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.     

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.     

Loss Aversion and Bargaining

We consider bargaining situations where two players evaluate outcomes with reference-dependent utility functions, analyzing the effect of differing levels of loss aversion on bargaining outcomes. We find that as with risk aversion, increasing loss aversion for a player leads to worse outcomes for that player in bargaining situations. An extension of Nash's axioms is used to define a solution for bargaining problems with exogenous reference points. Using this solution concept we endogenize the reference points into the model and find a unique solution giving reference points and outcomes that satisfy two reasonable properties, which we predict would be observed in a steady state. The resulting solution also emerges in two other approaches, a strategic (non-cooperative) approach using Rubinstein's (1982) alternating offers model and a dynamic approach in which we find that even under weak assumptions, outcomes and reference points converge to the steady state solution from any non-equilibrium state.     

Risk aversion and the relationship between Nash's solution and subgame perfect equilibrium of sequential bargaining

This article presents some new, intuitive derivations of several results in the bargaining literature. These new derivations clarify the relationships among these results and allow them to be understood in a unified way. These results concern the way in which the risk posture of the bargainers affects the outcome of bargaining as predicted by Nash's (axiomatic) solution of a static bargaining model (Nash, 1950) and by the subgame perfect equilibrium of the infinite horizon sequential bargaining game analyzed by Rubinstein (1982). The analogous, experimentally testable predictions for finite horizon sequential bargaining games are also presented.     

Two-person sequential bargaining behavior with exogenous breakdown

We examine bargaining behavior in a noncooperative game in which players alternate in making and responding to proposals over the division of a given surplus. Although the number of bargaining periods is unlimited and time is not discounted, the bargaining is subject to exogenous breakdown at each period with a fixed probability which is common knowledge. We manipulate three probabilities of break-down in a between-subjects design that allows comparison with previous studies of two-person bargaining with time discounting. Assuming that subjects maximize expected utility, and this utility is measured by monetary payoffs, our results reject both the subgame perfect equilibrium and equal split solutions. Data analyses reveal that a substantial percentage of subjects behave adaptively in that they systematically search for the highest acceptable demands.     

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.     

Alternating-Offer Bargaining and Common Knowledge of Rationality

This paper reconsiders Rubinstein's alternating-offer bargaining game with complete information. We define rationalizability and trembling- hand rationalizability (THR) for multi-stage games with observed actions. We show that rationalizability does not exclude perpetual disagreement or delay, but that THR implies a unique solution. Moreover, this unique solution is the unique subgame perfect equilibrium (SPE). Also, we reconsider an extension of Rubinstein's game where a smallest money unit is introduced: THR rules out the non-uniqueness of SPE in some particular case. Finally, we investigate the assumption of boundedly rational players. Perpetual disagreement is excluded, but not delay. Furthermore, we cannot use the asymmetric Nash bargaining solution as an approximation of the alternating-offer bargaining model once the players are boundedly rational ones.     

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.     

Uncertainty,the bargaining problem,and the Nash-Zeuthen solution

The Zeuthen bargaining model occupies a prominent place among those theories of the bargaining process that have been formulated and expounded by economists. Its solution to the bargaining problem is essentially economic, since invariant utility functions based on economic factors alone determine the outcome. However, this paper shows that a necessary condition for reaching the Zeuthen solution (shown by Harsanyi to be mathematically equivalent to the game-theoretic solution of Nash's theory) is that bargainers initially take up positions on opposite sides of the outcome that maximizes their utility product. Whether utility functions are mutually known or unknown, inherent in the bargaining situation itself is the requirement that bargainers be at least initially uncertain as to each other's subsequent concession behaviour. With uncertainty, von Neumann-Morgenstern rationality implies that each bargainer would make an initial demand that maximizes the expected gain from holding fast. Therefore, even if Zeuthen's concession criterion should subsequently dictate concession behaviour, expected utility maximization within the context of subjective uncertainty may well yield initial demands that are inconsistent with reaching the Nash-Zeuthen solution. Finally, a general methodological conclusion that emerges from the analysis is that, since the bargaining process necessarily proceeds from a context of subjective uncertainty, greater emphasis needs to be placed on its role as a device for affecting expectations.     

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 \(\delta _A+\delta _B < 1\).     

Interpersonal comparisons of utility in bargaining: evidence from a transcontinental ultimatum game

This article presents the experimental results of a “Transcontinental Ultimatum Game” implemented between India and France. We use a standard ultimatum game, but in one treatment, Indian subjects made offers to French subjects (ItoF treatment) and, in another treatment, French subjects made offers to Indian subjects (FtoI treatment). We observed that FtoI treatment bargaining mostly ended up with unequal splits of money in favor of French, while nearly equal splits were the most frequent outcome in ItoF treatment interactions. The experimental results are organized through a standard social reference model, modified for taking into account the different marginal value of money for bargainers. In our model, bargaining is driven by relative standing comparisons between players, occurring in terms of real earnings (that is monetary earnings corrected for a purchasing power factor) obtained in the game. The norm of equity behind the equalization of real earnings is called local equity norm, and contrasted to a global equity norm which would encompass the wealth of players beyond the game. According to what we observed, no beyond-game concern seems to be relevantly endorsed by subjects.     

