1862—1943 / 1928—2015 / Göttingen → Princeton

NASH & HILBERT

One asked what mathematics can be. The other asked what rationality must be. Together they show that infinity has a geometry and strategy has a fixed point.
NASH 1950-51

1. Equilibrium — Fixed Points of Self-Interest

Nash's 21-year-old thesis redefined games: an \(n\)-person non-cooperative equilibrium is a profile where each strategy is a best response to the others. Existence comes from Kakutani's fixed-point theorem, a generalization of Brouwer that Hilbert's topological work made central.

\[ \sigma^{*}=(\sigma_1^{*},\dots,\sigma_n^{*})\ \text{is Nash}\iff u_i(\sigma_i^{*},\sigma_{-i}^{*})\ge u_i(\sigma_i,\sigma_{-i}^{*})\ \forall \sigma_i,\ \forall i,\quad BR_i(\sigma_{-i})=\arg\max_{\sigma_i}u_i \]

2×2 Payoff Lab

Edit payoffs. Player 1 rows Up/Down, Player 2 cols Left/Right. Cell = (A,B).
A:

B:

GNU OCTAVE
function [pure,mix]=nash_2x2(A,B)
% NASH_2X2 pure and mixed Nash for 2x2
% A,B 2x2 payoff matrices for Row, Column
pure=[];
for i=1:2
 for j=1:2
  if A(i,j)==max(A(:,j)) && B(i,j)==max(B(i,:))
    pure=[pure; i j];
  end
 end
end
% mixed: indifference
a=A(1,1); b=A(1,2); c=A(2,1); d=A(2,2);
e=B(1,1); f=B(1,2); g=B(2,1); h=B(2,2);
q = (h-f)/(e-f-g+h); % prob Column plays Left
p = (d-b)/(a-b-c+d); % prob Row plays Up
mix.p = p; mix.q = q;
fprintf('pure equilibria rows:'); disp(pure);
fprintf('mixed p=%.3f q=%.3f\n',p,q);
end
% Example:
A=[3 0;5 1]; B=[3 5;0 1]; nash_2x2(A,B)
NASH 1950 / COURNOT 1838

2. Bargaining & Duopoly

The bargaining problem asks which point in a feasible convex set \(S\) rational bargainers pick given disagreement \(d\). Nash's four axioms force the product maximizer. The same fixed-point logic gives Cournot duopoly equilibrium as intersection of reaction functions.

\[ \max_{u\in S,\ u\ge d}(u_1-d_1)(u_2-d_2),\qquad q_1^{*}=\frac{a-c_1}{2b}-\frac{q_2}{2},\ q_2^{*}=\frac{a-c_2}{2b}-\frac{q_1}{2} \]

Nash Bargaining

Drag red d. Green = Nash product max. Hyperbolas = level sets.

Cournot Duopoly

a 20
c1 2 c2 3
GNU OCTAVE — BARGAINING
function s=nash_bargaining(S,d)
% S : Nx2 polygon vertices, d : 1x2 disagreement
% maximize log((u1-d1)*(u2-d2))
obj=@(u) -sum(log(max(u-d,1e-9)));
% start at centroid
u0=mean(S,1);
% simple fminsearch inside feasible via penalty
Sfun=@(u) inpolygon(u(1),u(2),S(:,1),S(:,2));
pen=@(u) obj(u) + 1e4*(~Sfun(u));
s=fminsearch(pen,u0);
fprintf('Nash bargaining s=(%.3f,%.3f)\n',s(1),s(2));
end
S=[0 0;10 0;7 7;0 10]; d=[1 1]; nash_bargaining(S,d)
GNU OCTAVE — COURNOT
function [q1,q2]=cournot_nash(a,b,c1,c2)
% Solve q1=(a-c1)/(2b)-q2/2, q2=(a-c2)/(2b)-q1/2
A=[2*b b; b 2*b]; rhs=[a-c1; a-c2];
q=A\rhs; q1=q(1); q2=q(2);
fprintf('q1*=%.3f q2*=%.3f price=%.3f\n',q1,q2,a-b*(q1+q2));
end
cournot_nash(20,1,2,3)
NASH 1954-56 / MOSER 1966

