Bargaining power in crisis bargaining

Abstract

A large body of game-theoretic work examines the process by which uncertainty can lead to inefficient war. In a typical crisis bargaining model, players negotiate according to a pre-specified game form and no player has the ability to change the rules of the game. However, when one of the parties has full bargaining power and is able to set the rules of the game on her own, the game itself becomes an endogenous decision variable. I formulate this problem in a principal-agent framework. I show that both the likelihood of costly war and the exact mechanism that yields it depend on the nature of the informational problem and the identity of the informed player.

Appendices

Appendix

This appendix provides the analyses of the two equilibria of the take-it-or-leave-it offer game in the main text. It also provides the analyses of private values with uninformed principal, private values with informed principal, common values with uninformed principal and common values with informed principal. The first three are straightforward applications. I include them for the sake of completeness of the theory.

Take-it-or-leave-it offer

1.1 Signalling equilibrium

First, I will show that D’s strategy is optimal given his beliefs.

Rejecting any offer $t < t_{h}$ is optimal for D since he can guarantee at least $t_{h}$ by rejecting it.

Suppose that D receives an offer of $t \ge t_{l}.$ Then he believes that $p=p_{l}$ with probability 1. Accepting the offer is optimal for him because rejecting it provides him with an expected payoff of $t_{l}=1-p_{l}-c_{D}$ given his beliefs.

If D receives an offer of $t\in (t_{h},t_{l})$, then he believes that $p=p_{l}$ and his expected payoff from rejecting t is $t_{l}$, which is greater than t. So rejecting t is optimal.

If D receives an offer of $t_{h},$ then he believes that $p=p_{h}$ with probability 1. Then he is indifferent between accepting and rejecting the offer, so any mixed strategy is optimal for him.

By the definition of $\alpha$, $S_{l}$ is indifferent between offering $t_{l}$ and $t_{h}$:

$$\begin{aligned} 1-t_{l} = \alpha (p_{l} - c_S) + (1 - \alpha )(1-t_{h}). \end{aligned}$$

So it is optimal for $S_{l}$ to offer $t_{l}$. Since $p_{h}>p_{l}$, this equality implies

$$\begin{aligned} 1-t_{l} < \alpha (p_{h} - c_S) + (1-\alpha )(1-t_{h}) \end{aligned}$$

so that it is optimal for $S_{h}$ to offer $t_{h}$. D’s beliefs are consistent with S’s strategy on the equilibrium path.

1.2 Bluffing equilibrium

First, I will show that D’s strategy is optimal given his beliefs. Suppose that D receives an offer of $t \ge t_{l}.$ Then he believes that $p=p_{l}$ with probability 1. Accepting the offer is optimal for him because rejecting it provides him with an expected payoff of $t_{l}=p_{l}-c_{D}$ given his beliefs.

If D receives an offer of $t\in (t_{h},t_{l})$, then he believes that $p=p_{l}$ and his expected payoff from rejecting t is $t_{l}$, which is greater than t. So rejecting t is optimal.

If D receives an offer of $t_{h},$ then he believes that $p=p_{h}$ with probability $\phi .$ To reject this offer with probability $\alpha \in (0,1),$ he must be indifferent between accepting and rejecting the offer:

$$\begin{aligned} t_{h}=\phi (1-p_{h}-c_{D})+(1-\phi )(1-p_{l}-c_{D}), \end{aligned}$$

where the left hand side is D’s payoff from accepting the offer and the right hand side is his expected payoff from rejecting it given his beliefs. This is satisfied by the definition of $t_{h}$, so it is optimal for D to mix between accepting and rejecting the offer of $t_{h}$.

If D receives an offer of $t < t_{h}$, he believes that $p=p_{h}$ with probability 1 and his expected payoff from rejecting t is $t_{h}$, which is greater than t. So rejecting t is optimal.

Next I will show that S’s strategy is optimal given D’s strategy and beliefs. Since D accepts any offer $t \ge t_{l}$ with probability 1, offering $t > t_{l}$ is not optimal for any type of S.

