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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? enter image description here

Thanks for your help.

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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 $a$.

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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 $a=0$ $$ u(a) = \begin{cases} 1 & \text{if}\ a<0,\\ [1,2] & \text{if}\ a=0,\\ 2 & \text{if}\ a>0.\\ \end{cases} $$

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    $\begingroup$ It is a consequence of Berge's theorem of the maximum that the argmax correspondence is generally upper hemi-continuous but not necessarily lower hemi-continuous, and that's the basis for most continuity properties in economic models. Your indirect payoff function $u(a)$ isn't a function and it isn't discontinuous at $a=0$; it is a uhc correspondence that inherits the uhc property because a composition of uhc correspondences is uhc, and a continuous function (the direct payoff function) is a trivially uhc correspondence. $\endgroup$
    – user762914
    Apr 28 '20 at 17:43
  • $\begingroup$ I figured that explaining the very simple intuition of why it happens would be more useful than giving the name of a theorem. I guess it is a matter of taste. In any case, my function $u$ is a function from the parameter space to the power set of R, that is why I wrote 'utilities' instead of 'utility,' but maybe I should have clarified. However, I disagree with you about it being continuous. It is uhc but not lhc, just like the example from OP. I'm not sure OP is very interested anymore. So, I'm not sure it is worth the time to edit my answer. $\endgroup$ Apr 29 '20 at 18:20
  • $\begingroup$ I didn't say $u(a)$ was continuous, I said the direct payoff function is continuous (the expected utility function mapping the mixed strategy simplex into $\mathbb{R}$), and when you substitute a uhc correspondence ($s^*$) into a continuous function, you get a uhc correspondence. I spend a lot of my time thinking about correspondences, so I am unreasonably interested in these details. I like the answer in general, I just wanted to point out a couple spots where there's a general principle (e.g., Berge's) or language might be confusing (e.g., a function that fails the vertical line test). $\endgroup$
    – user762914
    Apr 30 '20 at 16:03

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