Marshawn Lynch Was Optimal, But So Was a Quick Slant

Posted on February 1, 2015 | Leave a comment

It seems that social media has lashed out at Pete Carroll for not giving the ball to Marshawn Lynch on second and goal with less than a minute to go. The idea is that Marshawn Lynch is #beastmode, an unstoppable force that would have assuredly scored and won Super Bowl XLIX.

The problem is, the argument makes absolutely no sense from a game theoretical standpoint. The ability to succeed on any given play is a function of the offense’s play call and the defense’s play call. Call a run against a pass blitz with deep coverage, and the offense is in great shape. Run deep routes versus that same defense, though, and you are in trouble. Thus, once you strip everything down, play calling is nothing more than a very complex guessing game. The Seahawks want to guess the Patriots’ play call and pick the correct counter. Vice versa for the Patriots.

Game theory has killed countless trees exploring the strategic properties of such games. Fortunately, there is a simple game that encapsulates the most important finding. It is called matching pennies:

The premise is that we each have a penny and simultaneously choose whether to reveal heads or tails. I win $1 from you if the coin faces match, while you win $1 from me if the coin faces mismatch.

You should quickly work out that there is a single best way to play the game: both of us should reveal heads 50% of the time and tails 50% of the time. If any player chooses one side even slightly more often, the other could select the proper counter strategy and reap a profit. Randomizing at the 50/50 clip guarantees that your opponent cannot exploit you.

In terms of football, you might think of you as the offense and me as the defense. You want to mismatch (i.e., call a run play while I am defending the pass) and I want to match (i.e., defend the pass while you call a pass). What is interesting is that this randomization principle neatly extends to more complicated situations involving hundreds of strategies and counterstrategies. Unless a single strategy is always best for you regardless of what the other side picks, optimal strategy selection requires you to randomize to prevent your opponent from exploiting you.

What does this tell us about the Marshawn Lynch situation? Well, suppose it is so plainly obvious that Pete Carroll must call for a run. Bill Belichick, who many see as the god of the football strategy universe, would anticipate this. He would then call a play specifically designed to stop the run. By that I mean an all-out run blitz, with linebackers completely selling out and cornerbacks ignoring the receivers and going straight for the backfield. After all, they have nothing to lose—the receivers aren’t getting the ball because Lynch is assuredly running it.

Of course, it doesn’t take much to see that this is also a ridiculous outcome. If the Patriots were to certainly sell out because the Seahawks were certainly handing the ball to Lynch, Pete Carroll would switch his strategy. Rather than run the ball, he would call for a pass and an easy touchdown. After all, a pass to a wide-open receiver is a much easier touchdown than hoping Marshawn Lynch can conquer 11 defenders.

The again, Belichick would realize that the Seahawks were going to pass and not sell out on his run defense. But then Carroll would want to run again. So Belichick goes back to defending the run. But then Carroll would pass. And Belichick would call for pass coverage. And so forth.

There is exactly one way to properly defend in this situation: randomize between covering a run and covering a pass. There is also exactly one way to properly attack in this situation: sometimes run the ball and sometimes pass it. This is the only way to keep your team from being exploited, regardless of whether you are on offense or defense.

Okay, so we have established that the teams should be randomizing. What does that say about the outcome of Super Bowl XLIX? Well, clearly the play didn’t work out for the Seahawks. But to judge the play call, we can’t account for what happened. We can only account for what might happen in expectation. And in expectation, passing was optimal in this situation.

If you aren’t convinced, imagine we all hopped into a time machine to second and goal with the knowledge of what happened. Would Pete Carroll call a run? Maybe. Would Bill Belichick sell out on the run? Maybe. But maybe not—Carroll might call a pass precisely because Belichick is anticipating him running the ball. We are back in the guessing game before. And as before, the only way to solve it is to randomize.

That’s the magic of mixed strategy Nash equilibrium. Even if your opponent knows what you are about to do, there is nothing he or she can do to improve your score.

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How to Get a Ball at a Game, AKA the Best Thing I Will Ever Write

Posted on December 25, 2014 | 1 comment

Right before I left San Diego for Rochester, I wrote a post in one of the Los Angeles Angels’ fan message boards. On the surface, it explains how to catch baseballs at baseball games. In practice, it was a recap of the first 22 years of my life. It apparently struck a chord and popped up on the site’s front page later that night.

(Ironically, I wasn’t home when it was featured—I was at a Padres game.)

I run into it every year or so, and I end up drawing the same conclusion every time: even though it predates all the Game Theory 101 stuff by more than a year, it is the best thing I have ever written and probably the best thing I will ever write. As such, I am preserving it here so I will never lose it.

Enjoy.

