Strategic ambiguity about military capacity with multiple adversaries

Abstract

I present a model in which a nation must decide whether to reveal its military capacity when the nation faces two possible adversaries. One adversary would be inclined to attack if the country has a weak military capacity, and the other adversary would be inclined to attack preemptively if the country is developing a strong military capacity. I derive conditions under which it is an equilibrium for the nation to be ambiguous about its military capacity as a function of the hawkishness of the adversaries and the accuracy of the adversaries’ national intelligence.

Keywords

Military capacity multiple adversaries strategic ambiguity

Introduction

In the years preceding the 2003 invasion of Iraq, Iraqi leader Saddam Hussein generally maintained a policy of ambiguity about whether Iraq possessed weapons of mass destruction. In 1998, Iraq asserted that United Nations weapons inspectors would never be allowed in Iraq again, and Hussein refused to invite United Nations weapons inspections in Iraq for roughly 4 years. Even when the United States warned of war if the Iraqi government refused to comply with a United Nations Resolution requesting disarmament, the Iraqis delayed cooperation with the United Nations in a way that led chief U.N. Weapons Inspector Hans Blix to declare that ‘Iraq appears not to have come to a genuine acceptance of the disarmament which was demanded of it’ (Blix, 2003). Revelations after the invasion that Iraq indeed had no weapons of mass destruction thus surprised some Americans, including U.S. President George W. Bush (Milbank, 2004). Bush evidently believed that Hussein would have taken every opportunity to demonstrate that Iraq had no weapons of mass destruction so the United States would realize it was not necessary to invade to disarm Iraq.

While Bush may have expected Hussein to make every effort to demonstrate Iraq had no weapons of mass destruction if this were the case, Hussein may have perceived another strategy as preferable. According to interviews and documents collected by the Federal Bureau of Investigation as well as other sources, Hussein feared that revealing that Iraq had no weapons of mass destruction would encourage Iraq’s other enemies, such as Iran, to attack (Johnston, 2004; Meek, 2009; Shane, 2009; Woods et al., 2006). By being ambiguous about Iraq’s military capacity, Hussein hoped to discourage the Iranians from attacking by leaving open the possibility that Iraq may have weapons of mass destruction while simultaneously discouraging the Americans from attacking by leaving open the possibility that Iraq might not have weapons of mass destruction (Schrange, 2003).¹

While it is not uncommon for a country to make strategic decisions when it has to worry about the actions of more than one possible adversary, to the best of my knowledge, there has no been no work that considers whether a country would have an incentive to be ambiguous about its military capacity when that nation faces multiple possible adversaries. This paper fills this gap in the literature by analyzing a model in which a nation must decide whether to reveal its military capacity when the nation faces two possible adversaries in an analogous setting to that faced by Iraq prior to being invaded in 2003.

In the model I consider, if the nation successfully reveals its military capacity, then both adversaries learn the country’s precise military capacity, but if the nation never attempts to reveal its military capacity, then the adversaries rely on their imperfect national intelligence to determine if they should attack. I assume it is common knowledge that one adversary is relatively more willing to attack if the country has a weak military capacity, and the other adversary is relatively more willing to attack preemptively if the country is developing a strong military capacity, but I also assume that the two adversaries have private information about their own willingness to attack. The incentives faced by the adversaries thus correspond to those faced by the United States and Iran prior to the 2003 Iraq War.

If the country reveals that it has weapons, then the adversary that is inclined to attack preemptively attacks if and only if it is hawkish, whereas the other adversary never attacks. And if the country reveals that it has no weapons, then the adversary that is inclined to attack weak countries attacks if and only if it is hawkish, whereas the other adversary never attacks. But if the country is ambiguous about its military capacity, then hawkish adversaries attack if and only if their imperfect national intelligence indicates that the first country has a military capacity that makes it worthwhile for them to attack.

When the adversaries follow these strategies, I find there is an equilibrium in which the first country is ambiguous about its military capacity if and only if the two adversaries are roughly equally likely to be hawkish. The equilibrium in which the country is ambiguous about its military capacity holds for a non-generic set of parameters, and ambiguity is more likely to be an equilibrium if the adversaries are relatively more hawkish. I also find that ambiguity is more likely to be an equilibrium if the adversaries have relatively more accurate national intelligence.

There has been a large literature on models of international conflict that analyzes games in which pre-play communication precedes a possible military conflict (e.g. Banks (1990), Fearon (1995), Fey and Ramsay (2010, 2011), Filson and Werner (2002, 2004), Leventoglu and Tarar (2008), Morgan (1984), Morrow (1989), Powell (1987, 2004), Smith and Stam (2004), and Wagner (2000)). This literature includes analysis of models in which parties may attempt to negotiate a truce after a military conflict begins (Filson and Werner, 2002, 2004; Smith and Stam, 2004; Wagner, 2000), models where a mediator attempts to facilitate negotiations prior to a possible military conflict (Fey and Ramsay, 2010), models in which there are several counteroffers made prior to a war (Leventoglu and Tarar, 2008), as well as many other considerations that could potentially influence international conflict. However, these papers typically focus attention on different issues than whether a state has an incentive to be ambiguous about its military capacity.

