RATIONAL AND SOCIAL CHOICE Part 6 ppt

290 carlos alós-ferrer and karl h. schlag M ≥ 2, then the dynamics will circle close to and around the Nash equilibrium if sufficiently few individuals observe the play of others between rounds. 11 Pollock and Schlag (1999) consider individuals who know the game they play, so uncertainty is only about the distribution of actions. They investigate conditions on a single sampling rule that yield a payoff monotone dynamics in a game that has a cyclic best response structure as in Matching Pennies. They find that the rule has to be imitating, and that the continuous version of the population dynamics will have—like the standard replicator dynamics—closed orbits around the Nash equilibrium. They contrast this with the finding that there is no rule based only on a finite sample of opponent play that will lead to a payoff monotone dynamics. This is due to the fact that information on success of play has to be stored and recalled in order to generate a payoff monotone dynamics. Dawid (1999) considers two populations playing a battle-of-the-sexes game, where each agent observes a randomly selected other member of the same pop- ulation and imitates the observed action if the payoff is larger than their own and the gap is large enough. For certain parameter values, this model includes PIR. The induced dynamics is payoff monotone. In games with no risk-dominant equilibrium, there is convergence towards one of the pure-strategy coordination equilibria unless the initial population distribution is symmetric. In the latter case, depending on the model’s parameters, play might converge either to the mixed- strategy equilibrium or to periodic or complex attractors. If one equilibrium is risk-dominant, it has a larger basin of attraction than the other one. 11.3.2 Imitating your Opponents In the following we consider the situation where player roles are not separated. There is a symmetric game, and agents play against and learn from agents within the same population. Environments where row players cannot be distinguished from column players include oligopolies and financial markets. Here it makes a difference whether we look for rules that increase average payoffs or those that induce a better reply dynamics. Consider, first, the objective to induce a better reply dynamics. Rules that we characterized as being improving in decision problems have this property. To in- duce a (myopic) better reply dynamic means that, if play of other agents does not change, an individual agent following the rule should improve payoffs. Thus this condition is identical with the improving condition for decision problems. Specifically, a rule induces a better reply dynamic if and only if it is improving in decision problems. The condition of bounded payoffs translates into considering the set of all games with payoffs within these bounds. The decision setting with 11 Cycling can have a descriptive appeal, for such cycles might describe fluctuations between costly enforcement and fraud (e.g. see Cressman et al. 1998). imitation and learning 291 idiosyncratic payoffs translates into games where all pure strategies can be ordered according to dominance. Now turn to the objective of finding a rule that always increases average payoffs. Ania (2000) presents an interesting result showing that this is not possible unless average payoffs remain constant. The reason is as follows. When a population of players is randomly matched to play a Prisoner’s Dilemma, in a state with mostly cooperators and only a few defectors, increase in average payoffsrequiresthatmore defectors switch to cooperate than vice versa. However, note that the game might just as well not be a Prisoner’s Dilemma, but one in which mutual defection yields a superior payoff to mutual cooperation. Then cooperators should switch more likely to defect than vice versa. Note that the difference between these two games does not play a role when there are mostly cooperators, and hence the only way to solve the problem is for there to be no net switching. Thus, the strategic framework is fundamentally different from the individual decision framework of, for example, Schlag (1998). Given