Envy Freeness in Experimental Fair Division Problems

Envy is sometimes suggested as an underlying motive in the assessment of different economic allocations. In the theoretical literature on fair division, following Foley [Foley, D. (1967), Yale Economic Essays, 7, 45–98], the term “envy” refers to an intrapersonal comparison of different consumption bundles. By contrast, in its everyday use “envy” involves interpersonal comparisons of well-being. We present, discuss results from free-form bargaining experiments on fair division problems in which inter-and intrapersonal criteria can be distinguished. We find that interpersonal comparisons play the dominant role. The effect of the intrapersonal criterion of envy freeness is limited to situations in which other fairness criteria are not applicable.      

EVOLUTION AND ULTIMATUM BARGAINING

Empirical research has discovered that experimental subjects in ultimatum bargaining situations generally fail to play the decision-theoretic optimum strategy, and instead play something between that strategy and a fair split. In evolutionary dynamics, fair division and nearly fair division strategies often go to fixation and weakly dominated strategies can do quite well. Computer simulations were done using three different ultimatum bargaining games as determinates of fitness. (1) No tendency toward the elimination of weakly dominated strategies was observed, with or without mutation. (2) Strategies making fair demands had sizable basins of attraction. (3) In a system where five different demands can be made, demands closest to (approximately) 91% had the largest basins of attraction. (4) If the strategies have thresholds for acceptable demands, rather than individuated responses to each demand, the apparent optimum demand may be quite low: 64% for one set of trials.     

Simultaneous moves multi-person continuous time concession game

The alternating offers strategic bargaining approach (Rubinstein, 1982) suggest a unique perfect equilibrium. In this equilibrium the bargaining ends immediately and the outcome is Pareto optimal. In this paper we consider a bargaining game in which players move simultaneously. Since we allow players only to lower their demands over time we denote this game as a concession game. We demonstrate that in this game there is a perfect equilibrium in which the bargaining lasts long enough so that the pie shrinks to zero. We show that we can generalize this game to a multi-player concession game in a straightforward manner and so avoiding the difficulties of generalizing the alternating offers game to the multi-player case.     

To be fair,generous, or selfish: The effect of relationship on Chinese children’s distributive allocation and procedural application

Previous research has found that children’s sharing with others relies on fairness norms, but also varies according to their social relationships. The current study focuses on the conflict between fairness and relationship, exploring their impacts across two resource allocation contexts. We used a parallel work task to explore the effect of relationship with different recipients (friend, stranger, or disliked peer) on three allocation patterns (generous, fair, or selfish), when children directly allocated resources (distributive allocation), or applied different procedures to recipients (procedural application). Participants consisted of 123 Chinese children between the ages of 6 and 12. We found that in the distributive allocation context, in which participants directly decided the outcome, children primarily considered their relationship with recipients when dividing resources, not fairness. However, in the procedural application context, in which children could choose different allocation procedures for recipients, children primarily preferred fairness, regardless of social relationship. Moreover, when making distributive allocations, 6‐ to 8‐year‐olds were more selfish toward their disliked peers, whereas 9‐ to 12‐year‐olds tended to be more fair and generous toward their friends and strangers. These findings shed light on the link between social relationship and fairness within different allocation contexts among children of Chinese cultural background.     

Driving a hard bargain is a balancing act: how social preferences constrain the negotiation process

Theory and Decision - We investigate the haggling process in bargaining. Using an experimental bargaining game, we find that a first offer has a significant impact on the bargaining outcome even if...     

Ultimatum decision-making: A test of reciprocal kindness

While fairness is often mentioned as a determinant of ultimatum bargaining behavior, few data sets are available that can test theories that incorporate fairness considerations. This paper tests the reciprocal kindness theory in Rabin (1993 Incorporating fairness into game theory and economics, The American Economic Review 83: 1281-1302) as an application to the one-period ultimatum bargaining game. We report on data from 100 ultimatum games that vary the financial stakes of the game from $1 to $15. Responder behavior is strongly in support of the kindness theory and proposer behavior weakly in support of it. Offer percentages and past offers influence behavior the most, whereas the size of the pie has a marginally significant effect on offer percentages. The data is more in support of reciprocal kindness than alternative theories of equal-split or learning behavior, although the data also weakly support a minimum percentage threshold hypothesis. As a whole, our results together with existing studies suggest that, for smaller stakes games, fairness considerations dominate monetary considerations. This has implications for more complicated naturally occurring bargaining environments in which the financial stakes can vary widely.     

