Boring basketball game

FiveThirtyEight has an interesting column, Riddler column, which I follow with great interest. In this post we will take a look at another statistics puzzle published as Riddler Classic puzzle for April 29th.

Puzzle

In short (see the original post for a more detailed formulation), we have two teams (e.g., "red" and "blue") playing a game of \( 200 \) possessions (\( 100 \) possessions for each team). Let us allow only two-pointers in this game. Furthermore let us assume that if the teams are tied, scoring probability for either team is \( 0.5 \), but if one team is ahead then loosing team concentrates and starts shooting better: scoring with \( 0.5 + x \) probability. Likewise winning team looses concentration and starts shooting worse: scoring with \( 0.5 - x \) probability.

Organizer, who knows \( x \) (from empirical observation?), has done the math and came to conclusion that after \( 200 \) possessions there is \( 50 \% \) chance that the game will be tied. How big is the losers advantage \( x \)? Under these assumptions is there first possession (dis)advantage?

Solution

Analytical solution is possible, but will merit its own post. Here let us share interactive app, which may be used to solve the problem. The input parameters are self-explanatory, while the plot shows progression of a single game. Overall record is shown in the top label of the interactive app below.

Can you find the \( x \)?