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 xB be some fraction between 0 and 1, then
an NE takes the form (A) proposes: (A) 1- xB and (B) xB
(B) accepts anything ≥ xB , rejects otherwise
What happens in experiments?
Typically, offers range from 50-50 to 70-30
(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%
Finite horizon bilateral bargaining
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 δt-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
1st case: T is an odd number
When T is odd, (1) makes the final offer. (2) is indifferent between accepting zero or not,
- but must accept if (1) is to have a BR, otherwise (2) will get arbitrarily close to zero
The unique equilibrium in the final subgame: (2) accepts any non-negative number
Back-up (1) offers a (V,0) split and the resulting payoffs would be   t 1V , 0 
Now consider the bargaining in stage T-1:
(2) knows (1) will accept any offer   t 1V , because in stage T , (1) can get  T 1V
(2) offers  V , 1   V  , (1) accepts, game ends w/ payoffs  T 1V ,  T  2 1   V 
Continue backing up until period 1
- as t gets smaller the pie grows, so the player making the offer can always increase his
payoff, while holding the other player's payoff at his reservation value
  1   T 
(1) gets a payoff of V 
 ;
  1   

 1   T 
(2) gets payoff of V  V 

1





2nd case: T is an even number
When T is an even number, (2) makes the final offer.
A similar analysis demonstrates that

 1   T 1  
(1) gets a payoff of V   V 
 ;
 1   


 1   T 1  
(2) gets payoff of  V 

1





For the game from T-1 on it is just like the T odd case except with role reversal.
 1   T 1 
Player (2) can guarantee himself  V 
in the T period
T 
1




So (1) has to give him that in period T-1, and so on
“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
One defender: D
benefits: Ba
Bd
resources: Ra
Rd
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 Ui( Bi , Ri ) 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 (Raw > Ral); (Rdw> Rdl)
Rawl 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 (dh or dl)
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: dh> dl (dh is more likely to win battles)
To identify conditions necessary for Conflict Termination,
-Begin at the conclusion of the game
-Let γht and γlt denote the highest values of γt acceptable to types dh and dl in stage t
-Type dl is willing to make a greater concession to A in order to avoid a fight than is dh
Therefore, γht ≤ γlt
1) any proposal: γ ≤ γht is acceptable to both types
2) any proposal: γht < γ ≤ γlt is acceptable to type dl, but unacceptable to type dh
3) any proposal: γlt ≤ γ is unacceptable to both types
3 possible choices
At most three relevant options for A in each stage
1) proposes γht and the conflict ends
2) proposes γlt and depending on D’s type conflict may continue or end
3) ask for more than γlt and conflict will definitely continue
l- accept, but h-reject
both types accept
 3h
both types reject
 3l
To analyze the game start at the end and work backwards
Fig. 3 Continuation after A retreats
When will D accept?
Fig. 3, type d l will accept if
U d  Rdl , 1   3  Ba + Bd  
1-d l U d  Rdll ,
Bd   d lU d  Rdlw , Ba + Bd 
A will increase  3l until this inequality binds
There is a similar inequality for the high type
A's proposal  3  0,1
only 3 possible choices have any relevance  3l ,  3h ,
and something greater than  3l that both types reject
Rule out: γ ≥ γlt
R1) A never provokes a fight with both types. This implies that A proposes either  3l or  3h ;
we can rule out something greater than  3l
Intuition: 1) fighting is costly
2) both sides are risk averse
If A proposes  3h both types accept. Then A's payoff is U a  Raw ,  3h Ba 
If A proposes  3l , then A's payoff is
1-4 U a  Raw ,  3l Ba   4 d hUa  Rawl , 0   1  d h U a  Raww , Ba 
Ruling out  >  3l : the payoff would be
1-4  d lU a  Rawl , 0   1  d l U a  Raww , Ba   4 d hUa  Rawl , 0   1  d h U a  Raww , Ba 
Proof from Appendix B
Jensen's inequality is often used to explain why players are avoiding risk
For concave functions U d  E  x    E U d  x   ;
Raw  Raww  Rawl
Rather have (1 - d l )Ba  for sure than the average of  d lU a (0)+ (1-d l )U a ( Ba )
We need to show that (1 - d l )   3l , note that if  3l  1 then this holds
if  3l  1, then defender is indifferent between accepting and rejecting


U d Rdl , 1 -  3l  Ba  Bd  d lU d  Rdlw , Ba  Bd   (1 - d l )U d  Rdll , Bd 
Note: if type d l fights she receives benefits Ba  Bd w/prob. d l and benefits Bd
w/prob. (1 - d l ), her expected amount of benefits is d l Ba  Bd .
Jensen’s inequality
By Jensen's inequality U d  Rdl , d l Ba  Bd   d lU d  Rdl , Ba  Bd   (1 - d l )U d  Rdl , Bd 
Given that Rdl  Rdlw  Rdll ,
- this implies that U d  Rdl , d l Ba  Bd   d lU d  Rdlw , Ba  Bd   (1 - d l )U d  Rdll , Bd 
Jensen inequality is weak when we ho ld resources constant, but strict when we allow resources
to vary, fighting being costly.


Because the right-hand side of this inequality is equal to U d Rdl , 1 -  3l  Ba  Bd ,
this implies that d l Ba  1 -  3l  Ba or that 1-d l    3l , which was what was to be proved.
A similar argument establishes that U a  Raw ,  3h Ba   d hU a  Rawl , 0   (1 - d h )U a  Raww , Ba 
A proposes  3h if type d h is the only type left. QED
Result 1
R1) In equilibrium A never provokes a fight with both types
At each negotiating opportunity,
- if type dl remains then A prefers proposing γlt to any γt > γlt
- if type dh remains then A prefers proposing γht to any γt > γht
- 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 h3 , γ l3
- trade-off γ h3 < γ l3 high risk-return payoff γ l3
This result generalizes to n types
Result 2
R2) A is more likely to propose γt = γlt
- if A is optimistic that D is type dl
- if A anticipates low resource losses from fighting
- if dl 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 dl’s probability of victory is, the tougher a demand A can make because
type dl 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 h1 and both types accept. War does not start.
•
A proposes γ l1 , type dl accepts and type dh rejects.
If D rejects, then A attacks. If A wins the first battle, she proposes γ h2 and D accepts.
•
A proposes γ l1 type dl accepts with probability (0 < w < 1), and type dh rejects.
If D rejects, then A attacks. If A wins the first battle, she proposes γ h2 and D accepts,
and does not fight a second battle.
•
A proposes γ l1 , type dl accepts with probability (0 < w1 < 1) and type dh rejects.
If D rejects A attacks. If A wins the first battle, she proposes γ l2, type dl accepts with
(0< w1 ≤ 1) and type dh 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
7)
We may want to abandon equilibrium conditions and perfect foresight
or use equilibrium refinements to allow for two-sided incomplete information
- signaling vs. bargaining
- more dynamics
d X A , XD ;
A knows X A , D does not;
We could also use d  R A , R D , D's type
D knows X D , A does not
 and compute numerical results
Other extensions
8)
Provide a richer description of benefits:
B  B1  B 2  B 3  ...additive
indivisible
or
9)
B=  B1  B 2  B 3 
multiplicative effect
Consider decision making within a nation
a. a state is not a unitary actor
b. a coalition of supporters needs to be kept happy
c. some kind of aggregation of preferences occurs
d. regime type may affect aggregate preferences
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.