this negative result, it is natural to investigate directly the connection between imitation dynamics and Nash equilibria. The following dynamics, which we will refer to as the perturbed imitation dynamics, has played a prominent role in the literature. Each period, players receive revision opportunities with a given, exogenous probability 0 < 1 − ‰ ≤ 1; that is, ‰ measures the amount of inertia in individual behavior. When allowed to revise, players observe either all or a random sample of the strategies used and payoffs attained in the last period (always including their own) and use an imitation rule, e.g. Imitate the Best. Additionally, with an exogenous probability 0 <  < 1, players mutate (make a mistake) and choose a strategy at random, all strategies having positive probability. Clearly, the dynamics is a Markov chain in discrete time, indexed by the mutation probability. The “long-run outcomes” (or stochastically stable states) in such models are the states in the support of the (limit) invariant distribution of the chain as  goes to zero. See Kandori et al.(1993)orYoung(1993) for details. The first imitation model of this kind is due to Kandori et al.(1993), who show that when N players play an underlying two-player, symmetric game in a round- robin tournament, the long-run outcome corresponds to the symmetric profile where all players adopt the strategy of the risk-dominant equilibrium, even if the other pure-strategy equilibrium is payoff-dominant. A clever robustness test was performed by Robson and Vega-Redondo (1996), who show that when the round- robin tournament is replaced by random matching, the perturbed IB dynamics leads to payoff-dominant equilibria instead. We concentrate now on proper N-player games. When considering imitation in games, it is natural to restrict attention to symmetric games: that is, games where the payoff of each player k is given through the same function (s k |s −k ), where s k is the strategy of player k, s −k is the vector of strategies of other players, all strategy spaces are equal, and (s k |s −k ) is invariant to permutations in s −k . 292 carlos alós-ferrer and karl h. schlag The consideration of N-player, symmetric games immediately leads to a depar- ture from the framework in the previous sections. First, DMs imitate their oppo- nents, so that there is no abstracting away from strategic considerations. Second, the population size has to be N; that is, we are dealing with a finite population framework, and no large population limit can be meaningfully considered for the resulting dynamics. It turns out that the analysis of imitation in N-player games is tightly related to the concept of finite population ESS (Evolutionarily Stable Strategy), which is different from the classical infinite population ESS. This notion was devel- oped by Schaffer (1988). A finite population ESS is a strategy such that, if it is adopted by the whole population, then any single deviant (mutant) will fare worse than the incumbents after deviation. Formally, it is a strategy a such that (a|b, a, N−2 ,a) ≥ (b|a, N−1 ,a)foranyotherstrategyb. An ESS is strict if this inequality is always strict. Note that, if a is a finite population ESS, the profile (a, ,a) does not need to be a Nash equilibrium. Instead of maximizing the pay- offs of any given player, an ESS maximizes relative payoffs—the difference between the payoffs of the ESS and those of any alternative “mutant” behavior. 12 An ESS a is (strictly) globally stable if (a|b, m ,b, a, N−m−1 ,a.)(>) ≥ (b|b, m−1 ,b, a, N−m ,a) for all 1 ≤ m ≤ N − 1; that is, if it resists the appearance of any fraction of such experimenters. We obtain: Proposition 5. For an arbitrary, symmetric game, if there exists a strictly globally stable finite population ESS a, then (a, ,a) is the unique long-run outcome of all perturbed imitation dynamics where the imitation rule is such that actions with maximal payoffs are imitated with positive probability and actions with worse payoffs than one’s own are never imitated, e.g. IB or PIR. Alós-Ferrer and Ania (2005b) prove this result for IB. However, the logic of their proof extends to all the rules mentioned in the statement. The