On Robbers and Games (Column 197)

With God’s help

In column 20 I dealt a bit with the question of what rationality is, and from there I also touched on the meaning of game theory. We saw there that a person chooses a utility function for himself, and his rationality is assessed in light of it. Statements about people as though they were acting irrationally usually miss this point. Someone who acted in a way you do not understand has not necessarily made a mistake. He may be pursuing goals different from yours. Thus, for example, more than once I have heard criticism of a person who buys a lottery ticket for ₪10 when he has a one-in-a-million chance of winning ₪1,000, and the claim was that he is behaving irrationally because his expected return is negative. But there is no necessity whatsoever to assume that his utility function is tied to expected monetary return. He may be willing to pay for the hope of winning, or even for the very chance of winning itself (because ₪1,000 could help him a great deal, while an expenditure of ₪10 is insignificant for him). Some would argue that the very fact that he is willing to pay means that it is apparently worth it to him, and then it turns out that there is no such thing as an irrational person at all.

From this one can derive both the role and the limits of game theory. This theory assumes completely rational players, each of whom aims to maximize his personal profit. And again the criticism arises here that maximizing profit requires a definition of what, for that particular person, counts as profit, and there is no mathematical answer to that. Each person chooses his profit function (or utility function) according to his understanding, inclinations, and values. Only given a specific utility function can there be a mathematical answer to the question of how one ought to act (and even then, not always).

An irrational person is someone who posits a certain utility function and nevertheless does not act optimally in order to attain it. That is the more precise definition of irrational people. Game theory can expose such a person, provided that it accurately knows his utility function. If that person could have acted better in order to attain the utility he expects (or to attain greater utility), then he acted irrationally.

Yet there is still room for the following subtle argument, which undermines even this definition: it may be that the person decides that it is not worth investing time in thought or study in order to maximize his profit. He prefers to save the time and effort, even at the price of a chance of earning less or even losing. His utility function also includes the desire to save time and effort. If so, then even in such a case he is acting rationally. As I already mentioned, according to this line of thought it seems that every person always behaves rationally, since if he acted as he did, then apparently that is what he wanted and it was worth it to him.

Perhaps a more precise definition of irrationality would be the following: a person who behaves irrationally is someone who, when presented with the “correct” solution (that is, some alternative solution), would himself retract and admit that he was mistaken. Only such a case tells us with certainty that his behavior did not match his own utility function. But note: if he tells us that he was indeed mistaken with respect to the result, but he was right not to check because he preferred to save time and effort, this means that he is not really retracting his decision. In such a case he can still be regarded as having acted rationally.

This past weekend I took part in some conference, and one of the people there presented me with a nice riddle. I wanted to discuss it in order to illustrate a few more questions of this type.

and the plunderer shall divide the spoil: The riddle

Five robbers want to divide among themselves a loot of 100 gold coins. They decide on the following mechanism: the youngest (Robber A) proposes some division. If there is a majority in favor of his proposal, it is carried out and everyone goes on his way. If the proposal is not accepted (that is, there is no majority in its favor—tie or less), then they kill him and repeat the process: the youngest among those remaining (Robber B) proposes a way to divide all the coins into four parts, and so on. The question is what the outcome will be. That is: who will live and who will die (not in his allotted time), and how much money will each one have?

The assumption, of course, is that they are all completely rational players and each one’s sole aim is to maximize his profits and, of course, remain alive. Remaining alive takes precedence over monetary profit (not because life is important, Heaven forbid, but because if someone dies that means he has no money. His monetary profit is minimal.[1] I remind you that we are speaking here about rational agents who pursue monetary gain alone).

and the plunderer shall divide the spoil: The intuitive solution

The initial intuition says that the oldest robber (Robber E) will receive the greatest amount of money, since he has the power to oppose until he gets what he wants. Nor is he ever under threat of death from his comrades. For him there is only the question of monetary profit, so he is more relaxed. By contrast, the youngest (Robber A) has the least power, and so we would expect him to die or at best remain alive without money (and, as is well known, a poor man is considered as good as dead).

