The Quantitative Peace

When losing is the winning strategy: Game Theory, Badminton, and the 2012 Summer Olympics

The Badminton World Federation disqualified four female badminton teams today from the London Olympics for unsportsmanlike behavior. These teams purposefully tried to lose their first match in a round robin event in order to be paired against easier teams in subsequent rounds (video recap here). These teams were obviously cheating as purposely losing in this manner is against the badminton rules. So why would so many teams blatantly cheat in a very public and very important sporting event? Because they thought they could get away with it. Prior to this mass banning of teams, several Chinese teams had a history of purposefully trying to lose in badminton matches and they were never punished for their behavior.

Being able to cheat and get away with it gave the Chinese teams a huge advantage in the Olympics. Hence, other teams followed the example of the Chinese teams and tried to lose, so they would also have an advantage in later rounds of the competition. All the cheating teams believed there existed a low probability of being caught and/or punished for cheating given past examples.

I created a game to demonstrate why it was in the best interest of all the badminton teams to try to lose their first badminton matches, assuming there was a very low probability that they would be caught.

The Players

There are two players in this game, Team 1 and Team 2. Team 1 is the better team. They have a 60% chance of winning the match if both teams play fairly. Team 2 is the weaker team. They only have a 40% chance of winning the match if both teams play fairly.

Both teams want to lose the match (while the A team is the better team in this particular round, they would rather have easier competition in subsequent rounds of tournament play). If a team loses, they receive 10 points. Alternatively, if a team wins, they receive 0 points.

Nature also plays a role in this game. Nature represents both the probability that the team will be found cheating and punished for their behavior.  If Nature decides that a team played unfairly, Nature will punish the team, but only 5% of the time.  Given that punishment has been rare in the past, this probability is reasonable, but can be altered to change the risk/reward ratio for cheating. If a team is caught cheating, they receive -10 points.

The Moves (See Figure 1)

Team 1 goes first. They can “Cooperate” and try to win the game, or they can “Defect”, and purposefully try to lose.

Team 2 moves next. They observed Team 1’s move. Team 2 can also “Cooperate” and try to win the game, or “Defect”, and purposefully try to lose.

The Outcomes

There are four outcomes.

1. Both teams Cooperate

For this outcome, both teams played fairly and tried to win the game. Team 1 wins 60% of the time when this occurs, and Team 2 wins 40% of the time. A team receives 0 points for winning and 10 points for losing. Hence, the expected utility for each team if this outcome occurs is:

([(0.6*0)+(0.4*10)] , [(0.4*0)+(0.6*10)]) = (4, 6)

2. Team 1 Cooperates, Team 2 Defects

For this outcome, Team 1 tried to win and Team 2 tried to lose. Team 1 wins with a 100% probability and receives 0 points for winning.

Since Team 2 cheated, there is a probability that Nature will punish them. Nature punishes Team 2 5% of the time. If Team 2 is punished, they receive -10 points. The other 95% of the time, Team 2 gets away with it and receives 10 points for losing.

The expected utility for each team if this outcome occurs is:

(0, [(0.95*10)+(0.05*-10)]) = (0, 9)

3. Team 1 Defects, Team 2 Cooperates

For this outcome, Team 1 tried to lose and Team 2 tried to win. Team 2 wins with a 100% probability and receives 0 points for winning.

Since Team 1 cheated, there is a probability that Nature will punish them. Nature punishes Team 1 5% of the time. If Team 1 is punished, they receive -10 points. The other 95% of the time, Team 1 gets away with it and receives 10 points for losing.

The expected utility for each team if this outcome occurs is:

([(0.95*10)+(0.05*-10)], 0) = (9, 0)

4. Team 1 Defects, Team 2 Defects

For this outcome, both teams tried to lose. Since both teams tried to lose, there is a 50% chance that any team could win.

If a team wins, and Nature does not catch them cheating (which happens 95% of the time), the team receives 0 points.

If a team loses, and Nature does not catch them cheating (which happens 95% of the time), the team receives 10 points.

If Nature catches a team cheating (which happens 5% of the time), regardless of whether the team won or lost, the team receives -10 points.

The expected utility for each team if this outcome occurs is:

([(0.5*(0.95*0))+(0.5*(0.95*10))+(0.05*-10)], [(0.5*(0.95*0))+(0.5*(0.95*10))+(0.05*-10)]) = (4.25, 4.25)

The Nash Equilibria (See Figures 2 and 3)

There is one Nash Equilibrium derived from backwards induction, {Defect; Defect, Defect}.

