Category Archives: Game Theory

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.

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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.

Jeopardy’s Game Theory Irony

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.

The Game Theory of Mario Kart 8

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!

Game Theory and Bargaining on The Good Wife

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.

Park Place Is Still Worthless: The Game Theory of McDonald’s Monopoly

McDonald’s Monopoly begins again today. With that in mind, I thought I would update my explanation of the game theory behind the value of each piece, especially since my new book on bargaining connects the same mechanism to the De Beers diamond monopoly, star free agent athletes, and a shady business deal between Google and Apple. Here’s the post, mostly in its original form:

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McDonald’s Monopoly is back. As always, if you collect Park Place and Boardwalk, you win a million dollars. I just got a Park Place. That’s worth about $500,000, right?

Actually, it is worth nothing. Not close to nothing, but absolutely, positively nothing.

It helps to know how McDonald’s structures the game. Despite the apparent value of Park Place, McDonald’s floods the market with Park Place pieces, probably to trick naive players into thinking they are close to riches. I do not have an exact number, but I would imagine there are easily tens of thousands of Park Places floating around. However, they only one or two Boardwalks available. (Again, I do not know the exact number, but it is equal to the number of million dollar prizes McDonald’s want to give out.)

Even with that disparity, you might think Park Place maintains some value. Yet, it is easy to show that this intuition is wrong. Imagine you have a Boardwalk piece and you corral two Park Place holders into a room. (This works if you gathered thousands of them as well, but you only need two of them for this to work.) You tell them that you are looking to buy a Park Place piece. Each of them must write their sell price on a piece of paper. You will complete the transaction at the lowest price. For example, if one person wrote $500,000 and the other wrote $400,000, you would buy it from the second at $400,000.

Assume that sell prices are continuous and weakly positive, and that ties are broken by coin flip. How much should you expect to pay?

The answer is $0.

The proof is extremely simple. It is clear that both bidding $0 is a Nash equilibrium. (Check out my textbook or watch my YouTube videos if you do not know what a Nash equilibrium is.) If either Park Place owner deviates to a positive amount, that deviator would lose, since the other guy is bidding 0. So neither player can profitably deviate. Thus, both bidding 0 is a Nash equilibrium.

What if one bid $x greater than or equal to 0 and the other bid $y > x? Then the person bidding y could profitably deviate to any amount between y and x. He still wins the piece, but he pays less for it. Thus, this is a profitable deviation and bids x and y are not an equilibrium.

The final case is when both players bid the same amount z > 0. In expectation, both earn z/2. Regardless of the tiebreaking mechanism, one player must lose at least half the time. That player can profitably deviate to 3z/8 and win outright. This sell price is larger than the expectation.

This exhausts all possibilities. So both bidding $0 is the unique Nash equilibrium. Despite requiring another piece, your Boardwalk is worth a full million dollars.

What is going wrong for the Park Place holders? Supply simply outstrips demand. Any person with a Park Place but no Boardwalk walks away with nothing, which ultimately drives down the price of Park Place down to nothing as well.

Moral of the story: Don’t get excited if you get a Park Place piece.

Note 1: If money is discrete down to the cent, then the winning bid could be $0 or $0.01. (With the right tie breaker, it could also be $0.02.) Either way, this is not good for owners of Park Place.

Note 2: In practice, we might see Park Place sell for some marginally higher value. That is because it is (slightly) costly for a Boardwalk owner to seek out and solicit bids from more Park Place holders. However, Park Place itself is not creating any value here—it’s purely the transaction cost.

Note 3: An enterprising Park Place owner could purchase all other Park Place pieces and destroy them. This would force the Boardwalk controller to split the million dollars. While that is reasonable to do when there are only two individuals like the example, good luck buying all Park Places in reality. (Transaction costs strike again!)

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Now time for an update. What might not have been clear in the original post is that McDonald’s Monopoly is a simple illustration of a matching problem. Whenever you have a situation with n individuals who need one of m partners, all of the economic benefits go to the partners if m < n. The logic is the same as above. If an individual does not obtain a partner, he receives no profit. This makes him desperate to partner with someone, even if it means drastically dropping his share of the money to be made. But then the underbidding process begins until the m partners are taking all of the revenues for themselves.

In the book, I have a more practical example involving star free agent athletes. For example, there is only one LeBron James. Every team would like to sign him to improve their chances of winning. Yet this ultimately results in the final contract price to be so high that the team doesn’t actually benefit much (or at all) from signing James.

Well, that’s how it would work if professional sports organizations were not scheming to stop this. The NBA in particular has a maximum salary. So even if LeBron James is worth $50 million per season, he won’t be paid that much. (The exact amount a player can earn is complicated.) This ensures that the team that signs him will benefit from the transaction but takes money away from James.

Non-sports business scheme in similar ways. More than 100 year ago, the De Beers diamond company realized that new mine discoveries would mean that diamond supply would soon outstrip demand. This would kill diamond prices. So De Beers began purchasing tons of mines to intentionally limit production and increase price. Similarly, Apple and Google once had a “no compete” informal agreement to not poach each other’s employees. Without the outside bidder, a superstar computer engineer would not be able to increase his wage to the fair market value. Of course, this is highly illegal. Employees filed a $9 billion anti-trust lawsuit when they learned of this. The parties eventually settled the suit outside of court for an undisclosed amount.

