MinMax Principle in Game Theory

1
MinMax Principle in Game Theory continued.
PARTH
2
MinMax principle

• Consider a computation problem P
• Let A be finite set of deterministic algorithms
  for solving P
• Let I be set of all possible inputs of size n for
  P

3
MinMax principle

• Matrix of running times

Algorithms
A
Running time of A for input I C(A,I)
Inputs
I
Worst case running times for As (highest of
respective column of matrix)
Gives the optimal algorithm A
Deterministic lower bound min max CA,I
A
I
4
MinMax principle

• Lower bound is actually, Vc in Von Neumanns
  principle
• Vr Vc -gt statement of Von Neumanns prin.
• It can be restated as
• max min EC(A,I) min max EC(A,I)
• I from p A from q

A from q I from p
Where p and q are payoff matrices
5
MinMax principle

• Loomis theorem can be restated as
• max min EC(A,I) min max EC(A,I)
• I from p A A
  A from p I I
• Yaos min max principle
• Take p on left and any distribution q on the
  right
• min EC(A,I) lt max EC(A,I)
• A A I I

6
MinMax principle
7
AND-OR tree problem - revisited

• Replace all AND and OR gates with equivalent
  combinations of NOR gates
• For any Las Vegas algorithm, Exponential time gt
  n0.6
• Consider a distribution which sets each leaf
  independently
• Leaf 1 -gt probability p
• Leaf 0 -gt otherwise