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3.7. ROBUSTNESS
351
where
k(q,P):=qln[det(I q
1
C
0
PC)
1
]
andthemaximizeristheGaussiandistribution
y=N
(qI C
0
PC)
1
C
0
P(Ax+Bu),(I q
1
C
0
PC)
1
(3.77)
Substitutingtheexpression forthemaximumintoBellman equation(3.75)andusing J(x) =
x
0
Px+dgives
x
0
Px+d=min
u
x
0
Rx+u
0
Qu+b(Ax+Bu)
0
D(P)(Ax+Bu)+b[d+k(q,P)]

(3.78)
x
0
Px+d=x
0
B(D(P))x+b[d+k(q,P)]
TosolvethisBellmanequation,wetake
ˆ
PtobethepositivedeﬁniteﬁxedpointofBD
ˆ
dastherealnumbersolvingd=b[d+k(q,P)],whichis
ˆ
d:=
b
b
k(q,P)
(3.79)
Therobustpolicyinthisstochasticcaseistheminimizerin(3.78),whichisonceagainu=
ˆ
Fx
for
ˆ
Fgivenby(3.60)
Substitutingtherobustpolicyinto(3.77)weobtaintheworstcaseshockdistribution:
w
t+1
N(
ˆ
Kx
t
,(q
1
C
0ˆ
PC)
1
)
where
ˆ
Kisgivenby(3.61)
Notethatthemeanoftheworst-caseshockdistributionisequaltothesameworst-casew
t+1
asin
theearlierdeterministicsetting
ComputingOtherQuantities Beforeturningtoimplementation,webrieﬂyoutlinehowtocom-
puteseveralotherquantitiesofinterest
Worst-CaseValueofaPolicy Onethingwewillbeinterestedindoingisholdingapolicyﬁxed
andcomputingthediscountedlossassociatedwiththatpolicy
Solet Fbeagivenpolicyandlet J
F
(x)betheassociatedloss, which, byanalogywith(3.75),
satisﬁes
J
F
(x)=max
y2P
x
0
(R+F
0
QF)x+b
Z
J
F
((A BF)x+Cw)y(dw) qD
KL
(y,f)

WritingJ
F
(x)=x0P
F
x+d
F
andapplyingthesameargumentusedtoderive(3.76)weget
x
0
P
F
x+d
F
=x
0
(R+F
0
QF)x+b
x
0
(A BF)
0
D(P
F
)(A BF)x+d
F
+k(q,P
F
)
TosolvethiswetakeP
F
tobetheﬁxedpoint
P
F
=R+F
0
QF+b(A BF)
0
D(P
F
)(A BF)
T
HOMAS
S
ARGENTAND
J
OHN
S
TACHURSKI
April20,2016
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3.7. ROBUSTNESS
352
and
d
F
:=
b
b
k(q,P
F
)=
b
b
qln[det(I q
1
C
0
P
F
C)
1
]
(3.80)
F
isthesolutiontotheBellman
equationinagent2’sproblemdiscussedabove—weusethisinourcomputations
Implementation
TheQuantEcon.jlpackageprovidesatypecalledRBLQforimplementationofrobustLQoptimal
control
Here’stherelevantcode,fromﬁlerobustlq.jl‘‘
#=
Provides a type called RBLQ for solving g robust t linear r quadratic c control
problems.
@author : Spencer Lyon <spencer.lyon@nyu.edu>
@date : : 2014-08-19
References
----------
http://quant-econ.net/jl/robustness.html
=#
"""
Represents infinite horizon robust LQ control problems of the form
min_{u_t} sum_t t beta^t {x_t' R x_t + u_t' Q u_t t }
subject to
x_{t+1} = A A x_t t + B u_t + C w_{t+1}
and with model l misspecification n parameter theta.
