Take the last token game

Time for some more game theory! Our kids found this super interesting, and we actually had a lot of fun seeing them work it out.

For this activity you will need:

• Six tokens. We used some Connect 4 tokens, but anything will do
• Something small to incentivize your learners (there is very little our children will not do for a skittle)
• Two children

Playing the game

Place all six tokens in a pile. Players take turns removing either one or two tokens (they can choose) from the pile. The player who takes the last token (either on its own or by taking the last two tokens) wins.

That's it!

Explain it to your children, then have them play it over and over again, swapping who goes first. Reward the winner of each game with one of your incentives. Actually, I don't think our children needed the incentives, but as an experimental economist, I couldn't bring myself to run this un-incentivized.

Understanding the game

After a while, someone will probably work out that if it is their turn when there are three tokens left, they will lose. For us, this was marked by loud exclamation by our four-year-old:

No fair! Either way [my brother] is going to win! 

This is a great time to have a chat with your players about why this is the case. Note that if it is my turn and there are three tokens left in the pile:

• If I take one token, there are two left. My opponent will take both of these in the next turn, and so I will lose
• If I take two tokens, then there will be one left for my opponent. Again, I lose.

Get them to play for a bit longer. Once our understood that having three tokens in the pile was bad, it didn't take long for them to realize that they could win for sure if there were four or five token in the pile on their turn. That is:

• If there are four tokens, I take one away.
• If there are five tokens, I take two away.

Either way, my opponent has three tokens left on their turn, and so they must lose (and I win).

Soon after they realized this, it didn't take them that long to realize that if they were the first mover, then they will lose. This is because the second mover will always have five or six tokens in the pile, and we've already worked out that they can win for sure from that position.

Nice job! Your children have found the Nash equilibrium of this game! Actually, they've found the subgame perfect Nash equilibrium, which is a special kind of Nash equilibrium.

Want to do some more?

This is a very simple version of the game Nim. Some interesting extensions to the one they just played could be to work out:

• What happens if you start with seven tokens (instead of six)?
• What happens if the player who removes the last token loses, rather than wins?

Both of these questions can be answered using the same kind of reasoning, called "backward induction", that we used to work out what would happen in the original game.