You're in a position where you cannot waste scoring opportunities. You need to think offense more than defense. The opponent might win with three hands (crib then first count), but he will win with six. You need to win with four (first count, crib, first count). It's a risk, but 78 might be the right throw. 75 and 84 may be more palatable, but you reduce your chance at a big score.

Excellent post and thought process. That seems to be the prevailing wisdom.

My thoughts at the time: I don't want to give the dealer a shot at a huge crib with first count next hand. Minimize those chances, while giving myself a solid shot at the W. Hence, I tossed 57.

Never really played cribbage seriously but in the position you describe, I probably throw 83. If my opponent had 10 more points I probably throw 87 because I need a big hand to win in most cases.

Spending a few minutes looking at online strategy sites, including the one Aicirt linked a few posts above, it seems that 87 is better than all other throws.

Throw 87	Cut	Score	Distribution	EV
3345	A	10	0.086956522	0.869565217
3345	2	12	0.086956522	1.043478261
3345	3	21	0.043478261	0.913043478
3345	4	20	0.065217391	1.304347826
3345	5	20	0.065217391	1.304347826
3345	6	14	0.086956522	1.217391304
3345	7	14	0.065217391	0.913043478
3345	8	14	0.065217391	0.913043478
3345	9	12	0.086956522	1.043478261
3345	10	12	0.347826087	4.173913043
			1	13.69565217
Throw 83	Cut	Score	Distribution	EV
3457	A	7	0.086956522	0.608695652
3457	2	8	0.086956522	0.695652174
3457	3	14	0.043478261	0.608695652
3457	4	12	0.065217391	0.782608696
3457	5	12	0.065217391	0.782608696
3457	6	9	0.086956522	0.782608696
3457	7	10	0.065217391	0.652173913
3457	8	7	0.065217391	0.456521739
3457	9	5	0.086956522	0.434782609
3457	10	7	0.347826087	2.434782609
			1	8.239130435
Throw 75	Cut	Score	Distribution	EV
3348	A	6	0.086956522	0.52173913
3348	2	12	0.086956522	1.043478261
3348	3	12	0.043478261	0.52173913
3348	4	12	0.065217391	0.782608696
3348	5	14	0.065217391	0.913043478
3348	6	6	0.086956522	0.52173913
3348	7	8	0.065217391	0.52173913
3348	8	12	0.065217391	0.782608696
3348	9	8	0.086956522	0.695652174
3348	10	6	0.347826087	2.086956522
			1	8.391304348

Don't know if this is helpful but I think I got the distribution and scoring right. I haven't checked it though.

At this point, you don't really need a big hand, you just need average hands. You can afford (to a point) the risk of your opponent getting some extra points, as long as you are not missing out on your points. Keeping 3345 guarantees you 10 points, with a fair shot at much more. Even a huge score by your opponent probably won't be game losing.

I think it's important to note that we count last next hand though. If they get a huge crib this hand, and count first next hand......

Ok, it's closer than I thought. I hadn't accounted for pegging on the third deal. An extra-big-scoring hand could be enough for the opponent. OTOH, we get to peg, too, and we will be much closer. Actually, it would be to our advantage to keep pegging low for this deal and the next. If it gets to a third deal, we should be about 1 or 2 away with the lead.

We need 32 and opp needs 48. After this deal, we should need about 15 (assuming we throw 78), and the opp should need around 20-28. Opp could get a miracle hand next deal, otherwise we should win easily. Even if opp only needs 16 after this deal, we are still a narrow favourite, I think.

7 8 is the throw that maximizes EV[Your hand - Opp Crib], but yeah that's not necessarily what you're trying to do here.

These spots are things I should think about more. At what point do you stop trying to maximize the EV of the hands and start worrying about minimizing the probability your opponent gets a big hand or maximizing the probability you get a huge hand. It's kind of obvious that once you're over 100 that this holds, but it's probably correct to consider stuff like that a lot earlier.

