Tag Archives: Game Theory

Chimps Aren’t Better than Humans at Game Theory

(At least the evidence doesn’t match the claim.)

“Chimps Outsmart Humans When It Comes To Game Theory” has been making the social media rounds today. Unfortunately, this seems to be a case of social media run amok–the paper has some interesting results, but that interpretation is horribly off base.

Below, I will give four reasons why we shouldn’t conclude that chimps are better at game theory than humans. But first, let’s quickly review what happened. A bunch of chimps, Japanese students, and villagers played some basic, zero-sum, simultaneous move games like matching pennies. The mixed strategy algorithm derives equilibrium predictions of what each player should do under these scenarios. As it turns out, chimps played strategies closer to the equilibrium predictions. Therefore, the proposed conclusion is that chimps are better game theorists than humans.

So what’s wrong here? Well…

The Sample Size Is Lacking
Who participated in the study? Six chimps, thirteen female Japanese students, and twelve males from Guinea. We can’t generalize differences between these groups in a meaningful way without a larger sample size.

The Chimps Aren’t a Random Sample
From the study:

Six chimpanzees (Pan Troglodytes) at the Kyoto University Primate Research Institute voluntarily participated in the experiment. Each chimpanzee pair was a mother and her offspring…all six had previously participated in cognitive studies, including social tasks involving food and token sharing.

There are a couple of problems here. First, the pairs of chimps that played are related. It stands to reason that a mother who is good at these games would produce offspring that is also good at these games. So we really aren’t looking at six chimps so much as three. Ouch.

(It should be noted that the Japanese students aren’t really random either since they all come from the same university. However, this is true for many studies of this sort, so I’m going to overlook it.)

Second, these aren’t even your regular chimps. They have played plenty of games before!

Combined, this is like taking a group of University of Rochester Department of Political Scientist (URDPS) students and comparing their results to a group of random Californians. The URDPS group is “related” (they all go to the same school) and they all have plenty of experience playing games (at least three semesters’ worth). They would undoubtedly play more rationally than the random group from California. But you can’t use this to claim that New Yorkers play more rationally than Californians. Yet you are seeing the analogous claim being made.

They Aren’t Playing the Game the Researchers Are Testing Against
Only the researchers knew the game that the players were playing. In contrast, the players only knew their payoffs, not their opponents’. The mixed strategy algorithm only makes predictions about how players should play given that all facets of the game are common knowledge. That’s clearly not the case here.

Instead, the real game here is spending a number of iterations of the game trying decipher what your opponent’s payoffs are and then figuring out how to strategize accordingly. It’s not clear how to interpret the results in this light, though it is interesting that the (small, biased sample of) chimps figured this out more quickly.

The Game Was Not Inter-species
If you want to say that chimps are better at these games than humans, you need to have chimps playing humans. You would then have them play some number of iteration and see who received more apples/yen by the end of the game. Instead, it was chimps versus chimps and humans against humans. With that data, you cannot claim one party is better than the other.

Nash Equilibrium Isn’t a Good Baseline
“Fine,” you might say in response to the last point, “but the chimps still played closer to the Nash equilibrium strategies than humans. Therefore, chimps are better game theorists than humans are.” That’s still not want we want to know, though. Who cares if the players were playing Nash? If I played this game tomorrow, would I play Nash? Yes–if I thought the other player was clever enough to do the same. If not, I would try to beat them.

This is a nuanced problem, so let’s look at an example. If you ask people about soccer penalty kicks, they will likely tell you that you should kick more frequently to your stronger side as it becomes more and more accurate. This is wrong: you should increase your reliance on your weaker side. Knowing this, if I played the role of the goalie, I would start diving to the kicker’s stronger side more frequently. The kicker would do poorly and I would do very well.

How would the study interpret this? It would say that we are both bad at game theory! But that’s not what’s going on here. We have one bad strategist and one sophisticated one. The interpretation of the study would get half of it right but completely blow the other half. Worse, a sophisticated goalie taking advantage of the kicker’s incompetence would outperform a goalie who played Nash instead.

Nash equilibrium is useful for many reasons; testing whether one species is better than another with it as a baseline is not one of them.

I’d have to go through the paper more closely than I have so far to give an overall impression of it. However, even without that, it is clear that the way social media is describing the results is very questionable.

Game Theory Is Really Counterintuitive

Every now and then, I hear someone say that game theory doesn’t tell us anything we don’t already know. In a sense, they are right—game theory is a methodology, so it’s not really telling us anything that our assumptions are not. However, I challenge someone to tell me that they would have believed most of the things below if we didn’t have formal modeling.

