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.

Sampling Variation

Our tiny humans are well motivated by sugar, so today we are going look at some issues related to sampling variation, motivated by a sample....err ...snack.

For whom:This activity is good for kiddos learning how to sort and count, up to late elementary kids who are data savvy and don't look at you funny when you say "make me a histogram".

Supplies: 
For this lesson you will need something multicolored to sort: This might be something like a large bag of MnMs candies (or skittles) and a tablespoon,  4 snack pack sized fun bags, or even multi colored beads in a craft stash. (For the record, our home is light on craft stashes. Also, I would be lying if I said these weren't left over from Halloween, so feel free to judge!)

What to do: 

Start with one small bag of candy (or one spoonful/ scoop). Sort the candy by color. Draw a histogram(pictured below) to represent each color. As a pro-tip, if you use lined paper, it is easier for younger kids to have one line represent one candy and keep their columns to scale. In our first sample, you can see there were 4 oranges, 4 greens, one yellow, one red, one brown, and three blues (though one of the blue MnMs is hidden from this angle). 

Some things to point out/ questions to ask: 
Are the colors all represented the equally? Why not? If we opened another bag, would we get the same mix of colors? (Note the foreshadowing here!)

Now, try it again for your second bag:

I recommend making your next histogram above your first one and again use the lines to help keep your scale consistent.

So when we compare our first and second sample, in our second sample we got SO MANY yellows, but not a single brown or red! Spurred on by our variation, we pressed on, opening two more mini bags, and recording our sample a la histogram. After the samples were faithfully documented there was some rebellion in the ranks and the 4 year old retorted "I'm eating my sample so I don't contaminate my next one!"  Halloween candy in March. What could go wrong?

So...what do we see in our data? There is a tremendous amount of variability in the colors that occur in any one pouch. Some pouches don't even *have* all the colors. Some have a ton of one color and a scant few of another.
We decided to make a 'totals' graph for all the MnMs in all 4 of our bags together.

Based on our four bag sample, it's a little hard to believe that they make those colors in equal number. Maybe they don't! If you try this at home, send us a picture of yours in the comments so we can collect more data!


So far, I have shown you the data for MnMs, but, for the sake of science, we repeated the experiment with Skittles. You're welcome. It seems like skittles are far more committed to you 'tasting the rainbow' than MnMs. As you can see in the photo on the right, over a three bag sample, we see pretty consistent numbers of each color. If you are trying to show how samples converge to underlying parameters, I might recommend using skittles, but if you are #teamchocolate, just be prepared for more conversation about sampling!

The big idea of the lesson: Samples are different than their underlying populations. When we look at an individual bag (or tablespoon) of the candies, we see a different pattern of colors when we look at all the candies together. Sometimes colors are evenly distributed and sometimes not, but the larger our sample (or more little samples we put together) the closer we get to the 'true' underlying population.

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.

Probability and statistics: Dropping things onto a target

Today we decided to teach our littles some probability and statistics.

We tested this on our two children, who are aged 4 and 8. The 4-year-old was able to participate in the experiment, but (not too surprisingly) couldn't do the multiplication required to make the predictions. The 8-year-old could do the math. Both said they wanted to do something like this tomorrow.

For this activity you will need:

1. Something high up for your budding statisticians to stand on and drop things off. We used the top of our stairs.
2. Nine pieces of paper, all the same size
3. A lot of soft balls to drop from the high place onto the pieces of paper. E.g. scrunched up pieces of paper

Setup

1. Mark one of the nine pieces of paper with an O
2. Lay out the nine pieces of paper in a rectangle on the floor below your dropping place, like this:
   
   

   	O

Ours actually ended up looking like this, and we plan to use it for other activities:

The numbers are not important for this activity, but it sure did let the kiddos get excited and yell "ooh, I got a five!"

Taping it to some cardboard made it hold together better, but as long as your learners have a target to aim at (i.e. the circle with the 5 in it), you're all good to go.

We scrunched up some pieces of packing paper to make the soft balls. We made about 15 of them, and put them in a box.

Before the experiment

Tell your learner that they will be dropping balls onto the piece of paper marked O. Have them take a few practice drops to understand what they will be doing.

If you think the task is either too easy or too hard for them, move the target. We started our practice drops at the top of the stairs (aiming at the floor below), and found that our kids (4 and 8) were not hitting any of the pages most of the time. No problem, we had them move down the stairs a bit, and dropped over the side of the stairs. For those of you that don't have stairs like this, you could always have them stand on a chair, and make the target smaller (for example, cut the pages in half before assembling the target) if they are easily hitting the middle rectangle each time. Basically, it should be a bit of a challenge for them to hit it, but you want them to be successful often enough that it is fun.

If they do this 50 times, how many times do they think that:

1. The ball lands on the paper marked with an O
2. The ball lands on one of the other pieces of paper
3. The ball does not land on any piece of paper

By "lands on the paper", we mean that we are interested in the first part of the target it hits. If it (say) hits one page, and then skids on to another, you want to count this as hitting the first page, and not the second. 