3. Embedding & The Hard Theorem

Every Riemannian \(n\)-manifold admits an isometric \(C^1\) embedding into \(\mathbb R^{2n}\) and \(C^k\) into \(\mathbb R^{n(3n+11)/2}\). Nash's proof introduced hard implicit function iteration with smoothing, later abstracted as Nash-Moser.

\[ \forall (M^n,g)\ \exists u:M^n\to\mathbb R^N\ \text{s.t.}\ \partial_i u\cdot\partial_j u=g_{ij} \]

Nash-Kuiper Corrugation

Wrinkle 20
Base length preserved while high-frequency low-amplitude waves add embeddability. Arc length ≈ constant.
GNU OCTAVE — CORRUGATION
function nash_crug_demo(k,amp)
% k frequency, amp amplitude ~ 1/k preserves length
t=linspace(0,2*pi,2000);
base=sin(t); % smooth base curve y=sin x roughly
y=base+amp*sin(k*t)/k;
L_base=trapz(t,sqrt(1+cos(t).^2));
L_new =trapz(t,sqrt(1+(cos(t)+amp*cos(k*t)).^2));
fprintf('L base=%.4f L corrugated=%.4f ratio=%.4f\n',L_base,L_new,L_new/L_base);
plot(t,y); title('Corrugated embedding preserves length');
end
nash_crug_demo(30,0.8)
HILBERT 1899 / 1924

4. Axioms — Ground & Grand Hotel

Grundlagen der Geometrie rebuilt geometry from 21 axioms in 5 groups. To show infinity's strangeness, Hilbert imagined a hotel with \(|\mathbb N|\) rooms.

\[ |\mathbb N|=|\mathbb N\cup\{0\}|=|\mathbb N\times\mathbb N|,\quad n\mapsto n+1,\ n\mapsto2n,\ (p,q)\mapsto2^{p}3^{q} \]

Hilbert's Hotel

GNU OCTAVE — HOTEL
function hilbert_hotel()
% Bijection demos
n=1:10;
fprintf('n -> n+1: '); disp(n+1);
fprintf('n -> 2n (evens free odds): '); disp(2*n);
% Cantor pairing for N x N
cantor=@(p,q) (p+q)*(p+q+1)/2+q;
fprintf('pair (2,3) -> %d\n',cantor(2,3));
% Hotel occupancy as sequence
rooms=zeros(1,20); rooms(1:10)=1:10;
fprintf('Shift for 1 guest: '); disp([0 rooms(1:9)]);
end
hilbert_hotel()
HILBERT 1906 / FOURIER 1807

5. Hilbert Space — Infinity Made Geometric

\(\ell^2\) and \(L^2\) are complete inner product spaces. Every \(x\) has coordinates \(\langle x,e_n\rangle\) in an orthonormal basis. Fourier series is just Pythagoras in infinite dimensions.

\[ \ell^2=\{x:\sum|x_n|^2<\infty\},\ \langle x,y\rangle=\sum x_n\bar y_n,\ x=\sum\langle x,e_n\rangle e_n,\ \|f-S_N\|_2\to0 \]

Fourier \(L^2\) Explorer

N 5
GNU OCTAVE — FOURIER L2
function [err]=fourier_l2(N)
% Approximate square wave in L^2[-pi,pi]
x=linspace(-pi,pi,2000); f=sign(sin(x)); % square
S=zeros(size(x));
for k=1:2:2*N-1
  S+=4/pi/k*sin(k*x);
end
err=sqrt(trapz(x,(f-S).^2)/ (2*pi));
fprintf('N=%d L2 error=%.4f\n',N,err);
plot(x,f,x,S); legend('f','S_N'); grid on;
end
fourier_l2(15)
HILBERT 1904-10 / EINSTEIN-HILBERT 1915

6. Spectral & Variational

For compact self-adjoint \(A\), \(L^2=\overline{\bigoplus \ker(A-\lambda I)}\). Hilbert's variational principle \(J[y]=\int L\,dx\) gives Euler-Lagrange, and \(S=\int R\sqrt{-g}\,d^4x\) gives Einstein.

