On eliciting beliefs in strategic games

Several recent studies in experimental economics have tried to measure beliefs of subjects engaged in strategic games with other subjects. Using data from one such study we conduct an experiment where our experienced subjects observe early rounds of strategy choices from that study and are given monetary incentives to report forecasts of choices in later rounds. We elicit beliefs using three different scoring rules: linear, logarithmic, and quadratic. We compare forecasts across the scoring rules and compare the forecasts of our trained observers to forecasts of the actual players in the original experiment. We find significant differences across scoring rules. The improper linear scoring rule produces forecasts closer to 0 and 1 than the proper rules, and these forecasts are poorly calibrated. The two proper scoring rules induce significantly different distributions of forecasts. We find that forecasts by observers under both proper scoring rules are significantly different from the forecasts of the actual players, in terms of accuracy, calibration, and the distribution of forecasts. We also find evidence for belief convergence among the observers.     

Stated beliefs versus inferred beliefs: A methodological inquiry and experimental test

Belief elicitation in game experiments may be problematic if it changes game play. We experimentally verify that belief elicitation can alter paths of play in a two-player repeated asymmetric matching pennies game. Importantly, this effect occurs only during early periods and only for players with strongly asymmetric payoffs, consistent with a cognitive/affective effect on priors that may serve as a substitute for experience. These effects occur with a common scoring rule elicitation procedure, but not with simpler (unmotivated) statements of expected choices of opponents. Scoring rule belief elicitation improves the goodness of fit of structural models of belief learning, and prior beliefs implied by such models are both stronger and more realistic when beliefs are elicited than when they are not. We also find that “inferred beliefs” (beliefs estimated from past observed actions of opponents) can predict observed actions better than the “stated beliefs” from scoring rule belief elicitation.     

Belief elicitation in experiments: is there a hedging problem?

Belief-elicitation experiments usually reward accuracy of stated beliefs in addition to payments for other decisions. But this allows risk-averse subjects to hedge with their stated beliefs against adverse outcomes of the other decisions. So can we trust the existing belief-elicitation results? And can we avoid potential hedging confounds? We propose an experimental design that theoretically eliminates hedging opportunities. Using this design, we test for the empirical relevance of hedging effects in the lab. Our results suggest that hedging confounds are not a major problem unless hedging opportunities are very prominent. If hedging opportunities are transparent, and incentives to hedge are strong, many subjects do spot hedging opportunities and respond to them. The bias can go beyond players actually hedging themselves, because some expect others to hedge and best respond to this.     

Stated beliefs versus inferred beliefs: A methodological inquiry and experimental test

Belief elicitation in game experiments may be problematic if it changes game play. We experimentally verify that belief elicitation can alter paths of play in a two-player repeated asymmetric matching pennies game. Importantly, this effect occurs only during early periods and only for players with strongly asymmetric payoffs, consistent with a cognitive/affective effect on priors that may serve as a substitute for experience. These effects occur with a common scoring rule elicitation procedure, but not with simpler (unmotivated) statements of expected choices of opponents. Scoring rule belief elicitation improves the goodness of fit of structural models of belief learning, and prior beliefs implied by such models are both stronger and more realistic when beliefs are elicited than when they are not. We also find that “inferred beliefs” (beliefs estimated from past observed actions of opponents) can predict observed actions better than the “stated beliefs” from scoring rule belief elicitation.     

Gurus and belief manipulation

We analyze a model with two types of agents: standard agents and gurus, i.e. agents who have the ability to influence the other investors. Gurus announce their beliefs and act accordingly. Gurus are strategic: they take into account the impact of their announced beliefs on the other agents, hence on prices. Standard agents observe gurus' performances, choose a guru and follow her/his recommendations. Prices are determined through a classical Walras mechanism. The competition among gurus for attracting followers among standard agents is governed by the level of accuracy of their predictions. The strategic behavior leads to belief subjectivity and heterogeneity among the gurus even when gurus' initial beliefs coincide with the objective belief. Optimism as well as pessimism can both emerge. We find a positive correlation across the agents between pessimism and risk tolerance. The representative agent belief, or the consensus belief is pessimistic. As a consequence of the pessimistic bias at the aggregate level, the risk premium is greater than in the standard rational expectations equilibrium. We show on an example that this impact is significative. In a multi-asset framework, the impact is stronger on the riskier assets.     

