Effectiveness of Counter-proliferation Measures and their Impacts on Security.

Transcription

1 Effectiveness of Counter-proliferation Measures and their Impacts on Security. Kai Hao Yang March 217 Abstract Since the outset of development of nuclear technology, nuclear proliferation has been a widespread concern of the international society. States perceive threats as others attempt to acquire nuclear weapons. Responses toward proliferators thus have become a crucial component their security strategies. Possible responses include granting incentives, sanctions and preventive strikes. However, there are often trade-offs among these options and the consequences of induced interactions are often unclear. This article investigates the strategic choices and strategic interactions between counter-proliferators and proliferators as well as their impacts on international security and stability. A game theoretic model is established and its implications are discussed. Furthermore, this article also attempts to establish a general theory of strategic interactions during the process of nuclear proliferation and their impacts by characterizing equilibria in a class of models without detailed specifications in a game-free style and analyzing the determinants of equilibrium outcomes, with particular attentions on international stability and likelihood of successful development. It thus yields some results and implications that are robust to model forms and specifications and provides several generalizations and insights to the studies of effects of various counter-proliferation measures as well as the consequences of nuclear proliferation. Department of Economics, University of Chicago; khyang@uchicago.edu 1

2 2 1 Introduction Since the outset of development of nuclear technologies, the issue of nuclear proliferation has been highly concerned in the international society. Following the end of Cold War and the bipolar system, this issue has become even more salient (Sims, 1991; Maerli & Lodgarrd, 27:3; Rauchhaus et al., 211:3). Discussions about the regulations and related international regimes have also been increasing (Simpson, 1994; Feaver & Niou, 1996; Forland, 27; Goldblat, 27). Although there remains distinct arguments about the effects of nuclear proliferation on the stability of international system, nuclear proliferation does threaten some certain countries by increasing the potentiality of possessing nuclear weapons of some non-official nuclear states (Feaver & Niou, 1996; Schneider, 1994: 21). As a result, possible counteractions of the threatened countries (so called the counter-proliferators ) against the proliferators have become a crucial component of their security strategies. There are several possible reactions, both in theory and in practice, including tolerating the proliferation and granting economics incentives, sanctions and preventive strikes (Dunn, 1991; Martel, 21: 11; Feaver & Niou, 1996; Conway, 23; Benson & Wen, 211; Baliga & Sjostrom, 28; Schneider, 1994; 24). 1 However, there are often trade-offs among different strategies. The less unlikely that conflicts would occur the more the strategy is tolerance-orientated, but the likelihood of successful developments of proliferators will then be higher. On the contrary, the less likely of successful developments the more tougher the strategy is, but the probability of unnecessary conflicts is increased. (Feaver & Niou, 1996; Forland, 27; Moriarty, 24; Baliga & Sjostrom, 28; Benson & Wen, 211). Thus, the orientations of reactions of the counter-proliferators as well as the strategies and responses of the proliferators are often quite different during the nuclear proliferation processes. In the existing literature, although there are numerous discussions on particular 1 Conventionally, there are some distinctions between preventive strikes and preemptive strikes. As the former refers to conducting attacks regardless of whether the proliferator has the intension to threat, the latter refers to strikes under an imminent threat. Under the nuclear proliferation context, although the terminology used in the speech given by president George W. Bush in 22 was using preemptive surgical strikes, the preconditions of strikes did not include salient military preparedness and thus is effectively more similar to preventive strikes (Forland, 27:43). In general, some arguments posit that due to the characteristic of weapons of mass destruction that the time period between preparedness and actual attacks is relatively short, there is no need to distinguish preventive and preemptive strikes (Goldstein, 26: 169). Therefore, the term preventive strike will be used throughout this article for consistency.

3 3 cases and general analysis focusing on the policies, the theoretical basis of strategic choices and interactions and hence the impacts on the international system are relatively insufficient. Moreover, the existing theories are also limited and cannot be fully applied to the context described above. Therefore, in this article, we discuss the causal mechanism that determines the strategic interaction between proliferators and conter-proliferators, in order to generalize the explanations and hence provide theoretical basis to the study of processes of nuclear proliferation. The remaining of this article are divided as follows: In the first section we investigate several related literature. In the following section a game theoretic model is established, an equilibrium will be characterized, and its implications will be discussed. In the third section we will generalize the theory and discuss both the stability and likelihood of successful development under a more general framework. Finally, some implications, both theoretical and empirical, will be discussed. 2 Related literature At the macro level of nuclear proliferation, one of the most important issues is the impact of nuclear proliferation on the stability of international system. There are mainly two distinct perspective, namely the optimism and the pessimism. The optimism posits that possessions of nuclear weapons of one country will force other countries to be more cautious, as it was presented in the strategic stability relationship between the U.S. and Soviet Union during the Cold War era. Therefore, the more stable the international system is the more countries posses nuclear weapons, since the likelihood of conflicts is reduced because of cautions of countries (Feaver, 1993; Karl, 1996; Goldstein, 26: ; Waltz, 199; 23; Sagan & Waltz, 1995; Bueno de Mesquita & Riker, 1982; Berkowitz, 1985; Measheimer, 1993; Asal & Beardsley, 27; Gartzke & Jo, 211). On the other hand, the pessimism posits that since the background and context during the Cold War era are significantly different from the post-cold War era and therefore the strategic stability argument is not applicable nowadays (Dunn, 1982; Kaiser, 1989; Karl, 1996; Goldstein, 26:15-2; Woods, 22; Asal & Beardsley, 27). Furthermore, factors such as miscalculation, irrationality of proliferators, crisis instability, lack of second-strike ability during initial phases of nuclear developments, the mutual first strike incentives due to asymmetry and the possibility of preventive strikes induced by nuclear proliferation, some specific form of nuclear proliferation

