(1988)
Extended
Deterrence and the Prevention of War, New Haven, CT: Yale University Press.
Schwarz - Committments
ring.
In a normal usage the world i`blackmaili^ in not an appropriate description of a manager threatening a worker with O?
ring unless the worker puts forth good e?
ort.
However, from the modeling preceptive, it is no di?
erent from blackmail.
In the previous sections, we considered an inO? nite game between a potential aggressor and a victim where the aggressor can exercise the threat and end the game and the victim can make a single transfer or a sequence of transfers to the aggressor in order to appease him. If carrying out the threat is costly for the potential aggressor, the unique subgame perfect equilibrium entails no transfers (This statement is made formal in Proposition 1 and is also emphasized by Shavell and Spier, 2002). We show that a threatener is able to extract a stream of positive payments if the threat is probabilistic or inO? nitely i? divisible. i?
Before proceeding to the model let us illustrate i`probabilistic threatsi^with the story of a French diplomat Eon de Beaumont who blackmailed the French king Louis XV. 25 During his service as a diplomat in London, Eon de Beaumont found himself involved in a feud with the French ambassador. When di? Eon was asked by his superior to resign from em- bassy and return to France, he decided to resort to blackmail. Chevalier di? Eon had in his possession a copy of the diplomatic correspondence that was so secret that even the French ambassador to England did not know about its existence26. Chevalier di? Eon threatened to reveal the diplomatic correspondence to the Englishmen; among other things, the cor- respondence contained information about Louis XV secret preparation for invasion of the British Islands. di? Eon demanded from the French king money and a permission to remain in England. However, the king was not the only party who stood to loose from revelation of the secret correspondence. Publishing the correspondence could have made di? Eon the most despised traitor on both sides of the channel. At the O? rst glance di? Eon did not have a credi-
25For a complete account see Broglie (1879).
26An institution known as the i`kingi? s secreti^was a unique feature of the french diplomacy during the
reign of Louis XV. The members of the kingi? s secret, of which Chevalier di? Eon was one, received special directives from the kingi? s secret. The rest of the embassy sta? including the ambassador were not aware of the existence of this secret correspondence. A detailed account of the working of the kingi? s secret can be found in Broglie (1879).
? 15
ble threat; understandably, the crown found di? Eon demands ridiculous and was reluctant to o? er concessions.
To persuade the king of the seriousness of his intentions, di? Eon published a pamphlet that included small bits of the text from the secrete correspondence and vague hints about its existence. That prompted France to grant di? Eoni? s request: he was allowed to stay in England and received the allowance from the embassy. The bargaining power of di? Eon was enhanced by his ability to ina? ict arbitrary small amounts of harm by leaking small bits of secrete correspondence. The actions of di? Eon were a sort of brinkmanship. Each public hint about the correspondence could have tipped of the English that di? Eon stored the secret documents outside the embassy, in which case, the English might have tried to steal the documents.
To model brinkmanship, we modify the model of the previous section to allow for proba- bilistic threats. In the previous section, the only threat available to the aggressor was to start an all out war, that would impose a cost x on him and a cost x + L on the weaker party, where L > 0. We model divisible threats by allowing the aggressor, A; to use probabilistic threats. SpeciO? cally, the aggressori? s may take an observable action that leads to war with some positive probability p; p < 1:27 We will show that in this case the bargaining power of the aggressor is much greater than in the case where probabilistic threat is not available. Indeed, if the only threat the blackmailer has is to launch a war, he could not do better than extracting the expected gain from the war, which is zero if x < 0. In stark contrast to Proposition 1, with probabilistic threats, the strong side can extract the whole surplus from the victim. 28
There are two players, the blackmailer, A; and the victim, B. For the sake of simplicity, let agents be risk-neutral, and the outcome of war to be deterministic and if a war occurs, the Ai? s instantaneous consumption thereafter is x and the Bi? s is y = x L: We consider the case where war is an unambiguously bad outcome for both sides that is x < 0 and y < 0:29 As above, the time is continuous. In each period t 2 [0; +1); the victim makes a
27Alternatively, one might assumes that the blackmailer has a possibility to distribute harm between
di? erent periods instead of launching a war at once.
