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I political scientist specializing in international relations and formal theory. I received a PhD from the University of Rochester in 2015. If you want to know more, feel free to look around, download my CV, email me at williamspaniel@gmail.com, or use the links below as a cheat sheet:

Manuscripts

 

The Game Theory of the Cardinals/Astros Spying Affair

The NY Times reported today that the St. Louis Cardinals hacked the Houston Astros’ internal files, including information on the trade market. I suspect that everyone has a basic understanding why the Cardinals would find this information useful. “Knowledge is power,” as they say. Heck, the United States spends $52.6 billion each year on spying. But game theorists have figured out how to quantify this intuition is both interesting and under-appreciated. That is the topic of this post.

Why Trade?
Trades are very popular in baseball, and the market will essentially take over sports headlines as we approach the July 31 trading deadline. Teams like to trade for the same reason countries like to trade with each other. Entity A has a lot of object X but lacks Y, while Entity B has a lot of object Y but lacks X. So teams swap a shortstop for an outfielder, and bad teams exchange their best players for good teams’ prospects. Everyone wins.

However, the extent to which one side wins also matters. If the Angels trade a second baseman to the Dodgers for a pitcher, they are happier than if they have to trade that same second baseman for that same pitcher and pay an additional $1 million to the Dodgers. Figuring out exactly what to offer is straightforward when each side is aware of exactly how much the other values all the components. In fact, bargaining theory indicates that teams should reach such deals rapidly. Unfortunately, life is not so simple.

The Risk-Return Tradeoff
What does a team do when it isn’t sure of the other side’s bottom line? They face what game theorists call a risk-return tradeoff. Suppose that the Angels know that the Dodgers are not willing to trade the second baseman for the pitcher straight up. Instead, the Angels know that the Dodgers either need $1 million or $5 million to sweeten the deal. While the Angels would be willing to make the trade at either price, they are not sure exactly what the Dodgers require.

For simplicity, suppose the Angels can only make a single take-it-or-leave-it offer. They have two choices. First, they can offer the additional $5 million. This is safe and guarantees the trade. However, if the Dodgers were actually willing to accept only $1 million, the Angels unnecessarily waste $4 million.

Alternatively, the Angels could gamble that the Dodgers will take the smaller $1 million amount. If this works, the Angels receive a steal of a deal. If the Dodgers actually needed $5 million, however, the Angels burned an opportunity to complete a profitable trade.

To generalize, the risk-return tradeoff says the following: the more one offers, the more likely the other side is to accept the deal. Yet, simultaneously, the more one offers, the worse that deal becomes for a proposer. Thus, the more you risk, the greater return you receive when the gamble works, but the gamble also fails more often.

 

Knowledge Is Power
The risk-return tradeoff allows us to precisely quantify the cost of uncertainty. In the above example, offering the safe amount wastes $4 million times the probability that the Dodgers were only willing to accept $1 million. Meanwhile, making an aggressive offer wastes the amount that the Angels would value the trade times the probability the Dodgers needed $5 million to accept the deal; this is because the trade fails to occur under these circumstances. Consequently, the Angels are damned-if-they-do, and damned-if-they-don’t. The risk-return tradeoff forces them to figure out how to minimize their losses.

At this point, it should be clear why the Cardinals would value the Astros’ secret information. The more information the Cardinals have about other teams’ minimal demands, the better they will fare in trade negotiations. The Astros’ database provided such information. Some of it was about what the Astros were looking for. Some of it was about what the Astros thought others were looking for. Either way, extra information for the Cardinals organization would decrease the likelihood of miscalculating in trade negotiations. And apparently such knowledge is so valuable that it was worth the risk of getting caught.

Why Are the NBA Finals on Sundays and NHL Finals on Saturdays?

A simple answer: iterated elimination of strictly dominated strategies.

The NBA and NHL have an unfortunate scheduling issue: their finals take place at roughly the same time, and having games scheduled at the same time would hurt both of their ratings. But this isn’t a simple coordination game. Everyone wants to avoid playing on Fridays, which is the worst night for ratings. This forces one series to play games on Sundays, Tuesdays, and Thursdays, with the other on Saturdays, Mondays, and Wednesdays. The first series is far more favorable for ratings and advertisements: it avoids the dreaded Friday ans Saturday nights entirely and also hits the coveted Thursday night slot.[1]

So who gets the good slot and why?

Well, the NBA wins because of its popularity. Some sports fans will watch hockey or basketball no matter what, but a sizable share of the population would be willing to watch both. Sadly for the NHL, though, those general sports fans break heavily in favor of the NBA. This allows the NBA to choose its best choice and forces the NHL to be the follower.

A more technical answer relies on iterated elimination of strictly dominated strategies. In my textbook, I have analogous example between a couple of nightclubs, ONE and TWO.[2] Both need to decide whether to schedule a salsa or a disco theme. (This is like deciding whether to schedule games on Saturdays or Sundays.) More patrons prefer salsa to disco. However, ONE has an advantage in that it is closer to town, giving individuals a general preference for it. Thus, TWO really wants to avoid matching its choice with ONE.