There are two Nash Equilibria derived from the strategic form game, {Defect; Cooperate, Defect} and {Defect; Defect, Defect}.

This game suggests that as long as the Badminton World Federation rarely punishes players for trying to lose in the first match of a round robin even, it is in a team’s best interest to try to cheat early on in order to improve their chances of winning in subsequent rounds. To change this outcome, the Badminton World Federation should either detect and punish cheating more often, or change the rules of the tournament where losing in an early round is not so advantageous.

(For another interesting sports match- the 1994 Caribbean Cup provides a classic example of how rules fundamentally change the expected behavior of players in an environment where the object of the game appears simple. By the end of the game, it was in the best interest of both teams to try to score points for the other team).

 

On the Utility of IR Research

At the Duck of Minerva, Josh Busby has a post on the gap between political science research/IR and the policymaking community. I don't have a whole lot to say about the specific content of Busby's post, aside from the fact that I think this is an interesting an important debate to have. (Also, please note that the links in the quotations were provided in Busby's original post).

Using private information to beat Death

 

 

In Bill and Ted's Bogus Journey (not to be confused with their most Excellent Adventure), the duo are killed and sent to Hell.  On their way to an eternity of punishment, Death gives them the ability to escape their fate if they can best him in a game of their choosing.  The wager seems to be an obvious one to take: If you lose, you are stuck in hell for eternity; if you win, you can leave.  However, if you do not play the game, then you are still stuck in Hell without the ability to leave. In a previous dialogue (not shown in the above clip), Death boasts that he has never lost a game. 

If Death really never lost a game, it is reasonable to assume he has played billions of such games (assuming billions have encountered him and wanted to leave).  A variety of games are likely to have been proposed including athletic competitions (perhaps the most common up until this point in history), strategic games with full information, and even gambles.  Given this, Death would have to be some sort of super athlete that makes the right move every time (perfectly rational), and wins all lotteries. Even in fair lotteries, Death could only win so many coin tosses in a row before it appears the game is rigged in his favor.  As an aside, it is likely that no single player would know that he has played coin tosses before and won so many in the past, but his boast out to tip players off.  Thus, unfair lotteries biased in favor of the condemned are likely to still be skewed in their actual results for Death.

Given these constraints established by the movie, the question for the viewer becomes obvious: What game would you choose to play against Death?

 

After some pondering, the optimal game would be one of private information--where the player can control elements outside of Death's control.  As such, Death's ability to win depends on his ability to guess what the player has chosen. Given this proposition, then it appears that Bill and Ted stumbled upon the right course of action in their initial choice: Battleship.  In a world where you are playing the best strategy possible without cheating, the average win using a decent algorithm averages 40 moves. If we posit that Death plays perfectly strategically without knowing what the players know, then there is room for the other player to do better than Death.  So, Bill and Ted knowing he would put his ship in the J row may work out as a better strategy given their glimpse into Death's psychology. Clue, again, affords them room for private information in guessing which character committed the murder of Mr. Body.  Twister, on the other hand, I lack a convincing interpretation as it is clearly a game of athletics.  Perhaps Gymnasts have not challenged Death before.  Though, of course, I am sure the script was written with the idea of seeing an ancient mythological being playing modern games against two relatively airheaded dudes and not with a mind towards game theory.

Given this understanding of how to beat death, and while Bill and Ted succeed in finding the right path to victory, they allow too much for error.  If Death can lose at games of private information, then the best strategy is to make it a game of asymmetric private information instead of a symmetric one (technically, a symmetric, asymmetric information game) like Battleship.  Simply, offering Death a game of "Guess What Number I Am Thinking Of" and making the possible set of numbers contain all numbers would make Death's ability to guess the number indistinguishable from zero.  The game is much less fun, but the certainty it provides makes it an optimal choice given the symmetric alternative.  For this solution to be accurate, then it means that either people have never played an asymmetric information game against Death, that his strategy was superior to his opponent's in stategy previous games and could overcome the information asymmetry, or, perhaps, he was simply boasting to dissuade would-be challengers.  Also, this is what happens when you leave Comedy Central on during the day while doing house chores. 

Math's PR Problem OR Changing our Educational System

I was listening to the BBC's Global News Podast this morning and found an interesting segment on the utility of mathematical modeling. Professor John Adam's book (which can be found here) looks at how one can model various aspects of urban life—pollution, traffic congestion, number of dentists offices, etc. I've not read the book, but Adam's message in his interview was something that I can sympathize with. In particular, Adam talks about the degree to which the educational process divorces math from any kind of applied usage. Learning math in this way, he goes on, is partially responsible for why more children/people are not as interested in mathematics as they might otherwise be. 

