An analysis of dice mechanism by Elias Barry

compute.py

outcomes = ["AS", "AF", "NS", "NF", "DS", "DF"]

# Game played with N-sided dice
# Player may reroll R times
# Success threshold is max of D dice.
def prob(N, R, D):
    P = {}
    EV = {}
    advantage = range( 2 * N // 3 + 1, N+1 )
    disadvantage = range( 1, N // 3 + 1 )
    neutral = range( N // 3 + 1, 2 * N // 3 + 1 )
    
    for a in range( 1, N+1 ):
        pS = (a * 1.0 / N) ** D
        pF = 1.0 - pS
        for b in range( 1, N+1 ):
            for o in outcomes:
                P[(o,a,b,0)] = 0.0
                
            if b in advantage:
                P[("AS",a,b,0)] = pS
                P[("AF",a,b,0)] = pF                
            elif b in disadvantage:
                P[("DS",a,b,0)] = pS
                P[("DF",a,b,0)] = pF                
            else:    
                P[("NS",a,b,0)] = pS
                P[("NF",a,b,0)] = pF
                
    for r in range( 1, R+1 ):
        for a in range(1,N+1):
            for b in range(1,N+1):
                EV[(a,b,r-1)] = \
                    4 * P[("AS",a,b,r-1)] + \
                    3 * P[("NS",a,b,r-1)] + \
                    2 * P[("DS",a,b,r-1)] + \
                    2 * P[("AF",a,b,r-1)] + \
                    1 * P[("NF",a,b,r-1)]

        for b in range( 1, N+1 ):
            #print( f"EV[*,{b},{r-1}] = 1/{N} * (", end='' )
            #for a in range(1, N+1):
            #    print( f"{EV[(a,b,r-1)]} + ", end='' )
            #print( ")")
            EV[("*",b,r-1)] = sum(EV[(a,b,r-1)] for a in range(1,N+1)) / N

        for a in range(1,N+1):
            for b in range(1,N+1):
                # if EV( a, b, 0 ) < EV( *, a, r-1 ) then reroll
                if EV[(a,b,0)] < EV[("*",a,r-1)]:
                    #print( f'{a},{b} {r} {EV[(a,b,0)]} vs {EV[("*",a,r-1)]} reroll' )
                    # Reroll moves a to b and replaces a with unform outcome [1,N]
                    for o in outcomes:
                        P[(o,a,b,r)] = sum( P[(o,k,a,r-1)] for k in range(1,N+1) ) / N
                else:
                    #print( f'{a},{b} {r} {EV[(a,b,0)]} vs {EV[("*",a,r-1)]} stop' )
                    for o in outcomes:
                        P[(o,a,b,r)] = P[(o,a,b,0)]

    # Initial state probabilities
    init = {}
    for o in outcomes:
        init[o] = sum( P[(o,a,b,R)]
                       for a in range(1,N+1)
                       for b in range(1,N+1) ) / (N ** 2)

    return init, P, EV

def show_decision(N, r, EV):
    print( f"[{r} rerolls left]" )
    print( "a\\b " + " ".join( str(x) for x in range(1,N+1) ) )
    for a in range( 1, N+1):
        print( f"{a:3} ", end='') 
        for b in range(1, N+1):
            if EV[(a,b,0)] < EV[("*",a,r-1)]:
                print( "R ", end='')
            else:
                print( "x ", end='')
        print()
    
def main():
    N = 6
    R = 10
    for D in range(1,11):
        print( f"### Difficulty {D}")
        init, p, ev = prob(N, R, D)
        for o in outcomes:
            print( f"{o} {init[o]}" )

        print()
        for r in range(1, R+1):
            show_decision(N, r, ev)
        print()
        
if __name__ == "__main__":
    main()