The role of intelligence services in deterrence inspection games

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The role of intelligence

services in deterrence inspection games

Wilfred de Graaf

Student number: 10769730

Date of final version: 21 November 2015 Master’s programme: Econometrics

Specialisation: Free Track

Supervisor: Dr. T.A. Makarewicz Second reader: Prof. dr. J. Tuinstra

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Statement of Originality

This document is written by Wilfred de Graaf who declares to take full responsibility for the contents of this document.

I declare that the text and the work presented in this document is original and that no sources other than those mentioned in the text and its references have been used in creating it.

The Faculty of Economics and Business is responsible solely for the supervision of completion of the work, not for the contents.

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Introduction

From 1979 until the outbreak of the Second Gulf War (2003) Saddam Hussein was the fifth president of Iraq. Under his government Iraq transformed into a dictatorship with a closed and authoritarian society. Therefore it became difficult for US Intelligence to assess the military aspirations of Iraq. There was no parliament which exercised effective control over the governe-ment (especially Saddam), there was no free press and hardly any other possibility to check whether the statemens of Saddam were true.

When Saddam refused to provide accurate information on his weapons or to open his facilities, the US began to suspect Iraq from possessing weapons of mass destruction. The role of US intelligence in this suspicion has been the subject of an intense debate.1 Anyhow, the US had the possibility to use their intelligence to spy on Iraq in order to get a (noisy) signal about the existence of weapons of mass destruction. Even when an invasion by a US-led coalition was im-minent, Saddam refused to use his last chance to open his facilities. What happened thereafter is well known: Iraq lost the Second Gulf War, Saddam got arrested and tranferred to the Iraq interim-government which sentenced him to death. Weapons of mass destruction were never found and George Bush and Tony Blair eventually admitted that they probably had not existed.

If we suppose that there were no weapons of mass destruction in Iraq, then we might wonder: What was the influence of intellligence on the strategy of the US? Did intelligence indicate that weapons of mass destruction existed? Or did it (correctly) detect that there were no weapons of mass destruction, but was it still rational for the US to attack? Or did the strategy of the US include some irrational aspects?

1_{Many critics state that the US wanted to invade Iraq anyhow, whether intelligence indicated that Saddam}

possessed weapons of mass destruction or not. For literature about the cause of the Second Gulf War, see Brands and Palkki [2011] and Woods, Lacey, and Murray [2006].

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This thesis may serve as a framework to answer such questions, because it investigates the role of intelligence in models of interaction in conflicts like the Second Gulf War. We restrict ourselves to a special class of games: deterrence inspection games, which are a combination of deterrence games and inspection games. Deterrence games model situations in which one player wants to deter the other player from doing something (in example: from building weapons). Inspection games deal with situations where an inspector (in example: US) tries to check whether an agent (in example: Iraq) adheres to specific rules. Eventually we want to answer our research question:

What is the role of intelligence services in deterrence inspection games?

A cornerstone in the game-theoretical analysis of conflicts is Schelling [1960], who illustrates amongst other things the role of threats in deterrence games. Schelling shows that only credible threats have a deterring potential, since rational players do not take into account incredible threats when determining their own strategy. He argued that players can only make a credible threat, if they have the ability to commit themselves to promises or threats. Harsanyi and Selten [1988] continues on this by stressing ’the great strategic importance of the ability, or inability, (of players) to make firm commitments’ in noncooperative games. Selten [1965] makes similar statements about the ability of commitment. Myerson [1991] uses the term rational threat to emphasize that a player will only carry out threats if these maximize his expected payoff given the strategy (including threats) of the other players. He shows that the existence of rational threats can be proved by an argument using the Kakutani fixed-point theorem.

Fudenberg and Tirole [1991] scrutinize some commitments which can be necessary to make credible threats. They mention their ’paradoxical’ value: a player can gain utility by reducing his action set or by decreasing his payoff to some outcomes, provided that the other players are aware of this change. Aumann and Schelling [2005] explain that a credible threat does not necessarily imply committing to certain retaliation. It can also consist of threatening to let the situation ’slip out of hand’, or as Schelling [1960] states, make ’threats that leave something to chance’. The reason is that a modest probability of escalation (war) can be enough to deter the other player. An advantage of uncertain retaliation is that a credible threat is easier to attain, because the expected retaliation cost is lower.

Myerson [2009] mentions that the only rational predictions of players’ behaviour in a (deter-rence) game are Nash equilibria, but shows that predictions become difficult in case a game has multiple equilibria. Then players need to coordinate their behaviour. In this context the notion of a focal-point, introduced by Schelling [1960], can be useful. This means that anything in a game’s environment that focuses the players’ attention to one equilibrium may lead them to expect it, and so rationally to play it. Myerson [2009] argues that this notion opens the door for environmental factors to influence the rational behaviour of players of a deterrence game. For

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more literature about focal points, see Sugden [1995], Cooper et al. [1990], Mehta et al. [1994] and Kreps [1990].

Deterrence games have two popular applications in the literature: entry deterrence and nuclear deterrence. Entry deterrence deals with incumbent firms that try to discourage potential en-trants from entering into competition in their market. Tirole [1988] gives an extensive overview of the literature in this field. Nuclear deterrence deals with nations which want to prevent a conflict by threatening to use nuclear weapons if another nation initiates a (nuclear) conflict. Because we focus on conflicts instead of business strategies, nuclear deterrence is more relevant for this thesis. Shubik et al. [1987] discusses the advantages and disadvantages of several models which deal with nuclear deterrence. Kugler and Zagare [1990] argue that in a game where both players have a vast deterring potential (for example a nuclear bomb), deterrence will lead to a stable equilibrium in which a conflict is prevented, as long as the players shun serious risks. They also show that the risk for instability and conflicts is increasing in the risk propensity of the players. An interesting analysis of a deterrence game, which neither deals with entry deterrence nor with nuclear deterrence, is given in Smith and Price [1973]. They use a deter-rence game to model animal behaviour and show that potentially dangerous weapons are rarely used in conflict, because the use of these weapons would lead to escalation, in which case both players are worse off.

Inspection games have a wide range of applications. Avenhaus and Canty [2012] provide an extensive survey of the literature concerning inspection games and work out examples of arms control, environmental regulation and financial auditing. In these games the inspector employs Bayesian methods to find out whether the agent adheres to the rules. The agent adopts a strategy to send an optimal noisy signal to the inspector. Diamond [1982] points out that the strategy of the inspector depends on his loss function, where a measure for his loss can be the period for which the agent does not adhere to the rules. An illustrative example of an inspection game is the negotiations between the US and the USSR over nuclear arms limita-tions during the 1960s, in which the US wanted to determine the optimal scheme for on-site inspections. Both Avenhaus and Canty [2012] and Maschler et al. [2013] analyze this game and emphasize that the agent will always send a noisy signal to the inspector. The inspector takes this into account and therefore wants to limit the ability of the agent to send noisy signals.

As Andreozzi [2004] notices, many inspection games can be characterized by their counter-intuitive Nash equilibria. He illustrates this by analyzing an inspection game consisting of two players: the police (inspector) and a criminal (the agent). He came to the remarkable result that increasing inspector’s incentives to enforce the law increases the frequency of law infractions. For other examples of inspection games with counter-intuitive equilibria, see Tsebelis [1989] and Holler [1993].

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Note that the research question mentions intelligence services. This refers to the concept of an Intelligence Service which was introduced by Biran and Tauman [2009] and can be conceived as a formalization of the idea that in a conflict, enemies use intellligence to spy on each other. The Intelligence Service is a spying device, operated by one of the players, to spy on the other player. It important to recognize that the Intelligence Service is a machine, and not a human assessor of information. After the spying, the player who operates the Intelligence Service receives a noisy signal about which action the other player has played.

Biran and Tauman [2009] introduce the Intelligence Service in a nuclear deterrence game. Also Jelnov, Tauman, and Zeckhauser [2015] use the concept of an Intelligence Service in the setting of nuclear deterrence. Both papers assume that the mere purpose of the Intelligence Service is to detect whether a player is building a nuclear bomb. To our best knowledge there are no other papers which deal with the concept of an Intelligence Service. There is a body of literature about games which formalize the role of intelligence in conflicts, but they use other concepts than an Intelligence Service, see for instance O’Neill [1986], O’Neill [1990], Brams [2011] and Kamien, Tauman, and Zamir [1990].