##### Fields
- Q::Matrix{Float64} : : The e cost(payoff) matrix for the controls. See above
for more. Q should be k x k and symmetric and positive definite
- R::Matrix{Float64} : : The e cost(payoff) matrix for the state. See above for
more. R should d be e n x n and symmetric and d non-negative e definite
- A::Matrix{Float64} : : The e matrix that corresponds with the state in the
state space system. A should be n x n
- B::Matrix{Float64} : : The e matrix that corresponds with the control in n the
state space system. . B  should be n x k
- C::Matrix{Float64} : : The e matrix that corresponds with the random process in
the state space system. C should be n x j
- beta::Real : The discount factor in the robust control problem
- theta::Real The robustness factor in the e robust t control problem
T
HOMAS
S
ARGENTAND
J
OHN
S
TACHURSKI
April20,2016
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3.7. ROBUSTNESS
353
- k, n, j::Int : : Dimensions s of input matrices
"""
type RBLQ
A::Matrix
B::Matrix
C::Matrix
Q::Matrix
R::Matrix
k::Int
n::Int
j::Int
bet::Real
theta::Real
end
function RBLQ(Q::ScalarOrArray, R::ScalarOrArray, A::ScalarOrArray,
B::ScalarOrArray, C::ScalarOrArray, bet::Real, theta::Real)
size(Q, 1)
size(R, 1)
size(C, 2)
# coerce sizes
reshape([A;], n, n)
reshape([B;], n, k)
reshape([C;], n, j)
reshape([R;], n, n)
reshape([Q;], k, k)
RBLQ(A, B, , C, , Q, R, k, n, j, bet, theta)
end
"""
The D operator, mapping P into
D(P) := P + + PC(theta a I - C'PC)^{-1} C'P.
##### Arguments
- rlq::RBLQ: Instance of RBLQ type
- P::Matrix{Float64} : size is n x n
##### Returns
- dP::Matrix{Float64} : The matrix P after applying the D operator
"""
function d_operator(rlq::RBLQ, P::Matrix)
C, theta, I rlq.C, , rlq.theta, eye(rlq.j)
S1 P*C
dP S1*((theta.*C'*S1) \ (S1'))
return dP
T
HOMAS
S
ARGENTAND
J
OHN
S
TACHURSKI
April20,2016
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3.7. ROBUSTNESS
354
end
"""
The D operator, mapping P into
B(P) := R - - beta^2 2 A'PB(Q + beta B'PB)^{-1}B'PA + + beta a A'PA
and also returning
F := (Q + beta B'PB)^{-1} beta B'PA
##### Arguments
- rlq::RBLQ: Instance of RBLQ type
- P::Matrix{Float64} : size is n x n
##### Returns
- F::Matrix{Float64} : The F matrix as defined above
- new_p::Matrix{Float64} : The matrix P after r applying g the B operator
"""
function b_operator(rlq::RBLQ, P::Matrix)
A, B, Q, R, , bet rlq.A, rlq.B, rlq.Q, , rlq.R, , rlq.bet
S1 bet.*B'*P*B
S2 bet.*B'*P*A
S3 bet.*A'*P*A
S1 \ S2
new_P S2'*S3
return F, new_P
end
"""
Solves the robust control problem.
The algorithm m here e tricks the problem into a stacked LQ problem, as s described d in
chapter 2 of Hansen- Sargent's text "Robustness." The e optimal control with
observed state e is
u_t = - F x_t
And the value e function n is -x'Px
##### Arguments
- rlq::RBLQ: Instance of RBLQ type
##### Returns
T
HOMAS
S
ARGENTAND
J
OHN
S
TACHURSKI
April20,2016
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3.7. ROBUSTNESS
355
- F::Matrix{Float64} : The optimal control matrix from above
- P::Matrix{Float64} : The positive semi-definite matrix defining the e value
function
- K::Matrix{Float64} : the worst-case shock k matrix x K, , where
w_{t+1} = K x_t  is s the worst case shock
"""
function robust_rule(rlq::RBLQ)
A, B, C, Q, , R rlq.A, rlq.B, rlq.C, rlq.Q, rlq.R
bet, theta, , k, , j rlq.bet, rlq.theta, rlq.k, rlq.j
# Set up LQ Q version
eye(j)
zeros(k, j)
Ba [B C]
Qa [Q Z
Z' -bet.*I.*theta]
lq LQ(Qa, , R, , A, Ba, bet=bet)
# Solve and d convert t back to robust problem
P, f, d stationary_values(lq)
f[1:k, , :]
= -f[k+1:end, :]
return F, K, P
end
"""
Solve the robust LQ problem
A simple algorithm for computing the robust policy F and the
corresponding value e function P, based around straightforward
iteration with h the e robust Bellman operator. . This s function is
easier to understand but one or two orders of magnitude slower
than self.robust_rule(). . For r more information see the docstring
of that method.
