Did you check out the ranges I posted earlier in this thread (post 62)? I'm guessing you have if you are 3-betting QTs.

Anyways, I feel pretty comfortable calling 4-bets OOP (especially with hands like QTs) but you just need to get in the mind set that even though the call is +EV relative to folding it's still a bad situation to be in. In other words, whereas you may lose 10BB overall if you fold after 3-betting with QTs, if you call and play well maybe you'll on average only lose 8BB. Long story short it basically just sucks to face a polarized range (which the 4-betting range almost certainly is).

I've played more HU recently than 6-max and I am regularly calling 4-bets OOP with hands like KJ, KQ, AJ, AT, QTs, QJs, etc. For 6-max it will depend on the positions and you can call 4-bets tighter, but I would not be afraid to call with medium strength hands getting such a great price. If you feel you can't play well enough OOP or your opponents are not 4-bet bluffing enough, then I'd just fold pre-flop and not worry too much about it until you feel more comfortable. There's nothing wrong with avoiding tough spots (at least a little bit) while you improve if you think playing in those tough spots will be -EV.

Just checked and you are correct, sorry about any confusion that caused. It's also a correction that's been put on the front page.

How do you feel about open-limping? Don't think I saw it mentioned in the pre-flop section.

hey Matt, love your book and videos. you really bring it for every piece of content you make.

having trouble with a spot, for figuring how many bluff raises to add for every draw raise

(X)(.47) + (1-X)(.2) = 389 ; X=.7 ------- would need 2.3 bluffs for every draw

however if i make draw equity lower, like .3, it would be:

(X)(.3) + (1-X)(.2) = 389 ; X=1.89

meaning for every draw, i need -2.1 bluffs.

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

say in a spot i raise 5 value, 10 bluffs.

if my draws have .47%, and i make it 5 value and 1 draw, i would need 12.3 bluffs.

if my draws have .3%, and i make it 5 value and 1 draw, i would need 7.9 bluffs?

also if you change the draw equity around, .4 needs 17 bluffs/draws. .35 needs -4.8 bluffs/draw. something is a bit off about this.

i feel like draws could just be treated with the amount of equity they have. like if a draw was 50% to hit the next street, the next street half the time it would be nuts, half the time it would still be a bluff. so it would be treated as half a value hand, so you would need to add one bluff. (this 50% to hit next street draw is just to make it easy)

if you start with 5 value hands, 10 draws, adding this made-up bluff makes it:

5 value hands, 1 50% to hit turn draw, 11 bluffs.

if the draw was 25%, it would be like 1/4 of a value hand.

5 value hands, 1 25% to hit turn draw, 10.5 bluffs.

I imagine you would open with multiple bet sizings in theory (the same way I think in theory you'd have many different bet sizings post-flop but that's easier to show), so it seems reasonable to limp some of the time.

I really try to emphasize easier strategies over more complex ones though. That's why I gave us even flatting opens in the small blind (explained in post #62).

As I just said in the above post, I tend to just emphasize easier strategies rather than get too bogged down with the numbers. I went through a phase where I tried to "calculate" things a bit more precisely (not that you can get very precise when using a model anyways), and I just don't think it's worth it.

I basically just bluff a little more if my draws are really good and bluff a little less if they're pretty weak. Even in my videos I never calculate the equity of my draws, but I'm not opposed to someone doing it if they find that works for them and they feel it justifies the effort.

hard to be confident about the rest of the book when that equation doesnt make much sense

What about the equation doesn't make sense? If your draw has very little equity then it's going to be a "bluff" itself, as all "bluffs" are is basically low equity hands (which is why people usually call gutshots, backdoor flush draws and backdoor straight draws, etc "bluffs" and not "draws"). So if you assign a "draw" or "value" hand such little equity the equation is going to give you some weird numbers (like negative numbers) as your "draw" is really just a bluff itself.

Just skip the book then

from .39 equity to .38, it goes from super high (189 bluffs/draw) to negative (-21 bluffs/draw).

