Damn, I was sort of hoping to see "best math words ever played," but I'll take this content, too! I once was able to phony with ARCSINED because my opponent knew that I was a mathematician and that arcsine was "a math word of some kind".
haha, maybe this'll be a video later on! ARCSINE comes up a lot (anagrams ARSENIC/CARNIES) as it's a very high prob 7. Fun anagrammatical math pair: INTEGRAL/TRIANGLE. I even got EIGENMODE down once as a 9!
2 (very nuanced) corrections: 1. nCr is only correct as the denominator if you empty the bag here, if Chris plays only 1 tile, nPr (i.e., double nCr with 2 in the bag) would be appropriate since the order of the 2 tiles matter too (Chris would draw the first, Matt would draw the second) 2. Often, it is better to reweigh some of these based on the probability Matt kept specific tiles given his last play. To do this precisely, we'd need to invoke joint/conditional probability laws and Bayes rule (P( 2 tile letter combo in bag | FOO last turn) = \sum_{i in 4 tile combos} P( 2 tile combo in bag | Matt kept 4 tile combo "i")P(Matt kept 4 tile combo "i" | FOO last turn) = \sum_{i in 4 tile combos} P( 2 tile combo in bag | Matt kept 4 tile combo "i")P(FOO last turn | Matt kept 4 tile combo "i")P(Matt kept 4 tile combo "i")/(P(FOO last turn | Matt kept 4 tile combo "i")P(Matt kept 4 tile combo "i")+P(a play other than FOO last turn | Matt's seven tiles are "i"+FOO)P(Matt kept 4 tile combo "i"))). Practically, it is more common to eliminate certain combinations of letters (in this case, for instance, it is highly unlikely Matt kept 3 Rs, and likelier than not that he kept a blank), as calculating that probability statement is extremely hard in a small amount of time. So for making "statistically optimal" plays that lose, a substantial subset of them are not optimal with good ability to infer what sorts of combinations are likeliest to be in the bag, and this inference ability often separates world-class ones from "just" extremely good pre-endgame players.
Thanks Morris! You're absolutely correct on both of these -- as for (1) it's thankfully easy to see that playing off one tile isn't correct without diving deep into nPr vs nCr since RIN is such a likely and annoying possibility if Matt doesn't bingo immediately. I guess I didn't mention the possibility of P(I) instead of DIS(P)END if Chris wants to block the P, but again so often Matt just fishes with RIN and draws EPIGRAm/REAPInG or RAnGIER/REARInG that it's clearly worse than emptying the bag with DISPEND. As for (2) I don't think I've ever actually used Bayes' theorem in a real position, mostly because as you said it's completely impractical in the amount of time one usually has at that stage of the game. It also ends up being a lot of guesstimating both of the odds of your opponent keeping a certain letter combo given their play and also the odds that they'd actually make the given play with a particular rack (both of which may depend in part on the skill level of your opponent). I totally agree in this case it's highly unlikely Matt kept 3 Rs, not sure I agree it's more likely than average he kept a blank (he's rarely bingoing on this board with FOO + ? + 3 tiles, and I could see FOO with a lot of leaves similar to AERP) but the blank part seems kind of irrelevant anyway in terms of ranking plays since Matt has no available bingos anywhere without a blank. The inference that he didn't keep 3 Rs I suppose makes LARRUPER even unlikelier than the unbiased 1/36, which may slightly change some of my numbers, but blocking the P is still clearly correct (part of why I didn't bother going into the inference factor -- it doesn't materially change the overall analysis and there's enough complications already for this video!). That being said, you're 100% correct that inferences can sometimes turn a statistically optimal play into a suboptimal play, if I come across a good example would love to make a video on that (if you have one feel free to send my way)!
@@mackmeller I have one pre-endgame off the top of my head where I decided my play based on inference, though realized after it also just generally leads to slightly more wins as one of the pulls I thought would win for me in the inferior option would actually lose for me. I can send along.
