From Sperner to Brouwer

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From Sperner to Brouwer

• Raghavendra Prasad

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PPAD Definition

• Every input x defines a graph Gx through the
  machine M
• 0n is a source node
• Find a sink or another source node, and compute
  a polynomial time function of x and u

PPAD Problem
x
Sink or Another source
Polynomial Time Function
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Closure of PPAD under Reductions
g
x
PPAD Problem
x
Some Function
M
h
Gx
f(u)
h(x) f(x,u)
f
Gx must consist of vertices f(u), for vertices u.
But it might be hard to find out if (u,v) is an
edge from values of f(u),f(v).
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Closure of PPAD under reductions
x
g
PPAD Problem
g
y
M
y
PPAD Problem
x
Some Function
M
M
h
Gy
Gy
u
u
f(x,p(y,u))
h(x)
f(x,g(y,u))
p(y,u)
f
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Brouwers fixed point theorem

• Given any continous function f from the
  triangle on to itself, there is a fixed point,
• f(p) p
• for some point p in the triangle

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Sperners Lemma

• Every Sperner triangulation contains a
  achromatic triangle

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From Sperner to Brouwer
f(p)
p
A point p is assigned the color of the vertex it
is moving away from
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Brouwer from Sperner
A vertex is RED does not move closer P1 GREEN
does not move closer to P2 BLUE does not
move closer to P3 Take finer and finer
triangulations, the limit point does not move
closer to any of the three vertices! It stays
where it is!
P2
P1
P3
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3-D Sperner
Dividing a tetrahedron in to smaller tetrahedrons
is messy Therefore we state the theorem for a
cube. The theorem for a cube, is just a
corollary of the fact for the tetrahedron.
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3D Sperner

• No vertex colored 2 on face 013
• No vertex colored 1 on face 023 and no vertex
  colored 3 on 012
• No vertex colored 0 on other faces
• There is a cubelet with all four colors

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3D-Sperner is PPAD complete

• INPUT
• A machine M, that computes for each input x, and
  each node v its parent child pair M(x,v)

• OUTPUT
• A cube divided in to
• n X n X n grid for some n.
• A machine M that gives the color of any vertex

A solution to the original problem can be
obtained from a tetrachromatic cubelet
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The Cube
3
0
2
2
1
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The Cube

• Red, Yellow, Blue are rare colors
• P is the only point on surface with all three
  rare colors
• A trichromatic tube starting at P

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More tubes
P

• A small vertical tube for each vertex
• A small horizontal tube for each pair of vertices
• Connect the vertices and edges, by tubes.

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Computing the Colors

• To compute the color of the point P
• Not in any of the tubes or surface, then it is
  green.
• If inside or on some tube then, it is either red,
  blue, yellow

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Colors in a tube

• The colors in a tube constantly rotate, so that
  it connects their two endpoints.
• Given the position of the point, one can
  determine the color in polynomial time.

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Conclusion

• 3D Sperner is PPAD- complete
• Computing Nash equilibria is in PPAD
• PPAD is equivalent to finding a sink or source in
  a implicitly specified directed graph

Sperners Theorem
Brouwers Fixed point
Existence of Nash Equilibria