and the plunderer shall divide the spoil: Between mathematics and physics

A mathematician approaching such a problem looks for a solution for n robbers, where n is any number at all—even a complex number J—and then one easily solves the problem for n=5. Piece of cake. Except that first he must find a solution to the general problem, which is somewhat far from being a piece of cake. Here the physicist comes to his aid. Every beginning physicist knows that in order to solve a complicated problem, one should begin with the simplest problem of the same type (for example, when physicists want to understand the behavior of a donkey, they begin with a point-donkey), and then generalize.

What is the simplest problem of our type? A problem with two robbers. This problem is very easy to solve: any amount less than one hundred coins that the younger offers the elder will lead to his death (because that way the elder will take all the coins for himself). Our young fellow has no choice but to offer the elder all the coins from the outset, and thus at least remain alive. The assumption is that they do not murder for no reason, but only for the noble purpose of greedy profit.[2]

and the plunderer shall divide the spoil: Between generalization and recursion

All right, so two robbers divide the loot as (0,100). What is the solution to the problem of five robbers? The generalization is not immediate, so the physicist’s methodology does not really work. But fortunately, here the mathematician returns to the picture. He understands that there is no need to generalize from a simple case to a complex one, but simply to continue the recursive process—that is, to proceed and build the complex structure out of the simple case (which is a precise mathematical process and not speculative scientific generalization).

Let us now decide that the two robbers for whom we found the solution are in fact Robbers D and E. If so, we have described the end of the process; that is, we have understood what happens if the first three lose their lives and only the last two remain. Now the situation looks more promising, since one can begin to build a recursive process that assumes this solution and asks what happens when one more young robber joins the game.

and the plunderer shall divide the spoil: The full solution

Assuming that the three youngest have been killed and the two oldest remain, we already know that the solution is that the old one (Robber E) takes 100 and the younger one (Robber D) gets 0. Now Robber C joins the game. As stated, everyone already knows what will happen if Robber C dies (in the terminology of game theory, this is a “game with complete information”). As we saw, in that case the division will be (0,100). Therefore Robber C must entice one of them with a better offer so that he joins him and together his proposal will pass by majority. Only in that case will they not kill him. He is of course looking for the offer that leaves him alive and with maximum profit. He has no chance of convincing Robber E unless he gives him all 100 coins (since that would have to be more profit than E would get if he killed him and only two remained). But then Robber D can oppose, because he knows that in the next step he too gets 0. Beyond that, Robber C can simply bribe Robber D cheaply and give up on the more expensive Robber E. Therefore it is clearly more optimal for him to offer Robber D one coin and thus recruit him to his side, and to offer Robber E zero. Robber E will of course oppose the proposal, but he will find himself in the minority against the two younger robbers. Thus Robber C’s proposal wins a majority, and the outcome now is: (99,1,0). This is the solution for three rational players with complete information.

Let us continue the recursion. Now Robber B arrives, and he has to make an offer that will entice two more to join him. By now it is already clear to all of us that he must give up on the expensive Robber C and offer him 0. Robber C will of course oppose the proposal, but Robber B is about to recruit the other two to his side. How does he do that? He offers each of them one coin more than in the previous solution, so that it will be worthwhile for them to accept his proposal and stop here.[3] Therefore the solution at this stage is: (97,0,2,1).

Now we have arrived at our real problem, when Robber A, the youngest of the colleagues (the youngest of the group), joins the game. What should he offer his comrades in order to entice two of them to join him? Again, he will of course give up on the expensive Robber B and offer him zero. He needs two others on his side, and the two cheapest are Robber C, who is enticed by an offer of one coin, and Robber E, who is enticed by two. He can, of course, give up on Robber D as well (he too is relatively expensive). Thus the solution to our problem is: (97,0,1,0,2). His proposal has a majority, since Robbers C and E will vote with him.