intuition is as follows. If the dynamics starts at (a, ,a), any mutant will receive worse payoffs than the incumbents, and hence will never be imitated. However, starting from any symmetric profile (b, ,b), a single mutant to a will attain maximal payoffs, and hence be imitated with positive probability. Thus, the dynamics flows towards (a, ,a). Schaffer (1989) and Vega-Redondo (1997) observe that, in a Cournot oligopoly, the output corresponding to a competitive equilibrium—the output level that maximizes profits at the market-clearing price—is a finite population ESS. That is, a firm deviating from the competitive equilibrium will make lower profits than 12 An ESS may correspond to spiteful behavior, i.e. harmful behavior that decreases the survival probability of competitors (Hamilton 1970). imitation and learning 293 its competitors after deviation. Actually, Vega-Redondo’s proof shows that it is a strictly, globally stable ESS. Additionally, Vega-Redondo (1997) shows that the competitive equilibrium is the only long-run outcome of a learning dynamics where players update strategies according to Imitate the Best and occasionally make mistakes (as in Kandori et al. 1993). Possajennikov (2003) and Alós-Ferrer and Ania (2005b) show that the results for the Cournot oligopoly are but an instance of a general phenomenon. Consider any aggregative game, i.e. a game where payoffs depend only on individual strategies and an aggregate of all strategies (total output in the case of Cournot oligopolies). Suppose there is strategic substitutability (submodularity) between individual and aggregate strategy. For example, in Cournot oligopolies the incentive to increase individual output decreases, the higher the total output in the market. Define an aggregate-taking strategy (ATS) to be one that is individually optimal, given the value of the aggregate that results when all players adopt it. Alós-Ferrer and Ania (2005b) show the following: Proposition 6. Any ATS is a finite population ESS in any submodular, aggregative game. Further, any strict ATS is strictly globally stable, and the unique ESS. This result has a natural counterpart in the supermodular case (strategic com- plementarity), where any ESS can be shown to correspond to aggregate-taking optimization. 13 As a corollary of the last two propositions, any strict ATS of a submodular aggregative game is the unique long-run outcome of the perturbed imitation dy- namics with e.g. IB, hence implying the results in Vega-Redondo (1997). These results show that, in general, imitation in games does not lead to Nash equilibria. The concept of finite population ESS, and not Nash equilibrium, is the appropriate tool to study imitation outcomes. 14 In some examples, though, the latter might be a subset of the former. Alós-Ferrer et al.(2000) consider Imitate the Best in the framework of a Bertrand oligopoly with strictly convex costs. Contrary to the linear costs setting, this game has a continuum of symmetric Nash equilibria. Imitate the Best selects a proper subset of those equilibria. As observed by Ania (2008), the ultimate reason is that this subset corresponds to the set of finite population ESS. 15 13 Leininger (2006) shows that, for submodular aggregative games, every ESS is globally stable. 14 For the inertia-less case, this assertion depends on the fact that we are considering rules which depend only on the last period’s outcomes. Alós-Ferrer (2004) shows that, even with just an additional period of memory, the perturbed IB dynamics with ‰ = 0 selects all symmetric states with output levels between, and including, the perfectly competitive outcome and the Cournot–Nash equilibrium. 