Theoretical REsearch in Neuroeconomic Decision-making
(www.neuroeconomictheory.org)
Resource Allocation in the Brain
Ricardo Alonso
Isabelle Brocas
USC Marshall
USC and CEPR
Juan D. Carrillo
USC and CEPR
April 2011
Neuroeconomic 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)
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)
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 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
U l  xl ,l   
1
l
 xl  l 
2
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
The model
CES
central executive system
x0  x1  x2  k
U 0  U1  U 2
x0
S0
(lifting)
U 0 ( x0 ,  0 )  
1
0
( x0   0 ) 2
x1
x2
S2
(spelling)
S1
(rotation)
U1 ( x1 ,1 )  
1
1
( x1  1 ) 2
U 2 ( x2 ,  2 )  
1
2
( x2   2 ) 2
Motor function
Cognitive functions 1 and 2
0 “public”
1 and 2 “private”
Benchmark case: full information
max
x0 , x1 , x2
1
0
U 0  x0 1 ,  2  ,  0  
1
1
U1  x1 1 , 2  ,1  
s.t. x0 1 ,2   x1 1 ,2   x2 1 ,2   k
1
2
x0 1 ,2   0, x1 1 ,2   0, x2 1 ,2   0
U 2  x2 1 , 2  , 2 
(R)
(F)
Benchmark case: full information
max
x0 , x1 , x2
1
0
U 0  x0 1 ,  2  ,  0  
1
1
U1  x1 1 , 2  ,1  
s.t. x0 1 ,2   x1 1 ,2   x2 1 ,2   k
1
2
x0 1 ,2   0, x1 1 ,2   0, x2 1 ,2   0
U 2  x2 1 , 2  , 2 
(R)
(F)
Solution under full information (assuming (R) binds and (F) does not):
xlF  l 
l

  k 
Distribute k according to needs (θ0, θ1, θ2) weighed by importance (0, 1, 2)
U lF  xlF ;l   
l
  
 k
2 
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)
Roadmap
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
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 problem
Optimal allocation of resources when needs of systems 1 and 2 are unknown
and CES can use any mechanism. Using the revelation principle:
max
x0 , x1 , x2
U 0  x0 1 ,  2  ,  0  
1
 
0
1
1
U1  x1 1 ,  2  , 1  

 
~
s.t. U i xi i , j ,i   U i xi i , j ,i
1
2
U 2  x2 1 ,  2  ,  2  dF1 (1 )dF2 ( 2 )
~
i, j {1,2}, i ,i , j
(IC)
x0 1 ,2   x1 1 ,2   x2 1 ,2   k 1 ,2
(R)
x0 1 ,2   0, x1 1 ,2   0, x2 1 ,2   0 1 ,2
(F)
The solution
• Optimal mechanism M:
- resource capx1(θ2) for S1
- resource capx2(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 capsx1(θ2) andx2(θ1) ?
The 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 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
2
E 2 |  2  y2 (k ' )  y2 (k ' ) 
y
1
0
 0  k ' y2 (k ' )
• Under IHR, the optimal cap:
- is continuous in resources k’
- satisfies resource monotonicity
y2 k ' / k ' (0,1]
y2 (k’)
k
k’
Step 2a: P1-Mechanism
• 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:
θ2
y
θ’’2
y2 (k’)
y2 (k - θ1) [P1]
θ'2
k
k’
θ'1
k
θ1
Step 2b: P2-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
θ2
y
k
y1 (k - 2) [P2]
y1(k’)
k
k’
θ1
Step 3: optimal resource allocation
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)
θ2
y1 (k - 2) [P2]
y2 (k - θ1) [P1]
k
θ1
Optimal Resource Allocation
Optimal capsx1(θ2) andx2(θ1) are first strictly decreasing and then
constant in the needs of the other system
θ2
x1(θ2) = y1(k - θ2) if θ2 < k2
k1
if θ2  k2
x2(θ1) = y2(k - θ1) if θ1 < k1
k2
if θ1  k1
k2
(θ1+θ2=k)
k1
θ1
Optimal Resource Allocation
Optimal capsx1(θ2) andx2(θ1) are first strictly decreasing and then
constant in the needs of the other system
1
1
E1 | 1  k1   k1  
1
2
E 2 |  2  k2   k2  
1
0
θ2
k2
(θ1+θ2=k)
k1
θ1
 0  k  k1  k2 
Optimal Resource Allocation
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) )
θ2
(θ1,y2(k-θ1))
(k1,k2)
k2
(θ1,θ2)
(y1(k-θ2),θ2)
k1
k
θ1
Properties
• Equilibrium is unique (under Increasing Hazard Rate)
• k1 , k2 , k - k1 - k2 are guaranteed resources for S1, S2 , S0
• 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* 
Implication 1. Let 1 = 2 . Fix θ1 + θ2 with θ1 > θ2
- Full information:
U1F = U2F
- Private information: U1* < U2*
Better performance in easy tasks than in difficult tasks
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
Implementation
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
Implementation
θ2
k2
If θ1 > k1 and θ2 > k2
If θ1 = θ’1 and θ2 = θ’2
θ’2
If θ1 > x1(θ’2) and θ2 = θ’2
k1 θ’
x11(θ’2)
θ1
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
Application 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|it) satisfies MLRP: high it implies that si is likely to be
high which implies that it+1 is also likely to be high
Application 1: task inertia
Assume si unknown and compare public info. (θ1t,θ2t known by CES at t)
with private info. (θ1t,θ2t 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 x2t (θ1t,θ2t) , x1t (θ1t,θ2t) , x0t (θ1t,θ2t) 
 consistent with neuroscience evidence on “task switching cost”.
Application 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)
Conclusions
• 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.