Favorite Harvard Game Memory

[Towerroad]

This is a really unstable equilibrium, because order matters.

If I pull my goalie, I have a higher chance of scoring, whether you pull yours or not. So it seems like a Nash equilibrium.

But if you pull your goalie first, though, I'll take my chances.

Q: So... do you get in a situation where I pull, then you pull, then I send back, then you send back, and so on??
A: No. Because if I pulled my goalie, you wouldn't pull yours. You'd take the 50% chance of winning over a 10% chance.

Given that you and I are identivally motivated, I don't think you would pull your goalie.

Assuming this is OT (sudden death) and a loss = a tie:
A. Both sides are incented to pull the goalie initially (10% chance of winning > 5% chance of winning)
B. Both sides are incented to not pull the goalie if the other side already did. (50% > 10%)
C. Whoever pulls their goalie first is the probable loser. So nobody does. Except A says that someone should.

Given that hockey coaches are famously subtle game theoreticians, perhaps it doesn't happen in reality because they've worked this out.

This is really a very simple problem assuming you behave based on the probability of scoring a goal.

In a regular game you pull the goalie because the consequences of losing by 1 or losing by 2 are the same and you increase the odds of a tie improve. The logic here is unassailable because the consequences of the other team scoring an ENG are immaterial.

In the case that was suggested is not the traditional case. If you score you win, If the other team scores you lose. From my experience I think that team that pulls their goalie are scored on more often than they score.

If this were not true, coaches would always pull their goalies in an OT tie. We would have played 3 periods against Wisconsin without a goalie.

[Chris '03]

If this were not true, coaches would always pull their goalies in an OT tie. We would have played 3 periods against Wisconsin without a goalie.

Except against Wisconsin the game being tied forever was better than allowing a goal and losing.

[Rosey]

Given that hockey coaches are famously subtle game theoreticians, perhaps it doesn't happen in reality because they've worked this out.

If they haven't scored a goal in 6-5 play in a while, they'll both probably simultaneously think "We're due."

[Swampy]

I was wondering if Clarkson would pull the goalie in OT knowing they needed one more point than Cornell. Don't get the sense they did.

CHN box says they didn't.

I was wondering during the game whether there was any scenario where both Cornell and Harvard would need a win to get over the cliff to the next third, provoking both teams to pull their goalies simultaneously in the final minute of OT.

Interesting idea but I don't think the game theory works. If you believe that the side that pulls its goalie is scored on more than it scores (which I believe is true but am too lazy to prove it) then pulling your goalie is better for the other team since it increases its chances of victory. In this case both teams leave their goalie on the ice.

I don't think  that's quite the right analysis.  Pulling a goalie increases the chance of someone scoring.  It increases the chance of the attacking team scoring 6 on 5 and it increases the chance of the defending team scoring an ENG.  If a team needs a goal in a short period of time then this is a good strategy. In the hypothetical Cornell-Harvard situation, it's a good strategy if a win has significant value (reaching the next playoff band) and a loss is identical to a tie in terms of league standings. Neither team would be impacted by what happens to the other guy so the increased chance of the other scoring isn't really a factor.

Adding PWR into the mix is the factor that makes this improbable for Cornell because a loss would hurt more than a tie.  but if were far away from the bubble it might've been plausible.  And fun!

I was operating under the following Hypotheses:

1. For both teams only a victory had any value, the costs associated with a tie or loss were inconsequential.

2. The probability that a team that does not pull its goalie will score when the other team does is greater than the probability that the team that pulled its goalie will score. I believe this is true but have not looked at the data.

Under these 2 assumptions neither team pulls its goalie if they are rational.

I disagree.  The 4 permutations might look something like this:

1) We don't pull goalie, they don't pull goalie (i.e. status quo): 5% we score, 5% they score, 90% nobody scores
2) We pull goalie, they don't pull goalie: 10% we score, 50% they score, 40% nobody scores
3) We pull goalie, they pull goalie: 40% we score, 40% they score, 20% nobody scores
4) We don't pull goalie, they pull goalie: 50% we score, 10% they score, 40% nobody scores

Sure, our best chance is if they pull the goalie and we don't (scenario 4, 50%).  But we can't control that.  If time is winding down and they haven't pulled, then pulling our goalie still improves our probability from 5% (scenario 1) to 10% (scn 2), so it is still a rational choice to do so.