Suppose $p=p_{l}.$ If $S_{l}$ offers $t\in [0,t_{l})\backslash \{t_{h}\}$ then D fights with probability 1 so $S_{l}$’s payoff from offering t is $p_{l}-c_{S}.$ Her payoff from offering $t_{l}$ is $p_{l}+c_{D},$ since D accepts it with probability 1. So offering $t\in [0,t_{l})\backslash \{t_{h}\}$ with positive probability cannot be optimal for $S_{l}$. This implies that $S_{l}$ offers either $t_{l}$ or $t_{h}.$ For $S_{l}$ to bluff by offering $t_{h}$ with positive probability of $\beta \in (0,1)$, she must be indifferent between offering $t_{h}$ and $t_{l}$:

$$\begin{aligned} p_{l}+c_{D}=\alpha (p_{l}-c_S)+(1-\alpha )(1-t_{h}), \end{aligned}$$

where the left hand side is $S_{l}$’s payoff from offering $t_{l},$ which D accepts with probability 1, and the right hand side is her expected payoff from offering $t_{h},$ which D rejects with probability $\alpha .$ This equality holds by definition of $\alpha$ so it is optimal for $S_{l}$ to bluff with positive probability.

Suppose that $p=p_{h}.$ If $S_{h}$ offers $t_{h},$ her payoff is

$$\begin{aligned} \alpha (p_{h}-c_S)+(1-\alpha )(1-t_{h}), \end{aligned}$$

which is greater than $1-t_{l}$ by the construction of $\alpha$ and $p_{h}>p_{l}$. So $S_{h}$ does not offer any t such that $t \ge t_{l}$. Since D rejects any offer $t\in [0,t_{l})\backslash \{t_{h}\}$, $S_{h}$’s expected payoff from any such offer is $p_{h}-c_S$. Then offering $t_{h}$ is optimal for $S_{h}$ if

$$\begin{aligned}&\alpha (p_{h}-c_S)+(1-\alpha )(1-t_{h}) \ge p_{h}-c_S \\&\quad \iff 1-t_{h} = p_{l} + c_D + \phi (p_{h}-p_{l}) \ge p_{h}-c_S \\&\quad \iff c_S + c_D \ge (1-\phi )(p_{h}-p_{l}) = \frac{(1-\pi )\beta }{\pi + (1-\pi )\beta }. \end{aligned}$$

This inequality holds for small enough $\beta$.

Finally, I confirm the consistency of D’s beliefs on the equilibrium path. S offers $t_{l}$ only when $p=p_{l},$ so it must be the case that $p=p_{l}$ after observing $t_{l}.$ Since both types of S offer $t_{h}$, D’s belief after receiving $t_{h}$ must follow Bayes’ rule:

$$\begin{aligned} \Pr (p=p_{h}|t=t_{h})=\phi =\frac{\pi }{\pi +(1-\pi )\beta }, \end{aligned}$$

where, given the equilibrium strategies, the numerator is the ex ante probability that $t_{h}$ will be offered by $S_{h}$ and the denominators is the ex ante probability that $t=t_{h}$ will be offered by either type. Thus, D’s beliefs are consistent with the Bayes’ rule on the equilibrium path. Since no offer of $t\in [0,1]\setminus \{t_{l},t_{h} \}$ will be made on the equilibrium path, D’s beliefs after receiving such an offer can be arbitrary.

Private values: cost of war

Assume that D’s cost of war is his private information. It is common knowledge that it is either low $c_{D}=c_{l}$ with probability $\pi$, or high $c_{D}=c_{h}>c_{l}$ with probability $1-\pi .$ I will refer to D as $D_h$ if $c_{D}=c_{h}$ and as $D_l$ if $c_{D}=c_{l}$.

1.1 Private values with uninformed principal

For the time being, I assume that S’s cost of war is common knowledge. However, I will argue later that the analysis and the results remain the same when S’s cost of war is her private information. Therefore, the identity of the informed player does not affect either the methodology or the results in the case of private values.

