Game Theory

List of Continuous Games

A curated and standardised library of continuous-game examples

We present a compilation of parametrized continous games drawn from both recent and classical literature. Analogous to the Gamut test set of finite games, our suite should provide a convincing and comprehensive benchmark for continous-game solvers. Specifically, we compiled multiplayer games with continuous utility functions and compact strategy sets and, where possible, with known equilibria.

Game table

Standardization

Players are identified by successive natural numbers $1,2,\dots$ , while their variables are named consistently by $x$ followed by two indices indicating the controling player first and the variable number second. For example, variable 3 of player 4 is refered to as $x_{34}$ .

tan-ridge game — Consider the following **Tangent Ridge Game** with two players, each with a single variable on the interval
$x_{i} \in [0,1] \quad \forall i \in \{1,2\}.$
The utility functions are defined by:
$\begin{aligned} u_1(x_1,x_2) &= -\sqrt{ |x_1-x_2| } \\ u_2(x_1,x_2) &= -\sqrt{ |x_1-\tan(x_2)| } \end{aligned}$
The game has an equilibrium at $x^\star=(0,0)$ .

Note that continuous games are complicated, and one game may have many representations. A strategy set as simple as $z\in[-1,1]$ may also be writen as $z^2\le1$ or $(-1\ge z \land z \le1)$ , with all three having potentially different implementations in your solver.

In addition to the game description in a canonical format we also provide a simple and reusable format, where we formulate the constraints for each player in terms of non-negative and equal-to-zero relations. The incentives of players are described by utility functions. In our simplified format the Tangent Ridge game is equivalent to:

NNEG1: x11 >= 0, 1-x11 >= 0
NULL1: 0 = 0
NNEG2: x21 >= 0, 1-x21 >= 0
NULL2: 0 = 0
U1: -sqrt(abs(x11-x21))
U2: -sqrt(abs(x11-tan(x21)))

Browser

TBD