Utility, Games and Narratives

Abstract

This chapter provides a general overview of theories and tools to model decision-making. In particular, utility maximization and its application to collective decision-making, i.e. Game Theory, are discussed in detail. The most important exemplary games are presented, including the Prisoner’s Dilemma, the Game of Chicken and the Minority Game, also known as the El Farol Bar Problem. After discussing the paradoxes and pitfalls of utility maximization, an alternative approach is introduced, which is based on seeking coherence between competing interpretations. An assessment of the pros and cons of competing approaches to modelling decision-making concludes the chapter.

Appendices

Further Reading

Game Theory is a huge subject. Relevant handbooks are Aumann and Hart (1992, 1994, 2002) and, at a more introductory level, Rasmusen (2007). However, agent-based modellers should keep in mind that a substantial part of Game Theory has been developed around equilibrium states, which are generally not a main concern for them. Evolutionary games, thoroughly discussed in the above handbooks, are possibly closer to agent-based modelling. For other evolutionary mechanisms, see Chap. 21 in this volume (Chattoe-Brown and Edmonds 2012).

Neural networks are a huge subject as well. This field is currently split in two streams: on the one hand, research on neural networks as a model of cognitive processes in the brain and on the other hand, research on neural networks as an engineering tool for signal processing. A handbook oriented towards cognitive problems is Arbib (2002). Handbooks oriented towards engineering problems are Hu and Hwang (2002) and Graupe (2007). Specifically, unsupervised neural networks are often employed in pattern recognition. A comprehensive treatment of pattern recognition techniques is Ripley (1996).

All other tools and issues discussed in this chapter are in their infancy, so no generic reading can be mentioned. Interested scholars are better advised to start with the original papers mentioned in the bibliography, tracking developments on recent publications and working papers.

Arbib M. A. (Ed.). (2002). The handbook of brain theory and neural networks (2nd ed.). Cambridge, MA: MIT Press.

Aumann, R., & Hart, S. (Eds.). (1992). Handbook of game theory with economic applications (Vol. 1). Amsterdam: North-Holland.

Aumann, R., & Hart, S. (Eds.). (1994). Handbook of game theory with economic applications (Vol. 2). Amsterdam: North-Holland.

Aumann, R., & Hart, S. (Eds.). (2002). Handbook of game theory with economic applications (Vol. 3). Amsterdam: North-Holland.

Chattoe-Brown, E., & Edmonds, B. (2012). Evolutionary mechanisms. doi:https://doi.org/10.1007/978-3-319-66948-9_21.

Graupe, D. (2007). Principles of artificial neural networks. Singapore: World Scientific.

Hu, Y. H., & Hwang, J. N. (Eds.). (2002). Handbook of neural network signal processing. Boca Raton, FL: CRC Press.

Rasmusen E. (2007). Games and information: An introduction to game theory (4th ed.). Malden: Blackwell.

Ripley, B. D. (1996). Pattern recognition and neural networks. Cambridge: Cambridge University Press.

Weibull, J. W. (1997). Evolutionary game theory. Cambridge, MA: MIT Press.

Reasoned Bibliography

This chapter covered too many topics to be able to provide detailed references. Henceforth, a few basic publications will be listed that may be used by interested readers as a first orientation to each of the topics mentioned in this chapter.

1.1 Utility and Games

Utility maximization was pioneered by Frank Ramsay and Bruno De Finetti in the 1930s and subsequently refined by Leonard Savage in the 1950s. Savage still provides the most comprehensive explanation of this approach to uncertain reasoning.

Game Theory was initiated by John Von Neumann and Oskar Morgenstern in the 1940s. It subsequently developed into a huge research field within economics, with several specialized journals. Today, Game Theory is a field characterized by extreme mathematical sophistication and intricate conceptual constructions.