15 Alós-Ferrer and Ania (2005a) study an asset market game where the unique pure-strategy Nash equilibrium is also a finite population ESS. They consider a two-portfolio dynamics on investment strategies where wealth flows with higher probability into those strategies that obtained higher realized payoffs. Although the resulting stochastic process never gets absorbed in any population profile, it can be shown that, whenever one of the two portfolios corresponds to the ESS, a majority of traders adopt 294 carlos alós-ferrer and karl h. schlag The work just summarized focuses mainly on Imitate the Best. As seen in Proposition 5, there are no substantial differences if one assumes PIR instead. The technical reason is that the models mentioned above are finite population models with vanishing mutation rates. For these models, results are driven by the existence of a strictly positive probability of switching, not by the size of this probability. Behavior under PIR is equivalent to that of any other imitative rule in which imitation takes place only when observed payoff is strictly higher than own payoff. Whether or not net switching is linear plays no role. Rules like IBA and SPOR would produce different results, though, although a general analysis has not yet been undertaken. We would like to end this chapter by reminding the reader that our aim has been to concentrate on learning rules, and in particular imitating ones, that can be shown to possess appealing optimality properties. However, we would like to point out that a large part the literature on learning in both decision problems and games has been more descriptive. Of course, from a behavioral perspective we would expect certain, particularly simple rules like IB or PIR to be more descriptively relevant than others. For example, due to its intricate definition, we think of SPOR more as a benchmark. Huck et al.(1999) find that the informational setting is crucial for individual behav- ior. If provided with the appropriate information, experimental subjects do exhibit a tendency to imitate the highest payoffs in a Cournot oligopoly. Apesteguía et al. (2007) elaborate on the importance of information and also report that the subjects’ propensity to imitate more successful actions is increasing in payoff differences as specified by PIR. Barron and Erev (2003) and Erev and Barron (2005) discuss a large number of decision-making experiments and identify several interesting behavioral traits which oppose payoff maximization. First, the observation of high (foregone) payoff weighs heavily. Second, alternatives with the highest recent payoffsseemto be attractive even when they have low expected returns. Thus, IB or PIR might be more realistic than IBA. References Alós-Ferrer,C.(2004). Cournot vs. Walras in Oligopoly Models with Memory. Interna- tional Journal of Industrial Organization, 22, 193–217. and Ania,A.B.(2005a). The Asset Market Game. Journal of Mathematical Economics, 41, 67–90. it in the long run. The dynamics can also be interpreted as follows: each period, an investor updates her portfolio. The probability that this revision results in an investor switching from the first portfolio to the second, rather than vice versa, is directly proportional to the difference in payoffs between the portfolios. That is, those probabilities follow PIR. imitation and learning 295 (2005b). The Evolutionary Stability of Perfectly Competitive Behavior. Economic Theory, 26, 497–516. and Schenk-Hoppé,K.R.(2000). An Evolutionary Model of Bertrand Oligopoly. Games and Economic Behavior, 33, 1–19. Ania,A.B.(2000). Learning by Imitation when Playing the Field. Working Paper 0005, Department of Economics, University of Vienna. (2008). Evolutionary Stability and Nash Equilibrium in Finite Populations, with an Application to Price Competition. Journal of Economic Behavior and Organization, 65/3, 472–88. Apesteguía, J., Huck, S., and Oechssler,J.(2007). Imitation—Theory and Experimental Evidence. Journal of Economic Theory, 136, 217–35. Balkenborg,D.,andSchlag,K.H.(2007). On the Evolutionary Selection of Nash Equi- librium Components. Journal of Economic Theory, 133, 295–315. Bandura,A.(1977). Social Learning Theory. Englewood Cliffs, NJ: Prentice-Hall. Banerjee,A.(1992). A Simple Model of Herd Behavior. Quarterly Journal of Economics, 107, 797–817. Barron,G.,andErev,I.(2003). Small Feedback-Based Decisions and their Limited Cor- respondence to Description-Based Decisions. Journal of Behavioral Decision Making, 16, 215–33. Bessen,J.,andMaskin,E.(2007). Sequential Innovation, Patents, and Imitation. The Rand Journal of Economics, forthcoming. Björnerstedt,J.,andSchlag,K.H.