Sunday 3:32 AM

to continue yesterday's discussion about how the bot calculates our % chance of winning, while it's not clear to me that the Hal bot does it this way, in general there are only 14,641 (121 numbers between 0 and 120 for each player) possible starting positions before each game.

This is a very small number in computational terms, and we could look at actual outcomes e.g. here

https://blog.cribbagepro.net/2012/11/cribbage-strategy-and-board-position.html

where we see that based on millions of *actual* games played, we can know that at 90-90, there is a 50/50 chance of either player winning. Indeed at any of the 14,641 possible positions there will be a fixed % chance that we win. And near the end of the game, there will be a significant difference between say 110-110 and 109-109.

For a bot, we could ignore those numbers and instead use 'best possible play', where we calculate the chance of winning based on our play. For example, at 116-116 there is a 100% chance of the game ending this deal. At 100-100 that is not so, but if we have already calculated that the chances from 116-116 are, say 70% pone, 30%, dealer, 0% new deal, the exact winning % for us at 100-100 simply becomes a matter of recursing through the possible outcomes, weighted by the chance that they occur, and then using our already calculated win % for each possible score outcome at the end of the deal.

Here we might note that if we are playing against humans, then humans make mistakes, so it might be in some ways better for the bot to use the human winning %, for matches between players who win overall (don't look at the results for poor players), in order to determine outcomes.

Then when we have a hand like today's A248910, it follows that for each of the 15 possible discards/hands, then each one will have a distribution of outcomes after the deal.

For example, if we toss 29 today, then we have at least 4 points, while pone may have any number of points. If play were to proceed 4K8A, then there is some chance the bot plays AA for 7.

Overall, there will be for any given toss therefore a table of possible outcomes, together with the likelihood they occur.

E.g.

4-12 occurs 1.6% of the time for the toss 29


If we multiply the frequency of the chance of a given post-hand score occurring by our known (calculated above) binary chances of winning or losing from any given position, it follows that we can calculate the exact chance to win or lose with any given toss, providing we peg the same way as the bot.

At 0-0, there should be a relatively straightforward correlation between the difference between our points and pone's points and our winning chance, but clearly as with yesterday as we approach critical score zones, then both players for instance adding 4 points will benefit one player more than another and indeed we might profit by scoring average 8.2 to pone's 10.2 points (-2 points) rather than say 12.2 points to 12.2 points (+0 points), in that position is important.

I do not think this is particularly important at 0-0, but some seem to think it is. I have not seen numbers supporting this though?!

Anyway, with today's hand we have trash. A4 tends to peg well, i.e. AX4. No 6 though. 489T pegs poorly.

So I'm holding A249 and will try to out-peg pone. The cut is interesting - if pone has a 4, he will certainly pair us. However 74% of the time, he was not dealt a 4, so the lead of a 4 looks like a conservative lead and we may get the X back for our 15-2. If we lead the A then pone probably won't pair, but he may spot the trap and reply with 7,8, etc.

So will lead the 4, and then we have 4-9-2, 4-X-A, etc.

Hope to get a better hand next time.

   

Sunday 3:33 AM

It's a toss-up between Toss (A T) and Toss (8 T) for me this morning.

Holding the RUN in *any* form seems toxic but if forced, I would perhaps consider Toss (A 2) to be the safest.

If I feel adventurous, I might try leading the 8 Card, and 'go fishing' for a Trey response.

   

Sunday 3:38 AM

First hand as pone is usually offense strategy. Start with 3 points and a hand that has more cuts for improvement over 2-4-8-9 (second choice).

   

Sunday 3:38 AM

This to me the most offensive keep with an eleven count in the hand that might influence the pegging. Eight lead here for me. dec

   

Sunday 4:33 AM

9 lead?

Maybe: 9(9)-5(14)-A(15/2)...
Or: 9(9)-X(19)-4(23)-5(28)-2(30)-go-A(31/2).

   

Sunday 5:58 AM

Great puzzle by Ras. Quite a few options to consider today with this subpar hand. I decided to keep the ‘5’ in my hand (A-4) along with the touching 9-10. Only the 3 or 7 cut do not help.

   

Sunday 6:07 AM

Tough call today. Do not like 48910, a lead of the 4 can get caught in a 3- card run with the 3 middle cards. Will go with a toss of 28 instead of A2, A4 or 24.

   

Sunday 7:03 AM

Kept the 8-9-10, hoping for help. Tossed the A-2 because it probably won't go *Boom*.