_____________________________________________________

I have been an Angels fan since the tragedy known as the 1995 season. I grew up in the northern part of Los Angels (sic) County, so I don’t have a very good reason why I wear red instead of blue. It just is what it is. The downside was that I virtually never went to Angels games as a kid due to the fact that my parents did not like sports and we lived a pretty long distance away.

But the rare times I did went, I always dreamed of catching a ball—a foul ball, batting practice ball, home run ball, a ball flipped up to the stands by a groundskeeper, any ball. Of course, we always had cheap seats too far away to get anything during a game. And a batting practice ball? That would have required getting to the game early—and the bottom of the first inning does not qualify.

So I went through childhood with zero, zilch, nada. Undeterred, I went to college. Armed with my own car and my own money, I could go to a lot of games as early as I wanted to. Now I was bigger, faster, and stronger. And, dammit, I wanted a ball.

I kept striking out.

Junior year rolled by, and my then girlfriend bought us tickets to a game. I took her to batting practice. Maybe my luck would change. Maybe I could get a ball. Maybe I could impress her.

And with one flip from a groundskeeper by the bullpen, it did.

Unfortunately, one isn’t satisfying. I thought it would be, but it’s definitely not. You get a rush from getting your first, and you immediately want to get another. So I kept going to batting practice in search of a second high.

It never came.

In college, I studied political science. I was introduced to a tool known as game theory midway through my junior year. Rather than trying to craft a more clever argument than the next guy, you can use game theory to construct models of the political interactions you are trying to describe. The neat part is that, once you have solved the game, your conclusions are mathematically true. If your assumptions are true, then the results must follow as a consequence.

The other cool part is that game theory is applicable to more than just political science. Life is a game. Game theory is just trying to solve it. The trick is figuring out how to properly model situations and what assumptions to make. Take care of those things, and you can find an answer to whatever question you want.

Baseball is a game, but so is hunting down baseballs as a fan. We all want to get them. The question is how to optimally grab one when everyone else is trying to do the same thing.

Fast forward to Opening Day of my senior year. I was standing there, hoping like hell a ball would find its way into my glove. If I stayed there long enough, I am sure one would have eventually gone right to me. But batting practice is short, and I would hate to only get one ball every 100 games I go to.

Then I noticed something a little revealing. It seemed like there would always be a couple people who would get three or four balls every time I went to the ballpark. I would always hear people say “lucky” with a hint of disdain the second, third, and fourth times they caught a baseball. But let’s be honest—it would take a tremendous amount of luck to get four baseballs in a single game if unless you were doing something everyone else wasn’t. You are lucky just to get one. But four? Skill.

That’s when the game theorist slapped the naïve young boy inside me. The people who were getting all of the balls weren’t game theorists, but they sure did understand the game being played better than everyone else there, myself included. I figured out that batting practice isn’t some sort mystical game of luck, it’s a spatial optimization game. Spatial optimization games can be solved. I did some work, came up with an equilibrium (game theoretic jargon for “solved the game”), and came up with a plan. In sum:

Since then, I have never left a session of batting practice with fewer than three balls.

Why am I telling you this? After all, the more people who know the secret, the harder it will be for me to catch a ball.

Well, here is the sad part. It turns out that I am a half-decent game theorist, so the University of Rochester accepted me into their PhD program. I leave on Monday. Yesterday was my last game. But it was a successful day:

That’s Barbara, my favorite usher in Angels Stadium. I can’t count how many times I have heard her tell parents to stop dangling their five year olds over the railing trying to siphon a ball off a fielder. (It baffles me why parents take such a risk in the first place. I’m pretty sure it is because the parents want the ball for themselves more than they want it for their kids.) I couldn’t leave California without getting a picture with her.

What do I do with my collection? I don’t have one. During my initial college years of ball-catching failure, I read an article about the (presumed) record holder for most balls grabbed ever. He keeps all of them. I think he is a jerk. As a kid, it was my dream to get a ball. As an adult, getting a ball is a novelty—a story to relay to your friends, take pictures of, and write silly little posts about on baseball forums. After reading the article, I swore I would give the first ball I caught to a kid trying to live the dream.

That moment had to wait for my junior year. The groundskeeper flipped the ball into my glove. I showed it to my girlfriend and found a mother with her five year old son sitting a few rows behind us. I asked if she would take a picture of us with the ball. She obliged. Although he was clueless, her poor son had no hope of getting a ball. So I thanked her for snapping the photo and tossed the ball over to her son. If that wasn’t the best day of his life so far, it has to rank pretty high.

I have kept that tradition alive all the way to today. As I pack my car this weekend, there won’t be any baseballs in it. I have no batting practice ball collection. I haven’t kept a single ball. I will never be able to make my dream as a kid come true—it’s too late for that—but I can get close every time I toss a ball to someone who reminds me of me as a kid. Perhaps that will be my son one day.

