Dynamic Pricing Decisions and Seller-Buyer Interactions under Capacity Constraints*Games* **2018**, *9*(1), 10; doi:10.3390/g9010010 - 22 February 2018**Abstract **

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Focusing on sellers’ pricing decisions and the ensuing seller-buyer interactions, we report an experiment on dynamic pricing with scarcity in the form of capacity constraints. Rational expectations equilibrium solutions are constructed and then tested experimentally with subjects assigned the roles of sellers and

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Focusing on sellers’ pricing decisions and the ensuing seller-buyer interactions, we report an experiment on dynamic pricing with scarcity in the form of capacity constraints. Rational expectations equilibrium solutions are constructed and then tested experimentally with subjects assigned the roles of sellers and buyers. We investigate behavior in two between-subject conditions with high and moderate levels of capacity. Our laboratory market exhibits strategic sophistication: the price offers of sellers and the buyers’ aggregate responses largely approximate equilibrium predictions. We also observe systematic deviations from equilibrium benchmarks on both sides of the market. Specifically, in our experiment the sellers are boundedly strategic: their prices often exhibit strategic adjustments to profit from buyers with limited strategic sophistication, but they are also often biased towards equilibrium pricing even when that would not be ex-post optimal.
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Game of Thrones: Accommodating Monetary Policies in a Monetary Union*Games* **2018**, *9*(1), 9; doi:10.3390/g9010009 - 19 February 2018**Abstract **

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In this paper, we present an application of the dynamic tracking games framework to a monetary union. We use a small stylized nonlinear three-country macroeconomic model of a monetary union to analyze the interactions between fiscal (governments) and monetary (common central bank) policy

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In this paper, we present an application of the dynamic tracking games framework to a monetary union. We use a small stylized nonlinear three-country macroeconomic model of a monetary union to analyze the interactions between fiscal (governments) and monetary (common central bank) policy makers, assuming different objective functions of these decision makers. Using the OPTGAME algorithm, we calculate solutions for several games: a noncooperative solution where each government and the central bank play against each other (a feedback Nash equilibrium solution), a fully-cooperative solution with all players following a joint course of action (a Pareto optimal solution) and three solutions where various coalitions (subsets of the players) play against coalitions of the other players in a noncooperative way. It turns out that the fully-cooperative solution yields the best results, the noncooperative solution fares worst and the coalition games lie in between, with a broad coalition of the fiscally more responsible countries and the central bank against the less thrifty countries coming closest to the Pareto optimum.
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A Game-Theoretic Approach for Modeling Competitive Diffusion over Social Networks*Games* **2018**, *9*(1), 8; doi:10.3390/g9010008 - 13 February 2018**Abstract **

In this paper, we consider a novel game theory model for the competitive influence maximization problem. We model this problem as a simultaneous non-cooperative game with complete information and rational players, where there are at least two players who are supposed to be

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In this paper, we consider a novel game theory model for the competitive influence maximization problem. We model this problem as a simultaneous non-cooperative game with complete information and rational players, where there are at least two players who are supposed to be out of the network and are trying to institutionalize their options in the social network; that is, the objective of players is to maximize the spread of a desired opinion rather than the number of infected nodes. In the proposed model, we extend both the Linear Threshold model and the Independent Cascade model. We study an influence maximization model in which users’ heterogeneity, information content, and network structure are considered. Contrary to previous studies, in the proposed game, players find not only the most influential initial nodes but also the best information content. The proposed novel game was implemented on a real data set where individuals have different tendencies toward the players’ options that change over time because of gaining influence from their neighbors and the information content they receive. This means that information content, the topology of the graph, and the individual’s initial tendency significantly affect the diffusion process. The proposed game is solved and the Nash equilibrium is determined for a real data set. Lastly, the numerical results obtained from the proposed model were compared with some well-known models previously reported in the literature.
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Linear–Quadratic Mean-Field-Type Games: A Direct Method*Games* **2018**, *9*(1), 7; doi:10.3390/g9010007 - 12 February 2018**Abstract **