To answer our research question, we build a deterrence inspection game with two versions: a version without Intelligence Service and a version with Intelligence Service. For a detailed de-scription we refer to Chapter 2. The only difference between the versions is the existence of the Intelligence Service. All other factors are the same for both versions, so that we can make a fair comparison between the two versions of the game. Besides the concept of an Intelligence Service, we include two other concepts in our model: A provocateur-type of agent and the role of intermediary weapons.

The concept of a provocateur-type of agent is introduced in Jelnov, Tauman, and Zeckhauser [2015]. If the type of the agent is a provocateur, the agent wants the inspector to punish him in case he adheres to the rules, which causes the punishment to be unjustified. This leads to so much support from other agents or the outside world, that it outweighs the damage caused by the punishment. An empirical example of this may be the outbreak of the Second Gulf War. It is very well possible that Saddam refused to open his facilities, because he expected that an unjustified attack would raise so much support from other Arab countries, that it would outweighed the damage caused by such an attack. Many inspection games only deal with agents who always prefer not to be punished, so that their only type is a so called deterrer. Therefore the concept of a provocateur-type of agent is a valuable contribution to our model. To our best knowledge there is no other literature about games with this exact definition of a provocateur-type of agent. Nevertheless, there is some literature about agents who may be willing to provoke. See for instance Brams [1980] and O’Brien et al. [2002].

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The role of intermediary weapons, like semi-automatic weapons or tanks, in deterrence games is introduced by Chassang and Miquel [2010]. Nuclear deterrence games usually assume that an agent either builds weapons of mass destruction (a nuclear bomb) or does not build any weapons. Chassang and Miquel [2010] show that if the agent has an ’in-between option’, namely building intermediary weapons, the structure of the equilibria of the deterrence game can change enor-mously. For example in the Second Gulf War, an explanation for the attack by the US may not be the existence of weapons of mass destruction, but a build-up of intermediary weapons, which could disrupt the balance of power in the Middle East. This illustrates the relevance of including the possibility of building intermediary weapons into our model.

We find that if we include an Intelligence Service in our deterrence inspection game, the num-ber of equilibria reduces drastically. Instead of a multiplicity of equilibria in the case without Intelligence Service, we find a unique equilibrium in the case with Intelligence Service. Besides, using an Intelligence Service allows an inspector to use a more differentiated strategy, because each different signal constitutes a different information set for the inspector. We also find that the decision of the agent to build intermediary weapons is more sensitive to the precision of the intelligence service than the decision to build weapons of mass destruction. Similarly, the decision of the inspector to attack the agent is more sensitive to the signal that the agent is building intermediary weapons than to the signal that the agent is building weapons of mass destruction. Furthermore, we find that if inspector starts to use an intelligence service, the strategy of a deterrer-type of agent will tend to become more peaceful, while the strategy of a provocateur-type of agent will become more aggressive.

These findings apply not only to the military setting of the game which is investigated in this thesis, but also to other sort of games, as long as these have a similar interaction structure. An example is the entry-deterrence game described by Vickers [1985], in which Entrant wants to enter a market, but he does not know whether Incumbent’s goal is to maximize profits or to maintain market share. If we would extend this game such that Entrant could use an Intelli-gence Service, which would give him a noisy signal about the action played by Incumbent, the game would have an interaction structure which is similar to the one we analyze in this thesis. Besides entry-deterrence games, monitoring games and principal-agent games (see for example Maschler et al. [2013]) are also typical examples of games which may have a similar interaction structure if we extend them with an Intelligence Service.

Our findings can also contribute to the analysis of signalling games. In a usual signalling game the sender can determine which signal he will send to the receiver. This happens for example in the Job Market Signalling Game introduced by Spence [1973] or the Limit Pricing Game discussed by Milgrom and Roberts [1982]. In the game we consider in this thesis, both the

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sender (agent) and the receiver (inspector) have to cope with a noisy signal which they cannot control for 100%. This is clearly a different way of modelling a signalling game. Therefore this thesis can give insight in modelling signalling games in other fashions than the traditional one.

Hereafter, Chapter 2 will deal the game-theoretical model we constructed. Chapter 3 is an introduction to Chapter 4, which deals with the version of the game without Intelligence Service. Chapter 5 is about the version of the game with IS. Chapter 6 will deal with the comparison between the different versions of the game. In Chapter 7 we give our conclusion. In the Appendix one can find the best-reply functions, the proofs of the propositions and some other derivations.

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Chapter 2

The Model

To model a deterrence inspection game which integrates the concepts of an Intelligence Service (hereafter: IS), a provocateur-type of agent and an intermediary level of weapons, we construct a game with two versions: a version without IS and a version with IS. We will first explain the version without IS. Consider a game constisting of two players, called Agent and Inspector (see also Figure 2.1).

Figure 2.1: Game tree for the version without IS

Agent has the capability of building intermediary weapons (like semi-automatic weapons or tanks) and of building weapons of mass destruction (like a nuclear bomb) and would like to

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possess both these kinds of weapons. Inspector wants to prevent Agent from possessing those weapons, and he has the capability to attack Agent, which will lead to the destruction of all weapons of Agent.

The game starts with a move by Nature, which determines the type of Agent. He either becomes a Provocateur or a Deterrer, so if t denotes the type of Agent, then: t_{∈ {Provocateur, Deterrer}.} The types only differ with respect to their preference relations, which cause different payoffs for the different types. Apart from this difference, the types are completely the same. The likeli-hood that Agent becomes a Provocateur is β, which is common knowledge. When his type is determined, Agent moves. Agent has the same action set for both types: _{{NBO, NB, B}1, B2}.

In case Agent plays N BO or N B, he does not build weapons. The difference between those actions is that in case he plays N BO he also opens his facilities to show that he is not building weapons, while if he plays N B he keeps his facilities closed. Remark that this last action cor-responds to the behaviour of Saddam prior to the Second Gulf war: not building weapons (at least not weapons of mass destruction), while keeping the facilities closed. If Agent plays B1, he

only builds intermediary weapons. If he plays B2, he builds both weapons of mass destruction

and intermediary weapons.

Denote the probability that the Deterrer plays B1 by xDeter1 , the probability that the Deterrer

plays B2 by xDeter2 , the probability that the Provocateur plays B1 by xP rov1 and the probability

that the Provocateur plays B2 by xP rov2 . It will turn out that for both versions of the game there

exists no equilibrium, either in pure strategies or in mixed strategies, for which the Deterrer plays N B or the Provocateur plays N BO.

After Agent has played his action, Inspector moves. He can only observe whether Agent has opened his facilities or not, regardless of Agent’s type. This leads to two information sets for Inspector: one for which Agent has opened his facilities and one for which Agent has kept his facilities closed. If Agent has opened his facilities, Inspector knows for sure that Agent is not building weapons, so that attacking does not make sense anymore. Hence, we assume from now on that the game ends if Agent opens his facilities (by playing N BO). If Agent keeps his facilities closed, Inspector has to decide whether to attack, A, or not attack, N A. Therefore Inspector has the following action set: _{NAC, AC}, where the subscript ’C’ refers to the

infor-mation set in which Agent has kept his facilities closed. From now on we denote the probability that Inspector plays N AC by q.

The Provocateur prefers to be attacked above not to be attacked when he has played neither B1 nor B2. In the introduction we have explained why such preferences can make sense: the

(unjustified) attack causes so much (domestic or international) support that it outweighs the damage caused by the attack. On the other hand the Deterrer prefers not to be attacked

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above to be attacked when he has played neither B1 nor B2. Apart from this difference, the

two types have the same preferences. Given that they are not attacked, they want to build as big weapons as possible. Therefore they both consider (B2, N AC) as the best outcome and

(B1, N AC) as the second best outcome. Furthermore, their damage is the biggest if they build

a weapon whereupon it is destroyed. This holds even more for weapons of mass destruction than for intermediary weapons. Therefore they consider (B1, AC) as the second worst outcome

and (B2, AC) as the worst outcome.

