ChoosingBetweenLotteries: RemarkableCoordinationWithout Communication

The current research examines tacit coordination behavior in a lottery selection task. Two hundred participants in each of three experiments and 100 in a fourth choose to participate in one of two lotteries, where one lottery has a larger prize than the other. Independent of variations in the complexity of the mechanism of prize allocation, the prize amounts, and whether the lottery is the participant’s first or second choice, we typically find that the percentage of participants who choose the high versus low-prize lotteries does not significantly differ from the equilibrium predictions. This coordination is achieved without communication or experience. We additionally find that participants with an analytical thinking style and a risk-averse tendency are more likely to choose the low-prize lottery over the high-prize lottery. This tendency seems to be stable across choices. The pattern of our results suggests that to achieve tacit coordination, having a subset of individuals who attend to the choices of others is sufficient. Copyright © 2012 John Wiley & Sons, Ltd. key words coordination; lottery; choice; risk taking; uncertainty; thinking style Many decisions in everyday life involve choosing between two mutually exclusive alternatives in which one alternative seems more attractive than the other, and in which individual outcomes depend on the number of decision makers who choose each alternative. For example, when driving from the city to the beach, should one take the route with 80miles of highway or the alternative route with 90miles of highway? When applying for student housing that is distributed by a lottery, should one apply for the most attractive dorm? When the jackpot of a lottery increases, should one invest more money than usual? We investigate these choice problems of tacit coordination that arise when a decision maker can choose only one of two or more independent and mutually exclusive alternatives and when the outcome of that choice depends on the number of decision makers who choose the same alternative. Hence, the optimal behavior depends not only on the utility of each outcome but also on the estimated number of participants who choose each alternative. Despite the prevalence and importance of coordination problems, empirical research on this topic is scarce. One reason could be the mistaken belief that communication makes this type of problem easy to solve, and thus research is less needed. As pointed out by Camerer (2003, p. 337), however, “communication is not really a solution because simple coordination games with few players who do not talk are really meant as microscale reduced-form models of large social processes in which players cannot all talk at the same time.” Particularly, when large numbers of agents are operating simultaneously, communication can easily become impossible (e.g., regulatory constraints), improbable (e.g., lack of incentives), inefficient (e.g., lack of communication mechanisms), and/or unreliable (e.g., trust concerns). The current research examines behavior in this type of tacit coordination task through the use of a lottery selection task. Specifically, many individual participants must choose between entering one of two lotteries whose jackpots vary. The basic task is as follows: participants are told that N participants, including themselves, have the opportunity to choose to participate in one of two lotteries, either H (a highprize lottery) or L (a low-prize lottery). The winner of each lottery is the participant who guesses correctly a number between 0 and 9. This lottery task can be modeled as a noncooperative game with N players. The Nash equilibrium solution is the distribution of participants between the two lotteries for which participants receive no advantage from changing their choices. The Nash equilibrium, therefore, is the distribution of participants for which the expected values of the two lotteries are equivalent. The expected value of lottery (Lot) can be defined as follows: E Loti 1⁄2 1⁄4 1 1 p ð Þi Prize Ni (1) where Ni is the number of participants who participated in lottery i, p is the probability of being one of the winners (i.e., guessing correctly the number between 0 and 9), and Prize is the amount of the prize (see Appendix A for the mathematical development of this equation). A critical feature of each lottery is that the number of participants who choose each lottery impacts the size of the prize or the probability of winning it. Thus, insufficient consideration of the presence and the decisions of others when deciding how to act may lead to inferior outcomes. *Correspondence to: Yoella Bereby-Meyer, Department of Psychology, Ben-Gurion University, Beer Sheva 64105, Israel. E-mail: [email protected] Copyright © 2012 John Wiley & Sons, Ltd. Journal of Behavioral Decision Making, J. Behav. Dec. Making, 26: 338–347 (2013) Published online 10 July 2012 in Wiley Online Library (wileyonlinelibrary.com) DOI: 10.1002/bdm.1769 In the current research, we focus on a one-shot version of the lottery selection task, precluding any adaptation, learning, or feedback derived from multiple trials. Previous studies found an ability to coordinate in repeated games that do allow for learning and feedback between trials. For example, Rapoport, Chung Lo, and Zwick (2002) asked groups of 18 participants to repeatedly choose to participate in one of three lotteries with a range of prizes (e.g., $14; $12; $10). For each trial, one winner was randomly selected from each of the three lotteries. Hence, the probability of winning a prize in the lottery