# Game Theory on a Number Line

Another week, another Riddler. Here is a fun problem:

Ariel, Beatrice and Cassandra — three brilliant game theorists — were bored at a game theory conference (shocking, we know) and devised the following game to pass the time. They drew a number line and placed \$1 on the 1, \$2 on the 2, \$3 on the 3 and so on to \$10 on the 10.

Each player has a personalized token. They take turns — Ariel first, Beatrice second and Cassandra third — placing their tokens on one of the money stacks (only one token is allowed per space). Once the tokens are all placed, each player gets to take every stack that her token is on or is closest to. If a stack is midway between two tokens, the players split that cash.

How will this game play out? How much is it worth to go first?

To solve this we have to assume each player is a perfect logician – then we work in reverse: say we know exactly where Ariel and Beatrice have placed their token. If this is the case we can find the optimal place for Cassandra to put her token to maximize her earnings. Using this we can back out the optimal place for Beatrice to put her token after each of Allice’s moves — it would be where Cassandra’s best move is the worst. Using this we can back out Allice’s best move — its where even if Beatrice and Cassandra use their best moves its the best for Ariel. Here’s an example:

```
#Of the form:

# [[A's token place (0 indexed), B's token place, C's token place], [A's winnings, B's winnings, C's winnings]]

[[1, 5, 2], [1, 49, 5]]
[[1, 5, 4], [2.0, 47.0, 6.0]]
[[1, 5, 5], [3, 45, 7]]
[[1, 5, 6], [4.5, 10.5, 40]]
[[1, 5, 7], [4.5, 13.5, 37.0]]
[[1, 5, 8], [4.5, 16.5, 34]]
[[1, 5, 9], [4.5, 20.0, 30.5]]
[[1, 5, 10], [4.5, 23.5, 27]]

```

From this we can figure out if A places token on 1 and B places on 5 then C will always place on 6. Here is an example for B’s decision:

```
#Of the form:

# [[token places (1 indexed)], [winnings]]

[[1, 2, 3], [1, 2, 52]]
[[1, 3, 4], [2.0, 4.0, 49]]
[[1, 4, 5], [3, 7, 45]]
[[1, 5, 6], [4.5, 10.5, 40]]
[[1, 6, 7], [6, 15, 34]]
[[1, 7, 8], [8.0, 20.0, 27]]
[[1, 8, 7], [8.0, 27, 20.0]]
[[1, 9, 8], [10, 19, 26]]
[[1, 10, 9], [12.5, 10, 32.5]]

```

Here we determine that if A places on 1, then it is in B’s best interest to place on 8. We do this a final time and determine the players will play:

```#Of the form:
# [[token places (1 indexed)], [winnings]]

[[5, 9, 8], [21, 19, 15]]

```

And this is the answer to our problem.