28The same logic applies to wars: a i? bullyi? state that has only two options to either start a large war, or
not to start a war at all has much less bargaining power than a state that can start a tiny war or a large
war or anything in between.
29In particular, this assumption means that there is no impact of transfers on balance of power.
? 16
non-negative transfer bt to the blackmailer. Prior to the cona? ict the consumption of parties A and B is bt and bt respectively. At each period party A (the blackmailer) chooses action at 2 fW; P g. If at = W; then the nature moves: with probability p a war occurs and the game ends, and with probability 1 p the game continues. 30 An important assumption is that the history of actions is fully observable. The following proposition shows that availability of a probabilistic threat allows the potential aggressor to extract a share of surplus close to one.
Proposition 5 Suppose that (1 p)y px: For any 0 < b < y there exists a sub-game perfect equilibrium where along the equilibrium path party B transfers resources to A at rate b and party A never starts a cona? ict.
Let us sketch the intuition behind this result. There exists an equilibrium where party A does not attack B as far as it receives transfers at rate b: If party B ever deviates party A takes action W to punish party B: Party A continues to repeat the punishment at frequent intervals as long as party B continues to deviate. If a deviation of party B remains unpunished then party B stops transferring resources to party A. There are no beliefs in the model because the solution concept here is a subgame perfect Nash equilibrium; however, to understand the intuition behind the equilibrium it is useful to talk in terms of beliefs. If any deviation of party B remains unpunished, party B comes to a conclusion that future deviations will not trigger punishments from party A. Thus, party A will punish deviations of B in order to make sure that party B will continue to transfer resources in the future.
Intuitively, the crucial feature of the above blackmail game is that the harm is inO? nitely- divisible. The condition px + (1 p)y 0 means that the aggressor, A; prefers gambling on war (i. e. choosing at = W ) that yields a stream of losses at rate x with probability p and a stream of transfers at rate b with probability (1 p) over receiving zero transfers from now on. Thus, it is the best response for party A to punish deviations by B and it is the best response for B to transfer resource to A at rate b:
It is worth noting that the blackmail game has multiple equilibria. A war never occurs on the equilibrium path in the equilibrium sketched above. However, there are many econom- ically plausible equilibria where wars occur with positive probability because probabilistic threats or i`brinkmanshipi^ occurs on the equilibrium path. For instance, the equilibrium strategy of player A may call for taking action W (engaging in brinkmanship) every O? ve
30We omit the formal description of the space of histories because it is analogous to the previous section. 17
? years. If party A deviates from this strategy it looses credibility as a potential aggressor and
never receives transfers in the future. There is an equilibrium where along the equilibrium
path the weak party occasionally reduces transfer level and the strong party immediately
takes action W; with subsequent resumption of transfers at the normal level. Note that ac-
tion W is not a punishment for a deviation, the victim was supposed to reduce the transfers
and the aggressor is supposed to i`punishi^this equilibrium behavior. The intuition for this
result is as following. According to Proposition 4 there exists a subgame perfect equilibrium
with transfers 9 y and there is an equilibrium with no transfers. It is easy to construct an 10
equilibrium where transfers are 1 y per period and once a year transfers by B are reduced to 2
zero for one month. At the end of that month A plays action W and B resumes transfers at rate 1y. If aggressor fails to take action W both players switch to playing the equilibrium
2
without transfers in the remaining subgame. If party B fails to reduce transfer to zero at the end of the year both parties play the equilibrium where transfers are 9 y in the remaining
10
subgame.
