Tag Archives: Game Theory

Are Weapons Inspections about Information or Inconvenience?

Posted on August 20, 2013 | 3 comments

Abstract: How do weapons inspections alter international bargaining environments? While conventional wisdom focuses on informational aspects, this paper focuses on inspections’ impact on the cost of a potential program–weapons inspectors shut down the most efficient avenues to development, forcing rising states to pursue more costly means to develop arms. To demonstrate the corresponding positive effects, this paper develops a model of negotiating over hidden weapons programs in the shadow of preventive war. If the cost of arms is large, efficient agreements are credible even if declining states cannot observe violations. However, if the cost is small, a commitment problem leads to positive probability of preventive war and costly weapons investment. Equilibrium welfare under this second outcome is mutually inferior to the equilibrium welfare of the first outcome. Consequently, both rising states and declining states benefit from weapons inspections even if those inspections cannot reveal all private information.

If you are here for the long haul, you can download the chapter on the purpose of weapons inspections here. Being that it is a later chapter from my dissertation, here is a quick version of the basic “butter-for-bombs” model:

Imagine a two period game between R(ising state) and D(declining state). In the first period, D makes an offer x to R, to which R responds by accepting, rejecting, or building weapons. Accepting locks in the proposal; R receives x and D receives 1-x for the rest of time. Rejecting locks in war payoffs; R receives p – c_R and D receives 1 – p – c_D. Building requires a cost k > 0. D responds by either preventing–locking in the war payoffs from before–or advancing to the post-shift state of the world.

In the post-shift state, D makes a second offer y to R, which R accepts or rejects. Accepting locks in the offer for the rest of time. Rejecting leads to war payoffs; R receives p’ – c_R and D receives 1 – p’ – c_D, where p’ > p. Thus, R fares better in war post-shift and D fares worse.

As usual, the actors share a common discount factor δ.

The main question is whether D can buy off R. Perhaps surprisingly, the answer is yes, and easily so. To see why, note that even if R builds, it only receives a larger portion of the pie in the later stage. Specifically, D must offer p’ – c_R to appease R and will do so, since provoking war leads to unnecessary destruction. Thus, if R ever builds, it receives p’ – c_R for the rest of time.

Now consider R’s decision whether to build in the first period. Let’s ignore the reject option, as D will never be silly enough to offer an amount that leads to unnecessary war. If R accepts x, it receives x for the rest of time. If it builds (and D does not prevent), then R pays the cost k and receives x today and p’ – c_R for the rest of time. Thus, R is willing to forgo building if:

x ≥ (1 – δ)x + δ(p’ – c_R) – (1 – δ)k

Solving for x yields:

x ≥ p’ – c_R – (1 – δ)k/δ

It’s a simple as that. As long as D offers at least p’ – c_R – (1 – δ)k/δ, R accepts. There is no need to build if you are already getting all of the concessions you seek. Meanwhile, D happily bribes R in this manner, as it gets to steal the surplus created by R not wasting the investment cost k.

The chapter looks at the same situation but with imperfect information–the declining state does not know whether the rising state built when it chooses whether to prevent. Things get a little hairy, but the states can still hammer out agreements most of the time.

I hope you enjoy the chapter. Feel free to shoot me a cold email with any comments you might have.

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GRE Family Fued? The Game Theory of Standardized Test Grading

Posted on August 5, 2013 | Leave a comment

I can tell you when I began losing faith in standardized testing: August 11, 2009. That was the date of my GRE. My writing prompt was as follows:

Explain the causes of war.

Wow! This could not have been any more perfect. Here I was taking the GRE so I could go to grad school and write a dissertation on the causes of war. The College Board threw me a softball!

Then I received my score: 4.5. While not terrible, the 4.5 corresponded to the 63rd percentile. According to the College Board (roughly), more than a third of GRE takers could write my dissertation better than I could!

Maybe the essay was not that great. Maybe I am a terrible writer. Maybe I don’t understand the causes of war. The academic job market will likely be the ultimate arbiter of my abilities. But until then, the 4.5 seems silly.

