On carrying Momentum...

Frequently, the importance of carrying momentum over an intermission can be heard being talked about. I thought it were possible to measure this harmony with algebra, so I tried to do that. I choose to analyze a very specific question:

If the regulation of a game ends in a tie, other than 0-0, how frequently would the team that tied the game with the last regulation goal win in overtime. 

We would define the team that tied the game as the one having the momentum. We would define the other team as the one trying to show resilience. For answering the question, we analyzed the outcome of games of seasons 2007/08-2016/17 (including the ongoing playoffs). We discard the games that end in a shootout, because their outcome depend truly more on the skill of the shooting players/goaltenders rather than the whatever momentum might've been accrued.

The results of the analysis are displayed in the table below, per season, per the time frame during which the last tying goal was scored: in the last two, five, or ten minutes, in the last period, or in one of the first two. The numbers show the percentage of wins by the team having the momentum and the number of games falling into that specific segment. Also we display a separate column and a separate row for playoffs game, although a finer granularity is not really possible because of the sample size.

Season   2        5        10       20       1st/2nd  total     totalPO

2007     54.2/24  57.9/19  52.9/34  53.8/13  52.6/38  53.9/128  31.2/16

2008     43.5/23  48.1/27  45.2/31  53.8/13  40.0/40  44.8/134  25.0/16

2009     42.9/28  56.5/23  72.7/22  64.7/17  53.7/41  56.5/131  58.8/17

2010     48.6/37  54.2/24  47.1/34  40.7/27  56.8/44  50.0/166  59.1/22

2011     50.0/24  45.8/24  43.5/23  72.0/25  47.7/44  51.4/140  37.5/24

2012     62.5/16  33.3/15  50.0/22  50.0/14  57.9/19  51.2/86   53.8/26

2013     58.1/43  43.5/23  34.6/26  45.5/22  44.1/34  46.6/148  70.8/24

2014     51.7/29  65.2/23  55.3/38  46.7/15  60.5/43  56.8/148  57.9/19

2015     60.0/40  46.7/30  44.4/36  45.8/24  39.6/48  47.2/178  52.6/19

2016     43.6/39  50.0/28  60.5/38  48.1/27  61.8/68  54.5/200  63.2/19

totalPO  61.4/44  46.7/30  55.8/43  68.4/19  40.9/66  52.0/202 52.0/202

total   51.5/303 50.4/236 50.7/304 51.8/197 51.8/419 51.3/1459 52.0/202

We see that there is no specific "momentum" nor "resilience" capability overall, there is practically no indication on how the OT would end based on which team scored the last GTG. The only two moderate exceptions with decent sample sizes are the second and the sixth columns of the penultimate row. The GTG-scoring team is 27-17 (61.4%) in case it scored the tying goal in the last two minutes, however if the GTG was scored before the last period, as it happened in 66 games, the momentum would obviously not carry over two or more intermissions, and the tying team is 27-39 (40.9%) in these games.

Here is how it looks on a graph:

We can see all lines wobbling slightly above the 50 mark. Insufficiently above. Even if we observe the extra 1.3% chance overall (2.0% in playoffs) - wouldn't it be more related to the home/away advantage? I haven't looked at this aspect yet. Maybe another time.

The Real Life[TM] took a bit of a toll on this blog... But we resume, with resilience and hoping to generate momentum!

On Intangibles (a small addendum)

One "intangible" being tossed around is "motivation" of the players. Which brings memories of an episode I was witness to.

In 2003/04, in the Israeli Top Tier Chess League (which is indeed no slouch) our club managed to assemble an outstanding team, featuring, among others, a former Champion of Russia and a former Champion of Europe. I was part of the management team, and orchestrated bringing the first of the two, who also happened to be my childhood friend back in Leningrad, Soviet Union.

And so, in round III we were to face our main rival for the title, and the club's GM (also a pedestrian chess player) gathered the team and carried out a pronounced motivational speech, how we have to beat the team we're facing, and so on, and so on.

We lost 1½-4½ without winning a single game and lost any chance for the championship we could have.

On Intangibles. Carpe Jugulum.

