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.     

Delay in a bargaining game with contracts

In a multilateral bargaining game where a proposer and responders can set up a “principal–agent” relationship by means of binding cash-offer contracts, we show that there is a Markov SPE with a delay in reaching an agreement. We also show that all the individually rational and efficient payoffs can be supported by SPE.      

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.     

A bargaining model with players' perceptions on the retractability of offers

This paper studies a generalization of Rubinstein's bargaining model with retractable offers. The model incorporates and parameterizes the bargainers' perceptions on the retractability of offers. Our key result characterizes the limiting set of perfect equilibria as the time interval between two consecutive offers tends to zero. In this limit, for any possible players' perceptions on the retractability of offers such that at least one of the players perceives that there is at least a small chance that offers may be retractable, the bargaining game possesses a continuum of perfect equilibria.     

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 \(\delta _A+\delta _B < 1\).     

A Theory of Rational Choice under Ignorance

This paper contributes to a theory of rational choice for decision-makers with incomplete preferences due to partial ignorance, whose beliefs are representable as sets of acceptable priors. We focus on the limiting case of `Complete Ignorance' which can be viewed as reduced form of the general case of partial ignorance. Rationality is conceptualized in terms of a `Principle of Preference-Basedness', according to which rational choice should be isomorphic to asserted preference. The main result characterizes axiomatically a new choice-rule called `Simultaneous Expected Utility Maximization'. It can be interpreted as agreement in a bargaining game (Kalai-Smorodinsky solution) whose players correspond to the (extremal) `acceptable priors' among which the decision maker has suspended judgment. An essential but non-standard feature of Simultaneous Expected Utility choices is their dependence on the entire choice set. This is justified by the conception of optimality as compromise rather than as superiority in pairwise comparisons.     

Fairness motivation in bargaining: a matter of principle

In this paper, we study the role of fairness motivation in bargaining. We show that bargaining between two strongly fairness motivated individuals who have different views about what represents a fair division may end in disagreement. Further, by applying the Nash bargaining solution, we study the influence of fairness motivation on the bargaining outcome when an agreement is reached. In particular, we show that the bargaining outcome is sensitive to the fairness motivation of the two individuals, unless they both consider an equal division fair. We argue that our results accommodate existing experimental and field data on bargaining.     

Bargaining and Strategic Demand Commitment

On occasion, in multilateral negotiations, interested parties make unilateral demands. Certain agreements need unanimity. However, a lesser degree of consensus may be feasible. In this paper, an alternating demand bargaining game among n players is proposed, which envisages varying consensus requirements and commitment, both crucial in generating a unique and efficient outcome of the bargaining process.     

Ambiguity when playing coordination games across cultures

Cultural differences can be a source of ambiguity in coordination games. As players are likely to experience more ambiguity when playing a different culture, we expect players to choose safer strategies. We run experiments with a stag hunt and bargaining coordination game. Using a between-subjects design, we vary the identity of the opponent between someone of the same culture or a different culture. We compare the responses of British and East Asian students at the University of Exeter and show the cultural identity of the opponent by physical appearance. While we find no significant differences between treatments for East Asians, the British choose the safer option in the stag hunt and demand more of the pie in the bargaining game when faced with an opponent from a different culture.

     

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.     

Endogenizing the order of moves in matrix games

Players often have flexibility in when they move and thus whether a game is played simultaneously or sequentially may be endogenously determined. For 2 × 2 games, we analyze this using an extended game. In a stage prior to actual play, players choose in which of two periods to move. A player moving at the first opportunity knows when his opponent will move. A player moving at the second turn learns the first mover's action. If both select the same turn, they play a simultaneous move subgame.If both players have dominant strategies in the basic game, equilibrium payoffs in the basic and extended games are identical. If only one player has a dominant strategy or if the unique equilibrium in the basic game is in mixed strategies, then the extended game equilibrium payoffs differ if and only if some pair of pure strategies Pareto dominates the basic game simultaneous play payoffs. If so, sequential play attains the Pareto dominating payoffs. The mixed strategy equilibrium occurs only when it is not Pareto dominated by some pair of pure strategies.In an alternative extended game, players cannot observe delay by opponents at the first turn. Results for 2×2 games are essentially the same as with observable delay, differing only when only one player has a dominant strategy.     

On the evaluation of solution concepts

It is proposed that solution concepts for games should be evaluated in a way that is analogous to the way a logic is evaluated by a model theory for the language. A solution concept defines a set of strategy profiles, as a logic defines a set of theorems. A model theoretic analysis for a game defines a class of models, which are abstract representations of particular plays of the game. Given an appropriate definition of a model, one can show that various solution concepts are characterized by intuitively natural classes of models in the same sense that the set of theorems of a logic is characterized by a class of models of the language. Sketches of characterization results of this kind are given for rationalizability, Nash equilibrium, and for a refinement of rationalizability —strong rationalizability — that has some features of an equilibrium concept. It is shown that strong rationalizability is equivalent to Nash equilibrium in perfect information games. Extensions of the model theoretic framework that represent belief revision and that permit the characterization of other solution concepts are explored informally.     

