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king_horse_speed = (2,4,6)

tian_horse_speed = (1,3,5)



#the order to race the horses

strats = (

 (0,1,2),

 (0,2,1),

 (1,0,2),

 (1,2,0),

 (2,0,1),

 (2,1,0)

)



bonus = (3,2,1) # traditional: (1,1,1)



def gain(king_strat, tian_strat):

   g = 0

   for i,j,k in zip(king_strat, tian_strat, range(3)):

      if king_horse_speed[i] > tian_horse_speed[j]:

         g += bonus[k]

      else: 

         g -= bonus[k]

   return g





A = [ [gain(a,b) for a in strats] for b in strats ]





from pymprog import *



beginModel('racing')

v = var('game_value', bounds=(None, None))

x = var('prob', strats)

minimize(v)

st( sum( x[s] for s in strats ) == 1 )

rr = st( 

   v >= sum( A[i][j]*x[strats[j]] for j in range(6) ) 

                                  for i in range(6) 

)



solve()

print(v)

for s in x: 

   if x[s].primal > 0: print(x[s])

for r in range(6):

   if rr[r].dual > 0:

      print(strats[r], rr[r].dual)

end()