And if you thought my days of getting baseballs was over, think again. The Angels play the Rangers in Arlington on Thursday. I will be driving through Texas that day. Rangers fans won’t stand a chance.

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Do Elite Academic Institutions Make Fewer Football Mistakes?

Posted on November 29, 2014 | Leave a comment

To bury the lede, and by Betteridge’s law of headlines, it appears the answer is no.

This question was the result of some inane football commentary during Saturday’s Stanford/UCLA game. UCLA was down by a lot and attempted a fake field goal from Stanford’s 30 yard line. Stanford intercepted the pass in the end zone, resulting in a touchback. According to the color commentator (paraphrasing), “You just can’t run fake field goals against Stanford. Stanford is an elite academic institution—they don’t make mental mistakes.”

Now, there are two inherently boneheaded parts of this claim. First, immediately after the commentator gave us this knowledge smackdown, he then discussed how the Stanford defensive back should have batted down the pass rather than intercepted it. With the original line of scrimmage being the 30 yard line and a touchback putting the ball on the 20, the interception yielded a 10 yard loss. In other words, on the same play where Stanford was incapable of making a mental mistake because they are Stanford, Stanford made a mental mistake. Hmm.

Second, I would dare say that UCLA is an elite academic institution and they should therefore be immune from making mental mistakes. That being the case, they should have have internalized the fact that Stanford is an elite academic institution incapable of making mistakes and therefore not tried the fake punt. Or did UCLA, an elite academic institution incapable of making mistakes, make a mistake?

But I digress. As a friend pointed out to me, this type of inane commentary is nothing new. I know this all to well from living in Buffalo’s TV region and being repeatedly subjected to lectures on how Ryan Fitzpatrick is infallible because he went to Harvard.

Or something.

In any case, the commentators are giving us an answerable question: do elite academic school football teams make fewer mistakes than their…umm…more average brethren? I decided to test this proposition. Of course, there are all sorts of challenges to teasing out relationship. I cannot hope to give a comprehensive analysis in a single blog post. I can, however, give a first-cut answer using readily available data.

There are two obstacles standing in our way. First, what is an elite academic institution? Fortunately, university rankings has to be a million dollar industry. There simply is no shortage of them. I chose to look at US News and World Report’s rankings this time. Yes, these rankings are horribly flawed for a number of reasons. Yes, you may prefer Washington Monthly’s system.[1] But USNWR’s rankings are the standard bearer that all other rankings are compared to. I can thus measure academic quality by using USNWR’s cardinal score. Note the score is different from the ranking and captures how much better one school (allegedly) is than another.

There are a couple of important caveats here. First, USNWR features a wide variety of lists. To keep from comparing apples to oranges, I am only looking at schools on the national university surveys. Regional schools, liberal arts schools, and military schools that nevertheless have FBS programs are thus excluded. (I’m only looking at FBS schools, again to compare apples to apples.) In addition, USNWR does not give specific scores to any school earning a value below 25 on their 100 point scale. I have also excluded them from the analysis below.

Second, and perhaps more difficult, is that we need a way to measure mental mistake propensity. I can conceptualize “mistakes” in a variety of ways, but the trouble is hammering out something that is (1) somewhat objective and (2) quantitatively available. Penalties seem like an appropriate choice because they are (usually) mistakes and often leave everyone shaking their heads.[2] If these elite schools really have their academic prowess rub off on football players, we would expect them to incur fewer penalties. It’s not the same type of mental mistake that occurred during the UCLA/Stanford game, but it’s a decent substitute.

(If you have any alternative measures in mind, please post them in the comments!)

So, to recap, we can plot the academic quality scores against penalties. If there is a negative correlation, we would have confirmation that elite academic schools make fewer mistakes on the gridiron. The problem is, no matter whether you look at penalties or penalty yards, there is no relationship whatsoever. Here is the plot for yards:

penpergame

And penalties:

penpergame2

Both of those red trendlines are as flat as they get. There simply is no relationship.[3] And why would there be? Football is specialized knowledge. Players are recruited for that knowledge, not their ability to master biochemistry. And biochemistry is not going to teach you whether grabbing an opponent’s face mask is appropriate behavior.

Also worth noting: the two most penalized teams in the country are UC Berkeley and UCLA, two of the best public schools in the country. (I am happy to report that UCSD, the third highest ranked school in the UC system, has not committed a single penalty all season.[4])

In sum, unless penalties are a poor way to measure mistakes, academic prowess has nothing to do with football IQ. What we have here is a case of commentators filling airtime with silly platitudes.

[1] Which is obviously superior based on the #1 school on that list.