Once we have pulled our goalie, they would of course be foolish to follow suit, since at that point THEY would be in their own best case scenario #4. So it becomes a game of chicken - you're best off if the other guy pulls first, but pulling first is still better than if nobody pulls at all.

Sounds like Prisoner's Dilemma, except these are probabilities instead of payoffs.

[Trotsky]

Sounds like Prisoner's Dilemma, except these are probabilities instead of payoffs.

It is close to PD if you consider the expected value of the outcome (probability x impact) to be the payoff.

The main difference is of course both coaches have perfect information, where in PD there is partial information.

[RatushnyFan]

1990? 1991? Cornell wins a draw in the Harvard end with about a minute to play and a Tim Vanini slapshot from the point ties it up. Don't know why this means more that the Paolini/McRae heroics but college memories are sweeter, I guess.

It was 2/9/91.  Remember how badly Harvard beat us in prior games:

12/4/88 Cornell 1, @Harvard 9
2/10/89 @Cornell 2, Harvard 4
3/11/89 Cornell 3, Harvard 6 (ECAC consolation game)
12/3/89 @Cornell 0, Harvard 5
2/9/90  Cornell 2, @Harvard 5

They completely dominated us until the 1990 ECAC tourney quarterfinals as Beej cites, but then we turned around and got hammered in Cambridge by an 8-3 score in November 1990.  I had such high expectations for that 90-91 team.........

[Towerroad]

Sounds like Prisoner's Dilemma, except these are probabilities instead of payoffs.

It is close to PD if you consider the expected value of the outcome (probability x impact) to be the impact.

The main difference is of course both coaches have perfect information, where in PD there is partial information.

The original question was "Is there ever a time where both teams would pull their goalies" I suggested that there was not a "rational" reason except in some bizarre corner solution (One of your defenders was the size of the goal mouth).
 
The question was framed in the context of only a victory mattered  for both teams, a loss or a tie were of no consequence to each team. Take the following scenario:

Two Aged Rich Alumnae meet one from Harvard and one from Cornell. The Harvard Alum, stinking of cheap gin and Geritol begins to brag about the Harvard Hockey Team. The virtuous Cornell Alum immediately defends the honor of the Big Red. They agree to make a bet. Each will put $500,000,000 up and the two teams will play a special game under regular season rules. The winning team's school gets $1,000,000,000. The loser gets nothing. If there is a tie neither school gets anything. Only victory by one or the other will suffice.
 
Further assume that there is no collusion and that the mere thought of losing to Sucks is sufficient to motivate the Noble Red to play their level best.
 
The game is played, and in spite of their on ice heroics the Red find that, Sucks had bribed the officials, and the game is tied at the end of regulation. With 5 min of overtime do you pull your goalie knowing that this act will increase Sucks chances of scoring (therby ending the game) by more than it will increase your chances of scoring and winning?  I think not. Do you hope that Harvard's avarice will cause them to pull their goalie? I think so.

[RatushnyFan]

Prisoners dilemma with a Harvard twist and a different payoff matrix?

[Robb]

Sounds like Prisoner's Dilemma, except these are probabilities instead of payoffs.

It is close to PD if you consider the expected value of the outcome (probability x impact) to be the impact.

The main difference is of course both coaches have perfect information, where in PD there is partial information.

The original question was "Is there ever a time where both teams would pull their goalies" I suggested that there was not a "rational" reason except in some bizarre corner solution (One of your defenders was the size of the goal mouth).
 
The question was framed in the context of only a victory mattered  for both teams, a loss or a tie were of no consequence to each team. Take the following scenario:

Two Aged Rich Alumnae meet one from Harvard and one from Cornell. The Harvard Alum, stinking of cheap gin and Geritol begins to brag about the Harvard Hockey Team. The virtuous Cornell Alum immediately defends the honor of the Big Red. They agree to make a bet. Each will put $500,000,000 up and the two teams will play a special game under regular season rules. The winning team's school gets $1,000,000,000. The loser gets nothing. If there is a tie neither school gets anything. Only victory by one or the other will suffice.
 
