## Code for Figure 6.4

### main.m

```
1
2
3
4
5
6
7
8
9
10
11
12% A 2-variable steady-state problem.
lams = {[0.5, 0.5], [0.2, 0.8], [0.8, 0.2], [0.5, 0.5]};
iters = [10, 25, 25, 25];
names = {'equal', 'first', 'second', 'swapped'};
swaps = [false(), false(), false(), true()];

% Simulate with different values of lambda.
data = struct();
for i = 1:length(lams)
data.(names{i}) = noncoopvscoop(lams{i}, iters(i), names{i}, swaps(i));
end

```

### noncoopvscoop.m

```
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78function data = noncoopvscoop(lam, Niter, name, swap)
% Simulates non-cooperative vs. cooperative optimization.
narginchk(3, 4);
if nargin() < 4
swap = false();
end

% Unstable for about m1 < 0.58, lam1 = lam2 = 0.5.
m1 = 0.5;
m2 = -1/m1;
g  = [-m1, 1; -m2, 1];
if swap
g = [0, 1; 1, 0]*g;
end

ysp = [1; 1];
usp = g\ysp;
ude = [ysp(1)/g(1,1); ysp(2)/g(2,2)];

% Noncooperative problem.
Lnc = [1-lam(1), -lam(1)*g(1,2)/g(1,1); -lam(2)*g(2,1)/g(2,2), 1-lam(2)];
Knc = diag([lam(1)/g(1,1), lam(2)/g(2,2)]);

ssnc = abs(eig(Lnc));

unc = zeros(2, Niter + 1);
unc(:,1) = ude;

for i = 1:Niter
unc(:,i+1) = Knc*ysp + Lnc*unc(:,i);
end

% Cooperative problem.
Q = diag(lam);
H = [Q, zeros(2,1); zeros(1,2), 0];
G1 = g(:,1); G2 = g(:,2);
D = [eye(2), -G1];
M = [H, -D'; -D, zeros(2)];
Minv = inv(M);
K1 = Minv(3,1:2)*Q;
L1 = -Minv(3,4:5)*G2;
D = [eye(2), -G2];
M = [H, -D'; -D, zeros(2)];
Minv = inv(M);
K2 = Minv(3,1:2)*Q;
L2 = -Minv(3,4:5)*G1;
Kco = [lam(1)*K1; lam(2)*K2];
Lco = [(1-lam(1)), lam(1)*L1; lam(2)*L2, (1-lam(2))];
ssco = abs(eig(Lco));

uco = zeros(2, Niter);
uco(:,1) = ude;

for i = 1:Niter
uco(:,i+1) = Kco*ysp + Lco*uco(:,i);
end

% Make lines for plotting.
ld = min(unc');
ru = max(unc');
lr = max(-ld(1), ru(1));
ud = max(-ld(2), ru(2));
sslines = [-lr, lr, NaN(), -ud/m2, ud/m2;
-m1*lr, m1*lr, NaN(), -ud, ud];
sslines = sslines + repmat(usp, 1, 5);

% Make plot.
figure();
plot(sslines(1,:), sslines(2,:), '-k', ...
unc(1,:), unc(2,:), '-sr', ...
uco(1,:), uco(2,:), '-og');
title(sprintf('%s (lambda = [%g, %g])', name, lam));
legend('Steady State', 'Non-cooperative', 'Cooperative');
xlabel('u_1');
ylabel('u_2');

% Package data.
data = struct('ss', sslines', 'noncoop', unc', 'coop', uco');

```