Updates to Picker

Based on the math feedback from Mark Sherry, I made some changes to the core rules of Picker. Mainly this involved resolving this strange bug where being a Picker was actually more detrimental to each player's score. Based on the revised rules, the simulations show that being a Picker gives you a slight percentile advantage, which better fits the spirit of the game.

Here are the basic changes:

• The dice are different colors, which are now the basis of set bonuses, independent of the results.
• There is no more SHAFT role. It was actually a greater detriment to all AIs and led to lower final scores.
• The PICKER role always picks herself to be first player. Again, this balanced out the anti-picker bug a bit and streamlined the rules.
• The set bonus is streamlined to a linear +3, with a maximum of +15 for a set of 6-of-a-kind.
• Clarified that you can score bonus points for multiple sets, so long as each are a different color.

I've updated the main page with these new rules. Test it out and see what you think! The two-player game still doesn't feel substantial enough. How would you tweak it?

Analyzing the Math behind Picker

Whenever I have a math question, I call upon resident math whiz Mark Sherry for analysis. Seems like a lot of game designers have some background in advanced math, but I ended up taking the art path, so I'm very lucky to have Mark around.  I asked Mark to run some simulations for Picker, the pub dice game I posted last week.

Mark set up some AI players who each stick to one PICK strategy and one CHOOSE strategy throughout the game. Then he ran thousands of simulations to find out which strategy, if any, is the optimum strategy. If it was clear that one strategy would win, I'd tweak the rules, he'd adjust his AI, and run the simulations again. It's a nice, fast process.

PICK is who they pick to be first player when they're Picker. Mark tested these strategies.

• Always pick the trailing player
• Always pick yourself
• Pick random player

CHOOSE is what they do when its their turn to choose a die. Mark tested these strategies.

• Always keep the highest
• Always keep the lowest
• Never keep the highest
• Never keep the lowest
• Never keep the highest or lowest
• Pick random

Always choosing the minimum result was so obviously a bad strategy, that Mark removed that AI almost immediately. Mark coded four AIs with the following strategies.

• Alice picks a random player and chooses the highest die.
• Bob picks the player to his right and chooses the highest die.
• Carol picks a random player and chooses the highest result that won't make her the Shaft.
• Doug picks the player to his right and chooses the highest result that won't make him the Shaft. 

Here's how often each player came in first, second, third and fourth place.

        1st Place    2nd Place    3rd Place    4th Place
Alice   3105         2923         2766         1206
Carol   1452         2106         2615         3827
Doug    1709         2195         2380         3716
Bob     3734         2776         2239         1251

Clearly Alice and Bob's strategies are dominant. Choosing the highest die, even if it makes you the Shaft, is pretty powerful. We kept tweaking, tweaking, tweaking. I ended up deciding on the following rule changes.

• Game should be longer: 2 rounds per player. (A 4 player game goes 8 rounds.)
• Each d6 in the game is unique. (Maybe players bringing their own die?)
• For the sake of simulation, we assume the dice are different colors.
• You get a set bonus when you keep the same die color, not result.
• Bonuses are as follows: 0, 1, 2, 3, 6, 8, 10, 11, 12, 12 repeating.

The hope was that decoupling sets from results would create a few more options for play. Mark coded two new AIs and added them to the simulations.

• Eve picks a random player and chooses the color that builds her set.
• Frank picks the player to his right and chooses the color that builds his set.

And he ran the new numbers.

        1st     2nd     3rd     4th     5th     6th
Frank   21.38%  24.06%  20.72%  16.37%  11.25%  6.22%
Bob     16.48%  18.84%  20.49%  19.81%  14.79%  9.59%
Carol   17.07%  13.98%  14.76%  14.94%  18.54%  20.71%
Eve     13.34%  13.98%  15.50%  16.31%  18.88%  21.99%
Doug    21.53%  17.31%  14.87%  15.25%  14.89%  16.15%
Alice   10.20%  11.83%  13.66%  17.32%  21.65%  25.34%

Promising! But there is  a mystery here. Mark observes that, in theory, Frank and Eve pick the die that maximizes their score increase for that turn. Yet Frank comes in 2nd place and Eve comes in 5th place. Who knows why? Any math whizzes have some ideas?

EDIT: Mark notes in the comments below that I missed mentioning a bug he found in those last results. He adds the following simulations and results:

"To make things fairer for the other players, I added two new algorithms. They're both based off of the one Eve uses, but one will give a negative weighting of 1 point to getting the shaft, while the other gives a positive weighting of 1 to becoming picker.

The scores are closer. Picking the prior player instead of a random player when picking gives a 1.5-2% improvement. Using Eve and Frank as a baseline (algorithms unchanged), trying to avoid being Shaft lowers win percentage by about 2 points. Preferring becoming Picker lowers it by 3-5% (pick strategy becomes much more important.)

Here's the (probably poorly formatted) table:

Alice 16.05% 15.79% 15.85% 17.31% 17.07% 17.93%
Bob   17.67% 17.95% 18.21% 17.12% 16.25% 12.80%
Carol 12.92% 12.98% 14.02% 15.58% 17.85% 26.65%
Doug  15.84% 16.59% 17.08% 17.78% 17.44% 15.27%
Eve   18.27% 17.27% 16.77% 15.93% 15.96% 15.80%
Frank 19.25% 19.42% 18.07% 16.28% 15.43% 11.55%

Pick strategies alternate: pick random, or pick the player before you in the sequence.
Choose strategies are: Avoid shaft (Alice & Bob), Prefer pick (Carol & Doug), No preference (Eve & Frank)

As with the table Daniel posted above, the first column lists the percentage of time that that player comes in first, etc.

Given that 'pick random' is clearly worse, I think a good replacement would be 'pick player (other than one's self or shaft) with the lowest score', either with ties being broken by whichever gets you your turn sooner, or by weighting the selection somehow."

Picker - A Simple Pub Dice Game

Here's a super simple pub game to play with normal dice, inspired by the customized dice drafting mechanic in Seasons. It's so simple, in fact, that I could fit all the rules onto one graphic. Neat, eh?

3 or more Players
5-10 Minutes


EACH ROUND SETUP
Roll 1d6 per player. Each die is a different color.

Resolving Ties: If all the dice come up with the same result, roll them all again until there is at least one different result.

In the first round, youngest player goes first. Thereafter, each round's turn order changes.


ON YOUR TURN
Keep one result. Score that many points. Keep track of your chosen color each round.

If you are first to keep the lowest result, you will be first player in the next round.

Clarification:  If there are ties for the lowest result, only the first player to keep that result gets this effect. For example, if there are two 1s, and you keep a 1 first, then you're the Picker. Whoever keeps the second 1 gets no benefit.

Turns proceed clockwise for the rest of the round.


ENDGAME
Play 2 rounds per player.

You will score bonus points if you keep the same color across multiple rounds.

2: +3
3: +6
4: +9
5: +12
or 6: +15 (max)

Of course, you may continue keeping dice of the same color after the sixth, but you won't get any further benefit.

You may score several sets if each are a different color. For example, if you kept the red die three times and yellow die five times, you’d earn 18 bonus points total.


VICTORY
The player with the most points wins!