The model described above may shed some light on economics of terror. For decades,
the Irish Republican Army terrorizes the British government and people. A distinct feature of the IRA terror tactics is that acts of violence usually cause relatively small death toll, but, in retrospect, each attack might have caused much more human loss. The British making slow concessions and IRA engaging in occasional terrorist acts is consistent with an equilibrium path of play. Occasional bomb threats may be necessary to make the terroristsi? threat credible and thus induce more concessions from the British. Of course, there exists a more e? cient equilibrium where concessions are obtained without violence.
5 Conclusion
The bargaining game described herein captures some essential features of relationships were neither party has commitment power. A self-enforcing agreement is feasible if parties are either risk neutral or transfers do not change the balance of power. A successful peace agreement is likely to consists of a sequence of concessions. We show that if both parties are risk-averse and if transfers change balance of powers an ine? cient cona? ict may be inevitable.
The model of brinkmanship is our main contribution. We model brinkmanship as an ability to take an observable action that with large probability have no consequences and
? ? ? ? 18
with small probability ends the game with payo? s that are very undesirable for both sides. Examples of brinkmanship can be found situations ranging from international relations to Hollywood movies were gangster shoot under the feet to force a victim to cooperate. We show that if the action space of the potential aggressor allows him only two actions such as i? cooperatei? or i? go to the ultimate threat point,i? his bargaining power is far less than in a case where the potential aggressor can make probabilistic threats.
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21
Appendix The setup
The set of players is N = fA;Bg: The set of histories is denoted H and ; 2 H. A vector function h, ht = (at; kt) belongs to the set of non-terminal histories if and only if the support ofhissomReinterval[0;T[;andforanyt2[0;T[wehaveat =Pandbt 0where bt = k0 + 0t ktdt: There is a natural interpretation of kt: we assumed that the function representing transfer proO? le is di? erentiable and thus continuous. Consequently, although player B can change the transfer level at any instant his choice must be consistent with continuity assumption, essentially, moment by moment player B chooses the derivative of transfer proO? le, which we denote kt: (We slightly abuse notation by denoting agenti? s choice of the level of transfer at time zero by k0. )
Let Z denote the set of all terminal histories. The set Z consists of two types of terminal histories: inO? nite (these where a cona? ict never occurs) and O? nite (these that end in a war). A vector function ht = (at;kt) belongs to the set of inO? nite histories if and only if h has support [0; +1[ and for all t; at = P and bt 0: A vector function ht = (at; kt) belongs to a set of O? nite terminal histories if and only if it has support [0; ], bt 0 and at = P for all t < ; and w( ) = W:
The choice correspondence P(h) = fA;Bg for all h 2 HnZ: In other words following any non terminal history both players take action simultaneously.
The preferences of players are given by the discounted sum of consumption utilities. The consumption proO? les and utilities corresponding to terminal nodes remain as deO? ned in Section 2.
Proofs
Proof of Proposition 1.
If R+1 U(x)g(xj0)dx U(0); then it is an eRquilibrium for B to send no transfers and 1 +1
for A not to start a war. On the other hand if 1 U(x)g(xj0)dx > U(0) playeri? s A best response is war if B chooses a strategy where transfers are zero. R +1
Starting a war is a strictly dominated action if and only if 1 U(x)g(xj0)dx < U(0). Consequently, if the inequality is strict player A will never choose to start a war and playeri? s B unique best response is to send no transfers: bt = 0:
22
DeO? nition. A strategy of the agent B is a B-strategy, if, given that B follows , Ai? s pay-o? is the same regardless of the strategy A plays.
Proof of Proposition 2.
First, we prove that there exists a B-strategy. Let bt be a strategy of the player B: Starting a war at the moment (from the stand-point of today, which is t = 0 for simplicity), brings A the utility of
Z 1
V (bt;a =W)= U(bt)e tdt+ E[U(Xjb )]e :
0
The condition that Ai? s expected pay-o? from starting a war is the same at any means that dZ 1
d U(bt)e tdt+ E[U(Xjb )]e =0 0
? ? ? for all : Re-writing the above equation (and dropping e t), one gets U(b ) EU(Xjb )+1EdU(Xjb ) db =0: (1)
At =0;wehaveb0 deO? nedby
b0 = 1E[U(Xjb0)]: (2) There exists a unique solution to the above di? erential equation (1) satisfying the initial
condition (2).