In the international relations literature, only a few other papers focus on matters related to strategic ambiguity about military capacity. Baliga and Sjöström (2008) present a model in which a strong state that is known to possess advanced weapons faces a weak state that does not initially possess such weapons but may attempt to acquire them. Strategic ambiguity then arises because this deterrence by doubt can be a substitute for a costly weapons program. This differs from my model in which a country faces multiple adversaries, and strategic ambiguity arises from considering a country’s strategic interactions with multiple adversaries.

Finally, Meirowitz and Sartori (2008) have presented a model in which two competing states endogenously decide how much to invest in developing their military capacity before bargaining and then possibly going to war. The authors show that the equilibrium of this game generally consists of mixed strategy equilibria in which states endogenously create ambiguity about their true military capacity that may in turn create risks of war. My model again differs from this in that I consider an environment with more than two states, and in that strategic ambiguity can actually be used to decrease a country’s probability of being attacked in my model.

The model

There are three countries, $A$ , $B$ , and $C$ . Country $A$ has private information about its military capacity, and countries $B$ and $C$ have private information about their own willingness to attack country $A$ . In particular, I assume that country $A$ has a type known privately to $A$ , $τ_{A} \in {w, n}$ , where $τ_{A} = w$ indicates that the country has weapons and $τ_{A} = n$ indicates that the country does not have weapons. Country $B$ also has a type known privately to $B$ , $τ_{B} \in {d, h}$ , where $τ_{B} = d$ indicates that country $B$ is a dove and $τ_{B} = h$ indicates that country $B$ is a hawk. Similarly, country $C$ also has a type known privately to $C$ , $τ_{C} \in {d, h}$ , where $τ_{C} = d$ indicates that country $C$ is a dove and $τ_{C} = h$ indicates that country $C$ is a hawk.

While each country knows its own type, countries do not initially know the types of the other players with certainty. Instead I assume that country $A$ ’s adversaries share a common prior that $τ_{A} = w$ with probability $π_{A}$ and $τ_{A} = n$ with probability $1 - π_{A}$ . Countries $A$ and $C$ share a common prior that $τ_{B} = h$ with probability $π_{B}$ and $τ_{B} = d$ with probability $1 - π_{B}$ . And countries $A$ and $B$ share a common prior that $τ_{C} = h$ with probability $π_{C}$ and $τ_{C} = d$ with probability $1 - π_{C}$ .

At the start of the game, countries $B$ and $C$ receive private signals about country $A$ ’s military capacity from their private national intelligence. I assume throughout that each country’s national intelligence is accurate more often than not. In particular, I assume that country $B$ receives a signal $s_{B} \in {w, n}$ and country $C$ receives a signal $s_{C} \in {w, n}$ , where the signals $s_{B}$ and $s_{C}$ are independent draws from a distribution satisfying $\Pr (s = w | τ_{A} = w) = p$ , $\Pr (s = n | τ_{A} = w) = 1 - p$ , $\Pr (s = n | τ_{A} = n) = p$ , and $\Pr (s = w | τ_{A} = n) = 1 - p$ for some $p > \frac{1}{2}$ . Thus a country is more likely to obtain a private signal indicating that country $A$ has weapons if country $A$ has weapons, and a country is more likely to obtain a private signal indicating that country $A$ does not have weapons if country $A$ does not have weapons.

After countries $B$ and $C$ receive their private signals, country $A$ decides whether to attempt to publicly reveal its military capacity. If country $A$ attempts to reveal its military capacity, the country may or may not succeed. For instance, country $A$ might try to reveal that it has weapons of mass destruction by conducting a public test of its weapons. If the test is successful, then other countries will know that country $A$ has weapons, but if the weapons fail to function during the public test, then other countries will still face uncertainty about whether country $A$ really has weapons. Similarly, if country $A$ tries to reveal that it does not have weapons, it may do so by inviting weapons inspectors into the country to verify that country $A$ does not have weapons. The inspectors might be able to ascertain that country $A$ does not have weapons from the inspections, or the inspectors may finish the inspections with no definitive conclusions, in which case other countries will still not know whether country $A$ has weapons.

Formally, I assume that if country $A$ has weapons, then country $A$ can either take an action $a$ , indicating that country $A$ remains ambiguous about its military capacity, or take an action $r_{w}$ , indicating that the country attempts to reveal that it has weapons. Similarly, if country $A$ does not have weapons, then country $A$ can either take action $a$ or take an action $r_{n}$ , indicating that the country attempts to reveal that it does not have weapons.

After each possible choice of action by country $A$ , nature publicly reveals the realization of a random variable $θ_{A}$ that potentially gives information about whether country $A$ has weapons. $θ_{A}$ may either take on the values $w$ , $n$ , or $u$ , indicating either that country $A$ has weapons $(w)$ , does not have weapons $(n)$ , or that country $A$ ‘s military capacity is uncertain $(u)$ . If country $A$ takes action $a$ , then $θ_{A} = u$ holds with certainty, and nature does not reveal any information about whether country $A$ has weapons. If country $A$ takes action $r_{w}$ , then $θ_{A} = w$ with probability $π_{w} \in (0, 1]$ and $θ_{A} = u$ with probability $1 - π_{w}$ . And if country $A$ takes action $r_{n}$ , then $θ_{A} = n$ with probability $π_{n} \in (0, 1]$ and $θ_{A} = u$ with probability $1 - π_{n}$ .