My immediate instinct looking at this board was to play DISPEND as it is significantly better scoring/leaving than any bottom blocks, the lane on the bottom is significantly worse in general (and even more with the unseen pool), and it blocks a second marginal bingo line to the D. Doing the math like you did certainly confirms this, but doesn't even seem super necessary in this situation (as much as that can be for endgame situations). To me, the most instructive part of this video was why leaving 1 in the bag is worse than emptying, so great job on that!
Thanks Billy! Agreed that in a game (especially if time pressure was a factor) I probably wouldn't have done the exact math, it's not too hard to see the P is more dangerous without getting into the details.
my favorite part of scrabble is the pre-endgame math and probabilities. But I never have enough time to map it out and calculate it all in a real game so I usually have to go by intuition.
1 (very nuanced) correction: 14/36 is a probability, expressed as [# of outcomes in which Matt plays a bingo after DINO] divided by [# of possible outcomes after DINO]. The odds of Matt playing a bingo after DINO are 14:22 (or 14/22), expressed as [# of outcomes in which Matt plays a bingo after DINO] to (or divided by) [# of outcomes in which Matt does not play a bingo after DINO].
I loved the mathematical deep dive. I would have liked to know how this plays out in practice. You can't sit at the board for an hour and calculate every possibility. I'm guessing there is a hierarchy of how to spend your time in a competitive Scrabble game (I'm just a casual player).
Yep, there are absolutely positions where you don't have time to rigorously calculate every mathematical possibility (after all you only have 25 minutes for the whole game, which means you'll maybe have 10-15 left by this point, if not less). In that case you have to rely a lot on intuition and simplifying assumptions. For instance in this position, if I didn't have as much time I'd try to quickly scan through and once I saw there were a lot more possibilities through the P than on the bottom I'd block the P without going into the formalities of the math. But in general I always try to save a decent amount of the time for the preendgame and endgame, as mistakes at that point are far more costly than earlier in the game (as if you mess up early, there is still a whole game to make up for it).
I see you answered in the comments, but please tell us how the game actually played out in future videos like this. I was definitely curious after all that analysis
1 (very nuanced) correction: The odds need to be weighted by the likelihood that Chris has each of the letters. His previous play of FLEE allows inferences.
Hmm, not sure why Chris's last play is a factor? Chris is the one on move so it doesn't really matter what he had before, only what he has now. It's certainly true that Matt's last play (FOO) is a factor, though as Morris and I discussed above it's not nearly significant enough here to affect the ranking of the options.
Morris raises a good point about the last play FOO affecting the chances of Matt holding certain racks. I was wondering if HastyBot changes its strategy if the opponent does something suspicious such as fishing, squandering an S for two extra points, or (heaven forbid) keeping QQ in a game of giant scrabble when the bag is almost empty 😉 A similar situation exists in Bridge. If Declarer has nine cards in a suit missing the Q the math says playing the A-K is better than finessing, but not my much. In practice there will usually be something else about the hand (e.g. the auction, actions on previous tricks or even body language) that outweighs the math. And unlike Scrabble, most Bridge players will not dump the Q immediately unless it’s the only way to avoid a revoke! 😊
Thanks for your comment, and that's a cool analogy with Bridge! As for HastyBot, it doesn't alter strategy based on opponent plays, it's a fully equity-driven bot so it actually just plays the best play for points/leave without any strategic or dynamic considerations. Even more advanced bots like Quackle championship player don't take inferences based on opponent play into account (I believe), but they very much do simulate and take into account what they might give back to their opponent, whereas HastyBot will throw an E into the triple-triple lane for one extra point. But hopefully there will be even better engines soon!
Great video as usual. Two questions. One, my understanding is that, unlike tournament chess, players in Scrabble tournaments are allowed to write out notes during the pre-endgame and endgame analysis. Is this correct? Second question, even for players at the highest level, this in-depth analysis and calculations will take time. How much time is it wise to leave on the clock if a game appears to be headed for a cliffhanger endgame?