Unfortunately I don't think there is data that gives crib distributions. Just have EV.

Edited to remove data generated by buggy code.

It looks like we have different EVs for each of the discards you posted. I sure hope I don't have a bug in my program, lol.

Okay so I did have a bug in my program. I was allowing the discarded cards to be possible cut cards in the calculation. Oops. Bug is fixed, and my numbers now match Aaronk's for the 78 discard but they still differ slightly from the other two.

Here's what my program that calculates EV[Our hand - Crib] says. Seems like the only throw other than 78 that could possibly be in the discussion is throwing 48, but even that sacrifices 4.6 points of EV in our own hand to save 2.7 points in the opponents crib on average.

So 78 is probably the best throw.

------------------

EV of 3D 3C is -1.7234782608695651 (Hand: 4.956521739130435, Crib: -6.68)
EV of 3D 4H is 0.931304347826087 (Hand: 6.891304347826087, Crib: -5.96)
EV of 3D 5S is -0.28000000000000025 (Hand: 6.5, Crib: -6.78)
EV of 3D 7H is 2.9030434782608694 (Hand: 7.913043478260869, Crib: -5.01)
EV of 3D 8C is 3.0804347826086955 (Hand: 8.130434782608695, Crib: -5.05)
EV of 3C 4H is 0.931304347826087 (Hand: 6.891304347826087, Crib: -5.96)
EV of 3C 5S is -0.28000000000000025 (Hand: 6.5, Crib: -6.78)
EV of 3C 7H is 2.9030434782608694 (Hand: 7.913043478260869, Crib: -5.01)
EV of 3C 8C is 3.0804347826086955 (Hand: 8.130434782608695, Crib: -5.05)
EV of 4H 5S is -0.9991304347826082 (Hand: 6.260869565217392, Crib: -7.26)
EV of 4H 7H is -0.05391304347826065 (Hand: 4.826086956521739, Crib: -4.88)
EV of 4H 8C is 4.06 (Hand: 9.0, Crib: -4.94)
EV of 5S 7H is 1.5652173913043477 (Hand: 8.565217391304348, Crib: -7.0)
EV of 5S 8C is -1.3434782608695652 (Hand: 4.956521739130435, Crib: -6.3)
EV of 7H 8C is 6.065652173913043 (Hand: 13.695652173913043, Crib: -7.63)

Optimal Throw:
7H 8C
EV: 6.065652173913043

I think aaronk56 is looking at the scoring potential of our cards, whereas you seem to be looking at our value less the value to our opponent. Or something along those lines.

Looking at EV, as you have described it, is an early and middle game consideration. In the end game, we need to be looking at maximizing our score (sometimes minimizing our opponent's score), which is not always the same as maximizing the difference in scores.

I think the best way to express what I'm trying to say, is that we switch from strategic decisions and play, to tactical decisions and play, when we get near the end.

I think he is actually looking at EV of our hand for those three discards. The second column is each possible cut card. The third column is the actual score if that cut card comes up. The fourth column is the probability that that card is cut.

And I did find some minor errors in Aaronk's table (3457 2 is 6 points, 3457 7 is 9 points, 3457 8 is 9 points, and 3348 A is 8 points). Fixing those matches my code so I think the data I dumped should be correct EVs.



So when does the end game start would you say?

Hard to give a precise answer. Think about it this way: once you have hit the half-way point (60), there will be about four deals left in the game. Whoever deals next will score six hands to the other player's four before the other player's fifth and sixth. Look at your positions and see who has the advantage. This should guide whether you need to play defensively, offensively, or safely. Re-evaluate after each deal, of course. If the next dealer's six hands aren't going to be enough, the non-dealer will have seven hands instead of four. The fact that hands scored goes up by three means that you really need to be on top of where the advantage lies.

I hope this explains my position, I'm not always so good with that. I may not be correct, of course, but this is how I see it.

