Gaming Frivolity
This past week, the Senate Judiciary Committee heard testimony on whether to overrule two recent Supreme Court decisions that made it much harder for plaintiffs to bring civil lawsuits. The decisions, Bell Atlantic v. Twombly and Iqbal v. Ashcroft, held that a court may dismiss a suit if it finds that the plaintiff’s claims are “implausible” even before the plaintiff has had an opportunity to obtain information from the defendant to prove its case.
Critics have slammed the decisions as barring the courthouse door to meritorious suits in which crucial information is solely in the hands of the defendant. An example of such a suit might be one alleging a price-fixing. In such a case, evidence showing that the defendant’s prices were the result of a conspiracy rather than simply meeting competition is not likely to be available to a plaintiff unless the defendant is compelled to produce it in a lawsuit. But under Twombly and Iqbal, a court could dismiss a price-fixing lawsuit before the plaintiff had the opportunity to obtain that information on the ground that the defendant’s actions appeared equally consistent with fair competition.
Supporters of these decisions counter that requiring a plaintiff to identify facts at the outset that tend to exclude innocent explanations is warranted in order to weed out frivolous lawsuits. They claim that otherwise, plaintiffs may blindly file meritless lawsuits that defendants will be coerced to settle rather than face the cost of litigation.
While the cases and editorial pages are replete with references to the scourge of frivolous litigation, it turns out that there is very little empirical support for the claim that courts are awash in it. And there are reasons to believe that defendants settle too few rather than too many of the kind of suits that Twombly and Iqbal seek to eliminate.
The game theory of frivolous litigation is nicely modeled in a 1997 article by law professor Robert G. Bone.* Among the many scenarios he analyzes is the case, like our price-fixing example, where a defendant knows at the outset whether a plaintiff’s case has merit, but the plaintiff can only find out before filing suit, if at all, though a very expensive investigation (bribing an insider, for example).
In this case, it turns out that the defendant’s best strategy is to fight every unmeritorious case, and offer to settle some but not all meritorious cases at the outset. The plaintiff’s best strategy is drop its case some of the time when the defendant declines to settle at the outset, not knowing whether or not its case has merit. Thus defendants never pay to settle unmeritorious cases, but plaintiffs sometimes unknowingly drop meritorious ones. The result is a transfer of wealth from uncompensated meritorious plaintiffs to guilty defendants, not from innocent defendants to frivolous plaintiffs.
Twombly and Iqbal, it turns out, are a solution in search of a problem. They should be repealed.
* Modeling Frivolous Lawsuits, 145 U. Pa. L. Rev. 519 (1997).
Interview with Ben Polak
I learned of Ben Polak through his course Econ 159, available online through Open Yale Courses (see my review). In addition to being a superb teacher, Polak is an expert on decision theory, game theory, and economic history. His work explores economic agents whose goals are richer than those captured in traditional models. His contributions to game theory range from foundational theoretical work on common knowledge, to applied topics in corporate finance and law and economics.
Most recently, he has made contributions to the theory of repeated games with asymmetric information. Other research interests include economic inequality and individuals’ responses to uncertainty. Professor Polak is currently engaged in an ambitious empirical project that tackles questions of industrial organization in the setting of industrial revolution in England. For a list of his achievements, awards, and selected papers visit his Yale School of Management page.
Polak was kind enough to answer a few questions by e-mail.
Austin Frakt (AF): The overarching hook of Econ 159 was to build toward a game theoretic model of human cooperation. In doing so, more and more aspects of human behavior were discussed and incorporated. What is the current and historical relationship between game theory and behavioral economics? How has one influenced the other?
Ben Polak (BP): Some of the best work in behavioral economic theory is directly about games, Matt Rabin’s work for example. Behavioral economics naturally introduces some new game-theoretic issues. For example, in many behavioral models, the agents’ preferences are not “dynamically consistent”. This means that what they may want to do tomorrow may not be what they want today for themselves to do tomorrow. If the agents anticipates this, then they are going to be playing a game with themselves. Many of the ideas from the class – such as backward induction – take on a new importance. The agent now has to anticipate and roll back what she will do tomorrow to decide what she should do today. And ideas like commitment take on a new role: agents may want to commit themselves not just influence others choices (like burning the boats) but simply to control the scope of their future selves’ choices.