Great questions. As for Q1, yes -- you can write anything you want on your scoresheet during the game. I do this all the time when going through options in complicated endgames or pre-endgames. For Q2, there's no concrete answer, and it's something every player struggles with. As a rule of thumb, I personally try to leave myself 10 minutes for when there are fewer than 7 in the bag. It's also important not to use all your remaining time at that point on the pre-endgame, since often playing out the endgame will be nontrivial even if you make the right pre-endgame play. A big reason it's good to budget more time for the end is that while early-game decisions are important, they're usually way less consequential from a win % perspective than late-game decisions. For instance, let's look at the example from this video -- DISPEND wins 83%, DINO 61%. 22% is a huge difference in win %, and it comes simply from choosing to block one bingo line from another that late in the game. Early in the game, you'd have to probably miss a bingo or make some other huge equity mistake to have that large a loss in win %, simply choosing a wrong lane to block would maybe cost you a few %. So early in the game, it's not worth agonizing over a decision like that for 5 minutes when those 5 minutes could make the difference in you choosing the right decision in a pre-endgame like this one. An example I often give, and has come up in my HastyBot series before, is say your opening rack is RETAINS or something with tons of bingos. The biggest mistake you can make isn't where you play your bingo (as long as you don't do something crazy like STEARIN with the S on the star), it's spending too long to decide. You should spend at most a minute, and preferably less, in my opinion -- the difference that early on is minor enough that the time is way better kept for later.
Damn, I was sort of hoping to see "best math words ever played," but I'll take this content, too! I once was able to phony with ARCSINED because my opponent knew that I was a mathematician and that arcsine was "a math word of some kind".
haha, maybe this'll be a video later on! ARCSINE comes up a lot (anagrams ARSENIC/CARNIES) as it's a very high prob 7. Fun anagrammatical math pair: INTEGRAL/TRIANGLE. I even got EIGENMODE down once as a 9!
2 (very nuanced) corrections:
1. nCr is only correct as the denominator if you empty the bag here, if Chris plays only 1 tile, nPr (i.e., double nCr with 2 in the bag) would be appropriate since the order of the 2 tiles matter too (Chris would draw the first, Matt would draw the second)
2. Often, it is better to reweigh some of these based on the probability Matt kept specific tiles given his last play. To do this precisely, we'd need to invoke joint/conditional probability laws and Bayes rule (P( 2 tile letter combo in bag | FOO last turn) = \sum_{i in 4 tile combos} P( 2 tile combo in bag | Matt kept 4 tile combo "i")P(Matt kept 4 tile combo "i" | FOO last turn) = \sum_{i in 4 tile combos} P( 2 tile combo in bag | Matt kept 4 tile combo "i")P(FOO last turn | Matt kept 4 tile combo "i")P(Matt kept 4 tile combo "i")/(P(FOO last turn | Matt kept 4 tile combo "i")P(Matt kept 4 tile combo "i")+P(a play other than FOO last turn | Matt's seven tiles are "i"+FOO)P(Matt kept 4 tile combo "i"))). Practically, it is more common to eliminate certain combinations of letters (in this case, for instance, it is highly unlikely Matt kept 3 Rs, and likelier than not that he kept a blank), as calculating that probability statement is extremely hard in a small amount of time. So for making "statistically optimal" plays that lose, a substantial subset of them are not optimal with good ability to infer what sorts of combinations are likeliest to be in the bag, and this inference ability often separates world-class ones from "just" extremely good pre-endgame players.
Thanks Morris! You're absolutely correct on both of these -- as for (1) it's thankfully easy to see that playing off one tile isn't correct without diving deep into nPr vs nCr since RIN is such a likely and annoying possibility if Matt doesn't bingo immediately. I guess I didn't mention the possibility of P(I) instead of DIS(P)END if Chris wants to block the P, but again so often Matt just fishes with RIN and draws EPIGRAm/REAPInG or RAnGIER/REARInG that it's clearly worse than emptying the bag with DISPEND.