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In this work, a multi-person mean-field-type game is formulated and solved that is described by a linear jump-diffusion system of mean-field type and a quadratic cost functional involving the second moments, the square of the expected value of the state, and the control

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In this work, a multi-person mean-field-type game is formulated and solved that is described by a linear jump-diffusion system of mean-field type and a quadratic cost functional involving the second moments, the square of the expected value of the state, and the control actions of all decision-makers. We propose a direct method to solve the game, team, and bargaining problems. This solution approach does not require solving the Bellman–Kolmogorov equations or backward–forward stochastic differential equations of Pontryagin’s type. The proposed method can be easily implemented by beginners and engineers who are new to the emerging field of mean-field-type game theory. The optimal strategies for decision-makers are shown to be in a state-and-mean-field feedback form. The optimal strategies are given explicitly as a sum of the well-known linear state-feedback strategy for the associated deterministic linear–quadratic game problem and a mean-field feedback term. The equilibrium cost of the decision-makers are explicitly derived using a simple direct method. Moreover, the equilibrium cost is a weighted sum of the initial variance and an integral of a weighted variance of the diffusion and the jump process. Finally, the method is used to compute global optimum strategies as well as saddle point strategies and Nash bargaining solution in state-and-mean-field feedback form.
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Optimal Incentives in a Principal–Agent Model with Endogenous Technology*Games* **2018**, *9*(1), 6; doi:10.3390/g9010006 - 5 February 2018**Abstract **

One of the standard predictions of the agency theory is that more incentives can be given to agents with lower risk aversion. In this paper, we show that this relationship may be absent or reversed when the technology is endogenous and projects with

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One of the standard predictions of the agency theory is that more incentives can be given to agents with lower risk aversion. In this paper, we show that this relationship may be absent or reversed when the technology is endogenous and projects with a higher efficiency are also riskier. Using a modified version of the Holmstrom and Milgrom’s framework, we obtain that lower agent’s risk aversion unambiguously leads to higher incentives when the technology function linking efficiency and riskiness is elastic, while the risk aversion–incentive relationship can be positive when this function is rigid.
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Examining Spillovers between Long and Short Repeated Prisoner’s Dilemma Games Played in the Laboratory*Games* **2018**, *9*(1), 5; doi:10.3390/g9010005 - 31 January 2018**Abstract **

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We had participants play two sets of repeated Prisoner’s Dilemma (RPD) games, one with a large continuation probability and the other with a small continuation probability, as well as Dictator Games (DGs) before and after the RPDs. We find that, regardless of which

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We had participants play two sets of repeated Prisoner’s Dilemma (RPD) games, one with a large continuation probability and the other with a small continuation probability, as well as Dictator Games (DGs) before and after the RPDs. We find that, regardless of which is RPD set is played first, participants typically cooperate when the continuation probability is large and defect when the continuation probability is small. However, there is an asymmetry in behavior when transitioning from one continuation probability to the other. When switching from large to small, transient higher levels of cooperation are observed in the early games of the small continuation set. Conversely, when switching from small to large, cooperation is immediately high in the first game of the large continuation set. We also observe that response times increase when transitioning between sets of RPDs, except for altruistic participants transitioning into the set of RPDs with long continuation probabilities. These asymmetries suggest a bias in favor of cooperation. Finally, we examine the link between altruism and RPD play. We find that small continuation probability RPD play is correlated with giving in DGs played before and after the RPDs, whereas high continuation probability RPD play is not.
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Incentive Magnitude Effects in Experimental Games: Bigger is not Necessarily Better*Games* **2018**, *9*(1), 4; doi:10.3390/g9010004 - 18 January 2018**Abstract **

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In experimental games, task-related incentives are payments to experimental subjects that vary according to their strategy choices and the consequent outcomes of the games. Limited evidence exists regarding incentive magnitude effects in experimental games. We examined one-off strategy choices and self-reported reasons for

