You ate ALL MY PIZZA!

September 4, 2009 11:58 AM

Is there a mathematical, economic, logical or game-theory name/description for the following scenario?

Let's say you have two pairs of people: Pair A and Pair B.

Both pairs are ordering pizza. Pair A will not eat meat toppings, whereas Pair B wants meat toppings. Collectively all four decide to order two pizzas: one with no meat toppings (pizza #1) and one with meat toppings (pizza #2).

When the pizzas arrive, Pair A of course will only eat pizza #1. But Pair B are not committed carnivores; in addition to pizza #2 (which only they prefer) they also want to eat slices from pizza #1.

At the end of the meal, Pair A have had to struggle to get a "fair" share of pizza #1, whereas Pair B have helped themselves to both pizzas.

What's this called? (Besides "being jerks"?)

posted by contessa to grab bag (11 comments total)

Mind you, I slept through most of my game theory course. That said, you could be looking at a scenario where A's choices of topping are dominated (or strictly dominated) by B's choices.

If you introduce a third option of some topping which A likes but B does not (perhaps something gross, like pineapple or onions,) you could then reach an equilibrium where each pair eats just their own pizza.

They type of equilibrium would depend on just how much each pair hates meat/onions. That would likely determine your payoff matrix for the game.
posted by Wulfhere at 12:14 PM on September 4

I think that's just called "eating someone elses pizza".

Let' simplify the problem a little. Instead of A and B each being a pair of people, can we call each one player? I think it's functionally pretty similar, and makes it easier to analyze.

So: Player A and B sit down to order pizza. A is vegerarian,and orders a vegetarian pizza. B is not, and orders a meat Pizza.

My question is: Where does the expectation arise that the pizzas are to be shared?

Why would player A agree to an agreement "You and I will order two pizzas, and share them, and (presumably) split the costs. And one of the pizzas will be inedible to player A". How does that make sense?
posted by ManInSuit at 2:57 PM on September 4

(Which is to say- I don't think there's a game-theory problem here, just a problem of unclear expectations between friends...)
posted by ManInSuit at 2:58 PM on September 4

Commitment problem. Decision to order certain toppings is contingent upon expectations about Pair B's behavior, but they have no incentives to honor the agreement once the pizzas actually arive. Pair A should anticipate this and refuse to bargain. The solution is to increase the cost of reneging in period 2 ex ante. Adding toppings that Pair B dislikes would do the job, as would imposing some form of social cost.
posted by shadow vector at 3:07 PM on September 4

This isn't a game, it's an optimization problem. There is a scarce resource (pizza) which comes in two varieties, M and V. Following ManInSuit's useful simplification, there are two uses to which each variety of pizza may be put, feeding A and feeding B. What allocation of pizza maximizes happiness? The answer depends on how one defines the total happiness of A and B. This will surely be a complicated mathematical function involving hunger, vegetarianism, fairness, and whether there's any beer in the fridge.

One piece of jargon you could pull out of the optimization literature is the utility monster who enjoys something so much that it must be done, even if it is harmful to the other participants. From the point of view of the vegetarian, the carnivore's love of meat makes it a utility monster.
posted by drdanger at 3:19 PM on September 4 [1 favorite]

This is a pretty interesting variant of what's called a "Fair Division Problem". Your standard fair division problem deals with taking something like a cake (generally nonhomogeneous to make things interesting) and dividing it between several people.

In this problem, you're taking the bare pizzas and dividing the real estate upon them according to the preferences of the players.

Neat question!
posted by TypographicalError at 3:57 PM on September 4

Dang it, I have this same problem in real life with soda. I like Sprite, and I use about 4 oz at a time as a mixer for juice occasionally over a week and I don't really like Coke; hubby likes Coke to drink in humongous glasses seemingly as fast as possible, but will also drink Sprite if no Coke is available. Each week we buy one 2-liter bottle of each soda. His Coke is gone in 1.5 days, while I have only had one 4oz serving of Sprite. He doesn't want to buy more Coke each week - he's trying to "cut down".

Do I watch my Sprite disappear over the next few days until they are both gone, or do I pitch a fit and demand he go without soda when I still have almost a full bottle of soda? I think adding the marriage/relationship factor to your pizza puzzle might be interesting.

If, in your pairs, A1 is married to B1 and A2 is married to B2, then who gets the vegetarian pizza?? What a fun puzzle.
posted by CathyG at 7:48 PM on September 4

I have to say, I seems to me like there isn't really an interesting fair division problem or optimization problem here. If player (or pair) A cannot eat meat pizza, and player (or pair) B wants meat pizza, the "solution" is pretty simple, uncomplicated, and uninteresting: Each player (or pair) should order and eat their own pizza. I can't see how any other form of sharing makes sense in this situation.
posted by ManInSuit at 7:20 AM on September 5

Extra variables...

The more people are present, the more likely the problem will occur.

The further removed the pizza-eaters are from the pizza ordering process, the more likely the problem will occur.

The more variety amongst the vegetarian pizzas, the more likely the problem will occur.

The more variety amongst the meat pizzas, the less likely the problem will occur.


This is based off...

1. The supposition that a person confronted with two appealing pizzas will take a slice of each.

2. The supposition that omnivores don't really think about the existence of vegetarians unless vegetarianism is specifically brought to their attention.

3. Years of working at that one company where 'Friday is pizza day!' aka 'The_Latin_Mouse and the two other vegetarian employees better be first in line or they won't get any lunch day'.
posted by the latin mouse at 8:11 AM on September 5

Each player (or pair) should order and eat their own pizza. I can't see how any other form of sharing makes sense in this situation.

Ahhh....but see, that's exactly the solution that is being applied, and yet the resource of the pair that has a preference_against is consumed in part by the pair that didn't reveal that they actually have a preference_all. And then there's a social dynamic issue where you can't exactly ban sharing, when the assumption is being made by at least half the people that sharing won't occur in the first place.

I guess the example in my question could be used for any type of consumable "customizeable" resource, where one player or set of players will have strict preferences and the other set has more flexible preferences. It just seems to happen a lot with pizza. A LOT A LOT.
posted by contessa at 12:24 PM on September 5

It sounds like a freerider problem to me. The meat-eaters get a free ride off of the cheese-eaters at no extra cost, and the cheese-eaters have a hard time stopping them and protecting their comparatively scarce resources.
posted by RobotNinja at 8:19 AM on September 8