Spacetime games

We have seen in the previous section that games in normal form and games in extensive form are two side of the same coin. Indeed, games in normal form can be converted to an equivalent game in extensive form, but the price to pay is the introduction of imperfect information: the dashed lines.

The reason is that in a game in normal form, an agent is not informed about the other agents' decision. The agents are in different rooms and do not communicate with each other.

This is in contrast with a game in extensive form with perfect information, in which the players are together and play one after each other. Thus, the player who plays second (say, at chess) is informed about the previous move of the other player.

As it turns, special relativity has names for making this crucial difference: namely, spacelike separation and timelike separation:

• To imagine two decisions that are spacelike-separated, imagine them very far away from each other. So far away, that no information can be sent from one to the other. For example, one is made on the Earth and one on Mars, but with just a few seconds of difference (knowing that light takes several minutes to go from Earth to Mars).

• And to imagine two decisions that are timelike-separated, imagine that they are taken close enough to each other that there is communication, and one of them unambiguously is made before the other one.

Now, we are able to say that a game in normal form involves spacelike-separation between the players' decisions. And a game in extensive form with perfect information involves timelike-separation between the players' decisions.

We assume that you, the reader, are not familiar with special relativity, but that you may still have heard that space and time are part of the one and same continuum: spacetime, and that time is not absolute.

Thus, in special relativity, events happen "somewhere in spacetime", which you can see as a point "in spacetime". The position (in space) and time of this point may not be the same depending on who is looking (the observer).

A spacetime game is basically a generalization of games in normal form (only spacelike-separated decisions) and of games in extensive form with perfect information (only timelike-separated decisions), in which decisions are at any points in spacetime. And in the same spacetime game, some pairs of points can be timelike-separated, some other pairs can be timelike-separated, in the same game. This is what makes spacetime games considerably more general than a game in normal form or than a game in extensive form with perfect information.