Thursday, 29 September 2016

Golden balls and weakly dominant strategies

In ECON100 this week we covered game theory, in which the most famous game is the prisoners' dilemma. One of the tutors then sent me some YouTube links to clips from the game show, Golden Balls. This isn't quite the prisoners' dilemma, but it is close. In this game, the two players have to choose to split or steal. If they both choose split, they share the prize pool. If they both choose to steal, they both go home with nothing. If one chooses to steal and the other chooses to split, the one who steals gets the whole prize pool and the other gets nothing. Take a look:

Why would anyone steal? Stealing is actually a weakly dominant strategy - it's a strategy that is never worse than the other strategy (splitting), but is sometimes better. [*] Most rational players would choose to steal. To see why, consider the game in normal (payoff table) form below.

Consider Sarah's choice first. If Steven chooses to split, Sarah is better off choosing to steal, because £100,150 is better than £50,075. If Steven chooses to steal, it doesn't matter what Sarah does because her winnings will be zero regardless. So, stealing is a weakly dominant strategy for Sarah (it is never worse than splitting, and if Steven chooses to split then stealing is better for Sarah).

Now consider Steven's choice. If Sarah chooses to split, Steven is better off choosing to steal, because £100,150 is better than £50,075. If Sarah chooses to steal, it doesn't matter what Steven does because his winnings will be zero regardless. So, stealing is also a weakly dominant strategy for Steven (it is never worse than splitting, and if Sarah chooses to split then stealing is better for Steven).

So, I would expect the players to choose to steal. Unless they can find some way of changing the game. Which brings me to this clip:

Nick realises that if he chooses to split, he is at risk of going home with nothing if Ibrahim chooses the weakly dominant strategy (steal). So he attempts (successfully) to change the payoffs in the game by convincing Ibrahim that he will choose steal no matter what (and will split the prize pool with Ibrahim afterwards). The game changes to look like this:

Note that neither player has a dominant strategy now. In fact, Nick's strategy choice becomes irrelevant because Nick will get £6,800 if Ibrahim chooses to split, and nothing if Ibrahim chooses to steal, regardless of what he (Nick) chooses. Ibrahim also doesn't have a dominant strategy here. If Nick chooses to split, Ibrahim is better off to steal (because £13,600 is better than £6,800). But if Nick chooses to steal, Ibrahim is better off to split (because £6,800 is better than nothing). Ibrahim does have a minimax strategy to choose split. Minimax is where you look at the worst outcome from each strategy, and choose the strategy that has the worst outcome that is best. In this case, the worst outcome from choosing steal is going home with nothing, and the worst outcome from choosing split is £6,800 (provided Nick agrees to split the prize pool afterwards).

Ultimately though, it comes down to whether Nick's threat to choose steal no matter what (and split the prize pool afterwards) is credible (believable) or not. Once Ibrahim is convinced, then Nick chooses split as well. It's not an equilibrium solution to the game, but by changing the nature of the game Nick managed to make them both better off (than if they had both chosen their weakly dominant strategy).

[HT: My ECON100 tutor, Jae]


[*] Note that this is why this game isn't a classic prisoners' dilemma game. In the prisoners' dilemma, both players have dominant strategies (not weakly dominant strategies), and when both players choose their dominant strategy (which they will do if they are rational) then both players are made worse off.

Wednesday, 28 September 2016

The study strategies of top students

The headline of this Washington Post story from a couple of weeks ago is pretty provocative: "A telling experiment reveals a big problem among college students: They don’t know how to study". The problem is that the story doesn't actually tell us enough about the experiment to be able to judge. Here's the closest that the story gets:
Instead of highlighting, posing and answering questions as they read forces students to think about meaning, and helps them recognize whether they really understand. To prepare for a test, self-quizzing actually boosts memory more than studying does. For example in one recent experiment, college students read 36 facts taken from a freshman biology course. Then some took a quiz on the facts while others restudied them for an equivalent amount of time. Memory for the facts was tested two days later and those who took the quiz outperformed the re-studiers, 61 percent versus 39 percent.
Unfortunately, I can't find the study referred to in that paragraph (given a few minutes of Google Scholar searching), and the link refers to a different (and much older) study. Reading that WaPo article wasn't a total bust though. The paragraph before the one quoted above is of interest too:
Some students leave college because classes just aren’t going well. However, most students have never been taught how to study and the strategies they devise on their own don’t work. For example, they highlight their textbooks to signal what they should review later, but if you’re reading difficult material for the first time you probably can’t identify what’s important. When preparing for an exam, students reread their highlighted textbook and their lecture notes, but rereading doesn’t make information stick because it’s so easy to repeat something mindlessly. Think of the last time you tried to remember someone’s name by saying it to yourself again and again.
 An ungated version of the paper linked in that paragraph is available here. It makes for interesting reading, although it isn't really helpful if you want to know the study strategies that are successful. In the paper the authors (Marissa Hartwig and John Dunlosky, both of Kent State University) interviewed 324 introductory psychology students about their study habits and their GPA. They then sort out which study habits are exhibited more by students with high GPAs, and which are exhibited more by students with low GPAs. Of course, this is correlation not causation - we have no idea whether the study strategies used by high-GPA students are effective for raising GPAs, only that the better students are using them.

