# Ivory Tower Arcana 5: goldfish1's Game

royalchicken
Posers and Puzzles 25 Aug '04 05:25
1. royalchicken
CHAOS GHOST!!!
25 Aug '04 05:25
Two players (A &amp; B) divide a standard deck of cards by color. One player takes all diamonds, the other takes hearts. Black cards are placed face down after shuffling. The winner of the previous game or a randomly selected player plays first.

Each player has a 'hand' of red cards, which she may look at, and a 'bank' (initially empty) of black ones, which are face down and out of the game directly they are acquired.

The move consists of placing the top black card face up. A may:

1. Give the black card in question to B, who puts it face down in his 'bank' and take the move

2. Place a red card face up from his hand, as a 'bid'

B must then either let A take the black card, or place a 'bid' of his own, higher than A's. The bidding continues: after his bid, A's total pile of red cards must exceed B's in value. This continues until one player stops. The black card is given to the highest bidder's 'bank' and the used red cards from both players are discarded, face down.

The winner of the 'auction' gets the move, and the cycle begins again. This continues until both players are without cards, whereupon the player with the largest 'bank' wins.

Play it; strategy will be discussed shortly. I also have a means worked out to play it over RHP.

2. Cribs
Moderately Offensive
25 Aug '04 17:282 edits
Originally posted by royalchicken
Two players (A & B) divide a standard deck of cards by color. One player takes all diamonds, the other takes hearts. Black cards are placed face down after shuffling. The winner of the previous game or a randomly selected player pla ...[text shortened]... ed shortly. I also have a means worked out to play it over RHP.
I want to play. I have a vague strategy that I cannot
yet formulate into words.

One observation is that you need at least 91 black card
points to not lose, and you have 91 red card points to
spend in the auctions. Likewise, your oppenent needs
92 black card points to win. So the problem is to spend
your 91 RCP in such a way that you expect to prevent
your oppenent from accumulating 92 BCP.

From this analysis, a couple heuristics are obvious.
One is that if you are spending more than 1 RCP/BCP,
then your opponent will expect to spend less than 1 RCP/BCP,
and thus he will expect to get the majority of the BCP.
Likewise, if you can get your opponent to spend more
RCP/BCP in an auction than you, and his RCP/BCP for that
auction is over 1, then you have actually won that auction
in the big picture.

Of course, these are just heuristics, and deterministic
analysis of particular game situations could show them
to be unoptimal. For example, if you have 90 BCP to your
opponent's 91, and there is 1 BCP left, you need to get it,
regardless of how much RCP you have to spend on it. But
hopefully the above heuristics, along with others to be

Dr. Cribs
3. royalchicken
CHAOS GHOST!!!
26 Aug '04 04:49
Originally posted by Cribs
I want to play. I have a vague strategy that I cannot
yet formulate into words.

One observation is that you need at least 91 black card
points to not lose, and you have 91 red card points to
spend in the auctions. Likewise, your oppenent needs
92 black card points to win. So the problem is to spend
your 91 RCP in such a way that you expect to preven ...[text shortened]... uristics, along with others to be

Dr. Cribs
That's a cool way of looking at it. Goldfish1 and I have worked out three distinct strategies; I'll see if they can be deduced from your heuristics.

We need a third person to make the game work by PM.
4. Acolyte
26 Aug '04 08:481 edit
Originally posted by Cribs
I want to play. I have a vague strategy that I cannot
yet formulate into words.

One observation is that you need at least 91 black card
points to not lose, and you have 91 red card points to
spend in the auctions. Likewise, your oppe ...[text shortened]... be

Dr. Cribs
There is a complication in that, because you only have 13 cards, you can't always bid exactly the amount you want to. In addition, the bidding rules mean that you can kill an auction if you have higher-value cards than your opponent. For example, if A bids 'King' then B has no reply.

Also, this is not a conventional auction, in that even the loser of the auction must give up any cards bid. In effect, at any one point you should pretend that the player with the lower bid has bid 0 and the player with the higher bid has bid the difference, and simply factor in the cards each player has in hand.

Instead of using 1 as the RCP/BCP value ratio, I would use (total RCP remaining)/(total BCP remaining), including the current auction (with the adjustment of bids mentioned above). Cards bid in previous auctions are 'dead', as if they never existed, and for the points won, one might as well see the trailing player as having 0 points, and the leading player the difference, with the points pool reduced accordingly. A strategy to simply maximise mean score wouldn't take into account the leader's points either, so we can ignore tose as well for a first approximation.

Here's an example to show what I'm talking about:

Game starts.
A reveals a King.
A bids Queen.
B bids King.
A bids King.
B concedes. (forced)

Now my analysis of this is that A has gained 13 BCP, at a net cost of 12 RCP, and there are 144 RCP and 169 BCP remaining.

13/182 = 0.71428; 12/156 = 0.76923

So this round is favourable to B.
5. royalchicken
CHAOS GHOST!!!
26 Aug '04 13:08
I forgot to mention the values of face cards:

Jack = 11
Queen = 12
King = 13
Ace = 14
6. 26 Aug '04 21:473 edits

If A bids 5 and B bids 7, can A respond with 3 (which makes his total bid higher than B's) or must he bid 8 or more, making it very much higher?
7. Cribs
Moderately Offensive
26 Aug '04 21:571 edit
Originally posted by royalchicken
I forgot to mention the values of face cards:

Jack = 11
Queen = 12
King = 13
Ace = 14
Then my numbers are off. I was counting 1 through 13, not 2 through 14.
8. royalchicken
CHAOS GHOST!!!
27 Aug '04 03:58
Originally posted by iamatiger