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The question may be too vague, but ultimately in search of various (counter)examples or theorems to exhibit the following:

Do continuous families $t\mapsto G_t$ of "games" (say each $G_t$ is a stable finite game with unique Nash equilibrium) have that the Nash equilibrium varies continuously?; or a fixed point?; or discontinuously? Here the family may be a 1-parameter $t\in[0,1]$ or simply an infinitesimal deformation $t\in(-\varepsilon,\varepsilon)^n$ of a given game $G_0(x_1,\ldots,x_n)$ depending on $n$ hyperparameters.

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Usually, one looks at the correspondence (set-valued map) that associates with each game its set of Nash equilibria. Under many conditions, this correspondence is upper hemicontinuous. For example, if one fixes finite action spaces for a fixed finite set of players, then the correspondence from payoff functions, represented as points in a Euclidean space whose dimension is the number of pure strategy profiles times the number of players, is upper hemicontinuous.

A function can be identified with a single-valued correspondence, and the function is continuous at a point if and only if the corresponding correspondence is upper hemicontinuous. So if you restrict yourself to parameters for which there is a unique Nash equilibrium, it will be a continuous function of these parameters.

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  • $\begingroup$ Is there an explicit/implicit example of a deformation of games where either: 1) a unique Nash equilibrium bifurcates into two, or 2) a finite set of Nash equilibria collide or die? $\endgroup$
    – Chris Gerig
    Commented Apr 7, 2023 at 0:17
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    $\begingroup$ Take a coordination game between players both having the purse strategies $L$ and $R$ available. The payoff is $0$ unless both choose the same action. Both get a payoff of $1$ if they choose $L$ and a payoff of $\epsilon$ if they both choose $R$. For $\epsilon>0$, there are three equilibria, for $\epsilon=0$, there are only two. $\endgroup$
    – Michael Greinecker
    Commented Apr 7, 2023 at 5:32
  • $\begingroup$ I see, there are two pure strategies {([1,0], same), ([0,1], same)} and one mixed strategy ([e/(1+e),1/(1+e)], [same]), so that the latter two collide. How about pure strategies bifurcating/colliding with just pure? Is there any result of the form "any 2d dynamical system with discrete fixed points can be translated into a family of 2-player games with pure Pareto-optimal strategies"? $\endgroup$
    – Chris Gerig
    Commented Apr 7, 2023 at 15:49
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As Michael wrote, if the game $G_t$ has a unique equilibrium for every $t$, then the upper semi-continuity of the Nash equilibrium correspondence implies that the unique equilibrium varies continuously.

If uniqueness is not guaranteed, then you should look at Browder's Theorem (Browder, F.: On continuity of fixed points under deformation of continuous mappings. Summa Bras. Math. 4, 183–191 (1960)). A simple version of this theorem states that if the function $t \mapsto G_t$ is continuous (for $t \in [0,1]$), then the set $E := \{ (t,x) \colon x {\rm~is~an~equilibrium~of~} G_t\}$ has a connected component whose projection to the first coordinate is $[0,1]$. You can find a simple proof of this result here (https://arxiv.org/abs/2107.02428), and a generalization for a general compact parameter space here (https://link.springer.com/article/10.1007/s11784-021-00926-5) or here (https://arxiv.org/abs/2210.16369).

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