Science Art: A four stage sequential game with a foresight bound, Marco Mantovani

28 July 2025 grant Featured, Science Art 0

Scientific illustration diagramming a particular kind of game theory based on decision-making ... resuling in a graph that looks like a series of gothic arches.
Scientific illustration diagramming a particular kind of game theory based on decision-making ... resuling in a graph that looks like a series of gothic arches.

This is a game theory diagram from the paper, “Limited backward induction: foresight and behavior in sequential games,” though I found it on Wikimedia Commons.

It’s used as an illustration in the Wikipedia article on “Backward Induction. This is when you figure out the best possible choices looking back from a desired endpoint. In other words, when you ask, “How did I get here?”

In game theory, it’s used to determine optimal choices when there are two or more decision-makers:

In this situation, it may still be possible to apply a generalization of backward induction, since it may be possible to determine what the second-to-last player will do by predicting what the last player will do in each situation, and so on. This variant of backward induction has been used to solve formal games from the beginning of game theory.

That article gives an example:

The application of backward induction in game theory can be demonstrated with a simple example. Consider a multi-stage game involving two players planning to go to a movie.

  • Player 1 wants to watch The Terminator, and Player 2 wants to watch Joker.
  • Player 1 will buy a ticket first and tell Player 2 about her choice.
  • Next, Player 2 will buy his ticket.

Once they both observe the choices, the second stage begins. In the second stage, players choose whether to go to the movie or stay home.

  • As in the first stage, Player 1 chooses whether to go to the movie first.
  • After observing Player 1’s choice, Player 2 makes his choice.

In this chart, Player 1 is at the top of the main gothic arch here, and Player 2 is at the top of the lower two arches, and the movie choices are the legs of the arches.

But this game is a little more complicated because one of the players can’t see all the choices that led to the final choice (which is what the “foresight bound” means), which makes determining the best course more difficult.

Although to me, it does make the whole decision-making process look a lot like the face of a gothic cathedral … all those vaulted arches resting one on top of the other.