The Bettman Point and the Prisoner’s Dilemma

hockey

Picture a quick game theory scenario. You are to play a game, but before you play the game, you have the choice of two scenarios with different utility outcomes based on the results of the game. In the first scenario, you will receive two utility points if you win and zero utility points if you lose. In the second scenario, you will receive two utility points if you win and one utility point if you lose. Which scenario would you choose?

This isn’t a trick question; of course you would choose the second scenario. No matter what your expected odds of winning the game are, the second scenario will always give you more average utility. Here are the obvious parallels we see with the NHL, which, if the game is decided in regulation, rewards two points in the standings to the winner and zero to the loser. If the game is decided in overtime, the winner still receives two points, but the loser instead receives one point. If waiting for overtime is clearly the dominant game theory solution, what incentive is there to play the game in regulation?

Of course, pure game theory and the NHL do not make a perfect comparison. The immediate criticism of this idea is that a dominant team would not prefer to play the overtime game, due to the fact that playing in regulation allows for a much larger time for the dominant team to establish its superiority and be more likely to achieve the 2 points. This is clearly true. But just how much better would a team have to be in order for it to prefer to play regulation rather than just wait for overtime? For this, we will assume that overtime/shootout has a 50/50 split every time for both teams, no matter their predicted win percentage in regulation (which, if anything, is more conservative for the OT payoff for a better team). With this, we can determine the win percentage needed for it to be more advantageous to play in regulation.

Regulation Expected Points:  (2*x) + (0*(1-x)) = 2x

Overtime Expected Points: 2*(.50) + 1*(.50) = 1.5

Breakeven: 2x=1.5

x = .75

A team attempting to be a point-maximizer would have to be at least 75% confident of winning in regulation in order for actually playing regulation to be advantageous over just waiting for overtime. It is rare in hockey for a team to be as much as a 75% favorite in any game. For example, during the 2011-12 season, the New York Rangers, the team that ended up with the most points in the Eastern Conference, played the New York Islanders, the team that ended up with the 2nd-least points in the Eastern Conference, six times and went 4-2, winning 67% of the time. Obviously, this is anecdotal, but the NHL has proven to be a league with a lot of competitive balance, so having 75% confidence in a team winning in regulation before the game starts is quite rare.

Of course, it is also possible that two teams might conspire to not score on each other or at least agree to a number of goals to tie at. Another more plausible idea would be the coaches and teams agreeing beforehand on a certain point in the game at which, if the game was tied, both teams would not attempt to score on the other, or at least to play much more conservatively through delaying the puck when in possession as opposed to attacking.

So why doesn’t any form of collusion for overtime like this happen? For one, there is the whole “spirit of the game” argument. As with many established sports, there tends to be the sense that using niche strategies like this would make the game look bad, at least from an internal perspective.  But that’s too clichéd an excuse and, from an outsiders perspective, an argument that doesn’t change the approach that should be taken in a point-maximizing strategy.

A stronger argument against this kind of collusion comes from looking at the argument from a game theory perspective. In fact, we can see the reason that teams don’t conspire or attempt delay tactics through the situation’s parallels with the classic “Prisoner’s Dilemma.” Let’s say that two teams do actually conspire before a game to not score on the other team if there are less than 5 minutes left in the 3rd period and the game is tied (the actual time left doesn’t really matter in our outlook, but this scenario appears one that would be the most likely). When that timeline hits, both of the teams have two choices. They can either live up to their prior agreement by not playing “for real” in an attempt to waste time and bring the game to overtime or they can renege on their agreement and actually play to win in the time period.

If both teams wait for overtime, then we get an outcome of, on average, of 1.5 points for either team. If both teams play to win immediately, the game still very likely goes to overtime, but the utility is overall lower for both teams because of the possibility that the game is decided in regulation (and less total utility is given out). If one team plays to win in regulation and the other team is caught just trying to delay the puck and wait for overtime, however, the utility swings in the direction of the player who plays to win in regulation, as they both have the same utility if the game goes into overtime, but the aggressive team has a much higher chance to win in regulation because of the fact that they are actually trying to score whilst the other team isn’t. Overall, we get a situation that is equivalent to the Prisoner’s Dilemma (note that the numbers here are just a quick estimation, more important is their relative values to each other).

Average Points (Player 1 Average, Player 2 Average)

Choice 1:

Wait For Overtime

Choice 2:

Play To Win In Regulation

Choice 1:

Wait For Overtime

1.5, 1.5

1.7, 1.2

Choice 2:

Play To Win In Regulation

1.2, 1.7

1.3, 1.3

The Nash Equilibrium here will be both teams playing to win in regulation. This is because if you knew that the other team is playing to win in regulation, it is always optimal to also play to win in regulation to maximize points. Of course, this situation (1.3 points on average for each team) is strictly inferior in average points for each compared to when both teams just wait for overtime (1.5 points for both). Unfortunately, however, individual pursuit of incentives work to have both teams arrive at this unfortunate solution, and we can see why conspiracy to go to overtime between two tied teams wouldn’t end up working out.

The only situation in which teams would be able to find equilibrium through both waiting for overtime (and thus both gaining more expected utility), would be through an infinitely repeated Prisoner’s Dilemma situation, which could occur if two coaches and teams got together and agreed that whenever the two teams played from then on out they would take the game to overtime in a situation like that. In this situation, teams would be more likely to reach the “both waiting for overtime” decision because there would be less incentive to cheat the agreement, as ruining the agreement would result in less utility for the cheating team the theoretical next time the teams played. Of course, hockey, like almost any sports league, has a tremendous amount of coach and roster turnover, and a situation like that is very unlikely to occur (not to mention the fact that hockey coaches and players are probably not thinking like game theorists).

The Bettman Point makes conspiracy to actively take a regular season hockey game to overtime between two point-maximizing teams look very appealing and strategically advantageous. However, we can see through the example of the Prisoner’s Dilemma that much of the potential bonus utility of exploiting the extra overtime point is lost when the two parties must compete over who takes the surplus.

Xavier Weisenreder
Georgetown Class of 2016

Image courtesy of The Washington Post

Follow Xavier on Twitter: @BeMoreChillNext
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