Riddle for the reader

We went from the end to the beginning, but that is the way to solve a riddle like this. The dear reader can now try to formulate, in this manner, a general solution for n robbers. I guess that it is not trivial, but not very complicated. I also think that here the physicist’s perspective would be helpful in understanding in advance what will happen if we continue the recursion again and again, and in proposing the generalization directly for the general solution.

The meaning of counterintuitive solutions

Notice that we obtained a solution in which specifically the first proposer, who ostensibly has the least power, receives most of the coins. All the others receive small and similar sums. This is contrary to the initial intuition I described above. The meaning of this is that if you were to try a move like this in practice, if you were Robber A and proposed such a division to your older comrades, there is a good chance that they would be offended and immediately cut off your head. You would not even have a chance to explain to them how attractive this offer really is for them. Incidentally, they would realize this themselves when they reached a point at which, even without being a game theorist, one understands the constraints and their significance. This certainly happens when there are two robbers, but perhaps already when there are three everyone can understand where things are heading. Some of them will lose what they could have gained from the first proposal, but by then it will already be too late.

Now the game theorist will come, explain to them (in the World to Come, of course) the move described here, and ask them why they did not adopt the optimal course. They will likely regret what they did and understand that they should have studied and calculated, rather than immediately killing the proposer. In the end everyone lost (see below, the discussion of coalitions). If so, according to my suggestion above, this is indeed a case in which they behaved in a clearly irrational way. Can their behavior be justified by means of another utility function? Only with difficulty. For example, they estimated that not very much depended on the right decision (at most they would lose a few coins), and therefore it was not worth their while to invest time in studying game theory and arriving at the optimal solution. The estimate was of course mistaken, but it was their estimate, and therefore that is what determines the judgment of their behavior as rational. But this is strained, as I said, because now they would certainly admit that it would have been worthwhile for them to learn, since some of their lives were at stake and not just a few gold coins. It is more plausible that this was stupidity/laziness and not merely a different utility function.

Is game theory useful?

Here we come to a common criticism of game theory. The claim is that the theory assumes completely rational players who act so as to maximize profit. But in real life this is not the case. Human beings are not rational players, and therefore game theory is not a good tool for predicting their behavior and deciding how I ought to act in a game played against other human beings. If I assume that they are completely rational, I will probably be mistaken, and then I will act incorrectly.

This is a very common criticism in practical fields. For example, the stock market does not always react according to purely economic considerations. There is a great deal of psychology involved. Someone who makes investment decisions on purely economic grounds will not necessarily profit, and in any case will profit less than someone who also takes psychology into account. The same is true in confrontations between states (North Korea, Syria, Iran): we try to predict their behavior on the basis of rational considerations of interests, but many times we are disappointed. Sometimes this is because we did not understand their utility function (and then they are actually rational and we are not)[4], but sometimes it is because they (like us) truly do not act rationally. These cases raise the question whether game theory has scientific value—that is, whether it is useful for predicting the actions of people or various agents in the playing field, and thereby for making our own decisions correctly.

On coalition considerations, agreements, and the social contract

When dealing with complex arenas (states, large companies), game theory is more applicable because such bodies tend to consult and act rationally. But in these cases one must also take possible coalitions into account, since they can change the solutions significantly. Think, for example, of the situation of our five robbers if coalitions were allowed. Player E knows that in the end, in the ordinary solution, he will gain only 2. So he offers players B and D, who stand to gain 0, to form a coalition, threaten the other proposers with death, demand all the money, and then divide the profit equally among themselves. Alternatively, and even more simply, the last four unite to oppose the first one’s proposal, kill him, and then divide the money equally. Everyone gains significantly relative to the “optimal” solution (within which they get 0, 1, or 2). This of course changes the picture entirely, and in life there is always the possibility of coalitions, and each player has to take those possibilities into account as well (in game theory, the search for solutions while taking coalitions into account is a field in its own right).