Note that these are reasonably subjective questions, but some things to think about are:

• The paper marked with O is the thing that they are aiming at
• There is only one page marked with an O, and eight without and O. If your learner's aim is spread out enough (i.e. on average your learner is a good dropper, but there is a lot of variation), then the ball should land on each of the nine pieces of paper about the same number of times. Therefore, the ball should land on the pages on the outside of the target eight times as often as it lands on the one marked with an O. That is, if the ball landed on the O page 5 times, then we would expect that the ball landed on other pages 5x8 = 40 times
• If their aim is better than this, then maybe the ball lands on the blank pages less than eight times as often than the O page. That is, if the ball landed on the O page 5 times, then we would expect that the ball landed on the blank pages less than 40 times

Running the experiment

Have your learner drop the balls many times (say, 50) from the same height (as economists, we are all in favor of providing some incentives for doing this, i.e. one m&m for every 5 successful drops). Have them record where the ball lands. Note that there are three possible outcomes:

1. Lands on the page marked with an O
2. Lands on one of the other pages
3. Misses all of the pages

Here is what our data look like:

That is, by the end of the experiment, our littles had hit the middle of the target 12 times, missed the middle but hit one of the other pages 26 times, and missed the target entirely 16 times.

Each row of this table before the total involved our kids throwing all of our balls off the stairs. When all balls were son the ground, they went down the stairs to pick them up, then started throwing again.

After the fourth iteration, the 4-year-old was trying to do trick shots off the wall (which I recorded as a miss), so it was a good time to stop.

Analyzing the data

Test your learner's predictions they made before the experiment against the results of the experiment. In particular, is the number of times the ball lands on one of the outside pages close to eight times the number of times it lands on the O page?

For our experiment, we had 12 balls land on the O page. If our learners were not particularly accurate, that means we would have expected 12x8 = 96 balls to land on the outside pages. In fact, only 26 did, and so from this we can conclude that our learners were more accurate than just spraying the target would predict.

Pi day!

Hello friends,
March 14th (3/14) is sometimes celebrated by us nerdy folks as pi day (since pi is 3.14159....). True enthusiasts like to be doing an activity at 1:59 to get the next three digits--3/14 1:59. If you are a purist and attempt this on pi day, I recommend 1:59 p.m. But if you can't contain your excitement and are striving for 1:59 a.m.,  I salute you! (From my bed, sleeping:)

Pi is pretty awesome as it is the ratio of the circumference (all the way around a circle) to the diameter (across a circle). The thing that is so darn neat is that this is true for big and small circles alike!

So for this activity you will need: 
A big piece of paper (newspaper or construction paper is great!)
A marker or pen
A few items that are round. I chose to use some different sized can goods (corn and a small can of tomatoes), a quarter,  and a bowl.

What to do: 
On your big piece of paper in the lower left corner draw a big L:

Label the short side of your paper diameter (that's right across the circle--don't worry, I will show you a picture in a minute). Label the long side of the 'L' circumference (that's all the way around the circle).

Now, grab your first circular item! Putting one side that the corner of the 'L' (we call that the origin in math-talk), measure across the widest point of your circle and mark it with a pen.

Next, flip your round object on it's side and roll it up the axis. (Pro tip! make a little mark on your object so you can see when you have rolled it exactly once. Here I used the label on the spice jar)

I know it's a little hard to see, but I rolled the side of the spice jar up the long side of the 'L'. For the next picture, tip your head to the left, you are rolling up the page.

Now make a dot at the intersection of the two measurements you made, like this:

Repeat this as many times as you like for as many circular objects as you can find or until you get tired or need to eat.  The dots will all fall on the same line! The steepness of the line (called the slope in math-talk) is the magical, never-ending number pi!

What's so cool about this activity is it doesn't matter if you use big or small circles, the ratio of the diameter to the circumference is always the same! That's why all the dots fit on the same line.

Want to learn some of the digits of pi? Here is a song to help you learn! And, apologies parents, I find this spectacular. https://www.youtube.com/watch?v=AtgfsRH3H3U

Interested in some more digits of pi? Lets listen to what pi would sound like if each digit were assigned a musical key:
listen here: https://www.youtube.com/watch?v=SNpab2PDquw

This one shows you the piano and other instruments, which I kinda like:
https://www.youtube.com/watch?v=wK7tq7L0N8E

And if you are still interested, you can find even more here! There are a bunch of different audio files on this page as well of a 5 minute explanation of pi and making music out of it:  https://uh.edu/engines/epi3030.htm  You need to scroll way down the page.
Warning, the first audio file link isn't working, and 'audio 2' is linked above. 


Thanks for learning about pi and we hope to see you again soon!

Welcome to reluctant homeschooling economists

Hi!
Welcome to our reluctant homeschooling blog. We are economists at Bowling Green State University and University of Toledo. We are in the unfamiliar situation of homeschooling our kiddos (aged 4 and 8) on very little notice due to COVID-19. However, we are super nerdy and love thinking about education, so if you are cooped up with your little people we will provide some in-home activities that you might like to try with your kiddos.

We will do our best to highlight what resources are needed at the start of each lesson and if we are really on top of our game, we will link the national learning standards, too, though if those make it in, we will consider it an A+ effort.

Happy learning!
-Amanda & James