_________

Years later, I came across this revealing article on standardized testing graders’ incentive structure. Many tests use some sort of consensus method. Begin by giving the test to two graders. If the marks are close, average the grades and move to the next exam. If the marks are not close, bring in a third grader and average the three grades in some pre-defined way.

Standardized testing companies are not in the business of giving correct grades–they are in the business of grading tests as quickly as possible. The potential for a third grader merely appeases a school’s desire to have some semblance of legitimacy. For the company, the third grader is a speed bump. Every test that reaches a third reader requires 50% more labor–and a non-negligible decrease in profitability for that particular test.

Realizing this problem, testing companies pay careful attention to their graders’ agreement rate. A low agreement rate is the mark of a bad employee. Supervisors might creatively limit the number of tests such an employee can grade (thus inflating the supervisor’s overall agreement rate for his or her team), while managers might fire him or her.

The testing companies deserve respect for creating mechanisms to keep employees in line with company goals. Unfortunately, those company goals do not comport to what we as consumers want out of the standardized testing scores we buy.

_________

I have loved game shows all my life. One of my earliest memories is of watching Family Feud. The game is simple: producers survey 100 people with a variety of questions. They then tally the responses. Contestants on the show must then guess which answer was most frequently given.

Note the emphasis here is on matching, not correctness. For example, suppose the prompt was “name a cause of war.” As a contestant, I would say “greed” or “irrationality” way before I said “private information” and “commitment problems.” The first two responses are terrible, terrible answers. The second two are fantastic. Yet, because your average survey taker has not read “Rationalist Explanations for War,” I would not expect many people to give sensible responses to the survey. So commitment problems would yield a smaller score than greed. In turn, I as a contestant pander to their ignorance and say the silly thing.

_________

Suppose you are a test grader. Better yet, say you are the test grader. God has endowed you with absolute authority in test grading matters. You know a 10 essay is a 10 essay and a 1 essay is a 1 essay. Whereas others struggle to see the difference, your observations are perfect.

But you are also broke and need a job. I hand you a test. You recognize it is a clear 10. What grade do you give it?

In a world of justice, your answer is 10. But in a world where you need to eat, you take a different route. Perhaps the essay did something strange, something you have never seen before–something like argue that bargaining problems cause war. You recognize that such an argument reflects scholarly consensus and would be the baseline for tenure at Stanford. But you also know that other graders will think that the argument is just plain bizarre. So you credit the writer for having decent organizational structure but not much substance and turn in a 7.

The other grader gives it a 6.5. The system counts it as a match and you do not get into trouble.

Grading systems are not grading systems–they coordination games with multiple equilibria. Like in the Family Feud, a reader should not give the grade the writer actually deserves but rather what he or she thinks other readers will give it.

But this leads to perverse equilibria. For example, if we all created the rule that essays beginning with a vowel are 10s and essays beginning with consonants are 2s, no one would want to break the system and risk the wrath of being labeled inefficient. So substance goes out the window. Graders instead look for focal points to coordinate their scores.

Those cues need not reflect anything of substance. To wit, consider the following review of a supervisor’s team:

[A] representative from a southeastern state’s Department of Education visited to check on how her state’s essays were doing. As it turned out, the answer was: not well. About 67 percent of the students were getting 2s.

That’s when the representative informed Farley that the rubric for her state’s scoring had suddenly changed.

“We can’t give this many 1s and 2s,” she told him firmly.

The scorers would not be going back to re-grade the hundreds of tests they’d already finished—there just wasn’t time. Instead, they were just going to give out more 3s.

3s magically appeared out of nowhere–partially because the testing company wanted a higher average, and partially because graders feared that giving a 1 or 2 would result in a mismatch and disciplining from the supervisor.

The article is full of other lovely anecdotes and worth the read. And it is also terrifying to think about.

_________

In retrospect, I should not have written about how bargaining problems cause war. From the point of view of a grader, it is just too bizarre. I deserve my 4.5 for not properly playing the game.

(I suppose three years of grad school has made me more cynical about life. Perhaps a better subtitle for this article is “How I Learned to Stop Worrying and Love Pandering.”)

