount factor between periods we are able to derive endogenously the stationary equilibrium value of *Wi*. For notational simplicity and wherever there is no risk of confusion, we omit time subscripts throughout the chapter.

It is useful to define the utility achieved by country *i* = *L*, *R*, in case the conflict is prolonged for *n* periods. This utility is given by:

$$d\_{\rm lit}(H\_{\rm li}, W\_{\rm i}) = -\left(\frac{1-\delta^n}{1-\delta}\right)H\_{\rm i} + \delta^n W\_{\rm i}.\tag{1}$$

The utility *din* represents the discounted conflict costs during *n* periods plus the discounted expected utility at the beginning of period *t* + *n* + 1. Observe that *din* is strictly decreasing in *n*. To see this, we just need to check that

$$\frac{\partial d\_{\rm in}(H\_{\rm i}, W\_{\rm i})}{\partial n} = \frac{\delta^n \ln \delta}{1 - \delta} H\_{\rm i} + \delta^n \ln \delta W\_{\rm i} < 0,\tag{2}$$

since *δ* < 1 and ln *δ* < 0. A quick inspection to Eq. (1) reveals that *din*(*Hi*, *Wi*) depends negatively on *Hi* and positively on *Wi*.

A summary of the situation analyzed can be stated as follows: if at the beginning of period *t* a negotiation process starts, the parties have the possibility of reaching an agreement on the division of land. In the event that negotiations do not start (probably because the public statements are too demanding), the conflict continues one more period and the payoff faced by each country is *di*1(*Hi*, *Wi*) = −*Hi* + *δWi*, where *δWi* is the discounted value of the expected utility at the beginning of period *t* + 1. In case negotiations take place, the final outcome obtained corresponds to the Generalized Nash Bargaining solution8. This is a classical axiomatic solution to bargaining problems, which we refer specifically to our context in the next section.

#### **3. Generalized Nash bargaining solution**

We denote as *α<sup>L</sup>* and *α<sup>R</sup>* the bargaining power of countries *L* and *R*, with *α<sup>L</sup>* + *α<sup>R</sup>* = 1. If a negotiation process takes place, either an agreement is reached, or negotiations end up with a disagreement that involves returning to conflict again. A bargaining breakdown once the parties have engaged in negotiations is a serious negative outcome. We assume that the

<sup>8</sup> See Binmore (1987), Muthoo (1999) and Osborne and Rubinstein (1990) for a detailed discussion of the Nash bargaining solution.

4 Will-be-set-by-IN-TECH

We define a threshold *T*, such that if *xL* − *xR* > *T*, negotiations don't take place and conflict continues. If *xL* − *xR* ≤ *T*, a bargaining process takes place whose final outcome is assumed to be given by the generalized Nash bargaining solution. In case negotiations don't start, the conflict is ongoing and each country *i* faces a cost *Hi* during the present period. At the beginning of the next period, countries can make again public statements thus opening the possibility of a negotiation in that period. The discount factor between periods is *δ* < 1.

We denote by *Wi* the expected utility of country *i* = *L*, *R* at the beginning of period *t*, and usually refer to it as the *social welfare*. As long as all periods are identical, and we consider an infinite time horizon, the game that starts in period *t* is identical to the game that starts in period *t* + 1. This allows us to truncate the infinite horizon game and restrict our analysis to the equilibrium strategies of any given period (say *t*). The equilibrium payoff of country *i* will also be *Wi* at the beginning of period *t* + 1. Hence, using the discount factor between periods we are able to derive endogenously the stationary equilibrium value of *Wi*. For notational simplicity and wherever there is no risk of confusion, we omit time subscripts throughout the

It is useful to define the utility achieved by country *i* = *L*, *R*, in case the conflict is prolonged

<sup>1</sup> <sup>−</sup> *<sup>δ</sup><sup>n</sup>* 1 − *δ*

The utility *din* represents the discounted conflict costs during *n* periods plus the discounted expected utility at the beginning of period *t* + *n* + 1. Observe that *din* is strictly decreasing in

since *δ* < 1 and ln *δ* < 0. A quick inspection to Eq. (1) reveals that *din*(*Hi*, *Wi*) depends

A summary of the situation analyzed can be stated as follows: if at the beginning of period *t* a negotiation process starts, the parties have the possibility of reaching an agreement on the division of land. In the event that negotiations do not start (probably because the public statements are too demanding), the conflict continues one more period and the payoff faced by each country is *di*1(*Hi*, *Wi*) = −*Hi* + *δWi*, where *δWi* is the discounted value of the expected utility at the beginning of period *t* + 1. In case negotiations take place, the final outcome obtained corresponds to the Generalized Nash Bargaining solution8. This is a classical axiomatic solution to bargaining problems, which we refer specifically to our context

We denote as *α<sup>L</sup>* and *α<sup>R</sup>* the bargaining power of countries *L* and *R*, with *α<sup>L</sup>* + *α<sup>R</sup>* = 1. If a negotiation process takes place, either an agreement is reached, or negotiations end up with a disagreement that involves returning to conflict again. A bargaining breakdown once the parties have engaged in negotiations is a serious negative outcome. We assume that the

<sup>8</sup> See Binmore (1987), Muthoo (1999) and Osborne and Rubinstein (1990) for a detailed discussion of the

*Hi* + *δnWi*. (1)

<sup>1</sup> <sup>−</sup> *<sup>δ</sup> Hi* <sup>+</sup> *<sup>δ</sup><sup>n</sup>* ln *<sup>δ</sup>Wi* <sup>&</sup>lt; 0, (2)

*din*(*Hi*, *Wi*) = −

*<sup>∂</sup><sup>n</sup>* <sup>=</sup> *<sup>δ</sup><sup>n</sup>* ln *<sup>δ</sup>*

*∂din*(*Hi*, *Wi*)

chapter.

for *n* periods. This utility is given by:

*n*. To see this, we just need to check that

negatively on *Hi* and positively on *Wi*.

**3. Generalized Nash bargaining solution**

in the next section.

Nash bargaining solution.

consequence of such a breakdown is to suffer conflict along *n* more periods, until the parties in conflict are again ready for dialogue. The number of periods of conflict, *n*, represents a measure of the disagreement costs once the negotiations are underway.

