Question: A sequence can be created where the 1st 3 terms are 1, 4, 5 and every term after that is the sum of all proceeding terms (e.g. 1, 4, 5, 10...) how many of such sequences beginning with 1 and containing the digits 2016 are there?

I'm not sure where to start. I found that the digits of the sequence are just double the digit before them and if you keep halving 2016 you end up with 63 but I'm not sure how that would be useful.

Answer: 
Formulate the problem as a two player game in which both players have the action space $\mathbb{R}^n$.

The generic action of player 1 is denoted by $x$, the generic action of player 2 is denoted by $y$.

Player 1 wants to match the action of player 1:

$$u\_1(x,y)=-\|x-y\|$$

Player 2 wants to match $f(x)$, the $f$-transformed action of player 1:

$$u\_2(x,y)=-\|f(x)-y\|.$$

It is clear that if $x$ is a fixed point of $f$, then $(x,x)$ is a Nash equilibrium.

Conversely, if player 1 plays a best response, then $x=y$ and if player 2 plays a best response, then $y=f(x)$, so in every Nash equilibrium, $x=y=f(x)$ and we get a fixed point of $f$.

The argument is taken from [here](https://theoryclass.wordpress.com/2012/01/05/brouwer-implies-nash-implies-brouwer/). A more sophisticated version of argument can be used to show the equivalence of the existence of Nash equilibria and the Kakutani fixed point paper, see [From Imitation Games to Kakutani](http://cupid.economics.uq.edu.au/mclennan/Papers/kakutani60.pdf) (working paper version) by McLennan and Tourky.
In order for you to find a game that generates this function $f$. This function cannot be arbitrary. Let $f=(f\_1,f\_2,\ldots, f\_n)$ be a function of $(x\_1,x\_2,\ldots,n\_n)$. You need that $\frac{\partial}{\partial x\_i}f\_i =0$, that is the ith-entry of the function does not depend on the ith-variable. In this case you can interpret $f\_i$ as a best response of player i to the actions of the others (players distinct from i).

Suppose you have that $\frac{\partial}{\partial x\_i}f\_i =0$ for all $i$. The $f\_i$ is player $i$'s best-response, $x\_i$ is i's action, and the payoffs are:

$$ u\_i(x) = -\left (x\_i-f\_i(x)\right)^2 $$

Of course that you can find an infinite number of games such that $f$ is the best-response mapping. The answer is just one example.