The centipede game

Figure 1: The 4-year-old drew a centipede!

Yesterday we learned a bit about game theory. A lot of fun was had, but we should warn you that this also caused a few arguments. There was also a lot of learning!

For this activity you will need:

• Two children
• Two cups
• 20 small things to use as tokens that fit into the cups. We used the tokens from a Connect 4 game.
• A piece of paper for recording results and scores
• A bag of very small treats to use as incentives. We used m&ms. 

Explaining the Centipede Game

Begin by explaining to your children that they will be playing a game for points. You will play the game over and over again (it is quick, we played it about 20 times in total). 

Tell them that for every (say) 5 points they earn, they will get one incentive (for us, one m&m). With this incentive scheme, our kids ended up getting about 25 m&ms each. For us, we're OK with that. If you want them to end up with (say) about 12 m&ms each, give them one m&m for every 10 points they earn.

Make sure they understand this, and make sure you follow through on the promise at the end! Experimental economists never lie to their participants, and if you want to do more of our game theory activities, it is important that your children believe that they will receive the incentives you promise!

Now it is time to explain the game to your children. We ran them through all the stages, and did a few practice rounds. Here's how you play:

Choose one child to take the first turn

1. Place three tokens in one cup (the blue cup in our diagram) and one token in the other cup (the red cup).
The first player can either take the blue cup, or pass it to the second player.

• If they take it, then the first player gets three points, the second player gets one point, and the game ends.
• If they pass it, then we move on to the second stage

2. Place another three tokens in the blue cup, and another one token in the red cup. There are now six tokens in the blue cup, and two in the red cup.
The second player can either take the blue cup, or pass it back to the first player.

• If they take it, then the second player gets six points, the first player gets two point, and the game ends.
• If they pass it, then we move on to the third stage

3. Place another three tokens in the blue cup, and another one token in the red cup. There are now nine tokens in the blue cup, and three in the red cup.
The first player can either take the blue cup, or pass it back to the second player.

• If they take it, then the first player gets nine points, the second player gets three point, and the game ends.
• If they pass it, then we move on to the second stage

4. Place another three tokens in the blue cup, and another one token in the red cup. There are now twelve tokens in the blue cup, and four in the red cup.
The second player can either take the blue cup, or pass. This bit is different, so make sure your children know what happens here:

• If they take the blue cup, the second player gets twelve points, and the first player gets four points.
• If they pass, they both get ten points.

This is the end of the game. Once our littles understood what they were doing, it took less than a minute to play.

Playing the game and recording data

Play the game about ten times, swapping who goes first each time. Record the result of the game, and how many points each child has. Note that there are five possible outcomes of the game:

1. Player 1 takes the cup in the first round. Player 1 gets three points, and Player 2 gets one point.
2. Player 2 takes the cup in the second round. Player 1 gets two points, and Player 2 gets six points.
3. Player 1 takes the cup in the third round. Player 1 gets nine points, and Player 2 gets three points.
4. Player 2 takes the cup in the fourth round. Player 1 gets four points, and Player 2 gets six twelve points.
5. Player 2 passes the cup in the fourth round. Both players get ten points.

Here is what our data look like:

That is, we played the game 11 times. The game stopped in the first round eight times, and stopped in the 2nd, 3rd, and at the end once each.

Now tell your children that they will play for another 10 rounds (or however many you choose), but with a slight twist.

Tell them that if the second player passes in the fourth round, they both get fifteen points (rather than ten). This change happens at the place highlighted in yellow above. Make sure they understand how the rules have changed, and then go ahead and play!

Here is what our data look like for both games:

Understanding what happened

Wow, that changed a lot! Note that we played both games 11 times, but got all the way to the end of the game much more often in the second game. In fact, we got to the end of the second game three of our last four rounds of play.

While you might get slightly different results, it is very likely that you will also see something like this. How can we know this, even before your kids play the game? 

For the first game, put yourself in the shoes of the second player, and suppose that you have made it to the fourth round of the game. If all you care about is your own points, then you have the following decision:

• Take the blue cup, and earn twelve points
• Pass, and earn ten points

You would probably take the cup to get twelve points. That's not so good for your sibling, but if m&ms are on the line, it might seem like a good choice.

Now put yourself in the shoes of the first player, making a decision in round 3. You know that your sibling will take the blue cup in the next round, so you will be thinking:

• If I take the blue cup now, I will get nine points.
• If I pass the blue cup, my sibling will take it in the next round, and so I will get four points.

And so if it ever gets to the third round, you will take the blue cup. In fact, this unravels all the way to the first round, and so the first player will know that taking the blue cup in the first round is their best option. Our children learned this very quickly, and this was the source of the arguments!

For the second version of the game, the second player can get fifteen points by not passing in the last round, compared to twelve points if they take the blue cup. If the players trust each other enough to not take in the first few rounds, then getting to the end is much easier in this version of the game.

In fact, our children worked this out very quickly, too. To paraphrase our 8-year-old:

Oh my goodness, little brother! If you keep taking in the second round, then I will have to take in the first round, and we will never get the fifteen points!

Do you want to share your data?

At Reluctant Homeschooling Economists, we are all for open science! If you would like to share your data, here is a form you can fill out. Maybe we will do a post using all of your data later.

For reference, this is how we entered our data into the form.