No More Learning

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:
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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
Z1
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 = ec 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.