Amnon Rapoport

Measuring the vague meanings of probability terms.

Can the vague meanings of probability terms such as doubtful, probable, or likely be expressed as membership functions over the [0, 1] probability interval? A function for a given term would assign a membership value of /ero to probabilities not at all in the vague concept represented by the term, a membership value of one to probabilities definitely in the concept, and intermediate membership values to probabilities represented by the term to some degree. A modified pair-comparison procedure was used in two experiments to empirically establish and assess membership functions for several probability terms. Subjects performed two tasks in both experiments: They judged (a) to what degree one probability rather than another was better described by a given probability term, and (b) to what degree one term rather than another better described a specified probability. Probabilities were displayed as relative areas on spinners. Task a data were analyzed from the perspective of conjoint-measurement theory, and membership function values were obtained for each term according to various scaling models. The conjoint-measurement axioms were well satisfied and goodness-of-fit measures for the scaling procedures were high. Individual differences were large but stable. Furthermore, the derived membership function values satisfactorily predicted the judgments independently obtained in task b. The results support the claim that the scaled values represented the vague meanings of the terms to the individual subjects in the present experimental context. Methodological implications are discussed, as are substantive issues raised by the data regarding the vague meanings of probability terms.

Provision of step-level public goods: Effects of greed and fear of being gypped

Abstract In an attempt to study the differential effects of two motives on contribution to the provision of public goods, 12 groups of five subjects each participated in three multiple-trial, social dilemma games. Each subject received a fixed endowment, e , and then decided independently and anonymously whether to contribute it to a monetary public good. The public good, r , was provided to all five group members if three or more contributed; it was not provided, otherwise. In the first condition both major motives for not contributing—a desire to free ride (greed) and a fear of wasting ones resources (fear)—were present. A second condition eliminated fear but maintained greed, whereas a third condition eliminated greed but kept fear. It was found that both fear and greed motivate noncooperative behavior, but that the latter motive predominated. Individual and group data support a model of public good provision proposed by A. Rapoport (1987, Psychological Review , 94 , 74–83).

Provision of Step-Level Public Goods The Sequential Contribution Mechanism

Groups of five players participated in a social dilemma game in which each player receives a monetary endowment and then chooses whether to contribute it to a monetary public good. The good is provided to all group members if at least three contributions are made; it is not provided, otherwise. Experiment 1 showed that the simultaneous protocol of play—where decisions are made privately and anonymously—is significantly less effective in solving the dilemma than the sequential protocol—where decisions are made sequentially with complete information about previous decisions in the sequence. Experiment 2 replicated this finding and, in addition, showed that the sequential protocol with only information about previous noncooperative choices is significantly more effective in solving the dilemma than the sequential protocol with only information about previous cooperative choices.

Mixed strategies in strictly competitive games: A further test of the minimax hypothesis

Abstract We report the results of two studies designed to investigate interactive behavior in two-person zero-sum games and assess the descriotive power of the minimax hypothesis. Standard statistical tests reject the particular multinomial model implied by the minimax model on the individual and group levels. On the other hand, the minimax model outperforms an equiprobable model predicting equal choice probabilities and a win-weighted model in which the probability of choosing each pure strategy is proportional to the number of opportunities to win associated with this strategy. The combined results of the two studies support strategic equivalence.

Experimental Business Research

Preface R. Zwick, A. Rapoport. Part I: Coordination & Dynamic Decision Making. 1. An Experimental Study of Saving and Investing in a Complex Stochastic Environment V. Anderhub, et al. 2. Dynamic Decision Making in Marketing Channels: An Experimental Study of Cycle Time, Shared Information and Customer Demand Patterns S. Gupta, et al. 3. Cooperation in Single Play, Two-Person Extensive Form Games Between Anonymously Matched Players K. McCabe, et al. 4. Coordination in the Aggregate without Common Knowledge or Outcome Information A. Rapoport, et al. 5. Behavioral Accounting Experiments in Market and Game Settings W.S. Waller. 6. Behavioral Strategies in Repeated Pure Coordination Games R. Zwick, et al. Part II: Auctions. 7. Externalities and Alternative Auction Rules for Access to Rail Networks: Some Evidence from the Laboratory P.J. Brewer. 8. Equilibrium Bidding Strategies under the English and the Second-Price Auctions S.H. Chew, N. Nishimura. 9. Physiological Anticipation and Endowment Effects: Inferred Brain Function in the Formation of Allocations in an English Auction J. Dickhaut, et al. 10. Price Disclosure, Marginal Abatement Cost Information and Market Power in a Bilateral Ghg Emissions Trading Experiment Y. Hizen, T. Saijo. Part III: Learning and Construction. 11. EWA Learning in Bilateral Call Markets C.F. Camerer, et al. 12. On the Application and Interpretation of Learning Models E. Haruvy, I. Erev. 13. Preference Construction and Reconstruction F.R. Kardes, et al. Part IV: Bargaining and Contracts. 14. Trust, Reciprocity, and Other-Regarding Preferences: Groups vs. Individuals and Males vs. Females J.C. Cox. 15. Work Motivation, Institutions, and Performance S. Gachter, A. Falk. 16. Using Experimental Data to Model Bargaining Behavior in Ultimatum Games H. Lin, S. Sunder. Author Index. Subject Index. The Authors.

