Bargaining, Efficiency, and
Rationalist Explanations of
War
Nature of Warfare
• Risky – might win, might lose
• Benefits: There is a subjective value
associated with winning and also with
losing (How much is it worth to you)
• Costs – military expenditures, people die,
resources/assets are destroyed
(Subjective: How much do the costs
cost you)
War Involves Risks
• p = probability of winning (p is between 0
and 1); then (1-p) = probability of losing.
• War can have multiple outcomes (e.g.,
Win, Lose, Draw) or continuous outcomes
(integrate over all possible outcomes).
• Some outcome must happen. Therefore
probabilities over outcomes MUST sum to
1: p+(1-p)=1; p1+p2+(1-(p1+p2))=1, etc.
Costs and Benefits
• Subjective values for each possible outcome is
measured as a UTILITY (U).
• Relative size of utilities matter, not the
numerical value per se.
• We cannot compare utilities across people. We
compare each person’s utility across outcomes.
• E.g., Let U(Win)=1, U(Lose)=0,
• 0<U(Status Quo)<1. U(Win)>U(SQ)>U(Lose)
• Costs (e.g., k>0) are expressed as utilities that
are subtracted. Benefits are added.
What To Expect if A & B Fight
Leader/country A’s EXPECTED UTILITY from
war = probability A wins (p) times (A’s gain
(1) minus cost of fighting (-k)) if A wins +
probability A loses (1-p) times (A’s utility for
defeat (0) minus A’s cost of fighting (-k)) =
p(1-k)+(1-p)(0-k) = p – pk – k + pk = p – k
B’s EXPECTED UTILITY from war =
(1-p)(1-k)+p(0-k) = 1 – k – p + pk - pk =
1–p-k
Predicting Action: Expected Utility
If for country/leader A: p-k > U(SQ) and for
B: (1-p-k) > U(SQ)
Then A & B each prefer to Fight rather than
Accept Status Quo.
If for A p-k<U(SQ) and for B (1-p-k)<U(SQ), then
they do not fight; they prefer the status quo.
If, for A, p-k > U(SQ) and for B (1-p-k) < U(SQ)
then we expect A to threaten B and B to give
in to A or they might find a compromise
solution.
Negotiation/Compromise
Can we find a Compromise that A and B Accept?
Let x be proposed as a settlement, 0 ≤ x ≤ 1,
where x is the award to A of a portion of land in
dispute. Then A wants more land (larger x), and
B wants A to get less land so B gets more (1-x).
Let A’s U(x) from proposed settlement x = x
Let B’s U(x) from proposed settlement x = 1-x
How to pick x?  An example of
ENDOGENOUS Choice: Is there a
compromise (x) that A and B prefer to War?
Finding Agreement:
Picking a Compromise
• Although wars generally end in compromise,
their start is a failure to find an agreement
• Before using force, antagonists seek ways to
reach an agreement. They often succeed:
War, after all, is a rare event.
• But the threat of war or some other form
of coercion (e.g., sanctions) can alter
calculations about how to settle
differences.
• Hence, before investigating war, we need to
understand a bit about its avoidance through
bargaining and compromise.
The Bargaining Problem
• Two sides seek to divide what each wants
(territory; policy differences; resources).
• Each wants the most it can get.
• The bargaining environment creates the
opportunity for “gains from trade,” such
as avoiding the destruction of resources.
• A negotiated agreement may create
surplus value compared to conflict.  War
is ex post inefficient!
Europe After EU Expansion
Through 2013
Bargaining Without Coercive Threat
y
Player
B
Utility
U(y)=y
A = NATO leadership;
B = Putin
Red Line = All Pareto Efficient Allocations. Can’t
allocate more but it is possible to allocate less,
making one or more actors worse off (e.g.,Think of
space between Q and red line as expected war
costs)
Player B rejects A’s
proposal and proposes to
capture all of the surplus
Nash Equilibrium:
Split Surplus 50/50
Player A proposes to
capture all of the
surplus (A gains & B
Q
does not lose)
1’s U at Status Quo
(Q)
Utility
U(x)=x
Player
A
x
Russia’s 2022 Proposed
Division of Europe
Reconsidering with “Outside” Option:
The Threat of War
y
Red Line = All Pareto
Efficient Allocations
Player
B
Utility
U(y)=y
B’s Threat is Credible
O2
O1
Q
Settle: Split Surplus,
e.g., 50/50
B Threatens but the threat
is not credible. 50/50
settlement is better for B
than the threat
1’s U at Status Quo Utility
(Q)
U(x)=x
Player
A
x
Russian troops make outside demand (1997 status quo) credible. NATO moves
troops to Ukraine’s border to offset Russia’s threat. Both sides operating under
uncertainty about each other’s true intentions.