In the above model, if country $A$ attempts to reveal its military capacity, then nature reveals the country’s true military capacity with some positive probability, but nature also possibly indicates that country $A$ ’s true military capacity is unknown. The probability that country $A$ ’s military capacity is revealed when the country attempts to reveal its military capacity may differ for different values of $τ_{A}$ , as it may be easier for country $A$ to successfully reveal that it has weapons than to successfully reveal that it does not have weapons. Another consequence of this model is that if country $A$ ’s adversaries observe that $θ_{A} = w$ , then these countries will know that country $A$ has weapons, and if country $A$ ‘s adversaries observe that $θ_{A} = n$ , then these countries will know that country $A$ does not have weapons.

After country $A$ decides whether to attempt to reveal its military capacity, all countries learn the value of $θ_{A}$ , and country $B$ decides whether to attack country $A$ .² If country $B$ attacks country $A$ , then payoffs are realized and the game ends. Otherwise, country $C$ decides whether to attack country $A$ . After country $C$ makes this decision, payoffs are realized and the game ends.

The payoffs of the players are as follows: If country $A$ is not attacked by either country $B$ or $C$ , then country $A$ obtains a payoff of $0$ . But if country $A$ is attacked by some other country, then country $A$ obtains a payoff of $- V_{Aw}$ for some $V_{Aw} > 0$ if country $A$ has weapons and country $A$ obtains a payoff of $- V_{An}$ for some $V_{An} > 0$ if country $A$ does not have weapons. Thus regardless of whether country $A$ has weapons, country $A$ ‘s payoff is decreasing in its probability of being attacked, and country $A$ ’s objective is to avoid being attacked.

If country $B$ does not attack country $A$ , then country $B$ obtains a payoff of $0$ . Otherwise, country $B$ obtains the following payoffs. First country $B$ pays a cost of $c_{BH} > 0$ to attack country $A$ if country $B$ is a hawk, and pays a cost of $c_{BD} > 0$ to attack country $A$ if country $B$ is a dove. If country $A$ has weapons, then country $B$ also obtains a benefit of $V_{B} > 0$ from attacking country $A$ , but country $B$ does not obtain any benefit from attacking country $A$ if country $A$ has no weapons. Throughout I assume that $c_{BH} < V_{B} < c_{BD}$ .

These assumptions on payoffs indicate that the cost from attacking a country $B$ always exceeds the potential benefit from attacking if country $B$ is a dove, but the cost from attacking country $A$ can be smaller than the benefit from attacking if country $B$ is a hawk. Hawks are thus relatively more willing to attack than doves. Also, country $B$ ’s payoffs are such that country $B$ is more likely to want to attack country $A$ if country $A$ possesses weapons. Country $B$ ’s incentives thus correspond to those faced by the United States prior to the 2003 invasion of Iraq.

Similarly, if country $C$ does not attack country $A$ , then country $C$ obtains a payoff of $0$ . Otherwise, country $C$ obtains the following payoffs. First country $C$ pays a cost of $c_{CH} > 0$ to attack country $A$ if country $C$ is a hawk, and pays a cost of $c_{CD} > 0$ to attack country $A$ if country $C$ is a dove. If country $A$ has no weapons, then country $C$ also obtains a benefit of $V_{C} > 0$ from attacking country $A$ , but country $C$ does not obtain any benefit from attacking country $A$ if country $A$ has weapons. Throughout I assume that $c_{CH} < V_{C} < c_{CD}$ .

As with country $B$ , country $C$ ’s payoffs are such that country $C$ is relatively more willing to attack if country $C$ is a hawk. However, unlike country $B$ , country $C$ is relatively more willing to attack country $A$ if country $A$ does not have weapons. Country $C$ ’s incentives thus correspond to those faced by the Iranians prior to the 2003 invasion of Iraq.

Results

Note that if $τ_{C} = d$ , then the cost of attacking country $A$ always exceeds any potential benefits that country $C$ may obtain from attacking country $A$ . Thus if country $C$ is a dove, then country $C$ has a dominant strategy of never attacking country $A$ . Similarly, if $τ_{B} = d$ and country $B$ is a dove, then country $B$ has a dominant strategy of never attacking country $A$ .

Now note that if $τ_{C} = h$ and country $C$ attacks country $A$ , then country $C$ obtains a payoff of $V_{C} - c_{CH} > 0$ if $τ_{A} = n$ and a payoff of $- c_{CH} < 0$ if $τ_{A} = w$ . Thus if country $A$ successfully reveals its military capacity, then country $C$ has an incentive to attack country $A$ if and only if country $A$ has no weapons. Also note that if $τ_{B} = h$ and country $B$ attacks country $A$ , then country $B$ obtains a payoff of $V_{B} - c_{BH} > 0$ if $τ_{A} = w$ and a payoff of $- c_{BH} < 0$ if $τ_{A} = n$ . Thus if country $A$ successfully reveals its military capacity, then country $B$ has an incentive to attack country $A$ if and only if country $A$ has weapons.