[2] The real concern here—as any game theorist would love to point out—is that penalties aren’t really against the rules so much as an alternative way of playing the game. For example, if an offensive lineman is beat off the snap has the choice between giving up a 10 yard holding penalty and having the quarterback be sacked for a loss of 12 yards, the “penalty” really isn’t much of a penalty at all. While I don’t discount that intentional penalties occur and are the result of smart play, these seem to be the exception to rule. Indeed, we rarely hear of penalties as being a good thing, and when we do, it’s precisely because it is so rare.

[3] For the statistically inclined, here is the regression output for penalty yards per game using OLS.

reg

I didn’t control for anything else, and I’m not sure there is anything to control for here anyway. Let me know if I’m wrong in the comments.

The observations deleted due to missingness are the schools that lack a USNWR score on the national universities list.

[4] Still undefeated!

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A Wii Bit of an Error? Price Matching as Price Fixing

Posted on November 17, 2014 | 2 comments

Yesterday, Sears made a wiiiiii bit of an error, selling a new Wii U for the bargain price of $60, a sharp markdown from the standard $300 price tag. People caught on and immediately bought as many as they could. That was smart. Sears eventually pulled it, though. So smarter people went one step further: they visited other retailers and bought $60 systems using price match guarantees, i.e., promises retailers make to sell like-goods at the lowest price of their competitors. Particularly crafty individuals allegedly went from Wal-Mart to Wal-Mart clearly out the storerooms.

This raised a moral question: is it right for consumers to take advantage of Sears’ mistake, manipulate the system, and trick (“trick”?) other retailers into also selling their products at a loss? Some people on Reddit felt that way:

I think I’d feel guilty doing that. But dang that’s a good deal.

Am I the only one who thinks it’s kinda [bad] to make another store pay for Sears error? You know it’s a error so why make it someone else’s problem?

Is there a difference between a legitimate offer / deal and taking advantage of a mistake? I’ll see you all in hell, which is where I’ll be down-voted into, where I can play Mario Kart with you thieving [lovely individuals].

I can certainly see their point. However, I don’t think that anyone should lose sleep for taking advantage of Target and friends for price matching. Why? Because one of the main reasons to create price match guarantees is to screw you over.

Wait, what? How can price matching possibly be bad for consumers? After all, it allows consumers to pay smaller prices. It could not possibly hurt consumers, could it?

Unfortunately, it can. Price matching is a form of price fixing, cleverly disguised as a nice gesture toward consumers. The key is how companies act in the bigger picture with price matching in place.

Imagine that you are a company and you have widgets to sell to consumers. You would like to charge your consumers a lot of money to pay for your widgets. However, there is a rival company that sells identical widgets. So if you charge a high price, all of the consumers will go to your rival, and you will make no money. Of course, your competitor has the exact same incentives. As such, you both end up charging very low prices. All of potential profit to be made from widgets has gone up in smoke.

In game theory, we call this situation a prisoner’s dilemma. Broadly speaking, this is a situation where both actors must individually choose whether to act kindly to the other (raise prices) or act uncooperatively (lower prices). Regardless of what the other side does, you have incentive to take the uncooperative action—this is because you can take all of the profits if the other side raises prices and still maintain parity in case they also lower theirs. However, the other side has the exact same incentives. So both of you take the uncooperative action even though this leaves you collectively worse off than if you both took the cooperative action.

If this is confusing, it might help to look at the problem visually:

 

Still with me? Okay. The point of the pricing prisoner’s dilemma is that it sucks up all of the revenue for widgets and leaves it in the pockets of consumers. This obviously makes those consumers very happy. But the companies would bend over backwards to figure out a way to collude to raise prices to monopoly levels. Yet successful collusion requires preventing the other side from undercutting one’s own price. After all, I don’t want to charge $10 for widgets if you are just going to screw me over by charging $9.

See where this is going yet? Price matching serves as this precise enforcement mechanism. Imagine that I announce that I will match any price you offer. I then charge $10 for my widgets. What are your incentives? Obviously, charging more than $10 is a bad idea, as I will take all of your business. So what if you undercut me instead? Well, you can’t. If you sell your product for $9, discerning customers have no reason to flock to your business because they can also get the widget for $9 from me thanks to my price match guarantee.

What to do? Well, you could also charge $10 and institute your own price match guarantee. For the same reasons as before, I don’t have incentive to undercut you either. We can both sustain the price of $10, well above what we would charging in a competitive environment.

So, despite appearances and Federal Trade Commission approval, price matching is a form of price fixing. It is intentionally designed to reduce competition and increase prices.

This makes the $60 Wii U price matching incident all the better: consumers used a policy designed to screw over consumers to screw over those who instituted the anti-competitive price fixing.

TL;DR: Karma.

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Bribery and Cartel Violence in Mexico

Posted on November 13, 2014 | Leave a comment

Mexico has a massive murder problem. 2012 alone saw more than 26,000 homicides in the country, the fourth most of any state in the world. Why?