39% is still a decent draw. still confused about the numbers, o well. i like the thing i came up with for raises, think its pretty logical.

this is the only thing i was confused about in the book, the rest of the math makes complete sense, and the practical advice is great

just skip life

You sure you're not reading it backwards? Like a 39% equity draw would say you need 189 draws per 1 bluff, which is basically just saying "If your "draw" has 39% equity then in this model you should count it as neither a "value bet" or "bluff" (which actually seems pretty reasonable).

i actually realized during my post but forgot to change, was still confused.

i think i get it now after the last explanation. also if you make it a 80% equity "draw", it would be .46 "draws" per bluff, which i think would mean for every one 80% "draw", you would need 2 bluffs? it would match the advice in your book anyways.

I don't want to interrupt the discussion too badly, but I just finished the book and I want to start doing a ton of work in flopzilla balancing ranges, like the hands listed in the sample hands at the end of the book. Would love to have a study partner to get feedback from as we do this sort of work together. PM me if interested.

And thanks for the book, Matthew. This one has enough substance to keep me busy for a while.

Hi Matthew: I couldn’t figure out how to post this directly to Sklansky – nobody seems to use the Discuss the Magazine Forum anymore. However, given that I wanted to discuss this issue with you anyway, the timing is propitious. In any event, if you agree with my analysis, hopefully there’s still time to pass this info along in time to do some good.

Please note that I spent a year consulting on the HM2 stats, wrote most of the current HM2 stat definitions, and I’m probably the only person alive who has written more algorithms and calculated EV more times than you have …

The problem is that Sklansky’s EV calculations in his December article are incorrect. He is calculating EWin rather than EV - a very common mistake. See Sklansky’s current article, Norman Zadeh’s Simple GTO Game, Part One (December 2013); contrast to EV calculated correctly - See http://www.twoplustwo.com/magazine/i...ploitation.php

In this current hypothetical “There is one round of betting, each player has contributed $1 to the pot and the bet is now $2, which is the remaining stack of at least one of the players. In other words, if there is a bet, it is the size of the pot and there’ll be no raise. If it goes check-check, there is a show down.” This is analogous to River Jam situations that you analyzed in Part Eleven: River Play.

I’m just going to look at the River Jam decision, beginning with the following basic algorithms:

Decn EV (River Top 14% Jam OOP HU) = (OppFold%)(RiverPotsize) + (OppCallWin%)( RiverPotsize + Betsize) - (OppCallLose%)( Betsize), p. 332, and
Decn EV (River Bottom 7% Bluff Jam OOP HU) = (OppFold%)(RiverPotsize) – (OppCall%) (Betsize). p. 353.

Note: Because of the way River Jams are calculated, it doesn’t make any difference to the EV calculation whether the dead money in the beginning River Pot comes from the players or from Santa. On the other hand, the amount of player’s dead money is required information to calculate EWin in this hypothetical.

Assumptions: RiverPotsize = $2; RiverBetAmount = $2 and RiverCallAmount = $2; FinalPotsize when called = $6.
Player A Bets and wins when OppFolds: Player wins $1 on the hand, but Player’s profit on the bet is $2.
Player A Bets and wins when OppCalls: Player wins $2 on the hand, but Player’s profit on the bet is $6 - $2 = $4.
Player A Bets and loses when OppCalls: Player loses $4 on the hand, but Player’s loss is limited to ($2).
Player A bluffs and wins $1 on the hand: but Player’s profit on the bet is $2.
Player A bluffs and loses $3 on the hand: but Player A’s loss on the bet is limited to ($2).

Summary of EV of Jamming (First 3 and last 2 calculations, Bet is $2 (Pot Bet):

....Jam ($2)............Freq%...Profit...EV
Top 14 v Top 14 Win... .0098..... 4.... .0392
Top 14 v Top 14 Lose.. .0098.... (2). (.0196)
Top 14 v 14 - 50 Win.. .0504...... 4... .2016
Top 14 v 50-100 Win... .0700..... 2... .1400
Bluff v Top 50... Lose.. .0350.... (2). (.0700)
Bluff v Top 50... Win... .0350..... 2.... .0700
...............................21.00% ....... .3612

Total Bet = $0.42 EV $0.3612 ROI = 86% Total Final Pot = $1.12

Sklansky’s EWin calculations:

....Jam ($2)............Freq%...Win...EWin
Top 14 v Top 14 Split .0196...... 0 ... .0000
Top 14 v 14 - 50 Win .0504...... 3 ... .1512
Top 14 v 14 - 50 Win .0700...... 1 ... .0700
Bluff .. v Top 50 Lose .0350..... (3). (.1050)
Bluff ...v Top 50 Win. .0350...... 1.... .0350
.............................21.00% ......... .1512