The problem, of course, is who can guarantee us that the partners to the agreement will indeed abide by it. After all, the rules of the game remain as they were. The three or four members of the coalition can agree, with a handshake, on the policy I described, and after killing the first one betray their partners and make another coalition (for example, the last three against Robber B), or vote according to naked interest and without coalition considerations (each according to his own calculations of advantage). None of them has any guarantee that the others will honor the agreement, so long as there is a possibility of doing something even more worthwhile. Think about it: if they are willing to murder in order to earn one more coin, there is no reason they would not violate an agreement for precisely the same thing. So one cannot really rely on coalitions that are not themselves built on advantage. Try putting your faith in robbers…

This reminds me of explanations that obligate moral behavior by virtue of a (fictive) agreement, what is called “the social contract.” The question is: what obligates me to abide by that contract? The agreement itself? If the assumption is that without the agreement I would murder, then how can one trust me to keep the agreement once I have signed it?! Violating an agreement is less serious than murder, no? Ostensibly, so long as I do not have a clear interest in keeping the agreement, there is reason to fear that I will not keep it. But if I do have such an interest—what need is there for an agreement? I will behave that way even without an agreement because of utility considerations. It would seem, then, that agreements have value only in that they bring the worthwhile policy to the attention of the parties and make sure that everyone understands that it is worthwhile for them. The agreement has no added moral value. On this view, the agreement merely conveys information to the parties and does not really obligate them.

On international law

If the agreement is made under the umbrella of a shared normative system (such as state law or international law), that is something else. There, there is someone who will enforce the agreements, and therefore the agreement has added value beyond the actual terms agreed upon. Incidentally, this itself will in many cases be good for all sides, since were it not for the fear that the other side will violate a worthwhile agreement, it would pay all of us to keep the agreement. We would all profit more. See on this in column 122, in the discussion of the prisoner’s dilemma and the categorical imperative.[5] But the theory of the social contract, in which it is presented as the most fundamental basis for morality, while there is no more general effective normative system that would enforce it, seems like ethical-logical folly.

Therefore international agreements, such as a peace agreement between states, or an agreement between us and the Palestinians, must also be based on interests and advantage and not only on fairness. Anyone willing to murder me for one gain or another will also be willing to violate an agreement for that very same thing.

As I already mentioned, people and states are not necessarily completely rational agents, and it turns out that in practice agreements do have force. Even without an interest, there is some weight to the fact that there is an agreement between us. Just think about a situation in which we or the Palestinians violate an agreement we signed. Immediately various claims are raised about the unfairness of doing so. But before the agreement we were literally killing each other, and that was fine. What we are doing now is only violating an agreement and not murder; by all accounts that is an act far less severe than murder, and yet claims of immorality and unfairness arise (“you can’t trust them”).

The status of agreements apparently derives from the international framework that ensures that one who violates agreements will pay a price for it. There is a normative system here underlying the agreement, and only because of it does the agreement have meaning. That gives you one consideration in favor of the existence of international law. It gives added value beyond the equilibrium that would emerge solely from the web of mutual interests.[6]

Is game theory a branch of psychology or of mathematics?

Experts in various fields tend to present their field as highly applicable. Thus one hears mathematicians explaining how their field is very useful in physics research, which in my experience is often untrue (physics uses branches of mathematics, but usually what mathematicians do is esoteric and not useful from the physicists’ point of view). Experts in game theory also tend to explain that their field is useful and applicable in many real-world contexts. But in my experience, the practical utility of game theory is really quite marginal.

In most cases, game theory arrives at the correct solution (in the descriptive sense—that is, it correctly predicts what will happen) only where the problem is simple enough, because then ordinary people as well understand the solution and act that way in practice, and then of course one does not need it. Moreover, even in the simple cases people will not always act in accordance with it (because they have additional biases and psychological motives). In complicated cases, game theory is almost not useful at all, either because the problem is not mathematically solvable, and even more because in reality people will not act according to its instructions. Exceptions to this are large bodies, such as governments or commercial companies. Such bodies can more readily afford to consult an expert, because they have enough resources and access to experts, and beyond that, in their case a great deal is at stake (and therefore it is worthwhile for them to consult).