Drug violence and the interaction between cartels is a major factor. In a new working paper, Paul Zachary (UCSD) and I argue that uncertainty about local leaders has a great impact on that cartel violence. Cartels benefit from using violence to eliminate rivals. Politicians, however, have a vested interest in limiting that violence. This causes tension between cartels and officials, which cartels often attempt to resolve by bribing the officials to look the other way.

Why does uncertainty matter here? Paul and I investigate a model involving two rival cartels—a status quo and a challenger—and a local politician. The cartels want to capture as much of the drug rents as feasible; the local politician wants to minimize violence, but he is willing to look the other way if he receives a large enough bribe. The game begins with the status quo cartel offering a bribe to the politician to minimize enforcement. If the politician rejects, he chooses an amount of effort to exert to reduce the effectiveness of violence, which undermines the status quo cartel’s ability to maintain its drug rents. After, both cartels choose an amount of costly violence, which determines what percentage of the drug rents each receives.

We find that successful bribes lead to higher levels of violence. This is for two reasons. First, and most obviously, additional enforcement intercepts an additional percentage of violence. But there is an important second-order effect as well. The interception of violence functionally increases the marginal cost of violence for the status quo cartel. Consequently, more enforcement not only quashes violence in action but deters some of its production as well.

The above logic leads us to investigate what might lead to bargaining failure during the bribery stage. We show that the status quo cartel’s knowledge of the politician’s level of corruption is key here. When the cartel knows the politician’s minimally sufficient bribe (which is a function of the level of corruption), it can very easily come to terms. But when the cartel can only guess from a wide range of possibilities, it might ultimately offer a bribe that isn’t big enough for the politician to bite. In expectation, this leads to higher levels of enforcement and less subsequent violence.

Our theoretical argument has a noteworthy empirical prediction. Uncertainty leads to less violence, and some work in IR indicates that there is more uncertainty about newer leaders than older leaders. Thus, in Mexico, we would expect municipalities that the same political party has controlled for longer periods to be more violent than municipalities with greater turnover. (In the absence of our argument, this would be odd: the retrospective voting literature would suggest that less violence should correlate with greater tenure, as voters should be rewarding politicians who keep the streets safe.) The data support our theory. Indeed, we estimate that an increase of one year in tenure is associated with roughly one additional murder within a municipality. Although this might not seem like much, with so many municipalities in Mexico, a countrywide increase of one year in tenure matches up with about 2300 more murders. This number is on par with the 2011 murder totals in France, Germany, United Kingdom, Netherlands, and Belgium combined but still only a fraction of the overall number of murders in Mexico in every given year.

Again, you can download the paper here. This is the abstract:

What role do politicians have in bargaining with violent non-state actors to determine the level of violence in their districts? Although some studies address this question in the context of civil war, it is unclear whether their findings generalize to organizations that do not want to overthrow the state. Unlike political actors, criminal groups monopolize markets by using violence to eliminate rivals. In the context of the Mexican Drug War, we argue that increased time in office increases cartels’ knowledge about local political elites’ willingness to accept bribes. With bribes accepted and levels of police enforcement low, cartels endogenously ratchet up levels of violence because its marginal value is greater under these conditions. We formalize our claims with a model and then test its implications with a novel dataset on violent incidents and political tenure in Mexico. We find that each additional year after an official initially takes office is associated with an additional 2,300 violent deaths countrywide.

It’s a working paper, so we’d love your feedback!

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Subtle Clues in Final Jeopardy

Posted on October 28, 2014 | 2 comments

Tonight’s Final Jeopardy had another excellent example of how the wording of the clue sometimes gives a hint of the answer. The category was The Bible. And the clue:

The first conversation recounted in the Bible is in Genesis 3, between these 2; it leads to trouble.

Given that conversation takes place early on in Genesis, you should be able to narrow it down to two possibilities. The players did; they were split between “Who are Adam and Eve?” and “Who are Eve and the serpent?” If you aren’t sure which is the correct response, go back and reread the clue.

Do you see it?

Pretend you are the writer for a moment and the correct response is “Who are Adam and Eve?” How would you write the clue? Probably like this:

The first conversation recounted in the Bible is in Genesis 3, between these 2 people; it leads to trouble.

But the clue doesn’t say people! It leaves it vague, and deliberately so. Jeopardy writers don’t like to be unnecessarily vague when they don’t have to. Here, however, they most certainly need to be. The correct response is “Who are Eve and the serpent?” But if they phrased the clue as:

The first conversation recounted in the Bible is in Genesis 3, between this person and this creature; it leads to trouble.

They can’t do that, though—it makes the clue painfully obvious. So they make it vague. But the fact that it is vague when it wouldn’t have to be otherwise actually makes it perfectly clear!

I’ve actually opined on this once before. About a year and a half ago, the clue was:

The circulation of the Times of New York & London totals about half the “Times of” this place, the largest of any English daily.