Proof of Proposition 3. We need to consider the case of E[U(X)] > U(0): Consider the following strategy proO? le: suppose the strategy of player B calls for B-proO? le transfer schedule (see the proof of Proposition 2). To formally deO? ne a strategy of player B, assume that if at some decision node the history of the game is inconsistent with the B-proO? le then player B stops transferring resources to A forever. The strategy of A is to start a war, if B ever deviates from B-proO? le. Let us show that this strategy proO? le constitute a peaceful sub-game perfect equilibrium. By construction of B-proO? le, it is the best response for player A to start a war if B ever deviates and to abstain from a war as long as B follows this strategy. Now let us show that Bi? s strategy is a best response to Ai? s strategy. The di? erential equation governing the B-proO? le is
U(b ) E[U(Xjb )] = 0; 23
? ? ? db d
? since d E[U(Xjb )] = 0 by assumption. Therefore, b = b1 for all ; and dm
U(b1) = E[U(Xjb1)]:
Since A is risk-averse, b1 E[Xjb1]: The choice of player B is between deviating and receiving a consumption stream of x L thereafter, or receiving b1: In expectation, the former action brings the utility of 1 E[U( X Ljb1)]; while the latter brings 1 U( b1): Since b1 E[Xjb1];
E[U( X Ljb1)] < U( b1):
Thus, Bi? s best response is indeed to follow the B-proO? le transfer schedule.
Proof of Proposition 4. By inspection.
Proof of Example 1. By inspection. Suppose b(t) is a transfer proO? le that leaves A indif-
ferent between starting and not starting a war, we call this B-proO? le, among all appeasement
strategies this one gives the highest possible payo? to the weaker side. To show that peaceful
equilibrium does not exist we O? rst characterize B-proO? le and then show that even B-proO? le
gives the weaker side lower payo? than immediate war. To make the argument more in-
tuitive we start with comparing a payo? to party A from starting a war at time t versus
waiting till time t + . We take a limit of ! 0 and thus neglect terms of order o(): If
party A starts a war at time t its payo? is UA(b(t)+2) if a war is postponed till time t + 1
the payo? of party A becomes UA(b(t)) + UA(b(t+)+2) ; equating the payo? from starting 1
? ? ? ? ? a war at time t and t + we obtain the following di? erential equation describes B-proO? le: UA (b(t) + 2) UA (b(t)) = UA0 (b(t)+2)b0 (t) , Substituting UA = e c and solving the di? erential
? 1
equation we O? nd that transfers are linearly increasing in time b(t) = (exp(2) 1)(1 )t. It
is easy to check that this transfer proO? le gives utility of negative inO? nity to player B: Indeed, VB = R01 e(exp(2) 1)(1 )t t dt = 1 whenever (exp(2) 1)(1 ) + ln > 0. Thus, opting for a war gives player B a higher utility.
Proof of Proposition 5. To prove the existence, we construct a sub-game perfect equilib- rium as follows. First, we describe player Ai? s equilibrium strategy. If the victim, B transfers all resources so that her own instantaneous consumption is equal to b to the blackmailer, A; the latter makes no harm to B: If B transfers less than b (i. e. Bi? s instantaneous consumption is larger than x); A chooses at = W: With probability p a war erupts, and so the Ai? s pay-o? is y: With probability 1 p; the Ai? s pay-o? is b. Choosing at = P brings 0 in the future because the strategy of player B is to stop transfers if a deviation remains unpunished). So,
24
provided that (1 p)x py; it is proO? table for A to choose at = W: For A; not triggering a war while B transfers at rate b is clearly the best response. It is straightforward to fully specify strategies of both players o? the equilibrium path and to show that the equilibrium is subgame perfect because after an action W is taken players either O? nd themselves at a terminal node or in a subgame that is essentially identical to the original game.
