Explaining fairness in complex environments

Politics, Philosophy & Economics

Volumn 7, Issue 1, 2008, Pages 81-97

  Zollman, Kevin J S ^a  

^a UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

evolutionary game theory; fairness; Nash bargaining game; ultimatum game

Indexed keywords

EID: 44349108414     PISSN: 1470594X     EISSN: None     Source Type: Journal    
DOI: 10.1177/1470594X07081299     Document Type: Article

References (26)

1. For discussion of the relationship between justice and the Nash bargaining game, see Jason Alexander and Brian Skyrms, ‘Bargaining with Neighbors: Is Justice Contagious?’ Journal of Philosophy 96 (1999): 588-98; Ken Binmore, Game Theory and the Social Contract Volume 2: Just Playing (Cambridge, MA: MIT Press, 1998); Ken Binmore, Natural Justice (Oxford: Oxford University Press, 2005).
2. For evolutionary explanations of cooperative behavior in the Nash bargaining game, see Jason McKenzie Alexander, ‘Evolutionary Explanations of Distributive Justice’, Philosophy of Science 67 (2000): 490-516; Brian Skyrms, Evolution of the Social Contract (Cambridge: Cambridge University Press, 1996); Brian Skyrms, ‘Signals, Evolution and the Explanatory Power of Transient Information’, Philosophy of Science 69 (2002): 407-28; H. Peyton Young, Individual Strategy and Social Structure: An Evolutionary Theory of Institutions (Princeton, NJ: Princeton University Press, 1998). For the ultimatum game, see John Gale, Kenneth G. Binmore and Larry Samuelson, ‘Learning to be Imperfect: The Ultimatum Game’, Games and Economic Behavior 8 (1995): 56-90; Werner Güth and Menachem Yaari, ‘An Evolutionary Approach to Explain Reciprocal Behavior in a Simple Strategic Game’, in Explaining Process and Change: Approaches to Evolutionary Economics, edited by U. Witt (Ann Arbor: University of Michigan, 1995), pp. 23-34; William Harms, ‘Evolution and Ultimatum Bargaining’, Theory and Decision 42 (1997): 146-75; Steffen Huck and Jörg Oechssler, ‘The Indirect Evolutionary Approach to Explaining Fair Allocations’, Games and Economic Behavior 28 (1999): 13-24.
3. While there have been several authors who have provided models that predict fair behavior in the ultimatum game, many of these models rely on unrealistic assumptions. They will be discussed in more detail in Section 2.
4. There is also a Nash equilibrium where both players demand the entire good. Recall that in order to be a Nash equilibrium it must be the case that neither player would do better by unilaterally changing his strategy. Since an individual player reducing his demand without cooperation of the other would not increase his payoff, this combination of strategies constitutes a Nash equilibrium.
5. Rudy V. Nydegger and Houston G. Owen, ‘Two-Person Bargaining, an Experimental Test of the Nash Axioms’, International Journal of Game Theory 3 (1974): 239-50; Al Roth and Michael Malouf, ‘Game Theoretic Models and the Role of Information in Bargaining’, Psychological Review 86 (1979): 574-94; John van Huyck, Raymond Battalio, Somesh Mathur, P. van Huyck and Andreas Ortmann, ‘On the Origin of Convention: Evidence from Symmetric Bargaining Games’, International Journal of Game Theory 34 (1995): 187-212; Menachem Yaari and Maya Bar-Hillel, ‘On Dividing Justly’, Social Choice and Welfare 1 (1981): 1-24.
6. Informally, a state is evolutionarily stable if it cannot be invaded by a mutant strategy. For a formal definition, see John Maynard Smith, Evolution and the Theory of Games (Cambridge: Cambridge University Press, 1982).
7. Skyrms, Evolution of the Social Contract; Skyrms, ‘Signals, Evolution and the Explanatory Power of Transient Information’.
8. See Alexander, ‘Evolutionary Explanations of Distributive Justice’; Skyrms, ‘Signals, Evolution and the Explanatory Power of Transient Information’; Young, Individual Strategy and Social Structure. However, adding realistic assumptions does not guarantee an increase. One study found that under one realistic modification, the basin of attraction of fair behavior shrinks. See Justin D'Arms, Robert Batterman and Krzyzstof Gorny, ‘Game Theoretic Explanations and the Evolution of Justice’, Philosophy of Science 65 (1998): 76-102.
9. There is also another Nash equilibrium where the first player proposes to keep the entire good and the second rejects all proposals which give her less than the entire good. The second would still receive nothing by accepting and the first cannot propose any split that the second would accept (and would result in her receiving more than zero).
