Bargaining. Divide the melting cake Often bargaining is similar to a game where players make alternating offers about how to divide a cake that is melting Sometimes this is referred to a divide the dollar game, where time is money so if you wait you have less to divide Consider just one period and two players (A) and (B) The game works as follows: (A) proposes a division; then (B) either agrees or not - if (B) agrees then they divide the cake according to (A)s proposal – otherwise it melts What does game theory predict? – think about subgame perfection or using backwards induction. (B) knows that if he turns down (A) s offer, he gets nothing Therefore, if (B) is playing optimally in every subgame he will accept any positive offer and would even be indifferent between getting nothing and having the cake melt

The SPNE. (B) will accept any offer; (A) will best respond by proposing the following division: (A) gets everything; (B) gets nothing SPE:(A) proposes: (A) everything and (B) nothing (B) accepts anything with a non-negative payoff, rejects otherwise What about NE? There are an infinite number of NE Let x B be some fraction between 0 and 1, then an NE takes the form (A) proposes: (A) 1- x B and (B) x B (B) accepts anything ≥ x B, rejects otherwise What happens in experiments? Typically, offers range from to (B) rejects offers below 30% of the pie The SPNE is not descriptive and is not prescriptive for (A) either if you are playing against someone who rejects offers below 30%

Bargaining with Complete information Two players bargain to determine the split of V dollars The game begins in period 1 Player (1) makes an offer of a split (a share of V for each player) If Player (2) accepts (1)s offer, bargaining ends and payoffs are realized. If Player (2) rejects (1)s offer, then (2) makes an offer Then (1) can accept or reject … The game ends in period T and if there is no agreement, both players get 0 Each player has a discount factor (0 < δ < 1) - so a dollar in period t is worth δ t-1 terms of period 1 dollars There is a unique SPNE in this game and the path has no bargaining process – player (1)s first offer is accepted and the game ends Finite horizon bilateral bargaining.

1 st case: T is an odd number.

2 nd case: T is an even number.

“Bargaining model of War & Peace”. Motivation: Build on the literature, which has several gaps 1) War is often modeled as a costly lottery - too simple, does not describe realistic conflict, too short, doesn’t allow for bargaining, negotiation, counter-offers, and defeats usually do not end in the utter defeat of one side 2) The literature treats war onset separately from war outcomes -ignores forward looking behavior that might affect conflict engagement decisions 3) When bargaining has been considered, it has been considered in an unrealistic way - Example: allow bargaining before war starts but once war begins the conflict is back to a costly lottery with no surrender Basic model structure -parties can bargain before fighting to avoid war; bargain in between battles to end war -the model yields factors that affect onset, duration, outcomes, and negotiated settlements

Approach. Introduce a game-theoretic bargaining model with one-sided incomplete information - this leads to a more realistic bargaining process - there maybe a delay in reaching an agreement By adding one-sided asymmetric information, (2) is now one of two possible types δ1, δ2 - when (1) determines his proposal, he may aim his proposal at the less demanding type - then (1) bears the risk that a war will begin Two-sided incomplete information is more realistic but it makes it more difficult to obtain clear predictions a.If an informed player makes a proposal, the game becomes a signaling game which can have multiple equilibria b.Alternating bargaining is important in situations where it is not clear who the attacker is. For example, India vs. Pakistan

Model. One attacker: A benefits: B a resources: R a One defender: D B d R d Benefits are what the players fight and bargain for and are completely divisible so that it is possible to compromise, while the resources are what the players fight with. Each player’s objective is to obtain as many benefits as possible while conserving resources. - players may be willing to sacrifice resources to acquire more benefits. Assume each utility function U i ( B i, R i ) is increasing, concave, twice differentiable in both arguments and that the cross-partials are greater than zero (like a Cobb-Douglas fn.). a) each player always wants more resources and more benefits b) marginal utilities of resources and benefits are diminishing c) marginal utility of benefits is increasing in resources d) marginal utility of resources is increasing in benefits -these conditions imply that a disputant’s willingness to risk resources in battle is increasing in his amount of resources and decreasing in size if his benefits

War & Uncertainty. War consists of an alternating sequence of negotiations and battles - If negotiations succeed war ends - If negotiations fail and A does not quit, then players fight another battle - In each battle D defeats A with probability d and looses with probability (1-d) - Fighting expends resources; the amount depends on who wins (R a w > R a l) ; (R d w > R d l ) R a wl represents A’s resources after she wins the first battle but loses the second -War ends as soon as one side’s resources fall below a minimal amount of resources necessary to continue fighting. Uncertainty: A only knows the probability distribution of D’s type (d h or d l ) 1)assume uncertainty is one-sided; D has complete info about A, but A does not have complete info about D. 2)assume that A’s uncertainty about D is confined to uncertainty about D’s military ability (the d parameter). All types of D are assumed to have the same utility function 3)A has beliefs that are updated as D rejects proposals and battle outcomes are observed.

Simplest game. At most two battles can occur: Assumption 1: war can extend to a second battle only if A wins the first battle Assumption 2: D can sustain two battle losses before his resources are insufficient to continue fighting, which limits the size of the game tree Assumption 3: there are only two types of D: d h > d l (d h is more likely to win battles) To identify conditions necessary for Conflict Termination, -Begin at the conclusion of the game -Let γ h t and γ l t denote the highest values of γ t acceptable to types d h and d l in stage t -Type d l is willing to make a greater concession to A in order to avoid a fight than is d h Therefore, γ h t ≤ γ l t 1)any proposal: γ ≤ γ h t is acceptable to both types 2)any proposal: γ h t < γ ≤ γ l t is acceptable to type d l, but unacceptable to type d h 3)any proposal: γ l t ≤ γ is unacceptable to both types

3 possible choices. both types accept l- accept, but h-reject both types reject At most three relevant options for A in each stage 1) proposes γ h t and the conflict ends 2) proposes γ l t and depending on D’s type conflict may continue or end 3) ask for more than γ l t and conflict will definitely continue

Fig. 3 Continuation after A retreats.

