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Bargaining and Signaling. Basic Set-Up. Two parties, A and B , bargain over the division of something of value. Division of territory Distribution of economic gains Policy (e.g., taxes) We often normalize this range of possible deals to [0,1]. A settlement is x [0,1]. Basic Set-Up. - PowerPoint PPT Presentation
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Bargaining and Signaling
Basic Set-Up
• Two parties, A and B, bargain over the division of something of value.– Division of territory– Distribution of economic gains– Policy (e.g., taxes)
• We often normalize this range of possible deals to [0,1].
• A settlement is x [0,1].
Basic Set-Up
• A prefers larger values of x; B prefers smaller ones:– UA(x) increasing, UB(x) decreasing– For simplicity, assume risk neutrality for most
examples: UA(x) = x and UB(x) = 1 – x.
Basic Set-Up• Each party has a minimal acceptable
settlement– “reservation value”– the deal that it sees as equivalent to no deal.
• The reservation value is determined by the expected value of the “outside option”:– the expected value of war– the expected value of a revolution or coup
• An actor can always guarantee its reservation value by implementing the outside option
The Reservation Value
• Most generic form: wA, wB
• We sometimes assume that conflict can be seen as a “costly lottery”:– let p denote the probability that A will win– assume that the winner imposes its most preferred
outcome– let cA, cB denote the expected costs
• Then,wA = p – cA
wB = 1 – p – cB
The Reservation Value
• Reservation points are then x such thatUA(x) = p – cA and UB(x) = 1 – p – cB
• With example utility functions,
0 1p – cA p + cB
A will acceptB will accept
Zone of Agreement
• All settlements between the two reservation points constitute the “zone of agreement”: the set of deals that both sides prefer to conflict.
• The zone of agreement is always non-empty if– Conflict is costly in aggregate
In our example: The zone of agreement is non-empty if p + cB > p – cA or cA + cB > 0 .
Note: This means that one actor could have negative costs for conflict, as long as wA, wB < 1.
– The actors are not too risk acceptant
Fearon, “Rationalist Explanations for War”
Motivation: If war is costly, there exist settlements that both sides should prefer to war. Why do states sometimes fail to reach ex post efficient bargains?
Proposed mechanisms:1. Asymmetric information about p, cA, and/or cB , combined with incentives to misrepresent.2. Commitment problems: Deals in the zone of agreement may be non-self enforcing due to
• First-strike advantages• Exogenous shifts in the power distribution• Endogenous shifts in the power distribution
3. The good is lumpy or indivisible.
Asymmetric Information• Assume that each actor is incompletely informed
about the other’s value for conflict– Most generic: wA, wB unknown
– Common assumption: p known, cA, cB unknown
[ , ] with c.d.f. A A Aw w w F
[ , ] with c.d.f. B B Bw w w G
[ , ] with c.d.f. A A Ac c c F
[ , ] with c.d.f. B B Bc c c G
“Take It or Leave It” Bargaining
BOffer x
A
Accept
Reject
x, 1 – x
p – cA, 1 – p – cB
Equilibrium Strategies
(Offer ) Pr( Accepts) (1 ) Pr( Rejects) (1 )Pr( ) (1 ) Pr( ) (1 )[1 ( )](1 ) ( )(1 )
B B
A A B
B
EU x A x A p cc p x x c p x p cF p x x F p x p c
accepts iff .AA x p c x
There exists a “risk-return tradeoff” in B’s decision:• Increasing x decreases the risk of war, F(p – x), but also decreases B’s return on the deal, 1 – x.• More profitable bargains can only be achieved by accepting a greater risk of war.•But it never makes sense to offer more than . Ap c
Equilibrium Strategies
If F(x) has a “monotone hazard rate,” ( ) 01 ( )
d fd F
which ensures that there exists solution to the first-order condition.
The optimal offer, x*, solves( *) 1
1 ( *) *B
f p xF p x p c x
In general, the optimal offer entails a positive probability of war—i.e., .* Ax p c
Equilibrium StrategiesIf A’s costs are distributed uniformly, then
* min ,2
A BA
c cx p p c
The equilibrium probability of war isPr( ) Pr( *)
Pr2
2max 0,2( )
A
A BA
A B A
A A
War c p x
c cc
c c cc c
Two Shortcomings
1. The TILI bargaining framework• does not allow counter-offers• artificially imposes a final move.
2. Most conflicts are preceded by efforts to signal resolve through threats and escalatory efforts.
Powell, “Bargaining in the Shadow of Power”
D
Offer
D
Accept
Attack
SOffer Reject
S
Accept
Attack
…Reject
t=0 t=1
Assumptions
0 1
D’s capital S’s capital
q
Existing border
• Until an agreement or war, D gets a per-period payoff of q and S gets a per-period payoff of 1 – q.
