MathDB

Three cats--TheInnocentKitten, TheNeutralKitten, and TheGuiltyKitten labelled

P_1, P_2,

and

P_3

respectively with

P_{n+3} = P_{n}

--are playing a game with three rounds as follows:
[*] Each round has three turns. For round

r \in \{1,2,3\}

and turn

t \in \{1,2,3\}

in that round, player

P_{t+1-r}

picks a non-negative integer. The turns in each round occur in increasing order of

t

, and the rounds occur in increasing order of

r

\newline \newline

[*] Motivations: Every player focuses primarily on maximizing the sum of their own choices and secondarily on minimizing the total of the other players’ sums. TheNeutralKitten and TheGuiltyKitten have the additional tertiary priority of minimizing TheInnocentKitten’s sum.

\newline \newline

[*] For round

2

, player

P_{2}

has no choice but to pick the number equal to what player

P_{1}

chose in round

1

. Likewise, for round

3

, player

P_{3}

must pick the number equal to what player

P_{2}

chose in round

2

\newline \newline

[*] If not all three players choose their numbers such that the values they chose in rounds 1,2,3 form an arithmetic progression in that order by the end of the game, all players' sums are set to

-1

regardless of what they have chosen.

\newline \newline

[*] If the sum of the choices in any given round is greater than

100

, all choices that round are set to

0

at the end of that round. That is, rules

2

3

, and

4

act as if each player chose

0

that round.

\newline \newline

[*] All players play optimally as per their motivations. Furthermore, all players know that all other players will play optimally (and so on.)

Let

A

and

B

be TheInnocentKitten's sum and TheGuiltyKitten's sum respectively. Compute

1000A + B

when all players play optimally.
Proposed by Harry Chen (Extile)

Problem Statement