A contract is specified by $\{\alpha _{h},t_{h},\alpha _{l},t_{l}\}$. When D reports his type as $\tau \in \{h,l\}$, $\{\alpha _{\tau },t_{\tau }\}$ is implemented. The optimal contract maximizes S’s expected payoff subject to individual rationality and incentive compatibility constraints, as described below.

Since D can unilaterally guarantee his war payoff, individual rationality requires that each type of D is offered at least his war payoff in a peaceful deal:

Incentive compatibility ensures that D will report his type truthfully:

The left hand side of $IC^{D}_{l}$ is the expected payoff of D if $c_{D}=c_{l}$ and he reports $c_{D}$ truthfully and the right hand side is his payoff if he reports $c_{D}=c_{h}.$ In the latter case, war will break out and he will collect his war payoff $1-p-c_{l}$ with probability $\alpha _{h},$ and he will be offered $t_{h}$ with probability $1-\alpha _{h}.$ If he is offered $t_{h}$, he will accept the offer if $t_{h}$ is at least as good as his war payoff, i.e. $t_{h}\ge 1-p-c_{l}.$ So, his payoff will be the maximum of $t_{h}$ and $1-p-c_{l}$ in that case.

Note that $IR^{D}_{l}$ ensures that $t_{l}\ge 1-p-c_{l}>1-p-c_{h}$ so that $t_{l}=\max \{t_{l},p-c_{h}\}$ in $IC^{D}_{h}.$

If D turns out to have $c_{D}=c_{\tau }$, $\tau \in \{h,l\},$ then the two states will go to war with probability $\alpha _{\tau }$, and they will reach the peaceful settlement $t_{\tau }$ with probability $1-\alpha _{\tau }$ where S receives $1-t_{\tau }.$ Then, S obtains an expected payoff of

$$\begin{aligned} \alpha _{\tau }(p-c_{S})+(1-\alpha _{\tau })(1-t_{\tau }). \end{aligned}$$

S does not know D’s type when she offers the contract $\{\alpha _{h},t_{h},\alpha _{l},t_{l}\}$ but she knows that $c_{D}=c_{l}$ with probability $\pi$, so S’s expected payoff from offering $\{\alpha _{h},t_{h},\alpha _{l},t_{l}\}$ is given by

$$\begin{aligned} V(\{\alpha _{h},t_{h},\alpha _{l},t_{l}\})&=\pi \left[ \alpha _{l}(p-c_{S})+(1-\alpha _{l})(1-t_{l})\right] \\&\quad +(1-\pi )\left[ \alpha _{h}(p-c_{S})+(1-\alpha _{h})(1-t_{h})\right] . \end{aligned}$$

Then S chooses a contract that solves the following maximization problem

$$\begin{aligned}&\underset{\{\alpha _{h},t_{h},\alpha _{l},t_{l}\}}{\max }V(\{\alpha _{h},t_{h},\alpha _{l},t_{l}\})\\&\text {subject to}\\&IR^{D}_{l},\text { }IR^{D}_{h},\text { }IC^{D}_{l},\text { }IC^{D}_{h}\\&\text {and }\alpha _{h},\alpha _{l},t_{h},t_{l}\in [0,1] \end{aligned}$$

where $\alpha _{h},\alpha _{l},t_{h},t_{l}\in [0,1]$ are the usual feasibility constraints on probabilities and shares.

A straightforward analysis of this problem yields the optimal solution. Let ${\hat{\pi }}=\frac{c_{h}-c_{l}}{c_{h}+c_{S}}\in (0,1)$. Then optimal contract is given by

$$\begin{aligned} \text {if }\pi&\ge {\hat{\pi }} \text { then }\alpha _{l}=\alpha _{h}=0\text { and }t_{h}=t_{l}=1-p-c_{l},\\ \text {if }\pi&< {\hat{\pi }} \text { then }\alpha _{l}=1,\alpha _{h}=0\text {, }t_{h}=t_{l}=1-p-c_{h}. \end{aligned}$$

This is effectively equivalent to Fearon’s optimal take-it-or-leave-it offer. If S is sufficiently confident that she is likely to face a high-resolve type D with $c_{D}=c_l$, $\pi \ge {\hat{\pi }},$ she solves her risk-return trade-off by offering $1-p-c_{l}$ to D and avoids war. Otherwise, she takes the risk of war against the high-resolve type by making a low offer of $1-p-c_{h}$.