This chapter did not focus on the assumptions and methods of Game Theory but rather aimed at presenting the main prototypical games that have been devised hitherto. A classical treatise by Duncan Luce and Howard Raiffa may introduce the subject more easily than Von Neumann and Morgenstern did. Luce and Raiffa were first to present the Battle of the Sexes as well as the Prisoner’s Dilemma, which they ascribed to Albert Tucker anyway.

Readers interested in evolutionary games may rather read the treatises written by Jörgen Weibull and Herbert Gintis, respectively. The former is more specific on evolutionary games and also more technical than the second one.

Robert Axelrod is the main reference so far it regards simulations of the iterated Prisoner’s Dilemma with retaliation strategies. The idea that the iterated Prisoner’s Dilemma could yield cooperation simply relying on tags is due to Rick Riolo.

The stag hunt and the Game of Chicken are classical, somehow commonsensical games. The Game of Chicken has been turned into the Hawk-Dove game by Maynard Smith and George Price. The Hawk-Dove game is not terribly different from the war of attrition, conceived by Maynard Smith and improved by Timothy Bishop and Chris Cannings.

The Traveller’s Dilemma and the dollar auction are recent games invented by Kaushik Basu and Martin Shubik, respectively. Pure coordination games have been discovered by Thomas Schelling.

Axelrod, R. M. (1984). The evolution of cooperation. New York: Basic Books.

Basu, K. (1994). The traveller’s dilemma: Paradoxes of rationality in game theory. American Economic Review, 84, 391–395.

Bishop, D. T., Cannings, C., & Smith, J. M. (1978). The war of attrition with random rewards. Journal of Theoretical Biology, 74, 377–389.

Gintis, H. (2000). Game theory evolving. Princeton, NJ: Princeton University Press.

Luce, R. D., & Raiffa, H. (1957) Games and decision: Introduction and critical survey. New York: Wiley.

Riolo, R. L., Cohen, M. D., & Axelrod, R. M. (2001). Evolution of cooperation without reciprocity. Nature, 414, 441–443.

Savage, L. (1954). The foundations of statistics. New York: Wiley.

Schelling, T. C. (1960). The strategy of conflict. Cambridge, MA: Harvard University Press.

Shubik, M. (1971). The dollar auction game: A paradox in noncooperative behavior and escalation. Journal of Conflict Resolution, 15, 109–111.

Smith, J. M., & Price, G. R. (1973) The logic of animal conflict. Nature, 246, 15–18.

Weibull, J. (1997). Evolutionary game theory. Cambridge, MA: The MIT Press.

1.2 Influence Games

Ernst Ising introduced his model in the 1920s. Since then, a huge literature appeared.

The Ising model is taught in most Physics courses around the world, so a number of good introductions are available on the Internet. A printed introduction by Barry Cipra is mentioned here for completeness.

Schelling’s model of racial segregation was developed independently of the Ising model. However, it may be considered a variation of it.

The El Farol Bar Problem was conceived by Brian Arthur. Renamed The Minority Game and properly formalized, it was introduced to physicists by Damien Challet and Yi-Cheng Zhang.

A huge literature on the minority game has appeared on Physics journals. Good introductions have been proposed, among others, by Esteban Moro and Chi-Ho Yeung and Yi-Cheng Zhang.

Arthur, W. B. (1994). Inductive reasoning and bounded rationality. The American Economic Review, 84, 406–411.

Challet, D., & Zhang, Y. C. (1997). Emergence of cooperation and organization in an evolutionary game. Physica A, 246, 407–418.

Cipra, B. A. (1987). An introduction to the Ising model. The American Mathematical Monthly, 94, 937–959.

Moro, E. (2004). The minority game: An introductory guide. In E. Korutcheva & R. Cuerno (Eds.), Advances in condensed matter and statistical physics (pp. 263–286). New York: Nova Science Publishers. (Also available online as arXiv:cond-mat/0402651v1).

Schelling, T. C. (1971). Dynamic models of segregation. Journal of Mathematical Sociology, 1, 143–186.