The pair (*dLn*(*HL*, *WL*), *dRn*(*HR*, *WR*)) is the disagreement or threat point<sup>9</sup> in this bargaining problem. The *generalized Nash bargaining solution* is then given by:

$$\widetilde{y}(\mathfrak{a}\_{L\prime}\mathfrak{a}\_{R}) = \arg\max\_{\{y\}} \left[ y - d\_{Ln} \right]^{\mathfrak{a}\_{L}} \left[ (1 - y) - d\_{Rn} \right]^{\mathfrak{a}\_{R}}.$$

Solving the maximization problem stated above yields the following outcome:

$$
\tilde{y}(\mathfrak{a}\_{L'}\mathfrak{a}\_R) = \mathfrak{a}\_L - \mathfrak{a}\_L d\_{Rn} + \mathfrak{a}\_R d\_{Ln}.\tag{3}
$$

The share of territory that goes to country *<sup>L</sup>*, *<sup>y</sup>*(*αL*, *<sup>α</sup>R*), depends negatively on country *<sup>L</sup>*� *s* conflict costs (*HL*) but positively on country *R*� *s* conflict costs (*HR*). The magnitude of such effects increases as the discount factor *δ* approaches one. The reason is that what the bargainers obtain if they fail to reach an agreement is proportional to the conflict costs. Hence, <sup>1</sup>−*δ<sup>n</sup>* 1−*δ Hi* can be seen as implicit gains (costs avoided) from reaching an agreement. The higher are the costs avoided (i.e., the higher is *Hi*) the lower share of land is obtained in the agreement. This effect is reversed when we consider the conflict costs of the opponent country. Moreover, *<sup>y</sup>* also depends positively on country *<sup>L</sup>*� *s* bargaining power (*αL*). A similar interpretation can be given to the share of the interval [0, 1] that goes to country *R*, <sup>1</sup> <sup>−</sup> *<sup>y</sup>*(*αL*, *<sup>α</sup>R*).

We consider that the bargaining power of country *i* = *L*, *R*, *αi*, is given by the relative weight of country *i* � *s* public statement with respect to the sum of both countries' statements. This assumption highlights the importance of pre-bargaining claims in further negotiation and is central to our analysis. Specifically, we assume that: *α<sup>L</sup>* = *xL xL*+1−*xR* , and *<sup>α</sup><sup>R</sup>* <sup>=</sup> <sup>1</sup>−*xR xL*+1−*xR* . Therefore, the Nash bargaining outcome as a function of the pre bargaining public statements is given by

$$y(\mathbf{x}\_L, \mathbf{x}\_R) = \frac{\mathbf{x}\_L}{\mathbf{x}\_L + 1 - \mathbf{x}\_R} (1 - d\_{Rn}) + \frac{1 - \mathbf{x}\_R}{\mathbf{x}\_L + 1 - \mathbf{x}\_R} d\_{Ln}.\tag{4}$$

A quick inspection to Eq.(4) reveals that, for any given *xR*, *HR*, *HL* and *δ*, function *y* is increasing and concave in *xL*. By symmetry, the same occurs to 1 − *y* with respect to 1 − *xR*.

#### **4. Bayesian Nash equilibrium statements**

This section is devoted to analyze the equilibrium values for the pre-bargaining claims made by the countries in conflict. These equilibrium values depend on parameters such as the conflict costs (*Hi*), the duration of the conflict if no agreement is reached after negotiation (*n*) and the discount factor (*δ*), and they also depend on the stationary value for the social welfare (*Wi*). In order to obtain closed solutions and simplify calculus, we compute the equilibrium outcome under the assumption that the players are symmetric.

<sup>9</sup> Observe that the threat point varies as a function of parameters *δ*, *H* and *n*. We are only concerned on the effects of such changes on the Nash bargaining solution. To see how the influence of the threat point on the bargaining outcome varies across different type of solutions, see Anbarci et. al (2002).

welfare at the beginning of the next period (*t* + 1). If changing the statements *xL* and *xR* could

Social Welfare and the Emergence of Negotiations 209

Solving Eq. (8) for *xL* we obtain *xL*(*<sup>x</sup><sup>R</sup>*), the reaction function of country *<sup>L</sup>* to every possible given statement *<sup>x</sup><sup>R</sup>*. A similar routine leads us to *xR*(*<sup>x</sup><sup>L</sup>*), the reaction function of country *<sup>R</sup>* to any given *<sup>x</sup><sup>L</sup>*. In a Bayesian Nash equilibrium, the pair of statements (*<sup>x</sup><sup>L</sup>*, *<sup>x</sup><sup>R</sup>*) is such that

In order to provide an explicit expression for the pair (*<sup>x</sup><sup>L</sup>*, *<sup>x</sup><sup>R</sup>*) we now make some symmetry assumptions. In particular, we assume that both countries are identical in their valuations of territory and in the conflict costs. Namely, *θ<sup>L</sup>* = *θ<sup>R</sup>* = *θ* and *HL* = *HR* = *H*. In this scenario, we have *<sup>x</sup><sup>L</sup>* <sup>=</sup> <sup>1</sup> <sup>−</sup> *<sup>x</sup><sup>R</sup>*. Then, it follows that *WL* <sup>=</sup> *WR* <sup>=</sup> *<sup>W</sup>* and hence *dLn* <sup>=</sup> *dRn* <sup>=</sup> *dn*. In a symmetric equilibrium we have *<sup>x</sup><sup>L</sup>* <sup>=</sup> *<sup>x</sup>*∗(*n*, *<sup>H</sup>*, *<sup>W</sup>*) and *<sup>x</sup><sup>R</sup>* <sup>=</sup> <sup>1</sup> <sup>−</sup> *<sup>x</sup>*∗(*n*, *<sup>H</sup>*, *<sup>W</sup>*). We use Eq. (8)

Next we analyze the dependence of *x*∗(.) on parameters *H* and *n*. We simplify notation by writing *x*∗ instead of *x*∗(*n*, *H*, *W*) and *dn* instead of *dn*(*H*, *W*). Using the expression for *x*∗ in

> *∂dn ∂H* Indirect effect

The indirect effect reflects the equilibrium reaction of the pre-bargaining claim to an increase in conflict costs through the influence that such costs have on the disagreement point. Under the mild assumption that *θ* + 2*H* > 2*δW* (i.e. if conflict costs are high enough, and/or the valuation of land is high enough, and/or the discount factor is low enough), the derivative

The direct effect operates in the opposite direction. If the conflict costs increase, and the disagreement point *dn* is low enough10, then the threat of a conflict dissuades from making aggressive statements (e.g. to claim the entire territory). In general, it cannot be established whether the direct effect predominates or not over the indirect effect. We show later, in a more restricted context, that the response of the equilibrium claims to a change in *H* is non monotone. Specifically, *x*∗ is decreasing if *H* is low enough and increasing if *H* is high enough. If the parties disagree in negotiations, the conflict lasts for *n* periods. Next we analyze the influence of parameter *n* in the behavior of the parties before negotiations. In particular, we

+

< 0, and *<sup>∂</sup>x*<sup>∗</sup>

*∂dn ∂n* .

<sup>2</sup> . Observe that for this inequality to hold it is sufficient that 1 + 2*H* > 2*δW*,

*<sup>x</sup>*∗(*n*, *<sup>H</sup>*, *<sup>W</sup>*) = *<sup>θ</sup>* [<sup>1</sup> <sup>−</sup> <sup>2</sup>*dn*(*H*, *<sup>W</sup>*)]

1−*δ<sup>n</sup>* 1−*δ* 

> *dx*∗ *dn* <sup>=</sup> *<sup>∂</sup>x*<sup>∗</sup> *∂dn*

*<sup>∂</sup>dn* is negative. Therefore, the indirect effect is positive provided that *<sup>∂</sup>dn*

*dx*∗ *dH* <sup>=</sup> *<sup>∂</sup>x*<sup>∗</sup> *∂dn*

<sup>2</sup> , *<sup>∂</sup>dn <sup>∂</sup><sup>H</sup>* = −

*<sup>i</sup>* throughout all periods. The players, who are aware of the dynamics of

2 [*θ*(1 − *dn*(*H*, *W*)) − *d*1(*H*, *W*)]

*∂x*∗ *∂H* Direct effect

> *<sup>∂</sup><sup>H</sup>* <sup>=</sup> <sup>−</sup>*θ*[1−2*dn*] 2[*θ*(1−*dn*)−*d*1]

2 .

