TURF WARS

Helios H. HerreraMichael M. Ting

Columbia UniversityJune 2012

Motivation: Examples

New York City emergency managementI FDNY, NYPD coordination failures

U.S. military branchesI Pearl Harbor, VietnamI Goldwater-Nichols Act and Unified Combat Commands

U.S. Intelligence reformI 17 intelligence agenciesI 9/11 Commission Report: more consolidation

DEA investigationsI “Buy money” as incentivesI Regional investigations

Motivation: Examples, cont’d

U.S. Medicare programI Small practicesI Fee-for-service reimbursementI Lack of coordination among service providers

Securities regulationI CFTC vs. SEC in derivatives regulation

Casual observation: Public sector examples seem morecommon

I Will come back to this later

Motivation: What is a Turf War?

Jurisdiction-minded behaviorI Contrast with policy, budget motivationsI Inefficiencies

I Failure to share key informationI Denying the application of needed resources

ModelI When do they occur?I How might they be prevented?I Won’t consider “unification”

I Difficult, politically and technologicallyI Intra-agency turf battles are sometimes just as bad

Literature

Not much

Literature

However, some related ideas

Contracting and competitionI Contracting and cooperation (e.g., Holmström and Milgrom

1990, Itoh 1991, Marx and Squintani 2009)I Rank-order tournaments (e.g., Lazear 1989, Chen 2003)

Information sharingI Congressional committees (Austen-Smith and Riker 1987,

King 1994)I Oligopolies (e.g., Gal-Or 1986, Okuno-Fujiwara et al. 1990)

Other settingsI “Conversations” among two competitors (Stein 2008)I Coordination, centralization and decentralization (Alonso,

Dessein and Matouschek 2008)

The Model: Three Pieces

Necessary and sufficient features for a turf war.

1. Joint productionCollective output is higher when agents collaborate.

2. CompetitionAgents receive some benefit from “winning,” or performingbetter than others.

3. Property rightsAt least one agent can determine whether others receivejurisdiction over the task in question.

The Model: Players

AgentsI A1 and A2I At least one agent will have jurisdiction over a task.I Each Ai generates an output xi ∈ [0,Θ] (Θ > 0) when she

has jurisdiction.I Prize β ≥ 0 for achieving the highest output.

I Career concerns, or exogenous reward.I Ai ’s utility from task outcome:

ui =

mi(x1 + x2) + β1xi>x−i if both agents have jurisdictionmixi + β if only Ai has jurisdictionmix−i if Ai does not have jurisdiction.

where mi > 0 is her “policy” motivation.

The Model: Outcomes

Effort and outputsI Conditional upon Ai having jurisdiction, output xi is

determined by:

x1 = θ1 ∼ U[0,Θ]

x2 ={θ2 ∼ U[0,Θ] if e2 = 10 if e2 = 0.

I θi ∈ [0,Θ] is Ai ’s productivity (private information).I A1’s output is her productivity.

I Reflects “sunk” or statutorily mandated investments.I A2’s output depends on binary effort (e2).

I Not working is like refusing jurisdiction.I Exerting effort costs k2 ≥ 0.

The Model: Sequence and Solution Concept

By default, A1 (the originator) has jurisdiction.

1. Nature draws agents’ productivity types.2. A1 decides whether to extend jurisdiction to A2

(equivalently, whether to exclude A2 from participation).3. If given jurisdiction, A2 chooses e2.

Perfect Bayesian equilibrium.

I An out of equilibrium information set can only beencountered if A1 shares when no type should haveshared.

Complete Information ExampleSuppose θ1 is known.If A1 shares, when and why does A2 work?

k2m2

k2 � �m2

⇥

0

✓2

✓1

⇥

Policy'Mo)va)on'

Victory'Mo)va)on'

Complete Information Example

When does A1 share?

k2m2

k2 � �m2

⇥

0

✓2

✓1

⇥Sharing'Autarky'

Equilibrium

“Interval” equilibria for θ1, θ2.I All equilibria must be this way.