Intergroup Competition for the Provision of Binary Public Goods

An experimental paradigm is proposed for investigating interpersonal conflicts under conditions of intergroup competition. The paradigm constitutes an n-person game with imperfect and incomplete information in which the n players arc divided into two groups, are each assigned monetary endowment, and must then decide independently and anonymously whether to keep the endowment or contribute it to the groups benefit. The group with the larger sum of contributions is provided with a monetary reward (binary public good), which is shared equally among all of its members regardless of their decision. Two alternative models are proposed and their testable implications are derived and discussed. Both assume maximization of expected utility, but their assumptions about the expectations each player holds about the decisions of the remaining players are different. The effects of predccisional communication arc examined and several extensions of the basic paradigm are outlined. Although the interest of social psychologists in intergroup relations has been growing rapidly (Billi~, 1976; Brewer, 1979; Tajfel, 1982), the experimental research in this area is still sparse, especially when compared with the vast amount of research on interpersonal relations (Rabbie, 1982). The difficulty of simulating complex intergroup relations in the laboratory is one reason for the fields slow development (Gerard & Miller, 1967; Rabbie, 1982). Another and perhaps more important reason appears to be the lack of an integrated theory. Thus, Rabbie (1982) wrote: Although the literature offers a number of hypotheses about intergroup relations, these hypotheses do not form an integrated theory. They are usually stated at one particular level of analysis, they are sometimes contradictory, and their supporting evidence varies widely. (p. 124) The development of an integrated theory has been especially hindered by the traditional distinction between the interpersonal and intergroup levels of analysis. This distinction has characterized the theoretical study and experimental investigation of social conflicts and competitions for many years (Tajfel, 1982). Despite recent attempts to simultaneously investigate the interpersonal and intergroup aspects of social conflicts in small groups (Brewer & Kramcr, 1986; Kramer & Brewer, 1984), the literature shows little success in closing the gap between these two largely hypothetical levels of analysis. One possible way to narrow the gap is by designing and conducting systematic and theory-driven research of social situations involving small groups, in which social conflicts occur simultaneously at both levels of analysis. Extending the paradigm

Choice behavior in an optional stopping task

Abstract Seven subjects participated in an optional stopping experiment, sampling sequentially independent and identically distributed offers from a normal distribution. On each trial a subject could either pay a fixed cost and sample another offer or stop the search and receive a payoff based on the values of the offers sampled thus far. The cost per observation, the form of the payoff function, and the maximum number of offers that could be sampled were varied. Dependent variables consisted of the number of offers actually sampled, the proportion of optimally solved problems, and the magnitude and number of errors. Most of the optional stopping problems were solved optimally. When discrepancies from the optimal stopping rule occurred, subjects tended to stop the search when they should have continued. A cut-off point model was tested and compared to the optimal stopping rule.

Ultimatums in two-person bargaining with one-sided uncertainty: offer games

In the ultimatum game with complete information a Sender proposes a division of a given amount of surplus (“pie”), which a Receiver can either accept (payoffs are distributed according to the Senders proposal) or reject (both players earn nothing). We study another version of the ultimatum game under incomplete informaton in which the pie is drawn randomly from a commonly known distribution, the Sender knows the exact size of the pie, but the Receiver only knows her share of the pie, not the residual share requested by the Sender. The basic results are that (1) as the support of the pie distribution increases in a mean-preserving spread, the Senders make lower offers that the Receivers are less likely to reject, (2) for a given support, Senders tend to offer a lower proportion of the pie to the Receivers as the pie size grows larger, and (3) although knowing only their share of the pie, Receivers estimate its size quite accurately.

Psychological models of deferred decision making

Abstract In a two-state deferred decision making task one of two mutually exclusive states of nature is responsible for generating a sequence of independent, identically distributed, and costly observations. After purchasing each observation, the decision maker must either (a) stop purchasing costly observations and make a terminal choice favoring one of the two states, or (b) continue purchasing at least one more observation. We describe a new method, called pattern analysis, for distinguishing alternative models of deferred decision making. Seven different psychological models are evaluated including the optimal stopping rule, fixed sampling, random walk, fixed forgetting, horse race or accumulator, runs, and hybrid stopping rules. Violations of basic properties implied by each of these seven models are reported. The most promising psychological model was a myopic stopping rule, which prescribes purchasing observations until the expected loss of making a terminal decision after purchasing n observations is less than or equal to the sum of the costs of purchasing n + 1 observations.

Experimental tests of the mean-variance model for portfolio selection

Abstract Statistically knowledgeable male and female undergraduate students participated in 40 portfolio selection problems with monetary payoff contingent on performance. The portfolio selection task included two independent risky assets with normally distributed returns. It provided access to information about previous returns, allowed borrowing and lending at a fixed interest rate, and forced on each decision period a choice between the two risky assets. The findings show a high percentage of inefficient mean-variance portfolios which does not decrease with practice, a high rate of requests for useless information, a large frequency of switches between the two risky assets, and sequential dependencies. Theoretical and practical implications of the findings are briefly discussed.