  • People often take aggressive postures that lead to mutually bad outcomes even though mutual cooperation is mutually preferable. Source.
  • Even if everyone agrees that an outcome is everyone’s favorite, they might not get that outcome. Source.
  • Sometimes having fewer options is better than having more options. Source.
  • On a penalty kick, soccer players might wish to kick more frequently toward their weaker side as their weaker side becomes increasingly inaccurate. Source.
  • In a duel, both gunslingers should shoot at the same time, even if one is a worse shot and would seem to benefit by walking closer to his target. Source.
  • There’s a reason why gas stations are on the same corner and politicians adopt very similar platforms. And it’s the same reason. Source.
  • Closing roads can improve everyone’s commute time. Source.
  • Fewer witnesses to a crime might be preferable to more. Source.
  • You should bid how much you value the good at stake in a second price auction. Source.
  • If you pay the value you think something is worth, you are going to end up with a negative net profit. Source.
  • Lighting money on fire is often profitable. Source.
  • Going to college can be valuable even if college doesn’t teach you anything. Source.
  • An animal might be better off jumping high in the air repeatedly than running away from a predator. Source.
  • Knowing just slightly more about the value of your car than a potential buyer can make it impossible to sell it. Source.
  • Nigerian email scammers should say they are from Nigeria even though just about everyone is familiar with the scam. Source.
  • Everyone might mimic everyone else just because two people chose to do the same thing. Source.
  • A biased media may be better than an unbiased media. Source.
  • Every voting system is manipulable. Source.
  • You might want to abstain from voting even though you strictly prefer one candidate to another. Source.
  • Unanimous jury rulings are more likely to convict the innocent than simple majority rule if jurors vote intelligently. Source.
  • The House of Representatives caters to the median member of the majority party, not the median member of the institution overall. Source.
  • Plurality, first-past-the-post voting leads to two-party systems. Source.
  • United Nations Security Council members sometimes do not veto resolutions even though they strongly dislike them. Source.
  • Without the ability to propose offers, you receive very few benefits from bargaining. Source.
  • Settlements always exist that are mutually preferable to war. Source.
  • Fighting wars removes the need for war. Source.
  • You might want to shoot to miss in war. Source.
  • Nonproliferation agreements can be credible. Source.
  • Weapons inspections are useful even if they never find anything. Source.
  • Economic sanctions are useful even though they often fail in application. Source.
  • Pitchers shouldn’t change their pitch selection with a runner on third base, even though curveballs are more likely to result in wild pitches. Source.
  • Sports teams can benefit from a lack of player safety in contract negotiations. Source.
  • You shouldn’t try to maximize your score in Words with Friends/Scrabble. Source.
  • In speed sailing, competitors deliberately choose paths they believe will be slower. Source.
  • The first player wins in Connect Four. Checkers ends in a draw. Source.
  • Chess has a solution, though we don’t know it yet. Source. (Or maybe not.)
  • Warren Buffett was never going to pay $1 billion the winner of the March Madness bracket challenge. Source.
  • Park Place is worthless in McDonald’s Monopoly. Source.
  • Losing pays. Source.
  • As drug tests become more accurate, they should be implemented less often. Source.

Am I missing anything?

Roger Craig’s Daily Double Strategy: Smart Play, Bad Luck

Jeopardy! is in the finals of its Battle of the Decades, with Brad Rutter, Ken Jennings, and Roger Craig squaring off. The players have ridiculous résumés. Brad is the all-time Jeopardy! king, having never lost to a human and racking up $3 million in the process. Ken has won more games than anyone else. And Roger has the single-day earnings record.

That sets the scene for the middle of Double Jeopardy. Roger had accumulated a modest lead through the course of play and hit a Daily Double. He then made the riskiest play possible–he wagered everything. The plan backfired, and he lost all of his money. He was in the negatives by the end of the round and had to sit out of Final Jeopardy.

Did we witness the dumbest play in the history of Jeopardy? I don’t think so–Roger’s play actually demonstrated quite a bit of savvy. Although Roger is a phenomenal player, Brad and Ken are leaps and bounds better than everyone else. (And Brad might be leaps and bounds better than Ken as well.) If Roger had made a safe wager, Brad and Ken would have likely eventually marched past his score as time went on–they are the best for a reason, after all. So safe wagers aren’t likely to win. Neither is wagering everything and getting it wrong. But wagering everything and getting it right would have given him a fighting chance. He just got unlucky.

