Resource Allocation in the Brain Ricardo Alonso USC Marshall Juan D. Carrillo USC and CEPR T heoretical RE search in N euroeconomic D ecision-making ()

Resource Allocation in the BrainResource Allocation in the Brain

Ricardo AlonsoRicardo AlonsoUSC MarshallUSC Marshall

Juan D. CarrilloJuan D. CarrilloUSC and CEPRUSC and CEPR

Theoretical REsearch in Neuroeconomic Decision-making(www.neuroeconomictheory.org)

Isabelle BrocasIsabelle BrocasUSC and CEPRUSC and CEPR

April 2011April 2011

Neuroeconomic TheoryNeuroeconomic Theory

Use evidence from neuroscience to revisit economic theories of decision-making

• Examples of neuroscience evidence include: existence of multiple brain systems, interactions between systems, physiological constraints, etc.

• Revisiting theories of decision-making includes:

- Build models of bounded rationality based not on inspiration but on the physiological constraints of our brain derive behavioral biases from brain limitations

- Provide “micro-microfoundations” for elements of preferences usually taken as exogenous (discounting, risk-aversion, etc.) and processes taken for granted (learning, information processing, etc.)

The brain is, so it should be modeled as, a multi-system organization

This paper (1)This paper (1)

Build a model of constrained optimal behavior in multi task decision-making

based on evidence from neuroscience:

i. Different brain systems are responsible for different tasks. Neurons in a system respond exclusively to features of that particular task.

ii. The brain allocates resources (oxygen, glucose) to systems. Resources are transformed into energy that make neurons fire (fMRI measure blood oxygenation, PET measure changes in blood flows, etc.).

iii. More complex tasks necessitate more resources. Performance suffers if resources needs are not filled.

iv. Resources are scarce: “biological mechanisms place an upper bound on the amount of cortical tissue that can be activated at any given time”.

This paper (2)This paper (2)

v. Central Executive System (CES) coordinates the allocation of resources:- Active when two tasks are performed simultaneously.- Not active if only one task, if two sequential tasks, or if two

tasks but subject instructed to focus on only one.

vi. Asymmetric information in the brain: neuronal connectivity is very limited information carried by neurotransmitters reaches some systems but not others.

The modelThe model• Three systems (0, 1, 2) perform three types of tasks:

System 0 (S0) controls motor skill functions. Needs θ0 are known

Systems 1 and 2 (S1 and S2) control higher order cognitive functions (mental rotation, auditory comprehension, face recognition)Needs θ1 and θ2 depend on task complexity and are privately known

Distributions F1(θ1) and F2(θ2) satisfy Increasing Hazard Rate (IHR)

• Another system, Central Executive System (CES), allocates resources {x0, x1, x2} to S0, S1, S2

• Performance of system l is with l {0,1,2}.

Systems are tuned to respond only to their task

• CES maximizes (weighted) sum of performances of systems: U0+U1+U2

• Resources are scarce and bounded at k: x0 + x1 + x2 k

• Each system requires a minimum of resources to operate: xl 0

21,l l l l l

l

U x x

The modelThe modelCES

central executive system

210 UUU

200

0000 )(

1),(

xxU

S0

(lifting)

Motor function

0 “public”

222

2222 )(

1),(

xxU

S2

(spelling)

211

1111 )(

1),(

xxU

S1

(rotation)

Cognitive functions 1 and 2

1 and 2 “private”

x0 x1 x2

kxxx 210

Benchmark case: full Benchmark case: full informationinformation

0 1 2

0 0 1 2 0 1 1 1 2 1 2 2 1 2 2, ,

0 1 2

1 1 1max , , , , , ,x x x

U x U x U x

0 1 2 1 1 2 2 1 2s.t. , , , (R)x x x k

0 1 2 1 1 2 2 1 2, 0, , 0, , 0 (F)x x x

Benchmark case: full Benchmark case: full informationinformation

Solution under full information (assuming (R) binds and (F) does not):

Distribute k according to needs (θ0, θ1, θ2) weighed by importance (0, 1, 2)

Utility of each system depends on total needs (sum of θl) but not on how

needs are distributed among the systems (θ1 v. θ2)

2

2; F F ll l lU x k

kx ll

Fl

0 1 2

0 0 1 2 0 1 1 1 2 1 2 2 1 2 2, ,

0 1 2

1 1 1max , , , , , ,x x x

U x U x U x

0 1 2 1 1 2 2 1 2s.t. , , , (R)x x x k

0 1 2 1 1 2 2 1 2, 0, , 0, , 0 (F)x x x

RoadmapRoadmap

1. Normative approach: optimal allocation given private information if CES could use any conceivable communication mechanism - General properties - Comparative statics