Yeung, C. H., & Zhang, Y. C. (2009) Minority games. In R. A. Meyers (Ed.), Encyclopedia of complexity and systems science. Berlin, Springer. (Also available online as arXiv:0811.1479v2.).

1.3 Some Pitfalls of Utility Maximization

The idea that probabilities measured on samples of size zero are somewhat awkward is quite old and evidently linked to the frequentist view of probabilities. Daniel Ellsberg circulated this idea among economists, where in the meantime the subjectivist view of probability judgements had become dominant. Subadditive probabilities were conceived by Bernard Koopman in the 1940s and popularized among economists by David Schmeidler in the 1980s.

Maurice Allais submitted his decision problem to Leonard Savage, who did not behave according to his own axioms of rational choice. Since then, Savage presented utility maximization as a normative, not as a descriptive, theory. Prospect theory was advanced by Daniel Kahneman and Amos Tversky; it comes in a first version (1953) and a second version (1992).

The preference reversals highlighted by Paul Slovic have triggered a huge literature. A recent book edited by Sarah Lichtenstein and Paul Slovic gathers the most important contributions.

Kenneth Arrow originally devised his paradox as a logical difficulty to the idea of a Welfare State that would move the economy towards a socially desirable equilibrium. However, it may concern any form of group decision-making.

Michael Mandler is the main reference for a possible conciliation of Slovic’s and Arrow’s paradoxes with utility maximization, provided that preferences are incomplete.

Allais, M. (1953). Le comportement de l’homme rationnel devant le risque: critique des postulats et axiomes de l’école americaine. Econometrica, 21, 503–546.

Arrow, K. J. (1950) A difficulty in the concept of social welfare. The Journal of Political Economy, 58, 328–346. (Reprinted in The Collected Papers of Kenneth J. Arrow. Oxford: Blackwell, 1984).

Ellsberg, D. (1961). Risk, ambiguity, and the savage axioms. The Quarterly Journal of Economics, 75, 643–669.

Kahneman, D., & Tversky, A. (1953) Prospect theory: An analysis of decision under risk. Econometrica, 21, 503–546.

Koopman, B. O. (1940). The axioms and algebra of intuitive probability. The Annals of Mathematics, 41, 269–292.

Lichtenstein, S., & Slovic, P. (Eds.). (2006). The construction of preference. Cambridge: Cambridge University Press.

Mandler, M. (2005). Incomplete preferences and rational intransitivity of choice. Games and Economic Behavior, 50, 255–277.

Savage, L. (1967). Difficulties in the theory of personal probability. Philosophy of Science, 34, 305–310.

Schmeidler, D. (1989). Subjective probability and expected utility without additivity. Econometrica, 57, 571–587.

Tversky, A., & Kahneman, D. (1992). Advances in prospect theory: Cumulative representation of uncertainty. Journal of Risk and Uncertainty, 5, 297–323.

1.4 Decision-Making by Coherence-Seeking

In 1974 and 1976, James March was first to point to the fact that human beings distort their memories of the past in order to construct coherent stories that guide them into the future. This point has been also made by Karl Weick, who wrote a lengthy treatise on this subject a few decades later.

A number of psychological experiments confirm this idea. Interested readers may start with the works of Daryl Bem, Michael Conway and Michael Ross and a book edited by Ulric Neisser and Robyn Fivush.

However, the trouble with the idea of human beings reconstructing the past is that they are not willing to concede that they do so. Thus it is extremely difficult to find case studies. The one by David Lane and Robert Maxfield is possibly the only exception, though they were not allowed to publish all the materials they gathered during their investigation (Lane, personal communication).

A final remark on lack of communication in this stream of research. James March, Karl Weick and David Lane worked independently, possibly unaware of one another, focusing on the same issue but employing different expressions.

