Three-Way Tie for Last

Saw an item this morning regarding last night’s episode of Jeopardy! Hear about this?

Heading into “Final Jeopardy” -- that game-ending last clue when players are allowed to wager whatever they like from their total on their response -- two players were tied for first with $13,800 apiece while the third had $6,000.

Players are told what the category for the final item is but not the clue itself before having to write down the amount they are willing to wager. I’m not sure what the category was, but the clue turned out to be a not-so-easy one: “A 1957 event led to the creation of a national historic site in this city, signed into law by a president whose library is now there too.”

Here, you can look at how it went:

That’s right -- three incorrect answers, and all three contestants went all in with their wagers, meaning a three-way tie for last. Or first. Or whatever. Each did get a consolation prize -- $2,000 each for the two players tied for first heading into “Final Jeopardy,” and $1,000 for the other player. But none gets to come back tonight.

Seems to me that when it came to calculating that final wager, the third-place player chose the only amount that was unequivocally incorrect here -- namely, to bet all $6,000. Disregarding results, anything from $0 to $5,999 would have been better, and probably equally as good. Meanwhile the decision for each of the two players tied for first was less cut-and-dry.

Remember, you only have to win by one dollar in order to take home whatever your final total turns out to be (and to come back the next night), and those finishing second and third don’t get any of the money they’ve accumulated in the show, only the consolation prizes for appearing.

Okay all you poker players (and ICM-smarties) -- what would you have recommended here for these three players when placing their wagers for “Final Jeopardy”?

Image: 3-Way Tie (For Last) (1990), The Minutemen.

Getting It Wrong

Watched that Thursday Night Football clash last night to the bitter end. Was a pretty poorly played game, with a lot of endgame weirdness to create an unexpected outcome and thus enough extra drama to keep me watching to the last.

A couple of weeks pack I shared a pleasure-pain ranking system for evaluating my Pigskin Pick’em picks. Last night’s game -- which I got wrong -- ended up way over on the extreme pain side of the spectrum.

I picked the Steelers, who thanks to an unreliable field goal kicker and some especially bone-headed coaching decisions allowed the Ravens to complete a 13-point comeback to win in overtime. Making things worse, many in the pool (including the leader) had taken Baltimore, making it doubly unpleasant to lose after having with a couple of minutes left prematurely congratulated myself for having picked a winner. (Amateur move.)

I mentioned how the Steelers’ field goal kicker, Josh Scobee, had a rough night, missing two in the fourth quarter. Both were very late, and both happened with Pittsburgh ahead 20-17. The first miss was a 49-yarder with 2:29 to go, and the second a 41-yarder with 1:06 left.

As I had a rooting interest, I wasn’t happy to see Pittsburgh missing field goals. I was less happy, though, to see the Steelers even attempt them. Why? Because I was convinced each time that trying a field goal lessened rather than increased their chances of winning the game.

If Pitt. had made either FG, they’d have gone up by six, thereby forcing Baltimore to drive the length of the field in the hopes of scoring what would be a winning TD. Down just three, Baltimore instead played for the tie and overtime, and after being unsuccessful on the first try did manage to do so when given a second opportunity.

Because Pittsburgh missed their FG attempts, Baltimore played for the tie and overtime. They failed the first time, but given a second opportunity the Ravens were able to get close enough to kick a long tying field goal at the end of regulation.

When Pittsburgh missed the first FG attempt, then, it more or less assured they would not lose the game in regulation, as the Ravens then went for the tie (they still could lose in OT, of course). If Pittsburgh had hit the FG, however, there would have been a non-zero chance they could lose in regulation. Weighing the chances of losing in overtime (after Balt. hit a game-tying FG following a Pitts. miss) versus losing in regulation (after Balt. scored a game-winning TD after a Pitts. make), I suppose hitting that first FG would have been marginally better than punting, although not by much.

But when the Steelers held Baltimore on downs, then faced a similar decision with less time on the clock (and Baltimore having used their timeouts), trying the FG again was surely a poor decision (especially considering the inconsistent Scobee had just missed one). Punting and pinning the Ravens inside the 20 would have been a much, much better choice. Doing so would have further reduced the chance of Baltimore driving for a winning TD (because of a longer field), although that likely wouldn’t have been the Ravens’ aim, anyway, since the tying FG would’ve been a primary goal for them.

A Twitter exchange at the time involving Grantland’s NFL guru Bill Barnwell succinctly summed up the situation. A follower asked him “why did Pitt go for FG?” adding “Wouldn’t you rather be up by 3 than 6 there? Have Balt play for FG and OT instead of TD and win?” (This is the point I’m making.)

Barnwell’s response delivered the same observation in a different way: “NFL coaches optimize decisions to put off losing for as long as possible, not to win.” This was a point he made in greater detail a couple of weeks ago in a column where he was describing almost exactly the same scenario in a different game:

“Trailing by three in a two-minute drill, coaches will almost [always] settle for a field goal to try to push the game into overtime,” explained Barnwell. “They optimize their decision-making to tie, which only improves their chances of winning to 50 percent (or whatever the implied odds were from the pregame spread), because they still have to win in overtime. Down six and without any other choice, they get aggressive and optimize their play calling to try to score a game-winning touchdown.”

I’m trying to think of a decent poker analogy here (and struggling a little). Looking at it from the perspective of the team that is behind, being down a FG and playing for a tie would be like being short-stacked enough to fold your way into the money. Meanwhile being down six forces a team’s hand (so to speak), kind of like being too short to take the passive line of folding into the money and instead having to go into shove-or-fold mode.