*<sup>∂</sup><sup>H</sup>* < 0.

*<sup>i</sup>* �= *Wi*, then the stationary value of the social

. (9)

change the equilibrium value of *Wi*, say to *W*�

the game, take the value of *Wi* as given.

*<sup>x</sup><sup>L</sup>* <sup>=</sup> *xL*(*<sup>x</sup><sup>R</sup>*) and *<sup>x</sup><sup>R</sup>* <sup>=</sup> *xR*(*<sup>x</sup><sup>L</sup>*).

welfare would be *W*�

above to compute

Eq. (9) we compute

*<sup>∂</sup>dn* <sup>=</sup> <sup>−</sup> *<sup>θ</sup>*<sup>2</sup>+2*θd*<sup>1</sup>

compute the sign of *dx*<sup>∗</sup>

<sup>10</sup> It is required that *dn* < <sup>1</sup>

2[*θ*(1−*dn*)−*d*1]

*dn* .

provided that *dn* < −*H* + *δW* for *n* ≥ 2.

where *<sup>∂</sup>x*<sup>∗</sup>

*∂x*∗

We assume that threshold *T* is uncertain. This assumption reflects the parties' lack of knowledge of each other's response to any given territory claim. The threshold is uniformly distributed over the interval [0, 1], and this distribution is common knowledge. Any given realization of *T* represents a measure of the ex-ante probability of negotiation abortion. For instance, if *T* is close to zero, the probability that negotiations don't take place in this period is close to one. However, the realization of *T* is not known to the parties before they announce their positions.

The setting described above can be analyzed as a Bayesian game, in which the players' strategies are the public statements and the payoff of player *i* = *L*, *R*, at the beginning of period *t* is given by *Ui*(*xL*, *xR*) = *vui* [*y*(*xL*, *xR*)] + (1 − *v*)*di*1, where *v* = 1 represents a bargaining process leading to outcome *y*(*xL*, *xR*), and *v* = 0 means that the conflict persists (at least) during period *t*. Provided that *T* ∼ *U* [0, 1], the probability of conflict in period *t*, after the public statements (*xL*, *xR*) are made, can be expressed as:

$$p\left(\mathbf{x}\_{L\prime}\mathbf{x}\_{\mathcal{R}}\right) = \Pr(\boldsymbol{\upsilon} = \boldsymbol{0} \mid \mathbf{x}\_{L\prime}\mathbf{x}\_{\mathcal{R}}) = \mathbf{x}\_{L} - \mathbf{x}\_{\mathcal{R}}.\tag{5}$$

The probability that negotiations are held in period *t* is then given by 1 − *p* (*xL*, *xR*).

Uncertainty about the exact value of the threshold induces the following strategic situation: more extreme positions tend to favour a better outcome in a negotiation process, but as the players move towards the extremes the probability of aborting the negotiation process increases. Each player faces a trade off similar to the one faced by the bidders in a first price auction: A lower price is better if one gets the good, but lowering the price also lowers the probability of obtaining the good. Therefore, if a player makes a very tough public statement (to create a stronger bargaining position), negotiations hardly take place. We seek to analyze the Bayesian Nash equilibrium statements that emerge in this context.

The timing of this game is as follows: (i) players simultaneously announce bargaining positions *xL* and *xR*; (ii) the threshold is realized; (iii) each player receives his payoff. We look for the Bayesian Nash equilibrium of this game, defined as a pair of statements (*<sup>x</sup><sup>L</sup>*, *<sup>x</sup><sup>R</sup>*) such that *<sup>x</sup><sup>L</sup>* <sup>=</sup> arg max{*xL*} *<sup>E</sup>* [*UL*(*xL*, *<sup>x</sup><sup>R</sup>*)] and *<sup>x</sup><sup>R</sup>* <sup>=</sup> arg max{*xR*} *<sup>E</sup>* [*UR*(*<sup>x</sup><sup>L</sup>*, *xR*)] , where

$$E\left[\mathcal{U}\_{L}\left(\mathbf{x}\_{L\prime}\hat{\mathbf{x}}\_{R}\right)\right] = p\left(\mathbf{x}\_{L\prime}\hat{\mathbf{x}}\_{R}\right)d\_{L1} + \theta\_{L}\left[1 - p\left(\mathbf{x}\_{L\prime}\hat{\mathbf{x}}\_{R}\right)\right]y\left(\mathbf{x}\_{L\prime}\hat{\mathbf{x}}\_{R}\right),\tag{6}$$

and

$$\mathbb{E}\left[\mathcal{U}\_{\mathbb{R}}(\widehat{\mathbf{x}}\_{L},\mathbf{x}\_{\mathbb{R}})\right] = p\left(\widehat{\mathbf{x}}\_{L},\mathbf{x}\_{\mathbb{R}}\right)d\_{\mathbb{R}1} + \theta\_{\mathbb{R}}\left[1 - p\left(\widehat{\mathbf{x}}\_{L},\mathbf{x}\_{\mathbb{R}}\right)\right]\left[1 - y\left(\widehat{\mathbf{x}}\_{L},\mathbf{x}\_{\mathbb{R}}\right)\right].\tag{7}$$

The first order condition obtained from maximizing *<sup>E</sup>* [*UL*(*xL*, *<sup>x</sup><sup>R</sup>*)] with respect to *xL* implies the following equation:

$$\left[\theta\_L \left[1 - p\left(\mathbf{x}\_L, \hat{\mathbf{x}}\_R\right)\right] \frac{\partial y(\mathbf{x}\_L, \hat{\mathbf{x}}\_R)}{\partial \mathbf{x}\_L}\right] = \left[\theta\_L y(\mathbf{x}\_L, \hat{\mathbf{x}}\_R) - d\_{L1}\right] \frac{\partial p\left(\mathbf{x}\_L, \hat{\mathbf{x}}\_R\right)}{\partial \mathbf{x}\_L}.\tag{8}$$

If country *L* reports a higher *xL*, the probability that negotiations are held decrease, but the piece of territory obtained in case negotiations take place is higher. Therefore, Eq. (8) above represents the equality between the marginal costs of increasing *xL* (right hand term) and the marginal benefits of doing so (left hand term). The first order condition *<sup>∂</sup>E*[*UR*(*<sup>x</sup><sup>L</sup>*,*xR*)] *<sup>∂</sup>xR* = 0 admits the same interpretation.