Of course, that is precisely the problem. The incentive system for grading is perverse and rewards students for writing safe–and not particularly insightful–essays. In contrast, the academic job market rewards the opposite approach. Write a dissertation that has been done a thousand times before, and you won’t find employment. Do something revolutionary, and have your pick at the top jobs.

The test companies do not have final say over the method of standardized testing grading. We do. It’s time we demand change in the system.

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How Uncertainty about Judicial Nominees Can Distort the Confirmation Process

Posted on August 1, 2013 | 1 comment

In standard bargaining situations, both parties understand the fundamentals of the agreement. For example, if I offer you a $20 per hour wage, then I will pay you $20 per hour; if I propose a 1% sales tax increase, then sales tax will increase by 1%. But not all such deals are evident. Senate confirmation of judicial nominees is particularly troublesome—the President has a much better idea of the true nominee’s ideology than the Senate does. Indeed, as the Senate votes to confirm or reject, the Senate may very well be unsure what it is buying.

This situation is the center of a new working paper from Maya Sen and myself. We develop a formal model of the interaction between the President and the Senate during the judicial nomination process. At first thought, it might seem as though the President benefits from the lack of information by occasionally sneaking in extremist justices the Senate would otherwise reject. However, our main results show that this lack of information ultimately harms both parties.

To unravel the logic, suppose the President could nominate a moderate or an extremist. Now imagine that the Senate is ideologically opposed, so it only wants to confirm the moderate. The choice to reject is not so simple, though, because the Senate cannot directly observe the nominee’s type but rather must make inferences based on a noisy signal. Specifically, the Senate receives a signal with probability p if the President chooses an extremist. (This signal might come from the media uncovering a “smoking gun” document.) The President suffers a reputation cost if he is caught in this manner. If the President selects a moderate, the Senate receives no signal at all. Thus, upon not receiving a signal, the Senate cannot be sure whether the President nominated a moderate or extremist.

With those dynamics in mind, consider how the President acts when the signal is weak. Can he only nominate an extremist? No–the Senate would obviously always reject regardless of its signal. Can he only nominate a moderate? No–the Senate would respond by confirming the nominee despite the lack of a signal, but the President could then gamble by selecting an extremist and hoping that the weak signal works in his favor. As such, the President must mix between nominating a moderate and nominating an extremist.

Similarly, the Senate must mix as well. If it were to always confirm, the President would nominate extremists exclusively, but that cannot be sustainable for the reasons outlined above. If the Senate were to always reject, the President would only nominate moderates to avoid smoking guns. But then the Senate could confirm the moderates it was seeking.

Thus, both parties mix. Put differently, the President sometimes bluffs and sometimes does not; the Senate sometimes calls what it perceives as bluffs and sometimes lets them go.

These devious behaviors have an unfortunate welfare implication–both parties are worse off than if they could agree to appoint a moderate. Since the Senate mixes, it must be indifferent between accepting and rejecting. The indifference condition means that the Senate receives its rejection payoff in expectation, which is worse than if it could induce the President to appoint a moderate. Meanwhile, the President is also mixing, so he must be indifferent between nominating a moderate and nominating an extremist. But whenever he nominates a moderate, the Senate sometimes rejects. This also leaves the President in worse position than if he could credibly commit to appointing moderates exclusively.

Further, we show that the President and Senate can only benefit from more information about judicial nominees when they are ideologically opposed. And yet there seems to be little serious effort to change the current charade of judicial nominee hearings. (During Clearance Thomas’s hearing, when asked whether Roe v. Wade was correctly decided, he unconvincingly replied that he did not have an opinion “one way or the other.”) Why not?

The remainder of our paper investigates this question. We point to the potential benefits of keeping nominee ideology secret when the Senate is ideologically aligned with the President. Under these conditions, the President can nominate extremists and still induce the Senate to accept. Keeping the process quiet allows the President to nominate such extremists without worrying about suffering reputation costs as a result. Consequently, the current system persists.