Bem, D. J. (1966). Inducing belief in false confessions. Journal of Personality and Social Psychology, 3, 707–710.

Bem, D. J. (1967). Self-perception: An alternative interpretation of cognitive dissonance phenomena. Psychological Review, 74, 183–200.

Cohen, M. D., & March, J. G. (1974) Leadership and ambiguity: The American College President. New York: McGraw-Hill.

Conway, M., & Ross, M. (1984). Getting what you want by revising what you had. Journal of Personality and Social Psychology, 47, 738–748.

Greenwald, A. (1980). The totalitarian ego: Fabrication and revision of personal history. American Psychologist, 35, 603–618.

Lane, D. A., & Maxfield, R. R. (2005) Ontological uncertainty and innovation. Journal of Evolutionary Economics, 15, 3–50.

March, J. G., & Olsen, J. P. (1976). Organizational learning and the ambiguity of the Past. In J. G. March & J. P. Olsen (Eds.), Ambiguity and choice in organizations. Bergen, Norway: Universitetsforlaget.

Neisser, U., & Fivush, R. (Eds.). (1994). The remembering self: Construction and accuracy in the self-narrative. Cambridge: Cambridge University Press.

Ross, M., & Newby-Clark, I. R. (1998). Constructing the past and future. Social Cognition, 16, 133–150.

Weick, K. E. (1979). The Social psychology of organizing. New York: Random House.

Weick, K. E. (1995). Sensemaking in organizations. Thousand Oaks, CA: Sage Publications.

1.5 Tools for Modelling Coherence-Seeking

This chapter did not deal with tools where categories pre-exist to the information that is being received, namely, supervised neural networks and Case-Based Decision Theory. Readers interested in supervised neural networks may start with the classical handbook by Rumelhart, McClelland and the PDP Research Group. Readers interested in Case-Based Decision Theory may refer to a series of articles by Itzhak Gilboa and David Schmeidler.

The earliest intuitions on the nature of mental categories date back to Ludwig Wittgenstein. A good explanation of the main features of mental categories, and why they are so different from our common idea of what a “category” is, is provided by George Lakoff in his Women, Fire, and Dangerous Things.

So far it regards unsupervised neural networks; the classic book by Teuvo Kohonen is still unrivalled for its combination of mathematical rigour and philosophical insight. Having been written at an early stage, it still keeps a strong link between artificial neural networks and the human brain.

Paul Thagard is the basic reference for constraint satisfaction networks. Constraint satisfaction networks appear in several contributions to the book The Construction of Preference, edited by Sarah Lichtenstein and Paul Slovic, mentioned in the section “Some Pitfalls of Utility Maximization”. Regarding the importance of focussing on two alternatives in order to arrive at a decision, see Fioretti (2012).

Evidence Theory started with a book by Glenn Shafer in 1976 and triggered a small but continuous flow of mathematical works since then. An article by Guido Fioretti explains it to social scientists, along with examples of applications to decision problems.

Fioretti, G. (2009). Evidence theory as a procedure for handling novel events. Metroeconomica, 60, 283–301.

Fioretti, G. (2012). Either, or: Exploration of an emerging decision theory. IEEE Transactions on Systems, Man and Cybernetics C, 42 (6): 854–864.

Gilboa, I., & Schmeidler, D. (1995). Case based decision theory. Quarterly Journal of Economics, 110, 605–639.

Kohonen, T. (1989). Self-organization and associative memory. Berlin: Springer.

Lakoff, G. (1987). Women, fire, and dangerous things. Chicago: University of Chicago Press.

Rumelhart, D. E., McClelland, J. L., & the PDP Research Group. (1986). Parallel distributed processing: explorations in the microstructure of cognition. Cambridge, MA: MIT Press.

Shafer, G. (1976). A mathematical theory of evidence. Princeton, NJ: Princeton University Press.

Thagard, P. (2000). Coherence in thought and action. Cambridge, MA: MIT Press.