2. Positive approach: can this allocation be implemented using a physiological plausible mechanism?

3. Applications- Task inertia- Task separation




The optimization problemThe optimization problem

Optimal allocation of resources when needs of systems 1 and 2 are unknown and CES can use any mechanism. Using the revelation principle:

0 1 2

0 0 1 2 0 1 1 1 2 1 2 2 1 2 2 1 1 2 2, ,

0 1 2

1 1 1max , , , , , , ( ) ( )x x x

U x U x U x dF dF

(IC),~

,},2,1{,,,~

,,s.t. jiiijiiiijiii jixUxU

0 1 2 1 1 2 2 1 2 1 2, , , , (R)x x x k

0 1 2 1 1 2 2 1 2 1 2, 0, , 0, , 0 , (F)x x x

The solutionThe solution

• Optimal mechanism M:

- resource capx1(θ2) for S1

- resource capx2(θ1) for S2

• Equilibrium allocation under M:

- x1*(θ1,θ2) = min { θ1,x1(θ2) }

- x2*(θ1,θ2) = min { θ2,x2(θ1) }

- x0*(θ1,θ2) = k - x1(θ1,θ2) - x2(θ1,θ2)

• What are the optimal capsx1(θ2) andx2(θ1) ?

The solutionThe solution

Sketch of proof

1. Derive optimal allocation with only 2 systems (1 with private info.)

2. Use it to derive “priority mechanism”:

P1 : optimal mechanism under the requirement that S1 always obtains the resources it requests

P2 : optimal mechanism under the requirement that S2 always obtains the resources it requests.

3. Compare both priority mechanisms. Optimum is a hybrid of both:it behaves like P1 for certain (θ1,θ2) and like P2 for some other (θ1,θ2).

Step 1: two systemsStep 1: two systems• Consider first a resource allocation problem between systems

S0 and S2 when resources are k’ :

Solution is to put a cap y2 (k’) on the resources awarded to S2

Cap solves

)'('1

)'()'(|1

200

22222

kykkykyE

• Under IHR, the optimal cap: is continuous in resources k’ satisfies resource

monotonicity ]1,0('/'2 kky

k’k

y2 (k’)

y

Step 2a: PStep 2a: P11-Mechanism-Mechanism

θ1k

y2 (k - θ1) [P1]

θ2

• Consider now a “priority” P1-mechanism that awards S1 all its needs θ1.

• Then, there are k’ = k - θ1 resources left to split between S0 and S2

• From the previous step, we have:

θ'1

θ'2

θ’’2

k’k

y2 (k’)

y

Step 2b: PStep 2b: P22-Mechanism-Mechanism

• We can perform the same analysis with a “priority” P2-mechanism that awards S2 all its needs θ2 and divides the rest between S0 and S1

k’k

y1(k’)

y

θ1

k

θ2

y1 (k - 2) [P2]

Step 3: optimal resource Step 3: optimal resource allocationallocation

Putting everything together

• Si must be at least as well as under Pj: xi(j) ≥ yi (k - j)

• Si cannot be better than under Pi :xi(j) i

If Sj is constrained, then Si can be as in Pi (and viceversa)

θ1k

θ2y1 (k - 2) [P2]

y2 (k - θ1) [P1]

Optimal Resource AllocationOptimal Resource Allocation

θ1

(θ1+θ2=k)

θ2

x2(θ1) = y2(k - θ1) if θ1 < k1

k2 if θ1 k1

x1(θ2) = y1(k - θ2) if θ2 < k2

k1 if θ2 k2

k1

k2

Optimal capsx1(θ2) andx2(θ1) are first strictly decreasing and then

constant in the needs of the other system


θ1

(θ1+θ2=k)

θ2

k1

k2

Optimal capsx1(θ2) andx2(θ1) are first strictly decreasing and then

constant in the needs of the other system

2100

22222

11111

1|

1|

1kkkkkEkkE


θ1k

θ2

k1

k2

(θ1,θ2

)

(θ1,y2(k-θ1))

(y1(k-θ2),θ2)

(k1,k2)