All too often, weaker Jeopardy! players make all the safest plays in the world, doing everything they can to keep themselves from losing immediately. They are like football coaches who bring in the punting unit down 10 with five minutes left in the fourth. Yes, punting is safe. Yes, punting will keep you from outright losing in the next two minutes. But there is little difference between losing now and losing by the end of the game. If there is only one chance to win–to go for it on fourth down–you have to take it. And if there is only one way to beat Brad and Ken–to bet it all on a Daily Double and hope for the best–you have to make it a true Daily Double.

Edit: Roger Craig pretty much explicitly said that this was the reason for his Daily Double strategy on the following night’s episode. Also, this “truncated punishment” mechanism also has real world consequences, such as the start of war.

Edit #2: Julia Collins in the midst of an impressive run, having won 14 times (the third most consecutive games of all time) and earned more money than any other woman in regular play. She is also fortunate that many of her opponents are doing very dumb things like betting $1000 on a Daily Double that desperately needs to be a true Daily Double. People did the same thing during Ken Jennings’ run, and it is mindbogglingly painful to watch.

Interpret Your Cutpoints

Here is a bad research design I see way too frequently.* The author presents a model. The model shows that if sufficient amounts of x exist, then y follows. The author then provides a case study, showing that x existed and y occurred. Done.

Do you see the problem there? I removed “sufficient” as a qualifier for x from one sentence to the next. Unfortunately, by doing so, I have made the case study worthless. In fact, such case studies often undermine the exact point the author was trying to make with the model!

Let me illustrate my point with the following (intentionally ridiculous) example. Consider the standard bargaining model of war. State A and State B are in negotiations. If bargaining breaks down, A prevails militarily and takes the entire good the parties are bargaining over with probability p_A; B prevails with complementary probability, or 1 – p_A. War is costly, however; states pay respective costs c_A and c_B > 0.

That is the standard model. Now let me spice it up. One thing that the model does not consider is the cost of the stationery**, ink, and paper necessary to sign a peaceful agreement. Let’s call that cost s, and let’s suppose (without loss of generality) that state A necessarily pays the stationery costs.

Can the parties reach a peaceful agreement? Well, let x be A’s share of a peaceful settlement. A prefers a settlement if it pays more than war, or x – s > p_A – c_A. We can rewrite this as x > p_A – c_A + s.

Meanwhile, B prefers a settlement if the remainder pays better than war, or 1 – x > 1 – p_A – c_B. This reduces to x < p_A + c_B.

Stringing these inequalities together, mutually preferable peaceful settlements exist if p_A – c_A + s < x < p_A + c_B. In turn, such an x exists if s < c_A + c_B.

Nice! I have found a new rationalist explanation for war! You see, if the costs of stationery exceed the costs of war (or s > c_A + c_B), at least one state would always prefer war to peace. Thus, peace is unsustainable.

Of course, my argument is completely ridiculous–stationery does not cost that much. My theory remains valid, it just lacks empirical plausibility.

And, yet, formal theorists too often fail to substantively interpret their cutpoints in this way. That is, they do not ask if real-life parameters could ever sustain the conditions necessary to lead to the behavior described.

Instead, you will get case studies that look like the following:

I presented a model that shows that the costs of stationery can lead to war. In analyzing the historical record of World War I, it becomes clear that the stationery of the bargained resolution would have been very expensive, as the ball point pen had only been invented 25 years ago and was still prohibitively costly. Thus, World War I started.

Completely ridiculous! And, in fact, the case study demonstrated the opposite of what the author had intended. That is, if you actually analyze the cutpoint, you will see that the cost of stationery was much lower than the costs of war, and thus the cost of stationery (at best) had a negligible causal connection to the conflict.

In sum, please, please interpret your cutpoints. Your model only provides useful insight if its parameters match what occurred in reality. It is not sufficient to say that cost existed; rather, you must show that the cost was sufficiently high (or low) compared to the other parameters of your model.

* This blog post is the result of presentations I observed at ISA and Midwest, though I have seen some published papers like this as well.

** I am resisting the urge to make this an infinite horizon model so I can solve for the stationary MPE of a stationery game.

The Game Theory of MPSA Elevators

TL;DR: The historic Palmer House Hilton elevators are terribly slow because of bad strategic design, not mechanical issues or overcrowding.

Midwest Political Science Association’s annual meeting–the largest gathering of political scientists–takes place at the historic* Palmer House Hilton each year. While the venue is nice, the elevator system is horrible. And with gatherings on the first eight floors, the routine gets old really fast.

Interestingly, though, the delays are not the result of an old elevator system or too many political scientists moving at once.** Rather, the problem is shoddy strategic thinking.

