# Category Archives: Game Theory

## How I Write Formal Articles

Suppose you want to write a formal theory paper. Below is the template I use to do this. I do not always follow these rules. But whenever I break them, I usually justify to myself first why it is a good idea to sidestep the norm.

The Introduction
My introductions usually have a set formula:

1. Begin with an anecdote that motivates the main point of the paper.
2. Generalize that main point.
3. Pivot to how existing work does not address that main point.
4. Describe the model setup.
5. Give the results and basic intuition.
6. Explain empirical content, if there is any. For quantitative work, this means describing the type of regression you are doing and one or two key substantive effects. For qualitative work, this means describing the case, how the central issues of the model were in effect, and how the outcome fits expectations.
7. A paragraph or two of related work. Note that this may not be necessary depending on the extent of the comparison in (3) and whether there is a motivation section below.

Of these, I think (5) is the biggest problem I see as a peer reviewer. There are way, way too many papers that will say things along the lines of “increases in income decrease the probability of terrorist attacks” full stop. The intuition explaining the connection will not appear for the first time until page 18 or so. This fundamentally misses the point of doing formal theory. We are not interested in the what. We are interested in the why. Formal theory helps elucidate mechanisms. If you are not elucidating the mechanisms in the introduction, you are not writing an effective paper.

To that point, I find it helpful as a reader (and a reviewer) when this part begins with “The model produces x main results. First, …” Then each subsequent x – 1 paragraphs then explains the other results. This gives a good benchmark to the reader of what to expect in the paper and think about what a model look like that would be good for addressing those issues.

Motivation, Sometimes
I have the most variance in what comes next. Sometimes, it is straight to the model. Other times, I give a deeper explanation for why I am building the model that I am.

An underappreciated aspect of formal theory is that it just an exercise in mapping assumptions to conclusions. As the saying goes “garbage in, garbage out.” If the assumptions you put into a model make little sense, then there is no reason to pay attention to whatever the model outputs. Thus, if readers may view the assumptions your model makes as controversial, this is the time to defend them.

Sometimes, this is unnecessary. For example, if the model takes an existing approach and adds uncertainty, then you probably only need a couple of citations in (3) from the introduction to take care of it. Otherwise, I think through the main critical assumptions from the model. I then begin the second section by listing them. The following paragraphs take each assumption and motivate them. Basically, this is an exercise in going through the existing literature to demonstrate that your assumptions have merit. Key places to draw from are:

1. existing models that use the assumption in a different context (e.g., models of war have uncertainty over resolve, but the standard models of terrorism do not)
2. quantitative literatures that establish stylized facts that the theoretical literature has not yet developed
3. qualitative studies that devote the entire work to motivating the same point you want to make

Of these, (3) is the most useful and the type I try to emphasize.

There are two important notes to this section. First, it is not a literature review. You are not just rehashing what the literature says about a particular subject. You are motivating assumptions. Everything you write should be geared toward that.

Second, this is a good way to come up with research ideas in the first place. As a general exercise, whenever I read through the literature, I think about what assumptions are out there and whether they appear in the more specific areas I work in. When there is a mismatch, it is worth spending some time to think about whether those alternative assumptions fundamentally alter existing ideas.

The Model
My modeling sections usually follow a basic formula:

1. Introduce the players, moves, and payoffs in that order. For most models worth exploring, drawing a game tree is often more cumbersome than it is helpful to the reader. Bulletpoint lists are often more useful for illustrating this.
2. Describe any conditions on parameter spaces. For example, corner solutions often complicate the math without providing any extra insight. If that is the case, describe what you are assuming, give the explicit mathematical expression (perhaps in a footnote), and explain why the reader should not care about this.
3. Give any baseline results that are necessary to understand what is to come. For example, if you are working on an incomplete information game, explain the results of the complete information game first. Sometimes, these will be so straightforward that you can do this in a couple of paragraphs without the need to have formal propositions. Do this if you can. Other times, the baseline results are themselves of theoretical interest. In this case, use the formula below.
4. Give a proposition. Propositions are usually if-then statements. The “if” part should be an intuitive meaning and parameter space. For example, “Suppose costs are sufficiently high (i.e., c > mk – d).” The “then” part is the strategy or outcome that is worth exploring.
5. Explain the intuition of the proposition. Do not get bogged down in the calculations. But at the same time, do not be afraid to explain the derivation of cutpoints. Some cutpoints appear to be incredibly complicated but are in fact straightforward comparisons. This can give the reader greater insight as to where the relationships are coming from.
6. Repeat (4) and (5) until equilibrium is exhausted.
7. Recap using an equilibrium plot. Almost every paper benefits from one of these.
8. Give the interesting comparative statics, either as propositions or remarks. Provide the intuition just as you would with the equilibrium. Plot the comparative static.

The plot part is the thing I see as the easiest way to improve papers. A good rule of thumb is to pretend every paper you are writing is going to be used as a job talk paper. Then think about what slides you would want to present to illustrate the key points. For example, if you had a slide that said “the probability of war increases in the cost of fighting,” you would not want to leave it as just that. You would want the next slide to show a plot with cost on the x-axis and the probability of war on the y-axis. After going through this mental exercise, every visualization of the results should go in the paper.

Empirical Evaluation
This section may or may not exist. Some models require so much space that doing any sort of empirical evaluation is not impossible given the 10,000 word limit you have to aim for to fit most outlets. Otherwise, there are two ways to go here.