Here's what they found:
In summary, low performers were especially likely to base their study decisions on impending deadlines rather than planning, and they were also more likely to engage in late-night studying. Although spacing (vs. massing) study was not significantly related to GPA, spacing was associated with the use of more study strategies overall. Finally, and perhaps most importantly, self-testing was a relatively popular strategy and was significantly related to student achievement.
Perhaps the most interesting bit of that is that self-testing (posing and answering questions while reading, doing practice problems, etc.) was more often used by better performing students. It would be good to know if this was causal rather than simply correlation (which is why I was disappointed with no link to the experiment mentioned in the WaPo article). Similarly, late-night studying was more prevalent among the lowest performers, but it would be good to know if there was anything causal in that.

The most surprising result in the Hartwig and Dunlosky study (to me at least) was that some 23 percent of those surveyed claimed to "usually return to course material to review it after a course has ended". That wasn't my experience as a student (except in a very few cases of economic theory or econometrics that I referred back to during postgraduate study!), although all teachers (including me) probably hope that students look back on things that we covered sometime after the course has finished.

[HT: David McKenzie at Development Impact]

Monday, 26 September 2016

Rugby, dementia, and the prisoners' dilemma

The prisoners' dilemma is probably the most famous game in game theory (I previously discussed it here). The outcome of the game is that both players, acting in their own self-interest, choose actions (strategies) that result in both of them being worse off. In the classic prisoners' dilemma, both prisoners confess to the crime, and both go to jail for a longer time than if they had both stayed silent.

There are many applications of the prisoners' dilemma in real life. Consider exiting a building in the event of a fire. Everyone would be more likely to get out safely if everyone walked to the fire exits in an orderly fashion. However, every person has an incentive to get to the exit as fast as possible to make sure they get out before being engulfed in flames. So, the end result is lots of running, with everyone competing for the exit, and a greater likelihood of people being hurt in the ensuing chaos.

Which brings me to dementia and rugby. The impact of repeated concussions on former rugby players has been in the news quite a bit this year (and follows similar news about NFL players over the last few years). See for example this front page story from the New Zealand Herald:
The first paper from the study into the health of retired rugby players was published in online journal Sports Medicine yesterday. "A Comparison of Cognitive Function in Former Rugby Union Players Compared with Former Non-Contact-Sport Players and the Impact of Concussion History" investigated the difference in brain function between rugby players who experienced concussion and those who didn't.
It found that players who experienced one or more concussions in their career performed worse in tests that measure cognitive flexibility, complex attention, executive function and processing speed. To put it in layman's terms, that is the ability to understand and process information quickly, to make rapid decisions, to switch attention between tasks and to track and respond to information over long periods of time.
How does this relate to the prisoners' dilemma? Consider two rugby players (Player One and Player Two), who have two strategies: (1) to play hard and risk concussion; and (2) to play softer and avoid head injuries. Both players would be better off if they both played softer, since both would avoid head injuries. However, knowing that the other player is playing softer, each player would be better off playing hard since that gives them a better chance of winning. However, if either player plays hard, then both players risk a head injury. The game (in normal form) is laid out in the payoff table below.

What happens in this game? It actually depends on how players evaluate the trade-offs in this game. Let's start by assuming that winning is paramount.

Consider Player One first. They have a dominant strategy to play hard. A dominant strategy is a strategy that is always better for a player, no matter what the other players do. Playing hard is a dominant strategy because the payoff is always better than playing softer. If Player Two plays hard, Player One is better off playing hard (because being unsafe is better than being unsafe with a worse chance of winning). If Player Two plays softer, Player One is better off playing hard (because being unsafe with a better chance of winning is better than being safe, because winning is paramount). So Player One would always choose to play hard, because playing hard is a dominant strategy.

Player Two faces the same decisions. They also have a dominant strategy to play hard. If Player One plays hard, Player Two is better off playing hard (because being unsafe is better than being unsafe with a worse chance of winning). If Player One plays softer, Player Two is better off playing hard (because being unsafe with a better chance of winning is better than being safe, because winning is paramount). So Player Two would always choose to play hard, because playing hard is a dominant strategy.