Do you see it? My analysis was in this video:

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Jeopardy’s Game Theory Irony

Posted on October 24, 2014 | Leave a comment

Tonight’s Jeopardy had a big high and a big low for game theorists.

The High
For most of the game, challenger Matthew LaMagna held a large lead. During Double Jeopardy, other challenger Angela Chuang hit a Daily Double in the “I Have a Theory” category. At only ~$4000 and facing Matthew at ~$18,000, Angela had only one option: make it a true Daily Double. She did. That part was sweet.

So was the clue (paraphrasing):

Beautiful Mind John Nash is credited with launching this field in economics.

Obviously, the correct response was “What is game theory?” Angela nailed it. Again, sweet. Maybe she knows game theory!

The Low
Now the sour part. Despite her best efforts, Matthew pulled away. The scores entering Final Jeopardy were $20,800 for Matthew, $8400 for Angela, and $1200 for the returning champion. Wagers are trivial at this point. Matthew has first place locked. Angela has second place locked as well because she doubles up the third place’s dollar figure. It does not take a game theorist to see this, but it helps.

(Critically, the end dollar figures are irrelevant for second and third place. Second receives a fixed $2000; third place, $1000.)

However, despite Angela’s familiarity with game theory, she wagers $8300. The returning champion wagers nothing. Final Jeopardy’s clue is triple stumper. Angela drops to $100 and third place, when all she had to do was write $0 and guarantee herself $2000. Instead, she went home with a check for $1000.

To be fair, there might be reason to not wager $0 here even though you can guarantee second place by doing so. Everyone’s favorite love-to-hate champion Arthur Chu famously wagered enough so that he would draw with second place if second place wagered everything. But Angela wasn’t even going for that. The $8300 wager could do nothing but harm her. That was sour.

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The Game Theory of Mario Kart 8

Posted on October 19, 2014 | 7 comments

Although racing games usually do not involve much strategic interaction*, Mario Kart 8—and its dastardly item blocks—require some thinking. Over the past few months of play, I have put my on my methodological hat and found at least three topics that game theory can help sort out. Let’s get to it.

(*Of course, just about all racing games require good strategy to beat opponents—things like knowing how to cut corners, boost properly, accelerate at the start of the race, etc. Because game theory is the study of how individuals interact with each other, I am focusing on the strategically interdependent decisions—i.e., those that require me to think about what you are doing and for you to think about what I am doing.)

Mario Kart Is a Defensive Game
Here is a common but critical mistake. You race toward an item block and pick up a red shell. A split second passes by. You see a helpless opponent directly in front of you and let it loose. The red shell strikes him, and you overtake his position.

A successful maneuver? Hardly. The opponent behind you has a red shell and does the exact same thing. You explode, losing two valuable seconds. Five people pass you, including the guy you shelled. Your net gain is negative four positions.

Ready to fire? You might want to wait.

The problem here is one of externalities. When you hit someone with a shell, you benefit some. But so does everyone other than your poor victim. Thus, you only internalize a fraction of the overall benefit; most of the benefits are external to you. Meanwhile, the target internalizes every last bit of the damage.

In turn, whenever you fire off a shell, you are gambling that the small bit of benefit you internalize from striking your target exceeds the potential loss you will suffer if a shell hits you because you no longer have protection. The odds are clearly stacked against you. Hence, Mario Kart is primarily a defensive game, at least when it comes to items.

Of course, that does not mean you should always keep your shells and peels in inventory. If you are in second and have a lot of space behind you, that red shell may be your only option to reach first place. Meanwhile, when you are not in first place, keeping a shell forever is worse than dropping it before the next item block, where you will hopefully roll a mushroom or something of the sort. So you should use your items; you just need to be judicious about the timing.

On that note, the previous paragraph reveals a good time to use a red shell against an opposing Mario holding some sort of protection. If Mario is going to get rid of it, it will be right before he hits the item blocks. Anticipating this, you can time the shell just right so that Mario dumps the protection before impact.

The Item “Duel”
A “duel” in game theory is as it seems: two gunslingers have one bullet and slowly move forward until one is ready to shoot at the other. Shooting from further away has the benefit of preempting the opponent but is more inaccurate and risks allowing the other party to take a clean shot at you later. Waiting is also potentially bad because the other side might kill you first.

What to do? Modeling the dilemma produces an interesting result: both parties shoot at the same time! Yet this is perfectly reasonable if you think about it. Imagine that you were planning on shooting slightly sooner than the other party. You will hit him with some degree of probability. But if you wait just a fraction of a second longer (but before he plans to shoot you), the probability you hit him increases slightly. So you should wait. But that logic recurs infinitely. As a result, when the gunslingers behave optimally, they will shoot at the same time.