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In experimental games, task-related incentives are payments to experimental subjects that vary according to their strategy choices and the consequent outcomes of the games. Limited evidence exists regarding incentive magnitude effects in experimental games. We examined one-off strategy choices and self-reported reasons for choices in eight 3 × 3 and four 4 × 4 normal-form games under task-related incentives of conventional magnitude and compared them with choices and reasons in the same games under incentives five times as large. Both strategy choices and self-reported reasons for choices were almost indistinguishable between the two conditions. These results are in line with earlier findings on individual decision making and with a parametric model, in which the incentive elasticity of effort is very small when compared with other factors, such as the complexity of the decision problem.
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Acknowledgement to Reviewers of *Games* in 2017*Games* **2018**, *9*(1), 3; doi:10.3390/g9010003 - 12 January 2018**Abstract **

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On the Query Complexity of Black-Peg AB-Mastermind*Games* **2018**, *9*(1), 2; doi:10.3390/g9010002 - 2 January 2018**Abstract **

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Mastermind is a two players zero sum game of imperfect information. Starting with Erdős and Rényi (1963), its combinatorics have been studied to date by several authors, e.g., Knuth (1977), Chvátal (1983), Goodrich (2009). The first player, called “codemaker”, chooses a secret code

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Mastermind is a two players zero sum game of imperfect information. Starting with Erdős and Rényi (1963), its combinatorics have been studied to date by several authors, e.g., Knuth (1977), Chvátal (1983), Goodrich (2009). The first player, called “codemaker”, chooses a secret code and the second player, called “codebreaker”, tries to break the secret code by making as few guesses as possible, exploiting information that is given by the codemaker after each guess. For variants that allow color repetition, Doerr et al. (2016) showed optimal results. In this paper, we consider the so called Black-Peg variant of Mastermind, where the only information concerning a guess is the number of positions in which the guess coincides with the secret code. More precisely, we deal with a special version of the Black-Peg game with *n* holes and $k\ge n$ colors where no repetition of colors is allowed. We present upper and lower bounds on the number of guesses necessary to break the secret code. For the case $k=n$ , the secret code can be algorithmically identified within less than $(n-3)\lceil {\mathrm{log}}_{2}n\rceil +\frac{5}{2}n-1$ queries. This result improves the result of Ker-I Ko and Shia-Chung Teng (1985) by almost a factor of 2. For the case $k>n$ , we prove an upper bound of $(n-2)\lceil {\mathrm{log}}_{2}n\rceil +k+1$ . Furthermore, we prove a new lower bound of *n* for the case $k=n$ , which improves the recent $n-\mathrm{log}\mathrm{log}(n)$ bound of Berger et al. (2016). We then generalize this lower bound to *k* queries for the case $k\ge n$ .
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The Effects of Excluding Coalitions*Games* **2018**, *9*(1), 1; doi:10.3390/g9010001 - 1 January 2018**Abstract **

One problem in cooperative game theory is to model situations when two players refuse to cooperate (or the problem of quarreling members in coalitions). One example of such exclusions is the coalition statements of parliamentary parties. Other situations in which incompatible players affect

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One problem in cooperative game theory is to model situations when two players refuse to cooperate (or the problem of quarreling members in coalitions). One example of such exclusions is the coalition statements of parliamentary parties. Other situations in which incompatible players affect the outcome are teams in firms and markets, for example. To model these exclusions in cooperative game theory, the excluded coalitions value (${\phi}^{E}$ value) was introduced. This value is based on the Shapley value and takes into account that players exclude coalitions with other players. In this article, we deduce some properties of this new value. After some general results, we analyze the apex game that could be interpreted as a team situation and the glove game that models markets where sellers and buyers deal. For team situations, we show that all employees have a common interest for cooperation. On asymmetric markets, excluding coalitions affect the market players of the scarce side to a higher extent.
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Public-Goods Games with Endogenous Institution-Formation: Experimental Evidence on the Effect of the Voting Rule*Games* **2017**, *8*(4), 52; doi:10.3390/g8040052 - 2 December 2017**Abstract **