We might imagine a payoff matrix like this:

bars

So TWO can still break even if it picks the same choice as ONE but needs to mismatch to make a profit.

How should TWO decide what to do? Well, it should observe that ONE ought to pick salsa regardless of TWO’s choice—no matter what TWO picks, ONE always makes more by choosing salsa in response. Deducing that ONE will pick salsa, TWO can safely fall back on disco.

In the NBA/NHL case, the NHL must recognize that the NBA knows it will draw uncommitted fans regardless of the NHL’s choice. This means that the NBA should pick Sunday regardless of what the NHL selects. In turn, the NHL can safely place hockey on Saturday. It’s not the perfect outcome, but it’s the best the NHL can do given the circumstances.

[1] Thursdays are the biggest day for ad sales because entertainment companies want to compete for leisure business (movies, theme parks, etc.) over the weekend.

[2] I used these names in the textbook not only because they represent Player ONE and Player TWO but also because Rochester (where I went to grad school) has a club called ONE. This led to an interesting conversation when the Graduate Student Association scheduled an open bar there. I was relatively new at the time and didn’t know much about the city. After hearing rumors about the vent, I asked a fellow grad student where it would be. “ONE,” she said.

“Yes, I know it’s at 1, but where is it?”

“ONE.”

The last two lines repeated more times than I would like to admit.

 

Can More Information Ever Hurt You?

The answer would seem to be no. After all, if information is bad for you, you could always ignore it, continue living your life naively, and do better. Further, it is easy to write down games where a player’s payoff increases with the amount of information he has, and there are plenty of applications positively connecting information to welfare, like Condorcet jury theorem.

In reality, the answer is yes. Unfortunately, you can’t always credible commit to ignoring that information. This can lead to other players not trusting you later on in an interaction, which ultimately leads to a lower payoff for you.

Here’s an example. We begin by flipping a coin and covering it so that neither player observes which side is facing up. Player 1 then chooses whether to quit the game or continue. Quitting ends the game and gives 0 to both players. If he continues, player 2 chooses whether to call heads, tails, or pass. If she passes, both earn 1. If she calls heads or tails, player 2 earns 3 for making the correct call and -3 for making the incorrect call, while player 1 receive -1 regardless.

Because player 2 doesn’t observe the flip, her expected payoff for calling heads or tails is 0. As such, we can write the game tree as follows:

game1

Backward induction easily gives the solution: player 2 chooses pass, so player 1 chooses continue. Both earn 1.

If information can only help, then allowing player 2 access to the result of the coin flip before she moves shouldn’t decrease her payoff. But look what happens when the coin flip is heads:

game2

Now the solution is for player 2 to choose heads and player 1 to quit. Both earn 0!

The case where the coin landed on tails is analogous. Player 2 now chooses tails and player 1 still quits. Both earn 0, meaning player 1 is worse off knowing the result of the coin flip.

What’s going on here? The issue is credible commitment. When player 2 does not know the result of the coin flip, she can credibly commit to passing; although heads or tails could provide a greater payoff, the pass option generates the higher utility in expectation. This credible commitment assuages player 1’s concern that player 2 will screw him over, so he continues even though he could guarantee himself a break even outcome by quitting.

On the other hand, when player 2 knows the result of the coin flip, she cannot credibly commit to passing. Instead, she can’t help but pick the option (heads or tails) that gives her a payoff of 3. But this results in a commitment problem, wherein player 1 quits before player 2 picks an outcome that gives player 1 a payoff of -1. Both end up worse off because of it.

Weird counterexamples like this prevent us from making sweeping claims about whether more information is inherently a good thing. I noted at the beginning that it is easy to write down games where payoffs increase for a player as his information increases. Most game theorists would probably agree that more information is usually better. But it does not appear that we can prove general claims about the relationship.

Amazon’s Clever Price Discrimination Strategy

Amazon likes to discount books. Here are some examples, starting with Game Theory 101: The Complete Textbook:

gt101

We are only looking at the print prices in this blog post. Originally $13.99, Game Theory 101 is yours for only $11.75.

It’s a similar story for The Rationality of War:

war

Down from $10.54, you can buy The Rationality of War for $9.30.

And finally, here’s Game Theory 101: Bargaining:

bargain

Originally $11.09, Bargaining now sits just under $10.

I suspect the average consumer is pleased to see these discounts. For authors who publish through CreateSpace, however, these discounts are incredibly confusing. We can set the original price. No matter how much Amazon discounts it, they pay us a set amount of money per sale. As such, we also like Amazon’s discounts. In fact, the larger discount is, the happier we are.

The problem is, the discounts are inconsistent. When you initially publish a book, Amazon will always tag it with the list price. Then, after some time and without any warning, Amazon might reduce the price. Or they might not. I have discussed this problem with other authors, and there doesn’t seem to be any explanation for what’s going on.