 
“Duels” like this have important applications, including helping to explain why two rival video game companies often release their new systems at the same time. (It also applies to competitive cycling sprints. If you have never seen this, it is very bizarre. Despite appearances, that is not slow motion instant replay.) The strategic dilemma also shows up in at least a couple situations in Mario Kart. First, imagine you are neck-and-neck for first place with an opponent and you receive the spiny shell warning. Suddenly, your incentives change. Rather than racing to first place, you should slam on the brakes and try to get into second place. That way, the shell hits him and you can move along.

Of course, your opponent has the exact same incentive. So in the split second you have to react, both of you end up pressing the brakes at the same rate, analogous to choosing to shoot at the same time. And just like a duel, sometimes you both end up dying because the explosion has such a large blast radius.

spiny

The cause of countless nightmares.

Less frequently, you might encounter a similar breaking situation around a block of items. Item blocks give better items as a player’s position increases. So if you are with a pack of four people neck-and-neck, there is a great incentive to gently press the brakes, fall back to fourth, and get three mushrooms instead of the banana peel instead. However, once again, all players have a similar incentive, resulting in the entire bunch slowing down (or at least those with the strategic wherewithal). Indeed, whoever goes into first might have a temporary advantage but will quickly fall behind due to inferior items.*

(*Item selection may be a bit trickier than what it says here. Check the comments below.)

The Game that Isn’t a Game
Finally, I want to talk about the game that isn’t strategic at all: course selection in online play. If you haven’t played online before, the system works like this. The game queues up to 12 players and randomly selects three courses. You choose one of these courses or a “random” option. After everyone has submitted their picks, the game randomly draws one player. If that player selected a course, everyone plays that track; if that player selected random, then the game randomly picks a course from the pool of all 32.

course select
The course selection screen. Optimists like this one will soon find all their hopes and dreams crushed.

How should the course selection mechanism affect what you enter into the lottery? As it turns out, you don’t have to do any real thinking. You should just pick the track that you like the best. Unlike a traditional voting system, you don’t have to worry about what everyone else will pick. After all, if the game randomly selects your choice, then you are best off picking your favorite track; and if it chooses anyone else, then your selection is irrelevant.

If the course selection isn’t strategic in any way, then why am I talking about it? Well, as it turns out, such a mechanism that compels everyone to truthfully pick their favorite track is exceptionally rare. Economists and political scientists care about these issues greatly because effective voting mechanisms are of vital importance for both corporations and democracies. Unfortunately, the scholarly results are decidedly negative. In fact, the Gibbard-Satterthwaite theorem says that individuals will have incentive to lie about their preferences unless a person is a dictator, some options can never be chosen as the winner (i.e., we never play Mount Wario), or the selection mechanism is non-deterministic.

To see what I mean, imagine that the three tracks to select from are Music Park, Royal Raceway, and Toad’s Turnpike. (I’m going to ignore the random option for simplicity.) A majority (or plurality) of votes win. Suppose there are four other players you are squaring off against. Further, imagine that two of these guys prefer Music Park to Royal Raceway to Toad’s Turnpike; the other two prefer Royal Raceway to Toad’s Turnpike to Music Park. Meanwhile, you prefer Toad’s Turnpike to Music Park to Royal Raceway.

Is it rational for everyone to vote for his or her favorite course? No. If everyone did, we would have two votes for Music Park, two votes for Royal Raceway, and one vote for Toad’s Turnpike. With the tie between Music Park and Royal Raceway, the game might break it with a coin flip. The result is a 50% chance of Music Park and a 50% chance of Royal Raceway.

But imagine you misrepresented your preferences by voting for Music Park instead. Now Music Park has a strict majority and becomes the course that everyone will play. That is better for you than a 50% chance at Music Park and a 50% chance at Royal Raceway (your least favorite course). So you should lie! This means a majority/plurality system forces you to think about what others will select rather than just focusing on your own preferences.

While it might not be surprising that I can craft an example where you have incentive to lie, what is shocking is that just about all voting mechanisms suffer from this problem. That is the magic of the Gibbard-Satterthwaite theorem—it jumps from examples of failures to saying that just about everything will fail. The only way to break out of the problem is to give someone dictatorial powers, eliminate some choices from winning under any circumstance, or have the voting mechanism choose non-deterministically. Nintendo’s selection system opts for the last resolution.

In sum, just pick what you want on the course selection screen. And thanks to the incentive to tell the truth, let me tabulate all of your selections to investigate the world’s favorite courses.

Happy racing!

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Peace Science Presentation!

Posted on October 10, 2014 | Leave a comment

Brad Smith and I are excited to be presenting our paper on sanctions (conditionally accepted at ISQ) at Peace Science today. Check out the manuscript here or see the slides here. See you at 3 pm in 218 Houston!