Option 1 is to do some sort of qualitative examination. Hein Goemans and I have written about this in Security Studies. If you want to go down this route, you should read that.

The main trap I see when papers take qualitative approach is matching outcome to outcome. For example, the model might predict that poor people commit terrorism, and then the case study talks about how poor people commit terrorism in a certain country.

This misses the point of doing formal theory. As I described above, models map assumptions to conclusions. Case studies should do the same. In other words, I take the three or so assumptions that are key to the model’s mechanism. I then motivate why those assumptions held in the particular case. Only then is the outcome variable worth mentioning. But the key here is to establish that the incentives that the model describes was key to the actors’ reasoning. (Or at least those incentives plausibly drove it. There are many cases where finding a smoking gun would be a ridiculous expectation. If that is the case, then you should make an argument about why it is ridiculous.)

Option 2 is a quantitative examination of a comparative static. Most of this follows the basic quantitative paper template, so there is not much more to say here. The only thing worth adding is that you need a subsection that pivots the comparative static to a hypothesis that you can test. (Comparative statics are true statements. Hypotheses are things that may or may not be true of data.)

Conclusion
I think conclusions are overrated, so I have a simple formula for this:

1. Recap the main findings.
2. Describe takeaways for policymakers.
3. Consider what extensions to the model might be interesting for future theoretical research.
4. Explain how empirical scholars might wish to address the findings.

## Bargaining Power and the Iran Deal

Today’s post is not an attempt to give a full analysis of the Iran deal.[1] Rather, I just want to make a quick point about how the structure of negotiations greatly favors the Obama administration.

Recall the equilibrium of an ultimatum game. When two parties are trying to divide a bargaining pie and one side makes a take-it-or-leave-it offer, that proposer receives the entire benefit from bargaining. In fact, even if negotiations can continue past a single offer, as long as a single person controls all of the offers, the receiver still receives none of the surplus.

This result makes a lot of people feel uncomfortable. After all, the outcomes are far from fair. Fortunately, in real life, people are rarely constrained in this way. If I don’t like the offer you propose me, I can always propose a counteroffer. And if you don’t like that, nothing stops you from making a counter-counteroffer. That type of negotiations is called Rubinstein bargaining, and it ends with a even split of the pie.

In my book on bargaining, though, I point out that there are some prominent exceptions where negotiations take the form of an ultimatum game. For example, when returning a security deposit, your former landlord can write you a check and leave it at that. You could try suggesting a counteroffer, but the landlord doesn’t have to pay attention—you already have the check, and you need to decide whether that’s better than going to court or not. This helps explain why renters often dread the move out.

Unfortunately for members of Congress, “negotiations” between the Obama administration and Congress are more like security deposits than haggling over the price of strawberries at a farmer’s market. If Congress rejects the deal (which would require overriding a presidential veto), they can’t go to Iran and negotiate a new deal for themselves. The Obama administration controls dealings with Iran, giving it all of the proposal power. Bargaining theory would therefore predict that the Obama administration will be very satisfied[2], while Congress will find the deal about as attractive as if there were no deal at all.

And that’s basically what we are seeing right now. Congress is up in arms over the deal (hehe). They are going to make a big show about what they claim is an awful agreement, but they don’t have any say about the terms beyond an up/down vote. That—combined with the fact that Obama only needs 34 senators to get this to work—means that the Obama administration is going to receivea very favorable deal for itself.

[1] Here is my take on why such deals work. The paper is a bit dated, but it gets the point across.

[2] I mean that the Obama administration will be very satisfied by the deal insofar as it relates to its disagreement with Congress. It might not be so satisfied by the deal insofar as it relates to its disagreement with Iran.

## Serial and Credible Threats

[Serial Podcast Season 1 spoilers below]

I’m going to assume you have gone through the first season of Serial and know most of the background. However, some important recap:

According to Jay’s testimony, Adnan strangled Hae and then solicited the help of Jay to dispose of the body. The police wondered why Jay, an acquaintance of Adnan, would ever go along with that. Jay stated that he initially refused. But Adnan threatened to go to the cops about Jay’s pot dealings. Wanting to avoid that, Jay becomes an accessory to murder.

To me, this makes no sense at all. I could understand why Jay might prefer burying the body to having to deal with the police over some (relatively minor) marijuana, but the latter scenario would never happen. Adnan simply does not have a credible threat here. If Adnan goes to the police to turn in Jay, Jay can easily plead out of the crime by handing them Adnan. Jay has all the leverage here. Adnan has none.

Did Jay not realize this? I can’t imagine that is true. Jay is supposed to be street-smart. He might not understand the difference between Nash equilibrium and subgame perfect equilibrium, but he certainly should understand the difference between a credible threat and an incredible threat.

Would Jay not be willing to snitch on Adnan if Adnan turned him in? If so, then Adnan would not be deterred from pointing the police to Jay, and so maybe Jay would go along with it. But I can’t imagine this is true either. First, it would require Jay to not want to rat on the guy who just ratted on him, even though it would likely mean that Jay’s charges would be dropped. Second, we in fact know Jay was willing to snitch on Adnan—because he did!

This leads me to conclude that Jay’s lying. I’m not sure why or what it means, but I think it’s important.

TL;DR: Jay’s story is not subgame perfect.

Am I missing something here?

## 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.

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.

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:

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.

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:

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:

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.

## Costly Signaling on House of Cards

[spoilers, obviously]

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.

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

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.