   

Sunday 9:41 AM

Really interesting puzzle today.

I saw 8-T as a "goldilocks" between a-2 or 2-4 (probably significantly more likely than 8-t to give up a huge crib) on the one hand and a-t (really bad hand) on the other.

But checking myself on discardpro and reading the comments I don't think I was right. I often overestimate the danger of a-2 it seems.

   

Sunday 10:03 AM

May not be the popular vote, but the small company is good company.

   

Sunday 11:09 AM

Agree with Mike F. Not too dangerous and I don't have to type War and Peace to explain.

   

Sunday 11:44 AM

kept the three points.

   

Sunday 3:02 PM

For a low-scoring hand such as this there are still several options - runs with 4-8-9-10 (A-2), A-8-9-10 (2-4), the pegger A-2-4-9 (8-10) and the defensive discard 2-4-8-9 (A-10):

4-8-9-10: 3pts - 5pts (Schell: 5.07) = -2pts

A-8-9-10: 3pts - 5¼pts (Schell: 5.33) = -2¼pts

A-2-4-9: 2pts - 5pts (Schell: 5.10) = -3pts

2-4-8-9: 2pts - 4½pts (Schell: 4.48) = -2½pts

Potential:

4-8-9-10: Improves with AAA, 222, 3333, 444, 5555, 6666, 7777, 888, 999, 101010, JJJJ = 38 cuts = 82.6% up to 6/8pts with 7777, 888, 999, 101010 = 13 cuts.

A-8-9-10: Improves with AAA, 444, 5555, 6666, 7777, 888, 999, 101010, JJJJ = 31 cuts = 31/46 = 67.4% up to 6/7/8pts with 5555, 6666, 7777, 888, 999, 101010 = 21 cuts.

A-2-4-9: Improves with AAA, 222, 3333, 444, 5555, 6666, 888, 999 + 15xXs = 42 cuts = 91.3% up to 6/8pts with AAA, 222, 3333, 444, 999 = 16 cuts.

2-4-8-9: Improves with AAA, 222, 3333, 444, 5555, 6666, 7777, 888, 999, 101010 = 34 cuts = 34/46 = 73.9% up to 5/6/7pts with 222, 3333, 444, 7777, 999, 101010 = 20 cuts.

Position:

As First Pone I'll play Offense and try to reach 18pts which is unlikely with this hand!

Pegging:

I think A-2-4-9 will peg best with three low cards and a high card while 2-4-8-9 should be next best.

Summary:

A-2-4-9 has the lowest starting value but has the most cuts for improvement and 16 cuts for 6/8pts. It also should peg best. So I'll throw the 8-10.

   

Sunday 3:45 PM

At 0-0* playing an Offense strategy for the pegging the dynamic expected averages and Win/Loss %s are:

__________________Our
Offense__Hand_Pegs__Crib___Total___W9 %____W10 %
2-4-8-9___4.46+1.70+(-4.26)=1.90____18.7____19.2
A-2-4-9___4.70+1.93+(-4.75)=1.88____20.6____19.7
4-8-9-10__5.24+1.30+(-4.84)=1.70____19.7____19.4
A-8-9-10__5.28+1.35+(-5.05)=1.58____19.9____19.2

Offense_____L9 %____L10 %
2-4-8-9______36.9____53.7
A-2-4-9______38.5____54.1
4-8-9-10_____38.0____54.0
A-8-9-10_____38.4____54.3

2-4-8-9 is very slightly best for expected averages by 0.02pt over A-2-4-9. However A-2-4-9 is slightly better for Win %s but slightly worse for Loss %s. As we're playing Offense I'll select the more Offense oriented keep and select 8-10 to discard.

After the 4 cut I'll lead the 4 and play Offense:

Lead_____________Our Pegging Pts.
4______________________2.22
A______________________2.03
9______________________1.88
2______________________1.33

   

Sunday 3:50 PM

I wanted to keep the run, looked at the discard tables and threw A 2

   

Sunday 5:55 PM

This hand was from a game with REX and have seen it a few times since in electronic and live play. The three consecutive small cards with high pegging value put holding the A-2-4-9 up just a hair. If you lack confidence in your pegging game, play the 2-4-8-9, which has the higher combined value. You can come real close to sorting out pegging value using this simple approach: A-5 have high pegging value, 6-7-8-9 have moderate pegging value, Jack has low pegging value, and 10,Q,K have none. Appreciate the positive comments relative to the quality of the puzzle. It's off., off., off., and lead the 4 keeping A-2 intact.