\[ Af=\lambda f,\quad \frac{d}{dx}\frac{\partial L}{\partial y'}-\frac{\partial L}{\partial y}=0,\quad S=\int R\sqrt{-g}\,d^4x \]

Spectral Ellipse

a 2 b 1 c 1
GNU OCTAVE — SPECTRAL & HILBERT MATRIX
function spectral_hilbert_demo(a,b,c)
A=[a b; b c]; [V,D]=eig(A);
fprintf('eig = '); disp(diag(D)');
theta=linspace(0,2*pi,400); circ=[cos(theta); sin(theta)];
ell=A*circ; plot(ell(1,:),ell(2,:)); axis equal; grid on;
% Hilbert matrix ill-conditioning
for n=[3 5 8 12]
 H=hilb(n); fprintf('n=%d cond(H)=%.2e\n',n,cond(H));
end
end
spectral_hilbert_demo(2,1,1)
HILBERT 1900 — 23 PROBLEMS

7. Problems & The Space-Filling Curve

1900 Paris: 23 problems set 20th century agenda. 1 Continuum, 2 Consistency (Gödel 1931: no), 8 Riemann (open), 10 Diophantine (Matiyasevich 1970: no algorithm). Hilbert's own curve shows \([0,1]\to[0,1]^2\) continuous surjection.

\[ V(I)=\emptyset\iff1\in I\quad\text{(Nullstellenssatz)},\qquad H_{n+1}=\text{4 copies of }H_n\text{ rotated} \]

Hilbert Curve Order

Order 4
Maps 1D interval onto 2D continuously. Length \(4^n-1\) segments.
GNU OCTAVE — HILBERT CURVE
function [x,y]=hilbert_curve(order)
% L-system Hilbert curve
seq='A'; for k=1:order
  seq=strrep(seq,'A','-BF+AFA+FB-');
  seq=strrep(seq,'B','+AF-BFB-FA+');
end
% Actually simpler: use recursion
  function [x,y]=hilb_rec(o,x0,y0,xi,xj,yi,yj)
    if o<=0, x=x0+(xi+yi)/2; y=y0+(xj+yj)/2; return; end
    [x1,y1]=hilb_rec(o-1,x0,y0,yi/2,yj/2,xi/2,xj/2);
    [x2,y2]=hilb_rec(o-1,x0+xi/2,y0+xj/2,xi/2,xj/2,yi/2,yj/2);
    [x3,y3]=hilb_rec(o-1,x0+xi/2+yi/2,y0+xj/2+yj/2,xi/2,xj/2,yi/2,yj/2);
    [x4,y4]=hilb_rec(o-1,x0+xi+yi/2,y0+xj+yj/2,-yi/2,-yj/2,-xi/2,-xj/2);
    x=[x1 x2 x3 x4]; y=[y1 y2 y3 y4];
  end
[x,y]=hilb_rec(order,0,0,1,0,0,1);
plot(x,y,'-'); axis equal off;
fprintf('order %d points %d\n',order,length(x));
end
hilbert_curve(5)

Convergence — The Fixed Point That Holds Infinity

Hilbert gave us the language: complete inner product spaces, axiomatic rigor, integral equations. Nash used Kakutani's fixed-point theorem — a direct descendant of Brouwer's theorem from Hilbert's Göttingen school — to prove every finite game has an equilibrium. Hilbert asked for a finite basis for invariants; Nash gave a finite list of best responses. Hilbert's program sought consistency; Nash showed selfish consistency always exists, even if not Pareto optimal. Nash's embedding uses the hard analysis Hilbert pioneered.

Hilbert → Nash
1906 Hilbert space → infinite strategy spaces, 1900 fixed-point ideas → Kakutani 1941 → Nash 1950
Nash → Hilbert
Equilibrium as geometry, embedding as game against curvature, both use smoothing and approximation to conquer loss of regularity.
Timeline: 1899 Grundlagen, 1900 23 Problems, 1906 Hilbert space, 1928 von Neumann minimax, 1950 Nash thesis & bargaining, 1954-56 embedding, 1966 Nash-Moser, 1994 Nobel (Nash), 2015 Abel (Nash & Nirenberg).