Although our focus is on judicial nominations, the same obstacles are likely present in other nominations processes. And coming from an IR background, I have been thinking about similar situations in interstate bargaining. In any case, please check out the paper if you have a chance. We welcome your comments on it.

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Game Theory 101: The Complete Textbook Update

Posted on July 18, 2013 | Leave a comment

Two years ago today, I published the first incarnation of Game Theory 101: The Complete Textbook. (It was incomplete back then, heh.) Every summer, I like to go through it and make changes where I can. This time around, I decided to add a new lesson on games with infinite strategy spaces, like Hotelling’s game, second price auctions, and Cournot competition. I have correspondingly added some content to the MOOC version. Videos below.

Initially, I was hesitant to add more material to the textbook because Amazon’s fee increases as the file size of the book increases. Yet, the size of the textbook shrunk because I cut down on unnecessarily wordy sentences. (Switching “is greater than” to “beats” probably chopped off 300 words from the book.)

The optimistic interpretation: Readers now learn more while reading less!

The pessimistic interpretation: I really, really need to work on writing shorter sentences.

 

gt101

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Optimal Flopping: The Game Theory of Foul Fakery

Posted on June 18, 2013 | 1 comment

I was watching the NBA Finals last night. While the series has been good, watching professional basketball requires a certain tolerance for flopping–i.e., players pretending like they got hit by a freight train when in reality the defender barely made incidental contact. Observe LeBron James in action:


And that’s just from this postseason!

No one likes flopping, but it is not going away anytime soon. This post explains the rationality of flopping. The logic is as you might think–players flop to dupe officials into mistakenly calling fouls. There is a surprising result, however. When flopping optimally, “good” officiating becomes impossible–referees are completely helpless in deciding whether to call a foul. Worse for the integrity of the game, a flopper’s actions force referees to occasionally ignore legitimate fouls.

The Model
This being a blog post, let’s construct a simple model of flopping. (See figure below.) The game begins with an opponent barreling into a defender. Nature sends a noisy signal to the official whether contact was foul worthy or not. If it is truly a foul, the defender falls to the ground without a strategic decision. If it is not a foul, the player must decide whether to flop or not.

The referee makes two observations. First, he receives the noisy signal. With probability p, he believes it was a hard foul; with probability 1-p, it was not. He also observes whether the defender fell to the ground. Since the defender cannot keep standing if the offensive player commits a hard foul, the referee knows with certainty that the play was clean if the defender remains standing. However, if the player falls, the referee must make an inference whether the play was a foul.

Payoffs are as follows. The referee only cares about making the right call; he receives 1 if he is correct and -1 if he is incorrect. The player receives 1 if the referee calls a foul, 0 if he does not flop and the referee does not call a foul, and -1 if he flops and the referee does not call a foul. Put differently, the defender’s best outcome is what minimizes the offense’s chance at scoring while his worst outcome is what maximizes the offense’s chance.

flopping

(click image to enlarge)

Equilibria
Since legitimately fouled defenders have no strategic choices, we only have to solve for the non-fouled defender’s action. Therefore, throughout this proof, “defender” means a defender who was not fouled. (Rare exceptions to this will be obvious.)

We break down the parameter space into three cases:

For p = 0
Flopping does not work, since the referee knows no foul took place. This is why players don’t randomly fall to the ground when the nearest opponent is ten feet away from them.

For p > 1/2
Note the the referee will call a foul if he believes that the probability the play was a foul is greater than 1/2. Thus, if the defender flops, he knows the referee will call a foul. As such, the defender always flops, and the referee calls a foul. This is intuitive: on plays that look a lot like a foul, defenders will embellish the contact regardless of how hard they are hit.

For 0 < p < 1/2
Because mixing probabilities are messy, I will appeal to Nash’s theorem to prove that both the defender and referee mix in equilibrium. Recall that Nash’s theorem says that an equilibrium exists for all finite games. Therefore, we can show both players mix by proving that neither can play a pure strategy in equilibrium. (In other words, we expect players to sometimes flop and sometimes not to, while the referees to sometimes call a foul and sometimes not to when they aren’t sure.)