4 4 will rather cause instability to the international system (Goldstein, 26: 15-2; Berkowitz, 1985; Feaver, 1993; Feaver & Niou, 1996; Karl, 1996; James, 2; Sagan, 23). As it can be seen from the distinct arguments, as a macro level problem, the impact of nuclear proliferation on international stability is determined by the micro-foundations such as the strategies of proliferators, reactions of couter-proliferators as well as the strategic interactions between them. Specifically, except the irrationality and miscalculation arguments, the main distinction between optimism and pessimism are first, the strategic instability due to asymmetric nuclear abilities after proliferation as well as the crisis instability due to poession of nuclear weapons, which are both consequences after successful nuclear proliferation. (Berkowitz, 1985). Second, the possibility of preventive strike during the proliferation process (Sagan, 23:53-63; Goldstein, 26: 14-26). While the former is the center of discussions about impacts of nuclear proliferation on international security, where most of the arguments in pessimism has their counter-arguments and evidences in optimism, the latter is relatively less discussed in the literature, especially in the optimism. As a result, under the realm of rational-actor axioms, the debate between optimists and pessimists seems to be relatively incomplete on the impacts of process of nuclear proliferation and hence is one of the main concerns of this article. On the other hand, at micro level, there are also various empirical studies about the motivations for the proliferators to acquire nuclear weapons. Singh & Way (24) indicated that the most important factor the determines whether a country will attempt to develop nuclear weapons is the external threats. Jo & Gartzke (27) then further showed that the threat from traditional military forces is more significant than nuclear threats. They also indicated that one s economic capability is also a crucial determinant. Gartzke & Jo (211) also pointed that it is more likely that one country will capitulate in a crisis when its opponent posses nuclear weapons. 2 As for the theoretic studies, Benson & Wen (211) showed that it is the threats together with the overriding interests that motivates one to develop nuclear weapons. Moreover, under certain conditions, the proliferators will adopt a form of strategy called strategic ambiguity. 3 Furthermore, Baliga & Sjostrom (28) 2 The coefficient of interest in their probit estimation is -.62 and is significant with level 1% 3 Benson & Wen showed that, when the ambiguous strategy is adopted (that is, in the mixed strategy equilibrium), it is more likely that the counter-proliferator will reconcile. Whether this equilibrium exists depends on the interests that proliferators could gain by acquiring nuclear weapons, how insecure the counterproliferators perceive, and the prior belief of the proliferator being zealous.

5 5 showed that the ambiguous strategy described above would yield a more stable outcome between the two. 4 However, as whether conflicts occur is determined by the interaction between counter-proliferators and proliferators, it is necessary to consider not only the motivations of proliferators but also the reactions taken by counter-proliferators as well as their impcat on the outcome of interaction between the two. As for the responses of counter-proliferators as well as the outcome of their interactions, the key factor to the likelihood of conflicts is the counter-proliferator s choice among tolerance-oriented strategies, sanctions and preventive strikes, as well as the reactions taken by the proliferators. Mazarr (1995) highlighted the futility of sanctions and the importance of package deals by investigating the case of North Korean nuclear crisis. Montgomery (25) also emphasizes that granting economic incentives to the proliferator in exchange for their reconciliation is more effective than preventive strikes. It is also noteworthy that both of them argued that the packege deal granting to the proliferators must be equivalent to the gains of possessing nuclear weapons in terms of strategic interests. In addition, Goldstein (26) argued that the asymmetry in nuclear power between proliferator and counter-proliferator will increase the likelihood of preventive strike and hence reduce the stability between the two. Feaver & Niou (1996) also showed that the determinants of whether proliferator would attempt to acquire nuclear weapons and whether the counterproliferator would assist their development or initiating preventive strike when counterproliferators started their attempts include the prior belief of both on each other s type, their diplomatic relationships, the discrepancy of their military power and the degree to which counter-proliferators have already developed. Baliga & Sjostrom (28) additionally considered the information about the completeness of development by showing that under certain parametric values counter-proliferator would not allow any incomplete information about proliferator s status of development of nuclear weapons and the tough type counter-proliferator will initiate a preventive strike in such case. It is noteworthy that among the above studies, Feaver & Niou (1996) focused on the choice between assisting and preventive strikes, rather than between tolerance and preventive strikes ; while Baliga & Sjostrom (28) put most of the emphasis on the effect 4 In Baliga & Sjostrom s model, strategic ambiguity refers to the uncertainty for counter-proliferators about the completeness of proliferator s development. They showed that such uncertainty could serve as a deterrence force of proliferators and could contribute to overall efficiency.

6 6 of information structure imposed on efficiency and whether uncertainty can be a source of deterrence. Benson & Wen (211) is the only one considered the trade-off between tolerance and preventive strike, but there are still some possible extensions in their model, including the duality of incomplete information, introducing dynamic interactions, and models describing the process of nuclear proliferation. As a result, in this article, we will focus on the trade-off between tolerance and preventive strike during the nuclear proliferation period and investigate the determinants of counter-proliferator s choice and their impact on the outcomes. 3 The Baseline Model 3.1 Set up In this section, we consider a specific form of model that describes the situation of nuclear proliferation. We will analyze an incomplete information game with 2 players, each of them can choose a certain value of time. For the counter-proliferator, the time he chooses (t 1 ) represents the time he would tolerate whereas for the proliferator, the time being chosen (t 2 ) represents the time she would keep developing nuclear weapons. If the time that proliferator is willing to tolerate is less than the time counter-proliferator would keep developing, then preventive strike occurs. On the contrary, if the time of tolerance is larger than the time of developing, then the counter-proliferator would capitulate, provided that she had not yet acquired the nuclear weapons. Finally, if the time of tolerance is large enough so that the proliferator could acquire nuclear weapons if she chooses to, then the development would be successful. Formally, the model is set up as follows: The set of players is {1, 2}, where player 1 represents counter-proliferator and player 2 represents proliferator. The action spaces of both players are positive real numbers R +, which stands for the time they choose. Before choosing their actions, nature draws two values, c and τ according to CDF F 1 and F 2 respectively and then reveal c to player 1 and τ to player 2. The value of c stands for the private information of player 1 about the cost of involving into a preventive strike, while the value of τ stands for the threshold of time to have a successful development, that is, player 2 will acquire nuclear weapons successfully if the time devoted into development if larger than τ, provided that preventive strike did not occur before τ. To model the costs and gains of tolerance, let π H ( ) and π L ( ) be two functions of time. For any t, π H (t) represents