Total Bet = $0.42 EWin $.1512 ROI = 36% Total Final Pot = $1.12

AT LAST! Irrefutable proof that it happens to everybody. ☺

Larger issues: Sklansky concludes that “In other words, the first position player loses an average of a bit more than $.08 per hand”. This is EWin, not EV. We know that EV (Player) <> -EV (Opponent) and that EV (Player) + EV (Opponent) = $2 for this hypothetical River Jam OOP HU decision. Without making the rest of the calculations, we also know that the EV River Jam OOP HU decision in this hypothetical is at least $.13 per decision (-.08 + (.3612 - .1512)). These are both great examples of how to calculate EV as a discrete random variable in a hypothetical. Once the EV calculations are fixed, Part II of the current hypothetical should be a great example of one way to use and apply decision theory for exploitative purposes to maximize the expected value of a discrete random variable. I call this methodology paragraph C of the Optimal Corollary to the Fundamental Theory of Poker.

Matthew: Note the subtle distinction between “bet, wager, play, decision or situation can be expected to win or lose on average” and “hand can be expected to win or lose on average.” The first is profit and EV. The second is EWin.

p.s. There a number of valid reasons why default EV stats don’t, and never will, calculate profit – they define and calculate EWin. You got blindsided – big-time. My apologies. Glad to share this very long and complicated story with you, just not in the forums. ☺

You might find these “unified” theoretically consistent definitions helpful:

Expected Win is NetAmountWon adjusted for Pot Equity (Total Adjusted Winnings).
Expected Value (decision or situation faced) tells us how much we expect to profit (adjusted for Pot Equity) on average and considers all money previously invested in the pot as dead money. This means folding at the point of calculation always has an expectation of zero (EV) regardless of whether we lose money overall on the hand (EWin).
Expected Value (decision or situation faced) is the weighted average profit (adjusted for Pot Equity) of making a decision to act (or not to act).
Expected Value (decision or situation faced) is the weighted average value of making a decision to act (or not to act).

Looking forward to your response …. PM would be terrific.

General info. See http://en.wikipedia.org/wiki/Expected_value

Not your typical source text, but it actually contains a nice overview of the core concepts of advanced EV analysis. IMHO ☺

EV is the weighted average of all possible values of a random variable. In other words, each possible value that the random variable can assume is multiplied by its assigned weight, and the resulting products are then added together to find the expected value.

Let X be a discrete random variable taking values x
1, x
2, ... with probabilities p
1, p
2, ... respectively. Then the expected value of this random variable is the infinite sum x1p1 + x2p2 + .... xnpn.

This is how stats are calculated. Example: Decn EV (BNATS) = BNATS OppFold%* Decn EV (BNATS OppFold) + BNATS OppCall%* Decn EV (BNATS OppCall) + BNATS Opp3Bet% * Decn EV (BNATS Opp3Bet). p. 57.

In the case of a continuous random variable (that is, a random variable that can assume a theoretically infinite number of values, such as the height of a person, (or the possible outcomes of a decision made or situation faced in a theoretical analysis or brick and mortar hand), expected value is the limit of the weighted sum, i.e. the integral. EV or optimal equations are generally not solvable or provable as continuous random variables.

These days, Players DB’s often contain 1,000,000+ hands – ergo, the law of large numbers applies. In probability theory, the law of large numbers (LLN) is a theorem that describes the result of performing the same experiment a large number of times. According to the law, the average of the results obtained from a large number of trials should be close to the expected value, and will tend to become closer as more trials are performed.

Hard to miss the significance of this:

In decision theory, and in particular in choice under uncertainty, an agent is described as making an optimal choice in the context of incomplete information. For risk neutral agents, the choice involves using the expected values of uncertain quantities, while for risk averse agents it involves maximizing the expected value of some objective function. (a.k.a. paragraphs A and B of the Optimal Corollary to the Fundamental Theory of Poker).

Note: In order to enable advanced EV analysis, “Unified” theoretically consistent definitions must be consistent with statistics, probability and decision theory AND both methods of calculation (discrete and continuous). Fortunately, it all fits together like a glove, and the results are spectacular. Too bad EV analysis of the decisions (hands in DB) is a complete waste of time unless one calculates EV, optimal and informational stats as discrete random variables.

