There are a few tools, either online or offline, that could solve (find equilibrium) in a game explicitly given as a real-valued matrix.

Such tools are Game Theory Explorer and Gambit.

However, as far as I know, these can not be used for finding equilibrium in a game with parameters.

For example, consider a two player game symmetric game, parametrized by some $x\in[0,1]$, which is given by the following payoff matrix for the row player (and $A^t(x)$ for the column player): $$A(x)= \left( \begin{array}{ccc} \frac{x}{2} & x \\ 1-x & \frac{1-x}{2} \\ \end{array} \right) $$

Each equilibrium in this game can be represented as $<p_a,p_b>\in[0,1]^2$ where $p_a$ is the probability of the first player playing the first strategy and $p_b$ is the probability the second player plays the first strategy.

Analytically we can find all the equilibriums for this game

$$EQ(x)= \begin{cases} \{<1,1>\}&\mbox{if } x>\frac{2}{3}\\ \{<3x-1,3x-1>,<1,0>,<0,1>\}&\mbox{if } \frac{1}{3}\leq x\leq\frac{2}{3}\\ \{<0,0>\}& \mbox{else} \end{cases} $$

I'm looking for a solver which is able to understand inputs like $A(x)$ and produce $EQ(x)$.

Is there a tool which solves parametric games?

  • $\begingroup$ How can the entries depend on $x$? $\endgroup$ Jan 1, 2015 at 21:20
  • $\begingroup$ @KristofferArnsfeltHansen - what do you mean? For every $x$, $A(x)$ is a normal game given in matrix form. A parametric solver would be able to produce the output with respect to $x$. $\endgroup$
    – R B
    Jan 2, 2015 at 7:41
  • $\begingroup$ I understand that. But how is the input specified? For instance, in your example, the entries are simple linear functions of $x$. $\endgroup$ Jan 2, 2015 at 10:43
  • $\begingroup$ @KristofferArnsfeltHansen - assume you can simply write the matrix as $<f_{1,1}(x),f_{1,2}(x);f_{2,1}(x),f_{2,2}(x)>$, where $f_{i,j}$ is a some function of $x$, e.g. $<x/2, x; 1-x, (1-x)/2>$. Such tool would be very useful (for me anyways) even for a single parameter, small matrices (up to $3\times 3$) and linear functions of the parameter. $\endgroup$
    – R B
    Jan 2, 2015 at 11:40
  • $\begingroup$ For small matrices you can simply enumerate over square submatrices and use the characterization of Shapley and Snow. $\endgroup$ Jan 2, 2015 at 12:39

1 Answer 1


Is there a tool which solves parametric games?

Not that I am aware of (I am a co-author of GTE and help with Gambit).

The best suggestion I have if you don't find such a tool (and I doubt one exists) is to do a parameter sweep and solve a bunch of individual instantiations and see what the resulting sets of equilibria say about $EQ()$. Gambit is very tightly integrated with python, so you can easily define your game as a python function with the parameters as arguments and solve a set of instances via a parameter sweep.

  • $\begingroup$ Can you combine Gambit with sympy to solve the game symbolically? $\endgroup$ Jan 3, 2019 at 19:22
  • $\begingroup$ For simple settings one might indeed be able to use sympy relatively easily. E.g., one could use solveset from sympy to do support enumeration for a bimatrix game with some parametrized payoffs, which would give (some parametric description of) the extreme NE. To find all NEs one then need to solve a maximal clique enumeration problem - probably tricky in this parametrized setting. For >= 3, the NE set can be any semi-algebraic set and we don't have robust enumerators for the non-parametric case, so unless the payoff parametrization is highly constrained, it may be hopeless. $\endgroup$ Jan 5, 2019 at 12:57

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