Often the general managers, the coaches and the players talk about "intangible values". Sometimes it's about the "locker room contributions". Sometimes it's about "passion". In my opinion, these two are actually negligible and in certain cases even harmful. I remember such references, especially the latter one, made about Israeli soccer players, and that usually meant that the player doesn't have a lot of talent to go along, but contributes a lot of passion into the game. While a passionate play can indeed ignite the play and carry the team along, more often it indicated dumb physical low-talent execution that actually harmed the team.

However, there is one intangible that I take my hat off in front. It's the one that I always admired, and myself did not have enough in my chess career. It's the ability to go for the throat of the opposition at even momentary display of weakness by it, or as Terry Pratchett put it one of its books, 'Carpe Jugulum¹'.

So what is it, in my understanding? It is the situation when your opponent puts itself into an inferior position in a volatile situation (for example, in a close score), such as by a penalty, or by an icing at the end of a long shift, or by allowing an odd-man rush, and you are able to capitalize on it, yanking any remains the carpet of security from under the feet of the opposition. And then, you continue to hammer the blows on the opposition until it collapses completely. Some also call it the 'killer instinct'. This blog (and this article too) sins with abundance of examples from chess, so let me plant one from tennis... Before the match between Lleyton Hewitt and Taylor Dent at the New York Open, 2005, the latter one complained: 'He displays a poor sportsmanship: taking joy in double errors at the opponent services as well as in unforced errors.' 'I don't care what Dent thinks about it', parried Hewitt, 'I always go for a win, and on the way to it many things are allowed.'

Machiavelli advised the rulers and the politicians, 'Don't be kind'. Winston Churchill also knew something about achieving the goals when he was recommending: 'If you want to get to your goal, don't be delicate or kind. Be rough. Hit the target immediately. Come back and hit again. Then hit again with the strongest swing you can...'

All the chess champions had it, the extremes going to Alexander Alekhine, Robert J. Fischer and Garry Kasparov. Many wonderful players that never got the title complained that they couldn't commit themselves to going for the throat of the opponent time after time.

These qualities were elevated to perfection by the two best teams of the first half of 2010s, by the Los Angeles Kings and the Chicago Blackhawks that split between themselves five cups out of six from 2010 to 2015. Even when both teams seem to be struggling and wobbling, they seemed to be able to instill some kind of uncertainty into their opponents - and certainty into the spectators that these teams are going to be able to make a fist out of themselves that is going to hammer their opponents once they display any kind, and minimal level of weakness. That capability was championed by their leaders, Anze Kopitar, Drew Doughty and Jeff Carter for the Kings, and Jonathan Toews, Patrick Kane and Duncan Keith for the Hawks. When the playoffs series between the Blackhawks and their opponents were tied 3-3, Chicago has always been the favorite to win the game 7 because of their Carpe Jugulum reputation. The Kings gained even more notoriety, first by burying their sword to the hilt into each and every opponent in 2012 en route from the #8 seed to their first Stanley Cup, and then from the reverse sweep they managed against the Sharks that started their 2014 Cup run - which included two more comings from behind, 2-3 and 1-3. And even in 2016, down 1-3 to the Sharks in the first round of the playoffs somehow fans around the league were not ready to commit to the Sharks as the favorites to win the series, because the Kings were a hair away from the Sharks' throat in game 4, from 0-3 to 2-3 in the 3rd period, and then in game 5, they indeed were able to erase the 0-3 deficit into a 3-3 tie.

Well, that tie didn't hold, the Sharks broke the stranglehold and got a boost that carried them all the way to their own first even Stanley Cup Finals, and that outcome got the Kings' reputation as the Carpe Jugulum team damaged to a degree. So did the Blackhawks' one, losing their game 7 to a team that - along with the Sharks and, for instance, the Washington Capitals - had a reputation of a somewhat nonplussed one - the St. Louis Blues.

It would be entertaining to see whether the Carpe Jugulum landscape changes this year in the league, and whether the teams who were able to overcome their "benign" reputation will be able to go all the way to the Cup Finals - through their opponents' throats.