We report experimental results on voluntary contributions to public-goods provision from situations in which parties can create institutions to impose a certain contribution level on its members. We focus on a public-goods game where the joint decisions inside the institution are made based

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We report experimental results on voluntary contributions to public-goods provision from situations in which parties can create institutions to impose a certain contribution level on its members. We focus on a public-goods game where the joint decisions inside the institution are made based on the plurality voting rule. We show that, comparing to the unanimity voting rule, the plurality rule results in a significant and large decrease in the institution initiation rate, along with a significant and large increase in the institution implementation rate. In the end, as the two effects cancel each other out, the choice of the voting rule does not significantly affect the average contribution level or efficiency.
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Polarization and Segregation through Conformity Pressure and Voluntary Migration: Simulation Analysis of Co-Evolutionary Dynamics*Games* **2017**, *8*(4), 51; doi:10.3390/g8040051 - 22 November 2017**Abstract **

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While conformity pressures people to assimilate in a community, an individual occasionally migrates among communities when the individual feels discomfort. These two factors cause segregation and cultural diversity within communities in the society. By embedding a migration dynamic into Kuran and Sandholm’s model

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While conformity pressures people to assimilate in a community, an individual occasionally migrates among communities when the individual feels discomfort. These two factors cause segregation and cultural diversity within communities in the society. By embedding a migration dynamic into Kuran and Sandholm’s model (2008) of preference evolution, we build an agent-based model to see how the variance of preferences in the entire society *quantitatively* changes over time. We find from the Monte-Carlo simulations that, while preferences assimilate *within a community*, self-selected migrations enlarge the diversity of preferences *over communities* in the society. We further study how the arrival rate of migration opportunities and the degree of conformity pressures affect the variance of preferences.
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Contribution-Based Grouping under Noise*Games* **2017**, *8*(4), 50; doi:10.3390/g8040050 - 17 November 2017**Abstract **

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Many real-world mechanisms are “noisy” or “fuzzy”, that is the institutions in place to implement them operate with non-negligible degrees of imprecision and error. This observation raises the more general question of whether mechanisms that work in theory are also robust to more

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Many real-world mechanisms are “noisy” or “fuzzy”, that is the institutions in place to implement them operate with non-negligible degrees of imprecision and error. This observation raises the more general question of whether mechanisms that work in theory are also robust to more realistic assumptions such as noise. In this paper, in the context of voluntary contribution games, we focus on a mechanism known as “contribution-based competitive grouping”. First, we analyze how the mechanism works under noise and what happens when other assumptions such as population homogeneity are relaxed. Second, we investigate the welfare properties of the mechanism, interpreting noise as a policy instrument, and we use logit dynamic simulations to formulate mechanism design recommendations.
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Computing Human-Understandable Strategies: Deducing Fundamental Rules of Poker Strategy*Games* **2017**, *8*(4), 49; doi:10.3390/g8040049 - 6 November 2017**Abstract **

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Algorithms for equilibrium computation generally make no attempt to ensure that the computed strategies are understandable by humans. For instance the strategies for the strongest poker agents are represented as massive binary files. In many situations, we would like to compute strategies that

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Algorithms for equilibrium computation generally make no attempt to ensure that the computed strategies are understandable by humans. For instance the strategies for the strongest poker agents are represented as massive binary files. In many situations, we would like to compute strategies that can actually be implemented by humans, who may have computational limitations and may only be able to remember a small number of features or components of the strategies that have been computed. For example, a human poker player or military leader may not have access to large precomputed tables when making real-time strategic decisions. We study poker games where private information distributions can be arbitrary (i.e., players are dealt cards from different distributions, which depicts the phenomenon in large real poker games where at some points in the hand players have different distribution of hand strength by applying Bayes’ rule given the history of play in the hand thus far). We create a large training set of game instances and solutions, by randomly selecting the information probabilities, and present algorithms that learn from the training instances to perform well in games with unseen distributions. We are able to conclude several new fundamental rules about poker strategy that can be easily implemented by humans.
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Game Theoretic Interaction and Decision: A Quantum Analysis*Games* **2017**, *8*(4), 48; doi:10.3390/g8040048 - 6 November 2017**Abstract **