Subgame perfect equilibrium in a bargaining model with deterministic procedures

Two players, A and B, bargain to divide a perfectly divisible pie. In a bargaining model with constant discount factors, \(\delta _A\) and \(\delta _B\), we extend Rubinstein (Econometrica 50:97–110, 1982)’s alternating offers procedure to more general deterministic procedures, so that any player in any period can be the proposer. We show that each bargaining game with a deterministic procedure has a unique subgame perfect equilibrium (SPE) payoff outcome, which is efficient. Conversely, each efficient division of the pie can be supported as an SPE outcome by some procedure if \(\delta _A+\delta _B\ge 1\), while almost no division can ever be supported in SPE if \(\delta _A+\delta _B < 1\).     

Process fairness,outcome fairness,and dynamic consistency: Experimental evidence for risk and ambiguity

Literature on fairness preferences distinguishes between outcome fairness, concerning the final allocation of payoffs, and process fairness, concerning the expected allocation of payoffs. It is not obvious, however, whether process fairness can consistently be implemented. Once uncertainty is resolved and outcomes are determined, the ex-ante procedurally fair decision maker may become consequentialist ex-post, and reconsider her choice on the basis of the observed outcomes. We present experimental evidence on dynamic consistency of social preferences under both known risk and ambiguity. A significant share of people subscribe to process fairness both before and after the resolution of uncertainty.     

Distributive justice of bargaining and risk sensitivity

In this paper we will point out some possibilities and limitations of the discussion of distributive justice by bargaining in the classical bargaining models.We start by considering a kind of bargaining situation where two persons with different risk aversions have to distribute a given quantity of a certain good. Then we define a model in which two bargaining situations are compared. In both situations two persons divide a quantity of a certain good; in the second situation one of the persons, say person 2, is replaced by a more risk averse person. From a well-known theorem of Kihlstrom, Roth and Schmeidler it follows that in the Nash solution, the Kalai-Smorodinsky solution and the Maschler-Perles solution person 1 prefers the situation with the more risk averse opponent.In both classes of problems the judgement of distributive justice is impossible because of an informational poverty of the classical bargaining model. We propose to integrate changes in the economic situation of the persons into the model.Therefore, in a third step, we compare two distributive situations, where differences in the situations are implied by changes in the initial endowments of the persons. Under the assumption that each person has a decreasing local risk aversion, we show that every reallocation of the initial endowments is enlarged or at least preserved by risk sensitive bargaining solutions. This fact has some significance for the discussion of distributive justice in social decision making by bargaining.     

Bargaining Solutions as Social Compromises

A bargaining solution is a social compromise if it is metrically rationalizable, i.e., if it has an optimum (depending on the situation, smallest or largest) distance from some reference point. We explore the workability and the limits of metric rationalization in bargaining theory where compromising is a core issue. We demonstrate that many well-known bargaining solutions are social compromises with respect to reasonable metrics. In the metric approach, bargaining solutions can be grounded in axioms on how society measures differences between utility allocations. Using this approach, we provide an axiomatic characterization for the class of social compromises that are based on p-norms and for the attending bargaining solutions. We further show that bargaining solutions which satisfy Pareto Optimality and Individual Rationality can always be metrically rationalized.     

On the existence of fair matching algorithms

We analyze the Gale-Shapley matching problem within the context of Rawlsian justice. Defining a fair matching algorithm by a set of 4 axioms (Gender Indifference, Peer Indifference, Maximin Optimality, and Stability), we show that not all preference profiles admit a fair matching algorithm, the reason being that even this set of minimal axioms is too strong in a sense. Because of conflict between Stability and Maximin Optimality, even the algorithm which generates the mutual agreement match, paradoxically, has no chance to be fair.We then relax the definition of fairness (by giving preference to Stability over Maximin Optimality) and again find that some preference profiles admit a fair matching algorithm, while others still do not, but the mutual agreement algorithm now is fair under this definition.The paper then develops a test, which determines, for a given preference profile, whether a fair algorithm exists or not.     

Overlapping coalitions,bargaining and networks

We model the formation of coalitions that are not necessarily disjoint. We propose a new bargaining game that yields an overlapping coalition structure as an outcome. Equilibrium does not always exist in pure strategies for such a game, but we show that it always exists with a mild degree of mixed strategies. We derive conditions for a complete duality between networks and overlapping coalitions, and we provide a new rationale for the sequential formation of networks.     

Fair division of indivisible items between two players: design parameters for Contested Pile methods

Contested Pile methods are two-phase procedures for the fair allocation of indivisible items to two players. In the Generation Phase, items over which the players’ preferences differ widely enough are allocated. “Contested” items are placed in the Contested Pile, which is then allocated in the Splitting Phase. Each phase can be carried out using several different techniques; we perform a comprehensive analysis of the resulting design variants using a computational model. The properties of fairness and efficiency, generally achieved in the Generation Phase, must be traded off against robustness to manipulation. We find that the recently developed Undercut procedure for the Splitting Phase outperforms alternative methods in both fairness and efficiency. In general, procedures that keep the Contested Pile relatively small and incorporate the Undercut procedure score well in both fairness and efficiency, but are prone to manipulation.     