Socially Structured Games

We generalize the concept of a cooperative non-transferable utility game by introducing a socially structured game. In a socially structured game every coalition of players can organize themselves according to one or more internal organizations to generate payoffs. Each admissible internal organization on a coalition yields a set of payoffs attainable by the members of this coalition. The strengths of the players within an internal organization depend on the structure of the internal organization and are represented by an exogenously given power vector. More powerful players have the power to take away payoffs of the less powerful players as long as those latter players are not able to guarantee their payoffs by forming a different internal organization within some coalition in which they have more power. We introduce the socially stable core as a solution concept that contains those payoffs that are both stable in an economic sense, i.e., belong to the core of the underlying cooperative game, and stable in a social sense, i.e., payoffs are sustained by a collection of internal organizations of coalitions for which power is distributed over all players in a balanced way. The socially stable core is a subset and therefore a refinement of the core. We show by means of examples that in many cases the socially stable core is a very small subset of the core. We will state conditions for which the socially stable core is non-empty. In order to derive this result, we formulate a new intersection theorem that generalizes the KKMS intersection theorem. We also discuss the relationship between social stability and the wellknown concept of balancedness for NTU-games, a sufficient condition for non-emptiness of the core. In particular we give an example of a socially structured game that satisfies social stability and therefore has a non-empty core, but whose induced NTU-game does not satisfy balancedness in the general sense of Billera.      

Bayesian boundedly rational agents play the Finitely Repeated Prisoner's Dilemma

In this paper a model of boundedly rational decision making in the Finitely Repeated Prisoner's Dilemma is proposed in which: (1) each player is Bayesianrational; (2) this is common knowledge; (3) players are constrained by limited state spaces (their Bayesian minds) in processing (1) and (2). Under these circumstances, we show that cooperative behavior may arise as an individually optimal response, except for the latter part of the game. Indeed, such behaviorwill necessarily obtain in long enough games if belief systems satisfy a natural condition: essentially, that all events consistent with the players' analysis of the game be attributed by them positive (although arbitrarily small) subjective probability.     

Implementing equal division with an ultimatum threat

We modify the payment rule of the standard divide the dollar (DD) game by introducing a second stage and thereby resolve the multiplicity problem and implement equal division of the dollar in equilibrium. In the standard DD game, if the sum of players’ demands is less than or equal to a dollar, each player receives what he demanded; if the sum of demands is greater than a dollar, all players receive zero. We modify this second part, which involves a harsh punishment. In the modified game \((D\!D^{\prime })\) , if the demands are incompatible, then players have one more chance. In particular, they play an ultimatum game to avoid the excess. In the two-player version of this game, there is a unique subgame perfect Nash equilibrium in which players demand (and receive) an equal share of the dollar. We also provide an \(n\) -player extension of our mechanism. Finally, the mechanism we propose eliminates not only all pure strategy equilibria involving unequal divisions of the dollar, but also all equilibria where players mix over different demands in the first stage.     

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.     

Finite Horizon Bargaining With Outside Options And Threat Points

We characterize equilibrium behavior in a finite horizon multiple-pie alternating offer bargaining game in which both agents have outside options and threat points. In contrast to the infinite horizon case the strength of the threat to delay agreement is non-stationary and decreases over time. Typically the delay threat determines equilibrium proposals in early periods, while the threat to opt out characterizes those in later ones. Owing to this non-stationarity both threats may appear in the equilibrium shares immediately agreed upon in the first period. When the threat to opt out is empty for both agents, the outcome corresponds exactly with the (generalized) Nash bargaining solution. The latter observation may prove useful for designing experiments that are meant to test economic models that include a bargaining stage.JEL CLASSIFICATION CODES: C78, C91.The useful remarks of an anonymous referee greatly improved the presentation of the material and are gratefully acknowledged. The usual disclaimer applies.     

Implementing egalitarianism in a class of Nash demand games

We add a stage to Nash’s demand game by allowing the greedier player to revise his demand if the demands are not jointly feasible. If he decides to stick to his initial demand, then the game ends and no one receives anything. If he decides to revise it down to \(1-x\), where x is his initial demand, the revised demand is implemented with certainty. The implementation probability changes linearly between these two extreme cases. We derive a condition on the feasible set under which the two-stage game has a unique subgame perfect equilibrium. In this equilibrium, there is first-stage agreement on the egalitarian demands. We also study two n-player versions of the game. In either version, if the underlying bargaining problem is “divide-the-dollar,” then equal division is sustainable in a subgame perfect equilibrium if and only if the number of players is at most four.     

Two simple characterizations of the Nash bargaining solution

We provide two alternative characterizations of the Nash bargaining solution. We introduce new simple axioms, strong undominatedness by the disagreement point, and egalitarian Pareto optimality. First, we prove that the Nash solution is characterized by symmetry, scale invariance, independence of irrelevant alternatives, and strong undominatedness by the disagreement point. Second, we replace the independence of irrelevant alternatives axiom with the sandwich axiom (Rachmilevitch in Theory Decis 80:427–442, 2016) and egalitarian Pareto optimality. We then demonstrate that the Nash solution is characterized by symmetry, scale invariance, strong undominatedness by the disagreement point, the sandwich axiom, and egalitarian Pareto optimality.