Beliefs and Public Good Provision with Anonymous Contributors

We analyze a static game of public good contributions where finitely many anonymous players have heterogeneous preferences about the public good and heterogeneous beliefs about the distribution of preferences. In the unique symmetric equilibrium, the only individuals who make positive contributions are those who most value the public good and who are also the most pessimistic; that is, according to their beliefs, the proportion of players who most like the public good is smaller than it would be according to any other possible belief. We predict whether the aggregate contribution is larger or smaller than it would be in an analogous game with complete information and heterogeneous preferences, by comparing the beliefs of contributors with the true distribution of preferences. A trade‐off between preferences and beliefs arises if there is no individual who simultaneously has the highest preference type and the most pessimistic belief. In this case, there is a symmetric equilibrium, and multiple symmetric equilibria occur only if there are more than two preference types.     

Theoretical incentive properties of contingent valuation questions: Do they matter in the field?

Green et al. (1998) [Green, D., Jacowitz, K.E., Kahneman, D., McFadden, D., 1998. Referendum contigent valuation, anchoring, and willingness to pay for public goods. Resource and Energy Economics 20 (2), 85−116] show theoretically that stated preference questions about public services can be framed in such ways that if the subjects accept the frame the payoff­maximizing behavior will be to answer truthfully. One key element of such a theoretically incentive­compatible framing is that the (hypothetical) decision rule specified in the survey instrument is understood to be a majority rule rather than the efficiency rule typically used in cost­benefit analysis. We conducted field experiments in Germany and Switzerland to test if a referendum framing as suggested by Green et al. is effective in reducing strategic misrepresentation in a contingent valuation setting. We did not find the expected effects of the framing treatments on stated willingness to pay or on individuals' (stated) beliefs about the social choice context. The results do not support hopes that a theoretically incentive compatible framing could be purposefully used to invoke the specific beliefs about the linkage between responses and policy implementation that would make stated preference questions incentive compatible.     

Stated Beliefs and Play in Normal-Form Games

Using data on one-shot games, we investigate whether players' actions can be viewed as responses to underlying expectations about their opponent's behaviour. In our laboratory experiments, subjects play a set of 14 two-person 3×3 games and state beliefs about which actions they expect their opponents to play. The data sets from the two tasks are largely inconsistent. Rather, we find evidence that the subjects perceive the games differently when they (i) choose actions and (ii) state beliefs—their stated beliefs reveal deeper strategic thinking than their actions. On average, they fail to best respond to their own stated beliefs in almost half of the games. The inconsistency is confirmed by estimates of a unified statistical model that jointly uses the actions and the belief statements. There, we can control for decision noise and formulate a statistical test that rejects consistency. Effects of the belief elicitation procedure on subsequent actions are mostly insignificant.     

Belief formation: an experiment with outside observers

In this paper we study the belief formation processes of a group of outside observers making predictions about the actions of a player involved in a repeated game. We document four main results. First, there is substantial heterogeneity in the accuracy of our observers, with average accuracy being quite poor. Second, while there is no difference between the most and the least accurate observer in their initial beliefs, there are striking differences in their belief updating rules. The most accurate observers have a well-formulated model of player behavior, are good at best responding and quickly incorporate new information to their beliefs. The worst observers behave in an opposite manner on all three fronts. Third, when the game does not converge, subjects look beyond historical actions to make predictions and place more emphasis on forgone payoffs. Finally, we document that a “collective wisdom” emerges when our data are pooled across subjects and analyzed. Specifically, the accuracy of the group estimates becomes much higher than that of the average observer.     

Unfair inequality,governance and individual beliefs

Unfair inequality arises when incentives are not fairly tied to effort or investment across the socio-economic spectrum. The actual limitations on economic activity from this failure may depend on whether people believe the system is unfair, and how well governing institutions safeguard fair-play. In this paper, I study whether unfair wealth inequality is correlated with beliefs about fairness, and whether good governance can be a substitute in belief formations for decreases in unfair inequality. I find a that people in countries with recent increases in unfair wealth inequality are less likely to believe that inequality is due to fair processes. This relationship holds when“fair” determinants of inequality include effort, as well as moral and meritocratic components. The relationship is strongest in countries with poor quality governance. In countries with high quality governance, people appear to be more tolerant of unfair inequality, as it is only weakly reflected in their beliefs about process fairness.     