That’s not really describing the perspective of the team that is ahead here, though, who makes a choice that seemingly provides a temporary benefit but isn’t the best decision long-term. That would be a little like risking too much to win a single tourney hand when doing so doesn’t really improve your chance at realizing the more substantial goal of winning the event.

All of which is to say, Pittsburgh’s choice to try a long field goal to go up six with less than two minutes to go -- a decision they ended making twice -- was at the very least questionable even if they’d had a more reliable kicker. And in fact, I tend to think it was just plain wrong (especially in the second instance).

I’m remembering as I write this a game from two years ago in which Pittsburgh similarly decided to try a long field goal with less than two minutes to go in a close contest. In that case the game was tied and the FG was 54 yards, i.e., a would-be career long for their then-kicker Shaun Suisham. Things went similarly badly for the Steelers there, too, and in a post here I surmised their chances of winning were decreased merely by the decision to kick a FG instead of punt.

Pittsburgh made some other bad choices, too, most glaringly with regards to a couple of fourth down calls in overtime. Indeed, I think last night it was obvious the team I’d picked had hurt their own chances of winning because of in-game decisions -- i.e., because of things they could control -- which definitely added to the pain of getting it wrong.

“Trust everybody, but cut the cards”

Found myself earlier today involved in one of those click-click-click loops that might’ve begun as “research” but devolved into the usual mix of curiosity and voyeurism internet surfing usually represents.

It wasn’t fruitless, however, as one of the items I ended up lingering over had to do with the origin of that often-cited pokerism “Trust everybody, but cut the cards.”

I’ve always liked that quote because of pithy way it combines both the social or “community building” aspect of poker and the fact that at its core poker is a game of self-interest. We have to “trust” others to some extent if only to get a game going, but once hands are dealt we necessarily are each looking out for number one.

You’ll see vague references to the quote as being an “old cowboy saying,” which is probably more or less a correct way to refer to the origins of a lot of similar folklore surrounding the game. In this case, though, the phrase actually should be credited to the American satirist Finley Peter Dunne.

Dunne became an editor at the Chicago Times in the 1880s, and not long after created a character named “Mr. Dooley” who would author short pieces that served as an entertaining brand of social commentary. In some respects Dunne was following the tradition of the old 18th century essayists Joseph Addison, Richard Steele, Jonathan Swift, and Samuel Johnson. He gets compared to Mark Twain a lot as well, of course, having been a contemporary whose concerns, use of humor, and even worldview tended to overlap with Dunne’s.

The stories touched on a wide variety of subjects, all told from the point of view of Dooley, a pub-owning Irish immigrant. In 1899 a collection of the stories was published as Mr. Dooley in Peace and War, and there would be about a half-dozen more such compilations as Dunne would ultimately write over 700 “Mr. Dooley” stories. It was Dunne’s character of Dooley who uttered the phrase, although I’ll need to click around some more to figure out the exact story in which it appeared.

The sentiment it expresses I’m also seeing linked occasionally to the famous “Prisoner’s Dilemma” puzzle I once took a crack at discussing here.

That gives me an excuse to point you to a neat article by Robert Woolley from a couple of days ago over on PokerNews titled “Got GTO? The Connection Between ‘A Beautiful Mind’ and Perfect Poker” in which he shows the recently-deceased John Nash’s relevance to game theory and poker -- check it out.

Meanwhile, thanks for clicking over here again. If I pinpoint the story in which Dunne’s Dooley utters the phrase, I’ll come back and share it. You can trust me.

Jacob on Jeopardy!

Tuned in tonight to watch Jeopardy! in order to see Alex Jacob, the poker pro (or former poker pro) who is currently enjoying a massive run on the show.

Most in the poker world first met Jacob nearly a decade ago when he won that United States Poker Championship in Atlantic City in 2006, an event that was televised and thus got him some notice even though he’d already made final tables at the World Poker Tour and World Series of Poker by then.

He’d continue to collect cashes up through 2012, earning over $2.6 million total. I feel like he never was a full-timer on the circuit, though, and over the last couple of years has been even less conspicuous, perhaps having stepped away entirely.

In any case, you’ve probably heard about him popping up again on Jeopardy! where he’s been crushing. Going into tonight he’d won five times in a row, winning $129,401 total with mostly dominating performances. At the start of this sixth try, host Alex Trebek noted how Jacob had gotten to Final Jeopardy four of five times with leads of greater than double the nearest competitor, meaning he’d already clinched the win.

I did watch the end of one of those shows and laughed at the end when during Final Jeopardy Jacob appeared to push “all in” by moving his hands forward when his final betting amount was revealed -- $0, actually. At the start of tonight’s show Trebek introduced Jacob as a currency trader, so I’m not even sure they’ve discussed his poker background at all.

Hearing about some of Jacob’s strategies -- e.g., not going top-to-botton with categories but jumping all around the board with his clue selections, and usually betting everything on the Daily Doubles -- I thought back to that fellow from about a year ago, Arthur Chu, who also got a lot of press and poker players’ attention during a run of 11 straight wins on the show.

Chu became known as the “Jeopardy Villain” because of both his unorthodox strategy and his humorous baiting of folks over Twitter during the time of his reign. Chu would also pick clues out of order and routinely go “all in” on the Daily Doubles. Jeopardy! legend Ken Jennings wrote an article for Slate at the time discussing Chu, explaining how that latter decision to go for “true Daily Doubles” was correct.

“Like a poker player trying to increase the size of the pot when he has a good hand,” Jennings wrote, “Jeopardy! contestants should maximize their upside when the odds are in their favor.”