Thus if country $A$ successfully reveals its military capacity when country $A$ has weapons, then country $A$ is attacked if and only if country $B$ is a hawk. And if country $A$ successfully reveals its military capacity when country $A$ does not have weapons, then country $A$ is attacked if and only if country $C$ is a hawk. Thus if country $A$ succeeds in revealing its military capacity when $τ_{A} = w$ , then the country is attacked with probability $π_{B}$ , and if country $A$ succeeds in revealing its military capacity when $τ_{A} = n$ , then the country is attacked with probability $π_{C}$ .

I now seek to determine the circumstances under which country $A$ is attacked if the country is always ambiguous about its military capacity. I seek to demonstrate that if country $A$ is always ambiguous about its military capacity, then natural circumstances give rise to an equilibrium in which a country attacks if and only if the country is a hawk and the country’s national intelligence indicates that country $A$ ’s military capacity is such that it is in the country’s interest to attack. Thus I seek to find an equilibrium in which country $B$ attacks if and only if $τ_{B} = h$ and $s_{B} = w$ and country $C$ attacks if and only if $τ_{C} = h$ and $s_{C} = n$ .

To address whether this can be an equilibrium, I first consider whether country $C$ would want to attack country $A$ if country $B$ does not attack country $A$ . Note that if players use the strategies I have proposed, country $A$ does not attempt to reveal its military capacity, and country $B$ chooses not to attack, then country $C$ can deduce that either country $B$ is a dove $(τ_{B} = d)$ or country $B$ ’s national intelligence indicated that country $A$ has no weapons $(s_{B} = n)$ . This information would in turn influence country $C$ ’s beliefs about the probability that country $A$ has weapons. In particular, I obtain the following result:

Lemma 3.1. $\Pr (τ_{A} = w | τ_{B} = d or s_{B} = n) = π_{A}^{'}$ , where $π_{A}^{'} \equiv \frac{(1 - π_{B} p) π_{A}}{(1 - π_{B} p) π_{A} + (1 - π_{B} (1 - p)) (1 - π_{A})} < π_{A}$ .

Thus if country $B$ chooses not to attack country $A$ , then this causes country $C$ to deduce that country $A$ is relatively less likely to have weapons. This result makes sense intuitively. If country $B$ chooses not to attack country $A$ , then it is relatively more likely that country $B$ ’s national intelligence indicated that country $A$ did not have weapons. This in turn means that it is relatively more likely that country $A$ did not have weapons, thereby giving the desired result.

The probability $π_{A}^{'}$ in Lemma 3.1 gives what country $C$ believes is the probability that country $A$ has weapons if country $B$ does not attack under country $B$ ’s proposed strategy before country $C$ uses the information from her national intelligence. When country $C$ also incorporates the information from her national intelligence, country $C$ believes the probability country $A$ has weapons if $s_{C} = w$ is $\frac{π_{A}^{'} p}{π_{A}^{'} p + (1 - π_{A}^{'}) (1 - p)}$ and believes the probability country $A$ has weapons if $s_{C} = n$ is $\frac{π_{A}^{'} (1 - p)}{π_{A}^{'} (1 - p) + (1 - π_{A}^{'}) p}$ .

It is worth noting that while when one conditions on the fact that country $B$ did not attack country $A$ , one deduces that country $A$ is relatively less likely to have weapons, if country $C$ also incorporates the fact that her private national intelligence indicates that country $A$ has weapons $(s_{C} = w)$ , country $C$ would deduce that country $A$ is relatively more likely to have weapons than before, as $\frac{π_{A}^{'} p}{π_{A}^{'} p + (1 - π_{A}^{'}) (1 - p)} > π_{A}$ always holds. Thus country $C$ effectively places more weight on her own national intelligence than on the information contained in the fact that country $B$ chooses not to attack country $A$ . This is logical since the fact that country $B$ did not attack does not give perfect information that country $B$ thought that country $A$ did not have weapons. Country $B$ may simply have not attacked because country $B$ was not a hawk.

With these preliminary results in place, I now derive circumstances under which it is an equilibrium for countries $B$ and $C$ to act on their national intelligence in the manner described above:

Proposition 3.2. Suppose that country $A$ is always ambiguous about its military capacity. If countries $B$ and $C$ have sufficiently accurate national intelligence, then it is an equilibrium for country $B$ to attack if and only if $τ_{B} = h$ and $s_{B} = w$ and country $C$ to attack if and only if $τ_{C} = h$ and $s_{C} = n$ . Formally, there exists some $p^{*} \in (\frac{1}{2}, 1)$ such that $p > p^{*}$ implies it is an equilibrium for country $B$ to attack if and only if $τ_{B} = h$ and $s_{B} = w$ and country $C$ to attack if and only if $τ_{C} = h$ and $s_{C} = n$ .

Proposition 3.2 indicates that if country $A$ is always ambiguous about its military capacity and countries $B$ and $C$ have accurate national intelligence, then countries $B$ and $C$ attack if and only if they are hawks and their national intelligence indicates that it is in their interest to attack. This result makes sense intuitively. If the national intelligence is sufficiently accurate, then these countries have an incentive to act on their intelligence and attack if and only if their intelligence indicates that country $A$ ’s military capacity is such that it is in their interest to attack.