Both players will choose their dominant strategy (to play hard), and both will be unsafe. But what if being safe was more important than winning? That changes things considerably. Now, playing hard is no longer a dominant strategy.

Go back to Player One. If Player Two plays hard, Player One is still better off playing hard (because being unsafe is better than being unsafe with a worse chance of winning). If Player Two plays softer, Player One is now better off also playing softer (because being safe is better than being unsafe with a better chance of winning, since safety is now paramount). Note that neither strategy is always better for Player One - Player One has no dominant strategy.

Now consider Player Two. If Player One plays hard, Player Two is still better off playing hard (because being unsafe is better than being unsafe with a worse chance of winning). If Player One plays softer, Player Two is now better off also playing softer (because being safe is better than being unsafe with a better chance of winning, since safety is now paramount). Note that neither strategy is always better for Player Two - Player Two also has no dominant strategy.

What is the solution to this game? We can find it using the best response method (which we've already described in the last two paragraphs). Any combination of strategies where both players are choosing their best response to the other player's strategy is a Nash equilibrium (named after John Nash). This occurs where both players play hard, and where both players play softer. We have a coordination game (a game with multiple equilibriums).

In a coordination game, if one of the equilibriums is clearly better than the other, we call that a Schelling Point (named after Thomas Schelling), and that equilibrium is the more likely equilibrium to obtain. In this case, both players playing softer is a Schelling Point. So, we could expect that both players would play softer (given our assumption that safety is paramount).

So what? The implication of this rough analysis is pretty clear. When rugby players view winning as paramount, then playing hard is a dominant strategy and all players will continually put their bodies (and minds) on the line in the pursuit of glory. If instead, safety becomes paramount then the outcome of the game changes, and it becomes likely that players will play softer and fewer injuries will result. If reducing rugby players' long-term concussion-related injuries is important, then the challenge for rugby authorities will be to change players' incentives to ensure that safety becomes a higher priority. Unfortunately, that is likely to be easier said than done.

Sunday, 25 September 2016

Enforcement is key for sustainable fishing quotas

Many economists agree that sustainable management of common resources like fish stocks can be achieved using transferable quotas. Unfortunately, New Zealand's quota management (or rather, mis-management) has been in the news quite a bit recently. Consider this New Zealand Herald editorial:
The beauty about New Zealand's quota management system for commercial fisheries was that it would be self-policing, in theory. Once fishing companies were allocated a quota that could be traded, they would have a financial interest in the future of the fishery.
Therefore, it would be in the interest of each owner of a quota to catch no more than it was allowed and thus ensure the resource could be sustained and the quota would hold its value. It was also in the interest if each of them to see that all obeyed the rules. So they jointly set up a company that is contracted to police their catch.
This system has come under severe criticism this year following academic research that estimates fishing companies are catching more than twice their quota of some species and dumping the excess at sea. Even dumping filmed by CCTV cameras on board boats had not resulted in prosecution of their skippers by the Ministry for Primary Industries...
Unsurprisingly, many are saying the whole system is flawed, questioning the wisdom of giving the policing task to an industry-owned company ("putting a fox in charge of the chicken coop") and calling for a ministerial review or a commission of inquiry into the quota management system...
I wrote about fishing quotas very recently, in a post about kereru (emphasis added):
Quotas regulate the number of fish that are allowed to be removed from the sea in a given period of time. The total quota is set by determining a total allowable catch for a year (in theory at least this is roughly equal to the growth in the fishery stock), with some allowance made for recreational fishing. Quotas work well because they make fish excludable (no quota means no fishing) and are backed up by monitoring and enforcement.
Transferable fishing quotas create a property right (the right to fish). In order to be efficient (welfare maximising), property rights need to be universal, exclusive, transferable, and enforceable. In the case of fishing quota, universality means that all fish need to be covered by quota, and all fisherman must have quota in order to fish. Exclusivity means that only those who have quota can fish, and that all the costs and benefits of fishing should accrue to them. Transferability means that the quotas must be able to be transferred (sold, or leased) in a voluntary exchange. Enforceability means that quotas should be able to be enforced by the government, with high penalties for fishermen who fail to comply with the system.

The current system is failing in the last criterion (enforceability), and probably the first (universality) as well. A fishing industry that polices itself has little incentive to ensure that it is following the agreed management system. Discarding fish that are caught, and not recording them, is something that must be penalised. All fish that are caught (including those that are dumped) must be considered for the universality criterion to be satisfied.

The Herald editorial notes that the Ministry for Primary Industries' director of fisheries management wrote in an email that "If we found the golden bullet to stop discarding we would probably put half of the inshore fleet out of business overnight." If that's what it would take in order to ensure that fishing is sustainable, then it is the price that must be paid. Otherwise, the fish stocks are on a short road towards collapse from over-fishing.