Change (discontinuity) in Nash Equilibrium with change in parameter

Question

I am solving for duopoly competition between two firms who decide a product characteristic and price.

I find that I get two different types of equilibria based on a parameter 'a' - with a discontinuity in firm profits as a result of the parameter value where equilibrium changes.

My question: Is such a shift in equilibrium (and resulting discontinuity in profits) based on parameter change possible? Are there any "simple" textbook examples of this situation for me to learn more?

Thanks for your help.

Answer 1

If you wrote the Cournot model down as something like $p = 1 - q_1 - q_2$, have $a$ be firm $1$'s marginal cost of production, but have firm $2$'s marginal cost be something like $$ c_2(a) = \begin{cases} a, & a < .25 \\ 0, & a \ge .25 \end{cases} $$ so the firms' costs are the same until $.25$, but then at $ a= .25$, firm 2's marginal costs crash to zero so its profits jump discretely while firm 1's profits drop because 2 has become so much more efficient. This will make the profit functions discontinuous.

Many models with the $\{$ cases with discontinuous jumps in the fundamentals will give you that kind of behavior, but you should be careful to make sure that the game is well-posed for each $a$.

Answer 2

The incentive constraints for well-behaved games (e.g., finite games) take the form $$f(s,a)\geq 0,$$ where $f$ is a continuous function, $s$ represents strategies, and $a$ represents a parameter of the game (following the notation in OP's graph). The way to read the condition is as follows, $s$ is an equilibrium given $a$ if and only if $f(s,a)\geq 0$.

The key observation is that the inequality is weak. To see why that matters take a convergent sequence of parameter values $a_n\to a^*$ and a strategy profile (s^*) and note the following:

1. Suppose that $s^*$ is an equilibrium for every parameter value in the sequence. Then $f(s^*,a_n) \geq 0$ for all $n$, and thus $f(s^*,a^*) \geq 0$. This means that equilibria don't suddenly disappear. If an equilibrium (or a type of equilibrium) exists on a set, then it also exists in its closure.
2. It could be the case that $f(s^*,a_n) < 0$ for all $n$, but $f(s^*,a^*) = 0$. This means that equilibria can suddenly appear.

Formally, these observations say that the Nash equilibrium correspondence is upper-semicontinuous but not necessarily lower-semicontinuous.

It is easy to construct simple examples of discontinuities of Nash equilibria based on this idea. For example, consider the following game $$\begin{array}{c|c|c} & \mathrm{L} & \mathrm{R} \\ \hline \mathrm{T} & (0,2) & (0,0) \\ \hline \mathrm{B} & (a,0) & (a,1) \\ \end{array}$$

The strategy profile $(\mathrm{B},\mathrm{R})$ is an equilibrium for $a \leq 0$, and the strategy profile $(\mathrm{T},\mathrm{L})$ is an equilibrium for $a \geq 0$. There are no other pure strategy equilibria except for the knife-edge case $a=0$. Let $u$ be the possible utilities that player 2 could obtain in equilibrium as a function of $a$. It is discontinuous at $a=0$ $$ u(a) = \begin{cases} 1 & \text{if}\ a<0,\\ [1,2] & \text{if}\ a=0,\\ 2 & \text{if}\ a>0.\\ \end{cases} $$