Strategies for playing the dice game ‘Toss Up’

1. Strategies for playing the dice game ‘Toss Up’ Roger Johnson South Dakota School of Mines & Technology April 2012

2. ‘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

4. 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

5. 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

6. 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:

7. Positive Expected Increase Strategy

8. Positive Expected Increase Strategy • Empirical game length with this strategy (100,00 trials): Average = 11.92, Standard Deviation = 1.50

9. 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

10. Expected Values Recursions

11. Expected Values Recursions

12. Solving the Recursion • Have • Used

13. 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

14. 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

15. Rough Approximation of Optimal Solution

16. Empirical Results

18. 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).

19. Questions?

20. Chances of Various Outcomes