BNATS = BN Attempt to steal or Raise First In. FEQ = Opp Fold or Folding Equity

Hey Guys – Unfortunately, what I’ve been trying to explain sounds like complete gibberish unless you know what I'm talking about – pretty much the ultimate Catch 22. Can’t explain it, but I certainly can illustrate the basics with a few screenshots – and I finally figured out how to get them posted. ☺

There are 2 major causes of confusion about advanced EV analysis:

1) The method of calculation of EV in statistics, probability or decision theory depends on whether it is calculated as a continuous or as a discrete random variable, and
2) The definitional issues related to EV aren’t limited to the confusion caused by players using EV and Equity interchangeably - that’s just the teeny, tiny tip of the iceberg. Insofar as decision-based advanced EV analysis is concerned, the problem is that everyone – and I do mean everyone - is unaware of, or confused by, the definitional issues related to what I refer to as “the unholy trinity of EV/EWin/PotEQ”.

THINK ABOUT IT: Anybody who has read Matt’s book knows that when BNATS and the blinds fold, EV (BNATS) = 1.5 bb (if both blinds posted). Furthermore, since EV (Fold) = 0, then EV (SB Fold) = 0 and EV (BB Fold) = 0. Numbers don't lie - you can see this for yourself.

Straight up – Nobody realizes that there’s a distinction between EV and EWin because of… lots of reasons. The common wisdom is that EV = EWin = EV bb/100. Unless one thinks that 0.00 = -.50 = -1.00 = 2.44 = -8.20, it should be clear that EV <> EWin (players are actually confusing at least 4 entirely different concepts under the single rubric of EV). Consider the fact that Matt and the brain trust at 2 + 2 haven’t figured it out yet – well, obviously, it’s not a simple matter. No wonder everybody seems to think I’m speaking in an alien language.

These “unified” theoretically consistent definitions should help players distinguish EV and EWin:

Expected Win is NetAmountWon adjusted for Pot Equity (Total Adjusted Winnings).
Expected Value (decision or situation faced) tells us how much we expect to profit (adjusted for Pot Equity) on average and considers all money previously invested in the pot as dead money. This means folding at the point of calculation always has an expectation of zero (EV) regardless of whether we lose money overall on the hand (EWin).
Expected Value (decision or situation faced) is the weighted average profit (adjusted for Pot Equity) of making a decision to act (or not to act).
Expected Value (decision or situation faced) is the weighted average value of making a decision to act (or not to act).

This is how stats are calculated. Example: Decn EV (BNATS) = BNATS OppFold%* Decn EV (BNATS OppFold) + BNATS OppCall%* Decn EV (BNATS OppCall) + BNATS Opp3Bet% * Decn EV (BNATS Opp3Bet). p. 57.
Math check: .519*1.50 + .361*.38 + .12*-2.06 = 0.67.

The method of calculation of EV in statistics, probability or decision theory depends on whether it is calculated as a continuous or as a discrete random variable.

Theoretical (or brick and mortar hands) analysis calculates EV as a continuous random variable: it looks at all the theoretically possible outcomes of a specific decision for a single hand. EV or optimal equations are generally not solvable or provable as continuous random variables.
Custom Decn EV – like all stats - are calculated as discrete random variables: they measure all the actual possible outcomes of a specific decision over a range of hands. Decn EV stats combine statistics, probability theory, the law of large numbers and DB programming techniques.
EV stats calculate the EV of a decision and the EV of all of its alternative lines – without exception or theoretical limitation. Preflop, Flop, Turn, River, Showdown, Multi-street – makes absolutely no difference. Optimal stats: Add decision theory. EV stats are the door to advanced EV analysis: optimal stats are the key. Informational stats: the bounty within – the untapped information in a player’s DB is truly staggering.

The accepted wisdom is that EV can’t be calculated is just plain wrong. It’s clearly the most damaging: but it’s only one of many misassumptions of theoretical analysis that has prevented EV analysis (hands in DB) from ever getting off the ground.

Note: Although Matt clearly suspected what was amiss – and avoided using the expression “EV can’t be calculated” like the plague – he couldn’t come right out and say so without mathematically sound justification. He also couldn’t know that all stats are calculated this way. Alas, only a handful of people do. Now everybody who actually reads this post will know.