There was one other quirk Chu exhibited during Final Jeopardy. When leading but not having more than twice the “stack” as his nearest foe, he’d bet exactly enough to tie should his opponent bet everything and both players answered the clue correctly.

For example, if Chu had $12,000 and his opponent had $7,000, Chu would bet exactly $2,000 -- not $2,001, as some tend to do -- to end with $14,000 if he were to be correct and tie with his opponent if that player bet everything and was correct as well. In fact, one time that’s what happened and after tying both he and his opponent were able to return for the next show.

That strategy, as I understand it, makes it just a tiny bit less likely Chu could lose should his opponent happen not to bet everything but a lesser amount. In the above example, for instance, if he bet $2,001 and was incorrect he’d end with $11,999, and if his opponent only bet $5,000 even and was correct that player would have $12,000 and win by a buck.

I noticed someone retweeting Chu commenting on Jacob, in fact, interestingly bringing up his own poker playing as he did.

Getting back to Jacob’s performance tonight, Trebek introduced things by alluding to Jacob’s recent run, advising Jacob’s opponents Nikhil and Scott to “get him early, and then try to get him late, too.” Jacob ruled during much of the first round, though, quickly building a big lead before the others finally were able to buzz in and start notching some correct answers.

Then Scott got the Daily Double and with just $600 -- several thousand behind Jacob -- he surprisingly bet only $5. The questioned turned out to be an easy one for him and he won the $5, but it felt a lot like a player too timid to bet without a sure thing.

By the second round Jacob was well in front and in fact went a long stretch without buzzing in at all while the totals of other two went up and down. Scott got another Daily Double, and this time said “Alex is too good, I gotta do it... true Daily Double.” Alas for him he got a tough one about an Italian painter and lost his stack.

A little later Nikhil got the other Daily Double at a point when he had $4,200 and Alex $7,200. Betting it all and being correct would put him in the lead, but he chose only to bet $2,000. The category was “In the Dictionary” and the clue “Fittingly it means ‘Empty Orchestra’ in Japanese.” Nikhil guessed “What is kabuki?” but the correct response was “What is karaoke?” and Jacob’s lead increased again.

There was a clue about “Manhattan prosecutor Preet Bharara” -- he of Black Friday fame -- that perhaps got poker players’ attention. It got Jacob’s attention, too, as he finally buzzed back in to guess correctly that Bharara had vowed sweeping reform of Rikers Island.

Jacob then rattled off a few more correct answers, seeming at one point to pause unnaturally after buzzing in to answer an easy one about a Halloween TV special featuring a character who instead of candy gets a rock.

“Who is... Charlie Brown?” he said, and for a moment I thought he might have been stalling a little as the round was winding down. But he immediately picked back up the pace thereafter, and the trio was able to complete the entire board with Jacob sitting with $17,400, Nikhil $10,200, and Scott $3,600.

The Final Jeopardy category was “Book Reviews” and the clue was an easy one (I thought) -- “A 2008 review of this novel, later filmed, compared it to ‘Battle Royale’ & said it’s ‘a future we can fear.’” -- although I guess it might not have been easy if I hadn’t read The Hunger Games.

Nikhil missed it, though, while Jacob got it correct. He bet $3,001 -- meaning he did not choose the Chu approach -- and ended with $20,401 to bring his six-day total close to $150K.

To be honest, while it’s probably safe to say Jacob is using some of the same skills and strategic thinking he honed at the poker tables while playing and winning at Jeopardy!, there wasn’t that much in his play that obviously recalled his poker background. In fact, it was the timidity of the other two players that made me think more of poker, as well as a kind of “tell” from Scott when he declared he didn’t like one category as he was selecting it.

I guess, though, at the very end of the show I was reminded again that Jacob played poker. “Smile, Alex... smile!” said Trebek to Jacob who remained stoic even as the audience applauded his victory. He reminded me of a seasoned player who has just won a big pot and who has trained himself not to show emotion afterwards.

Jacob may be a “currency trader” now, but he still has that poker face.

The Super Bowl of Second Guessing

I remember once playing in a rec basketball game as a teen in which we were trailing a stronger opponent heading into the latter part of the final quarter. The coach that night was my friend’s father, standing in for the regular head coach who had to miss the game for some reason.

My buddy’s Pop was either an assistant or just helped out in practices here and there, I can’t recall. In any event, I remember him calling a timeout and instructing us to start fouling the other team’s worst shooters, a fairly standard approach teams often take to try to get more possessions and stage a comeback should the other team miss enough free throws.

The strategy worked especially well, and within just a couple of minutes we’d whittled a double-digit lead down to just one. Then we fouled again, they missed again, and we took the lead with less than a minute remaining. The other team called a timeout, and as we huddled up my friend’s Dad had an idea.

Let’s foul again, he said, almost sounding like he was asking us when he did whether or not we thought it might be a good idea. The other team was struggling mightily from the line, he noted, and if we fouled they’d likely miss again, we’d get the board, and the game would be in the bag for us. I remember thinking it seemed like a goofy plan, but he was so enthusiastic about it we were all pretty easily convinced it was somehow a genius move.

You can probably guess how this story ends. We fouled, their player hit both free throws, and we ended up losing. It was initially disappointing, but ultimately the game became a fun, much-referenced collective experience we often talked about afterwards. I remember whenever it came up, my buddy’s Dad often saying with a touch of humility and a wide, mischeivous grin -- “It was such a great idea!”

