## Puzzle at the beach

A little younger, we used to play a game of skill in family at the beach. A simplified two players version is like this. Find two light stones. Draw a straight line in the sand. Move back a little, so that throwing your stone close to the line becomes difficult enough.

The objective of the game is to throw your stone on the good side of the line (a throwing beyond the line means you loose) and of course closer to the line than your opponent. It is played sequentially… which favours Player 1.

In the following outcome of the game, Player 2 wins.

The questions are:

- to what distance away from the line should Player 1 be aiming to throw his stone ? (cf picture bellow)
- what is his probability to win ?

The model can be the following: a throwing aimed at a given point is distributed as a standard Gaussian random variable around this point. It means in particular that both players are equally skillful, and that the distance does not alter the precision. Assume that if Player 1 throws beyond the line, then Player 2 wins without even throwing. Assume finally that once Player 1 has succesfully thrown his stone, then Player 2 rationally maximizes his profit by aiming his throw right in between the line and the shoot of Player 1.

Good luck. (answers in a few days)

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