# Converting a sequential game to a simultaneous move game

One can convert a sequential game into a simultaneous move normal form game by defining contingent plans of what to do for any possible sequence of observable moves and allowing players to choose over these strategies simultaneously.

It is known that the Nash equilibria (NE) of the resulting normal form game can describe some unreasonable situations (e.g., non-credible threats).

One way of avoiding this is to look at extensive-form refinements of NE and invoke backward induction (starting from the end of the game and computing strategies backward in time) leading to subgame perfect Nash equilibria (SPNE).

Question: Intuitively, it seems SPNE are a more constrained version of NE. Is it possible to formulate the problem as a normal form game but impose some additional constraints on the form of the strategies such that the resulting computed equilibria is a SPNE and not just a NE? Perhaps some of the rows/columns are ruled out of the game matrix, or the problem of computing SPNE becomes a constrained LP?

## 1 Answer

At the risk of misinterpretation, I'll reformulate your question as follows:

Is there an equilibrium refinement criterion for any normal form game such that this criterion selects an equilibrium that would constitute a subgame perfect equilibrium in all possible extensive form games that can be reduced to the normal form game in question?

Short answer: Yes.

For example, Kohlberg and Mertens (1986) prove that any proper equilibrium of a normal form game is part of a sequential equilibrium in any extensive form game that can be reduced to the said normal form. (Recall that the strategy profile of any sequential equilibrium also constitutes a subgame perfect equilibrium.)

They expressly disagree that "any equilibrium concept defined on the normal form will miss the essence of backwards induction in the extensive form."

More generally, Kohlberg and Mertens (1986) propose notions of "stable equilibrium" that satisfy several desirable properties, in particular the two below:

• Backwards Induction: A solution of a tree contains a backwards induction (e.g. sequential or perfect) equilibrium of the tree.
• Invariance: A solution of a game is also a solution of any equivalent game (i.e., having the same reduced normal form).

In later work, the authors refine the definition of stable equilibrium by replacing invariance with a more rigorous concept of ordinality.

• Excellent, thank you! I wonder: has such an idea been extended to imperfect information games, that is, perfect Bayesian Nash equilbria instead of subgame perfect equiilbria. – jonem Aug 15 at 18:27
• @jonem: To the extent that games with incomplete information can be modeled using the (Bayesian) normal form or the (Harsanyi) extensive form, the results in Kohlberg and Mertens (1986) would apply. In fact, Mertens (1987) explicitly considers this aspect. – Herr K. Aug 15 at 19:51
• Thanks! Can you link Mertens '87? – jonem Aug 15 at 22:16
• @jonem: jstor.org/stable/3689732 – Herr K. Aug 15 at 22:25