Based on the foregoing we can assume the following preference relation for the Provocateur: (B2, N AC) (B1, N AC) (NB, AC) (NB, NAC) (B1, AC) (B2, AC);

and the following preference relation for the Deterrer:

(B2, N AC) (B1, N AC) (NBO, NAC) (NBO, AC) (B1, AC) (B2, AC).

Inspector’s ultimate goal is to prevent Agent from getting weapons. He can achieve this goal by attacking Agent. However, attacking involves a cost, so he is even more content if he can attain his goal without attacking. If Inspector attacks, he wants his attack to be justified, which is only the case if it becomes clear that Agent was building weapons. Still he prefers an unjustified at-tack above letting Agent get weapons. The latter would lead to such a dangerous situation that all other scenarios are preferable for Inspector. An example of this is the Israel-Iran conflict. Iran and Israel are obviously enemies, so Israel wants to prevent Iran from possessing a nuclear bomb. Although an unjustified attack on the facilities of Iran would cause much negative media attention and support for Iran, Israel still prefers it above the situation that Iran would possess a nuclear bomb, which has a devastating potential.

The foregoing leads us to assume the following preference relations for Inspector:2 (N B, N AC) (B2, AC) (B1, AC) (NB, AC) (B1, N AC) (B2, N AC).

The above describes the version of the game without IS. The version of the game with IS is identically the same, except that it includes an IS. We will describe it in detail in chapter 5.

2_{Another interesting variation would be if we assumed: (B}

1, N AC) (N B, AC) (B2, N AC). This means

that Inspector only prefers an unjustified attack above letting Agent get weapons of mass destruction, but not above letting Agent getting intermediary weapons. It clearly gives more weight to the loss of reputation of Inspector caused by an unjustified attack. Because the model becomes already rather complicated, this is beyond the scope of this thesis, but it is certainly an interesting topic for further research.

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Chapter 3

A simplified example

In this chapter we solve a game (without IS) similar to the one in Chapter 4, apart from the fact that in this chapter we adopt a payoff structure for Agent such that B1 is weakly dominated. In

Chapter 4 we will show why it is more realistic if B1 is not weakly dominated, but that this also

leads to quite more complicated equilibria. Therefore this chapter could be seen as a simplified example of the game discussed in Chapter 4.

We choose for normalized payoffs, so for both players the payoff is always between 0 and 1. Given Agent is a Deterrer, we assume that playing N BO gives d2 utility to the Deterrer and

1 utility to the Inspector. If the Deterrer keeps his facilities closed, we assume the following payoff matrix:

Agent: type Deterrer

N B B1 B2

Inspector N AC 1, d2 r1, d3 0, 1

AC r2, d1 r3, 0 r3, 0

where 0 < d1< d2 < d3< 1 and 0 < r1 < r2< r3 < 1.

Given Agent is a Provocateur, we assume that playing N BO gives p1 utility to the Provocateur

and 1 utility to the Inspector. If the Provocateur keeps his facilities closed, we assume the following payoff matrix:

Agent: type Provocateur

N B B1 B2

Inspector N AC 1, p1 r1, p3 0, 1

AC r2, p2 r3, 0 r3, 0

where 0 < p1 < p2 < p3 < 1. Note that in the payoff matrices above Inspector is the row player

and Agent is the column player.

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We first consider the cases where Agent’s type is deterministic: β = 0 or β = 1.

Proposition 1.1: Suppose β = 0. Then Gβ has a unique Nash equilibrium such that Agent

(the Deterrer) plays N BO and that Inspector plays N AO and AC.3

Proof: See Appendix A.1

This proposition makes clear that if Inspector knows for sure that Agent is a Deterrer, he will act aggressively if Agent keeps his facilities closed. This causes Agent to decide not to build weapons and to open his facilities to show so. Therefore if Inspector knows that Agent is a Deterrer, he will end up in his best-case scenario: Agent does not build weapons and Inspector does not attack.

Proposition 1.2: Suppose β = 1. Then Gβ has a unique Nash equilibrium such that Agent

(the Provocateur) strictly mixes between N B and B2 and that Inspector strictly mixes between

N AC and AC. Furthermore the Nash equilibrium satisfies

xP rov₂ = 1− r2 1_{− r}2+ r3 , q = p2 1_{− p}1+ p2 ,

where q is the probability that Inspector plays N AC and xP rov2 is the probability that the

Provo-cateur plays B2.4

Observe that if Agent is a Provocateur there is a unique equilibrium such that both players mix their pure strategies, opposite to the case that Agent is a Deterrer. In this sense one might say that if Agent is a Provocateur, the equilibrium structure is more difficult than if Agent would be a Deterrer. This is also quite intuitive, since the preferences of the Deterrer are easier to cope with: The Deterrer always prefers not to be attacked above to be attacked, while for the Provocateur this depends on which action he is playing. If he is not building weapons, he prefers to be attacked. If he builds weapons, he does not want to be attacked.

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Strictly speaking we should also state what Agent would do in the counterfactual case that he was a Provo-cateur. Since this is a strictly counterfactual case (where the counterfactuality is caused by the common prior β and not by the strategy profile which constitutes the Nash equilibrium), this can be any possible action for the Provocateur. So strictly speaking there are infinitely many Nash equilibria.

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Let us now consider the incomplete information case where Inspector does not know the type of Agent, but assigns a (prior) probability β, 0 < β < 1, that Agent is a Provocateur.

Proposition 1.3: Suppose 0 < β < 1 and d2 6= ₁_−pp₁2_+p₂.5 Then the game has a unique

sequential equilibrium (hereafter: SE). It satisfies the following:

(i) Inspector strictly mixes his pure strategies N AC and AC. This implies that the Deterrer

will never play N B and the Provocateur will never play N BO.

(ii) If d2 > ₁_−pp₁2_+p₂ the Deterrer plays N BO and the Provocateur mixes between N B and B2

with xP rov₂ = 1−r2

2−r2. Inspector mixes between N AC and AC with q =

p2

1−p1+p2.

(iii) If d2 < _1−pp₁2_+p₂ and β < _1−rr₂3_+r₃, the Deterrer mixes between N BO and B2 with xDeter2 = β(1−r2)

(1−β)r3 and the Provocateur plays N B. Inspector mixes between N AC and AC with q = d2. (iv) If d2 < ₁_−pp₁2_−p₂ and β ≥ ₁_−rr₂3_+r₃, the Deterrer plays B2 and the Provocateur mixes

between N B and B2 with xP rov2 = 1−_β(1−rr3₂+r3). Inspector mixes between N AC and AC

with q = p2

1−p1+p2. Proof: See Appendix A.1

Remark that case (ii) requires d2> ₁_−pp₁2_−p₂. Then the payoff for the Deterrer for not building

weapons is relatively high compared to the payoff for the Provocateur for not building weapons. We can explain this as follows: if Agent does not build weapons with some probability, he prefers to do so by his Deterrer type. Therefore the Deterrer plays a pure N BO in this equilibrium, while the Provocateur does not play a pure N B. Nevertheless, the Provocateur always (also in the cases (iii) and (iv)) plays N B with a strictly positive probability. If he would not do so, Inspector could deduce from the mere fact that Agent kept his facilities closed, that Agent is building weapons. This would result in a pure attack of Inspector. Therefore only way for Agent to prevent this, is by letting the Provocateur play N B with a strictly positive probability. The strategy of the Inspector in case (ii) is playing q = p2

1−p1+p2. Note that this term is increasing in p1 and p2 which are exactly the payoffs for the Provocateur in case he plays N B. So if the

Provocateur gets more utility from playing N B, Inspector will act less aggressively. 5_{Remark: We assume d}

26=_1−pp₁2_−p₂ in order to prevent multiple equilibria. Although the case d2=_1−pp₁2_+p₂

has mass zero, we have worked it out in Appendix A.4 for the sake of completeness. We find that the more one type plays B2, the less the other type plays B2. We could interpret this as an indication that Agent can never

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The cases (iii) and (iv) require: d2 < ₁_−pp₁2_+p₂. This means that the payoff of the Deterrer

for not building weapons is relatively low compared to the payoff of the Provocateur for not building weapons. Therefore, for those cases the Deterrer will never play a pure N BO. He will at most mix it with B2.