Each elevator bay has three walls. The elevators along each wall have different tasks. Here’s the first one:


Elevators on this floor go from the ground floor to the 12th floor.

Here’s the second:


These go from the ground floor to the eighth floor or the 18th floor to the 23rd floor.

And the last wall:


These go from the ground floor to the eighth floor and the 13th floor to the 17th.

Now suppose you are on the ground level want to go to the 7th floor. What’s the fastest way to get there? For most elevator systems, you press a single button. The system figures out which elevator will most efficiently take you there and dispatches that elevator to the ground level.

But historic Palmer House Hilton’s elevators are not a normal system. Each wall runs independent of one another with three separate buttons to press. So if you really want to get to the seventh floor as fast as possible, you have to press all three–after all, you do not know which of the three systems will most quickly deliver an elevator to your position.

Unfortunately, this has a pernicious effect. Once the first elevator arrives, the call order to the other two systems does not cancel. Thus, they will both (eventually) send an elevator to that floor. Often times, this means an elevator wastes a trip by going to the floor and picking no one up. In turn, people on other floors waiting for that elevator suffer some unnecessary delay.

This is why (1) the elevator system takes forever and (2) you often stop at various floors and pick up no one. We would all be better off if people limited their choice to a single system, but a political scientist running late to his or her next panel does not care about efficiency.

(Let this sink in for a moment. The largest gathering of political scientists has yet to overcome a collective action that plagues it on an every day basis.)

Given the floor restrictions for the elevators, the best solution I can think of would be to install an elevator system where you press the button of the floor you want outside the elevator, and the system chooses which to send from the three walls. This would be mildly inconvenient but would stop all the unnecessary elevator movements.


*Why is it historic? I have no clue. But everyone says it is.

**The latter undoubtedly contributes to the problem, however.

The Nefarious Reason to Draw on Jeopardy

Arthur Chu, current four-day champion on Jeopardy!, has made a lot of waves around the blogosphere with his unusual play style. (Among other things, he hunts all over the board for Daily Doubles, has waged strange dollar amounts when getting one, and clicks loudly when ringing in.) What has garnered the most attention, though, is his determination to play for the draw. On three occasions, Arthur has had the opportunity to bet enough to eliminate his opponent from the show. Each time, he has bet enough so that if his opponent wagers everything, he or she will draw with Arthur.

It is worth noting that draws aren’t the worst thing in Jeopardy. Unlike just about all other game shows, there is no sudden death mechanism. Instead, both players “win” and become co-champions, keeping the money accumulated from that game and coming back to play again the next day. There is no cost to you as the player; Jeopardy! foots the bill.

Why is Arthur doing this? The links provided above give two reasons. First, there have been instances where betting $1 more than enough to force a draw has resulted in the leader ultimately losing the game. Betting more than the absolute minimum necessary to ensure that you get to stay the next day thus has some risks. Second, if your opponents know that you will bet to draw, it induces them to wager all of their money. This is advantageous to the leader in case everyone gets the clue wrong.

That second point might be a little complicated, so an example might help. Suppose the leader had $20,000, second place had $15,000, and third place died in the middle of the taping. If the leader wagers $10,000, second place might sensibly wager $15,000 to force the draw if she thought she had a good chance of responding correctly. If only one is correct, that person wins. If they are both right, they draw. If both are wrong, second place goes bankrupt and the leader wins with $10,000.

Compare that to what happens if the leader wagers $10,001 (enough to guarantee victory with a correct response) and second place wagers $5,000. All outcomes remain the same except when both are wrong. Now the leader drops to $9,999 and the person trailing previously wins with $10,000.

Sure, these are good reasons to play to draw, but I think there is something more nefarious going on. Arthur knows he is better than the contestants he has been beating. One of the easiest ways to lose as Jeopardy! champion is to play a game against someone who is better than you. So why would you want to get rid of contestants that you are better than? Creating a co-champion means that the producers will draw one less person from the contestant pool for the next game, meaning there is one less chance you will play against someone better than you. This is nefarious because it looks nice–he is allowing second place to take home thousands and thousands of dollars more than they would be able to otherwise–but really he is saying “hey, you are bad at this game, so please keep playing with me!”

In addition, his alleged kindness might even be reciprocated one day. Perhaps someone he permits a draw to will one day have the lead going into Final Jeopardy. Do you think that contestant is going to play for the win or the draw? Well, if Arthur is going to keep that person on the gravy train for match after match, I suspect that person is going to give Arthur the opportunity to draw.

It’s nefarious. Arthur’s draws could spread like some sort of vile plague.

Are Weapons Inspections about Information or Inconvenience?

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.