First, can the defender flop as a pure strategy? If he does, the referee’s best response would be to not call a foul, as the referee believes the probability a foul occurred is less than 1/2. But given that the referee is not calling a foul, the defender should deviate to not flopping, since he will not get the call anyway.

Second, can the defender not flop as a pure strategy? If he does, the referee’s best response is to call a foul if he observes the defender falling, as he knows that the play was a legitimate foul. But this means the defender would want to deviate to flopping, since he knows he will get the foul called. This exhausts the defender’s pure strategies, so the defender must be mixing in equilibrium.

Third, can the referee call a foul as a pure strategy? If he does, the defender’s best response is to flop. But then the referee would not want to call a foul, since his belief that the play was actually a foul is less than 1/2.

Fourth, can the referee not call a foul as a pure strategy? If he does, the defender’s best response is to not flop. But this means the referee should call a foul upon observing the defender fall, as he believes the only way this could occur is if the foul was legitimate. This exhausts the referee’s pure strategies, so the referee must mix in equilibrium.

Strategically, these parameters are the most interesting. In equilibrium, the defender sometimes bluffs (by flopping) and sometimes does not. Upon observing a fall, the referee sometimes ignores what he perceives might be a flop and sometimes makes the call.

The real loser is the legitimately fouled defender. He can’t do anything to keep himself from falling over, yet sometimes the referee does not make the call. Why? The referee can’t know for sure whether the foul was legitimate or not and must protect against floppers.

While this seems unfortunate, be glad the referees act strategically in this way–the alternative would be that defenders would always flop regardless of how incidental the contact is and the referees would always give them the benefit of the doubt.

Conclusion
One of game theory’s strengths is drawing connections between two different situations. Although this post centered on flopping in the NBA, note that the model was not specific to basketball. The interaction could have very well described other sports–particularly soccer. As long as fouls provide defenders with benefits, there will always be floppers waiting to exploit the referee’s information discrepancy.

Postscript
If I ever expand my game theory textbook to cover Bayesian games, I think I will include this one. This also makes decent fodder when random people ask “what can game theory do for us?”

gt101

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Preemptive War on the Walking Dead

Posted on February 17, 2013 | Leave a comment

The Walking Dead is cable’s most successful TV show, ever. I’m writing this after “Home,” and I’m going to assume you know what is going on by and large.

Here’s what’s important. As far as we care, there are only two groups of humans left alive. One, the good guys, have fortified themselves inside an abandon jail. The other lives in a walled town called Woodbury. They became aware of each other a few episodes ago, and they have various reasons to dislike each other.

War appears likely and will be devastating to both parties, likely leaving them in a position worse than if they pretended the other simply did not exist. For example, in “Home,” the Woodbury group packs a courier van full of zombies, breaches the jail’s walls, and opens the van for an undead delivery. Now a bunch of flesh-eaters are wandering around the previously secure prison.

Meanwhile, the jail’s de facto leader went on a mysterious shopping spree and came back with a truck full of unknown supplies. I suspect next episode will feature the jail group bombing a hole in Woodbury’s city walls.

All this leads to an important question: why can’t they all just get along? It’s the end of the world for goodness sake!

As someone who studies war, I am sympathetic to the problem. Woodbury and the jail group are capable of imposing great costs on one another merely by allowing zombies to penetrate the other’s camp. The situation seems ripe for a peaceful settlement, since there appear to be agreements both parties prefer to continued conflict.

This is the crux of James Fearon’s Rationalist Explanations for War, one of the most important articles in international relations in the last twenty years. Fearon shows that as long as war is costly and the sides have a rough understanding of how war will play out, then both parties should be willing to sit down at the bargaining table and negotiate a settlement.

However, Fearon notes that first strike advantages kill the attractiveness of such bargains. From the article:

Consider the problem faced by two gunslingers with the following preferences. Each would most prefer to kill the other by stealth, facing no risk of retaliation, but each prefers that both live in peace to a gunfight in which each risks death. There is a bargain here that both sides prefer to “war”…[but] given their preferences, neither person can credibly commit not to defect from the bargain by trying to shoot the other in the back.

