The Positive Sum Game for third graders

The Game Room
1

I’m teaching a group of mid-to-upper-level math students in third grade. I’ve got them for about half an hour a day, and we’re doing a unit now on game theory. Very basic stuff: what’s a solved game? How do you discover a solution to a simple game? (Example game: start with ten objects, take turns removing one or two objects, the person who removes the last object wins) How do you write a strategy for a game?

It occurred to me yesterday that all the games I’m showing them are zero-sum games: one winner, one loser. It’d be fun to teach a positive-sum game. Can y’all critique my idea, or better yet offer improvements?

Materials:
Five cards per player, printed with one of the following words:
-Eggs
-Flour
-Sugar
-Chocolate
-Butter

Setup:
-Each player gets five matching cards (all butter, or all sugar, etc.)

Play:

  1. Each card is worth 1 point. Determine how many points are in the room by having each player report his hand’s current value (i.e., 5); since there are 22 kids in the class, that’s value of 110 for the room.
  2. Post some recipes, e.g., Sugar+Butter=Frosting, or Eggs+Butter=Omelette. The ultimate recipe uses all five ingredients to make brownies.
  3. Post values for the recipes equal to the square of the number of ingredients(so frosting and omelettes earn 4 points; a brownie equals 25 points).
  4. Tell students that there are the following rules for the game: no lying to each other, no stealing from each other, no breaking a deal with each other, and no deliberately sacrificing yourself to another player. Other than that, they should play to get the highest score for their hand that they can.
  5. Set them loose for ten minutes, or until they wind down.

By the end of the game, if they’ve played perfectly, the whole-class score should go from 110 to 550, and we can examine how that happened, and how that reflects real-world positive-sum games such as free trade.

Thoughts? Any twists on the game would be great, but I don’t want it to be significantly harder either to set up or to play.

2

Game theory for third graders? Interesting to say the least.
I have no suggestions but I’m keeping an eye on this to see how it goes…

3

I think that’s pretty good - trade is the obvious example of positive sum exchanges, and the principles of the game are very straightforward. (It’s very reminiscent of Pit, except for the minor detail of Pit being zero-sum.)

I think what might make it better would be if the players didn’t all start from the same position. You probably wouldn’t want to hand out combos that contain complete recipes, but having some start with, e.g. flour and sugar only would a) make it easier for them to start trading by giving them obvious “surplus” and b) mean that different players start with different goals, (“I only need eggs and I’ve got an omelette”) and gradually work round to realising they can go for the full 25 points.

On the ruleset, you don’t explicitly say this is a trading game (obviously, heavily implied in rule 4). Do you want people to trade blindly or openly? (E.g. three cards for three cards, or my eggs for your flour). Are there any limits to the number of cards you can trade in one go? I can guess you want unlimited open trading, but it’s not explicit.
(Also - and I know zip about the practicalities of teaching anyone, still less 9(?) year olds - the idea of 22 of them trying to simultaneously trade with 21 others seems like a recipe for anarchy!)

4

Unless the number of students is a multiple of 5, you cannot get a perfect score if everybody starts with a 5-point hand, as the number of cards of each ingredient must be a multiple of 5, so you can’t have, in your case, 22 of everything.

I have a gut feeling this is going to turn into a popularity contest of some sort, where the popular kids in the class gang together to make sure they get 25 points each.

This reminds me of a game theory exercise (although it wasn’t presented as such, and probably wasn’t intended that way either) I had in sixth grade back in 1974:
There are N groups (in my case, 6)
Each group secretly votes “red” or “blue”
There is a 5-minute “discussion” period before the vote; the only communication allowed is to write things on a chalkboard
If everybody votes blue, everybody gains N-1 points
If everybody votes red, everybody loses N-1 points
If it is a split, everybody who voted red gains (2 x blue votes) points, and everybody who voted blue gains (2 x red votes) points

What happened was…In pretty much every group, somebody realized, “If we vote red, we can’t possibly do worse than ending up tied for the most points”; in the multiple rounds played, every group but one voted red every time. Since there was no punishment for finishing with a negative score, how well you did compared to everybody else seemed to be the only thing that mattered.
What game theory says to do is…[SPOILER]always vote red.

If everybody else votes red, you lose N-1 points for voting red, but you lose 2(N-1) points for voting blue, so red is better.
If everybody else votes blue, you gain N-1 points for voting blue, but you gain 2(N-1) points for voting red, so red is better.
If the others split, you gain points for voting red, but you lose points for voting blue, so red is better.[/SPOILER]

5

Thanks for the feedback!
Stanislaus, you’re right that I expect open unlimited trades and don’t make that explicit. I don’t want to tell them to trade, but probably I ought to have a quick strategizing session right before play begins, in which someone will certainly point out the option to trade. And honestly if someone wants to trade blindly, I see no reason to prevent it, but afterwards we can discuss whether that was an effective strategy. And yes, it was definitely inspired by Pit :).