AF: Nash is so closely associated with game theory. How much of the content of Econ 159 was pioneered by him? Which is due to the more recent work of others? Where is the frontier today?
BP: Nash introduced the notion of Nash equilibrium but, at the time he was working, most of the work was on cooperative game theory rather than non-cooperative game theory. So, for example, bargaining over a pie was analyzed on the basis of normative axioms rather than strategic choices. One of Nash’s major papers was on normative bargaining solutions but another paper introduced a game in the sense we discuss in the class, and showed that the outcome suggested by normative criteria would be the outcome that would result in this game. That paper set off a whole literature called “the Nash program”: the attempt to find games that produce particular ‘desirable’ outcomes. The last Nobel prize given to game theorists (that given to Maskin, Myerson and Hurwicz) was given for studying “mechanism design” which (in a sense) is the field that descended from Nash’s first paper. The early pioneers of game theory – Nash, Shapley, Shubik, von Neumann, Scarf – were (and are) extraordinary minds.
AF: The game theoretic problems approached in 159 are, by necessity, simple enough to understand and analyze “by hand.” No doubt there are vastly more complex problems and models for which one requires the aid of computer. What are the common computational packages and approaches? Are there parameter estimation algorithms that one could say are the analog of standard econometrics? What are some really big and important problems that are addressed with such things?
BP: Here I am a bit out of my field when it comes to specific packages. But one of the main advances in econometric economics in the last two decades has been the introduction of so-called structural models. Typically, these models are game-theory based and the econometric techniques use features of equilibria to help identify parameters in the model. Two major examples are the econometric models used to estimate demand first for items such as cars and second behind bids in auctions. These econometric models have revolutionized empirical studies of industry. Hand in hand with this is a literature on computing equilibria in (possibly complicated) games. Again, one use is in analyzing the data.
AF: Do you teach other classes at Yale and might they one day be available via Open Yale Courses? (Would it help if I begged?)
BP: Well, for my sins, I am about to take over as chair of economics at Yale. Once that is over, I would like to develop a new course and (if it works well) I would like to try to have it added to the open courses. In the meantime, Bob Shiller’s wonderful course on Financial Markets is available [reviewed on this blog]. And we are hoping that, one day soon, John Geanakoplos’s superb course on Financial Theory will be available. John is one of the most inspiring teachers at Yale, so I am looking forward to it.
Passing the Plate for Universal Health Insurance?
My Regular Libertarian Foil ™ (RLF – an acquaintance who often provokes me with unorthodox points of view) recently posed the following question: Why can’t all the liberals who think universal health insurance is a moral issue take up a collection and fund it themselves without raising my taxes?
This question put me in mind of a problem I encountered in Ben Polak’s game theory course. Suppose that a person can make an investment of ten dollars that will return fifteen dollars (net return of five dollars), only if ninety percent of everybody also invests. If less than ninety percent invest, an individual’s return on his or her ten dollar investment is nothing. When Professor Polak first posed this problem to his class of Yale students, only about half chose to invest, and they all lost their investments. A second round of the game produced very few investors, and a third round almost none. He then explained the dynamics of the game and its payoffs. If I believe that nearly everybody will invest, then my best response to their expected play is to also invest, and the same is true for everybody else who shares the same belief. In this case, the solution set of everybody investing is said to be a “Nash equilibrium,” because nobody can do better by not investing given their belief that others will as well.
But there is another Nash equilibrium in this game. If I believe that not enough people will invest, then my best response to others’ expected play is not to invest either, and the same is again true for everybody else who shares that belief. We saw actual play converge on this Nash equilibrium as the members of the class set their expectations in response to others’ play in previous rounds. This was unfortunate, however, because the payoffs associated with the Nash equilibrium of everybody investing are superior for everybody (technically, “Pareto superior“). Happily, solving the coordination problem that frustrated the superior equilibrium in the initial play was as simple as explaining the game, as everybody chose to invest during a final round of play at the end of the lesson.
