I wrote a bit of R code to explore the ‘parity question’ in the NHL; specifically, if all teams were more or less equal, what would we expect to see in terms of regular season point totals? You can find the code for generating a hockey schedule here and the simulation code here.
The scheduler is just a dirty optimizing algorithm that I came up with that is probably inferior to what the NHL uses, but it seems to work. You can set the schedule parameters and it seems to work fairly well provided the inputs result in a feasible solution; for example, don’t lower the max games too much or it won’t be able to solve. I had to create the scheduler to generate a realistic play schedule that the hockey simulator could use.
For a given schedule, the hockey simulator plays a season of hockey and sums up point totals. For the purpose of this experiment, I assume that all teams have an equal chance of winning. If a team loses, they have a x% chance at getting a loser point (by default, this is set at 11%).
I then run the simulation many times. One way to investigate the results is to plot out the distribution of top regular season team (the team with the most points). Here is a histogram of the maximum point totals over 1000 simulations:
How does this compare to real data? Well, it depends on the season, but generally speaking, the best teams in the real world do better than the best teams in the simulation. This suggests that the NHL has not yet reached the point of real parity. The far left values on the figure below tell us the mean, range (95%) and maximum (of the maximum points) over 1000 simulations. Tampa Bay outperformed the parity simulation even after 1000 simulations–suggesting that their regular season performance in 2018/2019 was far from a matter of luck.
Nevertheless, imagine if 2018/2019 Tampa Bay was an anomaly, and the trend of real NHL point maximums continues downward–where the best regular season team gets point totals that approach 107 or so in a few years. At that point, it might suggest that the NHL has reached some level of true parity, where the vagaries of year-to-year luck (like injuries) will play an increasingly large role in determining the success of a team, rather than true superiority of skill and tactics on the ice.
Here is a very short, fairly non-technical description of the Brexit crisis seen through the game theory lens.
I start with some assumptions about the strategic payoffs (not in standard form for those acquainted with game theory) of the EU-UK Brexit negotiations. The payoffs tells us the net benefit from the possible strategic outcomes of the Brexit negotiations. Here we assume four strategic outcomes based on two decisions from two parties: offer a compromise Brexit deal or do not offer a compromise Brexit deal. I use ‘UK’ to stand in for the United Kingdom, and ‘EU’ to stand in for the European Union. The smaller the number (in brackets), the lower (and worse) the payoff (e.g., -10 is worse than -7). Here is a payoff table:
No compromises (no-deal Brexit): -10 (UK), -10 (EU)
Deal with UK compromising: -8 (UK), -5 (EU)
Deal with EU compromising: -5 (UK), -8 (EU)
Deal with both parties compromising: -7 (UK), -7 (EU)
The exact numbers don’t matter for these purposes, simply their relative sizes. What this payoff table says is that a no-deal Brexit is the worst situation for both parties (both get -10). It’s hard to know if this is true in the real world, but it does not seem unreasonable. It also tells us that we have two ‘pure strategy’ Nash equilibria: #2 (UK compromise, No EU compromise) and #3 (No UK compromise, EU compromise). These are equilibria because if forced into one of these two situations, neither party would be better off opting out. For example, say the US forced the #3 deal somehow (the UK does not compromise, but the EU does). Once that agreement is in place, the UK is not better off by switching its position from no-deal to deal, since switching would result in a worse payoff of -7 (#4). Similarly, the EU does not benefit from switching from deal to no-deal, since that would result in a worse payoff of -10 (#1).
Unfortunately, there is no easy way to ‘force’ a deal, so we get a standoff; both parties seem to be waiting for the other to compromise. We also have a deadline; if a deal is not made before October 31, 2019 , then #1 happens by default. Both parties know that a no-deal Brexit is the worst-case scenario, but do not seem compelled to give an inch. This is not a surprise for game theorists because unlike #2 and 3, #1 and #4 are not the same kind of equilibria. If both parties were somehow nudged into a #4 deal, for example, they both would have an incentive to defect to another option that has a higher payoff–where they do not compromise but the other side does. In this standoff, #1 may not be a desirable strategy, or optimal in any sense, but could still happen if a deal isn’t struck by the deadline.