7 7 the amount of costs (gains if negative) to the counter-proliferator, as well as the amount of gains (costs if negative) for the proliferator if the total time of tolerance is t and there is no preventive strike at the end. On the other hand, π L (t) represents the costs/gains if the total time of tolerance is t and preventive strike occurs at the end. Since the sum of gains/costs to the counter-proliferator and the proliferator is zero, the values of π H (t), π L (t) can be interpreted as the amount of transfers during the proliferation process. Notice that the possiblity of different value of costs/gains in different outcomes means that the costs/gains when there is preventive strike can be different from those when there is no preventive strike at the end. Such flexibility is able to capture the feature that some of the costs/gains (e.g. economic aids or sanctions) can be retrieved when the counter-proliferator decides to initiate a preventive strike. Finally, let w > be the cost to counter proliferator when the proliferator acquires nuclear weapons successfully, s > denote the cost for proliferator of being attacked by the preventive strike and ν > denote the gain for the proliferator if she possesses nuclear weapons. Furthermore, we have the following assumptions: Assumption 1. The random vector (c, τ) is independent and follows a distribution F F 1 F 2. Furthermore, For i {1, 2}, F i has the following properties: F i has full support on R +. F i is absolute continuous and has density (i.e. the Radon derivative with respect to the Lebesgue measure) f i. F i induces a random variable with finite second moment. That is, c 2 df 1 (c) < and τ 2 df 2 (τ) <. Assumption 2. The functions π H and π L are continuously differentiable. To formalize the payoffs, let u 1 and u 2 be the (ex-post) payoffs of player 1 and 2 respectively. According to the scenarios described above, u 1 and u 2 are specified as follows: u 1 (t 1, t 2, c, τ) := π H (t 2 ) if t 2 τ and t 1 t 2 c π L (t 1 ) if t 2 τ and t 1 < t 2. (1) π H (τ) w if t 2 > τ and t 1 τ c π L (t 1 ) if t 2 > τ and t 1 < τ

8 8 u 2 (t 1, t 2, τ) := π H (t 2 ) if t 2 τ and t 1 t 2 π L (t 1 ) s if t 2 τ and t 1 > t 2 π H (τ) + ν if t 2 > τ and t 1 τ π L (t 1 ) s if t 2 > τ and t 1 < τ. (2) To sum up, given a value of τ, the decisions of two players constitute four cases. In the first case, t 2 τ and t 1 t 2, which means the proliferator chooses not to aim for successful development and the counter-proliferator tolerates until the proliferator reconciles. In such case, the proliferator will not acquire nuclear weapons nor will there be preventive strike. In the second case, t 2 τ and t 1 > t 2, which means that the proliferator is not aiming to successful development but the counter-proliferator will initiate preventive strike before the porliferator reconciles. In the third case, t 2 > τ and t 1 τ, which means that the porliferator is aiming for develop nuclear weapon and the counter-porliferator will tolerate until the development is complete. In forth case, t 2 > τ and t 1 < τ, which means that the proliferator is aiming for successful development and the counter-proliferator will initiate preventive strike before the proliferator successes. These four caes are illustrated by the figure below: Figure 1. t 1 t 2 τ Case 1: Preventive Strike t 2 t 1 τ Case 2: Reconciliation t 1 τ t 2 Case 3: Preventive Strike τ t 1 t 2 Case 4: Successful Development

9 9 3.2 Equilibrium In the following section we will characterize an equilibrium of this game and show the existence of such equilibrium. The equilibrium concept here is the Bayesian Nash equilibrium, which is defined as: Definition 1. A pair of strategies (t 1, t 2 ), where t i : R + R + for i {1, 2} is a measurable function, is a Bayesian Nash equilibrium of this game if for and c, τ, t E F2 [u 1 (t 1(c), t 2(τ), c, τ)] E F2 [u 1 (t, t 2(τ), c, τ)] E F1 [u 2 (t 1(c), t 2(τ), τ)] E F1 [u 2 (t 1(c), t, τ)], where E Fi denoted the expectation operator under CDF F i, for i {1, 2}. As it will be shown in the following proposition, there is a Bayesian Nash equilibrium (t 1, t 2 ) of the form described below: Proposition 1. Under assumption 1 and assumption 2, there exists functions γ < γ, real numbers c c, and t ˆt, with γ( c) = ˆt such that (t 1, t 2 ) is a Bayesian Nash equilibrium, where γ(c) if c c t 1(c) = γ(c) if c < c c, (3) ˆt if c > c t τ if τ ˆt 2(τ) =. (4) t if τ > ˆt Proof. To start off, fix T >, given any t ˆt < T, consider first the best response of player 1 when player 2 is following the strategy of form τ if τ ˆt t 2 (τ) = t if τ > ˆt. (5) Let U 1 (t 1, c t, ˆt) := E F2 [u 1 (t 1, t 2 (τ), c, τ)] denote the interim expected payoff of player 1

10 1 with type c when choosing t 1. Then, U 1 (t 1, c t, ˆt) = 5 ( c π L (t 1 ))(1 F 2 (t 1 )) + t 1 ( π L(τ) w)df 2 (τ) if t 1 < t ( π H (t ))(1 F 2 (ˆt)) + ( c π L (t 1 ))(F 2 (ˆt) F 2 (t 1 )) + t 1 ( π L(τ) w)df 2 (τ) if t t 1 ˆt π H (t )(1 F 2 (ˆt)) + ˆt ( π L(τ) w)df 2 (τ) if t 1 > ˆt Let v, v, v be the value functions of the three segments and γ, γ, γ be the associated optimal choices. That is: v(c t, ˆt) := v(c t, ˆt) := [ max ( c π L (t 1 ))(1 F 2 (t 1 )) + t 1 t max t <t 1 ˆt t1 t1 ] ( π L (τ) w)df 2 (τ) [ ( π H (t ))(1 F 2 (ˆt)) + ( c π L (t 1 ))(F 2 (ˆt) F 2 (t 1 ))+ ] ( π L (τ) w)df 2 (τ). and v (t, ˆt) := π H (t )(1 F 2 (ˆt)) + ˆt [ γ(c t, ˆt) argmax ( c π L (t 1 ))(1 F 2 (t 1 )) + t 1 t ( π L (τ) w)df 2 (τ), t1 ] ( π L (τ) w)df 2 (τ) [ γ(c t, ˆt) argmax ( π H (t ))(1 F 2 (ˆt))+ t <t 1 ˆt ( c π L (t 1 ))(F 2 (ˆt) F 2 (t 1 ))) + t1 ] ( π L (τ) w)df 2 (τ). Thus, the maximization problem is then equivalent to 5 When t = ˆt, the second term is dropped. max t U 1(t 1, c t, ˆt) max { v(c t, ˆt), v(c t, ˆt), v (t, ˆt) }.

