Friday; Septernb+ir 25, 1070 THE MICHIGAN DAILY Page Five' Friday7 September 25, 1 9 7 0 THE MICHIGAN DAILY Page Five Prisoner's Dilemma: An answer in game theory? A FEATURE REVIEW BY STAN KAPLOWITZ Anatol Rapoport and Albert M. Chammah, PRISONER'S DILEMMA, University of Michigan Press, $2.95 paper- back. In the late 1950's, a group of social scientists, convinced that war was not so much a clash between good and evil as a con- sequence of the existing inter- national system, formed a peace research movement. This niove- ment was based on the supposi- tion that if the mechanisms which cause conflict were bet- ter understood, the international system might be changed so as to make war less likely. Prisoner's Dilemma by Rapo- port ad Chammah (which was first published in 1965 and whose appearance in paperback this year is the occasion for this review) is one of the best and most important works to be stimulated by this movement. It analyzes conflict and ,coopera- tion through the scientific method of controlled experi- ment, logical rigor and mathe- matical precision. One clearly cannot do experiments on the international system. Instead, experiments were done on pairs e of subjects playing a game which represents the kind of choices relevant to conflict and cooperation in larger social sys- tems. The name Prisoner's Dilemma comes from the following anec- dote. Two prisoners held in- communicado are charged with the same crime. If neither con- fesses, there is not enough evi- dence for conviction and both are set free. If both confess they are given the same moder- ate sentence. If only one coh- fesses, he is rewarded for hav- ing turned state's evidence and is not only set free but given an additional reward. In this case, the prisoner who holds out is convicted and is given an es- pecially severe sentence. At this point I should bring in the concept of game theory. Contrary to the connotations of the name, game theory is not -especially concerned with sports or parlor games. It is a theory which looks at the underlying structure of a decision problem and attempts to find a rational procedure for making a deci- sion' In game theoretic terms we can represent the prisoners di- lemma with the following game matrix, where the prisoners will now be called players. ~~ To explain the matrix above, erich player has a chioice be- tween C co-operation ie. not 4confessing) and D (defection ,,{N e confessing)r J.Y Tha meansy. ~h there areifour possible out comes, each one represented by a box in the matrix. The sym- w hic th e p lay ers rece v fo r a particular outcome. The letter Sbefore the comma in figure 1 is the payoff to player 1 and the letter after the comma is the pa yoff toS playe 2 . Ea.lte has crti numerical valu more desirale the out omet th player e eivingd cibe the C cotio e cried on ther aedoerreouirelefot- lowing relationship among the payoffs: T is greater than R is greater than P is greater than S. A simple matrix fulfilling the necessary conditions is as fol- ' ~ ~ . ~~ .ows; t -. * * . If we look at figure 2, then we notice that regardless of what player 2 does, player 1 will, receive a higher payoff by choosing C. In the terminology of game theory, D is the domi- nating strategy for player 1. Since the situation is symmetric the same clearly holds true for player 2. But if both players are concerned only with their own self interest and rationally make the choice, D, which best serves that self-interest, they will both receive -1. If on the other hand, they had both made the "irrational choice, C, they would have each received ±1. It is this then, which is the core of the dilemma. Therefore, unlike Adam Smith's vision of a society in which men who rationally pur- sue their individual interests are "led by an invisible hand" to act in the best interest of all, in Prisoner's Dilemma the in- visibleshand leads a collection of rationally self-interested peo- ple to an unhappy fate. If the prisoner's dilemma were merely the figment of the imagination of some science fic- tion writer, R a p o p o r t and Chammah would not have done the studies reported in this book. The reason the book Pris- oneris Dilemma is worth writing is precisily because so many problems in this work have ele- ments of an actual prisoner's dilemma. Take the following examples: 1) Two countries must each de- cide whether or not to arm. Both nations would be better off spending their resources on im- proved living conditions than on armaments. But if one arms and the other doesn't, the arm- ed nation can subjugate the un- armed. In this case the armed nation would receive its best possible outcome and the un- armed nation would receive its worst possible outcome. If both are rational and choose the dominating strategy, both arm and both loose. 