The jail birds and Woodbury are in a similar position:

pd

This is a prisoner’s dilemma.[1] Both parties prefer peace to mutual war. But peace is unsustainable because, given that I believe you are going to act peacefully, I prefer taking advantage of you and attacking. This leads to continued conflict until one side has been destroyed (or, in this case, eaten by zombies), leaving both worse off. We call this preemptive war, as the sides are attempting to preempt the other’s attack.

In the real world, countries have tried to reduce the attractiveness of a first strike by creating demilitarized zones between disputed territory, like the one in Korea. But such buffers require manpower to patrol to verify the other party’s compliance. Unfortunately, the zombie apocalypse has left the world short of people–Woodbury has fewer than a hundred, and the jail birds have fewer than ten. As a result, I believe we be witnessing preemptive war for the rest of this season.

[1] Get it? They live in a jail, and they are in a prisoner’s dilemma![2]

[2] I’m lame.

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Noise about Noise: The Good Coach/Bad Coach Fallacy

Posted on January 4, 2013 | Leave a comment

It is 4th and inches from the 50 yard line. The defense lines up with nine in the box, with a cornerback covering the loan wide receiver and the safety playing a bit closer than usual. The quarterback snaps the ball. The safety breaks in to blitz. The running back executes a play fake. The quarterback bombs it to his wide receiver, who has the safety beat. Touchdown.

“What a great call!” exclaims the color commentator.

Your first reaction might be to agree. After all, the play worked. The safety blitzed, leaving the wide receiver with one-on-one coverage. The quarterback came through, delivering a well-placed ball for a quick score. Credit the offensive coach for the play, and discredit the opposing coach for choosing to blitz.

Well, maybe not.

Let’s investigate how perfect coaches would play this situation. To simplify the situation greatly, suppose the offense can choose whether to call a run or a pass. The defense can choose whether to defend the run or the pass. The defense wants to match, while the offense wants to mismatch. To further simplify things, suppose the defensive benefits for matching are the same whether it is pass/pass or run/run. Likewise, the offensive advantages for mis-matching are the same whether it is pass/run or run/pass.

(These are strong assumptions, but the claims I will make hold for environments with richer play calling and differing benefits for guessing correctly/incorrectly.)

If all that holds, then the game is identical to matching pennies:

In equilibrium, both players flip their coins. Note that as long as the opponent is flipping his coin, the other player earns a fixed amount (zero in this case) regardless of which strategy he selects.

This is a necessary condition to reach equilibrium. If one strategy was even slightly better in expectation given the opposing strategy, then the player would always want to play the superior strategy. For example, if running was even slightly better than passing given the offensive’s expectations about the defense, then the offense must choose to run. But then the defensive coach’s strategy is exploitable. He could switch to defending the run and expect to do better. But the defensive coach is supposed to be superhuman, so he would never do something so foolish.

As it turns out, the only strategies that don’t leave open the possibility of exploitation are the equilibrium strategies. Thus, the superhuman coaches should play according to equilibrium expectations.

Now consider how this situation looks to the observer. We only see outcome of one play. But note that all outcomes occur with positive probability in equilibrium! Sometimes the offense does well. Sometimes the defense does well. But any given outcome is essentially chosen at random.

This makes it impossible to pass judgment in favor or against any coach. Certainly all real world coaches are not perfect. But on any given play, one superhuman coach looks foolish while the other superhuman coach looks great. Consequently, on any given real world play, we cannot tell whether the result was a consequence of terrific coaching on one side (and bad coaching on the other) or just pure randomness.

Thus, we have the good coach/bad coach fallacy. Commentators are quick to praise the genius of the fortunate and lambast the idiocy of the unfortunate, but there simply is no way of knowing what is truly gone on given the information. On-air silence might be awkward, but it beats noise about…noise.

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War Exhaustion and the Stability of Arms Treaties

Posted on December 20, 2012 | Leave a comment

(Paper here.)