As for (2) I don't think I've ever actually used Bayes' theorem in a real position, mostly because as you said it's completely impractical in the amount of time one usually has at that stage of the game. It also ends up being a lot of guesstimating both of the odds of your opponent keeping a certain letter combo given their play and also the odds that they'd actually make the given play with a particular rack (both of which may depend in part on the skill level of your opponent). I totally agree in this case it's highly unlikely Matt kept 3 Rs, not sure I agree it's more likely than average he kept a blank (he's rarely bingoing on this board with FOO + ? + 3 tiles, and I could see FOO with a lot of leaves similar to AERP) but the blank part seems kind of irrelevant anyway in terms of ranking plays since Matt has no available bingos anywhere without a blank. The inference that he didn't keep 3 Rs I suppose makes LARRUPER even unlikelier than the unbiased 1/36, which may slightly change some of my numbers, but blocking the P is still clearly correct (part of why I didn't bother going into the inference factor -- it doesn't materially change the overall analysis and there's enough complications already for this video!). That being said, you're 100% correct that inferences can sometimes turn a statistically optimal play into a suboptimal play, if I come across a good example would love to make a video on that (if you have one feel free to send my way)!
@@mackmeller I have one pre-endgame off the top of my head where I decided my play based on inference, though realized after it also just generally leads to slightly more wins as one of the pulls I thought would win for me in the inferior option would actually lose for me. I can send along.
My immediate instinct looking at this board was to play DISPEND as it is significantly better scoring/leaving than any bottom blocks, the lane on the bottom is significantly worse in general (and even more with the unseen pool), and it blocks a second marginal bingo line to the D. Doing the math like you did certainly confirms this, but doesn't even seem super necessary in this situation (as much as that can be for endgame situations). To me, the most instructive part of this video was why leaving 1 in the bag is worse than emptying, so great job on that!
Thanks Billy! Agreed that in a game (especially if time pressure was a factor) I probably wouldn't have done the exact math, it's not too hard to see the P is more dangerous without getting into the details.
my favorite part of scrabble is the pre-endgame math and probabilities. But I never have enough time to map it out and calculate it all in a real game so I usually have to go by intuition.
1 (very nuanced) correction:
14/36 is a probability, expressed as [# of outcomes in which Matt plays a bingo after DINO] divided by [# of possible outcomes after DINO]. The odds of Matt playing a bingo after DINO are 14:22 (or 14/22), expressed as [# of outcomes in which Matt plays a bingo after DINO] to (or divided by) [# of outcomes in which Matt does not play a bingo after DINO].
I loved the mathematical deep dive. I would have liked to know how this plays out in practice. You can't sit at the board for an hour and calculate every possibility. I'm guessing there is a hierarchy of how to spend your time in a competitive Scrabble game (I'm just a casual player).
Yep, there are absolutely positions where you don't have time to rigorously calculate every mathematical possibility (after all you only have 25 minutes for the whole game, which means you'll maybe have 10-15 left by this point, if not less). In that case you have to rely a lot on intuition and simplifying assumptions. For instance in this position, if I didn't have as much time I'd try to quickly scan through and once I saw there were a lot more possibilities through the P than on the bottom I'd block the P without going into the formalities of the math. But in general I always try to save a decent amount of the time for the preendgame and endgame, as mistakes at that point are far more costly than earlier in the game (as if you mess up early, there is still a whole game to make up for it).
@@mackmeller Thank you for the thought out response!
I see you answered in the comments, but please tell us how the game actually played out in future videos like this. I was definitely curious after all that analysis
How did the game actually end?
Chris blocked on the bottom (I'm pretty sure with TI) and Matt bingoed with GRAPPlER on top and won
1 (very nuanced) correction:
The odds need to be weighted by the likelihood that Chris has each of the letters. His previous play of FLEE allows inferences.