I'm throwing the 83 here all day. I have a 16 point lead. I have 4 counts to their 3 (and my 4 are 3 hands and 1 crib which is important). I'm banking I get out needing 32 with 3 hands, 1 crib and 3 pegging opportunities well before my opponent can go out needing 48 total. I'm also pegging very defensively here.

Any program that randomizes the opponents likely discard is WRONG. Good players try to set themselves up in the crib and I strongly expect that 6-7-8-9 have higher frequencies as dealer discards, often in combination. 78 is suicide. Methodology???

I don't hate 84 as two of 3s are gone. I see the hand ev is higher for 84. I don't have time to think that through if that makes sense to me. I know I like 83 (3457) a little better than 84 (3357) for pegging---more options, I'm never taking a 3-3-3 peg unless its 29 or higher. Although rare, the 3-3-3-3 for 12 is really bad. I also in this case would prefer not to discard suited cards since its in the example.

Willing to be a little suboptimal in my hand to minimize any big scoring opportunities for my opponent.

Yeah this makes a lot of sense. Thanks.


Yeah my program is not taking into consideration pegging at all, which of course should be a factor.



This is definitely true, but those crib EV tables are not computed that way. People have put them together in different ways. Some have simulated your opponent making "optimal" throws and some have actually hand-recorded the crib values from actual high-level play. Interestingly they both produced answers that are pretty similar to each other. I feel pretty confident in using them.

3-1 adv if dealer at the 90 pt mark. Never really start thinking about before then usually. May adjust if WH/WB but otherwise just play straight up until getting close to the final quarter.

I like the high level play tables. Link? It's been years since I've looked.

Here's another question that I don't really know the answer to. Suppose two good players of equal skill are in this position. How big of a favorite is Pone with 89 over the opponent? I feel like it would be helpful to get a feel for this but I haven't thought about that much. Is Pone 85%+ to win here?

It's the Rasmussen tables on each of these pages. First is discarding to your own crib and the second is discarding to your opponents crib.

http://www.cribbageforum.com/YourCrib.htm

http://www.cribbageforum.com/OppCrib.htm

Yeah, the half-way point is a good place to take stock and see where you are and make an end game plan. Sort of like jockeying for position before the final turn. Once you know what you need to do, you can pull the trigger on changes when it's time.

Loving this discussion. Such a great game.

Also think a direct EV calc is insufficient. I think the frequency and magnitude of bad outcomes is more important (should be represented in the EV but could be some bias toward a lot of 8-10 type cribs out weighing a small number of 16-24s that make the 78 dangerous. Stand deviation, median, range and frequency of very high cribs probably need to be considered, Likely more than a weighted average (EV).

Poker is a binary game: winner-loser.

Cribbage is more analog.

Yeah I agree with all that. And on top of that, what I am computing isn't taking several other things into consideration (pegging potential, Crib EVs don't consider the 4 cards you're keeping in your hand).

That being said, I became way better at this game by looking at the EVs, and I would bet that 75%+ of players playing online would too. I think these EV calcs will give the correct play in a lot of scenarios (maybe not the one we are talking about now). I learned a lot about this game by looking at these EV calcs. Like I hand't really realized how good things like Q6 is in your opponent's crib (really any two cards >5 that can't be connected in a run). It makes sense after you see the table, but I don't think I ever would have got there on my own.

Now that being said, based on this discussion I'm turning my attention towards trying to figure out win percentages by simulating an AI against itself. It won't be perfect but it will be a lot closer than anything else I could come up with. Right now I have no clue.

Where are folks playing on line these days?

Almost all my play with family and it's certainly social (not for $), so I've rarely thought this deeply about the game, though 10s of thousands of games since age 6 (I'm 52), I think I have pretty good instincts. I do ad hoc evs for the very end game by estimating outs to get the score I need but otherwise it's mostly gut. And I'm pretty defensive with my discards.

I've always guesstimated that 1/4-1/3 of games are winners or losers based on cards dealt. The remaining 1/3-1/2 are determined by skill in the long run.