In case (iii) β < r3

1−r2+r3 means that β is relatively low, so Agent is relatively often a Deterrer. So then the Provocateur can ’allow himself’ to play a pure N B. The disadvantage that he does not build weapons as a Provocateur is outweighted by the oppurtunity of eliciting an unjusti-fied attack and the relatively high probability that he is a Detterer, in which case he does build weapons (because he mixes with N BO with B2). Furthermore, it is interesting to notice that

case (iii) shows that is possible to have a Nash equilibrium in which the Provocateur plays a pure N B. Despite his provocative nature, it is apparently possible to have a Nash equilibrium in which he plays the least provocative strategy possible: N B. Remark that while the Provo-cateur does not provoke, this does not mean that Agent does not provoke at all. The Deterrer plays B2 with positive probability, so in this Nash equilibrium Agent certainly provokes, only

not by his type the Provocateur. In case (iii) Inspector plays q = d2. This means that if the

Deterrer gets more utility from playing N BO, Inspector will act less aggressively.

In case (iv) β≥ r3

1−r2+r3 means that β is relatively high, so Agent is relatively often a Provoca-teur. An explanation for the fact that the Deterrer plays B2and the Provocateur mixes N B and

B2 is the following: Both types play B2 with positive probability in order to ’seduce’ Inspector

to attack. Next to B2 the Provocateur also plays N B with positive probability, so he hopes

that the attack of Inspector comes when he plays N B. Then the attack is unjustified and the Provocateur benefits from that. Note that in this case the Inspector plays q = p2

1−p1+p2, which is the same as in case (ii). Therefore also in this equilibrium Inspector will act less aggressively if the Provocateur gets more utility from not building weapons.

Remark that for all the cases of Proposition 1.3 Agent strictly mixes between not building weapons at all and building weapons of mass destruction, whether it is by his type Deterrer or by his type Provocateur. Also Inspector strictly mixes between attacking and not attacking for all the cases of Proposition 1.3. Therefore the simplified example which is analyzed in this Chapter, has a structure which is similar to the Matching Pennies Game.

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Chapter 4

The case without Intelligence Service

4.1 The payoff structure

In the last chapter we analyzed a game where B1 was weakly dominated by B2 for Agent.

This may be not very realistic, since it is reasonable that Agent loses less resources in case of (B1, AC) compared to (B2, AC). Therefore we will transform the game such that B1 is not

weakly dominated anymore. We will call this game Gβ.

We choose for normalized payoffs, so for both players the payoff is always between 0 and 1. Given Agent is a Deterrer, we assume that playing N BO gives d3 utility to the Deterrer and 1

utility to Inspector. If the Deterrer keeps his facilities closed, we assume the following payoff matrix:

Agent: type Deterrer

N B B1 B2

Inspector N AC 1, d3 r1, d4 0, 1

AC r2, d2 r3, d1 r3, 0

where 0 < d1< d2 < d3< 1 and 0 < r1 < r2< r3 < 1.

Given Agent is a Provocateur, we assume that playing N BO gives p2 utility to the Provocateur

and 1 utility to Inspector. If the Provocateur keeps his facilities closed, we assume the following payoff matrix:

Agent: type Provocateur

N B B1 B2

Inspector N AC 1, p2 r1, p4 0, 1

AC r2, p3 r3, p1 r3, 0

where 0 < p1 < p2 < p3 < 1. Note that in the payoff matrices above Inspector is the row player

and Agent is the column player.

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4.2 Assumptions

Before we go on to the best response functions and equilibria, we will state two assumptions: Assumption 1. d3−d1 d4−d1 ≤ d3≤ d1 1−d4+d1. Assumption 2. p1−p3 p1+p2−p3−p4 ≤ p3 1−p2+p3 ≤ p1 1+p1−p4. 6

Observe that the expression d3−d1

d4−d1 is decreasing in d1 and d4. So in order for

d3−d1

d4−d1 ≤ d3 to hold, d1 and d4 have to be relatively large compared to d3. Remark that _1−dd₄1_+d₁ is increasing

in d1 and d4, such that in order for d3 ≤ ₁_−dd₄1_+d₁ to hold, d1 and d4 have to be again relatively

large compared to d3. Therefore an interpretation of Assumption 1 is that the payoffs for the

Deterrer for playing B1 (namely d4 and d1) have to be relatively large compared to the payoffs

for not building weapons, while not being attacked, which is d3. With the same argument

as before we can derive an analog inpretation for Assumption 2: The payoffs for the Provoca-teur for playing B1 have to be relatively large compared to the payoffs for not building weapons.

Remark that Assumption 1 and Assumption 2 are independent from each other, because As-sumption 1 is only about the payoff for the Deterrer while Assuption 2 only deals with the payoff for the Provocateur. Therefore it it is possible that one assumption is true while the other is not. Hereafter, these assumptions will turn out to be essential.

4.3 Best response functions

Let BRDeterrer(q) be the best-response function of the Deterrer, where q is the probability that

Inspector plays N AC. In Appendix A.2 we derived that if Assumption 1 holds, this function is

given by: BRDeterrer(q) =            N BO for q _∈h0,d3−d1 d4−d1 i B1 for q ∈ h d3−d1 d4−d1, d1 1−d4+d1 i B2 for q ∈ h d1 1−d4+d1, 1 i

If Assumption 1 does not hold, this function is given by:

BRDeterrer(q) =    N BO for q_{∈ [0, d}3] B2 for q∈ [d3, 1]

6_{We exclude the cases} d3−d1

d4−d1 = d3 = d1 1−d4+d1 and p1−p3 p1+p2−p3−p4 = p3 1−p2+p3 = p1 1+p1−p4, which correspond

to the types of Agent mixing all their pure strategies. This is to prevent extremely cumbersome computations. Furthermore it is highly unlikely to have parameter values such that both equalities hold.

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Let BRP rovocateur(q) be the best-response function of the Provocateur. In Appendix A.2 we

derived that if Assumption 2 holds, this function is given by:

BRP rovocateur(q) =            N B for q ∈h0, p1−p3 p1+p2−p3−p4 i B1 for q ∈ h p1−p3 p1+p2−p3−p4, p1 1+p1−p4 i B2 for q ∈ h p1 1+p1−p4, 1 i

If Assumption 2 does not hold, this function is given by:

BRP rovocateur(q) =      N B for q _∈h0, p3 1−p2+p3 i B2 for q ∈ h p3 1−p2+p3, 1 i

The above makes clear that the Deterrer will never play B1 in equilibrium if Assumption 1

does not hold and that the Provocateur will never play B1 in equilibrium if Assumption 2 is

not true. This could be interpreted as follows: If the assumptions are not true, Agent will only play his ’extreme’ strategies, that is either building weapons of mass destruction or not building any weapons at all. This is also what we would expect from the previous paragraph, where we observed that the assumptions are only true if the payoffs for playing B1 are relatively high.

In the Appendix A.2 we will show that in any equilibrium the Inspector will strictly mix between attacking and not attacking if Agent keeps his facilities closed. This requires that the expected utility of attacking, given that Agent keeps his facilities closed, is equal to the expected utility of not attacking, given that Agent keeps his facilities closed. Using this equality, in combina-tion with the best response funccombina-tions of Agent, turns out to be sufficient to find the equibria. Therefore we will use this equality, and not the cumbersome best reply function of Inspector, to compute the equilibria in the next sections.

4.4 Equilibria if type of the Agent is known

The case β = 0 is straightforward. With exactly the same reasoning as in Proposition 1.1, we find for β = 0 a unique NE where Agent (the Deterrer) plays N BO and Inspector acts aggressively by always playing AC. The case β = 1 is somewhat more difficult.

Lemma 2.1. Suppose β = 1. If Assumption 2 is true, there is a unique SE where Agent (the Provocateur) strictly mixes between N B and B1 and Inspector strictly mixes between N AC and

AC such that

xP rov₁ = 1− r2 1_{− r}1− r2+ r3

q = p1− p3

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Remark that if Assumption 2 is true, the Provocateur will never play B2. This is an incentive

for Inspector to act relatively peaceful, because he knows for sure that Agent (the Provocateur) will never possess weapons of mass destruction.

