Linear Programming and zero-sum two-player games - PowerPoint

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					Linear Programming
and zero-sum two-
player games
By Warren Schudy
Outline
 Intro to Linear Programs (LPs)
 LP Duality
 Corollaries:
     Minimax  theorem via duality
     Solving two-player zero-sum games in
      polynomial time via LPs
   For more on LPs, take CS 149: Intro to
    Combinatorial Optimization
A Production Problem
 Town dump has 4 bike frames, 5 unicycle
  frames and 11 wheels
 Bicycles sell for $1, Unicycles for $2

                               y (unic.)


max x  2 y
x4         (bike frames)                           OPT
y5            (unic.frames)
                                    2
2 x  y  11     (wheels)
x, y  0
                                        0
                                            0   2
                                                      x (bikes)
Alternate view: competitive
equilibrium
   Suppose you want to pay the dump enough for
    the parts that your little brother won’t have an
    incentive to compete with you
   Rival has incentive to make:
     bicyclesif price(bike frame)+2price(wheel)<=1
     unicycles if price(unic. frame)+price(wheel)<=2
   Will show via duality: preventing competition
    forces you to pay the dump all of your revenue!
Primal and Dual
  max x  2 y    min 4 b  5 u  10 w
  x4
                  b  2 w  1
  y5
  2 x  y  10   u w  2
  x, y  0        b , u , w  0


  max c xT         min bT 
   Ax  b           AT   c
  x0               x0
                           max c T x           min bT 
Weak duality                Ax  b             AT   c
                           x0                 x0
 Intuition: competition will keep you from
  making money
 Theorem: For feasible x,  we have bT   cT x
 Proof:    bT    T b   T ( Ax )  ( AT  )T x  cT x



                   Uses                     Uses
              b  Ax and   0       AT   c and x  0
                                                      2 x  2 y  20
 Upper Bounds                                       Also implies UB 20
                                                         because

                 x  2 y  20                          x0
                Perpendicular to
max x  2 y    objective function
               and valid -> Upper
x4                 Bound

y5
2 x  y  10
x, y  0        Best Upper
               Bound of this
                   form             2



                                        0
                                            0   2
Equality Constraints
   Suppose the dump manager hates dealing
    with bike frames (too big), so he wants you
    to take all of them, even if he has to pay
    you
    max x  2 y
                    Equality
                              min 4 b  5 u  10 w
    x4
                                   b  2 w  1
    y5
    2 x  y  10                  u w  2
                   Price can go
    x, y  0         negative            u , w  0
LP for games and its dual
    Proves minimax theorem using LP duality!
 max v                                min z  0T q
 v  p T M j  0 j " q j "           z  M (i ) q  0 i            " pi "
1T p  1            " z"              0 z  1T q  1          " v"
 p0                                  q0

max x  2 y
                                 min 4 b  5 u  10 w
x4         " bike frames"
                                  b  2 w  1        " bikes"
y5            " unic. frames"
                                 u w  2          " unicycles"
2 x  y  10     " wheels"        b , u , w  0
x, y  0
Game from LP
 Prove LP duality from games
 A bit unintuitive; see notes for details
Conclusions
   Two-player zero-sum games are poly-time
    solvable using linear programming