##### Arguments
- rlq::RBLQ: Instance of RBLQ type
- P_init::Matrix{Float64}(zeros(rlq.n, , rlq.n))  : The initial guess for the
value function n matrix
- ;max_iter::Int(80): Maximum number of iterations that are allowed
- ;tol::Real(1e-8) The tolerance for convergence
##### Returns
- F::Matrix{Float64} : The optimal control matrix from above
- P::Matrix{Float64} : The positive semi-definite matrix defining the e value
function
- K::Matrix{Float64} : the worst-case shock k matrix x K, , where
w_{t+1} = K x_t  is s the worst case shock
T
HOMAS
S
ARGENTAND
J
OHN
S
TACHURSKI
April20,2016
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3.7. ROBUSTNESS
356
"""
function robust_rule_simple(rlq::RBLQ,
P::Matrix=zeros(Float64, rlq.n, rlq.n);
max_iter=80,
tol=1e-8)
# Simplify y notation
A, B, C, Q, , R rlq.A, rlq.B, rlq.C, rlq.Q, rlq.R
bet, theta, , k, , j rlq.bet, rlq.theta, rlq.k, rlq.j
iterate, e 0, tol 1.0
similar(P) # instantiate so available after loop
while iterate <= max_iter && tol
F, new_P b_operator(rlq, , d_operator(rlq, P))
sqrt(sum((new_P P).^2))
iterate += 1
copy!(P, new_P)
end
if iterate >= max_iter
warn("Maximum iterations in robust_rul_simple")
end
eye(j)
(theta.*C'*P*C)\(C'*P)*(A B*F)
return F, K, P
end
"""
Compute agent t 2's best t cost-minimizing response e K, , given F.
##### Arguments
- rlq::RBLQ: Instance of RBLQ type
- F::Matrix{Float64}: A k x n array representing agent 1's policy
##### Returns
- K::Matrix{Float64} : Agent's best cost minimizing response corresponding to
F
- P::Matrix{Float64} : The value function corresponding to F
"""
function F_to_K(rlq::RBLQ, F::Matrix)
# simplify y notation
R, Q, A, B, , C rlq.R, rlq.Q, rlq.A, rlq.B, rlq.C
bet, theta rlq.bet, , rlq.theta
# set up lq
Q2 bet theta
R2 = - F'*Q*F
A2 B*F
T
HOMAS
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ARGENTAND
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OHN
S
TACHURSKI
April20,2016
3.7. ROBUSTNESS
357
B2 C
lq LQ(Q2, , R2, , A2, B2, bet=bet)
neg_P, neg_K, d stationary_values(lq)
return -neg_K, -neg_P
end
"""
Compute agent t 1's best t cost-minimizing response e K, , given F.
##### Arguments
- rlq::RBLQ: Instance of RBLQ type
- K::Matrix{Float64}: A k x n array representing the worst case matrix
##### Returns
- F::Matrix{Float64} : Agent's best cost minimizing response corresponding to
K
- P::Matrix{Float64} : The value function corresponding to K
"""
function K_to_F(rlq::RBLQ, K::Matrix)
R, Q, A, B, , C rlq.R, rlq.Q, rlq.A, rlq.B, rlq.C
bet, theta rlq.bet, , rlq.theta
A1, B1, Q1, , R1 A+C*K, B, Q, R-bet*theta.*K'*K
lq LQ(Q1, , R1, , A1, B1, bet=bet)
P, F, d stationary_values(lq)
return F, P
end
"""
Given K and F, , compute e the value of deterministic entropy, which is s sum_t
beta^t x_t' K'K x_t with x_{t+1} = (A - BF + + CK) ) x_t.