Belief and higher-order belief in the centipede games: An experimental investigation

This paper experimentally explores the epistemic conditions behind people's non-equilibrium behaviour in the centipede games. We propose a novel design of laboratory experiment to elicit people's first- and second-order beliefs regarding their opponents' choices and beliefs. The measured beliefs, together with the choice data, help us to estimate people's level of rationality, belief of rationality and second-order belief of rationality. To examine how these epistemic variables are affected by the social-efficiency property of the classic increasing-sum centipede game, we revisit the constant-sum centipede and compare the measured epistemic conditions from the constant-sum with those from the classic centipede. We find that people's non-backward induction behaviour may be attributed to the diffusion of beliefs and higher-order beliefs in the increasing-sum centipede. We consider a behavioural model in which people's preferences for social efficiency are incorporated into the extended utility maximization problem. Our analytical and estimation results indicate that the presence of efficiency-oriented players and people's belief towards the uncertain portion of such type of players may play a part in the non-backward-induction outcomes in experimental centipede games.     

Diverse beliefs and time variability of risk premia

Why do risk premia vary over time? We examine this problem theoretically and empirically by studying the effect of market belief on risk premia. Individual belief is taken as a fundamental primitive state variable. Market belief is observable; it is central to the empirical evaluation and we show how to measure it. Our asset pricing model is familiar from the noisy REE literature but we adapt it to an economy with diverse beliefs. We derive equilibrium asset prices and implied risk premium. Our approach permits a closed form solution of prices; hence we trace the exact effect of market belief on the time variability of asset prices and risk premia. We test empirically the theoretical conclusions. Our main result is that, above the effect of business cycles on risk premia, fluctuations in market belief have significant independent effect on the time variability of risk premia. We study the premia on long positions in Federal Funds Futures, 3- and 6-month Treasury Bills (T-Bills). The annual mean risk premium on holding such assets for 1?C12?months is about 40?C60 basis points and we find that, on average, the component of market belief in the risk premium exceeds 50% of the mean. Since time variability of market belief is large, this component frequently exceeds 50% of the mean premium. This component is larger the shorter is the holding period of an asset and it dominates the premium for very short holding returns of less than 2?months. As to the structure of the premium we show that when the market holds abnormally favorable belief about the future payoff of an asset the market views the long position as less risky hence the risk premium on that asset declines. More generally, periods of market optimism (i.e. ??bull?? markets) are shown to be periods when the market risk premium is low while in periods of pessimism (i.e. ??bear?? markets) the market??s risk premium is high. Fluctuations in risk premia are thus inversely related to the degree of market optimism about future prospects of asset payoffs. This effect is strong and economically very significant.     

Public goods in networks

This paper considers incentives to provide goods that are non-excludable along social or geographic links. We find, first, that networks can lead to specialization in public good provision. In every social network there is an equilibrium where some individuals contribute and others free ride. In many networks, this extreme is the only outcome. Second, specialization can benefit society as a whole. This outcome arises when contributors are linked, collectively, to many agents. Finally, a new link increases access to public goods, but reduces individual incentives to contribute. Hence, overall welfare can be higher when there are holes in a network.     

Individual Characteristics and the Disposition Effect: The Opposing Effects of Confidence and Self-Regard

We conduct two experiments to examine potential causes of the disposition effect. In Experiment 1, we rule out beliefs in mean reversion as a cause of the disposition effect. Although a belief in the mean reversion of stock prices should be independent of whether an investor owns or only follows the stock, we show only investors who own the stock behave as though prices will reverse. In Experiment 2, participants buy and sell securities over multiple periods. We find that self-regard and investing confidence (two types of self-esteem) have opposing influences on investors’ tendency to hold losing investments. Investors with lower self-regard hold losing investments longer than those with higher self-regard, and investors with higher confidence hold losing investments longer than those with lower confidence. We focus on investors’ tendency to hold losing stocks too long because prior research suggests the gain versus loss sides of the disposition effect are driven by different biases.     