Observe that the expression xP rov₁ = 1−r2

1−r1−r2+r3 is increasing in r1 and decreasing in r2 and r3. This means that if Inspector’s payoff for not attacking (r1) increases, the Provocateur will

become more agressive, while if Inspector’s payoffs for attacking (r2 and r3) increase, the

Provo-cateur will become less agressive. We can also observe that the probability that Inspector does not attack, q = p1−p3

p1+p2−p3−p4, is increasing in the payoffs for the Provocateur for not building weapons (p2 and p3) and decreasing in the payoffs for the Provocateur for building intermediary

weapons (p1 and p4). Both observations above can be explained by the fact that if a player gets

a higher payoff for a certain action (ceteris paribus), he will get a stronger incentive to play that action, such that he will play it more often.

Lemma 2.2. Suppose β = 1. If Assumption 2 is not true, there is a unique SE where Agent (the Provocateur) strictly mixes between N B and B2 and Inspector strictly mixes between N AC

and AC such that

xP rov₂ = 1− r2 1− r2+ r3

q = p3

1_{− p}2+ p3

Observe that if Assumption 2 is not true, Provocateur will never play B1. Therefore Inspector

realizes that Provocateur will always play an ’extreme’ strategy, that is: building weapons of mass destruction or not building any weapons.

Remark that xP rov₂ = 1−r2

1−r2+r3 is decreasing in r2 and r3, which are the payoffs for Inspector for attacking. If these gets higher, Inspector will attack more often, which causes the Provocateur to be more cautious by reducing the probability that he builds weapons of mass destruction.

With the same argument as before one can show that q = p3

1−p2+p3 is increasing in p2 and p3, which are the payoffs for the Provocateur for not building weapons. This means that if these gets higher, the Provocateur will build less weapons, such that Inspector becomes less agressive.

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4.5 Equilibria if type Agent is unknown

In subsection 4.4.1 we will solve the Nash equilibria if the type of Agent is unknown, so for 0 < β < 1. This will be quite elaborate, so if the reader is less interested in technical details we advise to continue with subsection 4.4.2 where the results are briefly summarized.

4.5.1 Propositions

There are four classes of propositions (represented by Propostion 2.1-2.4), depending on whether Assumption 1, Assumption 2 or both are true or not. They are given below, together with a interpretation. We start with the proposition that gives the overview:7

Proposition 2: Suppose 0 < β < 1. Then Gβ has sequential equilibria such that for each SE

holds:

(i) Inspector strictly mixes his pure strategies N AC and AC. This implies that the Deterrer

will never play N B and the Provocateur will never play N BO. (ii) The Provocateur plays N B with a strictly positive probability

(iii) If Assumption 1 is true, the Deterrer never mixes between N BO and B2. If Assumption

1 is not true, the Deterrer will only mix between N BO and B2.

(iv) If Assumption 2 is true, the Provocateur will never mix between N B and B2. If

Assump-tion 2 is not true, the Provocateur will only mix between N B and B2.

How can we explain the fact that the Provocateur will always play N B with positive probability? First of all, not building weapons, while keeping the facilities closed, is the only pure strategy which can elicit an unjustified attack. This causes (domestic or international) support for the Provocateur, which gives him some utility. Nevertheless we saw that an unjustified attack was not his best case scenario, because he still prefers building weapons and not being attacked above an unjustified attack. Therefore, trying to elicit an unjustified attack might not be a sufficient explanation for always playing N B with a positive probability. Now another explanation for this strategy is that the Provocateur tries to get Inspector to abstain from attacking. If the Provocateur also builds weapons with a positive probability (smaller than 1, because he also plays N B), he thereby creates the possibility of building weapons, while not being attacked. Thus the Provocateur is choosing his strategy such that he tries to reach his best case scenario. 7_{Note that the propositions of this section are strictly speaking existence propositions. We can also prove that}

the equilibria we found are the only possible equilibria. A sketch of this proof in words is given in Appendix A.2 .

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Proposition 2.1: Suppose 0 < β < 1 and that both Assumption 1 and Assumption 2 are true. Then Gβ has at least one SE.

Proof: See Proposition 2.1.1, Proposition 2.1.2, Proposition 2.1.3, Proposition 2.1.4 and Propo-sition 2.1.5, which constitute together a constructive proof.

Proposition 2.1.1: Suppose 0 < β < 1, both Assumption 1 and Assumption 2 are true and

d3−d1

d4−d1 <

p1−p3

p1+p2−p3−p4. Then Gβ has at least one SE: Inspector mixes between N AC and AC with q = d3−d1

d4−d1, while the Deterrer plays B1 with probability x

Deter

1 = _(1−β)(r(1−r2₃)β_−r₁ and N BO with

probability 1_{− x}Deter₁ (which requires β _≤ r3−r1

1−r1−r2+r3) and the Provocateur plays N B. Proof: See Appendix A.2

Remark that xDeter₁ is increasing in r1 and decreasing in r2 and r3. This can be understood as

follows: r1 is the reward for Inspector if he plays N AC, while the Deterrer plays B1. If this

increases, Inspector is less afraid to face a B1, while he plays N AC, so he will play more N AC.

Now, Agent in turn will play more B1, because the probability that intermediary weapons will

be destroyed decreases. r2 is the payoff to Inspector if he plays AC, while the Deterrer plays

N B. If this increases, the costs of an unjustified attack will become smaller, so Inspector is willing to play more AC. The Deterrer in turn will become more cautious and decreases the

probability that he builds intermediary weapons. r3 is the payoff to Inspector is he attacks,

while the Deterrer plays B1. If this increases, Inspector gets more rewarded for acting

aggres-sively, so the Deterrer will respond to that by reducing the probability of building intermediary weapons.

Furthermore, xDeter₁ is increasing in β. The best interpretation for this is in terms of a tradeoff. Remember that Agent’s goal is to possess weapons. If the Provocateur does not build weapons at all (which is the case here, since the Provocateur plays a pure N B), the only way for Agent to possess weapons is via his type Deterrer. If β increases, Agent will become more often a Provocateur. This has to be ’compensated’ by the fact that if Agent becomes a Detterer, he will build weapons with a higher probability, so xDeter₁ increases. The behaviour of the Provocateur should be interpreted as trying to entrap Inspector. Because the Provocateur does not build any weapons, he tries to seduce Inspector to attack with a sufficiently low probability, such that if Agent becomes a Deterrer and builds weapons, he is not attacked. From the perspective of Inspector though, the Deterrer is the bait, which triggers him not to attack with a too low probability.

Note that this equilibrium is hybrid, since the Provocateur never opens his facilities, while the Deterrer mixes between opening and not opening his facilities. Furthermore we can observe that

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this equilibrium is rather peaceful: The Provocateur never builds weapons, while the Deterrer never builds weapons of mass destruction. As a consequence, Inspector does not demonstrate too aggressive behaviour.

Proposition 2.1.2: Suppose 0 < β < 1, both Assumption 1 and Assumption 2 are true and

d1

1−d4+d1 <

p1−p3

p1+p2−p3−p4. Then, depending on additional assumptions for β, Gβ has 3 other SE:

(i) Inspector mixes between N AC and AC with q = ₁_−dd₄1_+d₁, while the Deterrer mixes between

B1 and B2 with xDeter1 = β(1−rr1(β−1)2+r3)−r3 and x

Deter

2 = 1− xDeter1 and the Provocateur plays

N B. This case requires: r3−r1

1−r1−r2+r3 ≤ β ≤

r3

1−r2+r3.

(ii) Inspector mixes between N AC and AC with _1−dd₄1_+d₁ < q < _p₁_+pp1₂−p_−p3₃_−p₄, while the Deterrer

plays B2 and the Provocateur plays N B. This case requires: β = ₁_−rr₂3_+r₃.