In the previous sections, we considered an inO? nite game between a potential aggressor and a victim where the aggressor can exercise the threat and end the game and the victim can make a single transfer or a sequence of transfers to the aggressor in order to appease him. If carrying out the threat is costly for the potential aggressor, the unique subgame perfect equilibrium entails no transfers (This statement is made formal in Proposition 1 and is also emphasized by Shavell and Spier, 2002). We show that a threatener is able to extract a stream of positive payments if the threat is probabilistic or inO? nitely i? divisible. i?
Before proceeding to the model let us illustrate i`probabilistic threatsi^with the story of a French diplomat Eon de Beaumont who blackmailed the French king Louis XV. 25 During his service as a diplomat in London, Eon de Beaumont found himself involved in a feud with the French ambassador. When di? Eon was asked by his superior to resign from em- bassy and return to France, he decided to resort to blackmail. Chevalier di? Eon had in his possession a copy of the diplomatic correspondence that was so secret that even the French ambassador to England did not know about its existence26. Chevalier di? Eon threatened to reveal the diplomatic correspondence to the Englishmen; among other things, the cor- respondence contained information about Louis XV secret preparation for invasion of the British Islands. di? Eon demanded from the French king money and a permission to remain in England. However, the king was not the only party who stood to loose from revelation of the secret correspondence. Publishing the correspondence could have made di? Eon the most despised traitor on both sides of the channel. At the O? rst glance di? Eon did not have a credi-
25For a complete account see Broglie (1879).
26An institution known as the i`kingi? s secreti^was a unique feature of the french diplomacy during the
reign of Louis XV. The members of the kingi? s secret, of which Chevalier di? Eon was one, received special directives from the kingi? s secret. The rest of the embassy sta? including the ambassador were not aware of the existence of this secret correspondence. A detailed account of the working of the kingi? s secret can be found in Broglie (1879).
? 15
ble threat; understandably, the crown found di? Eon demands ridiculous and was reluctant to o? er concessions.
To persuade the king of the seriousness of his intentions, di? Eon published a pamphlet that included small bits of the text from the secrete correspondence and vague hints about its existence. That prompted France to grant di? Eoni? s request: he was allowed to stay in England and received the allowance from the embassy. The bargaining power of di? Eon was enhanced by his ability to ina? ict arbitrary small amounts of harm by leaking small bits of secrete correspondence. The actions of di? Eon were a sort of brinkmanship. Each public hint about the correspondence could have tipped of the English that di? Eon stored the secret documents outside the embassy, in which case, the English might have tried to steal the documents.
To model brinkmanship, we modify the model of the previous section to allow for proba- bilistic threats. In the previous section, the only threat available to the aggressor was to start an all out war, that would impose a cost x on him and a cost x + L on the weaker party, where L > 0. We model divisible threats by allowing the aggressor, A; to use probabilistic threats. SpeciO? cally, the aggressori? s may take an observable action that leads to war with some positive probability p; p < 1:27 We will show that in this case the bargaining power of the aggressor is much greater than in the case where probabilistic threat is not available. Indeed, if the only threat the blackmailer has is to launch a war, he could not do better than extracting the expected gain from the war, which is zero if x < 0. In stark contrast to Proposition 1, with probabilistic threats, the strong side can extract the whole surplus from the victim. 28
There are two players, the blackmailer, A; and the victim, B. For the sake of simplicity, let agents be risk-neutral, and the outcome of war to be deterministic and if a war occurs, the Ai? s instantaneous consumption thereafter is x and the Bi? s is y = x L: We consider the case where war is an unambiguously bad outcome for both sides that is x < 0 and y < 0:29 As above, the time is continuous. In each period t 2 [0; +1); the victim makes a
27Alternatively, one might assumes that the blackmailer has a possibility to distribute harm between
di? erent periods instead of launching a war at once.