Looking back, I’m reminded a little how the game represented what was perhaps one of the first times -- and in a thankfully low-stakes way -- I was exposed to the idea that adults didn’t always know the right thing to do. I also can’t help but think of how my buddy’s father continuing to argue for the plan despite the outcome might well have been an early lesson in the dangers of being results-oriented in one’s thinking.

His plan to foul when ahead was pretty obviously not a good one, regardless of the outcome. But his (half-joking, half-serious) insistence that the idea was still valid despite the way things turned out definitely forced a young Shamus to think about how results don’t necessarily confirm or deny the correctness of a strategy -- something the older, poker-playing Shamus came to understand even more clearly.

Obviously it was the stunning conclusion to last night’s Super Bowl XLIX that inspired this bit of reminiscing from me today. Seattle’s decision when down 28-24 to throw that second-and-goal slant pass with 26 seconds left and the clock running rather than run the ball was certainly a surprising choice, with the calamitous outcome of an interception inspiring instant second guessing that will continue unabated for as long as the game continues to be discussed.

New England’s decision not to use one of its two remaining timeouts prior to the play was itself especially odd, too, letting the clock run down from 1:00 to 0:26 and all but eliminating any chance to get back down the field for a tying field goal should Seattle punch it in as expected.

As a Carolina Panthers fan, I think back to Super Bowl XXXVIII in which New England got the ball with the score tied 29-all and used up almost all of the last 1:43 gaining enough yards to set up a winning FG. If Seattle scores on second down last night, NE has but 20 seconds with which to gain (likely) at least 40-45 yards to set up a tying kick.

Anyhow, I tend to think that NE not calling a TO last night perhaps led Seattle to think they needed to be wary about how they were going to use the one they had left. That is to say, had they run the ball and been stopped, they’d face a third-and-goal and thus would be forced to call their last timeout, which would then (essentially) take away the option to run on third down.

That’s what I’m led to believe, anyway, by Seattle coach Pete Carroll’s statements afterward about wanting “really to kind of waste that play.” That, of course, could have been accomplished by spiking the ball on second down, though that would’ve seemed an odd choice. In truth, they didn’t want to “waste” the down, but to run what seemed a low-risk play that would either stop the clock with an incompletion or get them in the end zone. But neither of those outcomes happened.

I don’t want to wade too deeply into analyzing the play or decision, though, something everyone else is doing ad infinitum today. And I’m sure there will be a few who -- like my buddy’s father long ago did with his unorthodox move -- will stubbornly build cases for why the pass call was not such a bad idea. Indeed, the quants at Five Thirty-Eight are already doing so, pointing out how NE letting the clock run down was a more egregious error than was calling that pass play.

But most are taking and will take the other view regarding the decision. And bolstered by the outcome will forever second guess.

Playing Cepheus

A short follow-up regarding Cepheus, the heads-up fixed-limit hold’em playing program developed by the Computer Poker Research Group at the University of Alberta I was writing about yesterday -- i.e., the program that is said to have essentially “solved” heads-up LHE insofar as (the researchers claim) “a human lifetime of play is not sufficient to establish with statistical significance that the strategy [employed by Cepheus] is not an exact solution.”

I mentioned how they’ve put Cepheus online for the curious to play against. Today after queueing up for a long time I managed to get a game against the program. We played 100 hands of 10/20 LHE, after which I managed to finish up 105 units -- just about five big bets.

I ran hot early on, so hot it almost seemed like things were rigged in my favor as I built up a lead of over 250 through the first 30 hands. Then things evened out between us and after 49 hands we were dead even, and for a hand or two after that I was down briefly. But I won three big pots in a row to zoom back up over 200, and ultimately never lost the lead again.

I played tight-aggressive throughout, becoming a little more conservative during the last dozen hands or so as I wanted to preserve my lead. Both Cepheus and I were mindful of position, with Cepheus raising almost every single button and folding otherwise (i.e., never limping). Meanwhile I also mostly raised or folded my buttons (folding more than Cepheus did), though I limped occasionally, too.

Cepheus would three-bet me fairly often before the flop when I did raise, and probably bet when checked to around 80-90% of the time (I don’t have a log of the hand histories, so can’t say for sure). After about 75 hands I had just begun to become aware of the fact that Cepheus hadn’t seemed to have check-raised me on either the turn or river, then the program did it twice within just a few hands, both times successfully earning extra bets as a result.

In the first case I was playing from the button with K-7-offsuit with the king of clubs, and the flop had come all clubs with the ace to give me a nut flush draw. Cepheus check-called me there, then a king fell on the turn and that’s where Cepheus check-raised me. The river was a blank, and Cepheus won the hand with K-Q.

The second instance also involved Cepheus connecting on the turn -- that’s a screenshot of that hand above (click to enlarge). I also managed to check-raise a couple of turns after making hands to get extra value.

Obviously the tiny sample established practically nothing regarding either Cepheus or myself. I will admit that toward the latter part of the session I felt my attention flag just a touch, enough to remind me of the difference between myself and my non-human competitor. If we’d gone on, say, to play 1,000 hands or more I imagine it would have been very difficult for me to continue to make correct decisions (not that all of the ones I made were correct).

If you happen to play Cepheus, let me know how it goes and what impressions you get from the program.

Cepheus and the “Solving” of Heads-Up LHE

Yesterday I was writing about the “New York Times 4th Down Bot” and related efforts by those studying stats to present math-based models for dictating how best to coach a football game. There was a related story last week regarding the team of researchers at the University of Alberta who have been working for several years on “solving” heads-up fixed-limit hold’em, with their new study published in the January 2015 issue of the journal Science “announc[ing] that heads-up limit Texas hold’em is now essentially weakly solved.”