(1996). On the Evolution of Imitative Behavior. Discussion Paper No. B–378, Sonderforschungsbereich 303, University of Bonn. Börgers, T., Morales, A., and Sarin,R.(2004). Expedient and Monotone Rules. Econo- metrica, 72/2, 383–405. Boylan,R.T.(1992). Laws of Large Numbers for Dynamical Systems with Randomly Matched Individuals. Journal of Economic Theory, 57, 473–504. Cho, I K. and Kreps,D.(1987). Signaling Games and Stable Equilibria. Quarterly Journal of Economics, 102, 179–221. Conlisk,J.(1980). Costly Optimizers versus Cheap Imitators. Journal of Economic Behavior and Organization, 1, 275–93. Cressman, R., Morrison,W.G.,andWen,J.F.(1998). On the Evolutionary Dynamics of Crime. Canadian Journal of Economics, 31, 1101–17. Cross,J.(1973). A Stochastic Learning Model of Economic Behavior. Quarterly Journal of Economics, 87, 239–66. Daw i d,H.(1999). On the Dynamics of Word of Mouth Learning with and without Antici- pations. Annals of Operations Research, 89, 273–95. Ellison,G.,andFudenberg,D.(1993). Rules of Thumb for Social Learning. Journal of Political Economy, 101, 612–43. (1995). Word of Mouth Communication and Social Learning. Quarterly Journal of Economics, 110, 93–125. Erev,I.,andBarron,G.(2005). On Adaptation, Maximization, and Reinforcement Learn- ing among Cognitive Strategies. Psychological Review, 112, 912–31. Fudenberg,D.,andLevine,D.K.(1998). The Theory of Learning in Games. Cambridge, MA: MIT Press. Hamilton,W.(1970). Selfish and Spiteful Behavior in an Evolutionary Model. Nature, 228, 1218–20. 296 carlos alós-ferrer and karl h. schlag Hofbauer,J.,andSchlag,K.H.(2000). Sophisticated Imitation in Cyclic Games. Journal of Evolutionary Economics, 10/5, 523–43. and Swinkels,J.(1995). A Universal Shapley-Example. Unpublished MS, University of Vienna and Washington University in St Louis. Huck, S., Normann,H.T.,andOechssler,J.(1999). Learning in Cournot Oligopoly—An Experiment. Economic Journal, 109,C80–C95. Juang,W T.(2001). Learning from Popularity. Econometrica, 69, 735–47. Kandori, M., Mailath,G.,andRob,R.(1993). Learning, Mutation, and Long Run Equi- libria in Games. Econometrica, 61, 29–56. Kreps,D.,andWilson,R.(1982). Reputation and Imperfect Information. Journal of Eco- nomic Theory, 27, 253–79. Lakshmivarahan,S.,andThathachar,M.A.L.(1973). Absolutely Expedient Learning Algorithms for Stochastic Automata. IEEE Transactions on Systems, Man, and Cybernetics, SMC-3, 281–6. Leininger,W.(2006). Fending Off One Means Fending Off All: Evolutionary Stability in Submodular Games. Economic Theory, 29, 71 3– 19. Maynard Smith,J.(1982). Evolution and the Theory of Games. Cambridge: Cambridge University Press. Morales,A.J.(2002). Absolutely Expedient Imitative Behavior. International Journal of Game Theory, 31, 475–92. (2005). On the Role of Group Composition for Achieving Optimality. Annals of Oper- ations Research, 137, 378–97. Oyarzun, C., and Ruf,J.(2007). Monotone Imitation. Unpublished MS, Texas A&M and Columbia University. Pingle,M.,andDay,R.H.(1996). Modes of Economizing Behavior: Experimental Evi- dence. Journal of Economic Behavior and Organization, 29, 191–209. Pollock,G.,andSchlag,K.H.(1999).SocialRolesasanEffective Learning Mechanism. Rationality and Society, 11, 371–97. Possajennikov,A.(2003). Evolutionary Foundations of Aggregate-Taking Behavior. Eco- nomic Theory, 21, 921–8. Robson,A.J.,andVega-Redondo,F.(1996). Efficient Equilibrium Selection in Evolution- ary Games with Random Matching. Journal of Economic Theory, 70, 65–92. Rogers,A.(1989). Does Biology Constrain Culture?. American Anthropologist, 90, 819– 31. Schaffer,M .E .(1988). Evolutionarily Stable Strategies for a Finite Population and a Variable Contest Size. Journal of Theoretical Biology, 132, 469–78. (1989). 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Weibull,J.(1995). Evolutionary Game Theory. Cambridge, MA: MIT Press. Young,P.