11 11 Under assumption 1 and 2, the objective functions are differentiable and by compactness of [, t ] [t, ˆt], v, v, v and γ, γ are well defined. Moreover, v and v are differentiable almost everywhere and v (c t, ˆt) = [1 F 2 (γ(c t, ˆt))] v (c t, ˆt) = [F 2 (ˆt) F 2 ( γ(c t, ˆt))], by the Envelope Theorem and hence v (c t, ˆt) < v (c t, ˆt) <, c. Therefore, there exist a unique c(t, ˆt) such that v( c(t, ˆt) t, ˆt) = v (t, ˆt) and a unique c(t, ˆt) such that v(c(t, ˆt) t, ˆt) = v(c(t, ˆt) t, ˆt), provided that v( t, ˆt) v( t, ˆt). then follows that : γ(c t, ˆt) if c c(t, ˆt) t 1 (c t, ˆt) = γ(c t, ˆt) if c(t, ˆt) < c c(t, ˆt) ˆt if c > c(t, ˆt) It 6 is player 1 s best response against t 2 (τ t, ˆt). On the other hand, for player 2, given t, ˆt and the strategy t 1 (c t, ˆt) obtained above, let U 2 (t 2, τ t, ˆt) := E[u 2 (t 1 (c t, ˆt), t 2, τ)] denote the interim expected payoff. Then: U 2 (t 2, τ t π H (t 2 )P (t 2 t 1 (c t, ˆt), ˆt)) + {c t = 2 >t 1 (c t,ˆt)} [π L(t 1 (c t, ˆt)) s]df 1 (c) if t 2 τ (π H (τ) + ν)p (τ t 1 (c t, ˆt)) + {c τ>t 1 (c t,ˆt)} [π L(t 1 (c t, ˆt)) s]df 1 (c) if t 2 > τ. Also, define ϕ(t t, ˆt) := π H (t)p (t t 1 (c t, ˆt)) + {c t>t 1 (c t,ˆt)} ψ(t t, ˆt) := (π H (t) + ν)p (t t 1 (c t, ˆt)) + Then is equivalent to max U 2(t 2, τ t, ˆt) t 2 {c t>t 1 (c t,ˆt)} max{max t τ ϕ(t t, ˆt), ψ(τ t, ˆt)}. [π L (t 1 (c t, ˆt)) s]df 1 (c), [π L (t 1 (c t, ˆt)) s]df 1 (c), Notice that arg max t [,T ] ϕ(t t, ˆt) and is nonempty by continuity of ϕ and compactness of [, T ]. Let x (t, ˆt) be selected from argmax t ϕ(t t, ˆt). It then follows that x (t, ˆt) is continuous in (t, ˆt) by the Theorem of Maximum and x T. 6 If v( t, ˆt) v( t, ˆt), then the first term is dropped.

12 12 In addition, define ˆx(t, ˆt) as follows: ϕ(x (t, ˆt t, ˆt)) = ψ(ˆx(t, ˆt t, ˆt)) ˆx(t, ˆt) x (t, ˆt) if ψ(ˆt t, ˆt) ϕ(x (t, ˆt t, ˆt)) ˆx(t, ˆt) := ˆt otherwise. By existence of x (t, ˆt) and continuity of ϕ and ψ, ˆx is well-defined, continuous in (t, ˆt) and x ˆx ˆt. Notice that by construction, t 2 (τ t, ˆt) argmax U 2(t 2, τ t, ˆt) if t 2 (τ t, ˆt) t 2 is of form: t 2 (τ t x, ˆt) (t, ˆt) if τ > ˆx(t, ˆt) =. τ if τ ˆt(t, ˆt) Finally, since the set {(t, ˆt) t ˆt T } is non-empty, compact and convex. Also, as shown above, (x, ˆx) is a continuous self-map on {(t, ˆt) t ˆt T }. Thus, by Brouwer s Fixed Point Theorem, there exist (t, ˆt) such that x (t, ˆt) = t, ˆx(t, ˆt) = ˆt. Such fixed point (t, ˆt) and the associated γ(c t, ˆt), γ(c t, ˆt) and c(t, ˆt), c(t, ˆt) then defines a Bayesian Nash equilibrium (t 1, t 2 ) Without further specifications about π H, π L and F 1, F 2, we can already see some properties and of the equilibrium strategies and outcomes as well as their implications. The first implication is that for both players, the chosen time for tolerance and development are always bounded by ˆt, ruling out the possibility of the counter-proliferator s appeasement (i.e. always tolerate). For any realized cost of attack c, time of tolerance adopted by the counter-proliferator is always no greater than ˆt, which is also the critical value for the proliferator between choosing to aim for successful acquirement (t 2 (τ) = τ) or not (t 2 (τ) = t ). Indeed, in equilibrium, with ˆt being a common knowledge for both players, it is suboptimal for the proliferator to tolerate more than ˆt since shifting from tolerating t < ˆt to ˆt makes no (ex-ante) difference for the counter-proliferator, regardless of the cost c. Second, it is noteworthy that the proliferator is essentially adopting a cut-off strategy. That is, when the realized value of technology threshold τ is less than the critical value ˆt, she will always aim for successful development (i.e. t 2 (τ) = τ), whereas when the threshold is greater than ˆt, she will not aim for successful development and always choose some optimal development time t that is less than the threshold τ and is independent of τ. This indicates that whenever t 7, there are some cases when the proliferator is not aiming for successful development, 7 In particular, when π H, π L and π H() = π L().