Does Boris Johnson have a way out?
Boris Johnson, the current British PM has been a belligerent on the question of Brexit, and has made a number of decisions that signal preparation for a no-deal Brexit–promising money to help mitigate the financial consequences, and shutting down debate in Parliament. To many, this may seem like insanity (particularly if the no-deal Brexit really is the worse case scenario), but it could also be a clever strategy.
Once elected as PM, Johnson may have reasoned that if he can create a ‘credible threat’ of a no-deal Brexit, he will force the EU to compromise and solve the deadlock. He can create this credible threat by seeming to proactively remove #2 and #4 from the list of negotiating options (leaving only #1 and #3). He did this by reducing opposition debate in the House of Commons through parliamentary prorogation. This was a rare parliamentary maneuver (though not unfamiliar to Canadians), that could have been intended to signal that a compromise deal (that would require Parliamentary discussion) is now impossible. Once a British compromise deal is removed as a possibility, then the EU would have no choice but to compromise themselves, leading to agreement #3. This is because if the UK doesn’t compromise, the EU would be worse off with a no-deal Brexit (-10) (#1).
So Johnson’s quick work to prorogue Parliament may have been a tactic to convince the EU that the UK was creating a situation where no-deal was inevitable, thereby forcing their hand…
Another layer in the game…
However, now with the recent no-confidence vote in British Parliament, the threat of no UK compromise suddenly seems less credible. The British Parliament has changed its signal, effectively saying that they do not want a no-deal Brexit (or, for many, that they don’t want Brexit at all, and are trying to force conditions that will roll back the plan altogether).
In response to this, Johnson is pushing for an election. Why might he be doing this? Well, it is possible that the EU might also see an election as a credible threat of no compromise, particularly if it looks like Johnson might win with more support than his government currently has. Once an election is underway, there will be no government to negotiate with the EU, and less time to come up with a compromise deal acceptable to the UK parliament. It’s risky for Johnson–his government could easily lose an election. Nevertheless, if the election creates the credible threat of no-deal Brexit, he might again provoke some EU concessions.
It’s hard to know definitively what is going to happen in the short or long term. It’s also possible that there is no actual strategic intent here–maybe Johnson is just bluffing his way through this without much thought. What is clear, however, is that game theory gives us some language for understanding possible ways to think about the problem, and perhaps anticipate how the future will unfold…
Barrett’s focus in these papers is on climate change treaties. Climate change treaties are a challenge because costs of CO2 pollution are externalized, but countries are inclined to ‘free ride’ since there is no global government to punish non-signatories, or countries that fail to meet their commitments. For this reason, many argue that global-scale climate treaties are doomed to fail.
These papers show that coordination can work to address this climate treaty problem. Coordination occurs when actors work in their self-interest towards a goal that is collectively beneficial. When coordination is possible, there is no need for an enforcement mechanism; the treaties work because adhering to commitments are in the private interests of actors.
A number of conditions must exist to see effective coordination in abating climate change, but the one of Barrett’s focus is the reduction of uncertainty at the threshold which a climate catastrophe would occur. In short, as uncertainty in the threshold gets smaller, the more likely that an actor will follow through with abatement measures–like reducing CO2 emissions. Provided the uncertainty is small enough (in proportion to the damages resulting from climate change, and inversely proportional to the difference between costly abatement and the benefits of climate change) then actors will choose to take measures to avoid catastrophe.
In addition to demonstrating this theoretically, Barret and his co-author use some hypothetical choice experiments to test this theory in the lab. The experimental results are fairly consistent with their theoretical findings; the authors found that most people would choose to abate when the catastrophic tipping point is certain, and nobody would abate when it is very uncertain.
What this research shows is the importance of reducing scientific uncertainty, and specifically, that reducing uncertainty about thresholds of climate catastrophes may be key to getting useful and effective climate treaties.