It is simple to construct a similar model for funding universal health insurance through a voluntary tax on liberals. Suppose that a voluntary tax on each liberal in the amount of X would be sufficient to fund universal health insurance, provided that ninety percent of liberals paid the tax. Suppose also that each liberal experiences an increase in utility (happiness, moral satisfaction) equivalent to (i.e, for which they would have willingly paid) 2X when there is universal health insurance. Clearly, each liberal is better off when all liberals choose to pay the tax than if no liberals do. But is there a Nash equilibrium in which all liberals pay the tax? There is not. If I pay the tax and all the others do too, my expected payoff is X — the 2X utility from having universal health insurance less the X I paid in tax. But if all the other liberals pay and I don’t, my payoff is 2X. So not paying is my best response to the others’ expected play if I believe that they will pay the tax. And of course, my best response to others’ expected play if I believe a sufficient number of them will not pay the tax is also not to pay the tax. The game has one Nash equilibrium in which no liberal pays the tax and there is no universal health insurance, even though liberals would be collectively better off if everybody paid the tax. It is, in short, a prisoner’s dilemma.
One potential solution to a prisoner’s dilemma is cooperation. We can all agree to pay the tax for our mutual benefit. But of course, each of us has an incentive to cheat and receive the benefit without paying a share of the cost. And studies of successive-round prisoner’s dilemma type games have shown that even players who are initially inclined to cooperate will revert to their equilibrium strategies as they perceive or fear that others are cheating. So the expected outcome even if liberals all agree to pay the tax is eventually the same without some enforcement mechanism to prevent cheating. We could make the liberal tax mandatory, but then we would have to identify the liberals. Though I’m sure my RLF would rejoin that he finds them easy enough to spot, you get the picture. Private charity, unsurprisingly, can’t solve the problem of universal health insurance.
Yale’s Econ 159: Game Theory Made Fun
There’s a reason I’ve posted a lot on game theory of late: I was taking a course on the topic, continuing my education by podcast. The Open Yale Courses (OYC) program makes it easy to turn your iPod or MP3 player into a classroom of sorts, with 15 Yale courses online and 30 more expected over the next three years. Each course includes lectures by video or audio, downloadable from the OYC website or available through iTunes.
So far I’ve listened through Econ 252, which I reviewed previously, and Econ 159. The latter is Yale’s basic game theory course taught by Ben Polak. I don’t need to say very much about the content of the course since I’ve already given readers a large dose of basic game theory. In fact, all of the content of my game theory posts has been inspired by that of Econ 159.
In this review I’ll comment on the style of the course and the quality of the instructor, Ben Polak. Polak is one of those virtuoso teachers. If you’re lucky you’ve had one or two such teachers in your life: the sort who entertains as he instructs, who presents his subject in such easy, digestible bites that one cannot help but learn it. In fact, Polak has won awards for his teaching. It helps that he’s also very funny and has a charming British accent.
Game theory can’t be taught without use of a blackboard. There’s a lot of drawing of diagrams, graphs, arrays of numbers, and so forth. So one might think it impossible to learn the subject aurally by podcast. However Polak speaks what he writes so clearly I did not find it difficult to follow along and to “see” the visual in my head.
I confess that there were details I could not keep track of (was that a “2, 1″ payoff in the upper right of the array or the lower left?), but it didn’t matter. It isn’t the details that are important, it is the main concepts. Those Polak makes as plain as day, and they’re emphasized more than once so one cannot miss them.
Polak told his students at the start of the class that the course was “moderately hard but moderately fun.” I think that’s fair. It requires a certain type of logical mind to enjoy game theory, and that is hard for some people. But Polak makes it about as fun as game theory is likely to ever be.
In my review of Yale’s Econ 252 I wrote that its instructor, Robert Shiller, had set the bar high. Well, Ben Polak far exceeded it and Econ 159 is one of the best courses I’ve ever taken (and I’ve taken a lot of them). If you’re interested in learning basic game theory and/or if you’ve enjoyed my game theory posts, give it a try. Even if you only make it through part of the course you’ll still have learned something and probably had a good time doing so.