1.2 Private values with informed principal

Suppose that D’s cost of war is known and only S observes her own cost of war. In this case, S’s private information does not affect D’s payoff, and this constitutes a private values case.

Maskin and Tirole (1990) develop a method to solve the problem of an informed principal in the case of private values. They assume that some types of principals may violate the individual rationality constraint for the agent, however the individual rationality constraint has to hold in expectations (see their problem $F_{*}^{i}$ on page 392). In contrast, a fundamental assumption in crisis bargaining is that no player can be forced to accept any deal that is worse than its war payoff. That is, the individual rationality constraint cannot be violated by any type of principal.⁵ Therefore, their approach reduces to the standard principal-agent framework in my crisis bargaining model.

Since S’s private information does not affect D’s payoff, it does not affect the set of individually rational and incentive compatible contracts, either. Thus, the analysis remains the same. If D’s cost of war is known, then S’s problem becomes trivial since there is no incentive compatibility constraint for D and the optimal contract is given by $t=1-p-c_D$ and $\alpha =0$.

Common values: distribution of power

In this section, I study the informational problem that concerns distribution of power between players. Assume that $c_{D}$ and $c_{S}$ are common knowledge, but p is equal to $p_{h}$ with probability $\pi$ and to $p_{l}<p_{h}$ with probability $1-\pi$. The distribution of p is common knowledge, only one of the players knows the true value of p. The identity of the informed player matters in this case. First, I consider the scenario that D holds private information.

1.1 Common values with uninformed principal

Assume that D privately knows the true value of p. I will refer to D as $D_h$ if $p=p_{h}$ and as $D_l$ if $p=p_{l}$. This case is similar to the previous one because S’s choice of contract does not transmit information from S to D, and so S’s problem is set up as in B.1. S chooses an individually rational and incentive compatible contract $\{\alpha _{h},t_{h},\alpha _{l},t_{l}\}$ that maximizes her expected payoff among all individually rational and incentive compatible contracts. Her problem is formulated as follows:

$$\begin{aligned} \underset{\alpha _{l},t_{l},\alpha _{h},t_{h}}{\max }V(\alpha _{l},t_{l},\alpha _{h},t_{h})&=\pi \left[ \alpha _{h}(p_{h}-c_{S})+(1-\alpha _{h})(1-t_{h})\right] \\&\quad +(1-\pi )\left[ \alpha _{l}(p_{l}-c_{S})+(1-\alpha _{l})(1-t_{l})\right] \end{aligned}$$

subject to

and the feasibility constraints $\alpha _{i}\in [0,1]$ and $t_{i}\in [0,1].$

The optimal contract is given as follows: Let $\pi ^{**}=\frac{c_{S}+c_{D}}{(p_{h}-p_{l})+(c_{S}+c_{D})}\in (0,1).$ Then

$$\begin{aligned} \text {if }\pi&\ge \pi ^{**}\text { then }\alpha _{l}=1,\alpha _{h}=0\text { and }t_{h}=t_{l}=1-p_{h}-c_{D},\\ \text {if }\pi&<\pi ^{**}\text { then }\alpha _{l}=\alpha _{h}=0,t_{h}=t_{l}=1-p_{l}-c_{D}. \end{aligned}$$

This contract is effectively equivalent to a take-it-or-leave-it offer. If S is sufficiently confident that she is likely to face a D with information $p=p_l$, $\pi <\pi ^{**},$ she resolves her risk-return trade-off by offering $1-p_{l}-c_{D}$ to D and avoids war. Otherwise, she takes the risk of war by making a low offer of $1-p_{h}-c_{D}$. Fearon’s prediction of risk-return trade-off arises in this case as well.

