No More Learning

The rest of the paper is organized as follows. Section 2 o? ers a description of a bargaining game that captures some essential features of wars. In Section 3, we analyze conditions that allows peace to prevail. The main result is in Section 4 that introduce an additional feature to the model that captures the notion of brinkmanship. Namely we consider a possibility that the potential aggressor can take an observable actions that will lead to war with small positive probability. Section 5 concludes.
18In some sense, the di? culty of settling a war i^in the middle of the cona? icti^relative to settling a labor dispute is quantitative. However, this distinction is so dramatic that it becomes qualitative. Naturally, as a semantic matter a war can never be stopped in the middle: By deO? nition, whenever an armed cona? ict is settled it is the end and not the middle of a war. But, aside from semantic casuistry, it is the lack of external contract enforcement again, that makes negotiating an end to a war extremely di? cult. Negotiating a self-enforcing peace accord at the time when active military actions are taking place is often impossible due to the nature of warfare technology. Litigation is sometimes described as a form of i^ritualized warfarei^, yet analogy is far from perfect because i^litigation ritualsi^ and rules are enforced externally. In order to foster e? ciency, a settlement during a cona? ict is encouraged by the legal system; the same cannot be said about wars.
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2 The Setup
Consider a world consisting of two countries, A and B. Country A is a potential aggressor, whose expected payo? from a war is greater than that of B. Countryi? s B expected payo? from a war is negative. Consequently, B may be willing to pay tribute (bribes) to A in order to preserve peace. At each instant of time t > 0 player A (the potential aggressor) chooses whether or not to start a war, at 2 fP; W g; while the player B can adjust the rate of transfer to A: 19 The game ends if a war starts. During the pre-war period, the consumption of both parties is determined by the history of transfers. If the war occurs, the post war consumption is determined by the war outcome, while the probability distribution of war outcomes may be in turn ina? uenced by pre-war wealth transfers. 20 Both countries maximize expected discounted utility of consumption.
In a reduced form, a strategy of the weaker country (player B) can be represented by
a non-negative di? erentiable function b : R+ ! R+; t 7! bt: In the event that the game
proceeds until time t without a war, bt represents the rate at which B transfers resources to
A at time t. To deO? ne an aggressori? s strategy, let us summarize the history of the game up to
time T by an element from the set HT of non-negative di? erentiable function with support
on [0;T[:21 Each strategy of player A prescribes her to start or not to start a war for each
history of transfers, i. e. a strategy of player A is a function F : HT ! fW;Pg. The payo?
to playerRi is the discounted utility of the players consumption proO? le cit; the payo? equals
toVi = 1 tU(c )dtwhereU0(:)>0andU00(:)0,i=A;Band isthediscountrate. 22 0iit i i
Instantaneous consumption cit equals to some base rate plus a transfer received. Thus, without loss of generality, we normalize total consumption to zero, so that consumption proO? le prior to war is given by cAt = bt and cBt = bt for t  T; where T denotes the time when agent A selects action W: If a war never occurs, we can say that T = 1. If a war
19The same situation might be modelled assuming full symmetry between countries.
20Gains/losses from a war tend to be spread out over time. For example, the party that prevails in
the cona? ict may receive a stream of reparation payments or a perpetual stream of beneO? ts from concurred
territories. Consequently, we assume that the post war consumption is a constant stream of beneO? ts/losses
determined by the outcome of the war.
21The assumption that the function representing transfers is di? erentiable is essentially equivalent to the
assumption that the agent chooses the level of transfers at time zero, and chooses the rate of change at each
subsequent moment.
22For the sake of simplicity of notations, we assume here that both players are described by the same
utility function U(:).
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occurs, the total consumptions is permanently reduced so that cAt + cBt = L for t > T , where L is the loss of resources due to war. The post-war consumption of players are cAt = x and cBt = x L for all t > T; where x is random variable that represents the outcome of the war. In order to allow for a possibility that transfers shift the balance of power in favor of the recipient we assume that x is distributed according to probability density function g(xjc) where c is the consumption of party A at the time immediately preceding the war, the distribution g(xjc) is atomless, continuous, and dE[xjc] > 0.
dc
We leave a more formal description of the game to the appendix. Throughout the paper, our analysis is conO? ned to subgame-perfect Nash equilibria. A peaceful equilibrium is any equilibrium in which a war occurs with zero probability.
3 Peace and Transfers
In this section, we use the model developed above to obtain insights into the origins of war and international peace treaties. Proposition 1 establishes necessary and su? cient conditions for maintaining peace without one-sided concessions or wealth transfers. Proposition 3 shows that a self-enforcing peace agreement is viable whenever transfers do not change the balance of powers. In such an equilibrium, all gains from peace are captured by the weaker party, i. e. leaving the potential aggressor indi? erent between starting and not starting a war. Even if transfers increase the beneO? t from a war to the potential aggressor, thus increasing his demands for more concessions, a self-enforcing peace agreement between risk neutral parties is feasible (see Proposition 2). However, if parties are risk-averse, a self-enforcing peace agreement may not be feasible.