When will D accept?.

Rule out: γ ≥ γ l t.

Proof from Appendix B.

Jensen’s inequality.

Result 1. R1) In equilibrium A never provokes a fight with both types At each negotiating opportunity, - if type d l remains then A prefers proposing γ l t to any γ t > γ l t - if type d h remains then A prefers proposing γ h t to any γ t > γ h t - A makes a proposal that some type of defender will accept War does not start/endure simply because A is aggressive or doggedly committed to fighting. - in this model, the attacker never just provokes a fight Battles are risky and both players are risk averse - factors that affect this choice - γ h 3, γ l 3 - trade-off γ h 3 < γ l 3 high risk-return payoff γ l 3 This result generalizes to n types

Result 2. R2) A is more likely to propose γ t = γ l t - if A is optimistic that D is type d l - if A anticipates low resource losses from fighting - if d l is low Intuitive, if A’s settlement proposals are viewed in terms of a risk-return tradeoff, it depends on how A assesses the value of the gamble 1)The more confident A is that the defender is the weak type, the less risky it is to make a tough demand because it is more likely that the demand will be accepted. 2)If A anticipates low resource losses from fighting, then it is not so costly if she is wrong in her assessment of D’s type 3)The lower type d l ’s probability of victory is, the tougher a demand A can make because type d l is willing to make more concessions the lower the probability of victory (making a demanding proposal is risky, but the large reward may make it worth the risk)

Result 3. R3) Beliefs evolve through Bayes rule in response to proposal rejections and battle outcomes - Proposal rejections and battle losses make A more pessimistic - Battle victories make A more optimistic This result implies that war is an important source of information to disputants In addition, the possibility that diplomacy continues throughout the war also provides a means of revealing information

Results 4 and 5. R4) A’s subjective belief about the likelihood of winning battles falls as the war progresses Implies that proposal rejections reduce A’s subjective belief about her likelihood of winning battles because weaker defenders are screened out as they accept proposals (R3). Since on average, ex post equal ex ante beliefs, we can focus on proposal rejections where the weaker types drop out R5) In expectation, A is more likely to propose terms acceptable to stronger types as the war continues. On average, war is more likely to end the longer it has endured. Note: in expectation, battles have no effect on A’s beliefs, because by Bayes’ Rule expected posteriors equal priors. In expectation, the effect of proposal rejections determines how A’s proposals evolve over time. Wars exhibit positive duration dependence; they are more likely to end the longer they have endured.

Analysis. The possible equilibrium paths from page 834 A is unwilling to attack in stage 1, so war does not start. It does not matter what A proposes in this case; all proposals are rejected. A proposes γ h 1 and both types accept. War does not start. A proposes γ l 1, type d l accepts and type d h rejects. If D rejects, then A attacks. If A wins the first battle, she proposes γ h 2 and D accepts. A proposes γ l 1 type d l accepts with probability (0 < w < 1), and type d h rejects. If D rejects, then A attacks. If A wins the first battle, she proposes γ h 2 and D accepts, and does not fight a second battle. A proposes γ l 1, type d l accepts with probability (0 < w 1 < 1) and type d h rejects. If D rejects A attacks. If A wins the first battle, she proposes γ l 2, type d l accepts with (0< w 1 ≤ 1) and type d h rejects. If D rejects, then A attacks again Formal results provide intuition about the conditions under which each path occurs.

Extensions. 1)Include third parties, more participants 2)What happens when we put more structure on benefits: physical vs. ideological benefits 3)Separate resources into labor/capital; people care more about labor (casualties) than capital 4)We could endogenize resource choices and allow for the production of military resources, GDP growth, etc. 5)We could allow transfers of benefits through fighting itself (taking territory, for example) 6)There are problems with negotiated settlements. How do you enforce the settlement?

Another extension.

Other extensions.

Conclusion. In the model, War starts 1)because the attacker believes her power affords her a greater share of the benefits than she currently has and thus demands some concessions from the defender 2)because the defender believes that he does not need to make as many concessions as the attacker has demanded and thus refuses to concede War continues So long as attacker continues to overestimate what defender will concede - or in the event of a retreat, underestimate what she must give up War ends - when the attacker and defender’s beliefs about the defender’s power converge sufficiently for the attacker to make a proposal acceptable to the defender - at each stage the attacker can obtain a concession only if her threat to attack is credible

Testable Hypothesis. 1)The attacker never retreats after winning first battle. In general, retreats follow losses not victories 2)The attacker is more likely to make proposals that involve a risk that a war begins or continues a.if she is optimistic that the defender is weak b.if she anticipates low resource losses from fighting c.if her probability of winning battles conditional on the defender being weak is high 3)Since battle victories make the attacker more optimistic we expect that attacker is more likely to offer an acceptable settlement after a loss than a victory 4)Early losses for the attacker are conducive to a short war 5)The failure of negotiations makes the attacker more pessimistic. We expect that on average the duration of war will positively affect the likelihood that the war will end and negatively affect the attacker’s prospects for victory 6)The final settlement responds to military variables. Diplomacy & force are linked.