Chess Grandmaster Gennady Sosonko wrote, 'A real professional, having thought about the situation on the board, acts most decisively. He knows, that during the game, there should be no place either for doubt, nor for compassion, because a thought which is not converted into action, isn't worth much, and an action that does not come from a thought isn't worth anything at all.'

And it's important to remember, Carpe Jugulum is a necessary key to success in a competitive environment only. Albert Einstein used to say that chess "are foreign to me due to their suppression of intellect and the spirit of rivalry."

¹Carpe Jugulum (lat.) - seize the throat.

On Players Evaluation - Part VII and Final (Bundling it all up)

Part I

Part II

Part III

Part IV

Part V

Part VI

Now that we obtained a way to estimate players' performances for a season, we can move on to estimate their performances for a specific game.

For the season of interest, we compute the average against for each teams, just like we computed the season averages. I.e. we calculate how many goals, shots, hits, blocks, saves are made on average against each team. Thus we obtain the team against averages T_avg. The averages are then further divided by the number of skaters and goalies (for respective stats) the team had faced.

After that we can calculate the "result" R_t of each season average stat in a chess sense, i.e. the actual performance on the scale from 0 to 1:

For Goalie Wins/Losses:

R_twins = 0.5 + T_avgwins/(T_avgwins+T_avglosses)

For Plus-Minus:

R_t+/- = 0.5 + (T_avg+/- - S_avg+/-) / 10 (10 skaters on ice on average)

For the rest:

R_stat = 0.5 + (T_avgstat - S_avgstat) / K

where K is a special adjustment coefficient that is explained in part VI (and, as we remind, describes the rarity of each event)

And from the result R_t we can produce teams' Elo against in each stat, just like we computed the players' Elos.

Then, the expected result R_p of a player against a specific team in a given stat is given by:

R_p = 1/(1 + 10^{(E_t - E_p)/4000})

where E_t is the team's Elo Against and the E_p is the player's Elo in that stat.

From the expected result R_p, we can compute the expected performance E_p just like in the previous article:

P_exp = (R_p - 0.5) * A * S_avg + S_avg

(See there exceptions for that formula).

Please note that we do not compute "derived" stats, i.e. the number of points (or SHP, or PPP), or the GAA, given the GA and TOI, or GA, given SA and SV.

Thus, if we want to project expected result of a game between two teams, since it's the expected amount of goals on each side, we compute the sum of the expected goals by each lineup (12 forwards and 6 defensemen):

S_home = SUM_F1..12(MAX(P_expG)) + SUM_D1..6(MAX(P_expG)) for the home team

S_away = SUM_F1..12(MAX(P_expG)) + SUM_D1..6(MAX(P_expG)) for the away team

while filtering the players that are marked as not available or on injured reserve. Please note that we assume the top goal-scoring cadre is expected to play, if we knew the lineups precisely, we would substitute the exact lineup for the expected one.

You can see the projections at our Daily Summary page. So far we predicted correctly the outcome of 408 out of 661 games, i.e. about 61.7% . Yes, we still have a long way to go.

Now to the different side of the question. Given that a player expectation overall is a vector of [E₁, E₂, ... E_n] for all the stats, what is the overall value of that player. And the answer is, first and foremost, who's asking.

If it's a statistician, or a fantasy player, then the value V is simply:

V = SUM_1..n(W_nE_n)

where W_n are the weights of the stats in the model that you are using to compare players. Fantasy Points' games (such as daily fantasy) are even giving you the weights of the stats - this is how we compute our daily fantasy projections.

Now, if you're a coach or a GM asking, then the answer is more complicated. Well, not really, mathematically wise, because it's still something of a form

V = SUM_1..n(f_n(E_n))

where f_n is an "importance function" which is a simple weight coefficient for a fantasy player. But what are these "importance functions"?

Well, these are the styles of the coaches, their visions of how the team should play, highlighting the stats of the game that are more important for them. These functions can be approximated sufficiently by surveying the coaches and finding which components are of a bigger priority to them, for example, by paired-comparison analysis. Unfortunately, there are two obstacles that we may run into: the "intangibles", and the "perception gap".

But that's a completely different story.