The Bargaining Problem in the Shadow
of the Threat of Coercion
• The efficient settlement depends on:
– Relative leverage (power, probability of winning)
to shape an agreement. (e.g., Adv Russia)
– Relative motivation (utility) to pursue the threat.
(e.g., Adv. Ukraine)
• Settlements generally exist to avoid conflict
– War is ex post inefficient because:
– Whatever war’s outcome, that outcome might
have been agreed to beforehand, avoiding the
costs of war. Hence there is a bargaining
surplus/gains from trade.
Dividing the Pie, Avoiding War,
Finding X
A
Status Quo
Threaten
B
Q
Compromise
War
1-Q
X
p(1-kA)+(1-p)(-kA)
1-X
p(-kB)+(1-p)(1-kB)
Recall that: p=probability A Defeats B
ki is the costs of war with i={A, B}. U(X) = X for A and 1-X for B.
For B War>Compromise if 1-p-kB> 1-X;
For B: Compromise > War if X<p+kB, (remember, UB(X)=1-X)
For A, Threaten > Status Quo if p - kA >Q if B will fight; if B will
compromise then for A Threaten > Status Quo if X>Q. (i.e., A
expects to benefit from threatening B)
War is Ex Post Inefficient
0,1
Red Line = All Pareto
Efficient Allocations
Q
B
1-p-kB
EU(War,p)
A
p-kA
1,0
Note: As p decreases (1-p increases), the “balance of power” shifts in B’s favor,
altering the feasible compromises. Puzzle: Why does war ever occur if it is
Pareto-Inefficient. (Assumes that for B Q>1-X but for A Q<X)
Rationalist Explanations of War
• Uncertainty re power, motivation, costs, patience
(future benefits) can make agreement difficult.
– Leaders may bluff/lie about what they will accept.
– But if proposals are exchanged in short order then
uncertainty (asymmetric information) does not
materially alter the expected division of the pie.
• Commitment problems
– Promises can be cheap talk.
– Self-interest may dictate violating an agreement even
though cooperation leaves all better off than fighting
(e.g., Prisoners’ Dilemma).
• Indivisibility: But generally side-payments can be used.
Competing Factions, Divisible Outcome
DPRK
Offer Compromise
Accept
Coerce (tests
Nuke)
US
Government
War
Give In
x
p(1-k)+(1-p)(-k)
1-x
p(-k)+(1-p)(1-k)
War
1
p(1-k)+(1-p)(-k)
0
p(-k)+(1-p)(1-k)
DPRK seeks Nukes (U(Nukes)=1); US Government seeks (No
Nukes, U(Nukes)=0); Policy is divisible (e.g., degree of weapons
inspection, number of weapons, number of delivery missiles, range)
US Government: War>Give In if 1-p-k>0; Accept>War if x<p+k
Calculation: (1-x > 1-p-k ) = -x > -p–k = x < p+k
DPRK Compromises > War if x>p-k
United States Government Accepts
Compromise X if and only if p+k ≥ X
US rejects
these deals
US accepts these deals
0
p+k
p-k
Policy Space, X
1
North Korea/Kim Jong-Un Offers Compromise X
if and only if X ≥ p-k
DPRK rejects
0
No
DPRK
Nukes
DPRK accepts
p+k
p-k
Policy Space, X
1
Nukes
Possible USA-DPRK Deals
Compromises Agreeable to US
DPRK Rejects
Compromise in
this Area, Prefers
to Risk US Attack
0
US and DPRK Agree to
Compromise in Gray
Area
US Rejects
Compromise in
this Area, Prefers
to Risk Attacking
DPRK
Compromises Agreeable to DPRK
p+k
p-k
Policy Space, X
1
Can all disputes be bargained away? No
Problems with finding bargaining solutions:
1) Indivisible policy space
No
Missiles
0
Possible divisions of the policy
space do not fall in the
agreement region
Missiles
p-k
p+k
Policy space, x
1
Cuban missile crisis: Either missiles or no missiles  But a deal was
found: Traded Cuban missiles for US missiles in Turkey + (Nonbinding) promise not to overthrow Castro. True indivisibility is rare
Is the dispute between Israel and Hamas over
something indivisible?
