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