Suppose A1 shares for θ1 ∈ [θ, θ]. Then A2 works if:

m2(E [θ1 | share] + θ2)− k2 + βω(θ, θ, θ2,e) > m2E [θ1 | share],

where ω(θ, θ, θ2,e) is the probability that A2 wins. At an interiorsolution (θ2 ∈ [θ, θ]):

ω(θ, θ, θ2,1) =θ2 − θθ − θ

.

Equilibrium, cont’dSo for any [θ, θ], there is a unique type θ̃2 such that higherproductivity types work and lower productivity types shirk:

θ̃2 =

{ k2m2

if θ ≥ k2m2min

{Θ, k2(θ−θ)+βθ

m2(θ−θ)+β

}if θ < k2m2

Only sufficiently productive partners will work.Positive Selection of A2 Types

q̃2 =

8<

:

c2+(bc2)q̃1/Qm2(1q̃1/Q)+b/Q

q̃1 <c2m2

c2m2

q̃1 >c2m2

Herrera & Ting (Columbia University) Turf Wars PUC-Rio, March 2012 27 / 48

Equilibrium, cont’d

Given θ̃2, the probability that A2 will work given [θ, θ] is:

φ(θ, θ) = 1− θ̃2Θ.

And given θ1, the probability that A2 wins conditional uponworking is:

ξ(θ1) =

{1 if θ1 ≤ θ̃2Θ−θ1Θ−θ̃2

if θ1 ≥ θ̃2.

Now A1 shares if:

m1

(θ1 + φ(θ, θ)

θ̃2 + Θ

2

)+ (1− φ(θ, θ)ξ(θ1))β ≥ m1θ1 + β.

Equilibrium, cont’d

I For A1, increasing returnsI Utility from sharing minus utility from not sharing

continuous, increasing in θ1I Higher types always more inclined to share, so θ = Θ

I Problem is then to find the threshold value, θ∗1, at which A1is indifferent between sharing and not sharing.

I Often we hit corners: full or no sharingI Otherwise:

θ∗1 =

(m1−2m2)βΘ+m1(k2+m2Θ)Θ−2β2

m1(k2−β)+m2(m1Θ−2β) ifk2m2

> Θ, ork2m2≤ Θ, m12

(k2m2

+ Θ)> β

Θ− m1(Θ2−(k2/m2)2)2β otherwise.

Intuition

Who shares, who works?

⇥0

✓1

Share&Don’t&Share&

⇥0 Work&Don’t&Work&

✓2

Agent&1&produc5vity:&

Agent&2&produc5vity:&

✓̃2

✓⇤1

Results on Sharing

No sharingI β high and m2 lowI m1, Θ low

Full sharingI β lowI m2 low and β moderate, due to competition.I k2, m1 high

Partial sharingI Unique cutoff θ∗1 decreasing in m1

I Socially bad: implies weakest types won’t share.

But sharing isn’t enough!

Example: Sharing

I Let k2 = 1 and Θ = 2; on left, m1 = 1, on right, m1 = 2.I Figure depicts ex ante probability of sharing as a function

of β (horizontal axis) and m2 (vertical axis).I High β: less sharing.I High m2: usually but not necessarily more sharing.

Example: Working and Collaborating

I Let k2 = 1 and Θ = 2, m1 = 2 (same as before).I Left: probability A2 works conditional upon sharing, as a

function of β (horizontal axis) and m2 (vertical axis).I Right: probability of collaboration: A1 shares and A2

works.I High m2 ⇒ more work, collaboration.

Results, cont’d

How do we get collaboration?I Unless k2 = 0, partner won’t always work.

I Even under full sharing, “never” get full effort by bothplayers.

I Often, a trade-off between originator sharing and partnerworking.

I High m2, moderate β appear to work.I Role of β is non-monotonic.