Product Market Integration and Environmental Policy Coordination in An International Duopoly

We analyse the effect of product market integration on environmental policy incentives in an international duopoly, where national policy makers act strategically. If traditional trade policy instruments are not available, environmental policies will typically be determined by the interaction of conflicting policy incentives. Contrary to popular belief, we find that international product market integration, in this particular setting, might reduce the need for transnational policy coordination, both from a purely environmental and from a social welfare perspective.     

Asset pricing and institutional investors with disagreements

We develop a dynamic asset pricing model with two institutional investors who have benchmark incentives and who disagree about the underlying economy. We derive semi-closed form expressions for all equilibrium quantities. We find that the benchmark stock price increases and the non-benchmark stock price decreases with the benchmark incentives. Furthermore, each stock price decreases with its own disagreement and increases with the other stock disagreement. We also show that there is a positive relationship between the co-movement of the stocks and the benchmark incentives, but that this co-movement is negative with the disagreements, owing to the endogenous risk-sharing mechanisms. Moreover, we find that, when one stock disagreement increases, the optimistic institutional investor always takes positions on this stock by shorting the other stock and the bond in order to hedge against the risk of market changes, in line with the pessimistic investor's beliefs.     

Loss aversion and the quantity–quality tradeoff

Firms face an optimization problem that requires a maximal quantity output given a quality constraint. But how do firms incentivize quantity and quality to meet these dual goals, and what role do behavioral factors, such as loss aversion, play in the tradeoffs workers face? We address these questions with a theoretical model and an experiment in which participants are paid for both quantity and quality of a real effort task. Consistent with basic economic theory, higher quality incentives encourage participants to shift their attention from quantity to quality. However, we also find that loss averse participants shift their attention from quality to quantity to a greater degree when quality is weakly incentivized. These results can inform managers of appropriate ways to structure contracts, and suggest benefits to personalizing contracts based on individual behavioral characteristics.     

Crime and punishment in the “American Dream”

We observe that countries where belief in the “American dream”(i.e., effort pays) prevails also set harsher punishment for criminals. We know that beliefs are also correlated with several features of the economic system (taxation, social insurance, etc). Our objective is to study the joint determination of these three features (beliefs, punitiveness and economic system) in a way that replicates the observed empirical patterns. We present a model where beliefs determine the types of contracts that firms offer and whether workers exert effort. Some workers become criminals, depending on their luck in the labor market, the expected punishment, and an individual shock that we call “meanness”. It is this meanness level that a penal system based on “retribution” tries to detect when deciding the severity of the punishment. We find that when initial beliefs differ, two equilibria can emerge out of identical fundamentals. In the “American” (as opposed to the “French”) equilibrium, belief in the “American dream” is commonplace, workers exert effort, there are high powered contracts (and income is unequally distributed) and punishments are harsh. Economists who believe that deterrence (rather than retribution) shapes punishment can interpret the meanness parameter as pessimism about future economic opportunities and verify that two similar equilibria emerge.     

Dynamic coordination with individual learning

We study coordination in dynamic global games with private learning. Players choose whether and when to invest irreversibly in a project whose success depends on its quality and the timing of investment. Players gradually learn about project quality. We identify conditions on temporal incentives under which, in sufficiently long games, players coordinate on investing whenever doing so is not dominated. Roughly speaking, this outcome occurs whenever players? payoffs are sufficiently tolerant of non-simultaneous coordination. We also identify conditions under which players coordinate on the risk-dominant action. We provide foundations for these results in terms of higher order beliefs.     

A test of the rational expectations hypothesis using data from a natural experiment

Data on contestants’ choices in Italian Game Show Affari Tuoi are analysed in a way that separates the effect of risk attitude (preferences) from that of beliefs concerning the amount of money that will be offered to contestants in future rounds. This separate identification is possible by virtue of the fact that, at a certain stage of the game, beliefs are not relevant, and risk attitude is the sole determinant of choice. The rational expectations hypothesis is tested by comparing the estimated belief function with the ‘true’ offer function which is estimated extraneously using data on offers actually made to contestants. We find a close correspondence, leading us to accept the rational expectations hypothesis. The importance of belief formation is confirmed by the estimation of a mixture model which establishes that the vast majority of contestants are forward looking as opposed to myopic.