“War of Attrition”: Interpretations and Applications
Last week I posed the war of attrition game, and earlier this week I analyzed it. Building on that analysis, in this post I provide some interpretations and applications for the mixed strategy Nash equilibrium solution we found. As a reminder, here’s a short summary of the game in more general notation than originally posed:
You and a competitor will battle in rounds for a prize worth V dollars. In each round each of you may choose to either fight or fold. The first one to fold wins $0. If the other player doesn’t also fold he wins the V dollar prize. In the case you both fold in the same round you each win $0. If you both choose to fight you both go on to the next round to face a fight or fold choice again. Moving on to each round after round 1 costs C dollars per round per player. Assume V > C.
Recall that what we found in the analysis was that there was a mixed strategy Nash equilibrium to fight with probability p=V/(V+C). In the case V=$5 and C=$0.75, p=0.87. What does this mean?
There are multiple ways to interpret mixed strategy Nash equilibria. One way is to interpret the probability as a statement about a population. Applied to the game of attrition this interpretation would say that proportion p of the population are fighters and the rest are folders. That’s certainly plausible. I bet that upon reading the statement of this problem last week some folks immediately thought “I will not fight even one round,” while other folks immediately thought, “I would fight forever.” Even if nobody actually thought the latter, experiments show that people will really fight a very long time, even to the point that the cumulative fight fees exceed the prize. There really are “fighter” and “folder” personality types in the population.
A second interpretation is that each individual will play a mixed strategy. That is, you yourself will “roll the dice” in your head and fight with probability p and fold otherwise. Notice that each round is an independent “roll of the dice.” Past fight fees have no bearing on your probability of fighting in the current round. They are sunk costs. With probability p you will fight on, and on, and on…
What is the probability that this fight will go to round 2? It is the probability that both you and your opponent fight in round 1, or p2. What is the probability the fight will enter round 3? It is the probability that you and your opponent both fight in round 1 and both fight in round 2. Those decisions are independent so the probability of entering round 3 is p4. In general, the probability of fighting to round n+1 is p2n. When p is large (i.e., V is large relative to C) some very long fights can occur. With each round there is hope of earning some money (if you win) so it is rational for you to continue precisely when your expected winnings of doing so are equivalent to those if you don’t. That’s exactly what p promises.
In fact, long wars of attrition have occurred in history, in warefare, in competition between firms, and in politics. Wars of attrition also occur in auctions. Each side is rational to continue the war but not because they wish to recoup past fight fees. Those are sunk costs, they cannot be recovered, and therefore they are irrelevant to current play. Each stage is independent of the next so players fight on because the expected benefit is equivalent to not fighting (in mixed strategy Nash equilibrium play). Eventually one side’s resources are exhausted and the war of attrition comes to an end.
My set of things to say about this game has also been exhausted so this series on the war of attrition game also ends here, at least for now.
Analysis of the “War of Attrition” Game
Last week I posed the following problem from game theory called “war of attrition.” It is a simple yet famous game that explains some strange real world behavior. More on that later; first the problem and analysis:
You and a competitor will battle in rounds for a prize worth $5. Each round you may choose to either fight or fold. So may your competitor. The first one to fold wins $0. If the other player doesn’t also fold he wins the $5. In the case you both fold in the same round you each win $0. If you both choose to fight you both go on to the next round to face a fight or fold choice again. Moving on to each round after round 1 costs $0.75 per round per player (that is, both players pay $0.75 per round in which they both choose to fight onward). How many rounds of fighting would you be willing to go? How would your answer change with the size of the prize? With the size of the per-round fee?
I will analyze this problem with a mostly intuitive approach, sidestepping some slightly more advanced game theoretic ideas. I claim that if this game is played under conditions of common knowledge of rationality there is a rational strategy to fight in each round with probability 0.87. That’s quite a specific claim. Let’s see if I can back it up.
First, let’s notice a few things. If you know with certainty that your opponent will fold in any round 1-7 then it is rational for you to fight because you will win $5 and pay less than $5 in fight fees ($0.75 per round after round 1). However, if your opponent intends to fold in any round then it is only sensible for him to do so in round 1. Why pay fight fees only to fold later? One can make the same argument with the roles of the players reversed.