Further assume that there is no collusion and that the mere thought of losing to Sucks is sufficient to motivate the Noble Red to play their level best.
 
The game is played, and in spite of their on ice heroics the Red find that, Sucks had bribed the officials, and the game is tied at the end of regulation. With 5 min of overtime do you pull your goalie knowing that this act will increase Sucks chances of scoring (therby ending the game) by more than it will increase your chances of scoring and winning?  I think not. Do you hope that Harvard's avarice will cause them to pull their goalie? I think so.

If I have $500M to bet on a hockey game, I pull the goalie, because as much as I loathe Harvard and would hate for them to get my money, the world is legitimately probably a better place if they have the money rather than its staying in Mr. Geritol's pocket.   I'd be much more motivated "not to lose" if you told me Mr. Geritol was going to get my money rather than Harvard.  All you're doing is shifting around the relative stakes for the 3 outcomes (win, tie, lose) to try to mess with our "emotions" for what we "feel" would be the likely outcomes.  But the stakes don't change the underlying probabilities - the math is the math.  Pulling your goalie first is *still* the better play for you (in terms of increased probability of getting a win), even if its ALSO the better play for your opponent.  If you really want to get me to agree not to pull the goalie, why don't you just go ahead and say, "If Cornell wins, Cornell gets the billion dollars, but if Harvard wins, they get the billion dollars and we kill you, your family, and all of your Facebook friends."  Under that scenario, I not only bolt Iles to the goalpost, but I stack all 5 other players in the crease like cordwood as well.

tl;dr: I'd rather any school (even Harvard) have the money than see it remain in the pockets of rich old farts.

[BMac]

Serious question: stacking players like cordwood. Would it actually take up the entire goalmouth? Is there anything actually stopping one from trying? How many players would it take? Is it the optimal strategy for a 5-on-3 PK?

(OK, OK, not that serious. But still.)

[ugarte]

tl;dr: I'd rather any school (even Harvard) have the money than see it remain in the pockets of rich old farts.

eh, six of one...

[Jeff Hopkins '82]

Serious question: stacking players like cordwood. Would it actually take up the entire goalmouth? Is there anything actually stopping one from trying? How many players would it take? Is it the optimal strategy for a 5-on-3 PK?

(OK, OK, not that serious. But still.)

With the goalie, it might just take up the entire goalmouth.  However, there's nothing keeping an opposing player from running into a defending player in the crease.  Therefore, the answer is: run into the pile, and once everyone is on the ice, toss the puck in over them.  ::crazy::

[Towerroad]

Serious question: stacking players like cordwood. Would it actually take up the entire goalmouth? Is there anything actually stopping one from trying? How many players would it take? Is it the optimal strategy for a 5-on-3 PK?

(OK, OK, not that serious. But still.)

With the goalie, it might just take up the entire goalmouth.  However, there's nothing keeping an opposing player from running into a defending player in the crease.  Therefore, the answer is: run into the pile, and once everyone is on the ice, toss the puck in over them.  ::crazy::

Is there any requirement that the players actually be alive? Frozen corpses dressed appropriately and stack with a little help from the goalies water bottle might just do the trick. Gives a whole new meaning to "Stiff"

[Trotsky]

Is there any requirement that the players actually be alive? Frozen corpses dressed appropriately and stack with a little help from the goalies water bottle might just do the trick. Gives a whole new meaning to "Stiff"

"Lifeless, lifeless..."

[Towerroad]

Is there any requirement that the players actually be alive? Frozen corpses dressed appropriately and stack with a little help from the goalies water bottle might just do the trick. Gives a whole new meaning to "Stiff"

"Lifeless, lifeless..."

If they are wearing Red I am still going to root for them. Clearly LGR is out, and Kill Red Kill might also be out. "Chill Red Chill"?