While the model being considered assumes that $B$ ’s payoffs are such that $B$ does not benefit from a war between $A$ and $C$ and $C$ ’s payoff is also not affected by a war between $A$ and $B$ , it is worth noting that Proposition 3.2 (and thus all remaining results) would continue to hold even if these countries’ payoffs could vary in this way. Thus the assumption that countries $B$ and $C$ do not benefit from wars between other countries is not critical for any of the results in this paper.

With Proposition 3.2 in mind, I now assume that if country $A$ is always ambiguous about its military capacity, then country $B$ follows the strategy of attacking if and only if $τ_{B} = h$ and $s_{B} = w$ , and country $C$ follows the strategy of attacking if and only if $τ_{C} = h$ and $s_{C} = n$ . Given these strategies for countries $B$ and $C$ , I obtain the following result about country $A$ ’s equilibrium strategy:

Proposition 3.3. There exists an equilibrium in which country $A$ is always ambiguous about its military capacity if and only if $| π_{B} - π_{C} | \leq π_{B} π_{C} p$ .

The condition in Proposition 3.3 indicates that it is an equilibrium for country $A$ to be ambiguous about its military capacity if and only if countries $B$ and $C$ are approximately equally likely to be hawks. To understand why this condition is necessary, note that if country $C$ is much more likely to be a hawk than country $B$ , then country $A$ would primarily have an incentive to take actions that would discourage country $C$ from attacking country $A$ . Thus if country $C$ is much more likely to be a hawk than country $B$ and country $A$ had weapons, then country $A$ would have an incentive to attempt to reveal its military capacity to dissuade country $C$ from attacking, and country $A$ would attempt to reveal its military capacity in equilibrium if country $A$ has weapons.

Similarly, if country $B$ is much more likely to be a hawk than country $C$ , then country $A$ has an incentive to attempt to reveal its military capacity if doing so is likely to dissuade country $B$ from attacking country $A$ , and country $A$ would attempt to reveal its military capacity in equilibrium if country $A$ does not have weapons. Thus it is necessary for countries $B$ and $C$ to be approximately equally likely to be hawks in order for it to be an equilibrium for country $A$ to always be ambiguous about its military capacity.

But if countries $B$ and $C$ are approximately equally likely to be hawks, then country $A$ cannot profitably deviate by attempting to reveal its military capacity in either of the circumstances considered above. To understand why, note that country $A$ wants to induce correlation between the circumstances in which countries $B$ and $C$ want to attack, since it is better for country $A$ to have a situation in which either both countries want to attack or neither country wants to attack than a situation in which exactly one country always wants to attack. Since being ambiguous about military capacity induces correlation in the circumstances under which country $A$ ’s adversaries want to attack, being ambiguous can benefit country $A$ for this reason.

Choosing not to be ambiguous can also affect country $A$ ’s payoff by changing the probabilities with which countries $B$ and $C$ want to attack. But if countries $B$ and $C$ are approximately equally likely to be hawks and country $A$ successfully reveals its military capacity, then this revelation decreases the probability that one country wants to attack by approximately the same amount as it increases the probability that another country wants to attack. Thus if countries $B$ and $C$ are approximately equally likely to be hawks, the benefit from inducing correlation between the circumstances under which country $A$ ’s adversaries want to attack outweighs any costs from changes in the probabilities with which the individual countries want to attack, and country $A$ will want to be ambiguous about its military capacity.

The condition in Proposition 3.3 illustrates that it is essential for there to be more than one adversary in order for $A$ to have an incentive to be ambiguous about its military capacity in equilibrium. The case in which $A$ only has one adversary who might attack is mathematically equivalent to the case in which one of $A$ ’s adversaries is guaranteed to be a dove who will never attack (i.e. the case in which either $π_{B} = 0$ or $π_{C} = 0$ ). And the condition in Proposition 3.3 guarantees that if either $π_{B} = 0$ or $π_{C} = 0$ , then $A$ will not have an incentive to be ambiguous about its military capacity in equilibrium. Thus the existence of multiple adversaries is critical in order for $A$ to have an incentive to be ambiguous about its military capacity.

It is worth noting that the condition in Proposition 3.3 indicates that it can be an equilibrium for country $A$ to always be ambiguous about its military capacity for a non-generic set of parameters about the probabilities with which countries $B$ and $C$ are likely to be hawks. This indicates that if country $A$ is initially ambiguous about its military capacity and another country becomes more hawkish, then this need not cause country $A$ to change its policy about strategic ambiguity. Thus events that might cause a shift in U.S. foreign policy such as the election of Bush to replace Clinton in 2000, or a major terrorist attack such as the September 11 attacks, need not cause Iraq to abandon its policy of being ambiguous about its military capacity.