I remember many years ago -- way back in the summer of 2007 -- speaking with Jonathan Schaeffer of the University of Alberta about Polaris, the computer program developed by his team of researchers that comprised the university’s Computer Poker Research Group (CPRG) in the dept. of Computing Science which he chaired. An LHE match between Polaris and two pros, Phil Laak and Ali Eslami, took place that summer in which the humans won, and for PokerNews I spoke to Schaeffer about Polaris and the CPRG’s goals.

“One of these days -- within 5 to 10 years -- two-person, limit hold’em will be solved,” he said to me. Here we are about seven-and-a-half years later, and it sounds like those continuing the work there at Alberta have fulfilled Schaeffer’s prediction.

The new heads-up LHE-playing program is called Cepheus and is in fact available to play against online, although when I went to the site there were “too many in queue” and I was invited to come back later to try.

With my academic affiliation I have been able to get a copy of the Science study and have read through it. Simply titled “Heads-up limit hold'em poker is solved,” it begins by pointing out that while certain “perfect-information” games like Connect Four and checkers have been solved, others like chess have not even if much-celebrated events like Deep Blue’s victory over Garry Kasparov has led some to suggest it has. “Defeating top human players is not the same as ‘solving’ a game,” the study’s four authors point out.

The study then notes how only “perfect-information” games have been solved thus far, making their claim regarding the “imperfect-information” game of heads-up limit hold’em groundbreaking. Reference is made to the work of game theory pioneer John von Neumann and the element of bluffing that distinguishes poker, to Michael Craig’s The Professor, the Banker, and the Suicide King (which details the high-stakes LHE games between by Andy Beal and “the Corporation”), as well as to the previous work of Schaeffer and his team at Alberta.

An explanation of what it means to “solve” a game follows, with the distinction “weakly solved” referring to a game in which “for the initial position(s), a strategy has been determined to obtain at least the game-theoretic value, for both players, under reasonable resources.” From that definition, the researches extrapolate that as far as heads-up LHE goes, it is safe to say that the game is “essentially weakly solved... if a human lifetime of play is not sufficient to establish with statistical significance that the strategy is not an exact solution.”

From there comes further refinement of what is meant by an “imperfect-information game” then an explanation of the programming of Cepheus and the “solution” ultimately found. This admittedly is where your humble scribbler feels especially humbled, not being versed in the various fields of the researchers.

Back in 2007 there was already lots of talk about “poker bots” and online poker, so I had to ask Schaeffer about Polaris and how it might relate to the online game. “I want to be clear,” he told me. “We do not play online poker. None of our software is enabled to play online poker on any of the sites.”

Nor has being able to “solve” heads-up limit hold’em ever been the endgame for the researchers at Alberta. They conclude their study by noting how “the breakthroughs behind our result are general algorithmic advances that make game-theoretic reasoning in large-scale models of any sort more tractable.” In other words, as often gets pointed out by those who study game theory as it applies to recreational games or sports, the findings there have value in other realms involving human decision-making.

As a longtime fan of LHE, I’m curious to learn more about Cepheus, including how it was created and what it can do. If you are also curious, here’s a good piece on the FiveThirtyEight site summarizing the team’s work and placing it in a broader context by Oliver Roeder called “Computers Are Learning How to Treat Illnesses By Playing Poker and Atari.” And I’m about to go listen to the latest episode of the Thinking Poker podcast on which Andrew Brokos had on as guests two of the study’s authors, Michael Bowling and Michael Johanson of the CPRG at Alberta, to talk about Cepheus.

In the meantime I’m going to keep queueing up to try to play Cepheus. Last night Vanessa Selbst was on PokerStars playing at some some 8-game play money tables, and I actually sat at one for a few orbits. Feeling equally intimidated to sit across from Cepheus, I think.

Go For It or Take the Points?

Was captivated like most yesterday by that Packers-Seahawks NFC Championship tilt which ended so crazily with several unlikely plays going Seattle’s way to earn them the comeback and eventual victory.

If you saw the game, you know the plays to which I’m referring, the fake field goal TD, the Hail Mary two-point conversion, and the muff-and-recovery of the onside kick the most vivid of the among them. Those two instances of Green Bay choosing to kick field goals rather than go for it on short fourth-and-goal situations early on also loomed large throughout the day.

Regarding the latter, check out “The New York Times Fourth Down Bot” and its page assessing all of the fourth down decisions made in that GB-SEA game. Applying statistical-based findings about how such calls affect win probabilities, the NYT 4th Down Bot offers judgments on all fourth down decisions as games are being played. The obvious poker-related analogy would be to an online poker player’s “HUD” (Heads-Up Display) offering real-time stats that can be interpreted to help a player make plus-EV decisions on the fly.

As explained here, those findings generally recommend going for it on fourth down much more frequently than most coaches actually choose to do so. Of course, the NYT 4th Down Bot -- like all Monday morning quarterbacks -- can opine at will from the sidelines, not having to face the real-life consequences coaches do when making decisions.

Green Bay’s unwillingness to go for those two first-quarter fourth-and-shorts certainly seemed overly conservative at the time, especially when playing on the road in a do-or-die game versus such a heavy favorite. Not too surprisingly, the NYT 4th Down Bot recommended going for it in both cases yesterday.

Almost every other fourth down decision made yesterday was judged a “Good call!” by the NYT 4th Down Bot. A couple weren’t so cut-and-dry, with the judgments for those being “It’s complicated” and “Too close to call.” Only one other fourth down decision -- also by Green Bay -- was considered incorrect by the Bot, the one at the very end when the Packers punted with four minutes to go when facing fourth-and-14 on their own 39.