From Eq. (8) it is implicit that *Wi* is treated as an exogenous parameter. The reason is that *Wi* represents the welfare at the beginning of the present period (*t*) and it also represents the 6 Will-be-set-by-IN-TECH

We assume that threshold *T* is uncertain. This assumption reflects the parties' lack of knowledge of each other's response to any given territory claim. The threshold is uniformly distributed over the interval [0, 1], and this distribution is common knowledge. Any given realization of *T* represents a measure of the ex-ante probability of negotiation abortion. For instance, if *T* is close to zero, the probability that negotiations don't take place in this period is close to one. However, the realization of *T* is not known to the parties before they announce

The setting described above can be analyzed as a Bayesian game, in which the players' strategies are the public statements and the payoff of player *i* = *L*, *R*, at the beginning of period *t* is given by *Ui*(*xL*, *xR*) = *vui* [*y*(*xL*, *xR*)] + (1 − *v*)*di*1, where *v* = 1 represents a bargaining process leading to outcome *y*(*xL*, *xR*), and *v* = 0 means that the conflict persists (at least) during period *t*. Provided that *T* ∼ *U* [0, 1], the probability of conflict in period *t*, after

The probability that negotiations are held in period *t* is then given by 1 − *p* (*xL*, *xR*).

Uncertainty about the exact value of the threshold induces the following strategic situation: more extreme positions tend to favour a better outcome in a negotiation process, but as the players move towards the extremes the probability of aborting the negotiation process increases. Each player faces a trade off similar to the one faced by the bidders in a first price auction: A lower price is better if one gets the good, but lowering the price also lowers the probability of obtaining the good. Therefore, if a player makes a very tough public statement (to create a stronger bargaining position), negotiations hardly take place. We seek to analyze

The timing of this game is as follows: (i) players simultaneously announce bargaining positions *xL* and *xR*; (ii) the threshold is realized; (iii) each player receives his payoff. We look for the Bayesian Nash equilibrium of this game, defined as a pair of statements (*<sup>x</sup><sup>L</sup>*, *<sup>x</sup><sup>R</sup>*) such that *<sup>x</sup><sup>L</sup>* <sup>=</sup> arg max{*xL*} *<sup>E</sup>* [*UL*(*xL*, *<sup>x</sup><sup>R</sup>*)] and *<sup>x</sup><sup>R</sup>* <sup>=</sup> arg max{*xR*} *<sup>E</sup>* [*UR*(*<sup>x</sup><sup>L</sup>*, *xR*)] , where

The first order condition obtained from maximizing *<sup>E</sup>* [*UL*(*xL*, *<sup>x</sup><sup>R</sup>*)] with respect to *xL* implies

If country *L* reports a higher *xL*, the probability that negotiations are held decrease, but the piece of territory obtained in case negotiations take place is higher. Therefore, Eq. (8) above represents the equality between the marginal costs of increasing *xL* (right hand term) and the marginal benefits of doing so (left hand term). The first order condition *<sup>∂</sup>E*[*UR*(*<sup>x</sup><sup>L</sup>*,*xR*)]

From Eq. (8) it is implicit that *Wi* is treated as an exogenous parameter. The reason is that *Wi* represents the welfare at the beginning of the present period (*t*) and it also represents the

*<sup>E</sup>* [*UL*(*xL*, *<sup>x</sup><sup>R</sup>*)] <sup>=</sup> *<sup>p</sup>* (*xL*, *<sup>x</sup><sup>R</sup>*) *dL*<sup>1</sup> <sup>+</sup> *<sup>θ</sup><sup>L</sup>* [<sup>1</sup> <sup>−</sup> *<sup>p</sup>* (*xL*, *<sup>x</sup><sup>R</sup>*)] *<sup>y</sup>*(*xL*, *<sup>x</sup><sup>R</sup>*), (6)

<sup>=</sup> [*θLy*(*xL*, *<sup>x</sup><sup>R</sup>*) <sup>−</sup> *dL*1]

*<sup>∂</sup><sup>p</sup>* (*xL*, *<sup>x</sup><sup>R</sup>*) *∂xL*

. (8)

*<sup>∂</sup>xR* = 0

*<sup>E</sup>* [*UR*(*<sup>x</sup><sup>L</sup>*, *xR*)] <sup>=</sup> *<sup>p</sup>* (*<sup>x</sup><sup>L</sup>*, *xR*) *dR*<sup>1</sup> <sup>+</sup> *<sup>θ</sup><sup>R</sup>* [<sup>1</sup> <sup>−</sup> *<sup>p</sup>* (*<sup>x</sup><sup>L</sup>*, *xR*)] [<sup>1</sup> <sup>−</sup> *<sup>y</sup>*(*<sup>x</sup><sup>L</sup>*, *xR*)] . (7)

*p* (*xL*, *xR*) = Pr(*v* = 0 | *xL*, *xR*) = *xL* − *xR*. (5)

the public statements (*xL*, *xR*) are made, can be expressed as:

the Bayesian Nash equilibrium statements that emerge in this context.

*<sup>θ</sup><sup>L</sup>* [<sup>1</sup> <sup>−</sup> *<sup>p</sup>* (*xL*, *<sup>x</sup><sup>R</sup>*)] *<sup>∂</sup>y*(*xL*, *<sup>x</sup><sup>R</sup>*)

*∂xL*

their positions.

and

the following equation:

admits the same interpretation.

welfare at the beginning of the next period (*t* + 1). If changing the statements *xL* and *xR* could change the equilibrium value of *Wi*, say to *W*� *<sup>i</sup>* �= *Wi*, then the stationary value of the social welfare would be *W*� *<sup>i</sup>* throughout all periods. The players, who are aware of the dynamics of the game, take the value of *Wi* as given.

Solving Eq. (8) for *xL* we obtain *xL*(*<sup>x</sup><sup>R</sup>*), the reaction function of country *<sup>L</sup>* to every possible given statement *<sup>x</sup><sup>R</sup>*. A similar routine leads us to *xR*(*<sup>x</sup><sup>L</sup>*), the reaction function of country *<sup>R</sup>* to any given *<sup>x</sup><sup>L</sup>*. In a Bayesian Nash equilibrium, the pair of statements (*<sup>x</sup><sup>L</sup>*, *<sup>x</sup><sup>R</sup>*) is such that *<sup>x</sup><sup>L</sup>* <sup>=</sup> *xL*(*<sup>x</sup><sup>R</sup>*) and *<sup>x</sup><sup>R</sup>* <sup>=</sup> *xR*(*<sup>x</sup><sup>L</sup>*).