1.2 Common values with informed principal

I only provide the solution of the optimal separating equilibrium here in the appendix. The solution of the optimal pooling equilibrium and the optimal equilibrium are in the main text.

Lemma 6

$\alpha _{h}^{sep}\ge \alpha _{l}^{sep}$.

Proof

Summing up $IC^{S}_{l}$ and $IC^{S}_{h}$ at the optimal solution yields $\alpha _{h}^{sep}(p_{h}-p_{l})\ge \alpha _{l}^{sep}(p_{h}-p_{l})$. Then $p_{h}>p_{l}$ implies $\alpha _{h}^{sep}\ge \alpha _{l}^{sep}.$ $\square$

Lemma 7

$t_{h}^{sep}\le t_{l}^{sep}$.

Proof

By individual rationality for S,  it must be the case that $1-t_{h}^{sep}\ge p_{h}-c_{S}.$ Then $\alpha _{h}^{sep}\ge \alpha _{l}^{sep}$ and $IC^{S}_{h}$ at the optimal solution imply that $1-t_{h}^{sep}\ge 1-t_{l}^{sep},$ which is equivalent to $t_{h}^{sep}\le t_{l}^{sep}.$ $\square$

Lemma 8

$\alpha _{l}^{sep}=0$.

Proof

Suppose that $\alpha _{l}^{sep}>0$. Then $\alpha _{h}^{sep}>0$ by Lemma 6. Decrease $\alpha _{l}^{sep}$ by some small $\delta >0$ and $\alpha _{h}^{sep}$ by some small $\epsilon >0$ such that

$$\begin{aligned} \delta ((1-t_{l}^{sep})-(p_{l}-c_{S}))=\epsilon ((1-t_{h}^{sep})-(p_{l}-c_{S})). \end{aligned}$$

(1)

Individual rationality for S implies $1-t_{l}^{sep}\ge p_{l}-c_{S}.$ Also $t_{h}^{sep}\le t_{l}^{sep}$ from Lemma 7 so that the coefficients of $\epsilon$ and $\delta$ are both non-negative. So, there exists such positive and small $\epsilon$ and $\delta$. Then $IC^{S}_{l}$ continues to hold since the right and left hand side of $IC^{S}_{l}$ changes by the same amount. This equality implies that

$$\begin{aligned} \delta ((1-t_{l}^{sep})-(p_{h}-c_{S})) \le \epsilon ((1-t_{h}^{sep})-(p_{h}-c_{S})). \end{aligned}$$

(2)

Equality 1 yields

$$\begin{aligned} \delta = \frac{(1-t_{h}^{sep})-(p_{l}-c_{S})}{(1-t_{l}^{sep})-(p_{l}-c_{S})} \epsilon . \end{aligned}$$

Substitute this in 2. Then 2 becomes

$$\begin{aligned} \frac{(1-t_{h}^{sep})-(p_{l}-c_{S})}{(1-t_{h}^{sep})-(p_{l}-c_{S})} \le \frac{(1-t_{h}^{sep})-(p_{h}-c_{S})}{(1-t_{h}^{sep})-(p_{h}-c_{S})}, \end{aligned}$$

which is implied by $p_h<p_l$ and $t_{h}^{sep}\le t_{l}^{sep}$ by Lemma 7. This inequality implies that $IC^{S}_{h}$ continues to hold. The individual rationality constraints are not affected. This change increases $V_{h}^{sep}$ and $V_{l}^{sep}$. This is a contradiction so $\alpha _{l}^{sep}=0.$ $\square$

Lemma 9

$IC^{S}_{l}$ holds with equality.