Sure, there are the ubiquitous "if a train leaves city A at rate X, and a train leaves city B at rate Z, when will their paths cross," kind of examples. But seriously, is that the best we can do? Now, I recognize that I'm far from being the strongest, or the quickest, when it comes to picking up a lot of the methods that I've been taught over the past few years. I, like many people I suppose, was not the biggest fan of math class in high school. Some of this was laziness on my part, but some of it was also the framework in which the subject matter was taught. Even in college it was far from clear that any kind of math beyond balancing a checkbook would play as big a part in my life as it currently does. I did well enough in these classes, but it was not until graduate school that I really had to seriously engage math and statistics.  

As it turns out, I've really enjoyed the quantitative side of my training here at Binghamton, and I'm far more willing to work at developing my understanding and skills now than I was when I was 16. Had I known in high school that I this stuff could be used to study things like war, human rights, etc., I might have cared a bit more. These are things that I've always been interested in. And this is perfectly rational behavior, right? If the expected utility of learning this material is zero, or even negative, then why expend the effort? Where is the payoff? If this is the case, then a major part of the reason that the expected utility of learning this material is so low is because the people that should be able to provide you with the relevant information to update your expectations simply do not do a very good job of that. In fact, I'd say that I didn't really get that at all until grad school. And being able to make that initial connection between math and statistics, and the substantive topics that I care about, has subsequently led me to develop a greater appreciation for the math itself that underlies the models I often use. 

I don't have any answers for how the educational system could be modified to address this issue. I also realize that you would have to find a way to incorporate applied mathematics into most subject fields, as not everyone has the particular set of interests that would make more math in history or government class a sufficient solution to the problem. Still, there must be a better way. With folks like Neil deGrasse Tyson going around trying to promote greater scientific literacy, and the recent attacks on NSF funding for political science, it seems like there is a space for us to modify how we educate our younglings and how science, math, history, etc., can be better integrated in this process. Maybe if people developed a better understanding of science as a process, or math as something with real world and substantive applications, rather than something that existed strictly from 1pm to 1:45pm during 8th period, people would be better able to develop a more robust appreciation for how mathematics and science fit into, and matter for, our daily lives. This is fairly obvious when phrased like that, but the fact is that it's not really what we do currently, is it?

Am I missing something?

Also—and this is a total aside—Brian Rathbun at the Duck of Minerva has a discussion of how Iron Maiden material can be used to teach political science. This is perhaps a good example of what I was talking about above, so perhaps this is not a total aside, but what we have here is essentially "applied awesomeness."

How Bruce Bueno de Mesquita buys a car

Bruce Bueno de Mesquita explains how to buy a car cheaply--something he mentioned on the Colbert Report this week:

 

 

The Colbert interview can be found here.

Stuff is still happening in (the sequel)...

I really don't know how many times I'll be able to use that title, but given that the American audience has become so distracted by the Washington debt ceiling battle, I decided to check on some old favorites and post a quick update on goings on elsewhere.  

Libya:  Apparently Libyan rebels have made "significant" gains in the past week, taking control of some key cities on the way to Tripoli.  NATO planes have damaged or destroyed over 150 targets in the past week alone.  However, who really knows what qualifies as a "military target" for these purposes.  Sources in the article also indicate that some of Ghadaffi's forces have begun consolidating in Tripoli and have begun removing their uniforms, apparently gearing up for a transition in tactics.  It will be interesting to see what the next month brings.  

Syria:  It sounds like the body count in Syria is continuing to rise steadily.  Syrian security forces are apparently combating "armed gangs" in an effort to protect the people.  This includes firing into Palestinian refugee camps as well it seems.  The article also mentions a Youtube video that has surfaced, purportedly showing some pretty graphic scenes from the crackdown.  I've not seen it, but it sounds pretty unsettling.  

Bahrain:  It sounds as though things have quieted down for the time being, but Bahrain is still dealing with the fallout from the previous protests and crackdowns.  Quiet doesn't necessarily mean good, however, as the BBC is reporting that hundreds of people are still in jail and there are reports of widespread torture.  Not terribly surprising, but it's important to keep in mind the fact that the violence carries on even once the streets have cleared.    

Class, Empathy, and Taxes

Several recent studies by three University of California psychologists, Michael Kraus, Paul Piff, and Dacher Keltner, suggest that individuals with lower class incomes are more empathetic and more altruistic than individuals with higher class incomes. They argue that since lower income individuals have to rely more on others to survive, they learn pro-social behaviors and learn to emphasize with others.