Private and public pension schemes: Elements of a comparative approach

The comparative approach offers improved understanding of what is at stake for enterprises, workers and States in developing private pension schemes to fill the gap left by public schemes. The features that distinguish private schemes result from the freedom with which they were designed — by enterprises themselves or through collective bargaining — and the limits imposed by growing state intervention in granting fiscal concessions or stipulating social guarantees. Depending on how these various components are combined, private schemes may offers workers added protection or form a source of insecurity.     

Finite Alternating-Move Arbitration Schemes and the Equal Area Solution

We start by considering the Alternate Strike (AS) scheme, a real-life arbitration scheme where two parties select an arbitrator by alternately crossing off at each round one name from a given panel of arbitrators. We find out that the AS scheme is not invariant to “bad” alternatives. We then consider another alternating-move scheme, the Voting by Alternating Offers and Vetoes (VAOV) scheme, which is invariant to bad alternatives. We fully characterize the subgame perfect equilibrium outcome sets of these above two schemes in terms of the rankings of the parties over the alternatives only. We also identify some of the typical equilibria of these above two schemes. We then analyze two additional alternating-move schemes in which players’ current proposals have to either honor or enhance their previous proposals. We show that the first scheme’s equilibrium outcome set coincides with that of the AS scheme, and the equilibrium outcome set of the second scheme coincides with that of the VAOV scheme. Finally, it turns out that all schemes’ equilibrium outcome sets converge to the Equal Area solution’s outcome of cooperative bargaining problem, if the alternatives are distributed uniformly over the comprehensive utility possibility set and as the number of alternatives tends to infinity. Journal of Economic Literature Classification Number: C72.     

Three Models of Conflict Resolution: Effects on Intergroup Expectancies and Attitudes

Gordon Allport (1960) once noted that there are many different approaches to conflict resolution, but that few empirical studies have been conducted to examine their effectiveness. This experiment was designed to evaluate the distributive bargaining, integrative bargaining, and interactive problem solving models of conflict resolution, using intergroup expectancies and attitudes as dependent measures. Jewish and Arab students were paired into mixed-ethnicity dyads to discuss the Israeli-Palestinian dispute about Jerusalem according to one of the three models. It was hypothesized that dyad members using the interactive problem solving model would become the least pessimistic about the conflict, and would show the greatest positive change in their attitudes toward members of the other ethnic group. The results support these predictions, and help to make up for the paucity of research on the assessment of conflict resolution strategies.     

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.     

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.     

On the existence of stable states in game problems

Each individual or group of individuals unified by some common interests often have to either accept an offer or reject it. The choice is made on the basis of not only individual or group preferences but on the basis of the reaction to this choice by other people whose preferences are dependent to some extent on this choice. The following question is to be answered: under what conditions is an offer stable, i.e. will it be accepted by each participant in a group with non-coincident interests? To answer this question is to specify in the set of all feasible offers (options or situations) those ones which would be rejected by no participant. According to this intuitive understanding of stable offers a sequence of weak and strong equilibria referred to as active equilibria was defined. A weak equilibrium satisfies the following conditions: (1) It defines stable situations which cannot be changed by a participant because of certain retaliatory actions by the other participants, which would worsen his initial situation. (2) It always exists. (3) Under comparatively wide conditions it has a non-empty intersection with the Pareto set. (4) It includes non-cooperative equilibria well known in the game theory and does not lead to any paradoxes (as, for example, the Nash equilibrium does). These conditions, and especially the third and fourth ones, indicate that the active equilibria can be successfully used in the game theory as well. Strong equilibria define a more stable situation but they do not necessarily meet all four conditions listed above.     

High Stakes and Acceptance Behavior in Ultimatum Bargaining:

This paper presents the results of a within-subject experiment testing whether an increase in the monetary stakes by a factor of 50 – which had never been done before – influences individual behavior in a simple ultimatum bargaining game. Contrary to current wisdom, we found that lowest acceptable offers stated by the responder are proportionally lower in the high-stake condition than in the low-stake condition. This result may be interpreted in terms of the type of utility functions which characterize the subjects. However, in line with prior results, we find that an important increase of the monetary stakes in the ultimatum game has no effect on the offers made by the proposer. Yet, the present research suggests that the reasons underlying these offers are quite different when the stakes are high.     

政策实施成本与上下级政府讨价还价的发生机制——基于四东县休禁牧案例的分析

讨价还价是中国上下级政府互动关系中很重要的一种形式。本文详细记述了四东县草原休禁牧政策实施过程中上下级政府的多轮讨价还价，县政府制定政策，政策严格落实导致政府与民众的冲突频发，乡镇政府以冲突为策略与县政府讨价还价，最终获得了相对宽松的执行空间。本文使用实施成本分析框架解释了讨价还价行为的发生机制，执行过程成本、上级施加成本、民众施加成本高昂，致使乡镇政府只能选择与上级讨价还价。随着社会冲突的加剧，乡镇政府的讨价还价能力相对提高，使得双方的协议点从严格执行端向宽松执行端移动。执行无僵局、协议点左右摆动、短期均衡与讨价还价的交替发生是政策执行中上下级政府讨价还价的基本特征。