Proof

Suppose that $IC^{S}_{l}$ is slack. If $\alpha _{h}^{sep}>0$ or $t_{h}^{sep}>1-p_{h}-c_{D},$ then slightly decreasing $\alpha _{h}^{sep}$ or $t_{h}^{sep}$ increases $V_{h}^{sep}$ without violating any of the constraints. So $\alpha _{h}^{sep}=0$ and $t_{h}^{sep}=1-p_{h}-c_{D}$ must hold. Then substituting $\alpha _{l}^{sep}=0$ from Lemma 8, $IC^{S}_{l}$ becomes $1-t_{l}^{sep}\ge p_{h}+c_{D},$ equivalently $t_{l}^{sep}\le 1-p_{h}-c_{D}.$ We also have $t_{l}^{sep}\ge 1-p_{l}-c_{D}$ by $IR^{D}_{h}$. Combining these two implies $p_{h} \le p_{l}$, a contradiction. This proves that $IC^{S}_{l}$ holds with equality. $\square$

Lemma 10

$IC^{S}_{h}$ is implied by other constraints so it can be ignored.

Proof

Substitute $\alpha _{l}^{sep}=0$ in $IC^{S}_{l}$ and $IC^{S}_{h}.$ Then

$$\begin{aligned} 1-t_{l}^{sep}&=\alpha _{h}^{sep}(p_{l}-c_{S})+(1-\alpha _{h}^{sep})(1-t_{h}^{sep})\\&\le \alpha _{h}^{sep}(p_{h}-c_{S})+(1-\alpha _{h}^{sep})(1-t_{h}^{sep}), \end{aligned}$$

where the equality is the binding $IC^{S}_{l}$ constraint from Lemma 9 and the inequality is implied by $p_{l}<p_{h}$ and $\alpha \ge 0$. But the inequality is $IC^{S}_{h}$ so $IC^{S}_{h}$ holds and can be ignored. $\square$

Lemma 11

$t_{l}^{sep}=1-p_{l}-c_{D}$.

Proof

If $t_{l}^{sep}>1-p_{l}-c_{D}$ then slightly decrease $t_{l}^{sep}.$ $IR^{D}_{l}$ continues to hold for a small enough decrease, $IR^{D}_{h}$ is not affected, and $IC^{S}_{l}$ becomes slack. So $IC^{S}_{h}$ also continues to hold and $V_{l}^{sep}$ increases. This is a contradiction so $t_{l}^{sep}=1-p_{l}-c_{D}.$

So the solution to $S_l$’s problem is given by $\alpha _{l}^{sep}=0$ and $t_{l}^{sep}=1-p_{l}-c_{D}.$

Then $S_h$’s problem becomes

Take the total differential of $IC^{S}_{l}$ with respect to $\alpha _{h}$ and $t_{h}$:

$$\begin{aligned} ((p_{l}-c_{S})-(1-t_{h}))d\alpha _{h}-(1-\alpha _{h})dt_{h}=0, \end{aligned}$$

which yields

$$\begin{aligned} dt_{h} = \frac{(p_{l}-c_{S})-(1-t_{h})}{1-\alpha _{h}})d\alpha _{h}. \end{aligned}$$

$t_{l}^{sep}=1-p_{l}-c_{D}$ implies $p_{l}=1-t_{l}^{sep}-c_{D}$. Substitute this in $dt_{h}$ above and obtain

$$\begin{aligned} dt_{h} = \frac{(t_{h}-t_{l})-(c_S+c_D)}{1-\alpha _{h}}d\alpha _{h}<0 \end{aligned}$$

since $t_h \le t_l$ by Lemma 7.

Take the total differential of the objective function with respect to $\alpha _{h}$ and $t_{h}$ and replace the above equality and substitute for $dt_{h}$:

$$\begin{aligned} d\text {(objective)}&=((p_{h}-c_{S})-(1-t_{h}))d\alpha _{h}-(1-\alpha _{h})dt_{h}dt_{l},\\&=(p_{h}-p_{l})d\alpha _{h}. \end{aligned}$$

So increasing $\alpha _{h}$ and decreasing $t_h$ increases the objective. Then the solution to $S_h$’s problem is given by $t_{h}^{sep}=1-p_{h}-c_{D}$. Substitute $t_{h}^{sep}$ in $IC^{S}_{l}$ to obtain

$$\begin{aligned} \alpha _{h}^{sep}=\frac{p_{h}-p_{l}}{(p_{h}-p_{l})+(c_{D}+c_{S})}. \end{aligned}$$