When I posted earlier that "In order to enable advanced EV analysis, “Unified” theoretically consistent definitions must be consistent with statistics, probability and decision theory AND both methods of calculation (discrete and continuous). Fortunately, it all fits together like a glove, and the results are spectacular. Too bad EV analysis of the decisions (hands in DB) is a complete waste of time unless one calculates EV, optimal and informational stats as discrete random variables. " I'm sure everyone thought I was exaggerating. But seriously, anybody who misses the significance of:

EV stats calculate the EV of a decision and the EV of all of its alternative lines – without exception or theoretical limitation. Preflop, Flop, Turn, River, Showdown, Multi-street – makes absolutely no difference - well ...

Alternative lines: Here’s EV (BNATS) and ~ 15 EV stats of its alternative lines.

When EV is calculated as a discrete random variable and Player is IP, Player has 3 alternative lines when facing a bet or raise (Fold, Call, or Raise). Otherwise, he only has 2 (Bet or Check). Opponent has the same 2 or 3 alternative lines, but I prefer referring to them as opponent’s tendencies rather than alternative lines.

Unfortunately, this is completely different than theoretical analysis; if you're trying to analyze your online hands the way you're used to, you are wasting your time. You're also missing out on all the advantages and information available in your DB.

This is a good way to utilize the replayer more effectively - if a player wants to use EV analysis to analyze a decision in a hand in their DB more effectively. Normally, in an All-In situation, I’d only include 2 or 3 EV stats, and I’d also include 10 -15 informational stats like opponent’s tendencies, potsize, pot odds, SPR, actions on previous streets – you know, the obvious factors that players would tend to find most useful in a specific situation (or to use as input into 3rd party software).

Math check: Final Pot = $104. Less Rake $101. Pot Equity = 67.65% or $68.33. Less: Amount to call $37.25. Therefore, EV (BNATS Face, Dfnd, or Call 5Bet) for this decision is $31.08 or 62.16 bb.
However, the EWin = SklanskyBucks = 33.18 bb.

Please also note that there is no such thing as an EV graph. These graph SklanskyBucks in $ or bb/100 and are, therefore, EWin (Expected Win-rate) graphs. Have to admit, I’ve had many of the most surreal conversations with players over the last several years trying to explain to them that Expected Win-rate graphs … graph Expected Win-rate rather than Expected Value, and everybody looks at me like I’m confused.

"I spent a year consulting on the HM2 stats, wrote most of the current HM2 stat definitions, and I’m probably the only person alive who has written more algorithms and calculated EV more times than you (Matt) has …" "Matt p.s. There are a number of valid reasons why default EV stats don’t, and never will, calculate profit – they define and calculate EWin. You got blindsided – big-time. My apologies." Ironically, I ended up writing the current HM2 EV stat definitions, which are EV = EWin. Best I could do at the time, but at least the definitions are consistent with their algorithms. Oops.

Here’s a basic BNATS Optimal stat report. Note that I’ve customized it to reflect Matt’s concepts, methodologies and calculations. p 57 – 59, and recommended Hand Charts p 85 (and by the way, the A4s – A2s are duplicated in the flatting and 4Bet range – I believe that they belong in the flatting range).

I posted earlier, "In decision theory, and in particular in choice under uncertainty, an agent is described as making an optimal choice in the context of incomplete information.
A - For risk neutral agents, the choice involves using the expected values of uncertain quantities, while for
B - risk averse agents it involves maximizing the expected value of some objective function."
C - Part II of Sklansky's current hypothetical should be a great example of one way to use and apply decision theory for exploitative purposes to maximize the expected value of a discrete random variable.

Gee, I wonder:
- if this hypothetical player’s Defending range against vs a 3Bet from the blinds is optimal? Is player risk nuetral? Paragraph A of the Optimal Corollary to the Fundamental Theory of Poker.
- If player Is not changing strategy and 4 Betting enough because player is risk adverse – is it costing him money? If so, any idea how to measure and get a good estimate of how much? Paragraph B.
- How about player’s exploitative range against unknown SB and BB players? Known opponents? Paragraph C.

Think Bayes’ Theorem and decision-tree analysis.