The results in this paper also indicate that if countries $B$ and $C$ are relatively more hawkish in the sense that if $π_{B}$ and $π_{C}$ are relatively larger, then it is relatively more likely to be an equilibrium for country $A$ to be ambiguous about its military capacity, as $π_{B} π_{C} p$ will be relatively larger and the inequality $| π_{B} - π_{C} | \leq π_{B} π_{C} p$ will be easier to satisfy. This makes sense intuitively. When countries $B$ and $C$ are relatively more hawkish, if country $A$ does not successfully reveal its military capacity, there is relatively greater correlation between the circumstances under which countries $A$ and $B$ want to attack since there is now a lower chance that a country is a dove and will never attack. This greater correlation between the circumstances under which countries $B$ and $C$ want to attack when country $A$ is ambiguous about its military capacity provides a stronger incentive for country $A$ to be ambiguous, making it more likely that there will be an equilibrium in which country $A$ is ambiguous about its military capacity.

Proposition 3.3 also indicates that it is more likely to be an equilibrium for country $A$ to be ambiguous about its military capacity if its adversaries have relatively more accurate national intelligence since the inequality $| π_{B} - π_{C} | \leq π_{B} π_{C} p$ is more likely to be satisfied for larger values of $p$ . To understand this, recall that when country $A$ decides whether to be ambiguous, country $A$ benefits by inducing correlation in the circumstances under which its adversaries want to attack, but potentially loses by changing the probabilities with which countries $B$ and $C$ want to attack. However, if country $A$ ’s adversaries have accurate national intelligence, then revealing true military capacity will have little effect on the probabilities with which these countries want to attack country $A$ , and the loss in correlation will be relatively more important compared to the changes in the probabilities with which these countries want to attack country $A$ . Thus if country $A$ ’s adversaries have accurate national intelligence, then it is relatively more likely that it will be an equilibrium for these countries to be ambiguous about their military capacity.

Conclusion

This paper has presented a model in which a country must decide whether to attempt to reveal its military capacity when the country faces two possible adversaries, one of which is more inclined to attack if the country is weak, and the other of which is more inclined to attack preemptively if the country is developing a strong military capacity. The main result indicates that it is an equilibrium for the first country to be ambiguous about its military capacity if and only if the country is roughly equally concerned about the threats posed by the two adversaries. Ambiguity holds in equilibrium for a non-generic set of parameters, and the country is more likely to have an incentive to be ambiguous if the adversaries are relatively more hawkish or the adversaries have relatively more accurate national intelligence.

Footnotes

Appendix: Proofs

Lemma 3.1. $\Pr (τ_{A} = w | τ_{B} = d or s_{B} = n) = π_{A}^{'}$ , where $π_{A}^{'} \equiv \frac{(1 - π_{B} p) π_{A}}{(1 - π_{B} p) π_{A} + (1 - π_{B} (1 - p)) (1 - π_{A})} < π_{A}$ .

Proof.

$\Pr (τ_{A} = w | τ_{B} = d or s_{B} = n) = \frac{\Pr (τ_{B} = d or s_{B} = n | τ_{A} = w) \Pr (τ_{A} = w)}{\Pr (τ_{B} = d or s_{B} = n | τ_{A} = w) \Pr (τ_{A} = w) + \Pr (τ_{B} = d or s_{B} = n | τ_{A} = n) \Pr (τ_{A} = n)} = \frac{(1 - π_{B} + π_{B} (1 - p)) π_{A}}{(1 - π_{B} + π_{B} (1 - p)) π_{A} + (1 - π_{B} + π_{B} p) (1 - π_{A})} = \frac{(1 - π_{B} p) π_{A}}{(1 - π_{B} p) π_{A} + (1 - π_{B} (1 - p)) (1 - π_{A})}$ . Thus $\Pr (τ_{A} = w | τ_{B} = d or s_{B} = n) = π_{A}^{'}$ , where $π_{A}^{'} \equiv \frac{(1 - π_{B} p) π_{A}}{(1 - π_{B} p) π_{A} + (1 - π_{B} (1 - p)) (1 - π_{A})}$ .

To see that $π_{A}^{'} < π_{A}$ , note that $π_{A}^{'} = \frac{(1 - π_{B} p) π_{A}}{(1 - π_{B} p) π_{A} + (1 - π_{B} (1 - p)) (1 - π_{A})} = \frac{π_{A}}{π_{A} + \frac{(1 - π_{B} (1 - p))}{(1 - π_{B} p)} (1 - π_{A})}$ . Now since $p > \frac{1}{2}$ , we have $1 - π_{B} (1 - p) > 1 - π_{B} p$ and $\frac{(1 - π_{B} (1 - p))}{(1 - π_{B} p)} > 1$ . Thus $π_{A}^{'} = \frac{π_{A}}{π_{A} + \frac{(1 - π_{B} (1 - p))}{(1 - π_{B} p)} (1 - π_{A})} < \frac{π_{A}}{π_{A} + 1 - π_{A}} = π_{A}$ . This gives the result. ⃞

Proposition 3.2. Suppose that country $A$ is always ambiguous about its military capacity. Then there exists some $p^{*} \in (\frac{1}{2}, 1)$ such that $p > p^{*}$ implies it is an equilibrium for country $B$ to attack if and only if $τ_{B} = h$ and $s_{B} = w$ and country $C$ to attack if and only if $τ_{C} = h$ and $s_{C} = n$ .