In order to provide an explicit expression for the pair (*<sup>x</sup><sup>L</sup>*, *<sup>x</sup><sup>R</sup>*) we now make some symmetry assumptions. In particular, we assume that both countries are identical in their valuations of territory and in the conflict costs. Namely, *θ<sup>L</sup>* = *θ<sup>R</sup>* = *θ* and *HL* = *HR* = *H*. In this scenario, we have *<sup>x</sup><sup>L</sup>* <sup>=</sup> <sup>1</sup> <sup>−</sup> *<sup>x</sup><sup>R</sup>*. Then, it follows that *WL* <sup>=</sup> *WR* <sup>=</sup> *<sup>W</sup>* and hence *dLn* <sup>=</sup> *dRn* <sup>=</sup> *dn*. In a symmetric equilibrium we have *<sup>x</sup><sup>L</sup>* <sup>=</sup> *<sup>x</sup>*∗(*n*, *<sup>H</sup>*, *<sup>W</sup>*) and *<sup>x</sup><sup>R</sup>* <sup>=</sup> <sup>1</sup> <sup>−</sup> *<sup>x</sup>*∗(*n*, *<sup>H</sup>*, *<sup>W</sup>*). We use Eq. (8) above to compute

$$\mathbf{x}^\*(\mathfrak{n}, H, W) = \frac{\theta \left[1 - 2d\_{\mathfrak{n}}(H, W)\right]}{2\left[\theta(1 - d\_{\mathfrak{n}}(H, W)) - d\_1(H, W)\right]}.\tag{9}$$

Next we analyze the dependence of *x*∗(.) on parameters *H* and *n*. We simplify notation by writing *x*∗ instead of *x*∗(*n*, *H*, *W*) and *dn* instead of *dn*(*H*, *W*). Using the expression for *x*∗ in Eq. (9) we compute

$$\frac{d\mathbf{x}^\*}{dH} = \underbrace{\frac{\partial \mathbf{x}^\*}{\partial d\_n}\frac{\partial d\_n}{\partial H}}\_{} + \underbrace{\frac{\partial \mathbf{x}^\*}{\partial H}}\_{}$$

 Indirect effect Direct effect

where *<sup>∂</sup>x*<sup>∗</sup> *<sup>∂</sup>dn* <sup>=</sup> <sup>−</sup> *<sup>θ</sup>*<sup>2</sup>+2*θd*<sup>1</sup> 2[*θ*(1−*dn*)−*d*1] <sup>2</sup> , *<sup>∂</sup>dn <sup>∂</sup><sup>H</sup>* = − 1−*δ<sup>n</sup>* 1−*δ* < 0, and *<sup>∂</sup>x*<sup>∗</sup> *<sup>∂</sup><sup>H</sup>* <sup>=</sup> <sup>−</sup>*θ*[1−2*dn*] 2[*θ*(1−*dn*)−*d*1] 2 .

The indirect effect reflects the equilibrium reaction of the pre-bargaining claim to an increase in conflict costs through the influence that such costs have on the disagreement point. Under the mild assumption that *θ* + 2*H* > 2*δW* (i.e. if conflict costs are high enough, and/or the valuation of land is high enough, and/or the discount factor is low enough), the derivative *∂x*∗ *<sup>∂</sup>dn* is negative. Therefore, the indirect effect is positive provided that *<sup>∂</sup>dn <sup>∂</sup><sup>H</sup>* < 0.

The direct effect operates in the opposite direction. If the conflict costs increase, and the disagreement point *dn* is low enough10, then the threat of a conflict dissuades from making aggressive statements (e.g. to claim the entire territory). In general, it cannot be established whether the direct effect predominates or not over the indirect effect. We show later, in a more restricted context, that the response of the equilibrium claims to a change in *H* is non monotone. Specifically, *x*∗ is decreasing if *H* is low enough and increasing if *H* is high enough.

If the parties disagree in negotiations, the conflict lasts for *n* periods. Next we analyze the influence of parameter *n* in the behavior of the parties before negotiations. In particular, we compute the sign of *dx*<sup>∗</sup> *dn* .

$$\frac{d\mathbf{x}^\*}{dn} = \frac{\partial \mathbf{x}^\*}{\partial d\_n} \frac{\partial d\_n}{\partial n}.$$

<sup>10</sup> It is required that *dn* < <sup>1</sup> <sup>2</sup> . Observe that for this inequality to hold it is sufficient that 1 + 2*H* > 2*δW*, provided that *dn* < −*H* + *δW* for *n* ≥ 2.

values for parameters *θ* and *δ*. Namely, *θ* = 1 and *δ* = 0.5. Then, we write:

costs *H*.

Pr(*Y* = *k*) = (*p*∗)

**5.1 Case 1 (***n* = 1**)**

Then, by Eq. (14) we obtain *W* = <sup>1</sup>

*p*∗ to changes in *H* follows a similar pattern.

provided that the equilibrium statements are equal to <sup>1</sup>

country is <sup>−</sup>*<sup>H</sup>* <sup>+</sup> *<sup>δ</sup>W*. The Nash bargaining outcome is *<sup>y</sup>* <sup>=</sup> <sup>1</sup>

2 .

probability one, and each country enjoys a social welfare equal to <sup>1</sup>

<sup>11</sup> Observe this probability could be greater than zero if we considered *θ* > *<sup>d</sup>*<sup>1</sup>

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

Social Welfare and the Emergence of Negotiations 211

Next we discuss how the social welfare varies when *n* changes. In particular we study the cases where *n* = 1 (case 1), *n* = 2 (case 2), and *n* → ∞ (case 3). Case 1 corresponds to a situation in which the players receive the payoffs associated with the status quo when they fail to reach an agreement. Cases 2 and 3 should be interpreted as if players had an outside option whose effects imply facing *n* additional periods of conflict. For each case, we compute both the equilibrium statement *x*∗ and the probability of conflict *p*∗ as a function of the conflict

We obtain that in case 1 the social welfare is constant, but in cases 2 and 3, it is strictly decreasing in *H*. The effect of *H* on *x*∗ and *p*∗ is non monotonic in cases 2 and 3. For low values of *H*, the statement *x*∗ is decreasing in *H*. It reaches a minimum, and then increases in *H* and converges asymptotically to a constant value. As long as *p*<sup>∗</sup> = 2*x*<sup>∗</sup> − 1, the response of

We can also calculate how long will the conflict last, for different values of *H*. Notice that, if *p*∗ is the probability of conflict in period *t*, the number of periods of conflict (until a negotiation process is undertaken) is a random variable *Y* that follows a geometric distribution. Specifically, the probability that there are *k* periods of conflict is given by

*<sup>k</sup>* (<sup>1</sup> <sup>−</sup> *<sup>p</sup>*∗), and the expected value of *<sup>Y</sup>* is *<sup>E</sup>*(*Y*) = *<sup>p</sup>*<sup>∗</sup>

*p*<sup>∗</sup> as a function of *H* for the cases *n* = 1, *n* = 2 and *n* → ∞, with *θ* = 1 and *δ* = 0.5. Then, we estimate the number of periods the conflict is expected to last as a function of *H* for each given value of *n*. Moreover, the expected cost of a conflict can also be computed as *E*(*Y*)*H*.

In this case, *dn* = *d*<sup>1</sup> = −*H* + *δW*. This means that the utility achieved if negotiations break down is equal to the utility achieved in the case that negotiations do not take place and the conflict situation persists along period *t*. The probability of conflict (Eq. (11)) is zero<sup>11</sup>

It is worth to mention that neither the pre-bargaining claims nor the social welfare are affected by the conflict costs *H*. This is due to the fact that such costs exert exactly the same influence on the two possible outcomes (conflict or negotiation) that may arise after *xL* and *xR* are announced and the threshold *T* is realized. In the first case (conflict), the payoff for each

parties would obtain *d*<sup>1</sup> = −*H* + *δW*. This means that, if the parties broke down negotiations, the disagreement payoffs would equal the cost of suffering conflict during one more period. In equilibrium, both parties claim one half of the territory, a negotiation process emerges with

shared and there is no conflict. The expected number of periods the conflict will last is zero, and the expected cost of the conflict is also zero. As we show below, both the likelihood of a

1 − *dn* − *d*<sup>1</sup>

2

. (14)

<sup>1</sup>−*p*<sup>∗</sup> . Below we compute

<sup>2</sup> (the reader can check it in Eq. (9)).