The O? rst two Propositions of this section replicate and generalize the result of Shavell and Spier (2002) for our setting. Unlike Shavell and Spier (2002), we consider a continuous time model and allow for risk aversion. The results of this sections are important for understanding the main result presented in the next section. Peace without transfers is possible only if no party expects to beneO? t from a war. The following straightforward proposition makes this claim formal.
Proposition 1 A peaceful equilibrium without transfers exists if and only if the agent Ai? s expected instantaneous utility from starting a cona? ict does not exceed her instantaneous utility
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under the status-quo, that is R +1 UA(x)g(xj0)dx  UA(0). If the inequality is strict, any 1
sub-game perfect equilibrium is peaceful.
Now let us consider a more interesting case: a possibility of peaceful equilibrium with positive transfers. A possibility of peace, especially when transfers change the balance of powers is not obvious, because in this case concessions will lead to demands for new and bigger concessions (see Rajan and Zingales, 2000, for an example of a model where ine? cient outcome obtains for exactly this reason). Hirshleifer (2001) argues that this might be the case to start a war as early as possible. The following proposition shows that risk-neutral parties can always negotiate a self-enforcing peace agreement.
Proposition 2 Suppose that players have a constant marginal utility of consumption, U(c) = c: Then there exists an equilibrium transfer stream such that a war does not occur on the (sub-game perfect) equilibrium path.
The idea of the proof is to construct a transfer proO? le that makes party A indi? erent between starting and not starting a war at any point in time. Such strategy gives all rents to B, and so we refer to such proO? le as B-proO? le. By construction Ai? s best response to B-proO? le is to refrain from starting the war. It is also an equilibrium for player B to follow B-transfer proO? le because B will be strictly worse o? if she deviates (because a war will result). Both parties recognize that transfers received by a potential aggressor in the present will allow her to gain more transfers (or wage a more proO? table war) in the future. Indeed, consumption stream resulting from starting a war at period t is the amount xt  E[xjbt]: Player A may be willing to postpone the war at time t even if the transfer that he receives at time t and immediately afterwards is substantially less than the post war consumption level would have been. If transfers shift balance of power, an increase in transfers to A will trigger even greater demands by the potential aggressor tomorrow. In equilibrium future transfers will increase over time to some limit denoted b1 where b1 = limt! 1 bt. For a given probability distribution of war outcomes g(xjc) the asymptotic value of transfer is given by the non-negative root of the following equation E[xjc = b1] = b1. 23
History provides us with a number of examples illustrating that a stream of transfers could
support peace better than lump-sum transfer. For decades since the last days of Chinguiz
23Forinstance,ifE[xjc]=1+ c thensolving1+ b1 =b1weobtainb1=c=1+1p5:Alsonote c+1 b1+1 22
that E[xjc = b1] = b1 only if b1 is O? nite.
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Khan, the Mongol Empire was the largest state in the history of Earth by size (and the largest of the medieval empires by population). Most of the empire parts were controlled by relatively small forces, which collected modest, but regular tributes from occupied lands. Mongols used a threat of war to extract tributes from the territories under their control. In the political domain, a concession by one party often changes the balance of powers that in turn leads to an escalating sequence of concessions. (The history of the civil-rights movement provide a vivid example of this. ) Making a small change in a political institution may shift the balance of power making an institution unstable and leading to a sequence of reforms. Laguno? (2004) investigates the set of stable political institutions in a static setting. Our model suggests that political institutions may also be in a dynamic equilibrium where stakeholders initiating reform (as well as these opposed to it) understand and correctly anticipate that the change will trigger future reforms.
Now let us turn to the case where transfers do not ina? uence the balance of power.
Proposition 3 Suppose that A and B are risk-averse, and that past transfers do not a? ect the probability distribution of war outcomes, i. e. g(xjc) = g(x) for any c: Then there exists a peaceful sub-game perfect equilibrium.
From Proposition 1, we know that there exists a peaceful equilibrium whenever E[UA(X)]  UA(0). Let us consider the case when E[UA(X)] > UA(0); and hence E[X]  0 by the risk- aversion assumption. Consider the following strategy proO? le: the strategy of player B calls for B-proO? le transfer schedule, i. e. the schedule that makes A indi? erent between starting a war and waiting a little. The strategy of A is to start a war, if B ever deviates from B-proO? le. These strategy proO? les constitute a peaceful equilibrium. By construction, 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. Indeed, the transfers should be constant as they do not a? ect the balance of power: b = b1 such that UA(b1) = E[UA(Xjb1)] for all : The choice of player B is between deviating and receiving the consumption stream of x L thereafter, or receiving b1: Since B is risk-averse, Bi? s best response is indeed to follow B-proO? le transfer schedule.