• War is a costly lottery. Let p = Pr(D wins), Let d and s denote per-period loss from having fought a war. Hence, per-period expected values of war are
• wD = p – d• wS = 1 – p – s
0 1
D’s capital S’s capital
q
• If both states are known to be satisfied, then neither will ever attack, and no serious bargaining will take place:
p – d p + s
0 1qp – d p + s
•If p – d > q, then D is dissatisfied. If 1 – p – s > 1 – q, or p + s < q, then S is dissatisfied.•It is easy to see that at most one state can be dissatisfied:
Assumptions
Assumptions
• To generate incomplete information,assume
• If , then D is potentially dissatisfied.• At most one state can be potentially
dissatisfied.
~ [ , ]~ [ , ]
d U d ds U s s
p d q
Key Result
Lemma. The potentially dissatisfied state never rejects an offer in order to make a counter-offer.
Hence, in equilibrium, the equilibrium outcome is the same as in the TILI bargaining game:
– S offers
– D either accepts or attacks
* min ,2
d sx p p d
Intuition• Conjecture that some dissatisfied type(s) of D
counters with an offer, x. Let r denote the most resolute type that does so.
• Possible outcomes– War in some future period
• But war now is better than a period of SQ followed by war.– D accepts some offer from S in future period
• But the most S will ever offer is p−r, which is equivalent to the war payoff. War now is better for type r.
– S accepts the counter-offer • But S can always reject x, leading to the SQ payoff in that
period, and then offer p−r, which it knows will be accepted. S will reject any offer which gives it less than (1−q)+(1−(1-p+r).
• But D of type r could get p−r>q immediately and in all future periods by attacking now. Hence, this type is not willing to make a counter-offer that S would accept.
The Relationship of Power and War
The Relationship of Power and War
q = 0.5q = 0.33
Leventoğlu and Tarar, “War and Incomplete Information”
D D
Accept
Attack
SReject
S
Accept
Attack
…Reject
t=0 t=1
Leventoğlu and Tarar, “War and Incomplete Information”
D D
Accept
Attack
S Reject
S
Accept
Attack
…Reject
t=0 t=1
S D
Attack Attack
Main Result
• If is sufficiently high, then there exists a “no risk” equilibrium in which D rejects a low initial offer and then makes a counter-offer which is accepted.
• This implies that incomplete information leads to war only when– the states are impatient, or– they fail to coordinate on the risk free equilibrium
Thoughts
• As the time between offers shrinks to zero, or →1, a peaceful equilibrium always exists.
• Failure of bargaining is not well explained by “pure” bargaining models.
• Key question: Given that the existence of an efficient deal is common knowledge, why would states ever walk away from the bargaining table?
Signaling
B
Offer
A
Accept
Reject
AMessage
A Simple Signaling Game A
ChallengeStatus Quo
B
A
Acquiesce Resist
StandFirm
SQA
SQB
ACQA
ACQB
BDA
BDB
WARA
WARB
BackDown
Assumptions:1. ACQA>SQA, BDA
2. BDB>ACQB
3. WARA has cdf F4. WARB has cdf G
The Risk-Return Tradeoff
• Even in this simple setting, B faces a risk-return tradeoff:– Assume BD is B’s first-best outcome
– If WARB > ACQB, then B has a dominant strategy to Resist
– If WARB < ACQB, then B faces a choice between • getting its second-best payoff for certain, and• a lottery between its first- and third-best payoffs.
• The odds of the lottery are determined by the posterior belief that A will fight.
The Risk-Return Tradeoff
• Let q denote B’s posterior belief that A will stand firm given that A has challenged.
• Then B will Acquiesce if
B B
B B
BD ACQqBD WAR
Informative Signaling
• Let p = 1 – F(BDA) denote prior probability that A will stand firm
• A’s challenge is informative if q > p.• For this to happen, the probability of a
challenge must be less than one.– Separation of types requires that BDA < SQA for
some types. – Otherwise, ACQA > SQA ensures that a challenge
weakly dominates the status quo for all types.