The picture I have presented up to this point assumes that game theory is a branch of psychology, since it expects it to describe and predict correctly what people do. As stated, in this respect it rather fails. But one can see it as a branch of mathematics, whose concern is solving pure mathematical problems, regardless of the question of what people would actually do in such situations. If game theory has no scientific-descriptive pretension—that is, no pretension to describe what happens in the actual world—then there is no problem in the fact that it does not succeed in doing so. In that sense it belongs not to psychology but to mathematics.

Advisory roles

Even if this is mathematics, however, game theory can still have a practical role. When someone comes for advice about some step he should take, game theory can tell him what may happen, or what the optimal step is that he can take in order to minimize damage or maximize profit. And again, even this is only with limited reliability, since usually the practical result will also depend on the other players, and on the question whether they will act as expected of them (rationally). Thus, for example, game theorists created an algorithm called “deferred acceptance” whose original purpose was to create stable matches, and they applied it to optimal matching between intern doctors or lawyers and internship placements, or candidates for an M.A. in psychology and academic departments.[7] Such a situation is one in which the decision of all the parties is made by the mathematical algorithm (which takes the preferences and qualifications as input), and then of course the problems I described do not arise. This is a natural and useful role for game theory. It turns a game that includes conflict into a coalition game in which everyone together tries to reach a common goal (which of course has to be defined). One must understand that not every intern will get the place he desires, and therefore the individual player pays a price for accepting the verdict of the calculations of the coalition as a whole. This yields a general solution for everyone in a way that has some advantage (it will not be bad), such as stability (a local advantage, in which it is not worthwhile for any player to move from his position) in the example of matching, but it is certainly not the optimal solution for each player individually.

Example: the meaning of the algorithm for the logical hermeneutic rules

When we developed the algorithm of the logical hermeneutic rules (see the first book in the Talmudic Logic series), a very confusing question arose.[8] Our algorithm starts from a table of data and infers from it the rule in the missing cell. For example, a table of an a fortiori inference looks like this:

Mechanism/rule	Marriage	Betrothal
Money	0	1
Canopy	1	?

We have a collection of known rules (in this case three), and we want to derive from them an unknown rule (whether a canopy effects betrothal). The a fortiori inference instructs us to fill the unknown cell with 1. We showed there that there is a single-valued algorithm that can fill any such cell in any table of any size. We showed that this algorithm matches our way of thinking and that of the Sages in all the cases that were tested (up to tables of 8 by 5, if I remember correctly, in all filling patterns).

And then the question arose: how is this different from deductive inference? Are analogies (such as binyan av) or inferences like a fortiori reasoning and argument from a common denominator, all of which are not logically necessary inferences, in fact necessary after all? After all, we have a closed algorithm that gives us an answer that is necessarily correct for every such puzzle. Ostensibly, this is full-fledged deduction, meaning that the conclusion follows necessarily and univocally from the premises.

The penny dropped for us in a lecture we gave at Tel Aviv University. In the discussion that followed it, someone mentioned the question I presented here: is the goal of game theory to describe human behavior or to arrive at the correct (optimal) result? This is in fact a reflection of the dilemma I described here: whether game theory is a branch of mathematics or of psychology. From here it was a short way to understand that even if our algorithm is deductive—that is, it has a compelled result for every data set and every question—that result is not necessarily the correct result, but rather the result to which a person who thinks like us would arrive. This algorithm is not a way to arrive at the correct result, but at the result to which a creature that thinks like human beings, but without local failures and biases that can mislead a flesh-and-blood person, would arrive.

