After reading an introductory post on game theory, specifically regarding Schelling Points, I noticed a paradox and I wonder if anyone here can shed some light on the subject.

The scenario I will use to illustrate this is the inverse of a frequently used example to describe popular Schelling points.

**The Scenario**

Suppose, in a dystopian future, you are on a game show with a partner. A list of numbers is presented to you:

[2, 5, 9, 25, 69, 73, 82, 96, 100, 126, 150]

You and your partner have to pick one number from the list each, without communicating with each other. You both know that if you pick the same number, a terrible fate looms. If you pick different numbers, you are both set free and live long and happy lives.

Assume that you are both human and computer unaided and therefore cannot choose truly random positions on the list.

Naturally, you would want to avoid Schelling points (special numbers or numbers in special positions in the list) to minimise the chance of picking matching numbers. In this case, the Schelling points are numbers which you would think your partner would be more likely to pick, for whatever reason. However, if you both rule out Schelling points, you make the list of numbers to choose from smaller, thus increasing the chance of you both picking the same number significantly. Therefore, if you both actively pick numbers which you think your partner is least likely to pick, assuming you both think rationally, you inadvertently increase the chance of picking the same number. Thus the notion of the Schelling point has become the numbers that are *especially insignificant*, and the cycle continues. This is the paradox.

**A real life example:**

Which bar do i choose to drink at on a Friday night if I want to avoid my ex (assuming she's actively avoiding me too)?

**The point of this post:**

Using game theory, what logical strategy would you employ in two-player avoidance games similar to the one above?

**The apology in advance:**

I'm new to the site and if any of this is convoluted or has been covered before, I apologise in advance.

Or actually randomize I suppose. Number all options 1-N, multiply random numbers in your head until you get stuck (eg 2×6×9×95×34 in order, then when you miltiply that by 37 you get stuck. Use the last number,) though obviously getting it wrong is fine. Modulo N, +1. Walla, your choice, randomized.