decreased as a function of the number of agents choosing it. Participants received feedback regarding the winner of each lottery after each trial. This study revealed a strong pattern of tacit coordination achieved through repeated trials, under the assumption of a risk-averse utility function. In a variation of this experiment, Zwick, Rapoport, and Chung Lo (2002) asked participants to repeatedly choose among three lotteries. The lotteries differed with regard to the number of prizes (i.e., winners), but prize value was constant across all. This experiment also found impressive coordination. The authors proposed a model with randomly perturbed probabilities, which accounts quite well for their data. Another class of coordination games that involves entering into markets and competitions obtained similar results. In a typical market entry game, N players are in a market with capacity C. Entrants earn a return that declines as a function of the number of entrants and becomes negative if more than C participants enter. Kahneman (1988) found the number of players who chose to enter the market in the market entry game was close to the number predicted by theory (i.e., around C entrants), even though all players made their choices simultaneously and could not communicate or learn from feedback. Since then, researchers have conducted several controlled experiments on the market entry game. In most of them, participants were told how many others decided to enter after each decision. Findings show the entry rate was close to equilibrium, with over-entry when the market capacity was low and under-entry when the market capacity was high (e.g., Rapoport, 1995; Rapoport, Seale, Erev, & Sundali, 1998; Sundali, Rapoport, & Seale, 1995). Huberman and Rubinstein (2000) found less impressive coordination. They asked subjects to self-select into one of two contests, “coin” or “die”. The winner in each of the contests was the person with the most correct guesses of 20 coin flips or 20 rolls of a die, respectively. Most subjects reported they believed most people would go to the “coin” but nevertheless incorrectly chose the “coin” contest. Ochs (1990) also reported failure to coordinate in a decentralized market game. This coordination game consisted of several locations and a set of agents. Trade could take place at any of the locations, and each agent was required to select a trading location in complete ignorance of the decisions of other agents. This game was repeatedly played with either the same or a different set of players. The degree of coordination achieved was much higher—yet not perfect—when the game was repeatedly played with the same players. Meyer, Van Huyck, Battalio, and Saving (1992) also reported difficulty in reaching perfect coordination in a decentralized market game. The authors found that entry decisions did not converge to equilibrium unless subjects were experienced in the game. Another example is the optic cable industry, which shows a failure in coordinating the production of high-capacity optic cable. Similarly, field studies of business entry and exit find most new businesses fail and usually fail rapidly (Camerer, 2003). Given the evidence regarding tacit coordination on the one hand and the failure of coordinating on the other hand, some important questions need to be addressed to better understand the conditions under which coordination may be achieved. First and most importantly, does tacit coordination emerge spontaneously when no opportunity for learning, experience, or repeated play exists? To determine the immediacy of tacit coordination, we mostly use a one-shot, rather than repeated trial, task. Second, is tacit coordination likely in large groups of people? To examine the independence of tacit coordination on small, easily visualized sample sizes, we recruit a large number of participants (n=200). Third, given the complexity of real-life situations, we aim to test whether tacit coordination depends on the complexity of the mechanism by which others’ choices affect one’s outcome. To do so, we vary the complexity of the mechanism by which the prizes are allocated. Finally, we examine the extent to which participants’ choices are stable and correlate with thinking style and risk-taking tendency. In the series of experiments we report in the succeeding text, we asked participants to choose one of two lotteries in which they wanted to participate. In all experiments, one lottery had a bigger prize than the other. We varied the complexity of the mechanism by which the prizes were allocated. In experiment 1, we randomly selected one of all the participants in each lottery to receive the prize money. In experiment 2a and b, we first asked participants to guess a number between 0 and 9. The winners were those who predicted the “lucky number” correctly, and the prize was equally divided among all winners. Like experiment 2, experiment 3 also had a preselection stage, but we randomly selected only one of the winners in each lottery to receive the prize money. In experiment 4, we replicated experiment 3 while trying to better understand the mechanism by which participants achieved coordination. The results of the current research are striking. We found no significant difference from the equilibrium for almost all treatments, despite variations in the sizes of the prizes, the prize-allocation mechanism, and the equilibrium. These exceptional findings, across multiple studies, took place among 200 (or 100 in experiment 4) decision makers, in the absence of any communication, and in both a single-shot game (experiments 1 through 3) and a repeated game (experiment 4).