(1993). The Evolution of Conventions. Econometrica, 61/1, 57–84. chapter 12 DIVERSITY klaus nehring clemens puppe 12.1 Introduction How much species diversity is lost in the Brazilian rainforest every year? Is France culturally more diverse than Great Britain? Is the range of car models offered by BMW more or less diverse than that of Mercedes-Benz? And more generally: What is diversity, and how can it be measured? This chapter critically reviews recent attempts in the economic literature to answer this question. As indicated, the interest in a workable theory of diversity and its measurement stems from a variety of different disciplines. From an economic perspective, one of the most urgent global problems is the quantification of the benefits of ecosystem services and the construction of society’s preferences over different conservation policies. In this context, biodiversity is a central concept that still needs to be understood and appropriately formalized. In welfare economics, it has been argued that the range of different life-styles available to a person is an im- portant determinant of this person’s well-being (see e.g. Chapter 15 below). Again, the question arises as to how this range can be quantified. Finally, the definition and measurement of product diversity in models of monopolistic competition and product differentiation constitute an important and largely unresolved issue since Dixit and Stiglitz’s (1977) seminal contribution. We thank Stefan Baumgärtner, Nicolas Gravel, and Yongsheng Xu for helpful comments and sug- gestions. diversity 299 The central task of a theory of diversity is properly to account for the similar- ities and dissimilarities between objects. In the following, we present some basic approaches to this problem. 1 12.2 Measures Based on Dissimilarity Metrics A natural starting point for thinking about diversity is based on the intuitive inverse relationship between diversity and similarity: the more dissimilar objects are among themselves, the more diverse is their totality. Clearly, this approach is fruitful only to the extent to which our intuitions about (dis)similarity are more easily accessible than those about diversity. In the following, we distinguish the different concrete proposals according to the nature of the underlying dissimilarity relation: whether it is understood as a binary, ternary, or quaternary relation, and whether it is used as a cardinal or only an ordinal concept. 12.2.1 Ordinal Notions of Similarity and Dissimilar ity Throughout, let X denote a finite universe of objects. As indicated in the introduc- tion, the elements of X can be as diverse objects as biological species, ecosystems, life-styles, brands of products, the flowers in the garden of your neighbor, etc. The simplest notion of similarity among the objects in X is the dichotomous distinction according to which two elements are either similar or not, with no intermediate possibilities. Note that in almost all interesting cases such binary similarity relations will not be transitive. Pattanaik and Xu (2000)haveusedthissimplenotionof similarity in order to define a ranking of sets in terms of diversity, as follows. A similarity-based partition of a set S ⊆ X is a partition {A 1 , ,A m } of S such that, for each partition element A i , all elements in A i are similar to each other. Clearly, similarity-based partitions thus defined are in general not unique. As a simple example, consider the universe X = {x, y, z} and suppose that x and y,aswellas y and z are similar, but x and z are not similar. The singleton partition (i.e. here: {{x}, {y}, {z}}) always qualifies as a similarity-based partition. In addition, there are the following two similarity-based partitions in the present example: namely, {{x, y}, {z}} and {{x}, {y, z}}. Pattanaik and Xu (2000) propose to take the minimal cardinality of a similarity-based partition of a set as an ordinal measure of its diversity. 1 For recent alternative overviews, see Baumgärtner (2006) and Gravel (2008). [...]... North-Holland Vane-Wright, R I., Humphries, C J., and Williams, P H (1991) What to Protect?— Systematics and the Agony of Choice Biological Conservation, 55, 235–54 Van Hees, M (2004) Freedom of Choice and Diversity of Options: Some Difficulties Social Choice and Welfare, 22, 253 66 Weitzman, M (1992) On Diversity Quarterly Journal of Economics, 107, 363 –405 (1998) The Noah’s Ark Problem Econometrica, 66 , 1279–98... 