That one seems less obvious on the surface, but the explanation suggests the lateness of the decision, being up by 12, and other factors make it so that “teams who go for it would win about 2% more often than teams who punt.” As it happened, GB only gained 30 yards of field position on the play, and it would take Seattle just two quick plays to more than make that yardage back.

I’ve written here before about how I occasionally like looking at these mathematical models that crunch the numbers from thousands of games to give instantaneous interpretations of win probability. The graph from that GB-SEA game was especially wacky, as plotted by Advanced Football Analytics (see left). Green Bay was sailing along at at WP of 80% or above from the late first quarter onwards, peaking at 98% with 3:07 left and then suddenly plunging down to just 14% with 1:33 to go (after Seattle grabbed the lead). Things evened out after the game was tied and went to OT, but Seattle wasn’t challenged thereafter.

Today I was listening to sports talk radio and hearing an ex-player (who years ago also lost a heartbreaking conference championship game to miss a Super Bowl opportunity) talking about how losing hurts much more than winning feels good, another idea with which most poker players are familiar. Interestingly, that same player was also defending Green Bay’s decisions to kick those first-quarter field goals and “take the points.”

I think the first of these ideas -- that losing or failing hurts more than winning or succeeding feels good -- is not unrelated to the second one encouraging the avoidance of risk and acceptance of small victories over the prospect of having to endure what are perceived to be large losses. In any case, as fascinating as the numbers can be, humans and their capacity to be non-rational -- or not bots -- with their decisions are even more so.

Page 2 of 366; or, Following the Crowd

Saw this odd-looking item among the trending topics on Twitter this morning. Took me a while to figure out what it meant. And once I did I decided to go along with the crowd and name today’s post after it. Because, well, the post is sort of about just that -- following others.

It’s a new year. Spent the first day of 2012 intently concentrating on those NFL games yesterday, for a couple of reasons.

One was because I’m a football fan and there were a ton of meaningful games, all piled onto the same day as each of the 32 teams finished out their regular seasons. The NFL does a smart thing with their schedule, having all 16 of these final games being within the division, thereby increasing the likelihood that they’d have some impact on playoffs. And as often happens there were a lot of fantastic finishes all day that further amped up the excitement.

The other reason I was so focused on football was this pool I’ve mentioned off and on here, the Pauly’s Pub one in which we pick all the games all year. No spreads to worry about, but still a not insignificant challenge each week to guess winners.

I’d been hovering near the top of the leaderboard since mid-season and entered yesterday tied for first, meaning I had a shot at winning the whole shebang. So I fretted all week over the games, deciding that this final week of the season was probably the hardest to handicap given all the different factors in play regarding injuries, playoff seedings, and teams’ relative need to win.

I was conscious of one other factor, too, something I’d been thinking about pretty much every week since I’d found myself either leading or tied for the lead in the pool -- namely, what picks others were making.

There had been a number of games over the last few weeks where I had picked a team not necessarily because I thought that team would win, but because I knew everyone else near the lead was going to pick that team as well. It might sound odd, consciously not picking a team I actually thought would win. Hell, it is odd. But by not going against the crowd and picking an underdog I could just about assure myself that the game wouldn’t affect the standings no matter how it turned out, which in several such cases was the option I took.

To give one example, a week ago the Houston Texans were at the Indianapolis Colts on Thursday night. Houston had already locked up its playoff spot and was coming off a bad outing at home in which they were beaten by the Carolina Panthers, while Indy had just avoided a winless season the week before by beating Tennessee. Was one of those games where even though Houston was a decent-sized favorite -- by 5 or 6 points, I think -- there were a few reasons why Indy actually seemed like the right pick.

I remember hearing some pundits say as much that day, including Michael Wilbon making a convincing case on “Pardon the Interruption” a couple of hours prior to kickoff for why Indy would win the game. But I took Houston anyway, and as expected the entire top half of the pool’s leaderboard did the same. We took the game out of play, essentially. And when the Colts won we all thought a moment about what might have been, then moved on to consider the weekend’s games.

In poker we sometimes talk about “game theory” entering into certain decisions, although most of the time a lot of us aren’t altogether clear on what exactly game theory is. I’m not going to pretend to understand all of the ins and outs of game theory, but I do get how it involves making decisions based on what we think our opponents have chosen to do.

I can see how game theory perhaps could be said to have come into play occasionally in the pool. Especially in a situation like this, where all of us at the top of the leaderboard faced having to pick that Houston-Indy game and thus found ourselves in a spot kind of like that represented by the prisoner’s dilemma, that classic example often used to illustrate game theory.

In that example, two prisoners are suspected of a crime. Each has an opportunity either to turn the other in or remain silent. If one turns the other in and the other remains silent, the one who did the turning in goes free while the other receives a maximum sentence, say a year. If both turn each other in, both get a lesser sentence, say three months. If both remain silent, both will serve even less time in jail, say one month.

To translate the example into “Pigskin Pick’em” terms, picking Indy would have been like remaining silent here, but we were all worried about everyone else “betraying” us and picking Houston. In other words, the fact that we were sure everyone else was going to pick Houston made picking Indy seem more risky than it would have been otherwise.

So we all took Houston, thus guaranteeing the game wouldn’t matter in terms of our relative standing in the pool. We avoided the longer sentence (i.e., the risk represented by going it alone and possibly dropping a game to everyone else), but took the lesser one (i.e., picking as we assumed everyone else would and accepting that the game wouldn’t help us, but wouldn’t hurt us, either).