<sup>2</sup> , and in case of disagreement the

1−*d*<sup>1</sup> .

<sup>2</sup> . The territory is equally

From Eq. (2) we know that *<sup>∂</sup>dn <sup>∂</sup><sup>n</sup>* <sup>&</sup>lt; 0. We also know that *<sup>∂</sup>x*<sup>∗</sup> *<sup>∂</sup>dn* < 0 whenever *θ* + 2*H* > 2*δW*. Hence, we can establish that *dx*<sup>∗</sup> *dn* > 0, i.e., the share of territory initially claimed is larger if *n* increases (and so do the costs from disagreement).

The likelihood of being in conflict during period *t* is also affected by changes in *H* and *n*. Evaluated at equilibrium, this probability is given by *p*<sup>∗</sup> = *p* (*x*∗, 1 − *x*∗) = 2*x*<sup>∗</sup> − 1. Hence, *dp*∗ *dH* <sup>=</sup> <sup>2</sup> *dx*<sup>∗</sup> *dH* and *dp*<sup>∗</sup> *dn* <sup>=</sup> <sup>2</sup> *dx*<sup>∗</sup> *dn* > 0. If the utility obtained in case of disagreement lowers (higher *n*), the probability of conflict increases. This effect becomes apparent in the next section, in which we investigate the relationship between *p*∗ and *H* under different values of *n*.

Let us now focus our attention on the equilibrium value for the generalized Nash bargaining outcome. First, observe that in our symmetric framework it holds that *<sup>x</sup><sup>L</sup>* <sup>=</sup> <sup>1</sup> <sup>−</sup> *<sup>x</sup><sup>R</sup>* <sup>=</sup> *<sup>x</sup>*∗. Then, the bargaining power of both countries is the same, i.e., *α*∗ *<sup>L</sup>* <sup>=</sup> *<sup>x</sup><sup>L</sup> <sup>x</sup><sup>L</sup>*+1−*<sup>x</sup><sup>R</sup>* <sup>=</sup> *<sup>x</sup>*<sup>∗</sup> <sup>2</sup>*x*<sup>∗</sup> <sup>=</sup> <sup>1</sup> <sup>2</sup> , and *α*∗ *<sup>R</sup>* <sup>=</sup> <sup>1</sup>−*<sup>x</sup><sup>R</sup> <sup>x</sup><sup>L</sup>*+1−*<sup>x</sup><sup>R</sup>* <sup>=</sup> *<sup>x</sup>*<sup>∗</sup> <sup>2</sup>*x*<sup>∗</sup> <sup>=</sup> <sup>1</sup> <sup>2</sup> . Plugging these values into Eq. (4) we obtain *<sup>y</sup>*(*<sup>x</sup><sup>L</sup>*, *<sup>x</sup><sup>R</sup>*) = <sup>1</sup> <sup>2</sup> (1 − *dn*) + <sup>1</sup> <sup>2</sup> *dn* <sup>=</sup> <sup>1</sup> <sup>2</sup> . Not surprisingly, the Nash bargaining solution evaluated at the equilibrium statements involves that the territory is equally shared between both countries.

It is worth mentioning that the equilibrium statements computed above eventually depend on the equilibrium value for *W*. We derive endogenously the stationary value of *W* in the next section.

#### **5. Social welfare, conflict, and the emergence of negotiations**

The purpose of this section is twofold: First, it is devoted to compute the welfare expected from the equilibrium play of the game at the beginning of each period. Secondly, it includes estimations on the way how *x*∗ and *p*∗ respond to *H* under different scenarios (*n* = 1, *n* = 2 and *n* → ∞).

The value for *W* is implicitly defined in the following expression:

$$\mathcal{W} = \mathbb{E}\left\{ \mathbb{U}\_L \left[ \mathbf{x}^\*(\mathfrak{n}, H, \mathcal{W}), 1 - \mathbf{x}^\*(\mathfrak{n}, H, \mathcal{W}) \right] \right\}. \tag{10}$$

In order to solve Eq. (10) above for *W*, we use the probability of conflict at equilibrium, given by:

$$p^\* = \frac{d\_1 - \theta d\_n}{\theta (1 - d\_n) - d\_1}.\tag{11}$$

Taking into account that *<sup>y</sup>*(*x*∗, 1 <sup>−</sup> *<sup>x</sup>*∗) = <sup>1</sup> <sup>2</sup> , we rewrite Eq. (10) as:

$$\mathcal{W} = p^\* d\_1 + (1 - p^\*) \frac{1}{2}. \tag{12}$$

Plugging the expression for *p*∗ in Eq. (11) into Eq. (12) we obtain:

$$\mathcal{W} = \frac{(d\_1 - 1 - \theta d\_n) \, d\_1 + \frac{\theta}{2}}{\theta (1 - d\_n) - d\_1}. \tag{13}$$

We just need to substitute *d*<sup>1</sup> = −*H* + *δW* and *dn* = − <sup>1</sup>−*δ<sup>n</sup>* 1−*δ H* + *δnW* in Eq. (13) and solve it for *W*. In order to provide a tractable expression for the social welfare, we assume specific 8 Will-be-set-by-IN-TECH

The likelihood of being in conflict during period *t* is also affected by changes in *H* and *n*. Evaluated at equilibrium, this probability is given by *p*<sup>∗</sup> = *p* (*x*∗, 1 − *x*∗) = 2*x*<sup>∗</sup> − 1. Hence,

*n*), the probability of conflict increases. This effect becomes apparent in the next section, in

Let us now focus our attention on the equilibrium value for the generalized Nash bargaining outcome. First, observe that in our symmetric framework it holds that *<sup>x</sup><sup>L</sup>* <sup>=</sup> <sup>1</sup> <sup>−</sup> *<sup>x</sup><sup>R</sup>* <sup>=</sup> *<sup>x</sup>*∗.

It is worth mentioning that the equilibrium statements computed above eventually depend on the equilibrium value for *W*. We derive endogenously the stationary value of *W* in the next

The purpose of this section is twofold: First, it is devoted to compute the welfare expected from the equilibrium play of the game at the beginning of each period. Secondly, it includes estimations on the way how *x*∗ and *p*∗ respond to *H* under different scenarios (*n* = 1, *n* = 2

In order to solve Eq. (10) above for *W*, we use the probability of conflict at equilibrium, given

*θ*(1 − *dn*) − *d*<sup>1</sup>

<sup>2</sup> , we rewrite Eq. (10) as:

1 2

2

<sup>1</sup>−*δ<sup>n</sup>* 1−*δ* 

*<sup>p</sup>*<sup>∗</sup> <sup>=</sup> *<sup>d</sup>*<sup>1</sup> <sup>−</sup> *<sup>θ</sup>dn*

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

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

it for *W*. In order to provide a tractable expression for the social welfare, we assume specific

*θ*(1 − *dn*) − *d*<sup>1</sup>

which we investigate the relationship between *p*∗ and *H* under different values of *n*.

statements involves that the territory is equally shared between both countries.

