strategies for playing the dice game ‘toss up’
DESCRIPTION
Strategies for playing the dice game ‘Toss Up’. Roger Johnson South Dakota School of Mines & Technology April 2012. ‘Toss Up’ Dice. Game produced by Patch Products (~$7) ( http://www.patchproducts.com/letsplay/ tossup.asp ) Ten 6-Sided Dice 3 sides GREEN 2 sides YELLOW 1 side RED - PowerPoint PPT PresentationTRANSCRIPT
Strategies for playing the dice game ‘Toss Up’
Roger JohnsonSouth Dakota School of Mines &
TechnologyApril 2012
‘Toss Up’ Dice
• Game produced by Patch Products (~$7)(http://www.patchproducts.com/letsplay/tossup.asp)• Ten 6-Sided Dice– 3 sides GREEN– 2 sides YELLOW– 1 side RED
• Players take turns– Each turn consists of (potentially) several rolls of the dice– First player to at least 100 wins
A Roll in ‘Toss Up’
• SOME GREEN add the number of green to your turn score; remaining (non-green) dice may be used on the next roll
• ALL YELLOW no change in turn score, all dice thrown on the next roll
• NO GREEN and AT LEAST ONE RED lose points accumulated in current turn; turn ends
A Turn in ‘Toss Up’
• After each roll:– If the player is not forced to stop - she may either
continue or voluntarily stop– With a voluntary stop, the score gained on the
turn is added to previously accumulated score
• If all the dice have been “used up”, then the player returns to rolling all 10 dice again
One Strategy
• Continue only when expected increase in score is positive.
• Suppose current turn score is s and d dice are being thrown. The expected increase is:
1
( Green) (Forced Stop)
1 10
2 2 3
d
g
d d
g P g s P
ds
Positive Expected Increase Strategy# Dice Being Tossed Continue rolling . . .
1, 2 Never!
3 . . . when turn score < 18
4 . . . when turn score < 40
5 . . . when turn score < 93
6, 7, 8, 9, 10 Always!
Positive Expected Increase Strategy
• Empirical game length with this strategy (100,00 trials):
Average = 11.92, Standard Deviation = 1.50
Second Strategy
• Minimize the expected number of turns (c.f. Tijms (2007))
• is the expected additional number of turns to reach at least 100 when
i = score accumulated prior to the current turn
j = score accumulated so far during the current turn
[ , ]E i j
Expected Values Recursions
10
1
[ ,0] [0,0 ] (1 [ ,0])
[0,0 ] [ ,0]
[0, ] [ , ]
FS
AY
k
E i p E i
p E i
p k E i k
Expected Values Recursions
10 '
1
' mod10, 0 :
[ , ] min 1 [ ,0],
[ ',0 ] (1 [ ,0])
[ ', ' ] [ , ]
[ ', ' ] [ , ]
FS
AY
j
k
j j j
E i j E i j
p j E i
p j j E i j
p j j k E i j k
Solving the Recursion
• Have
• Used
( )x f x
1
0
( )
0n nx f x
x
Minimal Expected Value
• 7.76 turns as opposed to about 11.92 turns for first strategy (~35% reduction)
• Simulation with optimal strategy, using 100,000 trials, gives an average of 7.76 turns with a standard deviation of 2.77 turns
Character of Optimal Solution
• Complicated• Not always intuitive• Some (weak) dependence on previously
accumulated score• Optimal solution at
http://www.mcs.sdsmt.edu/rwjohnso/html/research.html
Rough Approximation of Optimal Solution
# Dice Being Tossed Expected Increase Strategy: Continue Rolling when…
Rough Approx of Optimal Strategy: Continue Rolling when…
1,2 Never! …when turn score < 27
3 …when turn score < 18 …when turn score < 27
4 …when turn score < 40 …when turn score < 36
5 …when turn score < 93 Always!
6,7,8,9,10 Always! Always!
Empirical ResultsPositive Expected Increase Strategy
Rough Approximation of Optimal Strategy
Average (Optimal mean = 7.76)
11.92 7.81
Standard Deviation 1.50 2.80
Each column from a simulation of 100,000 trials
References• Johnson, R. (2012), “‘Toss Up’ Strategies”, The
Mathematical Gazette, to appear November.• Johnson, R. (2008), “A simple ‘pig’ game”,
Teaching Statistics, 30(1), 14-16.• Neller, T. and Presser (2004), “Optimal play of the
dice game Pig”, The UMAP Journal, 25, 25-47 (c.f. http://cs.gettysburg.edu/projects/pig/).
• Tijms, H. (2007), “Dice games and stochastic dynamic programming”, Morfismos, 11(1), 1-14 (http://chucha.math.cinvestav.mx/morfismos/v11n1/tij.pdf).
Questions?
Chances of Various Outcomes
# dice tossed Run Red Light (no green, at least one red)
Yellow Light (all yellow)
Gain Some Points (at least one green)
n
1 1
2 3
n n
1
3
n
1
12
n
10 0.00096 0.00002 0.99902 9 0.00190 0.00005 0.99805 8 0.00375 0.00015 0.99609 7 0.00736 0.00046 0.99219 6 0.01425 0.00137 0.98438 5 0.02713 0.00412 0.96875 4 0.05015 0.01235 0.93750 3 0.08796 0.03704 0.87500 2 0.13888 0.11111 0.75000 1 0.16666 0.33333 0.50000