The conclusion was that the result of our algorithm can be mistaken, and in that sense there is no deduction here. What the algorithm ensures is that the result is the result to which a person who thinks in the way we think ought to arrive, without errors of implementation. He generalizes and makes analogies as we do. Of course, generalizations and analogies are not necessary tools, and we make many mistakes, but this is the best we have. The algorithm ensures that we do indeed arrive at that best result. Whether that result is correct or not is already a question of how reliable our tools of thought are (for the fit between them and the world is neither certain nor sweeping).

A closing riddle

To conclude, I thought of a riddle that I will leave to the readers. Our five robbers are standing around the hundred coins. They need to decide on the order of the proposals. In the previous riddle it was from youngest to oldest, but now that is reopened. Each one is supposed to make a proposal regarding the order of proposals, and the proposal that wins a majority is accepted. He can propose a plan in which he himself is chosen to be the first proposer (of course everyone wants to be first, because as we saw, he gains 97 coins), and in return for that he is willing to distribute some sum to his colleagues so that they vote for him. If no proposal is accepted, they return to the mechanism I described above (from youngest to oldest). Assuming that all are rational players whose goal is to maximize profits, what will the order of the proposers be, and what will the outcome of the game be?

Notice that in principle they are all in the same position, and therefore we would expect a symmetrical solution (there would be no preferred one). But the rule that if no other proposal is accepted they return to the original proposal (proposals according to age order) breaks that symmetry. The default first proposer has no interest in accepting a different proposal even if he is placed first in it, because even if no proposal is accepted he returns to the original proposal, in which he too gains 97 coins.

I have not thought about this problem, but on its face it seems to me interesting and not simple. Good luck.

[1] And there is a kind of corroboration for this in the Talmudic passage in tractate Yoma that explains why saving a life overrides the Sabbath according to the reasoning of Rabbi Shimon ben Menasya: Profane one Sabbath for his sake so that he may keep many Sabbaths (Desecrate one Sabbath for him so that he may keep many Sabbaths). The meaning of the statement is that the value of life does not really override the Sabbath; rather, if he is saved he can keep many Sabbaths, and that is what overrides this particular Sabbath. A riddle for the reader: why is this not a necessary explanation of Rabbi Shimon ben Menasya’s reasoning?

[2] And there is likewise a kind of corroboration for this from the well-known story about Rabbi Chaim of Brisk, who once invited to him the head of the gang of Jewish robbers in Brisk. When he arrived, Rabbi Chaim asked him whether they break down a door when they need to. The man answered: “Of course, otherwise how would we get in?!” Rabbi Chaim continued and asked what they do if the homeowner resists. The man answered: “We smash his skull, of course.” And is there no limit on what they take in their loot? The robber answers: “Of course not. We came to make money, not to do charity and kindness.” Then Rabbi Chaim asked him: if you are hungry, do you open the refrigerator and take food? The robber was utterly shocked and answered: “Heaven forbid, Rabbi, the food there may not be kosher.” To Rabbi Chaim’s astonishment—how they permit themselves to break in, steal, impoverish a person, and even kill him when necessary, but do not eat non-kosher food—the man replied: “All those things are livelihood, but what permission do we have to eat non-kosher food?! What shall we answer on the Day of Judgment?!”

[3] One could also define the possibility of abstaining from the vote when the two outcomes are equivalent for someone. In such a case, it would have been enough for Robber B to offer one of them an equal amount so that he would abstain.

[4] Here is another example of irrational behavior that is hard to explain by means of a different utility function.

[5] See there the reference to the television program Golden Balls and the discussion of coalition considerations.

[6] As for the folly of the moral claim about not honoring agreements, this explanation will not help. The international framework is an interest-based reason to keep an agreement, but not a moral claim. True, in column 122 I linked morality (Kant’s) to consequential considerations of interest (the prisoner’s dilemma); see there.

[7] Thanks to Beni Shlomo for this example.

[8] See a discussion of this at the end of the aforementioned book.