118, 252 64 (2004b) Modelling Phylogenetic Diversity Resource and Energy Economics, 26, 205–35 Pattanaik, P K., and Xu, Y (2000) On Diversity and Freedom of Choice Mathematical Social Choice Sciences, 40, 123–30 (20 06) Ordinal Distance, Dominance, and the Measurement of Diversity Mimeo Polasky, S (ed.) (2002) The Economics of Biodiversity Conservation Burlington, VT: Ashgate Publishers Solow, A., and Broadus,... Econometrica, 52, 1 369 – 86 Simpson, E H (1949) Measurement of Diversity Nature, 163 , 68 8 Solow, A., and Polasky, S (1994) Measuring Biological Diversity Environmental and Ecological Statistics, 1, 95–107 320 klaus nehring and clemens puppe Solow, A., Polasky, S., and Broadus, J (1993) On the Measurement of Biological Diversity Journal of Environmental Economics and Management, 24, 60 –8 Van de Vel, M... Induction, Conceptual Spaces and AI Philosophy of Science, 57, 78– 95 Gaston, K J (ed.) (19 96) Biodiversity: A Biology of Numbers and Difference Oxford: Blackwell Gravel, N (2008) What is Diversity? In R Gekker and M van Hees (eds.), Economics, Rational Choice and Normative Philosophy London: Routledge, forthcoming Hill, M (1973) Diversity and Evenness: A Unifying Notation and its Consequences Ecology,... Consequences Ecology, 54, 427–31 Nehring, K (1999a) Diversity and the Geometry of Similarity Mimeo (1999b) Preference for Flexibility in a Savagian Framework Econometrica, 67 , 101–19 and Puppe, C (2002) A Theory of Diversity Econometrica, 70, 1155–98 (2003) Diversity and Dissimilarity in Lines and Hierarchies Mathematical Social Choice Sciences, 45, 167 –83 (2004a) Modelling Cost Complementarities in Terms... Measurement of Diversity Journal of Theoretical Politics, 15, 405–21 Dixit, A., and Stiglitz, J (1977) Monopolistic Competition and Optimum Product Diversity American Economic Review, 67 , 297–308 Faith, D P (1992) Conservation Evaluation and Phylogenetic Diversity Biological Conservation, 61 , 1–10 (1994) Phylogenetic Pattern and the Quantification of Organismal Biodiversity Philosophical Transactions of... Ashgate Publishers Solow, A., and Broadus, J (1993) Searching for Uncertain Benefits and the Conservation of Biological Diversity Environmental and Resource Economics, 3, 171–81 Renyi, A (1 961 ) On Measures of Entropy and Information In Proceedings of the Fourth Berkeley Symposium on Mathematical Statics and Probability, 547 61 Berkeley: University of California Press Rosenbaum, E F (2000) On Measuring Freedom... literature; see among others, Vane-Wright, Humphries and Williams (1991); Faith (1992, 1994); Solow, Polasky and Broadus (1993); Weitzman (1998); and the volumes edited by Gaston (19 96) and Polasky (2002) 302 klaus nehring and clemens puppe diversity is based on binary dissimilarity information, and to ask questions such as “When, in general, can diversity be determined by binary information?” 12.3.1... ( p) ; w diversity 319 Berger, W H., and Parker, F L (1970) Diversity of Planktonic Foraminifera in Deep Sea Sediments Science, 168 , 1345–7 Bervoets, S., and Gravel, N (2007) Appraising Diversity with an Ordinal Notion of Similarity: An Axiomatic Approach Mathematical Social Sciences, 53, 259–73 Bossert, W., Pattanaik, P K., and Xu, Y (2003) Similarity of Options and the Measurement of Diversity Journal... differ 308 klaus nehring and clemens puppe the second coordinate (the white front cube in Figure 12.2) By contrast, S1 always gives a choice between “0” and “1” in each coordinate 12.3.3 On the Application of Diversity Theory In the context of biodiversity a key issue is the choice of an appropriate conservation policy such as investment in conservation sites, restrictions of land development, anti-poaching . Econometrica, 69 , 735–47. Kandori, M., Mailath,G.,andRob,R.(1993). Learning, Mutation, and Long Run Equi- libria in Games. Econometrica, 61 , 29– 56. Kreps,D.,andWilson,R.(1982). Reputation and Imperfect. {x, y, z} and suppose that x and y,aswellas y and z are similar, but x and z are not similar. The singleton partition (i.e. here: {{x}, {y}, {z}}) always qualifies as a similarity-based partition Humphries and Williams (1991); Faith (1992, 1994); Solow, Polasky and Broadus (1993); Weitzman (1998); and the volumes edited by Gaston (19 96) and Polasky (2002). 302 klaus nehring and clemens