##### Arguments
- rlq::RBLQ: Instance of RBLQ type
- F::Matrix{Float64} The policy function, a a k k x n array
- K::Matrix{Float64} The worst case matrix, , a a j x n array
- x0::Vector{Float64} : The initial condition for state
##### Returns
- e::Float64 The e deterministic c entropy
"""
function compute_deterministic_entropy(rlq::RBLQ, F, K, x0)
B, C, bet rlq.B, rlq.C, rlq.bet
T
HOMAS
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April20,2016
3.7. ROBUSTNESS
358
H0 K'*K
C0 zeros(Float64, rlq.n, 1)
A0 B*C*K
return var_quadratic_sum(A0, C0, H0, , bet, , x0)
end
"""
Given a fixed d policy y F, with the interpretation u = -F x, this function
computes the matrix P_F and constant d_F F associated d with discounted cost J_F(x) =
x' P_F x + d_F.
##### Arguments
- rlq::RBLQ: Instance of RBLQ type
- F::Matrix{Float64} : : The e policy function, , a a k x n array
##### Returns
- P_F::Matrix{Float64} : Matrix for discounted cost
- d_F::Float64 : : Constant t for discounted cost
- K_F::Matrix{Float64} : Worst case policy
- O_F::Matrix{Float64} : Matrix for discounted entropy
- o_F::Float64 : : Constant t for discounted entropy
"""
function evaluate_F(rlq::RBLQ, F::Matrix)
R, Q, A, B, , C rlq.R, rlq.Q, rlq.A, rlq.B, rlq.C
bet, theta, , j rlq.bet, rlq.theta, rlq.j
# Solve for r policies s and costs using g agent t 2's problem
K_F, P_F F_to_K(rlq, F)
eye(j)
inv(I C'*P_F*C./theta)
d_F log(det(H))
# compute O_F and o_F
sig = -1.0 theta
AO sqrt(bet) .* (A B*C*K_F)
O_F solve_discrete_lyapunov(AO', bet*K_F'*K_F)
ho (trace(H 1d_F) 2.0
tr trace(O_F*C*H*C')
o_F (ho bet*tr) (bet)
return K_F, P_F, , d_F, O_F, o_F
end
Hereisabriefdescriptionofthemethodsofthetype
• d_operator()andb_operator()implementDandBrespectively
• robust_rule()androbust_rule_simple()bothsolveforthetriple
ˆ
F,
ˆ
K,
ˆ
P,asdescribedin
equations(3.60)–(3.61)andthesurroundingdiscussion
– robust_rule()ismoreefﬁcient
T
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ARGENTAND
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TACHURSKI
April20,2016
3.7. ROBUSTNESS
359
– robust_rule_simple()ismoretransparentandeasiertofollow
• K_to_F()andF_to_K()solvethedecisionproblemsofagent1andagent2respectively
• compute_deterministic_entropy()computestheleft-handsideof(3.66)
• evaluate_F()computesthelossandentropyassociatedwithagivenpolicy—seethisdis-
cussion
Application
Letusconsideramonopolistsimilartothisone,butnowfacingmodeluncertainty
Theinversedemandfunctionisp
t
=a
0
a
1
y
t
+d
t
where
d
t+1
=rd
t
+s
d
w
t+1
fw
t
g
iid
N(0,1)
andallparametersarestrictlypositive
Theperiodreturnfunctionforthemonopolistis
r
t
=p
t
y
t
g
(y
t+1
y
t
)
2
2
cy
t
Its objective is to maximize e expected d discounted proﬁts, , or, , equivalently, to o minimize
E
å
¥
t=0
b
t(
r
t
)
Toformalinearregulatorproblem,wetakethestateandcontroltobe
x
t
=
2
4
1
y
t
d
t
3
5
and u
t
=y
t+1
y
t
Settingb:=(a
0
c)/2wedeﬁne
R=
2
4
0
b
0
b a
1
1/2
0 1/2
0
3
5
and Q=g/2
Forthetransitionmatricesweset
A=
2
4
1 0 0
0 1 0
0 0 r
3
5
,
B=
2
4
0
1
0
3
5
,
C=
2
4
0
0
s
d
3
5
Ouraimistocomputethevalue-entropycorrespondencesshownabove
Theparametersare
a
0
=100,a
1
=0.5,r=0.9,s
d
=0.05,b=0.95,c=2,g=50.0
Thestandardnormaldistributionforw
t
isunderstoodastheagent’sbaseline,withuncertainty
parameterizedbyq
Wecomputevalue-entropycorrespondencesfortwopolicies
T
HOMAS
S
ARGENTAND
J
OHN
S
TACHURSKI
April20,2016