(iii) Inspector plays N AO and mixes between N AC and AC with q = _p₁_+pp1₂_−p−p3₃_−p₄, while the

Deterrer plays B2 and the Provocateur mixes N B and B1 with xP rov1 = β(1−r_β(1−r12+r−r23)−r+r33), xDeter₂ = 1− xDeter

1 . This case requires: β ≥ _1−rr₂3+r3. Proof: See Appendix A.2

Note that d1

1−d4+d1 is increasing in d1 and d4, which are the payoffs to the Deterrer if he plays B1. Because this proposition requires: _1−dd₄1_+d₁ < _p₁_+pp1₂−p_−p3₃_−p₄, we know that d1 and d4 have

to be relatively small, at least compared to p3 and p4, which are respectively the payoff to the

Provocateur in case of an unjustified attack and the payoff to the Provocateur if he builds in-termediary weapons, but is not attacked. A relatively small d1 and d4 means that the Deterrer

does not get much reward for playing B1. Therefore, the Deterrer never plays a pure B1 in

this proposition. If he plays B1 at all, he mixes it with B2. So if the Deterrer builds

interme-diary weapons, it is always combined with weapons of mass destruction. On the other hand the Provocateur never builds weapons of mass destruction, so his strategy counterbalances the more aggressive strategy of the Deterrer.

What is remarkable about this proposition is that the Inspector plays a more aggressive strategy if the Deterrer mixes B1 and B2 (case (i)) than if the Deterrer plays a pure B2 (case (ii) and

(iii)). Note that the Deterrer only mixes B1 and B2 if β is relatively low, so if Agent’s type is

relatively often a Deterrer, who plays a more aggressive strategy than the Provocateur. This in-fluence of the value of β is so strong, that is causes Inspector to act more aggressively, although the strategy of the Deterrer is less aggressive than in the other cases. This aggressiveness of Inspector means that it is optimal for the Provocateur to play N B, such that the probability of an unjustified attack, which gives him utility, increases and that it, as we mentioned before, counterbalances the strategy of the Deterrer.

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Note that for all equilibria of this proposition, both types of Agent never open their facilities. Therefore, these equilibria are all pooling equilibria: Inspector is always in the same information set, namely the one corresponding to the closed facilities.

Proposition 2.1.3 Suppose 0 < β < 1, both Assumption 1 and Assumption 2 are true and: (i) d3−d1

d4−d1 <

d1

1−d4+d1 ≤

p1−p3

p1+p2−p3−p4, then we have the SE: Inspector mixes between N AC and AC with d_d3₄−d_−d1₁ < q < ₁_−dd₄1_+d₁, while the Deterrer plays B1 and the Provocateur plays N B.

This case requires: β = r3−r1

1−r1−r2+r3 (ii) d3−d1 d4−d1 < p1−p3 p1+p2−p3−p4 < d1

1−d4+d1, then we have the SE: Inspector mixes between N AC and AC with q = _p₁_+pp1₂−p_−p3₃_−p₄, while the Deterrer plays B1 and the Provocateur mixes between

N B and B1 with xP rov1 =

(1−β)(r1−r3)

β(1−r1−r2+r3 (no further requirement on β). (iii) d1

1−d4+d1 =

p1−p3

p1+p2−p3−p4, then we have the SE: Inspector mixes between N AC and AC with

q = d1

1−d4+d1, while the Deterrer mixes between B1 and B2 with xDeter₁ = xP rov1 [β(1−r1−r2+r3)]−β(1−r2+r3)+r3

(1−β)r1 and x

Deter

2 = 1− xDeter1 and the Provocateur

mixes N B and B1 with xP rov1 =

xDeter

1 (1−β)r1+β(1−r2+r3)−r3

β(1−r1−r2+r3) , x

P rov

2 = 1− xP rov1 . This case

requires: r3−β(1−r2+r3) (1−β)r1 ≤ x Deter 1 ≤ 1, β(1_β(1−r−r2₁+r_−r3₂)+r−r33) ≤ x P rov 1 ≤ 1 + _β(1−rr1₁−r_−r3₂+r3) and β≥ r3−r1 1−r2+r3. Proof: See Appendix A.2

Since the cases of this proposition have different additional assumptions, we will interpret them separately:

(i): In order for d3−d1

d4−d1 <

d1

1−d4+d1 to hold d3 has to be relatively small compared to d1 and d4. Because d3 is the payoff for the Deterrer for playing N BO and d1 and d4 are the payoffs for

the Deterrer if he plays B1, this triggers him to to build intermediary weapons instead of not

building weapons and opening his facilities. In order not to evoke a too aggressive reaction of Inspector, the Provocateur compensates the aggressive strategy of the Deterrer by never build-ing any weapons.

(ii): In order for p1−p3

p1+p2−p3−p4 <

d1

1−d4+d1 to hold, d1 and d4 have to be relatively large, at least compared to p3 and p4. Since d1 and d4 are the payoffs to the Deterrer for playing B1, this

means that playing B1 is relatively beneficial for the Deterrer. Therefore the Deterrer plays a

pure B1in this equilibrium, that is: he always builds intermediary weapons. Again this strategy

is ’compensated’ by a less aggressive strategy of the Provocateur, in order not to evoke a too aggressive reaction of Inspector.

(iii): In this equilibrium Inspector attacks with a relatively low probability. This is caused by the the facts that Agent’s type is relatively often a Provocateur (β-value is high) and that the Provocateur plays, at least from the perspective of Inspector, a relatively ’harmless’ strategy:

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he mixes between not building weapons and building intermediary weapons. So of the three equilibria of this propostion, (iii) has the most peaceful character.

Note that for all equilibria of this proposition, both types of Agent never open their facilities. Therefore, these equilibria are all pooling equilibria.

Proposition 2.1.4 Suppose 0 < β < 1, both Assumption 1 and Assumption 2 are true and

d3−d1

d4−d1 =

p1−p3

p1+p2−p3−p4, then we have the SE: Inspector mixes between N AC and AC with q =

d3−d1

d4−d1, while the Deterrer mixes between N BO and B1 with x

Deter 1 =

xP rov

1 [β(1−r1−r2+r3)]−β(1−r2)

(1−β)(r1−r3) and the Provocateur mixes between N B and B1 with probability xP rov1 = β(1−r2

)+xDeter

1 (1−β)(r1−r3)

β(1−r1−r2+r3) , where the two latter mixing probabilities require: 0≤ xDeter

1 ≤ _(1−β)(rβ(r2−1)₁_−r₃) and 1−

r3−r1

β(1−r1−r2+r3 ≤ xP rov₁ _≤ 1−r2

1−r1−r2+r3(but no further requirement on β). Proof: See Appendix A.2

The best interpretation for d3

1−d4+d1 =

p1−p3

p1+p2−p3−p4 is that the payoffs for the Deterrer and the Provocateur are ’in balance’. They are not such that it is clearly better for one type to build weapons than for the other type. Therefore both types mix between building weapons or not building weapons. Since both types of Agent play a mixed strategy, the belief updating of Inspector becomes more complicated. The best interpretation for his strategy is that it is ’in balance’ between aggressive and not aggressive. So this proposition represents a ’balanced’ equilibrium.

Proposition 2.1.5 Suppose 0 < β < 1, both Assumption 1 and Assumption 2 are true and d3−d1

d4−d1 >

p1−p3

p1+p2−p3−p4, then we have the SE: Inspector mixes between N AC and AC with q = p1−p3

p1+p2−p3−p4 while the Deterrer plays N BO and the Provocateur mixes between N B and B1 with xP rov₁ = 1−r2

1−r1−r2+r3 Proof: See Appendix A.2

Note that for d3

1−d4+d1 >

p1−p3

p1+p2−p3−p4 to hold, d3, which is the payoff for the Deterrer for playing N BO, has to be relatively large. Therefore the Deterrer plays a pure N BO. The Provocateur ’compensates’ the completely harmless strategy of the Deterrer, by playing a more aggressive strategy, such that Agent still has the possibility of achieving his ultimate goal: possessing weapons. This is clearly a separating equilibrium, such that Inspector can deduce which type of Agent he faces. He knows that if Agent keeps his facilities closed, he faces the Provoca-teur type. Therefore, his probability of attack only depends on the payoff for the ProvocaProvoca-teur: q = p1−p3

p1+p2−p3−p4. The more utility the Provocateur gets from playing aggressive strategies (p1 and p4), the higher the probability that Inspector will attack. On the other hand, if the

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Provo-cateur’s payoffs for peaceful strategies (p2 and p3) increase, he will act less agressive, in case it

is favourable for Inspector to reduce his probability of attack. Remember after all that in case Agent does not build weapons, Inspector is better off by not attacking, because it saves him the cost of an attack.