The economic intuition behind the above propositions do not require the assumption that consumption proO? le is continuos. However, this assumption does have some interesting implications. For instance, in the context of Proposition 3, the following strategy proO? le
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constitutes a Nash equilibrium: party B transfers at a rate bt = E[X] (recall that we assume here that the distribution of X; the outcome of war, does not depend on transfers), and if it ever deviates, it rapidly decreases transfer rate to zero. And party A starts a war if and only if party B deviates. It is easy to see that this strategies form an equilibrium, where no war occurs on the equilibrium path. However, this equilibrium is not sub-game perfect. Indeed, if B deviates and transfers become slightly less then E[X]; it is not in the interest of party A to carry out the threat of immediately starting a war because party A is strictly better o? not starting a war as far as transfer level stays over b1 deO? ned by UA(b1) = E[UA(X)]: Indeed, even if party A believes that B is going to reduce transfers to zero very soon, there is no reason not to wait until transfer rate would drop to b1: Consequently, continuity implies that out of a large set of Nash equilibria, only the least favorable for A survives subgame perfection. The following simple proposition generalizes this argument.
Proposition 4 In any sub-game perfect equilibrium, the expected pay-o? of A is equal to the expected pay-o? of A in the case of war at t = 0: The pre-war consumption proO? le coincides with the B-proO? le.
Propositions 2 and 3 show that a carefully negotiated treaty may be a self-enforcing peace agreements. However, for some parameter values a self-enforcing peace agreement is not feasible. Suppose both agentsi? utility functions are strictly concave. In this case when consumption is large, the marginal utility is very small, and when consumption is negative, the marginal utility of consumption is large. If transfer shift the balance of power the weaker party will have to transfer more and more resources in order to appease the potential aggressor. Due to concavity of the utility function the weaker party may O? nd the prospect of making big transfers later more painful than O? ghting the war earlier. The example demonstrates that a self enforcing peace agreement between risk averse parties may not be viable if transfers shift the balance of power.
Example 1 Let ct denote the consumption level of A at time t: Suppose that the outcome of a cona? ict is deterministic: if a war occurs at time , the strong side, A; receives post cona? ict consumption a? ow at the rate of c +2, while post cona? ict consumption of the weak side, B;is c 2 L per period, L > 0. Let instaRntaneous utility of both parties be U (c) = ec and the total utility of either player is Vi = 01 0:9tU(ci(t))dt where i = A;B: Then there does not exist a sub-game perfect peaceful equilibrium.
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The above example shows that not only the violation of power parity may prompt a war, but the mere expectation of a future power shift may prompt a war. Indeed, a party whose relative power is expected to diminish if it were to pursue an appeasement strategy may prefer to O? ght a war sooner rather than later (provided that the discount factor is su? ciently high).
24 Fearon (1995) surveys a number of authors that support this conclusion, citing, among many others, Taylor (1954): i`Every war between the Great Powers [in the period of 1848-1918] started as a preventive war, not a war of conquesti^, and Carr (1964): i`The most serious wars are fought in order to make onei? s own country military stronger or, more often, to prevent another country from becoming military strongeri^. Within this logic, Example 1 formalizes the Hirshleifer (2001) assertion that if the opponent (A; the potential aggressor in our setup) is hostile and non-appeasable, the best possible strategy is to keep the aggressor poor, so that if he opts for a war, he has as little resources as possible.
4 Main Result: Brinkmanship and Blackmail
Unless the potential aggressor expects to beneO? t from a cona? ict, extracting any concessions from the weaker party is not subgame-perfect. In light of this, any blackmail is doomed to failure if the blackmailer su? ers some costs associated with ina? icting the damage on the victim. In this section, we demonstrate that this result hinges on the assumption that the potential aggressor has a small action set: previously, we assumed that the only possible action are peace or ending the game with an all-out war. Here, we show that the bargaining power of the potential aggressor is dramatically increased if action sets includes actions that are in between the two extremes mentioned above.
For instance, we argue that possession of nuclear weapons may not be useful for extracting concessions from weaker countries. A country with air force capable of delivering compara- ble destruction using conventional weapons may be able to extract concessions from other countries, because, unlike a nuclear bomb, conventional weapons are a i`divisible threati^. The importance of i`divisible threatsi^is not limited to international negotiations. We argue that the i`divisibility of a threati^matters as much in cona? icts and negotiations within or- ganizations as it does in international relations. The applicability of the model considered
24Formally, there are no beliefs in this model. However, it is useful to think of conditional strategies as those of based on expectations.
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in this section is not limited to blackmail in international negotiations. It captures essential features of blackmail in a wide range of situations. For instance, a manager who has the power to O? re a worker at any point in time is far less powerful than a manager who has a range of punishments less severe than O? 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).
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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.
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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
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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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