13 13 yet due to the nature that this intention is observationally unidentifiable for the counterproliferator in the sense that t and τ is unobservable to the counter-proliferator so that she cannot distinguish between t 2 (τ) = τ and t 2 (τ) = t by seeing only t 2 it is possible that the preventive strike is unnecessary in that the proliferator has no intension to complete the development but counter-proliferator still embark a preventive strike. In addition, if we further specify the properties of π H, π L and F 1, F 2, there will be more properties of the equilibrium strategies that are substantially relevant. To see this, consider the following further assumptions: Assumption 3. π H, π L are increasing and convex For any T >, i {1, 2}, the function F i(t ) F i (t) f i (t) is strictly increasing on [, T ]. 8 Notice that for any t, ˆt, the first order conditions give: ( ( ( ) ) γ(c t, ˆt) t ) 1 F2 π L(γ(c t (γ(c t, ˆt)), ˆt)) + w c =, f 2 (γ(c t, ˆt)) < γ(c t, ˆt) t, ( ( ) ) 1 F2 π L(γ(c t (γ(c t, ˆt)), ˆt)) + w c, f 2 (γ(c t, ˆt)) and ( γ(c t ˆt) ˆt ) ( π L( γ(c t, ˆt)) ( F2 (ˆt) F 2 ( γ(c ˆt, t )) f 2 ( γ(c ˆt, t )) ) ) + w c =, t < γ(c t, ˆt) ˆt, ( ( π L( γ(c t F2 (ˆt) F 2 ( γ(c ˆt, t, ˆt)) ) ) )) + w c. f 2 ( γ(c ˆt, t )) Under assumption 1, 2 and 3 it then follows that γ(c t, ˆt), γ(c t, ˆt) are increasing in c and that γ( c(t, ˆt) t, ˆt) = ˆt. In particular, this holds for the fixed point (t, ˆt). Therefore, the equilibrium strategy t 1 (c) is increasing and t 1 (c) = ˆt for any c c. Such equilibrium strategies are depicted in figure 2 8 The exponential distribution, for example, satisfies such property.

14 14 t 1 (c) t 2 (τ) ˆt ˆt t γ(c) t γ(c) 45 c c c Figure 2. t ˆt τ Furthermore, combing the setups and the equilibrium strategies under assumption 1,2 and 3, the type spaces for counter-porilferator and proliferator can be partitioned into parts as depicted in figure 3. The first region indicates the combination of (c, τ) where the proliferator successfully obtains nuclear weapons. Qualitatively, this is when the cost for counter-proliferator is relatively high or the technology threshold of the proliferator is relatively low. The second region gives preventive strike as an outcome. This occurs when the cost is relatively low or threshold is relatively high. It is also noteworthy that when the cost is low enough and the threshold is high enough (specifically, when c c and τ > ˆt), such preventive strike is unnecessary. The third region stands for the case when the proliferator capitulates and the deal is made, which occurs when both the cost and threshold are high. τ Reconciliation ˆt Preventive Strike t Successful Development c c c Figure 3. There are several implications from these equilibrium strategies. First, the tolerance

15 15 time adopted by the counter-proliferator is increasing in the cost of conflicts. This property corresponds to the argument in the literature that when the relative military power between the two is more asymmetric (reflected by lower c), it is more likely for the counterproliferator to initiate preventive strike. Indeed, not only the tolerance time is lower when c is lower, the (interim) probability of preventive strike is also higher when c is lower, as seen in figure 3. However, we should be cautious in generalizing this result in that the increasing property holds after introducing assumption 3. That is, when the cost (gain) of tolerating is increasing (decreasing) in time for the counter-proliferator and the distribution of types have some certain properties in their hazard rates, counter-proliferator s tolerance time will be increasing in cost in equilibrium, but it is still unclear if the similar property holds in more general environments without assumption 3. 9 Second, there exists a region, where c c, such that there would be no preventive strikes as an outcome. Whenever c c, the equilibrium outcome will be either that the proliferator acquires nuclear weapons successfully or the proliferator backs down. In such cases, the relationship between two sides is stable in the sense that there will be no conflicts in the end. Third, the unnecessary preventive strike can occur only when the cost is sufficiently small and threshold is sufficiently large (c c, τ > ˆt). Although the above analyses provide us some implications that are substantially relevant and enable us to further re-examine the arguments in the literature, there is a drawback, as commonly seen in most of the game-theoretic models in the study of international conflicts, that the implications above are derived under certain specification of the form of the modeled game and assumptions. For instance, one might argue that even if the counter-proliferator s tolerance time is less than the proiferator s development time, preventive strike might still not occur for sure since the two parties are able to negotiate or embark certain bargaining process at this moment. Alternatively, one might argue that the cost (gain) of tolerance, can also be determined endogenously, rather than being given by exogenous functions π H and π L, since in reality, the level of economic aids and/or the severity of sanctions are mostly results of bargaining and negotiations and hence both players should be able to (at least partially) affect these results. To address this problem, we will consider a more general setup in the next section and will derive some results that are robust to the particular forms 9 In fact, in more general settings, the interim probability of preventive strike is indeed decreasing in c, which will be shown in the next section.

16 16 and specifications of the game being modeled. 4 A General Model for Nuclear Proliferation 4.1 Mechanisms and the Revelation Principle In this section, we will generalize the forms and setups for the model and examine the implications that are robust to the details of the models for such proliferation scenario. To study such general problem, as commonly applied the literature in mechanism design, we will use the revelation prnciple to reduce the complexity inherited in the general setups without detailed specifications about the game forms. 1 The strategy is as follows: We will fist consider a class of games (in the literature, they might be called mechanisms) that capture the nature of scenario of nuclear proliferation. Specifically, we consider a class of games characterized by the determination rules of outcomes the level of cost/gain of tolerance, the probability of successful developments, and the probability of preventive strike. However, unlike the previous section, we will not specify how these outcomes are determined, nor the actions available to the players. Instead, we will study the implications that holds for all games with this form. As noted above, the seemingly complexity of this problem can be addressed by applying the revelation principle, which enables us to focus only on a (much smaller) class of incentive compatible direct mechanisms. Formally, for i {1, 2}, let A i denote the action space available for player i, T i denote the type space for player i and let A := A 1 A 2, T := T 1 T 2. Also, let O denote the set of all possible outcomes, ω : A T O is a map called outcome function, which determines the outcome given the player s type and their actions. Finally, player i s payoff is given by a von Neumann-Morgenstein utility function u i : T O R. Together, a mechanism is defined by a 5-tuple M := (A i, T i, u i, O, ω) 2 i=1. With the notations above, a direct mechanism is a mechanism with action spaces being identical with type spaces, denoted as D : (T i, u i, O, ω) 2 i=1. A direct mechanism D is called incentive compatible if truth-telling is a Bayesian Nash equilibrium, that is, the identity function is an equilibrium strategy for all players. The revelation principle is then stated as follows: Theorem 1 (Revelation Principle). For any mechanism M := (A i, T i, u i, O, ω M ) 2 i=1 and a 1 In some literature of international conflicts, this approach is called game-free analysis, as in Banks(199) and Fey & Ramsey (29, 211)