Then, the bargaining power of both countries is the same, i.e., *α*∗

**5. Social welfare, conflict, and the emergence of negotiations**

The value for *W* is implicitly defined in the following expression:

Plugging the expression for *p*∗ in Eq. (11) into Eq. (12) we obtain:

We just need to substitute *d*<sup>1</sup> = −*H* + *δW* and *dn* = −

*<sup>∂</sup>dn* < 0 whenever *θ* + 2*H* > 2*δW*.

*<sup>x</sup><sup>L</sup>*+1−*<sup>x</sup><sup>R</sup>* <sup>=</sup> *<sup>x</sup>*<sup>∗</sup>

. (11)

. (12)

. (13)

*H* + *δnW* in Eq. (13) and solve

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

<sup>2</sup> , and

<sup>2</sup> (1 −

*dn* > 0, i.e., the share of territory initially claimed is larger if *n*

*<sup>L</sup>* <sup>=</sup> *<sup>x</sup><sup>L</sup>*

*dn* > 0. If the utility obtained in case of disagreement lowers (higher

<sup>2</sup> . Plugging these values into Eq. (4) we obtain *<sup>y</sup>*(*<sup>x</sup><sup>L</sup>*, *<sup>x</sup><sup>R</sup>*) = <sup>1</sup>

*W* = *E* {*UL* [*x*∗(*n*, *H*, *W*), 1 − *x*∗(*n*, *H*, *W*)]} . (10)

<sup>2</sup> . Not surprisingly, the Nash bargaining solution evaluated at the equilibrium

*<sup>∂</sup><sup>n</sup>* <sup>&</sup>lt; 0. We also know that *<sup>∂</sup>x*<sup>∗</sup>

From Eq. (2) we know that *<sup>∂</sup>dn*

Hence, we can establish that *dx*<sup>∗</sup>

*dH* and *dp*<sup>∗</sup>

*<sup>x</sup><sup>L</sup>*+1−*<sup>x</sup><sup>R</sup>* <sup>=</sup> *<sup>x</sup>*<sup>∗</sup>

<sup>2</sup> *dn* <sup>=</sup> <sup>1</sup>

*dp*∗ *dH* <sup>=</sup> <sup>2</sup> *dx*<sup>∗</sup>

*α*∗

*dn*) + <sup>1</sup>

section.

and *n* → ∞).

by:

*<sup>R</sup>* <sup>=</sup> <sup>1</sup>−*<sup>x</sup><sup>R</sup>*

increases (and so do the costs from disagreement).

*dn* <sup>=</sup> <sup>2</sup> *dx*<sup>∗</sup>

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

Taking into account that *<sup>y</sup>*(*x*∗, 1 <sup>−</sup> *<sup>x</sup>*∗) = <sup>1</sup>

values for parameters *θ* and *δ*. Namely, *θ* = 1 and *δ* = 0.5. Then, we write:

$$\mathcal{W} = \frac{\left(d\_1 - d\_{\eta} - 1\right)d\_1 + \frac{1}{2}}{1 - d\_{\eta} - d\_1}.\tag{14}$$

Next we discuss how the social welfare varies when *n* changes. In particular we study the cases where *n* = 1 (case 1), *n* = 2 (case 2), and *n* → ∞ (case 3). Case 1 corresponds to a situation in which the players receive the payoffs associated with the status quo when they fail to reach an agreement. Cases 2 and 3 should be interpreted as if players had an outside option whose effects imply facing *n* additional periods of conflict. For each case, we compute both the equilibrium statement *x*∗ and the probability of conflict *p*∗ as a function of the conflict costs *H*.

We obtain that in case 1 the social welfare is constant, but in cases 2 and 3, it is strictly decreasing in *H*. The effect of *H* on *x*∗ and *p*∗ is non monotonic in cases 2 and 3. For low values of *H*, the statement *x*∗ is decreasing in *H*. It reaches a minimum, and then increases in *H* and converges asymptotically to a constant value. As long as *p*<sup>∗</sup> = 2*x*<sup>∗</sup> − 1, the response of *p*∗ to changes in *H* follows a similar pattern.

We can also calculate how long will the conflict last, for different values of *H*. Notice that, if *p*∗ is the probability of conflict in period *t*, the number of periods of conflict (until a negotiation process is undertaken) is a random variable *Y* that follows a geometric distribution. Specifically, the probability that there are *k* periods of conflict is given by Pr(*Y* = *k*) = (*p*∗) *<sup>k</sup>* (<sup>1</sup> <sup>−</sup> *<sup>p</sup>*∗), and the expected value of *<sup>Y</sup>* is *<sup>E</sup>*(*Y*) = *<sup>p</sup>*<sup>∗</sup> <sup>1</sup>−*p*<sup>∗</sup> . Below we compute *p*<sup>∗</sup> as a function of *H* for the cases *n* = 1, *n* = 2 and *n* → ∞, with *θ* = 1 and *δ* = 0.5. Then, we estimate the number of periods the conflict is expected to last as a function of *H* for each given value of *n*. Moreover, the expected cost of a conflict can also be computed as *E*(*Y*)*H*.

#### **5.1 Case 1 (***n* = 1**)**

In this case, *dn* = *d*<sup>1</sup> = −*H* + *δW*. This means that the utility achieved if negotiations break down is equal to the utility achieved in the case that negotiations do not take place and the conflict situation persists along period *t*. The probability of conflict (Eq. (11)) is zero<sup>11</sup> provided that the equilibrium statements are equal to <sup>1</sup> <sup>2</sup> (the reader can check it in Eq. (9)). Then, by Eq. (14) we obtain *W* = <sup>1</sup> 2 .

It is worth to mention that neither the pre-bargaining claims nor the social welfare are affected by the conflict costs *H*. This is due to the fact that such costs exert exactly the same influence on the two possible outcomes (conflict or negotiation) that may arise after *xL* and *xR* are announced and the threshold *T* is realized. In the first case (conflict), the payoff for each country is <sup>−</sup>*<sup>H</sup>* <sup>+</sup> *<sup>δ</sup>W*. The Nash bargaining outcome is *<sup>y</sup>* <sup>=</sup> <sup>1</sup> <sup>2</sup> , and in case of disagreement the parties would obtain *d*<sup>1</sup> = −*H* + *δW*. This means that, if the parties broke down negotiations, the disagreement payoffs would equal the cost of suffering conflict during one more period. In equilibrium, both parties claim one half of the territory, a negotiation process emerges with probability one, and each country enjoys a social welfare equal to <sup>1</sup> <sup>2</sup> . The territory is equally shared and there is no conflict. The expected number of periods the conflict will last is zero, and the expected cost of the conflict is also zero. As we show below, both the likelihood of a

<sup>11</sup> Observe this probability could be greater than zero if we considered *θ* > *<sup>d</sup>*<sup>1</sup> 1−*d*<sup>1</sup> .

the observed evidence that even if there is room for a mutually advantageous negotiation,

Social Welfare and the Emergence of Negotiations 213

Now we compute the equilibrium value of *x*∗ as a function of the conflict costs. By substituting

It is easy to check that *x*∗ is strictly decreasing in *H* across the range (0.125, 0.569 5). The minimum value of *x*∗ is 0.569 5, taken at *H* = 0.572 87. In this case, *p*∗ = 0.139, and the conflict is expected to last 0.161 44 periods. The expected cost of the conflict is 0.09 248 4. The maximum statement is made when *H* = 0.125. This statement is equal to 0.666 67, associated with a probability of conflict of 0.333 34. In this case, the expected number of periods of conflict is 0.5, and the expected cost of the conflict is 0.062 5. When *H* → ∞, we have *x*<sup>∗</sup> → 0.585 79. The probability of conflict is 0.171 58 and the expected duration of the conflict is 0.207 12.

