Imperfect information
Last updated
Last updated
We have previously discussed the main two forms of games in mainstream game theory: normal form (also called strategic games), and extensive form (also called dynamic games).
At first sight, these two forms seem unrelated. In the normal form, a game is a matrix and the players are in "separate rooms." In the extensive form, a game is a tree and the players are together and play in turns. And yet, they are two faces of the same coin, as I showed in 2019 that they are two special cases of a more general form called the spacetime form (this is the academic paper).
But before I explain the spacetime form, let us look at the relationship between the two forms, as there is an insight that has been known for several decades, even probably back to the 1950s. It requires the introduction of an additional concept: imperfect information vs. perfect information.
Let us go back to our example of game in normal form, the prisoner's dilemma.
The way to think about this is that Alice and Bob are in separate rooms and make their decisions separately. But let us think a bit more: imagine that Alice makes her decision at 1pm, and Bob at 2pm, but without communication between the two.
Now, if there had, hypothetically, been some communication between Alice and Bob, this would simply have been a game in which Alice plays first and then Bob (the payoff order on the outcomes is the same: Alice first and then Bob):
Bub this is a different game: in this game, Bob is informed of Alice's decision. The only difference with the original game, in normal form, is that Bob is not informed of Alice's decision. The rest is identical. There are four outcomes, and the payoffs correspond exactly to the four combinations of choices of Alice and Bob.
The trick to turn this into a game with the same meaning as the original game is to introduce what is called imperfect information. This is visualized with a dashed line between the two nodes played by Bob, like so:
The meaning of the dashed line is that Bob does not know, when making his decision between defecting and cooperating, whether he is the "left Bob" or the "right Bob". This is equivalent to saying that he does not know Alice's decision. This is, in fact, the same situation as in the original game. The Nash equilibrium is also the same: (1,1).
To be fair, the equivalence has been the topic of some debate in the game theory community in the sense that the players may reason in a different way depending on the form shown to them. But we side with those who think that these are the one and same game, and this is also true in non-Nashian game theory.
So, a game in extensive form with imperfect information is a tree, as we have seen before, but this tree has additional dashed lines that link nodes together. The only constraint is that the nodes connected with a dashed line must be played by the same agent, and the actions at their disposal must be the same set. You can see above that such is the case.
A natural question that arises is: we let Alice play "first" and then Bob. What if we had let Bob play "first' and then Alice? This leads to the following structure (where Alice's payoff is still on the left and Bob's payoff on the right).
The two versions are in fact equivalent.
With the same procedure as above, any game in normal form, with any number of players, can easily be transformed into a game in extensive form with imperfect information.
Does this work the other way round? No. In fact, the vast majority of games in extensive form with imperfect information are not equivalent to a game in normal form obtained in this specific way.
So, games in extensive form with imperfect information are more generic and more powerful than that. In fact, we will see in the next section that they are so powerful, that they can describe games played by agents making decisions in any position in (flat) spacetime. And this is the reason why games in extensive form with imperfect information are central to the extension theory of quantum physics that we are building and describing in this book.