28The same logic applies to wars: a i? bullyi? state that has only two options to either start a large war, or
not to start a war at all has much less bargaining power than a state that can start a tiny war or a large
war or anything in between.
29In particular, this assumption means that there is no impact of transfers on balance of power.
? 16
non-negative transfer bt to the blackmailer. Prior to the cona? ict the consumption of parties A and B is bt and bt respectively. At each period party A (the blackmailer) chooses action at 2 fW; P g. If at = W; then the nature moves: with probability p a war occurs and the game ends, and with probability 1 p the game continues. 30 An important assumption is that the history of actions is fully observable. The following proposition shows that availability of a probabilistic threat allows the potential aggressor to extract a share of surplus close to one.
Proposition 5 Suppose that (1 p)y px: For any 0 < b < y there exists a sub-game perfect equilibrium where along the equilibrium path party B transfers resources to A at rate b and party A never starts a cona? ict.
Let us sketch the intuition behind this result. There exists an equilibrium where party A does not attack B as far as it receives transfers at rate b: If party B ever deviates party A takes action W to punish party B: Party A continues to repeat the punishment at frequent intervals as long as party B continues to deviate. If a deviation of party B remains unpunished then party B stops transferring resources to party A. There are no beliefs in the model because the solution concept here is a subgame perfect Nash equilibrium; however, to understand the intuition behind the equilibrium it is useful to talk in terms of beliefs. If any deviation of party B remains unpunished, party B comes to a conclusion that future deviations will not trigger punishments from party A. Thus, party A will punish deviations of B in order to make sure that party B will continue to transfer resources in the future.
Intuitively, the crucial feature of the above blackmail game is that the harm is inO? nitely- divisible. The condition px + (1 p)y 0 means that the aggressor, A; prefers gambling on war (i. e. choosing at = W ) that yields a stream of losses at rate x with probability p and a stream of transfers at rate b with probability (1 p) over receiving zero transfers from now on. Thus, it is the best response for party A to punish deviations by B and it is the best response for B to transfer resource to A at rate b:
It is worth noting that the blackmail game has multiple equilibria. A war never occurs on the equilibrium path in the equilibrium sketched above. However, there are many econom- ically plausible equilibria where wars occur with positive probability because probabilistic threats or i`brinkmanshipi^ occurs on the equilibrium path. For instance, the equilibrium strategy of player A may call for taking action W (engaging in brinkmanship) every O? ve
30We omit the formal description of the space of histories because it is analogous to the previous section. 17
? years. If party A deviates from this strategy it looses credibility as a potential aggressor and
never receives transfers in the future. There is an equilibrium where along the equilibrium
path the weak party occasionally reduces transfer level and the strong party immediately
takes action W; with subsequent resumption of transfers at the normal level. Note that ac-
tion W is not a punishment for a deviation, the victim was supposed to reduce the transfers
and the aggressor is supposed to i`punishi^this equilibrium behavior. The intuition for this
result is as following. According to Proposition 4 there exists a subgame perfect equilibrium
with transfers 9 y and there is an equilibrium with no transfers. It is easy to construct an 10
equilibrium where transfers are 1 y per period and once a year transfers by B are reduced to 2
zero for one month. At the end of that month A plays action W and B resumes transfers at rate 1y. If aggressor fails to take action W both players switch to playing the equilibrium
2
without transfers in the remaining subgame. If party B fails to reduce transfer to zero at the end of the year both parties play the equilibrium where transfers are 9 y in the remaining
10
subgame.
The model described above may shed some light on economics of terror. For decades,
the Irish Republican Army terrorizes the British government and people. A distinct feature of the IRA terror tactics is that acts of violence usually cause relatively small death toll, but, in retrospect, each attack might have caused much more human loss. The British making slow concessions and IRA engaging in occasional terrorist acts is consistent with an equilibrium path of play. Occasional bomb threats may be necessary to make the terroristsi? threat credible and thus induce more concessions from the British. Of course, there exists a more e? cient equilibrium where concessions are obtained without violence.