Hence, if either player is not willing to fight forever he should fold in round 1. If the other player knows this to be the case, the other player should fight. It turns out these are the two pure strategy Nash equilibria (game theory jargon) in this game: (1) you fight, your opponent folds in round 1 and (2) you fold, your opponent fights in round 1. (You can think of a Nash equilibrium as a pair of strategies–one for each player–from which neither can profitably deviate. As I argued above, if one is to adopt a “fold” strategy one should do so in round 1. It is clear that the “fighter” cannot gain more by also folding.)
In real life (as well as in the game) neither player knows for certain that the other player is going to fold. So the above analysis isn’t complete. There is another Nash equilibrium, one that mixes the two pure strategies–fight and fold–probabilistically. To find it, we can use the argument I made in The Game (Theory) within the Game: a mixed strategy Nash equilibrium is one in which the pure strategies are mixed in such a way to make the pure strategy payoffs to each player equivalent. Put another way, one mixes precisely in a way to make one’s opponent indifferent to his options.
So, you fight with probability p such that your opponent receives the same expected payoff (expected dollar winnings) under each of his options. This is a symmetric game so your opponent is also going to fight with probability p as well. How do we find the value of p? It isn’t so hard, but it is worth doing it for a more general case than the one described in the game.
To make things more general, let’s call the prize V (=$5 in the game) and the per-round fight fee C (=$0.75 in the game). We can find p in terms of V and C by using the property of a mixed strategy Nash equilibrium given above: p makes one’s opponent indifferent to his options. So, the solution approach is to figure out your opponent’s payoffs under each option and set them equal to each other.
If your opponent fights in round 1 he expects to earn -C with probability p if you also fight, and he expects to earn V with probability 1-p if you do not fight. Therefore, his expected payoff if he fights is -pC+(1-p)V. On the other hand, if your opponent does not fight in round 1 he expects to earn $0 no matter what you do. Setting these two expected payoffs equal gives an equation with one unknown, p:
-pC+(1-p)V = 0
The solution is p=V/(V+C). Plugging in the values from the game we get that p=5/(5+0.75)=0.87 (rounded). Notice how p changes with V and C. It increases with increasing V and decreases with increasing C. This should be consistent with one’s intuition. The greater the prize the more likely one is to fight for it. The higher the fight fee the less likely one is to fight. Notice also that there is a chance that the fight could go on for a long time, even to the point that the cumulative sum of the fight fees are higher than the prize. In this fight, players are exhausting resources they can never recoup. In real life the player who runs out of resources (money to pay the fight fee) will have to fold, hence the name “war of attrition.”
I’ll provide some interpretation of this solution and discuss applications in my next post about the war of attrition game.
Later: Those really interested in the nitty-gritty mathematical details might want to read the comments to my original post of this problem.
War of Attrition
This is another game theory problem called “war of attrition”:
You and a competitor will battle in rounds for a prize worth $5. Each round you may choose to either fight or fold. So may your competitor. The first one to fold wins $0. If the other player doesn’t also fold he wins the $5. In the case you both fold in the same round you each win $0. If you both choose to fight you both go on to the next round to face a fight or fold choice again. Moving on to each round after round 1 costs $0.75 per round per player (that is, both players pay $0.75 per round in which they both choose to fight onward).
Example 1: In round 1 you and your competitor both fold. The game is over. Both you and your competitor incur no cost and win no money.
Example 2: In round 1 you fold but your competitor fights. The game is over. You lose nothing and gain nothing. Your competitor incurs no cost and wins the $5.
Example 3: In round 1 both you and your competitor fight. Therefore you move on to round 2 and each pay $0.75. In round 2 you both fight again. Therefore you move on to round 3 and each pay another $0.75. In round 3 you fight and your competitor folds. The game is over. You have spent $1.50 and win $5 for a net gain of $3.50. Your competitor has lost $1.50.
How many rounds of fighting would you be willing to go? How would your answer change with the size of the prize? With the size of the per-round fee? I’ll post analysis next week followed by interpretations and applications.
Nuclear Proliferation and the “Hierarchy of Hungry Cannibals” Game
Last week I posed a game called “Hierarchy of Hungry Cannibals.” The game is easy to grasp but takes too many words to state to repeat it here. This Monday I analyzed the game. In this post I attempt to relate it to a real world issue. Because the game is so structured and contrived it isn’t easy to find a good application in real life. Sometimes games are just games. However, I’m going to try anyway. The application is nuclear proliferation.