Earlier this month, I wrote about Iranian nuclear intransigence. In this post, I want to generalize the argument: war exhaustion sabotages long-term arms treaties.

This is part of my dissertation plan, so some background is in order. My main theoretical chapter shows that if declining states can’t threaten preventive war to stop rising states from proliferating, they can buy them off instead. The idea is that weapons are costly to develop. Rising states don’t have any reason to proliferate if they are already receiving most of the concessions they wish to obtain. Meanwhile, the declining state is happy to offer those concessions to deter the rising state from proliferating.

Let’s boil it down to the simplest version of the game possible. The United States has two options: bribe or not bribe. Iran sees the US’s move and decides whether to build a nuclear bomb. American preferences (from top to bottom) are as follows: not bribe/not build, bribe/not build, not bribe/build, bribe/build. Iranian preferences are as follows: bribe/not build, bribe/build, not bribe/build, not bribe/not build.

(I derive these utilities from a more general bargaining setup, so I suggest you look at the paper if you think these seem a little off. I personally wouldn’t blame you, since it seems strange that Iran prefers accepting bribes to taking bribes and proliferating anyway.)

Given that, we have the following game:

b4bgame

By backward induction, Iran builds if the US does not bribe but does not build if the US bribes. In turn, the US bribes to avoid having Iran build.

Great! Iran should not proliferate. But…yeah…that’s not happening at the moment. Why?

One problem is the reason why Iran prefers not building if the United States is bribing. The idea here is that bribes are permanent. By continuing to receive these bribes for the rest of time, Iran sees no need to proliferate since it is already raking in the concessions and nuclear weapons will only waste money.

But what if the United States had the power to renege on the concessions? In the future, the US will no longer be suffering from war exhaustion from Afghanistan and Iraq and will force Iran not to proliferate by threat of preventive war. At that point, the US can renege on the bribe without any sort of repercussions.

Again, boiling the argument down to the simplest game possible, we have this:

warexhaustion

Backward induction gives us that the US will renege (why give when you don’t have to?). So Iran builds regardless of whether the US offers a bribe (it’s a ruse!). Proliferation results today because the United States can treat Iran as essentially nuclear incapable in the future. Iran has a window of opportunity and must take it while it can.

This is neat because a commitment problem sabotages negotiations. Recovering from war exhaustion makes the United States stronger in the sense that it will be more willing to fight as time progresses. Yet, this additional strength causes bargaining to fail, since Iran fears that the United States will cut off concessions at some point down the line. More power isn’t always better.

In addition to discussing Iran, the chapter also talks about the Soviet nuclear program circa 1948, which is fascinating. We often take Moscow’s decision to proliferate as a given. Of course the Soviet Union wanted nuclear weapons–there was a cold war going on! But this doesn’t explain why the United States didn’t just buy off the Soviet Union and avoid the mess of the Cold War. Certainly both sides would have been better off without the nuclear arms race.

Again, war exhaustion sabotaged the bargaining process. The United States was not about to invade Russia immediately after World War II ended. Thus, the Soviets had a window of opportunity to proliferate unimpeded and chose to jump through that window. The U.S. was helpless to stop the Soviet Union–we had zero (ZERO!) spies on the ground at the end of WWII and thus had no clue where to begin even if we wanted to prevent. The same causal mechanism led to intransigence in two cases separated by about 60 years.

If this argument sounds interesting to you, I suggest reading my chapter on it. (Apologies that some of the internal links will fail, since the attachment contains only one chapter of a larger project.) I give a much richer version of the model that removes the hokeyness. Feel free to let me know what you think.

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Book Review: The Evolution of Cooperation

Posted on November 5, 2012 | 1 comment

Book: The Evolution of Cooperation by Robert Axelrod
Five stars out of five.

Suppose two generals each have two choices: attack or defend. The decisions are simultaneous and private. Military strategy favors the offensive, so both really want the other guy to defend while he attacks and really does not want to defend while the other guy attacks. On the other hand, war is extremely bloody. Both generals agree that mutual defense is better than mutual aggression. What should we expect the generals to do?