As one tied for the lead, I liked inching one game closer to season’s end like this without having to worry about the possibility of losing my lead. But I knew once we got to the final week it might not be such a simple thing to continue to play this way -- to try to pick games the same way I thought my closest challengers were picking -- unless I had somehow built a big enough lead and was reasonably sure I could predict my opponents’ picks with accuracy.

But neither was the case. As I mentioned above, I made it to the final week tied for first with another player, Johnnie Walker, with Dawn Summers just one game back in third. And it didn’t take me long when looking over yesterday’s schedule to realize we were probably going to go in a lot of different directions on these final games. There really were only a couple of “obvious” picks out of the 16 -- e.g., there was little doubt that Atlanta was gonna crush Tampa Bay, Philly was probably gonna whip the Redskins, etc. However, while there were some favorites here and there among the other games, most were genuine toss-ups where it wasn’t really possible to predict which way the others were going to go.

So, unlike the last several weeks, I picked all 16 this time without giving a single thought to what others were doing. And as it happened I did well with my picks, getting lucky in a few cases -- as I have all year -- and hitting 14 of them to take the sucker down. (The two I missed were picking Detroit to beat Green Bay and Oakland to beat San Diego.)

Like I say, I only have a general understanding of game theory, and realize I might be applying it a little too loosely or even incorrectly here. I also can’t help but think how worrying too much about what others are doing in a non-zero-sum game like this one is bad strategy, making me pick a team not because I think it will win but in order to avoid what appears to be a larger cost should I miss the pick.

Hard not to play it safe and follow the crowd sometimes, though. Speaking of, figured out that post title yet?

WSOP Final Table Hand No. 229: Split Decision

Okay, one more. All in all, ESPN’s pay-per-view was well done, I’d say. I can’t imagine sitting through the whole thing live, but watching it this way -- in easier-to-digest segments, after the fact -- has been a treat. In fact, as this here post exceeds all reasonable guidelines for length, let me suggest you approach it similarly. Don’t feel obligated to read all at once . . . take it in small chunks, if needed. The discussion comes in three parts: (1) a description of the action; (2) an attempt to assess one of the player’s decisions using game theory; (3) a last reflection.

I. The Action

This is the last hand to feature three players. Jamie Gold has 63,425,000 in chips. Paul Wasicka has 13,325,000. Michael Binger has 12,650,000. The blinds are still 200,000/400,000 (with the 50,000 ante). We watch the action from the bird’s-eye view of the overhead camera. Not following Jesus’s advice for playing the button while three-handed (described in the previous post), Gold limps from the button from over on the right-hand side of the table. On the left, Wasicka calls from the small blind. Binger -- in the middle -- puts in a raise of 1.5 million. Gold pauses and calls Binger’s raise. We are shown a lengthy close-up of Wasicka riffling his chips as he decides what to do. We hear Wasicka speaking. “You can’t have a hand every single time, Jamie . . . Jeez.” About twenty seconds go by, and Wasicka makes the call. The pot is 4,650,000.

“And here’s the flop,” calls out the tournament announcer -- . Wasicka very deliberately taps the felt with an extended finger. Phil Gordon points out how “it wasn’t an instant check . . . he did study it.” Binger reaches forward and pushes in a healthy 3,500,000 chip bet. Close to half of Binger’s chips are now in the pot, so he’s likely committed to go all the way here. Before he can even bring his hands back to his sides, Gold waves back-handedly and says “I’m all-in.”

Wasicka immediately groans and stands up from the table, obviously less than thrilled with Gold’s move. “This is sick,” he says through gritted teeth. Gold loudly cries out “This is it, guys!” “This is sick,” repeats Wasicka. “We all got a hand, let’s do it,” says Gold. “Let’s go all three . . . let’s get it over with, right now.”

Wasicka is fit to be tied. “Are you kidding me?” he says to no one in particular, sitting down and reexamining his cards. He stands again and asks Binger how much he has bet. Gold interrupts saying it doesn’t matter because he’s all-in, but Wasicka objects and Gold backs off. A moment passes and Wasicka again voices his displeasure. “This is disgusting,” he says. Gold nods as if in agreement. His nod evolves into a goose-like bobbing action, then he stands upright and claps his hands. Wasicka folds and Binger instantly calls.

Binger has for top-pair, top-kicker. Gold has for an open-ended straight draw. Gold claps his hands, then shakes Binger’s hand and wishes him good luck. Gordon announces Gold has eight outs, but soon we learn that he actually only has seven. “Paul, You didn’t have the best hand, did you?” Gold calls across the table. “I had the seven-eight of spades,” answers Wasicka. He has folded an open-ended straight flush draw. “Wow,” says Gordon. “I don’t know how you can fold that hand.”

The pot is 26,800,000. “Gimme a spade, at least,” asks Gold. The turn comes -- the . Even better. Gold is hugging Johnny Chan. And Binger is drawing dead.

Binger is smiling and very gracious. He tells a friend he’s curious to see what the river card is, and whether it would have helped Wasicka beat him (if Gold had stayed out of the pot). He walks over to Wasicka and they shake hands. “It’s been a pleasure, man,” says Wasicka. “I’m curious if I would’ve beaten your hand,” says Binger. He’s thinking about what would’ve happened if he had gone all-in first. Would Gold have folded? (Probably not.) If Gold had folded, would Wasicka have called him? “I probably would’ve,” says Wasicka. “One of us has to take [chips], you know.” The now-meaningless river card comes -- . Wasicka would’ve made his flush.

