Symbolic Computation of Sequential Equilibria

TL;DR

This work tackles the problem of symbolically computing all sequential equilibria in finite extensive-form games with imperfect information. It encodes both sequential rationality and consistency as a single finite system of polynomial equations and inequalities and solves it via cylindrical algebraic decomposition, aided by a novel, efficient treatment of extreme directions in polyhedral cones. The main contributions include an explicit construction of the linear system $A x = b$, a finite polynomial reformulation using extreme directions $\mathcal{E}^A$, and a practical symbolic solver for small games, representing all equilibria exactly and handling infinite families of equilibria. While the approach demonstrates exactness and pedagogical value, it faces scalability limits due to an exponential number of cones and the double-exponential CAD complexity, motivating future work on optimization and alternative consistency representations.

Abstract

The sequential equilibrium is a standard solution concept for extensive-form games with imperfect information that includes an explicit representation of the players' beliefs. An assessment consisting of a strategy and a belief is a sequential equilibrium if it satisfies the properties of sequential rationality and consistency. Our main result is that both properties together can be written as a single finite system of polynomial equations and inequalities. The solutions to this system are exactly the sequential equilibria of the game. We construct this system explicitly and describe an implementation that solves it using cylindrical algebraic decomposition. To write consistency as a finite system of equations, we need to compute the extreme directions of a set of polyhedral cones. We propose a modified version of the double description method, optimized for this specific purpose. To the best of our knowledge, our implementation is the first to symbolically solve general finite imperfect information games for sequential equilibria.

Figures (4)

• Figure 1: A non-credible threat in an extensive-form game. The actions that are played are marked in red. Each terminal node is annotated with the utilities of player 1 and 2.
• Figure 2: A game with an inconsistent assessment. Beliefs are depicted in blue, strategies in red. Although the assessment is locally sequentially rational, it is not sequentially rational.
• Figure 3: Five examples of game trees annotated with consistent assessments. Beliefs are marked in blue, strategies in red.
• Figure 4: All sequential equilibria of the game from Figure \ref{['fig:localsr']} as returned by the cylindrical algebraic decomposition. Note that, e.g., $\beta(b) = 1-\beta(a)$, $\mu(\left<b\right>) = 1-\mu(\left<a\right>)$ and $U_1^E(\left<\right>) = 0$.

Theorems & Definitions (13)

• definition 1: Nash equilibrium
• definition 2: Subgame Perfect Equilibrium
• definition 3: Sequential Equilibrium
• definition 4: Local Sequential Rationality
• theorem 1: One-Shot Deviation Principle, hendon1996one
• definition 5: Positive Approximate Solution of a Linear System
• theorem 2: Solution Existence for Linear Systems, kohlberg1997independence
• proposition 1
• proposition 2
• proposition 3