> <sup>1</sup>−*δ<sup>n</sup>* 1−*δ*

*<sup>W</sup>* <sup>=</sup> <sup>2</sup>*<sup>H</sup>* <sup>−</sup> <sup>2</sup>

<sup>1</sup>−*<sup>δ</sup>* in the disagreement utility.

inequality we find that it holds for values of *H* such that 6*H* > 2

3 2 3 2

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

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

somehow extreme situation in which breaking down negotiations involves that conflict will

*<sup>W</sup>* <sup>=</sup> 0.25*W*<sup>2</sup> <sup>−</sup> *<sup>H</sup>*<sup>2</sup> <sup>+</sup> <sup>1</sup>

<sup>1</sup>−*<sup>δ</sup> <sup>H</sup>* into Eq. (14) to obtain:

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

<sup>√</sup><sup>15</sup> <sup>−</sup> <sup>1</sup>

12

1 + 4*H*

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

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

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

*<sup>H</sup>* <sup>+</sup> *<sup>δ</sup>nW* <sup>=</sup> <sup>−</sup> <sup>1</sup>

2

8 1 2 − 1 

< 0. This suggests that the magnitude of the social welfare losses

whenever *H* > −0.5*W*. By substituting *W* (Eq. (21)) in the latter

<sup>1</sup> <sup>+</sup> <sup>3</sup>*<sup>H</sup>* <sup>−</sup> 0.5*<sup>W</sup>* . (20)

<sup>4</sup> . Moreover, for all *<sup>H</sup>* <sup>≥</sup> <sup>1</sup>

<sup>1</sup> <sup>+</sup> <sup>3</sup>*<sup>H</sup>* <sup>−</sup> 0.5*<sup>W</sup>* . (22)

<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

4 1 2 − 1 

(19)

<sup>1</sup>−*<sup>δ</sup> <sup>H</sup>*. We are dealing with the

. (21)

<sup>8</sup> it holds

− 1. This

4 1 2 − 1

<sup>1</sup> <sup>+</sup> 2. 285 7*<sup>H</sup>* <sup>+</sup> 0.285 72

<sup>1</sup> <sup>+</sup> 1. 428 5*<sup>H</sup>* <sup>+</sup> 0.428 57

equilibrium statements may abort the emergence of the process.

*W* in Eq. (17) into Eq. (18) we obtain:

**5.3 Case 3 (***n* → ∞**)**

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

In this third case, we have that lim*n*→<sup>∞</sup> <sup>−</sup>

*dH* 

last forever. We substitute *dn* <sup>=</sup> <sup>−</sup> <sup>1</sup>

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

It is easy to see that *dW*

of magnitude <sup>−</sup> <sup>1</sup>

We have that *x*<sup>∗</sup> ∈

*<sup>n</sup>*→<sup>∞</sup> <sup>&</sup>lt; *dW dH n*=2

The equilibrium statements are now:

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

that *dW dH* 

negotiation process to emerge and the social welfare are dependent on the conflict costs when *n* ≥ 2.

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

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 conflict must be positive.

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

$$\mathcal{W} = \frac{(d\_1 - d\_2 - 1) \, d\_1 + \frac{1}{2}}{1 - d\_2 - d\_1}. \tag{15}$$

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, we obtain:

$$W = \frac{(0.25W + 0.5H)\left(-H + 0.5W\right) + \frac{1}{2}}{1 + 2.5H - 0.75W} \tag{16}$$

There are two values of *W* that satisfy Eq. (16), but only one of them satisfies our assumption that *θ* + 2*H* > 2*δW*. This value is:

$$W = 1.428\,6H - 0.571\,43\left[\left(5H + 8H^2 - \frac{3}{4}\right)^{\frac{1}{2}} - 1\right].\tag{17}$$

Notice<sup>12</sup> that *dW dH n*=2 <sup>&</sup>lt; 0 for all *<sup>H</sup>* <sup>≥</sup> <sup>1</sup> 8 .

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 social welfare falls to a lower level.

Next we check that the equilibrium statements are higher than <sup>1</sup> <sup>2</sup> . For this purpose, we substitute the value for the social welfare in Eq. (16) into Eq. (9) to obtain:

$$\alpha^\* = \frac{1}{2} \frac{1 + 3H - 0.5W}{1 + 2.5H - 0.75W}. \tag{18}$$

From Eq. (18) it is straightforward that *x*<sup>∗</sup> ∈ <sup>1</sup> <sup>2</sup> , 1 provided that *H* > 0, *W* > 0 and <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> <sup>8</sup> . The fact that *<sup>x</sup>*<sup>∗</sup> <sup>&</sup>gt; <sup>1</sup> <sup>2</sup> implies *p*<sup>∗</sup> = 2*x*<sup>∗</sup> − 1 > 0. An important conclusion is that if *<sup>n</sup>* <sup>=</sup> 2, and *<sup>H</sup>* <sup>≥</sup> <sup>1</sup> <sup>8</sup> the probability that countries continue with the conflict instead of initiating a negotiation process is positive. This result is in accordance with

<sup>12</sup> A solution for *<sup>W</sup>* exists whenever *<sup>H</sup>* <sup>∈</sup>/ − 3 4 , 1 8 .

the observed evidence that even if there is room for a mutually advantageous negotiation, equilibrium statements may abort the emergence of the process.

Now we compute the equilibrium value of *x*∗ as a function of the conflict costs. By substituting *W* in Eq. (17) into Eq. (18) we obtain:

$$\chi^\* = \frac{1}{2} \frac{1 + 2.2857H + 0.28572\left(\left(5H + 8H^2 - \frac{3}{4}\right)^{\frac{1}{2}} - 1\right)}{1 + 1.4285H + 0.42857\left(\left(5H + 8H^2 - \frac{3}{4}\right)^{\frac{1}{2}} - 1\right)}\tag{19}$$

It is easy to check that *x*∗ is strictly decreasing in *H* across the range (0.125, 0.569 5). The minimum value of *x*∗ is 0.569 5, taken at *H* = 0.572 87. In this case, *p*∗ = 0.139, and the conflict is expected to last 0.161 44 periods. The expected cost of the conflict is 0.09 248 4. The maximum statement is made when *H* = 0.125. This statement is equal to 0.666 67, associated with a probability of conflict of 0.333 34. In this case, the expected number of periods of conflict is 0.5, and the expected cost of the conflict is 0.062 5. When *H* → ∞, we have *x*<sup>∗</sup> → 0.585 79. The probability of conflict is 0.171 58 and the expected duration of the conflict is 0.207 12.
