The theory behind the defensive shell game

Last week there was a twitter discussion on the merits of playing a defensive shell game by limiting scoring chances against but also limiting scoring chances for, even if it meant the ration of goals for to goals against gets worse. The two sides of the debate are as follows:

Argument 1: It is always best to play a game where you are expected to out score the opposition regardless of the goals for/against rates.

Argument 2: When playing with a lead late in the game it is more important to reduce the goals against rate than maintain the goals for rate, even if it means the goals for to goals against ratio drops significantly.

To test each theory I simulated a number of games between teams T1 and T2 according to the following theories:

1. During normal play between teams T1 and T2, T1 will score at a rate of 2.75 goals/60 minutes and T2 will score at a rate of 2.50 goals/60 minutes. During this play it is expected that T1 will score approximately 52.4% of all the goals that are scored.

2. During play between T1 and T2 when T1 has a lead and is playing in defensive shell mode T1 score at a rate of 2.00 goals/60 and T2 will score at the same 2.00 goals/60 rate.

From there I simulated 1,000,000 games in which T1 is protecting a 1 goal lead for the remaining 2.5, 5, 7.5, 10, 12.5, 15, 17.5 and 20 minutes of a game under both normal style play and defensive shell style play. Here are the results at the end of regulation play.

Normal play

Wins Losses Ties RegWin% OTL Pts% PlayoffWin%
2.5mins 911132 4471 99307 96.08% 93.60% 96.32%
5mins 847011 15230 187894 94.10% 89.40% 94.54%
7.5mins 799667 28880 268711 93.40% 86.68% 94.04%
10mins 764672 44692 340642 93.50% 84.98% 94.31%
12.5mins 738696 59869 405525 94.15% 84.01% 95.11%
15mins 717679 75094 464680 95.00% 83.38% 96.11%
17.5mins 702071 88968 518004 96.11% 83.16% 97.34%
20mins 690638 102013 565261 97.33% 83.20% 98.67%

Defensive Shell

Wins Losses Ties RegWin% OTL Pts% PlayoffWinRate
2.5mins 926241 3011 79934 96.62% 94.62% 96.81%
5 mins 868285 10599 153384 94.50% 90.66% 94.86%
7.5mins 821835 21109 221668 93.27% 87.73% 93.79%
10mins 785935 32888 283819 92.78% 85.69% 93.46%
12.5mins 755920 46048 341509 92.67% 84.13% 93.48%
15mins 733346 58874 392918 92.98% 83.16% 93.92%
17.5mins 713419 72115 442202 93.45% 82.40% 94.50%
20mins 697687 85092 486930 94.12% 81.94% 95.27%

Wins, losses, ties are T1’s record after 60 minutes and regulation win% is the standard regulation winning percentage using 2 points for a win, 0 points for a loss and 1 point for a tie. PlayoffWinRate is the winning percentage of T1 in a playoff game assuming that they would win 52.4% of all overtime games. OTL Pts% is the current regular season system where you get 1 point for an overtime loss, 2 points for a win of any kind and zero points for a regulation loss (under this system for simplicity sake I assumed a 50% chance of winning an overtime game since we don’t know odds of winning a shoot out).

That is a lot of numbers, so lets look at these in nicer easier to read charts.




Under this constructed scenario the break even point for when to go into a defensive shell and when to continue playing normal hockey is at about 7-7.5 minutes for regulation win % and playoff win % systems and about 13 minutes for the point for an overtime loss system currently used during the regular season.

For some people this may not make sense intuitively. How can it be better to stop playing a system in which you are expected to out score your opposition and start playing a system in which you are expected to score the same as your opponent. The reason is simple and it comes down to that over a short period of time your are essentially dealing with small sample size issues and randomness becomes more important than long term skill. The reality is, over a short time one team is almost as likely to score as the other so which team scored next is close to random, if any team scores at all. The most important thing when protecting a lead is simply reducing the likelihood that your opponent will score because the cost of your opponent scoring is far greater than the benefit if you scoring (it is irrelevant whether you win 3-1 or 2-1, a win is a win in the standings).

What is interesting is the effect of awarding the point for an overtime loss is in reality providing additional incentive for teams to play the defensive shell game for longer periods of time because the cost of giving up a goal is not as great in that system because a tied at the end of regulation guarantees you one point with the possibility of 2 where as in the other systems it does not. This means teams can play the defensive shell for twice as long as they could otherwise.

Of course, this is only looking at one side of the equation. Typically the trailing team will get more offensively aggressive even if it means increasing the possibility of having a goal scored against them. This is why teams pull their goalie late in the game. At that point scoring a goal is the only thing that matters so you may as well risk giving one up to score. Over the last 5-10 minutes or so it probably makes sense for the trailing team to take more high risk high reward plays in the offensive zone because at that point scoring a goal has more benefit than the cost of giving up a goal.



This article has 3 Comments

  1. I apologize if I missed it, but how do you determine that a team is in a defensive shell? Do you go by game tape, or is it an assumption that when a team is up in the 3rd period, they are in prevent mode?

    Also, I’m not so sure that the scoring rate would be even when teams go into a defensive shell. I can’t prove this, but due to the team being behind having their possession increase, causes them to have a higher chance of scoring (even though the quality of shots will be lower), likewise, the team in the defensive shell’s possession rate will be much lower, due to the focus on not allowing goals, rather than scoring. It might be close to 50/50 (as far as goal scoring rate), but I think there might be a slight edge towards the team that is down.

    5v5 Tied
    17/30 2013
    12/30 2012
    15/30 2011
    13/30 2010
    16/30 2009
    14/30 2008
    AVG: 14.5/30

    5v5 Trailing
    19/30 2013
    17/30 2012
    18/30 2011
    20/30 2010
    20/30 2009
    16/30 2008
    AVG: 18.3/30

    5v5 Leading
    15/30 2013
    11/30 2012
    17/30 2011
    10/30 2010
    13/30 2009
    16/30 2008
    AVG: 13.7/30

    I can’t find ones for periods, I’m sure it will look differently in the 3rd period, compared to the 1st and 2nd period.

    1. You present some interesting questions but the post was meant to be a theory post, not a reality post. The theory is, play conservative defensive hockey to protect the lead and presented a theoretical example of why teams might do it. In reality the effect might not be the same and there is always the trailing teams more aggressive style offensive game that needs to be factored in as well. We know that score effects exist so it is clear that teams do play differently based on the score. Determining how much of that is the leading team going into a defensive shell and how much of that is the trailing team playing a more high risk-high reward offensive style game is a difficult question to answer and of course, every team will be a little different.

      I think you are right though, the trailing team seems to have the edge which indicates teams probably go into a defensive shell too early on average, whether deliberately or otherwise.

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