Fair (and not so fair) division

Drawbacks of existing procedures are illustrated and a method of efficient fair division is proposed that avoids them. Given additive participants’ utilities, each item is priced at the geometric mean (or some other function) of its two highest valuations. The utilities are scaled so that the market clears with the participants’ purchases proportional to their entitlements. The method is generalized to arbitrary bargaining sets and existence is proved. For two or three participants, the expected utilities are unique. For more, under additivity, the geometric mean separates the prices where uniqueness holds and where it fails; it holds for the geometric mean except in one case where refinement is needed.      

中国公众的收入公平感:一种新制度主义社会学的解释

借助“嵌入性”和“合法性”概念,本文将关于社会公平感形成的基本共识理论与自我评价理论整合起来,提出了一种关于分配公平感的新制度主义社会学解释。本文认为,社会成员对自己所获得的社会资源是否公平的感受取决于社会成员的自我认知与社会共识之间达成一致的程度。当自我感知的地位低于社会评价的地位时,个体倾向于认为自己所得到的社会资源是不公平的;相反,当自我感知的地位与社会评价地位一致或更高时,个体则更有可能认为自己所得到的社会资源是公平的。基于上述命题,本文进一步提出研究假设,相对于主观阶层地位达到或高于其职业声望者,主观阶层地位低于其职业声望的人更倾向于认为自己的收入是不公平的。对2006年中国综合社会调查(CGSS2006)数据的分析结果较有力地支持了这一研究假设;研究结果表明,本文所提出的理论命题比基于理性选择的自利理论更好地解释了为什么社会成员,尤其是社会经济地位较低者,依然会认为自己的收入是公平的。     

BARGAINING AND DELAY: THE ROLE OF EXTERNAL INFORMATION

This paper examines the notion that delay in reaching agreement in bargaining may be caused by learning that is independent of the bargaining procedure. In particular, learning is not due to inference from the observed offers and responses of the opponent, but derives from observation of an exogenous, costly signal – we call this 'investigation'. First we observe that even if learning is costless and perfectly informative, investigation may not occur in equilibrium. Under more general conditions, however, uninformed agents typically have an incentive to try to manipulate their prior beliefs through investigation. The main result is that investigation by an uninformed agent may result in significant delay occurring before agreement is reached. We show that this delay may be sustained in the limit as the length of time period vanishes, and that this result depends crucially on the properties of the information production technology underlying investigation. Also, the delay we observe is shown to be robust to changes in the bargaining procedure.     

Effects of fixed costs in two-person sequential bargaining

Rubinstein (1982) considered the problem of dividing a given surplus between two players sequentially, and then proposed a model in which the two players alternately make and respond to each other's offers through time. He further characterized the perfect equilibrium outcomes, which depend on the players time preferences and order of moves. Using both equal and unequal bargaining cost conditions and an unlimited number of rounds, two experiments were designed to compare the perfect equilibrium model to alternative models based on norms of fairness. We report analyses of final agreements, first offers, and number of bargaining rounds, which provide limited support to the perfect equilibrium model, and then conclude by recommending a shift in focus from model testing to specification of the conditions favoring one model over another.     

Bargaining with reasonable aspirations

Consider bargaining situations with two persons, where both have a specific idea of what would be an equitable outcome. In case these ideas differ, a procedure is needed which leads to a compromise between the different views. In the present paper the axioms of restricted homogeneity and of relative monotonicity are introduced, each of which, together with other standard conditions, allows one to determine a solution for this class of bargaining problems. An extension of this solution to bargaining problems with more than two persons is formulated.     

Externalities in a Bargaining Model of Public Price Announcements and Resale

We study the one-seller/two-buyer bargaining problem with negative identity-dependent externalities with an alternating offer bargaining model in which new owners of the object have the opportunity of resale. We identify the generically unique subgame perfect equilibrium outcome. The resale opportunity increases the competition among the buyers and therefore benefits the seller. When competition between buyers is very fierce, the seller may prefer to respond to bids rather than to propose an offer herself: a first-mover disadvantage.     

Nullified equal loss property and equal division values

We provide characterizations of the equal division values and their convex mixtures, using a new axiom on a fixed player set based on player nullification which requires that if a player becomes null, then any two other players are equally affected. Two economic applications are also introduced concerning bargaining under risk and common-pool resource appropriation.