17 17 Bayesian Nash equilibrium σ = (σ1, σ 2 ) in M, there exists an incentive compatible direct mechanism D = (T i, u i, O, ω) 2 i=1 such that for any t = (t 1, t 2 ) T, the equilibrium outcome in D is the same as in M, that is: ω(t, t) = ω M (σ (t), t), t T. 4.2 Setups, Incentive Compatibility and Individual Rationality As noted at the end of previous section, there are some possible drawbacks in the baseline model. The results and implications derived above might subject to the specific form and assumptions in the settings. To generalize the model and derive some results that are robust to such specifications, we now consider a general class of models that describe the nature of scenario of nuclear proliferation as discussed in the previous section. As we can observe from the setups of the model in the previous section, the essence of the context of nuclear proliferation problem where a proliferator is attempting to develop nuclear weapons and a counter-proliferator can either tolerate, possibly in accompany with sanctions and/or granting economic incentives, with the hope that the proliferator would capitulate, or initiate a preventive strike, in order to eliminate the possibility of further developments can be characterized by a combination of outcomes consisting of, as modeled in the previous sections, the (ex-post) probability of conflicts, the (ex-post) probability of successful developments, and the cost/gain to the counter-proliferator/proliferator if there is or is not conflicts in the end, which is depending on total time passed by. Two critical assumptions we have here is that the game ends once the preventive strike is initiated or the proliferator capitulates and that whenever there is a preventive strike, proliferator cannot obtain nuclear weapons successfully. Thus, in any such model, the outcome can be partitioned into three scenarios preventive strike, successful development, and capitulate. To formally setup a general model, we let A 1, A 2 be some abstract (non-empty) action spaces for the players. For technical concerns, for each i {1, 2} we also let A i be a sigmaalgebra on A i and µ i be a measure on (A i, A i ). This allows us to capture any specifications about the strategic choices for the players, including the choices of tolerance and preventive strikes, the choices on the developments, as well as all the possible actions that can be taken in bargaining or negotiating processes. 11 For instance, in the baseline model A 1 = A 2 = R + are the sets of time chosen by players, A 1, A 2 are then the Borel algebra on R + and 11 Furthermore, by considering only an abstract action space, we allow the model to be an sequential game, by reducing its extended form into strategic form with action spaces A 1, A 2.

18 18 µ 1, µ 2 are both Lebesgue measures. One can also think of A i as R + B i, where for any (t i, b i ) A i, t i denotes the time chosen by player i and b i denotes the combination of all other actions adopted by player i in bargaining or negotiating, for i {1, 2}. We will not put any further specifications about the set A := A 1 A 2 for generality. However, we maintain the specifications about the information. That is, the type spaces of players are T 1 = T 2 = R +, where c denotes the cost of preventive strike to player 1 and τ denotes the technology constraint of player 2 and c is drawn from F 1 and then informed to player 1, τ is drawn from F 2 and then informed to player 2. We also maintain assumption 1 as basic assumptions on F 1, F On the other hand, consider three functions π H : A T 2 R, π L : A T 2 R and g : A T 2 ({1, 2, 3}), where ({1, 2, 3}) denotes the set of all probability measures on the finite set {1, 2, 3} that stands for the three possible outcomes: capitulate, successful development and preventive strike, respectively. It is sometimes useful to write g = (g 1, g 2, g 3 ) T [, 1] 3, with g 1 + g 2 + g 3 = 1. The outcome space of this class of nuclear proliferation games is then R 2 ({1, 2, 3}) and (π H, π L, g) is the outcome function. Under such notations, π H, π L then denote the amount of transfers and g 1, g 2, g 3 denote the probability of capitulate, successful development and preventive strike, respectively. Several implications are noteworthy in this setting. First, by allowing the range of g to be a probability distribution, we allow the possibility that the outcomes are randomly selected by a distribution, given a combination of players actions, rather than being deterministic. Second, by allowing the range of π H, π L being real numbers, both sanctions (negative) and economic incentives (positive) are captured. Third, by allowing the functions to depend on actions, it allows the amount of transfers, as well as the probability distribution of possible outcomes to be endogenously determined by the interactions between players and hence include all possible models with negotiations, bargaining and choice between sanctions and economic incentives. Finally, although it will not change all the results derived below if the outcome function depends also on T 1, we believe that it is more substantively reasonable to model π H, π L, g to be dependent only on T 2, in that T 1 is the cost to the proliferator in preventive strike, which is irrelevant to the outcomes whereas T 2 is the technology constraint of the proliferator, which can possibly affect the outcome of developments. To rule 12 Under such general structure, a Bayesian Nash equilibrium σ = (σ1, σ2) is defined analogously to definition 1, with measurability defined with respect to measure spaces (A 1, A 1, µ 1) and (A 2, A 2, µ 2).

