cicero wrote:Again re tahitiwahini's explanation of the Card Set Estimate ...
Having read pretty closely I think I now understand fully. The one feature that seems to be missing is incorporating the escalation factor in Escalating games.
Let's say there are 4 players all with 5 cards, does the calculation of the Card Set Estimate for the 2nd, 3rd and 4th players take into account that the set value will have changed by the time it is their turn?
Equally, if only 3 or 4 cards are held is the probability that they will cash in a set factored in to the value of the set for subsequent players?
Like I say, I think this isn't included yet. Can it be?
Cicero
You're right, it's not currently included in the Card Set Estimate.
The card set estimate originally started as an effort to quantify the armies that could be expected next turn in a flat rate bonus game. An assumption was made in order to simplify the calculation: that a player would cash in a card set as soon as one was formed. In a flat rate game that's actually a sound assumption because a strong case can be made that it's always an optimal decision to cash in a card set as soon as one is formed
in a flat rate game (yes, even if it's a red card set). The argument involves the discounted future value of armies and the negative effect going forward of retaining 3 cards of a single color in your hand. I don't mean to rehash the argument here, just to say that it is a reasonably sound assumption to make in a flat rate game.
So the calculation for the card set estimate developed for the flat rate game was merely extended to the escalating game keeping the same assumption that a player would cash in a set as soon as one is formed. That pretty clearly is not always a good assumption to make in an escalating game (especially in the early going in which no one really wants to be the first one to cash in a card set and receive only 4 armies for doing so). So the card set estimate in an escalating game becomes more complicated. We have to quantify not only whether a card set exists (which we already do) but the likelihood that a player will actually cash that card set.
What factors go into the decision about whether to cash a card set in an escalating game? Well clearly the current redemption value of cashing a card set is probably the most important. Theoretically it would be possible to develop a probability of cashing a card set for each redemption value (armies received).
For example,
Redemption Value: Probability of Cashing the Set
4: 20%
6: 40%
8: 75%
10: 90%
12: 100%
15: 100%
20: 100%
I'm not arguing these are the right numbers (since I don't play escalating for the most part I'm pretty sure they're incorrect), I'm just saying it could probably be estimated. Maybe in an escalating game it's always better to wait until you have to cash in a set, I don't really know.
What other factors are there? Well, to go to your example, if a player holds 5 cards we know he will be cashing a card set during his turn, which in turn will change the redemption value for the next player who takes his turn after him. I suppose that could be considered in the calculation, but it might not be easily done. Then there's the complication of a player holding 3 cards. Does he cash in or not? Maybe the player looks at the current redemption value? Maybe the player looks to see which other players could possibly cash in a set in their next turn after his if he cashes in his set in this turn? So it begins to get rather complicated to decide if a player is or is not going to cash in a set in an escalating game.
If someone wants to try to figure that out for the escalating game he should by all means give it a try, but I suspect it will be a rather difficult problem. Much more difficult than in the flat rate game. It may be that the way the card set estimate is currently calculated is not very useful for an escalating game (in which case it can be turned off in stocksr's script for those games), but it's not at all clear to me at this point what would replace it in the escalating game. I think I'll leave it someone who has a better grasp of the card cashing strategy in escalating games to work out.
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