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Game Theory and Bargaining on The Good Wife

Posted on October 5, 2014 | Leave a comment

Last week’s episode of The Good Wife (““Trust Issues”) was interesting for two reasons: it used a “ripped from the headlines” legal case that I discuss in my book on bargaining and the legal argument they use is essentially a trivial application of pre-play cheap talk in a repeated prisoner’s dilemma.

The $9 Billion Google/Apple Anti-Trust Lawsuit
First, the background of the real life version of the case. In the early 2000s, Google and Apple (along with Adobe and Intel) allegedly had a “no poaching” gentleman’s agreement. That is, each company in the group pledged to not attempt to hire employees at any of the other companies. The employees eventually figured out what was going on, filed a $9 billion lawsuit, and settled in April 2014 for an undisclosed amount.

Why is the practice illegal? It goes without saying that quashing competition among firms hurts the employees’ bargaining power, and the law is there to protect those employees. But what is not so clear is just how attractive a no poaching agreement is to the firms. In fact, when companies play by the rules, just about all of the potential for profit goes into the employees’ hands.

To see why, imagine that Google and Apple both wanted to hire Karen. Karen has impressive computer programming skills. And because Google and Apple value computing skills at a roughly equal rate, suppose that the most Google would be willing to pay her is $200,000 while Apple’s maximum is $195,000. Put differently, $200,000 and $195,000 represent the break even points for the respective companies. Put differently again, Karen will bring in $200,000 in profits to Google and $195,000 to Apple, so hiring her for any more than that will result in a net loss.

How will that profit ultimately be divided between Karen and her employer? You might think that Google should be the one hiring her. And you are right—she is worth $5000 more to Google than Apple. You might also think that Google will profit handsomely from her employment. However, as I discuss at length in the book, the logic of bargaining shows this to be untrue. If Google offers Karen any less than $195,000, she can always secure a job from Apple; this is because Apple values her at that amount, and so Apple would be willing to slightly outbid Google to hire her. Thus, the outbidding process ultimately ensures that Karen receives at least $195,000. She is the real winner. Although Google might still profit from her employment, its net gain will not exceed $5000 ($200,000 – $195,000).

Negotiating Collusion
So the firms have great incentive to collude, drive down wages, and secure more of the profits for themselves. What does that sort of collusion look like?

Well, we might think of it as a repeated prisoner’s dilemma. In this type of interaction, in any given year, each of us would maximize profits by trying to poach the rival firm’s employees regardless of what the other firm chooses to do. (If you don’t poach, then I make out like a bandit. If you do poach, I’m still better off poaching and not losing all the employees.) However, because each of us is poaching and driving up employee wages, both us are ultimately worse off than if we could enforce an agreement that required us to cooperate with each other and not poach.

Of course, anti-trust laws prevent us from explicitly contracting such an agreement in a legally enforceable manner. However, an informal and internally enforceable agreement is possible. Suppose we both start off by cooperating with each other by not poaching. Then, in each subsequent year, if both of us have consistently cooperated before, we continue cooperating. Otherwise, we revert to poaching.

Would anyone like to break the agreement? No. Although I could gain a temporary advantage against you by poaching your employees today, the higher wages over the long-term with mutual poaching are going to vastly outstrip that short-term benefit.

This is exactly the type of agreement Google and Apple struck. In fact, when a Google recruiter attempted to hire some Apple employees, Steve Jobs shot the following email to Google bigwigs: “If you hire a single one of these people, that means war.”

Alicia Florrick’s Defense
The episode of The Good Wife featured fictionalized versions of Google and Apple involved in the same affair. Like reality, employees caught on and sued.

The plaintiff’s lawyers thought they had the case in the bag. Indeed, they had turned one of the owners of a trust company against the defense. He went on record that the defense had negotiated the terms of the no poaching policy explicitly and was very happy to agree to the deal.

Alicia Florrick (the defense attorney and titular Good Wife) had a great defense: any discussion of such an agreement is not an unambiguous signal of plans to break the law. These repeated prisoner’s dilemmas have an interesting property in that regardless of whether you plan to cooperate with the other company or screw them over at the first possible moment, you always want to convince the other side that you will cooperate. If you plan to cooperate, then you want to tell the other side to cooperate as well so you can sustain that cooperation in the long term. If you want to follow the law and poach freely instead, you still want to convince the other side that you are going to cooperate so that they cooperate as well, allowing you to screw them over in the process.

So Florrick points out that this type of pre-play communication is meaningless. Regardless of the ultimate intend, the defendant would say the exact same thing. The testimony therefore proves nothing. The plaintiff promptly settled.

All told, I really appreciate two things about the episode: its sophisticated understanding of a potentially very complicated strategic situation and the how punny the “Trust Issues” title is.

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Posted in Bargaining, Game Theory