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An interaction system has a finite set of agents that interact pairwise, depending on the current state of the system. Symmetric decomposition of the matrix of interaction coefficients yields the representation of states by self-adjoint matrices and hence a spectral representation. As a

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An interaction system has a finite set of agents that interact pairwise, depending on the current state of the system. Symmetric decomposition of the matrix of interaction coefficients yields the representation of states by self-adjoint matrices and hence a spectral representation. As a result, cooperation systems, decision systems and quantum systems all become visible as manifestations of special interaction systems. The treatment of the theory is purely mathematical and does not require any special knowledge of physics. It is shown how standard notions in cooperative game theory arise naturally in this context. In particular, states of general interaction systems are seen to arise as linear superpositions of pure quantum states and Fourier transformation to become meaningful. Moreover, quantum games fall into this framework. Finally, a theory of Markov evolution of interaction states is presented that generalizes classical homogeneous Markov chains to the present context.
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Construction of Subgame-Perfect Mixed-Strategy Equilibria in Repeated Games*Games* **2017**, *8*(4), 47; doi:10.3390/g8040047 - 1 November 2017**Abstract **

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This paper examines how to construct subgame-perfect mixed-strategy equilibria in discounted repeated games with perfect monitoring. We introduce a relatively simple class of strategy profiles that are easy to compute and may give rise to a large set of equilibrium payoffs. These sets

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This paper examines how to construct subgame-perfect mixed-strategy equilibria in discounted repeated games with perfect monitoring. We introduce a relatively simple class of strategy profiles that are easy to compute and may give rise to a large set of equilibrium payoffs. These sets are called self-supporting sets, since the set itself provides the continuation payoffs that are required to support the equilibrium strategies. Moreover, the corresponding strategies are simple as the players face the same augmented game on each round but they play different mixed actions after each realized pure-action profile. We find that certain payoffs can be obtained in equilibrium with much lower discount factor values compared to pure strategies. The theory and the concepts are illustrated in 2 × 2 games.
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Shapley Value-Based Payment Calculation for Energy Exchange between Micro- and Utility Grids*Games* **2017**, *8*(4), 45; doi:10.3390/g8040045 - 23 October 2017**Abstract **

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In recent years, microgrids have developed as important parts of power systems and have provided affordable, reliable, and sustainable supplies of electricity. Each microgrid is managed as a single controllable entity with respect to the existing power system but demands for joint operation

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In recent years, microgrids have developed as important parts of power systems and have provided affordable, reliable, and sustainable supplies of electricity. Each microgrid is managed as a single controllable entity with respect to the existing power system but demands for joint operation and sharing the benefits between a microgrid and its hosting utility. This paper is focused on the joint operation of a microgrid and its hosting utility, which cooperatively minimize daily generation costs through energy exchange, and presents a payment calculation scheme for power transactions based on a fair allocation of reduced generation costs. To fairly compensate for energy exchange between the micro- and utility grids, we adopt the cooperative game theoretic solution concept of Shapley value. We design a case study for a fictitious interconnection model between the Mueller microgrid in Austin, Texas and the utility grid in Taiwan. Our case study shows that when compared to standalone generations, both the micro- and utility grids are better off when they collaborate in power exchange regardless of their individual contributions to the power exchange coalition.
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Representations of Political Power Structures by Strategically Stable Game Forms: A Survey*Games* **2017**, *8*(4), 46; doi:10.3390/g8040046 - 23 October 2017**Abstract **