Wasicka wanders around the table with a half-smirk, half-grin on his face as Harrah’s officials set the stage for heads-up play. Within seven hands, Gold will take the rest of the chips (between scoops of blueberries).

II. Wasicka’s decision

Gold’s all-in was bold. It is probably safe to say it altered the outcome of the hand. It arguably sealed his victory for the tournament as well. If Wasicka had made the call with his straight-flush draw, he would’ve won the hand and had around 38 million chips -- not too far from half of the chips in play. Let's forget about that spade on the river for a moment. Should he have called?

If we are talking strictly pot odds, the answer is yes. He need to put 12 million in to win a pot of nearly 27 million -- that’s 3.25-to-1. With a whopping 15 outs for the turn and the river, he’s looking at better than 2-to-1 to hit a winner. If this were a cash game, calling would be a no-brainer.

But this ain’t no cash game. If he calls and wins, terrific. However, if he calls and Gold still wins the hand, the tournament ends right there with Wasicka finishing in second place (since he began the hand with more chips than Binger). If Wasicka calls and Binger wins the hand, he will either be left with 675,000 chips (if his hand beats Gold’s), or will be out right there in third place (if Gold’s hand beats his) -- either way, he’s very likely destined to finish third. So calling could end well or disastrously. But folding also is something of a risk. Folding means either assuring himself at least second place or becoming the short stack by a fairly large margin (he’d be about 15 million behind Binger).

Put yourself in Wasicka’s position. What would you do?

How about this. Let’s say everyone was allowed to turn their cards face up after Gold went all-in. Now put yourself in Wasicka’s position. Let’s say you were even allowed to use CardPlayer's Texas Hold ’em Calculator. You know you’re 53.82% to win the hand. You know Binger is 29.01% to win. You know Gold is 17.17% to win. What would you do?

Here’s where we might actually use some of that game theory stuff we keep hearing people like Chris "Jesus" Ferguson answering questions about but most of us never really pay much heed. (If numbers ain’t yr bag, save yourself some grief and skip now to section III.)

Let’s say you make the call. Over half the time you end up heads-up with nearly half the chips. Let’s also say if that were to occur, you’d have a 50-50 chance of winning the whole ball of wax. About a third of the time you will either finish in third right here or within the next hand or so. Another 1/6 of the time you’ll finish in second place right here. So by calling . . .

-- 27% of the time you win $12 million (after winning a heads-up battle)
-- 43% of the time you win $6 million (after losing a heads-up battle or after busting out right here in second)
-- 30% of the time you win $4 million (after busting out right here or soon afterwards in third)

Let’s say you fold. Once you’ve folded, Binger is about 65% to win and Gold 35%. (I can’t be precise here, but that’s about where it stands given Gold has seven available outs plus the backdoor flush draw.) Let’s also say that if Binger wins the hand, 2/3 of the time you’ll end up playing for awhile and getting bounced out in third place. (I say that because he’d have a bit over twice your chips.) And if Gold wins (as happened), let’s make you a 6-to-1 dog to overcome the huge chip lead and somehow win. (I say that because Gold has over 6 times Wasicka’s chips when heads-up begins.) In other words, by folding . . .

-- 5% of the time you win $12 million (after winning a heads-up battle)
-- 52% of the time you win $6 million (after outlasting Binger and making it to second or after watching Binger lose here and then eventually losing a heads-up battle)
-- 43% of the time you win $4 million (eventually busting out in third)

Which is the better decision? Isn’t it obvious? (Ha ha.)

Knowing what we know, calling is going to be the more profitable play here. Let’s say we play out this scenario 200 times and we call 100 times and fold 100 times. The 100 times we call, here’s how we do:

Win $12 million 27 times = $324 million.
Win $6 million 43 times = $258 million.
Win $4 million 30 times = $120 million.
TOTAL = $702 million or an average of $7.02 million each time.

The 100 times we fold, here’s how we do:

Win $12 million 5 times = $60 million.
Win $6 million 52 times = $312 million.
Win $4 million 43 times = $172 million.
TOTAL = $544 million or an average of $5.44 million each time.

Thus, between the two choices, calling is clearly going to be more profitable than folding. So says game theory, anyway . . . .

III. Final thoughts

To be fair to Wasicka, this here experiment assumes full knowledge of everyone’s cards and all of the relevant percentages. Not precisely what happened in real life, to be sure. However, I think it is safe to say Wasicka pretty much knew where he stood when he made the decision. He knew how many chips everyone had. And he knew that all 15 of his outs were probably good. (He could’ve feared a higher spade draw, but I’d be willing to bet that possibility really didn't affect his thinking too greatly when he made his decision to fold.) [EDIT (added 9/18/06): Listening to Wasicka on CardPlayer's The Circuit this week, I discovered that, in fact, he was worried Gold was on a higher flush draw here! So I'd have lost that bet.] So even without knowing that the spade came on the river, Gordon was probably right when he suggested calling was the play to make here.

While it's doubtful Wasicka had all of this worked out to the nearest hundredth of a percentage point, he still knew (I'd argue) that he probably should call. And that he probably wasn’t going to. That’s why he is repeating how “sick” and “disgusting” the situation is. We’ve all been there (though never for these stakes). We know know know what is right, but we just can’t act on that knowledge and the pull the trigger.

In the final hand of the tournament, Wasicka did pull the trigger and make the call of an all-in. Only this time Gold had him beat. As I mentioned before, I’m gonna leave that hand to others to decipher. Watch it here, if you want. Meanwhile, for anyone who’s made it all the way through this monster of a post, it’s been a pleasure . . . .