2) The problem of whether or not to pollute is in many ways a commonplace prisoner's dilemma. To simplify the matter we can regard each industrial firm as having the choice of polluting or not pol- luting. It is in the individual interest of each firm to pollute because the firm saves money by polluting, whereas the social cost of the pollution is spread across a large section of the population. Therefore, those re- sponsible for any decision to pollute suffer only a small frac- tion of the total damage caused by that decision. But if every- one pollutes to his heart's con- tent, an area eventually be- comes unliveable and everyone is worse of f. The experimental study de- scribed in this book had pairs of subjects presented with a prisoner's dilemma matrix and had them play the game for a few hundred trials. Each player had to make his choice without consulting his partner and with- out k n o w i n g his partner's choice. When prisoner's dilemma is played for many successive trials (not just once which we have assumed in past discus- sion) it might seem that D is not necessarily the "rational" choice. For if player 1 always plays D he can expect that after a while player 2 will do likewise in which case player 1 will do worse than if he had played ac- cording to the principle "if you co-operate with me I'll co-oper- ate with you." But as it turns out, if the number of trials is known and finite with two players who are r a t i o n a 11 y and strategically sophisticated and assuming the other to be likewise, they will always play D. To see this, let us assume that the game will be played for 100 trials. Since both players are rational, each knows that the only reason to ever play C is to avoid provoking his partner to retaliate by choosing D subsequently. Player 1 might reason as follows. "Clearly, I should play D on trial 100, since player 2 will not be able to retaliate after that trial. But player 2 also knows this so he also will play D on trial 100. Therefore, no matter what I do on trial 99, the out- come on trial 100 will be DD. In this case I should choose D on trial 99 and since player 2 also realizes this, he too will choose D. Since the outcome of trial 99 will definitely be DD then I should choose D on trial 9 and so should he." This rea- soning ran be extended all the way back to trial 1 so that each player will always choose D. "Confronted with this para- dox," say the authors, game theoreticians have no answer. Ordinary mortals, h o w e v e r, when playing prisoner's dilem- ma many times in succession hardly ever play DD one hun- dren per cent of the time. To be sure long stretches of DD d6 occur but so do long stretch- es of CC choices, Evidently," t h ey add,. "the run-of-the- mill players are not strategical- ly sophisticated enough to re- alize that strategy DD is the only rationally defensible strate- gy. and this intellectual short- coming saves them from losing." Not all results can be regard- ed as simply common sense. To give an interesting and some- what significant example of a surprising result, in most con- ditions, subjects had the matrix in front of them as they played. In the concealed matrix condi- tion the matrix was not display- ed, though as in other condi- tions they were told the outcome and payoffs from each trial. The authors had expected a tingent propensities indicate someth.ing which we've already seen evidence of - that fear of being the lone co-operator is a more important cause of the DD lock-in than is the hope of being the lone defector. More- over the severe games (those with the least co-operation) are severe because subjects tend to be more afraid to take the risk necessary to break out of the DD lock-in. While the first section of the book is devoted to discussing the game and analyzing the can be briefly summarized. 1) To create a certain model of a process is not to say the pro- cess works like this but rather to say, "Let us assume it works like this, find out the conse- quences of our assumptions and see how well they agree with + the data. The degree to which the predictions of t h e model agree with some of the observ- ed features of our data will give us an idea as to what extent and in what ways the model has + or has not captured what is go- + ing on in the process. 2) All of the models present- ed are dealing with the proba- bility of a co-operative response ' from an average subject in a particular population. As such they attempt to predict regular- ities in gross data and probabil- ities of individual behavior but they very carefully avoid guar- anteeing a particular sequence + of choices for anyone. The kind of variation permitted by the laws of probability maintains the notion of causality in peo- ple's behavior while at the same time allowing for the kind of unpredictability and freedom that many people feel is such an important part of human na- ture. In the next section of the book the models are tested and while none seems to fully capture the process, the time the authors spent developing and testing the models and the time a reader would spend reading them is by no means wasted. At the time this book was written these mo- dels were fairly new and had been subjected to few tests. Two of them, the stochastic learning model and the markov model seemed, from other studies, to be especially promising. More- over the authors were able to learn some substantive informa- tion from the models. (eg: the main reason the markov model fails is that there is a greater tendency for a CC lock in then the model predicts). Finally, the state of