Instead of ten cannibals on an island we have ten countries in the world (or any other even number greater than two). Like the cannibals, name the countries 1, 2, 3, …, 10. Country 1 is the dominant superpower, 2 its chief rival but less powerful than 1, 3 the chief rival but less powerful than 2, and so on up to 10. Each country becomes interested in acquiring nuclear weapons only if its more powerful rival has them. But each country is more interested in preventing its next lower rival from acquiring nuclear weapons than in acquiring them itself.
This set up is almost identical to that of the hierarchy of hungry cannibals just with different words (”countries” for “cannibals”, “acquiring nuclear weapons” for “eating”, and so forth). So the solution is almost the same. There is one key difference. In the case of the cannibals, if number 1 passes on eating Meathead then he has no one to eat and therefore cannot eat. In the case of nuclear proliferation, country 1 can acquire nuclear weapons any time it likes. It doesn’t give up the option. If it figures out that country 2 will acquire nuclear weapons despite what it does, then country 1 might as well acquire them too.
Using backward induction in the same way used on to find the solution to the hierarchy of hungry cannibals game, we find that country 2 will indeed acquire nuclear weapons without risk of country 3 doing so. Thus, country 1 might as well too. Therefore, the preferred strategies of the countries are:
- Acquire
- Acquire
- Don’t acquire
- Acquire
- Don’t acquire
- Acquire
- Don’t acquire
- Acquire
- Don’t acquire
- Acquire
This is just like the cannibals case with “acquire” in place of “eat” with one difference. While country 1 would seem to have a preferred strategy of “don’t acquire” in order to prevent country 2 from acquiring, we know that country 2 will acquire them anyway because it is risk free for that country to do so. Country 3 cannot acquire them or else the next country–number 4–will (same goes for every odd country other than 1). Therefore, country 1 might as well acquire nuclear weapons.
Thus, the final solution is:
- Acquire
- Acquire
- Don’t acquire
- Don’t acquire
- Don’t acquire
- Don’t acquire
- Don’t acquire
- Don’t acquire
- Don’t acquire
- Don’t acquire
The two dominant countries have nuclear weapons, but nuclear weapons are prevented from proliferating to other countries. Once upon a time that was, in fact, the state of the world. But this correspondence is a bit of luck (or, to be honest, I rigged it this way). The solution would be different if we started with an odd number of countries. If you’ve read this far you can likely deduce that for yourself.
Analysis of the “Hierarchy of Hungry Cannibals” Game
Last week I posted a game theory problem I called “Hierarchy of Hungry Cannibals.” Though it is a simple idea the problem statement is long so I won’t re-post it here (go back and read it if you need to).
I think most people can put themselves in cannibal number 1’s shoes (or whatever) and make what seems to be the best choice. If cannibal 1 doesn’t eat Meathead he won’t fall asleep and he won’t be at risk of being eaten. That’s safe. In fact, every cannibal could make that choice. But, perhaps surprisingly, not all have to.
This is an iterative game. That is, the players (cannibals) do not all chose their strategy (eat or not eat) simultaneously. Each waits for his turn while other players (cannibals) make their moves. Cannibal 2 can’t decide what to do until cannibal 1 does so. 3 has to wait for 2, and so on.
There are two key ideas in solving iterative games. One, which applies to all problems in game theory, is to put yourself in the shoes (or whatever footwear) of each player (cannibal) in turn. The second key idea for iterative games is to start your analysis at the end of the game, with the choices of the last player to move. After thinking about the last move, work backward through the choices of the other players: cannibal 10, then 9, then 8, then 7, etc. This solution method is known as backward induction.
The last cannibal to make a decision is number 10. He has to wait until numbers 1, 2, …, 9 decide whether or not to eat, in turn. Since number 10 is the lowest in the hierarchy there is no cannibal below him who will eat him if he sleeps. He can eat without risk. Therefore, if given the chance, he eats.
