Analyzing My Favorite Game

I’ve discussed my favorite game before, which is played as follows:

• On a piece of paper, everyone playing writes down a positive integer.
• Show your number to a neighbor (for verification purposes only).
• The winner is the person who wrote down the smallest integer not written by anyone else.

I recently used this game with a group of 32 people at the end of a presentation. The first round was a sample round only, and folks didn’t know the rule for determining a winner before choosing their number. (People often find the rule confusing, so I often do a first round where I don’t tell folks the rule until after everyone has written down a number. Then I give the rule, and we determine the winner to provide an example of the rules in action.)

We played six rounds. I gave the winner of the third and sixth rounds a copy of Math Jokes 4 Mathy Folks; the other rounds were just for fun.

I’ve always been curious about the strategy that folks use when playing this game, so I asked folks to record their numbers for each round, and then I collected their choices. Geek that I am, I analyzed the results, and I thought I’d share them with you. (Don’t you feel special?)

In Round 1, choices were all over the charts. This is to be expected, since folks had no idea why they were choosing a random number. Choices ranged from 3 to 99, with a mean of 15.9.

After the rules were revealed, though, things got more interesting. The charts below show the choices during Rounds 2‑6. (Horizontal axis is the number chosen; vertical axis is the number of attendees who chose the number.)

Some observations about these results:

• The maximum number chosen by any player decreased in each round.
• The average value chosen for Rounds 2‑6 was 8.5, 7.5, 5.3, 7.7, and 6.4, respectively. It’s interesting that the average decreased to 5.3, then shot back up to 7.7. This might be explained by the trend in winning values. The winner in Rounds 2 and 3 chose 2. In Round 4, at least three players chose each of the numbers 1‑4, and the winner chose 5. Players may have assumed that too many players were tending to choose low numbers, so they chose slightly higher numbers in Round 5.
• No fewer than five players chose the number 7 in every round.
• Interestingly, the winner who chose 1 in Round 5 was also the winner who chose 1 in Round 6. She went to the well twice — and it paid off!

A few days later, I ran the same experiment with a different group of 35 people. The results were slightly different.

• The average value chosen decreased in every round, as follows: Round 1, 11.96; Round 2, 7.54; Round 3, 7.53; Round 3, 4.88; Round 4, 4.84; and Round 6, 4.70. As with the previous group, the average took a big dip from Round 3 to Round 4. However, unlike the previous group, the average did not shoot back up in Round 5.
• No fewer than four players chose the number 4 in every round, and it was chosen by 10 players in two different rounds.
• Even after learning the rules by playing a practice round in Round 1, several folks chose surprising numbers in Round 2. Among them: 100; 102; 1,000; 1,900; and 10¹⁰⁰.

I hope you enjoyed this diversion. We now return you to our regularly scheduled programming…