To sum up, matching is good for those in demand and bad for those in high supply. With that in mind, good luck finding that Boardwalk!

What Does Game Theory Say about Negotiating a Pay Raise?

A common question I get is what game theory tells us about negotiating a pay raise. Because I just published a book on bargaining, this is something I have been thinking about a lot recently. Fortunately, I can narrow the fundamentals to three simple points:

1) Virtually all of the work is done before you sit down at the table.
When you ask the average person how they negotiated their previous raise, you will commonly hear anecdotes about how that individual said some (allegedly) cunning things, (allegedly) outwitted his or her boss, and received a hefty pay hike. Drawing inferences from this is problematic for a number of reasons:

  1. Anecdotal “evidence” isn’t evidence.
  2. The reason for the raise might have been orthogonal to what was said.
  3. Worse, the raise might have been despite what was said.
  4. It assumes that the boss is more concerned about dazzling words than money, his own job performance, and institutional constraints.

The fourth point is especially concerning. Think about the people who control your salaries. They did not get their job because they are easily persuaded by rehearsed speeches. No, they are there because they are good at making smart hiring decisions and keeping salaries low. Moreover, because this is their job, they engage in this sort of bargaining frequently. It would thus be very strange for someone like that to make such a rookie mistake.

So if you think you can just be clever at the bargaining table, you are going to have a bad time. Indeed, the bargaining table is not a game of chess. It should simply be a declaration of checkmate. The real work is building your bargaining leverage ahead of time.

2) Do not be afraid to reject offers and make counteroffers.
Imagine a world where only one negotiator had the ability to make an offer, while the other could only accept or reject that proposal. Accepting implements the deal; rejecting means that neither party enjoys the benefits of mutual cooperation. What portion of the economic benefits will the proposer take? And how much of the benefits will go to the receiver?

You might guess that the proposer has the advantage here. And you’d be right. What surprises most people, however, is the extent of the advantage: the proposer reaps virtually all of the benefits of the relationship, while the receiver is barely any better off than had the parties not struck a deal.

How do we know this? Game theory allows us to study this exact scenario rigorously. Indeed, the setup has a specific name: the ultimatum game. It shows that a party with the exclusive right to make proposals has all of the bargaining power.

 

That might seem like a big problem if you are the one receiving the offers. Fortunately, the problem is easy to solve in practice. Few real life bargaining situations expressly prohibit parties from making counteroffers. (As I discuss in the book, return of security deposits is one such exception, and we all know that turns out poorly for the renter—i.e., the receiver of the offer.) Even the ability to make a single counteroffer drastically increases an individual’s bargaining power. And if the parties could potentially bargain back and forth without end—called Rubinstein bargaining, perhaps the most realistic of proposal structures—bargaining equitably divides the benefits.

As the section header says, the lesson here is that you should not be afraid to reject low offers and propose a more favorable division. Yet people often fail to do this. This is especially common at the time of hire. After culling through all of the applications, a hiring manager might propose a wage. The new employee, deathly afraid of losing the position, meekly accepts.

Of course, the new employee is not fully appreciating the company’s incentives. By making the proposal, the company has signaled that the individual is the best available candidate. This inevitably gives him a little bit of wiggle room with his wage. He should exercise this leverage and push for a little more—especially because starting wage is often the point of departure for all future raise negotiations.

3) Increase your value to other companies.
Your company does not pay you a lot of money to be nice to you. It pays you because it has no other choice. Although many things can force a company’s hand in this manner, competing offers is particularly important.

Imagine that your company values your work at $50 per hour. If you can only work for them, due the back-and-forth logic from above, we might imagine that your wage will land in the neighborhood of $40 per hour. However, suppose that a second company exists that is willing to pay you up to $25 per hour. Now how much will you make?

The answer is no less than $40 per hour. Why? Well, suppose not. If your current company is only paying you, say, $30 per hour, you could go to the other company and ask for a little bit more. They would be obliged to pay you that since they value you up to $40 per hour. But, of course, your original company values you up to $50 per hour. So they have incentive to ultimately outbid the other company and keep you under their roof.

(This same mechanism means that Park Place is worthless in McDonald’s monopoly.)

Game theorists call such alternatives “outside options”; the better your outside options are, the more attractive the offers your bargaining partner has to make to keep you around. Consequently, being attractive to other companies can get you a raise with your current company even if you have no serious intention to leave. Rather, you can diplomatically point out to your boss that a person with your particular skill set typically makes $X per year and that your wage should be commensurate with that amount. Your boss will see this as a thinly veiled threat that you might leave the company. Still, if the company values your work, she will have no choice but to bump you to that level. And if she doesn’t…well, you are valuable to other companies, so you can go make that amount of money elsewhere.

Conclusion
Bargaining can be a scary process. Unfortunately, this fear blinds us to some of the critical facets of the process. Negotiations are strategic; only thinking about your worries and concerns means you are ignoring your employer’s worries and concerns. Yet you can use those opposing worries and concerns to coerce a better deal for yourself. Employers do not hold all of the power. Once you realize this, you can take advantage of the opposing weakness at the bargaining table.

I talk about all of these issues in greater length in my book, Game Theory 101: Bargaining. I also cover a bunch of real world applications to these and a whole bunch of other theories. If this stuff seems interesting to you, you should check it out!