5 Conclusion
The bargaining game described herein captures some essential features of relationships were neither party has commitment power. A self-enforcing agreement is feasible if parties are either risk neutral or transfers do not change the balance of power. A successful peace agreement is likely to consists of a sequence of concessions. We show that if both parties are risk-averse and if transfers change balance of powers an ine? cient cona? ict may be inevitable.
The model of brinkmanship is our main contribution. We model brinkmanship as an ability to take an observable action that with large probability have no consequences and
? ? ? ? 18
with small probability ends the game with payo? s that are very undesirable for both sides. Examples of brinkmanship can be found situations ranging from international relations to Hollywood movies were gangster shoot under the feet to force a victim to cooperate. We show that if the action space of the potential aggressor allows him only two actions such as i? cooperatei? or i? go to the ultimate threat point,i? his bargaining power is far less than in a case where the potential aggressor can make probabilistic threats.
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Appendix The setup
The set of players is N = fA;Bg: The set of histories is denoted H and ; 2 H. A vector function h, ht = (at; kt) belongs to the set of non-terminal histories if and only if the support ofhissomReinterval[0;T[;andforanyt2[0;T[wehaveat =Pandbt 0where bt = k0 + 0t ktdt: There is a natural interpretation of kt: we assumed that the function representing transfer proO? le is di? erentiable and thus continuous. Consequently, although player B can change the transfer level at any instant his choice must be consistent with continuity assumption, essentially, moment by moment player B chooses the derivative of transfer proO? le, which we denote kt: (We slightly abuse notation by denoting agenti? s choice of the level of transfer at time zero by k0. )
Let Z denote the set of all terminal histories. The set Z consists of two types of terminal histories: inO? nite (these where a cona? ict never occurs) and O? nite (these that end in a war). A vector function ht = (at;kt) belongs to the set of inO? nite histories if and only if h has support [0; +1[ and for all t; at = P and bt 0: A vector function ht = (at; kt) belongs to a set of O? nite terminal histories if and only if it has support [0; ], bt 0 and at = P for all t < ; and w( ) = W:
The choice correspondence P(h) = fA;Bg for all h 2 HnZ: In other words following any non terminal history both players take action simultaneously.
The preferences of players are given by the discounted sum of consumption utilities. The consumption proO? les and utilities corresponding to terminal nodes remain as deO? ned in Section 2.
Proofs
Proof of Proposition 1.
If R+1 U(x)g(xj0)dx U(0); then it is an eRquilibrium for B to send no transfers and 1 +1
for A not to start a war. On the other hand if 1 U(x)g(xj0)dx > U(0) playeri? s A best response is war if B chooses a strategy where transfers are zero. R +1
Starting a war is a strictly dominated action if and only if 1 U(x)g(xj0)dx < U(0). Consequently, if the inequality is strict player A will never choose to start a war and playeri? s B unique best response is to send no transfers: bt = 0:
22
DeO? nition. A strategy of the agent B is a B-strategy, if, given that B follows , Ai? s pay-o? is the same regardless of the strategy A plays.
Proof of Proposition 2.
First, we prove that there exists a B-strategy. Let bt be a strategy of the player B: Starting a war at the moment (from the stand-point of today, which is t = 0 for simplicity), brings A the utility of
Z 1
V (bt;a =W)= U(bt)e tdt+ E[U(Xjb )]e :
0
The condition that Ai? s expected pay-o? from starting a war is the same at any means that dZ 1
d U(bt)e tdt+ E[U(Xjb )]e =0 0
? ? ? for all : Re-writing the above equation (and dropping e t), one gets U(b ) EU(Xjb )+1EdU(Xjb ) db =0: (1)
At =0;wehaveb0 deO? nedby
b0 = 1E[U(Xjb0)]: (2) There exists a unique solution to the above di? erential equation (1) satisfying the initial
condition (2).