That Don Guy, you’re right about the no-perfect-score logistics. I was trying to do this in my head, but I obviously should’ve written it down. I definitely think that’s okay, however, and was actually considering a variant: instead of distributing equal numbers of all cards, I was thinking of making it something like 6 hands of eggs, 6 hands of flour, 4 hands of butter, 4 hands of sugar, and 2 hands of chocolate. I’d give the chocolate to my sneakiest students and hope they’d realize that the scarcity of their product increased the price: they could convince someone to give up two or even three cards in exchange for a chocolate, perhaps. In order to make that work, I might have to give some different values to recipes such that 4-card recipes containing chocolate are more valuable than other 4-card recipes. I dunno.

I don’t think a popularity contest would really happen, though. The skills involved in making good trades don’t necessarily correspond to the skills involved in being popular, and in the game unit so far, the kids who consistently solve the games I offer aren’t necessarily the best-liked kids in the group.

As for your red vs. blue game in sixth grade, I’ve conducted a version of the Prisoner’s Dilemma with my classes for a number of years: I call it The Sharing Game, and it’s been very useful at discussing why treating people with kindness is a real-world winning strategy, not just an arbitrary school rule. (Read that writeup before you tell me about the Prisoner’s Dilemma’s normal implications).

6

Random thoughts –

Is the goal to maximize global utility or to “win” as an individual? These two goals lead to very different trading strategies, and if people aren’t on the same page goal-wise, it could introduce tension. (“No, dummy, it’s better to do Trade X!” “Nuh-uh, that gives you more points!”)

In fact, score optimization becomes an interesting topic in itself, as it has a different flavor in a 2-player game than a multi-player game, and it has yet another flavor entirely in cooperative scenarios. In any case, the overall goal should be clear. (On a re-read, I’m guessing the goal is to maximize the total classroom score.)

Will there be a chance to practice hand evaluation? It will probably not be obvious to most third graders what a good goal should be for their hand once some trades have happened (or if they start with jumbled sets). For instance, will they have a chance to discover that two triplets scores better than a single quad? Will they have a chance to think about the recipe “tree” to see which recipes are dead-ends and which offer opportunities to trade-up to better hands?

(Toward the last point, it might be good to have several cases where a recipe is a proper subset of another so that one can trade something away and still have something left.)

Part of me thinks the game is too complicated if the pedagogical aim is to deconstruct optimal and/or non-exploitable and/or exploitative strategies. However, part of me thinks the complexity is good, since there is plenty to talk about, and the richness of the game can be conveyed. So this paragraph is useless. :slight_smile:

Aside: if the goal is to maximze the global score, would it be a problem for you if a leader emerged and said, “Hey, guys. Just gimme all your cards, and I’ll sort them and hand them back as 25-point sets.”? Must trades be one-on-one, or can they be handled collectively?

7

I don’t get it. It seems like the team that voted blue won since 2x5 is bigger than 2x1.

8

The goal will be to maximize your personal score, not compared to everyone else, but compared to your beginning score. In other words, I’ll tell students, “Try to get the highest score that you can get.”

Since I’m going for general mathematical problem-solving strategies, and since these are mostly academically-gifted students, I’d rather go for a complicated chewy problem with no one answer, something that can lead to good discussions.

I’d be delighted if that happened. I was even considering a corollary (although I probably won’t do it): if the chocolate card-holders play well and if everyone else doesn’t play spitefully, the chocolate card holders could make out like bandits. But if the non-chocolate-card-holders band together, i.e., “unionize,” they can bargain collectively.

In any case, if the students come up with a way to game the system, that’ll be the kind of moment that makes my teacher heart leap.

9

Update: played the game with students and it worked quite well–although someone inexplicably ended up with a score of 4 at the end of the game (I suspect either theft or massive incompetence at counting). They were pretty amazed to see how the total score in the class rose when everyone was trying to maximize their individual score.

10

Did you go with equal distribution of ingredients in the end, or did you make chocolate scarce?

11

I printed equal numbers of ingredients, but I had (I think) 21 players, so I just put an extra set of eggs out there. Eggs were therefore slightly more common than everything else. In a more gamer-y group, that might have been significant, but for these guys it was the first time most of them had played a game with a free-trade dynamic, so they didn’t really notice that, and also didn’t make any surprising or conniving trades (“two eggs for a chocolate!”)

12

The team that voted blue lost 2x5 points. The five teams that voted red gained 2x1 points each.