Intuitively, you might think that mutual defense is a reasonable outcome since peace is an agreeable outcome. However, this fails to appreciate individual incentives. If one general knows the other will play defensively, he should take advantage of his rival’s cooperation and attack. As a result, mutual aggression is the only sustainable outcome. But war is worse for both parties. This is the tragedy known as the prisoner’s dilemma: both parties end up in a mutually despised outcome but cannot commit to the better result due to their selfish individual incentives.

The prisoner’s dilemma has been around since the 1950s. For the next three decades or so, game theorists speculated that repeated interaction could solve the cooperation problem. Perhaps war favors the aggressor, but only a slight degree. If so, the generals could agree to maintain the peace as long as the other guy did. But the moment one slips up, the generals will fight all-out war. The threat of a painful breakdown in peace might incentivize the generals to never start conflict, even if a surprise attack might yield short-term benefits.

However, the cooperative solution remained elusive…until Robert Axelrod’s The Evolution of Cooperation. For a three sentence summary, Axelrod shows that these generals can adopt a “grim trigger” strategy and credibly promise infinite punishment in the future to enforce cooperation in the present. Thus, even bitter rivals can maintain friendly relations over the long term. In essence, we can rationally expect cooperative relationships in even the worst of environments.

Despite how I glossed over all of the intricacies of the repeated prisoner’s dilemma, The Evolution of Cooperation is a must-read for that result alone. But the book is so much more. I first picked it up during my junior year of college. I hadn’t taken a math class in five years, and the grade in that class was a C. Yet, despite the sophistication of the argument, I understood exactly what was going on. Axelrod’s exposition of formal theory in this book is quite simply the best you will ever see.

The fourth chapter is nothing short of awesome. Axelrod takes cooperation to the limit in his study of the “live and let live” trench warfare system during World War I. For a significant chunk of the war, troops spent most of their time deliberately shooting to miss their enemies in the opposing trench. While shooting and killing an enemy soldier provided a marginal gain should a battle take place, said act of shooting risked sparking a larger battle which would cause great causalities on both sides. Thus, for the sake of self-preservation, armies avoided fighting. This culminated in the famous Christmas Truce, in which the troops actually got out of their trenches and began fraternizing with the so-called enemies. (In that vain, you should watch Joyeux Noel if you have not already.)

If there is one issue with the book, it is the emphasis on tit-for-tat. Tit-for-tat is a less aggressive way of responding to your opponent’s aggression than grim trigger; rather than punishing forever, you merely punish at the next available opportunity. Axelrod correctly identifies a bunch of nice properties of tit-for-tat, especially how well it plays nice with others. However, as every modern game theorist knows, tit-for-tat is not subgame perfect and thus is extremely questionable on theoretical grounds. Of course, we would not have found out about that had this book not existed, so this just further solidifies how important The Evolution of Cooperation is.

In sum, go out and buy it. The book has applications to game theory, economics, political science, sociology, evolutionary biology, and psychology. If you are reading this blog, you likely have an interest in one or more of those fields, so you should pick it up.

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Game Theory 101 MOOC Completed

Posted on November 2, 2012 | 3 comments

My Game Theory 101 MOOC (massive open online course) has been completed for Fall 2012. Conveniently, you can watch the entire series below, find the playlist on YouTube, or take the course via Udemy.

The course covers basic complete information game theory and has an accompanying textbook. Enjoy!

P.S. Here’s a (partial) list of the things it covers: prisoner’s dilemma, strict dominance, iterated elimination of strictly dominated strategies, pure strategy Nash equilibrium, best responses, mixed strategy Nash equilibrium, matching pennies, the mixed strategy algorithm, calculating payoffs, battle of the sexes, weak dominance, iterated elimination of weakly dominated strategies, infinitely many equilibria, extensive form games, game trees, backward induction, subgame perfect equilibrium, tying hands, burning bridges, credible commitment, commitment problems, forward induction, knife-edge equilibria, comparative statics, rock paper scissors, symmetric games, zero sum games. Okay, that was a fairly complete list.

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