Types of Signaling
1. “Slippery slope”: challenge creates an exogenous risk of war
2. “Tying hands”: challenge creates an “audience cost” for backing down
3. “Sunk costs” or “burning money”: A must pay an up-front cost to challenge
Slippery Slope A
ChallengeStatus Quo
B
A
Acquiesce Resist
StandFirm
SQA
SQB
ACQA
ACQB
BDA
BDB
WARA
WARB
BackDown
WARA
WARB
N1 –
Tying Hands A
ChallengeStatus Quo
B
A
Acquiesce Resist
StandFirm
SQA
SQB
ACQA
ACQB
BDA = SQA – a BDB
WARA
WARB
BackDown
Sunk Costs A
ChallengeStatus Quo
B
A
Acquiesce Resist
StandFirm
SQA
SQB
ACQA – m ACQB
BDA = SQA – m BDB
WARA – m WARB
BackDown
Equilibrium
• In general, for fixed , m, or a, the equilibrium strategies are defined by a set of cutpoints in the continuum of types:
WARA
WARB
ChallengeStand Firm
ChallengeBack Down
Status QuoBack Down
ResistAcquiesce
Schultz, “Do Democratic Institutions Constrain or Inform?”
• Questions: Does democracy influence crisis outcomes, and if so how?
• Competing Theories– Institutional constraints: democracy increases the political
costs of war– Informational: democratic institutions increase
transparency and/or increase audience costs– Realism (the null hypothesis): democracy doesn’t matter
• Problem: While it is relatively easy to determine whether democracy matters, it is much harder to distinguish competing arguments for why it matters.
The Theoretical Model A
ChallengeStatus Quo
B
A
Acquiesce Resist
StandFirm
(0,1)
(1,0)
(– a, 1) (wA, wB)
BackDown
Putting Democracy in the Model
• Institutional constraints– Democracy lower expected value for war on
average– Assume wA ~ [– CA – dZA, – dZA], where dA > 0 and
ZA = 1 if state A is a democracy
• Information– Democracy higher audience costs (a)– Transparency democracy generates complete
information about wA
Comparing Complete and Asymmetric Information
• Probability of a challenge– CI: A only challenges when wA > – a
– AI: A challenges when wA > – b , with b > a
• Probability of resistance– CI: B never resists conditional on a challenge– AI: B resists with nonzero probability for some parameters
• Probability of war– CI: Zero– AI: Nonzero for some parameters
Magnitude of constraint, dA
Prob
abili
ty in
Equ
ilibr
ium
0
1 B Resists|ChallengeA Challenges
War
Outcomes as a Function of dA
Magnitude of Audience Costs, a
Prob
abili
ty in
Equ
ilibr
ium
0
1
B Resists|Challenge
A Challenges
War
Outcomes as a Function of a
Predictions of the Two Views of Democracy
Predicted effect onprobability of...
If democracy in A means...Constraints InformationDecrease in
wA
Complete Information
Increase in a
A Challenges - - +
B Resists| Challenge
+ - -
War +/- - +/-
The Data• Dependent variable: Did the target resist?
– Data set: Militarized Interstate Disputes (MIDs)• 1654 disputes over period 1816-1980• arranged in dyads of initiator-target
– RECIP = 1 if target reciprocated the initiator’s action, and RECIP = 0 otherwise.
• Main independent variable: Regime type of the initiator– Data set: Polity III– DEMINIT = 1 if initiator is democratic (score of 7 or
higher on 21-point composite democracy scale), and DEMINIT = 0 otherwise.
Bivariate Correlation
Non-DemocraticInitiator
DemocraticInitiator
Not Reciprocated 617 (49.2) 219 (56.9)
Reciprocated 637 (50.8) 166 (43.1)
Pearson 2 = 6.95 Pr = 0.008
Initiator-TargetPower Status
Non-Democratic Initiator
Democratic Initiator
Major Power-Major Power
0.34 0.26
Major Power-Minor Power
0.34 0.25
Minor Power-Major Power
0.42 0.33
Minor Power-Minor Power
0.43 0.34
Predicted Probabilities of Reciprocation
Summary
• Use of model to – generate testable hypotheses and – identify a critical test between theories.
• Convinced?
Summary• Use of model to
– generate testable hypotheses and – identify a critical test between theories.
• Potential problems– Unmeasured factors
• Democracies select weak targets• Democracies make smaller demands
– Observed correlation could arise from more than one causal pathway (identification problem)
– Mismatch between data and model