Theoretically consistent EV and EV stats (Decn EV) must have the following basic properties, Profit, Profit, Profit, Profit, Profit, Profit:
Calculates weighted average profit (loss);
Includes Player’s dead money;
Calculated from a specific decision point (decision or situation faced);
Multiple decision points (and EV stats) possible for every hand;
Adjusted for Pot Equity;
Cannot be zero-sum because there is always dead money in pot;
Expressed in terms of currency or bb per (decision or situation faced);
Since Decn EV (Fold) = 0, then, by definition Decn EV (SB Fold) = 0 and Decn EV (BB Fold) = 0.

Theoretically consistent EWin and EWin stats must have the following basic properties:
Calculates the expected results (Net amount won (lost));
Does NOT include Player’s dead money;
EWin can be calculated for the hand overall
Adjusted for Pot Equity;
Must be zero-sum;
Expressed in terms of currency, bb, or per 100 hands;
EWin (SB Fold) = (0.5) bb and EWin (BB Fold) = (1.0).

Stats are simply algebraic equations: they calculate and express the results of mathematical formulas.
Ain’t no room for context: Since stats are equations, the stats themselves, as well as every variable used to calculate them, have to be defined very specifically, and, in toto, the stats and all the variables used in their formulas have to be logically consistent with each other – resorting to context is verboten.

IMO, it’s much less confusing to avoid the term EWin altogether. I use:

SklanskyBucks tells us the amount we expect to win or lose overall on a hand (adjusted for Pot Equity).
WR (Total Hands) - Expected Winrate or TotalAdjustedWinnings – SklanskyBucks for Total hands expressed in currency/100 hands or bb/100 hands.

WR (decision or situation faced, position, action, range, anything other than Total hands): Expected Win-rate Contribution. This is a “hybrid” concept in that the number of hands used in the numerator is not Total Hands. Extremely useful information stat, because it measures components of Win Rate, potential costs of mistakes, leaks, worth of exploitative play, etc. I use it as a companion stat to EV. Examples: WR (EP, MP, CO, BTN, SB, BB); WR (BNATS); WR (River Value Jam IP HU).

It's a simple matter to write ~98% of Matt's equations as stats. Except for the amounts posted in the blinds, we'll get the same results every time - his equations and formulas are not at issue.
My point is that unless you understand
- the distinction between EV and EWin (and a couple other definitional issues), and
- the how and why and the advantages of calculating EV, optimal and informational stats as discrete random variables,
- advanced EV analysis of the decisions (hands in DB) is a complete waste of time.

In light of the foregoing, it might be worth the time to revisit my comments re: Sklansky's calculations.

Assumptions: RiverPotsize = $2; RiverBetAmount = $2 and RiverCallAmount = $2; FinalPotsize when called = $6.

Player A Bets and wins when OppFolds: Player wins $1 on the hand, but Player’s profit on the bet is $2.
Player A Bets and wins when OppCalls: Player wins $2 on the hand, but Player’s profit on the bet is $6 - $2 = $4.
Player A Bets and loses when OppCalls: Player loses $4 on the hand, but Player’s loss is limited to ($2).
Player A bluffs and wins $1 on the hand: but Player’s profit on the bet is $2.
Player A bluffs and loses $3 on the hand: but Player A’s loss on the bet is limited to ($2).

Of course, I left off my favorite because its not related to Player A's jam. But Player A's EV (Check Fold) = -($1) is a dead giveaway.

From everybody else's perspective, it must seem presumptuous beyond belief that I'm sitting here pointing out to Matt and David things that they don’t know about EV. It's really hard for me to believe it as well. I chose David's EV (mis)calculations to illustrate how insidious and pervasive the confusion about EV and EWin is and to make sure that everybody understands that it's not related to knowledge of the subject matter. It has to do with the alternative methods of calculation of EV and basic programming skills, insider knowledge about HEM stats, sequential processing and DB analysis.

Fortunately, I CAN'T make these mistakes because theoretically consistent EV equations (stats) won't let me. Just take one look at the 1st screenshot.



Give me a break.
From my perspective, I’d have to be blind, deaf and terminally stupid if I didn’t know that EV <> EWin or didn't notice an EV (Fold) stat calculation that wasn’t 0.00.



Kinda hard to miss that stats are calculated as discrete random variables. Can't argue with certain characteristics of equations and numbers.

That's it for EV and Optimal Report stats 101. Would appreciate a little constructive feedback. Happy New Year.

Would appreciate a 5-10 line abstract of the thesis (the past dozen or so posts).