Proposition 2.2 Suppose 0 < β < 1 and Assumption 1 is not true, while Assumption 2 is true. Then Gβ has at least one SE.

Proof: See Proposition 2.2.1, Proposition 2.2.2 and Proposition 2.2.3.

We make the remark that Proposition 2.2-2.3.3 will not be interpreted because their interpre-tation is either equal to the one of proposition 2.1 or to the one of proposistion 2.4.

Proposition 2.2.1 Suppose 0 < β < 1, Assumption 1 is not true, while Assumption 2 is true and q = p1−p3

p1+p2−p3−p4 < d3 then we have one SE: Inspector mixes between N AC and AC with q, while the Deterrer pays N BO and the Provocateur mixes N B and B1 with xP rov1 = 1−r11−r−r22+r3 Proof: See Appendix A.2

Proposition 2.2.2 Suppose 0 < β < 1, Assumption 1 is not true, while Assumption 2 is true and q = d3

(i) For q _{∈ (0,} p1−p3

p1+p2−p3−p4) we have the SE: Inspector mixes between N AC and AC with q, while the Deterrer mixes N BO and B2 with xDeter2 = _(1−β)rβ(1−r2₃ and the Provocateur plays

N B. 0_{≤ x}Deter₂ _{≤ 1 requires: β ≤} r3

1−r2+r3.

(ii) For q = p1−p3

p1+p2−p3−p4 we have the SE: Inspector mixes between N AC and AC with q, while the Deterrer mixes N BO and B2 with xDeter2 = β(1−r2)−x

P rov

1 β[1−r1−r2+r3]

(1−β)r3 and the

Provocateur mixes N B and B1 with xP rov1 =

β(1−r2)−xDeter2 (1−β)r3

β(1−r1−r2+r3) . Proof: See Appendix A.2

Proposition 2.2.3 Suppose 0 < β < 1, Assumption 1 is not true, while Assumption 2 is true and q > d3.

(i) For q ∈ (0, p1−p3

p1+p2−p3−p4) we have the SE: Inspector mixes between N AC and AC with q, while the Deterrer plays B2and the Provocateur plays N B. This case requires:β = ₁_−rr₂3_+r₃.

(ii) For q = p1−p3

p1+p2−p3−p4 we have the SE: Inspector mixes between N AC and AC with q, while the Deterrer plays B2 and the Provocateur mixes N B and B1 with xP rov1 = β(1β(1−r−r21+r−r23)+r−r33). 0_{≤ x}P rov₁ < 1 implies: β _≥ r3

1−r2+r3. Proof: See Appendix A.2

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Proposition 2.3 Suppose 0 < β < 1 and Assumption 1 is true, while Assumption 2 is not true. Then Gβ has at least one SE.

Proposition 2.3.1 Suppose 0 < β < 1, Assumption 1 is true, while Assumption 2 is not true and q∈ (0, p3

1−p2+p3). (i) For q = d3−d1

d4−d1 we have the SE: Inspector mixes between N AC and AC with q, while the Deterrer mixes N BO and B1 with xDeter1 =

β(1−r2)

(1−β)(r3−r1 and the Provocateur plays N B. 0_{≤ x}Deter₁ _{≤ 1 implies: β ≤} r3−r1

1−r1−r2+r3.

(ii) For q ∈ (d3−d1

d4−d1,

d1

1−d4+d1) we have the SE: Inspector mixes between N AC and AC with q, while the Deterrer plays B1 and the Provocateur plays N B. This case requires: β =

r3−r1

1−r1−r2+r3.

(iii) For q = d1

1−d4+d1 we have the SE: Inspector mixes between N AC and AC with q, while the Deterrer mixes B1 and B2 with xDeter1 =

β(r2−1)+(1−β)r3

r1(1−β) and the Provocateur plays N B. 0≤ xDeter

1 ≤ 1 requires: ≥ _1−rr₁3_−r−r₂1+r3 ≤ β ≤

r3

1−r2+r3. (iv) For q_{∈ (} d1

1−d4+d1, 1) we have the SE: Inspector mixes between N AC and AC with q, while the Deterrer plays B2 and the Provocateur plays N B. This case requires: β = ₁_−rr₂3_+r₃.

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Proposition 2.3.2 Suppose 0 < β < 1, Assumption 1 is true, while Assumption 2 is not true

and q = p3

1−p2+p3. (i) For q _{∈ (0,}d3−d1

d4−d1) we have the SE: Inspector mixes between N AC and AC with q, while the Deterrer plays N BO and the Provocateur mixes N B and B2 with xP rov2 = 1−r1−r2+r2 3. (ii) For q = d3−d1

d4−d1 we have the SE: Inspector mixes between N AC and AC with q, while the Deterrer mixes N BO and B1 with xDeter1 =

β(1−r2)−xP rov2 β[1−r2+r3]

(1−β)(r3−r1) and the Provocateur mixes N B and B2 with xP rov2 = β(1−r2)−x

Deter

1 (1−β)(r3−r1)

β(1−r2+r3) . 0_{≤ x}Deter₁ _{≤ 1 and 0 ≤ x}P rov₂ < 1 require: 0_{≤ x}Deter₁ _≤ β(1−r2)

(1−β)(r3−r1) and

β(1−r2)−(1−β)(r3−r1)

β(1−r2+r3) , but no further requirements on β.

(iii) For q ∈ (d3−d1

d4−d1,

d1

1−d4+d1) we have the SE: Inspector mixes between N AC and AC with q, while the Deterrer plays B1 and the Provocateur mixes N B and B2 with

xP rov₂ = β(1−r2)−(1−β)(r3−r1)

β(1−r2+r3) . 0≤ x

P rov

2 < 1 require: β ≥ _1−rr₁3_−r−r₂1+r3 (iv) For q = d1

1−d4+d1 we have the SE: Inspector mixes between N AC and AC with q, while the Deterrer mixes B1 and B2 with xDeter1 =

xP rov

2 β(1−r2+r3)+β(r2−1)+(1−β)r3

(1−β)r1 and the Provoca-teur mixes N B and B2 with xP rov2 =

xDeter

1 (1−β)r1+β(1−r2)−(1−β)r3

β(1−r2+r3) . 0 _{≤ x}Deter₁ _{≤ 1 and 0 ≤ x}P rov₂ < 1 require: β(r2−1)+(1−β)r3

(1−β)r1 ≤ x Deter 1 ≤ _(1−β)rr3 1 and 1− r3 β(1−r2+r3) ≤ x P rov 2 ≤ β(1−r2)+(1−β)(r1−r3)

β(1−r2+r3) . This requires in turn: β≥

r3−r1

1−r1−r2+r3. (v) For q∈ ( d1

1−d4+d1, 1) we have the SE: Inspector mixes between N AC and AC with q, while the Deterrer plays B2 and the Provocateur mixes N B and B2 with xP rov2 = 1−β(1−rr32+r3. 0_{≤ x}P rov₂ < 1 requires: β _≥ r3

1−r2+r3. Proof: See Appendix A.2

Proposition 2.3.3 Suppose 0 < β < 1, Assumption 1 is true, while Assumption 2 is not true and q∈ (0, p3

1−p2+p3). Then Gβ does not have any SE.

Proof: In any SE the Provocateur should play N B with a positive probability, so this case leads to a contradiction.

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Proposition 2.4 Suppose 0 < β < 1 and both Assumption 1 and Assumption 2 are not true. Then Gβ has at least one SE.

Proposition 2.4.1 Suppose 0 < β < 1, both Assumption 1 and Assumption 2 are not true

and q = p3

1−p2+p3 < d3 we have the SE: Inspector mixes between N AC and AC with q, while the Deterrer plays N BO and the Provocateur mixes N B and B2 with xP rov2 = 1−r1−r2+r23.

