Purposefully Imperfect Information (Pig I)

I don't know much about game theory (just like every other branch of mathematics with a name, haha). Game theory was always an afterthought in the classes I took – tacked on to the end of discrete math, or mentioned briefly in a problem set. But I play a lot of games, and I see interesting mathematics arise from them. 

The kinds of games I’m talking about are very idealized. You’ve got players, who are perfectly logical and rational, you’ve got some sort of win condition, which might be different for each player, and you’ve got some type of allowed actions or events set to happen which offer the players a choice. Games can have perfect information, like chess, where every player knows everything possible about the state of the game, or not, like poker, where you don’t know what cards your opponent has. There’s competitive games, where each player is self-interestedly trying to win, and cooperative games, where players have to work in teams or all together to win. Cooperative games lead to “super-rationality” and other interesting concepts, and that’s what got me thinking about this hypothetical and a strange question.

Hypothetical: Let’s say you have a cooperative game between players with some high level of rationality. Perfectly rational, super-rational, theoretical rationalities, I’m not sure yet. This game features imperfect information, so certain players will know things other players will not. Finally, communication between players is a free action. 

Can you picture the game? Here’s a quick example. You and three others divide a deck of cards into four hands of thirteen cards each, and each player can see their hand. Each player takes a turn passing or playing a card onto the table. The goal is to play all cards, in order, into four separate stacks sorted by suit (like the goal of solitaire) with as few passes as possible. And you’re free to talk to the other players about your hand whenever you feel like it.

Simple enough, right? You and your friends are rational, you don’t have all the information, talking is free, and you’re working together to sort the deck. This is the kind of game I’m talking about.

Now, the question is: Is there a game that satisfies these rules in which it’s not optimal to give out all the information you know?

In the example above, the best strategy would just be to show everyone your cards, and decide together what to do. In that game, more information means you will always make better decisions, and the players are all identically rational, so everyone would agree on a best way to play, and play. But is there a game where more information, maybe even perfect information, could make a rational player play worse than a game with limited information?

I don’t actually have an answer. I’ve got some ideas, but no answer just yet. Interesting, isn’t it? Interesting enough for a name, I think. Such a game would have players purposefully hiding information about the game from their people who are supposed to be on their team... so... Omission? Censoring? Purposefully Imperfect Information? Oooh, I like that one, because Purposefully Imperfect Information Game abbreviates to PIIG, which "abbreviates" to PIG, with some artistic license! Henceforth, this shall be known as the Pig problem! Until next time!