19 19 out some pathological cases that are not substantively relevant, the following assumption rule out the case where π H and π L are infinite in expectation in equilibrium. Assumption 4. For each a 1 A 1, A 2 π H (a 1, a 2, τ) df 2 (τ)dµ 2 (a 2 ) < and A 2 π L (a 1, a 2, τ) df 2 (τ)dµ 2 (a 2 ) < For each a 2 A 2 and τ, A 1 π H (a 1, a 2, τ) dµ 1 (a 1 ) < and A 1 π L (a 1, a 2, τ) dµ 1 (a 1 ) < Additionally, due to the nature of nuclear proliferation scenarios, it is not reasonable to rule out the possibility that the counter-proliferator can always choose to initiate a preventive strike or not, and that the proliferator can always choose not to start developing. To capture this nature, we will impose the following assumption: Assumption 5. There exists ā 1, ã 1 such that, for any a 2 A 2, τ T 2, g 3 (ā 1, a 2, τ) =, g 3 (ã 1, a 2, τ) = There exists ā 2 such that, for any a 1 A 1, τ T 2, g 1 (a 1, ā 2, τ) = 1. We also maintain the structure of payoffs in the baseline model in previous section. That is, when the outcome is capitulate, the counter-proliferator gets π H and the proliferator gets π H ; when the outcome is successful development, the counter-proliferator gets π H w and the proliferator gets π H + ν; when the outcome is preventive strike, the counterproliferator gets π H c and the proliferator gets π L s, where w, s, ν >. With the notations above, the payoffs are then: u 1 (a 1, a 2, c, τ) := [π H (a 1, a 2, τ)g 1 (a 1, a 2, τ) + (π H (a 1, a 2, τ) + w)g 2 (a 1, a 2, τ) + (π L (a 1, a 2, τ) + c)g 3 (a 1, a 2, τ u 2 (a 1, a 2, τ) := π H (a 1, a 2, τ)g 1 (a 1, a 2, τ) + (π H (a 1, a 2, τ) + ν)g 2 (a 1, a 2, τ) + (π L (a 1, a 2, τ) s)g 3 (a 1, a 2, τ). 13 Although the setups in the baseline model presented above does not exhibit this property, it can be modified by extending the action space of player 1 to extended real number. That is, by including in player 1 s available choices. It is clear that this extension will not change the analyses given in proposition 1.

20 2 By using the fact that g 1 + g 2 + g 3 = 1, the payoffs can be rewritten into: u 1 (a 1, a 2, c, τ) = [(π L (a 1, a 2, τ) + c)g 3 (a 1, a 2, τ) + π H (a 1, a 2, τ)(1 g 3 (a 1, a 2, τ)) + wg 2 (a 1, a 2, τ)], u 2 (a 1, a 2, τ) = (π L (a 1, a 2, τ) s)g 3 (a 1, a 2, τ) + π H (a 1, a 2, τ)(1 g 3 (a 1, a 2, τ)) + νg 2 (a 1, a 2, τ) To sum up, with the notations introduced above, a nuclear proliferation game P with can be characterized by P = (A, π P H, πp L, gp ), satisfying assumption 4 and 5. Also, a direct mechanism in this environment can then be characterized by D = (π H, π L, g), with π H : T T 2 R, π L : T T 2 R and g : T T 2 ({1, 2, 3}). In the analyses below, given any nuclear proliferation game P and a Bayesian Nash equilibrium σ in P, it is useful to consider the interim expected payoffs. That is, U1 P(c σ ) := E F2 [u 1 (σ1 (c), σ 2 (τ), c, τ)] and U2 P(τ σ ) := E F1 [u 2 (σ1 (c), σ 2 (τ), τ)]. Under assumption 5, in any reasonable Bayesian Nash equilibrium, both players should have interim expected payoffs no less than the payoff they can obtain unilaterally always initiate preventive strikes or always not for counterproliferator and not to develop at all for the proliferator. This requirement, which is often referred as individual rationality, is defined formally as follows. Definition 2. A nuclear proliferation game P = (A, π P H, πp L, gp ), together with a Bayesian Nash equilibrium σ, is said to be individually rational if: and U P 1 (c σ ) c, c, U P 2 (τ σ ), τ. Moreover, as mentioned above, a direct mechanism is called incentive compatible if the identity function is an equilibrium strategy for each player. That is, each player would have no incentive to mis-report their type given that the other is telling the truth. This is also defined formally as follows: Definition 3. A direct mechanism D = (π H, π L, g) is incentive compatible if: and 14 E F2 [u 1 (c, c, τ, τ)] E F2 [u 1 (c, c, τ, τ)], c, c, E F1 [u 2 (c, τ, τ)] E F1 [u 2 (c, τ, τ)], τ, τ. 14 These two inequalities are often called incentive constraints

21 21 Similarly, in the analyses below, it is useful to consider the interim expected payoffs U 1 (c, c ) := E F2 [u 1 (c, c, τ, τ)] and U 2 (τ, τ) := E F1 [u 2 (c, τ, τ)], which denotes the interim expected payoffs of reporting their types as c, τ when their actual types are c, τ in the direct mechanism, respectively. We also write U1 (c) := U 1(c, c) and U2 (τ) := U 2(τ, τ), which is the interim expected payoffs when reporting their types truthfully, given that the other is reporting truthfully. With these notations, we notice that a direct mechanism D = (π H, π L, g) is incentive compatible and individually rational if and only if: U 1 (c) U 1 (c, c), U 1 (c) c,, c, c and U2 (τ) U 2 (τ, τ), U2 (τ), τ, τ. We now begin to analyze the equilibria in class of nuclear proliferation games satisfying assumption 4 and 5 that are individually rational. 4.3 Main Results and Implications As noted above, although it seems difficult examine the behavior of equilibria in a class of games P due to their complexity and generality, it is useful to apply the revelation principle and restrict our concerns on the class of incentive compatible direct mechanisms. Formally, for any nuclear proliferation game P = (A, πh P, πp L, gp ) and any Bayesian Nash equilibrium σ in P, by the revelation principle, there exists an incentive compatible direct mechanism D = (π H, π L, g) such that π P H(σ 1(c), σ 2(τ), τ) = π H (c, τ, τ) π P L (σ 1(c), σ 2(τ), τ) = π L (c, τ, τ) g P (σ 1(c), σ 2(τ), τ) = g(c, τ, τ), for all c, τ. It then follows directly that U1 P(c σ ) = U1 (c) and U 2 P(τ σ ) = U2 (τ), c, τ and hence (P, σ ) is individually rational if and only if the corresponding incentive compatible direct mechanism D is individually rational. Therefore, to study the class of nuclear proliferation games P and its equilibria σ that are individually rational is equivalent to study the class of direct mechanisms D that are incentive compatible and individually rational. This observation will be useful in the analyses below.