Note that p3

1−p2+p3 is increasing in both p2 and p3. Therefore, the requirement:

p3

1−p2+p3 < d3 implies that p2 and p3, which are the payoffs to the Provocateur for playing N B, are relatively

small compared to d3. Since d3 is the payoff for the Deterrer for playing N BO, this means that

if Agent does not build weapons, he prefers to do so by his Deterrer type and not by his Provoca-teurr type. Therefore, the Deterrer plays a pure N BO, while the Provocateur mixes N B and B2.

Note that xP rov₂ is decreasing in both r2 and r3. For r2 this is straightforward: r2 is the payoff

that Inspector receives if he attacks while Agent does not build weapons. If this increases, In-spector is less afraid for an unjustified attack, so he will play more AC. Agent will become more

cautious and will decrease the probability with which he builds weapons of mass destruction. If r3 increases, Inspector gets more utility if he plays AC, given that Agent builds weapons.

So he will act more aggressively and Agent will react to that by building less weapons of mass destruction.

Proposition 2.4.2 Suppose 0 < β < 1, both Assumption 1 and Assumption 2 are not true and q = d3.

(i) For q_{∈ (0,} p3

1−p2+p3) we have the SE: Inspector mixes between N AC and AC with q, while the Deterrer mixes N BO and B2 with xDeter2 = β(1−r_(1−β)r23) and the Provocateur plays N B. 0≤ xDeter

2 ≤ 1 require: β ≤ 1−rr23+r3. (ii) For q = p3

1−p2+p3 we have the SE: Inspector mixes between N AC and AC with q, while the Deterrer mixes N BO and B2 with xDeter2 = β(1−r2)−x

P rov

2 β(1−r2+r3)

(1−β)r3 and the Provocateur

mixes N B and B2 with xP rov2 =

β(1−r2)−xDeter2 (1−β)

β(1−r2+r3) .

0 _{≤ x}Deter₂ _{≤ 1 and 0 ≤ x}P rov₂ < 1 require:0 _{≤ x}Deter₂ _≤ β(1−r2)

1−β and 1− β(1−rr32+r3) ≤ xP rov₂ ≤ 1−r2

1−r2+r3. This in turn does not require further conditions on β. Proof: See Appendix A.2

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In Proposition 2.4.2, the requirement q = d3 means Inspector attacks with such a ’balanced’

frequency that the Deterrer is indifferent between not building weapons and building weapons of mass destruction.

In case (i) q∈ (0, p3

1−p2+p3) means that q is too low (so probability of Inspector attacking is too high) for the Provocateur to mix N B with B2. His best option is not to build weapons at all.

In case (ii) q = p3

1−p2+p3 means that q is high enough (so the probability of Inspector attacking is low enough) for the Provocateur to mix N B with B2. Still it is not so high that it allows the

Provocateur to play a strategy in which he only builds weapons.

Proposition 2.4.3 Suppose 0 < β < 1, both Assumption 1 and Assumption 2 are not true and q∈ (d3, 1).

(i) For q∈ (0, p3

1−p2+p3) we have the SE: Inspector mixes between N AC and AC with q, while the Deterrer plays B2 and the Provocateur plays N B. This case implies: β = ₁_−rr₂3_+r₃

(ii) For q = p3

1−p2+p3 we have the SE: Inspector mixes between N AC and AC with q, while the Deterrer plays B2 and the Provocateur mixes N B and B2 with xP rov2 = 1− _β(1−rr32+r3). 0≤ xP rov

2 < 1 require: β ≥ _1−rr₂3+r3 Proof: See Appendix A.2

The requirement q > d3 means that the probability that Inspector attacks that low, that the

Deterrer can afford himself to play a pure B2.

In case (i) q∈ (0, p3

1−p2+p3) means that q is too low (so probability of Inspector attacking is too high) for the Provocateur to mix N B with B2.His best option is not to build weapons at all.

In case (ii) q = p3

1−p2+p3 means that q is high enough (so the probability of Inspector attacking is low enough) for the Provocateur to mix N B with B2. Still it is not so high that it allows the

Provocateur to play a strategy in which he only builds weapons.

Note that if Assumption 1 and Assumption 2 do not hold, there is only one case where both types of Agent play a mixed strategy: Proposition 2.4.2 (ii). Here the Deterrer mixes N BO and B2 and the Provocateur mixes N B and B2. This is essentially the same Matching Pennies

structure as in the simplified example of Chapter 3. Furthermore it is interesting to observe that xDeter₂ , the probability that the Deterrer builds weapons of mass destruction, is decreasing in xP rov₂ , the probability that the Provocateur builds weapons of mass destruction, and vice versa. This clearly illustrates the interaction between the two types of Agent. It also shows that there is an upper bound for the overall probability that B2 is played by Agent. Inspector

takes this into account when he determines his own strategy: it is an incentive not to act too aggressively.

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4.5.2 Overview

Note that in each equilibrium of the case without IS, Inspector strictly mixes between attacking and not attacking, given that Agent keeps his facilities closed. However, the behaviour of Agent (in particular the strategies of both his types) highly depends on which assumptions are either true or not. Therefore we can categorize all equilibria by the different strategies of (the types of) Agent. In the table 4.1 we provide an overview.

Remember from section 4.2 that Assumption 1 can be interpreted as that the payoff for the Deterrer for playing B1 is relatively high, and that Assumption 2 can be interpreted as that

the payoff for the Provocateur for playing B1 is relatively high. This is reflected in table 4.1

where there is no equilibrium where the Deterrer plays B1, given that Assumpion 1 is not true,

or that the Provocateur plays B1 given that Assumption 2 is not true. More remarkable is that

if Assumption 1 is true, the Deterrer does not necessarily play B1 in equilibrium. Consider for

example the equilibrium in which the Deterrer plays N BO and the Provocateur mixes N B and B2, given that only Assumpion 1 is true. The same holds for the Provocateur: If Assumption 2

is true, the Provocateur does not necessarily play B1 in equilibrium. An example of this is the

equilibrium in which Deterrer mixes N BO and B2 and the Provocateur plays N B, given that

only Assumption 2 is true. The foregoing shows that if an assumption is true, it allows a type of Agent to play B1 in equilibrium, but it does not force him to do so. Sometimes there are

even better strategies possible, although playing B1 gives a relatively high payoff.

It is highly interesting that there is only one equilibrium category which holds for all the four possible combinations with respect to Assumption 1 and Assumption 2. That is the equilibrium in which the Deterrer always builds weapons of mass destruction, while the Provocateur does not build any weapons at all. Because it is the only kind of equilibrium which occurs for all the four possible combinations, it is the only equilibrium which existence is independent of Assumption 1 and Assumption 2. This equilibrium also has an interpretation which is quite natural. Agent’s ultimate goal is to possess weapons of mass destruction, which he tries to achieve via his Deterrer type. Since he does not want to be attacked too often, he uses his Provocateur type, who never builds weapons, as a trap in order to seduce Inspector not to attack with a too high probability. On the hand Agent uses the specific nature of the Provocateur by not letting him build any weapons, since he is the only type who can benefit from an unjustified attack. Because the equilibrium described above is the only one which existence is independent of Assumption 1 and Assumption 2 and since it has a natural interpretation, we think that this equilibrium is the focal point (see the Introduction for an explanation of this term) of the game without IS.

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Table 4.1: Overview categories equilibria case without IS

AS 1 and 2 true Only AS 2 true Only AS 1 true No AS true

D mixes NBO & B1

P plays NB 1 1 D plays B1 P plays NB 1 1 D mixes B1 & B2 P plays NB 1 1 D plays B2 P plays NB 1 1 1 1 D plays NBO P mixes NB & B1 1 1

D mixes NBO & B1

P mixes NBO & B1

1 D plays B1 P mixes NB & B1 1 D mixes B1 & B2 P mixes NB & B1 1 D plays B2 P mixes NB & B1 1 1

D mixes NBO & B2

P plays NB 1 1

D mixes NBO & B2

P mixes NB & B1

1 D plays NBO

P mixes NB & B2

1 1

D mixes NBO & B1

P mixes NB & B2 1 D plays B1 P mixes NB & B2 1 D mixes B1 & B2 P mixes NB & B2 1 D plays B2 P mixes NB & B2 1 1

D mixes NBO & B2

P mixes NB & B2

The role of intelligence services in deterrence inspection games