That said, I now have a theory. Amazon has found a clever form of price discrimination.

What Is Price Discrimination?
The maximum price any of us is willing to pay for a good or service can vary heavily. A lot of Americans will pay $10 or more to see Fifty Shades of Grey. Meanwhile, others may only be willing to pay $1 to see such a movie. We call such a maximum price an individual’s reservation value.

As a business owner, your dream is to charge everyone their reservation value. For example, suppose Fifty Shades’ potential audience consists of two people, one who is willing to pay $10 to see the film and the other who is willing to pay $1. If you could somehow charge the $10 person $10 and the $1 person $1, you would make $11. This makes you the most amount of money possible.

Of course, movie theaters cannot easily distinguish between those high value and low value types. As such, like most businesses, they offer a single blanket price of $10. The $10 person sees the film but the $1 person does not.

Despite the difficulty in price discriminating, businesses try it to varying degrees of success. Student and senior citizen discounts are perfect examples. Both of these groups live off of fixed (and small) incomes. Consequently, as a whole, they are less willing to pay high prices for entertainment. Businesses like movie theaters therefore offer cheaper prices to these groups than to people who tend to have larger disposable incomes.

Airplane flight prices work in a similar way. Vacation travelers are unwilling to pay $1000 for a flight across the United States. In contrast, many business travelers who need to get to New York on short notice are willing. Airlines thus charge relatively cheap prices on flights booked well in advance and massively jack up the prices on the days before takeoff.

Don’t let these discounts fool you. Although they may make it seem like the businesses are acting generously, the discounts exist to maximize profits.

Price Discrimination on Amazon
Broadly, people who publish through CreateSpace fall into one of two categories: vanity authors and what I will call profit makers. Vanity authors write books without the intention to make money. They simply want to “publish” a book so they can say they have. These authors will sell tens of books to friends and family, but their work will never catch on with a larger audience. They give self-publishing a bad name.

Profit makers use CreateSpace because they do not want to hand over a large share of revenue to a traditional publishing house. Vanity is not a concern here. They invest time in writing books and publishing through CreateSpace because they know their works will make a substantial amount of money.

Unfortunately for Amazon, it is very difficult to differentiate between vanity authors and profit makers. Further, there are substantially more vanity authors than profit makers out there. As such, Amazon’s best guess for any new book coming from CreateSpace is that the work is from a vanity author.

This is where I think Amazon’s price discrimination comes into play. Amazon suspects that every new book is vanity. Sales of vanity books do not operate like a normal market. Vanity authors are selling virtually all of their books friends and family. These individuals are willing to spend more money on these books because they know the author. Their reservation price is consequently higher than your average individual. In many cases, it may be substantially higher—a poorly edited vanity book is essentially worthless to the average consumer, but friends and family might be willing to spend $10 or $20 on the book.

If you are Amazon, what incentive do you have to cut the price? Any discount you offer directly hurts your bottom line, and these vanity books are not responding to standard supply and demand factors. Consequently, you don’t have any incentive to discount. The vanity books will be sold to the friends and family and no one else. No discount maximizes your profit.

Of course, Amazon suffers when the book is from a profit maker, not a vanity author. These books respond to supply and demand, so cutting prices by 10% can actually cause more people to buy them. So Amazon might want to reduce the price in these circumstances.

Put yourself in Amazon’s shoes for a moment. You want to discriminate here to maximize your profit. But how?

From my personal experience and discussion with other authors, I think Amazon has figured out a way. They start by offering no discount, under the assumption that the book is from a vanity author. They then wait. And wait and wait and wait. Vanity books will see their sales fall off a cliff after a month or two. Profit making books will see continued sales over the long term. This differentiates the type of book. Amazon thus cuts the price of books that sell, knowing that doing so will lead to even more sales.

To be clear, this is speculation backed up with some non-random observations. Still, I think there is a good chance that price discrimination explains Amazon’s strategy. Although the discounts may seem to be applied randomly, I can’t imagine a company with $88 billion in revenue is doing this without purpose. Price discrimination explains it.

Costly Signaling on House of Cards

[spoilers, obviously]

hoc1

hoc2

hoc3

Game theorists often talk about “burning money” metaphorically, but this is as close to reality as it gets. Doug Stamper wants President Frank Underwood to appoint him White House Chief of Staff. Frank is unsure whether Doug is a committed type or an uncommitted type. In the absence of any new information, Frank would be better off denying Doug the position, as it would give Doug the ability to feed sensitive information to Frank’s primary opponent. So Doug burns a scandalous journal entry that he could have sold for $2 million and notes that only a resolved type would be willing to forgo that gain. Frank hires him.

If you are wondering why political scientists like House of Cards so much, that’s why. Costly signaling at its finest.

Marshawn Lynch Was Optimal, But So Was a Quick Slant

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.

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

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.