We survey the results on representations of committees and constitutions by game forms that possess some kind of equilibrium strategies for each profile of preferences of the players. The survey is restricted to discrete models, that is, we deal with finitely many players

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We survey the results on representations of committees and constitutions by game forms that possess some kind of equilibrium strategies for each profile of preferences of the players. The survey is restricted to discrete models, that is, we deal with finitely many players and alternatives. No prior knowledge of social choice is assumed: As far as definitions are concerned, the paper is self-contained. Section 2 supplies the necessary general tools for the rest of the paper. Each definition is followed by a simple (but nontrivial) example. In Section 3 we give a complete account of representations of committees (proper and monotonic simple games), by exactly and strongly consistent social choice functions. We start with Peleg’s representations of weak games, and then provide a complete and detailed account of Holzman’s solution of the representation problem for simple games without veto players. In Section 4 we deal with representations of constitutions by game forms. Following Gärdenfors we model a constitution by a monotonic and superadditive effectivity function. We fully characterize the representations for three kinds of equilibrium: Nash equilibrium; acceptable equilibrium (Pareto optimal Nash equilibrium); and strong Nash equilibrium. We conclude in Section 5 with a report on two recent works on representations of constitutions under incomplete information.
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Moral Entitlements and Aspiration Formation in Asymmetric Bargaining: Experimental Evidence from Germany and China*Games* **2017**, *8*(4), 44; doi:10.3390/g8040044 - 17 October 2017**Abstract **

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Using a unique experimental data set, we investigate how asymmetric legal rights shape bargainers’ aspiration levels through moral entitlements derived from equity norms and number prominence. Aspiration formation is typically hard to observe in real life. Our study involves 15 negotiations from Germany

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Using a unique experimental data set, we investigate how asymmetric legal rights shape bargainers’ aspiration levels through moral entitlements derived from equity norms and number prominence. Aspiration formation is typically hard to observe in real life. Our study involves 15 negotiations from Germany and China. Over the course of the negotiation, bargainers discuss the distribution of an amount of money by alternating offers until they consent or break off. Legal rights are randomly assigned by asymmetric outside options. We videotape and code the in-group discussions. In total, verbal data from 30 groups, 1100 pages of transcripts, and 65 h of discussions are content-analyzed. Our main finding is that strong groups derive and defend moral entitlements from equity concerns with regard to their outside options. They strive for equitable but unequal distributions (e.g., proportional split and split the difference). Moral entitlements materialize in the recorded aspiration levels and final payoffs, which exceed the equal split. By contrast, weak groups aim at equality. Over the course of the negotiation, equity tends to lose, while the prominence of round numbers gains importance. Similarities between the subject pools are found in that equity and prominence are both decisive for the formation of aspiration levels. Chinese negotiations are characterized by long periods of stagnation, only minimal concessions, and the communication of false goals. By contrast, Germans steadily reduce their goals and make concessions.
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Social Preferences and Context Sensitivity*Games* **2017**, *8*(4), 43; doi:10.3390/g8040043 - 13 October 2017**Abstract **

This paper is a partial review of the literature on ‘social preferences'. There are empirical findings that convincingly demonstrate the existence of social preferences, but there are also studies that indicate their fragility. So how robust are social preferences, and how exactly are

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This paper is a partial review of the literature on ‘social preferences'. There are empirical findings that convincingly demonstrate the existence of social preferences, but there are also studies that indicate their fragility. So how robust are social preferences, and how exactly are they context dependent? One of the most promising insights from the literature, in my view, is an equilibrium explanation of mutually referring conditional social preferences and expectations. I use this concept of equilibrium, summarized by means of a figure, to discuss a range of empirical studies. Where appropriate, I also briefly discuss a couple of insights from the (mostly parallel) evolutionary literature about cooperation. A concrete case of the Orma in Kenya will be used as a motivating example in the beginning.
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