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) ( tossup.asptossup.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:

Positive Expected Increase Strategy # Dice Being TossedContinue rolling... 1, 2Never! 3... when turn score < when turn score < when turn score < 93 6, 7, 8, 9, 10Always!

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

Expected Values Recursions

Solving the Recursion Have Used

Minimal Expected Value 7.76 turns as opposed to about 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 research.html

Rough Approximation of Optimal Solution # Dice Being TossedExpected Increase Strategy: Continue Rolling when… Rough Approx of Optimal Strategy: Continue Rolling when… 1,2Never!…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 < 93Always! 6,7,8,9,10Always!

Empirical Results Positive Expected Increase Strategy Rough Approximation of Optimal Strategy Average (Optimal mean = 7.76) Standard Deviation 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), Neller, T. and Presser (2004), “Optimal play of the dice game Pig”, The UMAP Journal, 25, (c.f. Tijms, H. (2007), “Dice games and stochastic dynamic programming”, Morfismos, 11(1), 1-14 ( 1n1/tij.pdf). 1n1/tij.pdf

Questions?

Chances of Various Outcomes