Other questionsI Generalized/heterogeneous distributions of θiI Limited contracting

Remedies

The equilibrium suggests a few remedies for turf warsI i.e., conditions under which collaboration will increaseI Implicit assumption: a principal can’t “force” sharing

1. Reward (β)2. Policy motivations (mi )

Can derive conditions under which “switching” A1, A2 willprevent no sharing.In government agencies, could correspond to appointeesvs. career bureaucrats.

3. Ability, costs (Θ, k2)Increasing Θ (increasing salary?) generally helps sharing.Reducing k2 increases work but may decrease sharing.

Extension 1: Originator Effort and Referrals

Suppose that originator has to exert effort at the start of thegame

I For both agents, binary effort ei ∈ {0,1}, cost ki > 0,production is 0 if ei = 0

I Both agents know type when choosing whether to workI A1 can go alone, share, and (now) refer the case to A2

I Referral means none of A1’s productivity is realizedI Clearly, will occur iff A1 does not work

I Working/referral threshold θ̂∗1I Expected payoff to referrals is constantI By monotonicity of payoffs in basic game, refer iff θ1 < θ̂∗1

Complete Information Example

A2 has greater incentive to work under a referral

k2m2

k2 � �m2

⇥

0

✓2

✓1

⇥

Policy'Mo)va)on'

Victory'Mo)va)on'

Complete Information Example

Lowest types of A1 refer

k2m2

k2 � �m2

⇥

0

✓2

✓1

⇥Sharing'Autarky'Referral'

Only'A2

'works'

Originator Effort and Referrals, cont’d

What happens?

I If θ∗1 > θ̂∗1

I Original analysis carries through: no change in sharing,collaboration

I Under referral, if A2 works and θ2 > θ1, better performanceI But if A2 doesn’t work or θ2 < θ1, worse performance

I If θ∗1 < θ̂∗1

I Some types who would have shared now refer insteadI Sharing types “truncated” to [θ̂∗1 ,Θ]I Reducing the set of sharing originator types also reduces

the set of working A2 types, so bad for collaborationI Again, some gains under referrals possible if A2 works andθ2 > θ1

Extension 2: Reward Threshold

Suppose β is only awarded if output is at least T

Three effects1. Can’t improve A2’s incentive to work2. Can improve A1’s incentive to share . . .

I For θ1 just below T3. . . . or not

I For θ1 far below T

Effect on sharing depends on where T is relative to the originalsharing threshold

I Can the principal write a contract?

Complete Information Example

Reduces A2’s incentive to work

k2m2

k2 � �m2

⇥

0

✓2

✓1

⇥T

Lost'Effort'Under'Sharing'

Extension 3: Synergies (Decision-Theoretic)

SimplificationsI No effort, just sharingI Productivity parameters are common knowledge.

Probability of victory proportional to productivity θi .

GeneralizationsI n ≥ 2 agents, productivity θi ∈ (0,1),

∑θi = 1.

I CES production function for policy:

Eui = mi

(∑i

θρi

)1/ρ+ βθi .

I So, production “synergies” if and only if ρ < 1I Game theoretic model has ρ = 1

Synergies, cont’d

Relative payoff of sharing increases when:I Competition is reduced (β decreases)I Originator is more policy-minded (m1 increases)I Synergies improve (ρ decreases)

Effect of originator’s relative productivity (θ1) on sharing payoffdepends on ρ:

I If ρ < 1 (synergies), then increasing in θ1I Like the game theoretic modelI Bad, since society or a principal would want low productivity

types to shareI If ρ > 1 (no synergies), then decreasing in θ1I If ρ = 1, then no effect

Summary

A common problem with no literature

FrameworkI Three elementsI Incomplete information and effort

ResultsI Tension between inducing effort and winningI Strong originators more likely to share, strong partners

more likely to workI Relative rewards need to be intermediateI Pronounced inefficienciesI Remedies