Proof. First I show that if country $A$ is always ambiguous and country $B$ does not attack, then for sufficiently large $p$ it is a best response for country $C$ to attack if and only if $τ_{C} = h$ and $s_{C} = n$ . We have seen that country $C$ has a dominant strategy of never attacking country $A$ if $τ_{C} = d$ , so to show that this is a best response for country $C$ , it suffices to show that if $τ_{C} = h$ , then country $C$ has an incentive to attack if $s_{C} = n$ but not to attack if $s_{C} = w$ .

Note that if country $A$ is always ambiguous, country $B$ does not attack, and $s_{C} = w$ , then country $C$ believes the probability country $A$ has weapons is $\frac{π_{A}^{'} p}{π_{A}^{'} p + (1 - π_{A}^{'}) (1 - p)} = \frac{(1 - π_{B} p) π_{A} p}{(1 - π_{B} p) π_{A} p + (1 - π_{B} (1 - p)) (1 - π_{A}) (1 - p)}$ . And if country $A$ is always ambiguous, country $B$ does not attack, and $s_{C} = n$ , then country $C$ believes the probability country $A$ has weapons is $\frac{π_{A}^{'} (1 - p)}{π_{A}^{'} (1 - p) + (1 - π_{A}^{'}) p} = \frac{(1 - π_{B} p) π_{A} (1 - p)}{(1 - π_{B} p) π_{A} (1 - p) + (1 - π_{B} (1 - p)) (1 - π_{A}) p}$ . Now ${lim}_{p \to 1} \frac{(1 - π_{B} p) π_{A} p}{(1 - π_{B} p) π_{A} p + (1 - π_{B} (1 - p)) (1 - π_{A}) (1 - p)} = 1$ and ${lim}_{p \to 1} \frac{(1 - π_{B} p) π_{A} (1 - p)}{(1 - π_{B} p) π_{A} (1 - p) + (1 - π_{B} (1 - p)) (1 - π_{A}) p} = 0$ . Thus in the limit as $p$ goes to one, then country $C$ believes the probability country $A$ has weapons goes to 1 if country $A$ is always ambiguous, country $B$ does not attack, and $s_{C} = w$ , and country $C$ believes the probability country $A$ has weapons goes to 0 if country $A$ is always ambiguous, country $B$ does not attack, and $s_{C} = n$ .

Now if country $C$ attacks country $A$ and $τ_{A} = n$ , then country $C$ obtains a payoff of $V_{C} - c_{CH} > 0$ , and if country $C$ attacks country $A$ and $τ_{A} = w$ , then country $C$ obtains a payoff of $- c_{CH} < 0$ . Thus if country $C$ believes the probability that country $A$ has weapons is $π$ , then country $C$ obtains an expected payoff of $(1 - π) V_{C} - c_{CH}$ from attacking country $A$ . From this it follows that country $C$ prefers to attack country $A$ if $(1 - π) V_{C} - c_{CH} > 0$ or $π < 1 - \frac{c_{CH}}{V_{C}}$ and country $C$ prefers not to attack country $A$ if $π > 1 - \frac{c_{CH}}{V_{C}}$ .

But this means that if $π$ is sufficiently close to 1, then country $C$ prefers not to attack country $A$ , and if $π$ is sufficiently close to 0, then country $C$ prefers to attack country $A$ . Combining this with the results of the second paragraph of this proof shows that if $p$ is sufficiently close to 1, country $A$ is always ambiguous, and country $B$ does not attack, then it is a best response for a country $C$ with $τ_{C} = h$ to attack if and only if $s_{C} = n$ .

Now I show that if country $A$ is always ambiguous, then for sufficiently large $p$ it is a best response for country $B$ to attack if and only if $τ_{B} = h$ and $s_{B} = w$ . We have seen that country $B$ has a dominant strategy of never attacking country $A$ if $τ_{B} = d$ , so to show that this is the best response for country $B$ , it suffices to show that if $τ_{B} = h$ , then country $B$ has an incentive to attack if $s_{B} = w$ but not to attack if $s_{B} = n$ .

Note that if country $A$ is always ambiguous and $s_{B} = w$ , then country $B$ believes the probability country $A$ has weapons is $\frac{π_{A} p}{π_{A} p + (1 - π_{A}) (1 - p)}$ . And if country $A$ is always ambiguous and $s_{B} = n$ , then country $B$ believes the probability country $A$ has weapons is $\frac{π_{A} (1 - p)}{π_{A} (1 - p) + (1 - π_{A}) p}$ . Now ${lim}_{p \to 1} \frac{π_{A} p}{π_{A} p + (1 - π_{A}) (1 - p)} = 1$ and ${lim}_{p \to 1} \frac{π_{A} (1 - p)}{π_{A} (1 - p) + (1 - π_{A}) p} = 0$ . Thus in the limit as $p$ goes to 1, country $B$ believes the probability country $A$ has weapons goes to 1 if country $A$ is always ambiguous and $s_{B} = w$ , and country $B$ believes the probability country $A$ has weapons goes to 0 if country $A$ is always ambiguous and $s_{B} = n$ .