10. See Reinhard Selten, ‘Reexamination of the Perfectness Concept of Equilibrium in Extensive Games’, International Journal of Game Theory 4 (1975): 25-55.
11. Hessel Oosterbeek, Randolph Sloof and Gijs van de Kuilen, ‘Cultural Differences in Ultimatum Game Experiments: Evidence from a Meta-Analysis’, Experimental Economics 7 (2004): 171-88.
12. See Joseph Henrich, Robert Boyd, Samuel Bowles, Colin Camerer, Ernst Fehr and Herbert Gintis, Foundations of Human Sociality: Economic Experiments and Ethnographic Evidence from Fifteen Small-Scale Societies (Oxford: Oxford University Press, 2004); Joseph Henrich, Robert Boyd, Samuel Bowles, Colin Camerer, Ernst Fehr, Herbert Gintis, Richard McElreath, Michael Alvard, Abigail Barr, Jean Ensminger, Kim Hill, Francisco Gil-White, Michael Gurven, Frank Marlowe, John Q. Patton, Natalie Smith and David Tracer, ‘“Economic Man” in Cross-cultural Perspective: Behavioral Experiments in 15 Small-Scale Societies’, Behavioral and Brain Sciences 28 (2005): 795-855.
13. This assumption has already limited our ability to explain all the data on the ultimatum game. Henrich et al. observe that some hyper-fair offers (that is, larger than ½) are rejected in some societies. Since this is a relatively rare behavior that Henrich et al. suggest can be explained by a peculiar feature of a few cultures, I suspect its explanation resides outside of an explanation for more robust irrational behavior. See Henrich et al., Foundations of Human Sociality; Henrich et al., ‘“Economic Man” in Cross-cultural Perspective’. As an interesting aside, Oosterbeek et al. find that when the game is described this way in experiments the rejection rate is significantly higher. See Oosterbeek et al., ‘Cultural Differences in Ultimatum Game Experiments’.
14. Gale et al., ‘Learning to be Imperfect’.
15. Güth and Yaari, ‘An Evolutionary Approach to Explain Reciprocal Behavior in a Simple Strategic Game’.
16. Huck and Oechssler, ‘The Indirect Evolutionary Approach to Explaining Fair Allocations’.
17. Harms, ‘Evolution and Ultimatum Bargaining’.
18. Skyrms, Evolution of the Social Contract, p. 28.
19. The strategy of employing several different games in order to explain the emergence of norms governing behavior has been recently pursued by Bednar and Page. They study several common 2x2 games allowing players to adopt similar strategies across games. See Jenna Bednar and Scott Page, ‘Can Game(s) Theory Explain Culture? The Emergence of Cultural Behavior within Multiple Games’, Santa Fe Institute Working Paper 2004-12-039 (Santa Fe, NM: Santa Fe Institute, 2004).
20. Gale et al., ‘Learning to be Imperfect’, p. 59.
21. Miguel Costa-Gomes, Vincent P. Crawford and Bruno Broseta, ‘Cognition and Behavior in Normal-Form Games: An Experimental Study’, Econometrica 69 (2001): 1193-235.
22. This observation was first made by Roberto A. Weber, Colin F. Camerer and Marc Knez, ‘Timing and Virtual Observability in Ultimatum Bargaining and “Weak Link” Coordination Games’, Experimental Economics 7 (2004): 25-48.
23. I am indebted to Brian Skyrms for this suggestion.
24. We might have combined the game by forcing a player to adopt one value which functions as both a demand and as a rejection threshold. Our method seems more intuitive since a player may be making a demand unaware if the other is simultaneously making a demand or not, but not unaware that she is in a position of accepting and rejecting. This also allows us to analyze a situation in which the norm can maximally distinguish between the two games without adopting completely different strategies for each. In fact, this method is confirmed by the analysis of experimental subjects’ strategies in a similar game studied by Sven Fischer, Werner Güth, Wieland Müller and Andreas Stiehler, ‘From Ultimatum to Nash Bargaining: Theory and Experimental Evidence’, unpublished manuscript, 2003.
25. Strictly speaking, the Nash equilibrium of the expected return game is not a Nash equilibrium of the game as described. The Nash equilibrium of the expected return game is a Bayesian Nash equilibrium of the game as described. For simplicity's sake, I will use the two terms interchangeably here. For detailed discussion, see John Harsanyi, ‘Games with Incomplete Information Played by Bayesian Players’, Management Science 14 (1967): 159-82.
26. See Fischer et al., ‘From Ultimatum to Nash Bargaining: Theory and Experimental Evidence’.