Having figured out what cannibal number 10 will do, conditional on number 9’s choice, let’s work backward to cannibal number 9. He knows cannibal 10 will eat him for sure if he eats and sleeps (because as we argued above, cannibal 10 can eat risk free). So cannibal 9 will certainly not eat (and, so, not fall asleep), whether given the choice or not.
Now move back a step to cannibal 8. He knows cannibal 9 will certainly not eat him (he can’t or he’ll be eaten, as argued above). So cannibal 8 can eat risk free if given the chance, just like cannibal 10. So far we have determined that cannibal 10 will eat if given the option, cannibal 9 will not, and cannibal 8 will. We can continue the backward induction making the same arguments as above (exercise left to reader). What we will find is that the strategies the cannibals will adopt if given the choice are:
- Don’t eat
- Eat
- Don’t eat
- Eat
- Don’t eat
- Eat
- Don’t eat
- Eat
- Don’t eat
- Eat
So every even cannibal can eat without risk and will do so if the opportunity presents itself. Every odd cannibal cannot eat or he will be eaten. Thus, number 1 will pass on Meathead, and Number 2 will eat Meathead. Though number 2 will fall asleep, number 3 can’t eat number 2 or he will himself be eaten because that is the preferred strategy of number 4. Since number 4 isn’t given a chance to eat he can’t do so. The same goes for the rest of the cannibals 5, 6, 7, …, 10. None are given a chance to eat, even though some would if they could. That is, the final outcome is:
- Doesn’t eat
- Eats Meathead
- Doesn’t eat
- Doesn’t eat
- Doesn’t eat
- Doesn’t eat
- Doesn’t eat
- Doesn’t eat
- Doesn’t eat
- Doesn’t eat
Therefore one cannibal eats (number 2) and no others do, and all cannibals live (for today). My next post about this game will show how it is a (very) simple model of nuclear proliferation (far too simple, really).
Hierarchy of Hungry Cannibals
Here’s another fun problem the solution to which illustrates fundamental concepts in game theory. (Other game theory problems and posts are listed under the game theory tag.) This problem is not hard to grasp but it takes a lot of words to set it up:
Ten cannibals live on an island. Their culture is very hierarchical and highly values rationality. To make the hierarchy explicit they’ve named themselves 1, 2, 3, …, 10. The order of the hierarchy corresponds to their number-name. Number 1 is dominant, 2 is second, and so on down to the lowly number 10.
A week ago they ate the last humans on the island, and they aren’t interested in the other flora or fauna available. The cannibals are now very hungry.
A moment ago a human named “Meathead” showed up, his sailboat having blown off course. Being the dominant cannibal, number 1 has the first option of eating Meathead. No other cannibal is permitted to touch Meathead before number 1 makes a decision as to whether or not he eats Meathead himself.
If number 1 eats Meathead he will get drowsy and fall asleep. (We assume cannibals always sleep after eating and at no other time. So these cannibals are very tired, as well as hungry.) If number 1 eats and sleeps, the number 2 cannibal has an opportunity to eat the number 1 cannibal while his defenses are down. But if number 2 eats number 1 he himself will fall asleep and he may be prey to number 3, and so on to number 10. If a cannibal eats the next dominant one he risks being eaten himself by his immediate subordinate during his slumber.
However, number 1 may choose not to eat Meathead. In that case, number 2 gets to make a decision about eating Meathead. If he eats Meathead he will sleep. If he sleeps he may get eaten by number 3 (and number 3 may then be eaten by number 4, and so forth). If number 2 passes on Meathead then number 3 gets a crack at Meathead…and so on.
Just to be explicit, cannibal number n may only eat Meathead (if given the option) or eat cannibal number n-1 (and only if n-1 is asleep). Cannibal n is not permitted to eat any other cannibal.
Number 1 gets the first move. What does he decide? What are the decisions of the other cannibals? Which cannibal, if any, eats Meathead? Which other cannibals, if any, eat? Which cannibals, if any, are eaten? Remember, these are rational cannibals. In fact, you can assume they all know the others are rational. (If you like, assume common knowledge of rationality.) That is, each will eat if (s)he can do so risk free. None will eat if it means certain death. I’ll post analysis on Monday and discuss an application in a subsequent post.
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