Equilibrium allocation (x1*(θ1,θ2), x2*(θ1,θ2)) unconstrained for “small” needs and fixed for “large” needs (with x0*(θ1,θ2) = k - x1*(θ1,θ2) - x2*(θ1,θ2) )

PropertiesProperties

• Equilibrium is unique (under Increasing Hazard Rate)

• k1 , k2 , k - k1 - k2 are guaranteed resources for S1, S2 , S0

Implication 1. Let 1 = 2 . Fix θ1 + θ2 with θ1 > θ2

- Full information: U1F = U2

F

- Private information: U1* < U2*

Better performance in easy tasks than in difficult tasks

• Resource monotonicity: if θ1 , both x2 and x0

• Comparative statics. Same monotonicity principle:

- If 2 (S2 more important), then x2* , x1*, x0*

- If k (more resource), then x0* x1* x2*




ImplementationImplementation

So far, abstract revelation mechanism: “announce θ’i, receive xi(θ’i,θ’j)”

Can CES implement the optimal mechanism in a “simple” way and, most importantly, in a way compatible with the physiology of the brain?

• CES sends oxygen to S1, S2 , S0 at rates k1 / k, k2 / k, (k - k1 - k2) / k.

• Systems deplete oxygen to produce energy. CES observes depletion which is a signal that more resources are needed (autoregulation).

• If Si stops consumption, oxygen is redirected to Sj and S0 at a new

rate.

• If both Si and Sj stop consumption, the remaining oxygen is sent to S0.

Si grabs incoming resources up to satiation or up to constraint

Si doesn’t need to know needs or even existence of Sj

Si doesn’t need to know its own needs θi until they are hit

CES must be able to redirect resources and change the rates

ImplementationImplementation

θ1

θ2

k1

k2 If θ1 > k1 and θ2 > k2

If θ1 = θ’1 and θ2 = θ’2θ’2

θ’1

θ’2

x1(θ’2)

If θ1 > x1(θ’2) and θ2 = θ’2




Application 1: task inertiaApplication 1: task inertia

• CES has imperfect knowledge of the distribution of needs

• CES gradually learns the distribution through observation of past needs

• How should CES adjust current allocation rules based on past needs?

• Learning:

- Distributions Fi(i|si) depends on unknown but fixed state si

(is this task usually complex or simple?)

- Prior belief of state is pi (si)

- Realization of it conditionally independent across periods

- After Si reports it in period t, CES updates belief over si

- Assume Fi(i|si) satisfies MLRP: needs are likely to be high (i high) when task is usually complex (si high).

Lemma: Gi(it+1|i

t) satisfies MLRP: high it implies that si is likely to be

high which implies that it+1 is also likely to be high

Assume si unknown and compare public info. (θ1t,θ2

t known by CES at t)

with private info. (θ1t,θ2

t unknown by CES at t)

Implication 2. Inertia and path-dependence of the allocation rule.Under private info. and conditional on present needs, allocation of Si is higher if past needs were high:

If 2t-1 , then x2

t (θ1t,θ2

t) , x1t (θ1

t,θ2t) , x0

t (θ1t,θ2

t)

consistent with neuroscience evidence on “task switching cost”.

Application 1: task inertiaApplication 1: task inertia

Application 2: task separationApplication 2: task separation

• Is it better to have an integrated system responsible for tasks 1 and 2 or two separate systems each responsible for one task?

• Trade-off: - Integrated system allocates more efficiently its resources

between tasks 1 and 2- Separated systems require less “informational rents”: cap of

S1 can depend on announcement of S2

Implication 3.

- Integration of S1 and S2 dominates when motor task is important (“low” 0)

- Separation of S1 and S2 dominates when cognitive tasks are

important (“high” 0)

• The brain is a multi-system organization.• Bounded rationality model based not on inspiration but on physiological

constraints of the brain derive behaviors from brain limitations

• Optimal resource allocation: each system has guaranteed resources (k0, k1, k2). More resources are available only if others are satiated

• Physiologically plausible implementation. • Resource allocation under capacity constraint and asymmetric

information provides an informational rationale for (not a model built to explain):

- Better performance in easier tasks- Task inertia and task switching cost- Conditions for integration v. separation of functions

• Model can be straightforwardly applied to standard organization problems:

- Allocation of resources between research, marketing and production- Market split of colluding firms (no transfers!), etc.

ConclusionsConclusions

Documents

Resource Allocation in the Brain Ricardo Alonso USC Marshall Juan D. Carrillo USC and CEPR T heoretical RE search in N euroeconomic D ecision-making ()