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 that

UA(x) = p – cA and UB(x) = 1 – p – cB

• With example utility functions,

0 1p – cA p + cB

A will accept

B 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

B

Offer 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 c

c p x x c p x p c

F 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 f

d F

which ensures that there exists solution to the first-order condition.

The optimal offer, x*, solves

( *) 1

1 ( *) *B

f p x

F p x p c x

In general, the optimal offer entails a positive probability of war—i.e., .* Ax p c

Equilibrium Strategies

If A’s costs are distributed uniformly, then

* min ,2

A BA

c cx p p c

The equilibrium probability of war is

Pr( ) Pr( *)

Pr2

2max 0,

2( )

A

A BA

A B A

A A

War c p x

c cc

c c c

c 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 d

s 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

SReject

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 ACQq

BD 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

Pro

bab

ilit

y in

Eq

uil

ibri

um

0

1 B Resists|ChallengeA Challenges

War

Outcomes as a Function of dA

Magnitude of Audience Costs, a

Pro

bab

ilit

y in

Eq

uil

ibri

um

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 Information

Decrease 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