Proof of Proposition 3. We need to consider the case of E[U(X)] > U(0): Consider the following strategy proO? le: suppose the strategy of player B calls for B-proO? le transfer schedule (see the proof of Proposition 2). To formally deO? ne a strategy of player B, assume that if at some decision node the history of the game is inconsistent with the B-proO? le then player B stops transferring resources to A forever. The strategy of A is to start a war, if B ever deviates from B-proO? le. Let us show that this strategy proO? le constitute a peaceful sub-game perfect equilibrium. By construction of B-proO? le, it is the best response for player A to start a war if B ever deviates and to abstain from a war as long as B follows this strategy. Now let us show that Bi? s strategy is a best response to Ai? s strategy. The di? erential equation governing the B-proO? le is
U(b ) E[U(Xjb )] = 0; 23
? ? ? db d
? since d E[U(Xjb )] = 0 by assumption. Therefore, b = b1 for all ; and dm
U(b1) = E[U(Xjb1)]:
Since A is risk-averse, b1 E[Xjb1]: The choice of player B is between deviating and receiving a consumption stream of x L thereafter, or receiving b1: In expectation, the former action brings the utility of 1 E[U( X Ljb1)]; while the latter brings 1 U( b1): Since b1 E[Xjb1];
E[U( X Ljb1)] < U( b1):
Thus, Bi? s best response is indeed to follow the B-proO? le transfer schedule.
Proof of Proposition 4. By inspection.
Proof of Example 1. By inspection. Suppose b(t) is a transfer proO? le that leaves A indif-
ferent between starting and not starting a war, we call this B-proO? le, among all appeasement
strategies this one gives the highest possible payo? to the weaker side. To show that peaceful
equilibrium does not exist we O? rst characterize B-proO? le and then show that even B-proO? le
gives the weaker side lower payo? than immediate war. To make the argument more in-
tuitive we start with comparing a payo? to party A from starting a war at time t versus
waiting till time t + . We take a limit of ! 0 and thus neglect terms of order o(): If
party A starts a war at time t its payo? is UA(b(t)+2) if a war is postponed till time t + 1
the payo? of party A becomes UA(b(t)) + UA(b(t+)+2) ; equating the payo? from starting 1
? ? ? ? ? a war at time t and t + we obtain the following di? erential equation describes B-proO? le: UA (b(t) + 2) UA (b(t)) = UA0 (b(t)+2)b0 (t) , Substituting UA = e c and solving the di? erential
? 1
equation we O? nd that transfers are linearly increasing in time b(t) = (exp(2) 1)(1 )t. It
is easy to check that this transfer proO? le gives utility of negative inO? nity to player B: Indeed, VB = R01 e(exp(2) 1)(1 )t t dt = 1 whenever (exp(2) 1)(1 ) + ln > 0. Thus, opting for a war gives player B a higher utility.
Proof of Proposition 5. To prove the existence, we construct a sub-game perfect equilib- rium as follows. First, we describe player Ai? s equilibrium strategy. If the victim, B transfers all resources so that her own instantaneous consumption is equal to b to the blackmailer, A; the latter makes no harm to B: If B transfers less than b (i. e. Bi? s instantaneous consumption is larger than x); A chooses at = W: With probability p a war erupts, and so the Ai? s pay-o? is y: With probability 1 p; the Ai? s pay-o? is b. Choosing at = P brings 0 in the future because the strategy of player B is to stop transfers if a deviation remains unpunished). So,
24
provided that (1 p)x py; it is proO? table for A to choose at = W: For A; not triggering a war while B transfers at rate b is clearly the best response. It is straightforward to fully specify strategies of both players o? the equilibrium path and to show that the equilibrium is subgame perfect because after an action W is taken players either O? nd themselves at a terminal node or in a subgame that is essentially identical to the original game.