How about, Who gets control over a country, the government or
opponents who want to destroy the government?
Revolution: Possibly an Indivisible Outcome
Revolutionary
No Revolt
Revolt
0
1
Government
Fight
p(1-k)+(1-p)(-k)
p(-k)+(1-p)(1-k)
Give In
1
0
Note: p=probability Revolutionary wins/Government loses
k is the costs of fighting; No compromise that divides
government control is possible between these adversaries
Government: Fight>Give In if 1 – p – k > 0
Revolutionary: Revolt>No Revolt if p > k
Impediments to Deals:
Incomplete Information (Uncertainty)
• Disagree about probability of victory, p,
costs of fighting, k, or misjudge another’s
motivation, u (e.g., Risk Acceptance)
• What influences strength? Weapons,
morale, capability, distance.
• Communicate my information and beliefs
• In 1914 Germans should have asked Britain and
France if they were really strong enough and
committed enough to help the Russians.
• But they lie!
• Revelation of information requires a costly signal.
Examples of Incomplete information
Miscalculate incentives to fight
• Germans underestimated UK and Russia in WW1
• Japanese doubt US willingness to fight in 1941
• 1904 Russia vs. Japan
– Russia thought it was almost certain to win.
– Japan estimates its chance in war is 50/50. Japan fights when
Russia refuses to compromise.
• Spring 1990.
– Iraq demands reparations for oil they claimed Kuwait pumped
from under Iraq’s ground.
– Negotiation’s failure: Iraq doesn’t believe US will help Kuwait
• Falklands
– In 1979 British government had a bill to return Falklands to
Argentina. Minister resigns as bill rejected in House, but
Argentina thought UK was unprepared to fight
• Ukraine, 2022
– Russia (and int’l community) think Ukraine easily defeated
– Ukraine’s President unexpectedly motivates resistance
– US & Europe inspired to aid Ukraine
Risk Taking & War:
Russian Invasion of Ukraine
• Why wasn’t the Russian dispute with
Ukraine resolved without war?
• Could Vladimir Putin have credibly
committed to a compromise agreement?
• Was there significant uncertainty leading
up to the invasion?
• Might Putin’s apparent risk-acceptance
have mattered?
Modify an Assumption
• We assumed U(x)=x for A and U(x)=1-x for B.
• What if U(x)=x2 or U(x)=x1/2 etc.
• How does U(x) change as X changes? If it decreases at
an increasing rate as the offer X improves (dU(X)/d(X)>0
and d’’U(X)/d(X)>0), then the decision maker favors
gambles over a sure-thing of equivalent value; s/he is
risk-acceptant.
• If U(x) increases at a decreasing rate as X improves
((dU(X)/d(X)>0 and d’’U(X)/d(X)<0), then the decision
maker favors a sure-thing (equivalent value) over
gambling; s/he is risk-averse.
• What if Putin or Zelensky or both were risk-acceptant?
Some Thoughts: Russia-Ukraine Conflict
Risk Neutral Deal for A: p(1-k)+(1-p)(0-k)=X =p-k
(p+k acceptable to B). REMEMBER U(X) beed not equal X
Utility for Outcome
1
Risk-averse
Risk-acceptant
(e.g., U(x)=x2)
p+k
p-k
Bargaining
Space
0
0
p+k
p-k
Value for Outcome
Acceptable Utilities yield policy
concessions outside the
bargaining space
1
28
Impediments to Deals:
Commitment Problems
The Prisoner’s Dilemma
Attack
Negotiate
Attack
(2,2)
(4,1)
Negotiate
(1,4)
(3,3)
Commitment Problems
Future Demands
• Suppose deal x generates more power
(increases p)
• Today, I ask for a little, it makes me
stronger, so tomorrow I can demand more,
which makes me stronger still, so I can
demand yet more …………….
• Salami Tactics (Schelling)
• German Appeasement
• Peace in NI: Once IRA disarms what stops
the British from imposing new settlement?
Some Qustions for Recitation (Pick
any conflict)
•Once Israel gave up Gaza it became much harder
to defend Israel. Does Hamas ask for more?
•Once Hamas disarms what is to stop Israel from
reneging on future land concessions?
•How can a deal be struck that insures
compliance?
•What makes an agreement likely to last? What
makes it likely to fall apart?