mathematical model building has not, to my knowl- edge, progressed greatly beyond these kinds of models. Hence those who want to learn how models can be built and tested can learn much about the state of the field from this book. While these specific kinds of models are still important for . mathematical model building, I would agree with those who say that they have now been shown to be " sufficiently inadequate. Mathematical social scientists and social science-ish mathe- maticians should , be trying to construct new models. In par- ticular, they should try to in- vent new kinds of mathematics more appropriate to the espec- ially complex phenonmena of human behavior. It seems that nowadays there is such a proliferation of pub- lished material that it is im- possible to read more than a small fraction of it. Prisoner's Dilemma, however is a b o o k which many people should in- clude in that s m a 11 fraction which they read. One group of people for whom this book is worthwhile is those (whether in the social sciences or not) who are interested in the basis of conflict and the promise of co- operation. Even with minimal formal mathematical training, such a person can understand most of the book (except the chapters devoted to models) if he can handle numbers a n d read tables. The other major group for whom this book is of value is people in the social sciences and in mathematics who are inter- ested in the potentialities and limitations of mathematics for studying human behavior. This is by no means the only book in which mathematics is sub- stantially used by social scien- tists, but it is one of the best. It pursues a particular problem in unusual depth, applying mathematics to many different aspects. For example, it looks at the behavior of the process over a long time period as well as the effects of the immediately pre- ceding trial. More importantly, the book is extremely well- written (as is the case with oth- er works which Anatol Rapoport has authored and co-authored), Its style is rigorous and at the same time both clear and inter- esting. Among the important features which increase the readability of the book are the heuristic and intuitive interpre- tations which often accompany mathematical derivations and presentation of data. Events of the p a s t several years have made many social scientists less enthusiastic about peace research than w a s the case in the early 1960's. This change stems largely from the fact t h a t the peace research movement was based on undue optimism about the willingness of the U.S. government to ap- ply the findings of peace re- search towards building a more peaceful world. For this reason, many people interested in peace research a r e trying to orient their research towards findings which can be of use to institu- tions which are seriously com- mitted to peace. Other peace re- searcherss having become con- vinced that American institu- tions stand in the way of peace have switched to research aim- ed at understanding and ulti- mately changing powerful in- stitutions in American society. Even though Prisoner's Di- lemma is part of the earlier, more naive, stage of peace re- search, and is in t h a t sense somewhat dated, it is not use- less or irrelevant. The kind of research movement that it was part of was necessary to show us that we were naive and per- haps, if we can change the in- stitutions of this country, these findings may yet be applied to- ward creating a less violent world. To sum up - it ain't quick reading but its a damn good book - interesting, informative, relevant and thought provoking. b 0 0 k s k +;*, 'x ! f s c. _.d , RESERVATIONS FOR ROSH HASHANAH MEALS WILL NOT BE ACCEPTED AFTER MONDAY THE 28th RESERVE NOW ,.. . greater degree of co-operation in the concealed matrix condi- tion on the grounds that sub- jects who saw the matrix would be more likely to be aware that D is the dominating strategy. In fact the opposite was the case, suggesting that the pre- dominant function of the ma- trix is to make obvious the pos- sibilities for m u t u a l gain through co-operation. This sug- gests that fear of being shafted (receiving S) is a more impor- tant motivation for people to play D than is desire to profit at the expense of ones partner (by receiving T). Aside from looking at the gross data, which gives overall rates of co-operation for differ- ent conditions, the authors show that much can be learned by dissecting the data in various ways. Not surprisingly, a strong positive correlation (greater than .9) is found between the p e r c e n t a g e of co-operative choices of the two partners. These correlations clearly dem- onstrate that the process is not simply determined by the char- acteristics of the individual players but that there are strong interaction effects. One way of further under- standing what is happening is to look at the contingent pro- pensities. These are a set of probabilities, each one being a probability of a C response after a particular outcome on the previous trial. Examining these contingent propensities can be quite intriguing and helps us to understand the cause of the DD lock-in, in which both play- ers continuously lose. The con-