22 22 The first result for the general model of nuclear proliferation considers the property of equilibirum payoff of the counter-proliferator and the probability of preventive strike as well as the relationship between them. The following lemma is a key to these implications: Lemma 1. Let D = (π H, π L, g) be a direct mechanism. If D is incentive compatible, then: 1. U 1 (c) = U 1 () c E F 2 [g 3 (y, τ, τ)]dy for any c 2. E F2 [g 3 (c, τ, τ)] is decreasing in c. Conversely, if 1 and 2 holds, then U 1 (c) U 1(c, c ) for any c, c. Proof. For sufficiency, let D = (π H, π L, g) be an incentive compatible direct mechanism. Then for any c > c, from incentive compatibility, U 1 (c) U 1 (c, c ) = E F2 [ πl (c, τ, τ)g 3 (c, τ, τ) + π H (c, τ, τ)(1 g 3 (c, τ, τ)) + wg 2 (c, τ, τ) + cg 3 (c, τ, τ) ] = U 1 (c ) E F2 [g 3 (c, τ, τ)](c c ). Since c > c, it follows that U 1 (c ) U 1 (c) c c E F2 [g 3 (c, τ, τ)]. On the other hand, incentive compatibility also implies: U 1 (c ) U 1 (c, c) = E F2 [ πl (c, τ, τ)g 3 (c, τ, τ) + π H (c, τ, τ)(1 g 3 (c, τ, τ)) + wg 2 (c, τ, τ) + c g 3 (c, τ, τ) ] = U 1 (c) E F2 [g 3 (c, τ, τ)](c c), and hence Together, we have: U 1 (c ) U 1 (c) c c E F2 [g 3 (c, τ, τ)]. E F2 [g 3 (c, τ, τ)] U 1 (c ) U 1 (c) c c E F2 [g 3 (c, τ, τ)] (6) for any c > c and therefore E F2 [g 3 (c, τ, τ)] is decreasing in c. Furthermore, since g 3 [, 1], equation (6) then implies that U 1 is Lipschitz continuous, in particular, is absolutely continuous and hence is differentiable almost everywhere, with U 1 (c) = E F 2 [g 3 (c, τ, τ)]

23 23 for any c at which U1 is differentiable by the envelope theorem. The fundamental theorem of calculus for Lebesgue integral then gives: U 1 (c) = U 1 () c E F2 [g 3 (y, τ, τ)]dy, c Conversely, let D = (π H, π L, g) be a direct mechanism satisfying 1 and 2 in the assertion. Then for any c and any c > c, as above, U 1 (c, c ) = U 1 (c ) + E F2 [g 3 (c, τ, τ)](c c) = U 1 () = U1 (c) + U1 (c), c c c E F2 [g 3 (y, τ, τ)]dy + c c E F2 [g 3 (c, τ, τ)]dy ( EF2 [g 3 (c, τ, τ)] E F2 [g 3 (y, τ, τ)] ) dy where the last inequality follows from 2. Since c, c are arbitrary, the above inequality also implies that U 1 (c, c ) U 1 (c) for any c < c whenever c >. This completes the proof. From lemma 1 and the revelation principle, we are then able to characterize the equilibrium payoff of the counter-proliferator in any nuclear proliferation games. In addition, together with the individual rationality, it can be shown that the interim probability of preventive strike exhibits certain properties that corresponds to the results in the baseline model. Proposition 2. Let P = (A, π P H, πp L, gp ) be any nuclear proliferation game and σ be a Bayesian Nash equilibrium in P. Suppose that P satisfies assumption 4 and 5 and that (P, σ ) is individually rational. Then: 1. U1 P(c σ ) = U1 P( σ ) c E F 2 [g3 P(σ 1 (y), σ 2 (τ), τ)]dy, c. In particular, equilibrium payoff of player 1, U P 1 (c σ ), depends only on interim probability of preventive strike E F2 [g3 P ] in equilibrium up to a constant. 2. E F2 [g3 P(σ 1 (c), σ 2 (τ), τ)] is decreasing in c. 3. lim z z E F2 [g3 P(σ 1 (y), σ 2 (τ), τ)]dy =. Proof. Let P = (A, π P H, πp L, gp ) and σ be any pair of individually rational nuclear proliferation game and its Bayesian Nash equilibrium. By the revelation principle, there exists an incentive compatible and individually rational direct mechanism D = (π H, π L, g)

24 24 such that π H (c, τ, τ) = πh P (σ 1 (c), σ 2 (τ), τ), π L(c, τ, τ) = πl P(σ 1 (c), σ 2 (τ), τ) and g(c, τ, τ) = g P (σ 1 (c), σ 2 (τ), τ), for any c, τ and hence U 1 (c) = U P 1 (c σ ) for any c. lemma 1, since D is incentive compatible, From U 1 (c) = U 1 () c E F2 [g 3 (y, τ, τ)]dy, c. Assertion 1 then follows. Also, from lemma 1 and incentive compatibility of D, E F2 [g 3 (c, τ, τ)] is decreasing in c and hence assertion 2 follows. For assertion 3, since σ is a Bayesian Nash equilibrium, U1 P(c σ ) E F2 [u P 1 (a 1, σ2 (τ), c, τ)] for any a 1 A 1, in particular, by assumption 5, let ã 1 be such that g3 P(ã 1, a 2, τ) = for any a 2 A 2, τ, U P 1 (c σ ) E F2 [π H (ã 1, σ 2(τ), τ) + wg 2 (ã 1, σ 2(τ), τ)], c. Under assumption 4, since σ2 is A 2 measurable, for w := E F2 [πh P (ã 1, σ2 (τ), τ)+wgp 2 (ã 1, σ2 (τ), τ)], w <. By the revelation principle and assertion 1, we have: Hence, U 1 () + Since g 3, it then follows that c E F2 [g 3 (y, τ, τ)]dy w, c. E F2 [g 3 (y, τ, τ)]dy U 1 () + w <. lim z z E F2 [g 3 (y, τ, τ)]dy =. Assertion 3 then follows by the revelation principle. In the previous section, we showed that under assumption 1, 2 and 3, the interim probability of preventive strike is decreasing in the cost of counter-proliferator c and is approaching to as c increases (in particular, is whenever c c). Such result corresponds to the common argument in literature that the more asymmetric the military capability between the two is, the more likely there would be preventive strikes and hence the relationship is more unstable. Proposition 2 generalized this result and indicates that it is robust to all the Bayesian Nash equilibria in nuclear proliferation games. Specifically, assertion 2 implies that in any Bayesian Nash equilibrium of any nuclear proliferation game, the interim probability, as a function of c, is always decreasing. Moreover, assertion 3 implies that this interim probability is not only decreasing, but is approaching to zero quite