Friday, 18 January 2013

Guess Who? Beginnger Strategy Analysis

My daughter has recently taken a liking to hauling out all of my old board games. The favourite so far is Guess Who? Due to my consistently being demolished the majority of the time and having victory snatched away from me with one remaining piece by 1/8 Hail Mary chances many more than 1 out of 8 times, I decided to look at the numbers to try to gain an edge.

I'm choosing to ignore all of the potential meta-game, question combining and deceptive game play strategies available. She is not yet 5 years old after all. But I thought there must be some statistical edge to gain through the simple, common questions.

The Setup

  • 24 Characters.
  • 9 Common traits: Bald, Bearded, Eye Colour, Gender, Glasses, Hair Colour ,Hats, Moustachioed, and Nose Size.
  • Traits are split relatively evenly at 19:5: 5 with glasses, 19 without glasses, 5 females, 19 males, 5 with large noses, 19 with small noses.
  • There are two exceptions: Only 4 bearded characters and 4 with brown hair.
  • Traits are split unevenly between characters, ranging from 1 to 4 uncommon traits each.
  • Each player chooses one character at random out of a deck of cards.
  • *Note I am also ignoring the fact that, according to 4 year olds "Oh, I/you already had this one! That's silly! [draws a new card]." Also, some hair is distinctly yellow, not blonde.
  • The game proceeds by a process of elimination.

The result of inputting all of this data into a spreadsheet cross-referenced by character quickly shows that there is a sum of 63 uncommon traits among the group: (5 each among 7 of the non-hair groups (35), 4 of one (4), 4 hair colours of 5 each (20) and one hair colour of 4(4). Thus we can already determine that the first guess (assuming we always get a 'no') will yield 5 eliminations for every guess except questions about beards and brown hair which will result in 4 eliminations.

Keep in mind that your character will skew the results, essentially acting as blockers. Therefore we should avoid guessing character traits that our character possesses. For example, if our character has black hair and we ask if our opponent has black hair, we are only eliminating 4 characters in practice (again, ignoring the meta-game of optimal elimination vs bluffing strategies employed by experienced players).

However -- and this is where the true genius of Milton Bradley's board game creation ability shines by balancing luck and skill -- there are small, yet distinct statistical advantages to gain in the combination of your first two guesses.

Top 10 Ranked Questions:

Worst 10 Ranked Questions:

It becomes immediately obvious that it's not a good idea to ask first questions about beards or brown hair or the traits our character has blocked as they are all under represented, as well as second questions where the first eliminated some of those traits already, such as gender and hats, as a 'no' to female already eliminates two hats (Maria and Claire).

For the sake of being meticulous, lets look at an in practice example. You draw George with uncommon traits of white hair and a hat. Theoretically, if all cards were in the deck, asking about hats and white hair would yield a 24/9 result. However, in practice you have already eliminated 2 traits and 1 character from the group as blockers and have a true result of 22/8. This makes asking other 24/9 questions such as glasses and black hair superior to the question involving traits you own. Likewise, this drops starting questions about baldness and hats off the top of the starting list.

Understanding the best starting questions is a little more difficult as it's hard to know at a glance which combination of traits applies to the largest group of characters, especially as the game progresses into the 3rd and 4th turns. Therefore the best thought process seems to be thinking through and eliminating your worst options while memorizing the top five or ten theoretical guesses until you have a better understanding of the top starting guesses versus your character blockers.

I'm hoping this analysis will give me and all of the readers out there with unscrupulous pre-school opponents the edge we need to win. Good luck.

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