The results of these studies may explain why some individuals with a great deal of disposable income (1) are unaware of the magnitude of income inequality in the US, (2) believe they should pay the same tax rate as lower income individuals, and (3) want to reduce government spending on social programs. These studies can also explain the results of a recent CNN poll that suggests that individuals making under 50K a year are more likely than individuals making 50K a year or more to want the government to keep taxes high "so the government can use their money for programs to help lower-income people". 

 

Debt Ceiling

This information comes from the Office of Management and Budget's Historical Tables of the US Government's budget.  Just some basic information on the debt ceiling that might put the current political wrangling into greater context.  I will add the usual disclaimer that this is not my usual playground, so if anyone catches any mistakes please let me know.

Below is a quick graphic illustrating the points at which the debt ceiling has been raised since 1940.  These figures have not been adjusted for inflation.  The dots show the point at which the debt ceiling was raised, or a point at which a previous raise in the debt ceiling was reaffirmed/extended, while the line is intended to give the viewer a better ability to connect the dots (so to speak).  

This second graph adjusted the previous figures for inflation (billions of 2009 dollars).

Of note---If my count is right, President Reagan increased the debt ceiling 18 times during his administration, from a starting point of $985 billion to $2.8 trillion.  The Clinton administration increased the debt limit 6 times from $4.3 trillion to $5.95 trillion.  The second Bush administration 7 times from $6.4 trillion to $11.3 trillion.  Of the data available (it only runs through 2009 here) the Obama administration increased the debt limit from $11.3 trillion to $12.1 trillion.  

The real point at which the debt ceiling seems to take off is during the Reagan administration.  Prior to this point government seemed to keep closer tabs on the debt limit---although there are several dots on the graphics above prior to the Reagan administration, many of these cases are simply reaffirming or briefly extending previous raises to the debt limit.  Others can probably speak to this in greater detail, but this seems to make sense given that the Reagan administration was characterized by unusually high peacetime military spending.  The second point at which the debt ceiling rises seems to be during the second Bush presidency.  This also seems to make sense given the tremendous levels of spending for the war on terror.  

Republicans have been campaigning on the fiscal irresponsibility of Democrats, but a cursory inspection of these charts suggest that recent Republican presidencies don't have a great track record of keeping the debt ceiling down.  Earlier Republican presidencies (Eisenhower, Nixon, and Ford) seem to have been better at this.  That said, this does not take into account a host of other factors (international crises, which party controls Congress, divided government, etc.) so this is far from conclusive.  Still, if the argument is that the president is accountable for what happens on their watch then the fact remains that Republican presidents have presided over the greatest increases in the debt ceiling.    

Aid to Pakistan II

Apparently some US policymakers have decided to cut some of the US' aid to Pakistan.  I'll keep the commentary short on this one as I've already commented on the topic before at greater length.  I still have a suspicion that getting some distance between the US and Pakistan is not for the relationship's greater good---assuming that what is "good" entails keeping the Pakistani government afloat.  The government needs to maintain some semblance of authority or credibility amongst their own people, and publicly bickering with the US may be one way to do that.  Suffering some sort of tangible cost should enhance the credibility of their stand.  

Alternatively, there are surely very real reasons why US policymakers would be upset with Pakistan, and I also have to wonder if this provided an opportunity for some policymakers to start trimming some fat, as it were.  There were rumblings earlier in the year that Republican policymakers wanted to start cutting foreign aid.  Granted, the aid categories that Republicans were focusing on were really not military aid, but it's possible that some of these policymakers thought that we could save some money by cutting some military aid to countries who are, possibly, grossly overpaid relative to the kind/quality of assistance they've given us.  Take this with a large grain of salt, however, as this is wild speculation and I haven't really looked into this in further detail.  It's not unreasonable to think that military aid would be on the table, though---some prominent Republicans have come out in support of making cuts to the military within the past year.

It's moments like these when I really wish I could be a fly on the wall of some of these back room conversations.  Invariably there is going to be a public face and a private face to these kinds of events, and it would be nice to get a better grasp on just how close or far from one another those two faces actually are.   

The Targeting of Corporate Executives

I came across this article from Defense News yesterday, and the topic is something that I have not come across before.  The gist of the article is that a Pakistani branch of al-Qaeda, or at least with members affiliated with al-Qaeda, was planning the assassination of Lockheed Martin's CEO.

More after the jump...