Now if country $B$ attacks country $A$ and $τ_{A} = w$ , then country $B$ obtains a payoff of $V_{B} - c_{BH} > 0$ , and if country $B$ attacks country $A$ and $τ_{A} = n$ , then country $B$ obtains a payoff of $- c_{BH} < 0$ . Thus if country $B$ believes the probability that country $A$ has weapons is $π$ , then country $B$ obtains an expected payoff of $π V_{B} - c_{BH}$ from attacking country $A$ . From this it follows that country $B$ prefers to attack country $A$ if $π V_{B} - c_{BH} > 0$ or $π > \frac{c_{BH}}{V_{B}}$ and country $B$ prefers not to attack country $A$ if $π < \frac{c_{BH}}{V_{B}}$ .

But this means that if $π$ is sufficiently close to 1, then country $B$ prefers to attack country $A$ , and if $π$ is sufficiently close to 0, then country $B$ prefers not to attack country $A$ . Combining this with the results of the sixth paragraph of this proof shows that if $p$ is sufficiently close to one and country $A$ is always ambiguous, then it is a best response for a country $B$ with $τ_{B} = h$ to attack if and only if $s_{B} = w$ .

Putting this all together shows that if $p$ is sufficiently close to one and country $A$ is always ambiguous, then it is a best response for country $B$ to attack if and only if $τ_{B} = h$ and $s_{B} = w$ , and it is a best response for country $C$ to respond to these strategies by attacking if and only if $τ_{C} = h$ and $s_{C} = n$ . Thus there exists some $p^{*} \in (\frac{1}{2}, 1)$ such that $p > p^{*}$ implies it is an equilibrium for country $B$ to attack if and only if $τ_{B} = h$ and $s_{B} = w$ and country $C$ to attack if and only if $τ_{C} = h$ and $s_{C} = n$ . ⃞

Proposition 3.3. There exists an equilibrium in which country $A$ is always ambiguous about its military capacity if and only if $| π_{B} - π_{C} | \leq π_{B} π_{C} p$ .

Proof. To derive the conditions under which it is an equilibrium for country $A$ to always be ambiguous about its military capacity, it suffices to derive the conditions under which, if country $A$ is always ambiguous about its military capacity, then country $A$ cannot profitably deviate by successfully revealing its military capacity regardless of whether country $A$ has weapons.

Note that if country $A$ is always ambiguous about its military capacity, then we know from Proposition 3.2 that country $A$ is attacked if and only if either $τ_{B} = h$ and $s_{B} = w$ or $τ_{C} = h$ and $s_{C} = n$ . Now if $τ_{A} = w$ , then $τ_{B} = h$ and $s_{B} = w$ holds with probability $π_{B} p$ and $τ_{C} = h$ and $s_{C} = n$ holds with probability $π_{C} (1 - p)$ . And if $τ_{A} = n$ , then $τ_{B} = h$ and $s_{B} = w$ holds with probability $π_{B} (1 - p)$ and $τ_{C} = h$ and $s_{C} = n$ holds with probability $π_{C} p$ .

From this it follows that if $τ_{A} = w$ and country $A$ is always ambiguous about its military capacity, then the probability country $A$ is not attacked is $(1 - π_{B} p) (1 - π_{C} (1 - p))$ . And if $τ_{A} = n$ and country $A$ is always ambiguous about its military capacity, then the probability country $A$ is not attacked is $(1 - π_{B} (1 - p)) (1 - π_{C} p)$ .

Now we have seen that if $τ_{A} = w$ and country $A$ successfully reveals its military capacity, then country $A$ is not attacked with probability $1 - π_{B}$ . Thus if country $A$ is always ambiguous about its military capacity, then a country $A$ with $τ_{A} = w$ does not have an incentive to deviate if and only if $(1 - π_{B} p) (1 - π_{C} (1 - p)) \geq 1 - π_{B}$ or $1 - π_{B} p - π_{C} (1 - p) + π_{B} π_{C} p (1 - p) \geq 1 - π_{B}$ or $π_{B} (1 - p) \geq π_{C} (1 - p) - π_{B} π_{C} p (1 - p)$ or $π_{B} \geq π_{C} - π_{B} π_{C} p$ or $π_{C} - π_{B} \leq π_{B} π_{C} p$ .

And we have seen that if $τ_{A} = n$ and country $A$ successfully reveals its military capacity, then country $A$ is not attacked with probability $1 - π_{C}$ . Thus if country $A$ is always ambiguous about its military capacity, then a country $A$ with $τ_{A} = n$ does not have an incentive to deviate if and only if $(1 - π_{B} (1 - p)) (1 - π_{C} p) \geq 1 - π_{C}$ or $1 - π_{B} (1 - p) - π_{C} p + π_{B} π_{C} p (1 - p) \geq 1 - π_{C}$ or $π_{C} (1 - p) \geq π_{B} (1 - p) - π_{B} π_{C} p (1 - p)$ or $π_{C} \geq π_{B} - π_{B} π_{C} p$ or $π_{B} - π_{C} \leq π_{B} π_{C} p$ .

Combining the results in the previous two paragraphs, we see that there is an equilibrium in which country $A$ is always ambiguous about its military capacity if and only if $| π_{B} - π_{C} | \leq π_{B} π_{C} p$ . The result then follows. ⃞

Acknowledgements

This work was completed while the author was affiliated with Yahoo! Research.

Funding

This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.

Notes

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