Hmm, not sure why Chris's last play is a factor? Chris is the one on move so it doesn't really matter what he had before, only what he has now. It's certainly true that Matt's last play (FOO) is a factor, though as Morris and I discussed above it's not nearly significant enough here to affect the ranking of the options.
@@mackmeller Ah i mixed that up. Matt's last play then.
Morris raises a good point about the last play FOO affecting the chances of Matt holding certain racks. I was wondering if HastyBot changes its strategy if the opponent does something suspicious such as fishing, squandering an S for two extra points, or (heaven forbid) keeping QQ in a game of giant scrabble when the bag is almost empty 😉
A similar situation exists in Bridge. If Declarer has nine cards in a suit missing the Q the math says playing the A-K is better than finessing, but not my much. In practice there will usually be something else about the hand (e.g. the auction, actions on previous tricks or even body language) that outweighs the math. And unlike Scrabble, most Bridge players will not dump the Q immediately unless it’s the only way to avoid a revoke! 😊
Thanks for your comment, and that's a cool analogy with Bridge! As for HastyBot, it doesn't alter strategy based on opponent plays, it's a fully equity-driven bot so it actually just plays the best play for points/leave without any strategic or dynamic considerations. Even more advanced bots like Quackle championship player don't take inferences based on opponent play into account (I believe), but they very much do simulate and take into account what they might give back to their opponent, whereas HastyBot will throw an E into the triple-triple lane for one extra point. But hopefully there will be even better engines soon!
Great video as usual. Two questions. One, my understanding is that, unlike tournament chess, players in Scrabble tournaments are allowed to write out notes during the pre-endgame and endgame analysis. Is this correct? Second question, even for players at the highest level, this in-depth analysis and calculations will take time. How much time is it wise to leave on the clock if a game appears to be headed for a cliffhanger endgame?
Great questions. As for Q1, yes -- you can write anything you want on your scoresheet during the game. I do this all the time when going through options in complicated endgames or pre-endgames. For Q2, there's no concrete answer, and it's something every player struggles with. As a rule of thumb, I personally try to leave myself 10 minutes for when there are fewer than 7 in the bag. It's also important not to use all your remaining time at that point on the pre-endgame, since often playing out the endgame will be nontrivial even if you make the right pre-endgame play.
A big reason it's good to budget more time for the end is that while early-game decisions are important, they're usually way less consequential from a win % perspective than late-game decisions. For instance, let's look at the example from this video -- DISPEND wins 83%, DINO 61%. 22% is a huge difference in win %, and it comes simply from choosing to block one bingo line from another that late in the game. Early in the game, you'd have to probably miss a bingo or make some other huge equity mistake to have that large a loss in win %, simply choosing a wrong lane to block would maybe cost you a few %. So early in the game, it's not worth agonizing over a decision like that for 5 minutes when those 5 minutes could make the difference in you choosing the right decision in a pre-endgame like this one. An example I often give, and has come up in my HastyBot series before, is say your opening rack is RETAINS or something with tons of bingos. The biggest mistake you can make isn't where you play your bingo (as long as you don't do something crazy like STEARIN with the S on the star), it's spending too long to decide. You should spend at most a minute, and preferably less, in my opinion -- the difference that early on is minor enough that the time is way better kept for later.
Mack, thanks for your reply. Your comments on time management make a lot of sense.
@@mackmeller
1 (very nuanced) correction: you didn’t use Euler’s identity despite it being in the thumbnail!
lol! same with Stokes's Theorem I guess... I'll have to admit to click baiting slightly on this one :P
Did he really exchange 7 Es on the first turn?
Highly doubt it -- people will often input a dummy rack for an opponent exchange if they don't know what was exchanged
@@mackmeller OK, that makes more sense, since the odds of drawing 7 Es are less than 1 in 10 million.