Nash Demand Game Research Articles

A note on the risk dominance of the Nash demand game

A note on the risk dominance of the Nash demand game

The Map/Territory Relationship in Game-Theoretic Modeling of Cultural Evolution

AbstractThe cultural red king effect occurs when discriminatory bargaining practices emerge because of a disparity in learning speed between members of a minority and a majority. This effect has been shown to occur in some Nash Demand Game models and has been proposed as a tool for shedding light on the origins of sexist and racist discrimination in academic collaborations. This article argues that none of the three main strategies used in the literature to support the epistemic value of these models—structural similarity, empirical confirmation, and how-possibly explanations—provide strong support for this modeling practice in its present form.

Pledge-and-review bargaining

This paper presents a novel dynamic bargaining game where every party is proposing only its own contribution, before all pledges must be unanimously approved. I show that, with uncertain tolerance for delay, each equilibrium pledge maximizes an asymmetric Nash product. The weights on others' payoffs increase in the uncertainty, but decrease in the correlation of the shocks. The weights vary pledge to pledge, and this implies that the outcome is generically inefficient. The Nash demand game and its mapping to the Nash bargaining solution follow as a limiting case. The model sheds light on the Paris climate change agreement, but it also applies to negotiations between policymakers or business partners that have differentiated responsibilities or expertise.

Indirect Dynamic Negotiation in the Nash Demand Game

The paper addresses a problem of sequential bilateral bargaining with incomplete information. We proposed a decision model that helps agents to successfully bargain by performing indirect negotiation and learning the opponent’s model. Methodologically the paper casts heuristically-motivated bargaining of a self-interested independent player into a framework of Bayesian learning and Markov decision processes. The special form of the reward implicitly motivates the players to negotiate indirectly, via closed-loop interaction. We illustrate the approach by applying our model to the Nash demand game, which is an abstract model of bargaining. The results indicate that the established negotiation: i) leads to coordinating players’ actions; ii) results in maximising success rate of the game and iii) brings more individual profit to the players.

Positive feedback in coordination games: Stochastic evolutionary dynamics and the logit choice rule

We study the problem of stochastic stability for evolutionary dynamics under the logit choice rule. We consider general classes of coordination games, symmetric or asymmetric, with an arbitrary number of strategies, which satisfies the marginal bandwagon property (i.e., there is positive feedback to coordinate). Our main result is that the most likely evolutionary escape paths from a status quo convention consist of a series of identical mistakes. As an application of our result, we show that the Nash bargaining solution arises as the long run convention for the evolutionary Nash demand game under the usual logit choice rule. We also obtain a new bargaining solution if the logit choice rule is combined with intentional idiosyncratic plays. The new bargaining solution is more egalitarian than the Nash bargaining solution, demonstrating that intentionality implies equality under the logit choice model.

Articulating bargaining theories: movement, chance, and necessity as descriptive principles

The Nash Demand Game (NDG) has been one of the first models (Nash in Econometrica 21(1):128–140, 1953. https://doi.org/10.2307/1906951) that has tried to describe the process of negotiation, competition, and cooperation. This model has had enormous repercussions and has leveraged basic and applied research on bargaining processes. Therefore, we wonder whether it is possible to articulate extensive and multiple developments into a single unifying framework. The Viability Theory has this inclusive approach. Thus, we investigate the NDG under this point of view, and, carrying out this work, we find that the answer is not only affirmative but that we also advance in characterising viable NDGs. In particular, we found foundations describe the distributive Bargaining Theory: the principle of movement and the principle of chance and necessity. Finally, this initial work has many interesting perspectives. The probably most important idea is to integrate developments of the Bargaining Theory and thus capture the complexity of the real world in an articulated way.

A prospect theory Nash bargaining solution and its stochastic stability

We consider the long-run outcomes of bargaining games when players obey prospect theory. We extend the evolutionary bargaining model of Young (1993) to a two-stage Nash demand game. Two players simultaneously choose whether to exercise an outside option in the first stage and play the Nash demand game in the second stage, which will be reached only if neither player exercises the outside option. We address the influence on the stochastically stable division of reference-dependent preferences where the reference point is the value of the outside option. We show that the division consistently differs from the Nash bargaining solution under expected utility theory. Inspired by this, we propose a prospect theory Nash bargaining solution , which coincides with the stochastically stable division.

A Prospect Theory Nash Bargaining Solution and Its Stochastic Stability

We consider the long-run outcomes of bargaining games when players obey prospect theory. We extend the evolutionary bargaining model of Young (1993) to a two-stage Nash demand game. Two players simultaneously choose whether to exercise an outside option in the first stage and play the Nash demand game in the second stage, which will be reached only if neither player exercises the outside option. We address the influence on the stochastically stable division of reference-dependent preferences where the reference point is the value of the outside option. We show that the division consistently differs from the Nash bargaining solution under expected utility theory. Inspired by this, we propose a prospect theory Nash bargaining solution, which coincides with the stochastically stable division.

An implementation of the Nash bargaining solution by iterated strict dominance

Anbarcı (2001) presented a modification of the “divide the dollar” game, in which equal division is obtained by iterated removal of strictly dominated strategies. I extend his mechanism to a general Nash demand game and show that it implements the Nash bargaining solution.

Strategic uncertainty and equilibrium selection in discontinuous games

We introduce the new concept of prudent equilibrium to model strategic uncertainty, and prove it exists in large classes of discontinuous games. When the game is better-reply secure, we show that prudent equilibrium refines Nash equilibrium. In contrast with the current literature, we don't use probabilities to model players' strategies and beliefs about other players' strategies. We provide examples (first-price auctions, location game, Nash demand game, etc.) where prudent equilibrium concept removes most non-intuitive solutions of the game.

Collective risk-taking in the commons

The management of natural commons is typically subject to threshold effects. If individuals are risk-averse, some of the recent economic literature holds that uncertainty on the threshold may have a positive impact by lowering incentives to over-consume. The paper shows that, this intuitive result may unravel when uncertainty is modeled as a discrete or multimodal distribution. Using a variant of the Nash demand game with two thresholds, two types of Nash equilibria typically coexist: cautious (respectively, dangerous) equilibria in which agents coordinate on the low threshold (resp. the high threshold). When both types of equilibria coexist, the symmetric dangerous equilibrium is always Pareto dominated by the symmetric cautious equilibrium, and the latter is always Pareto efficient. We use an experimental setting to assess the severity of the coordination and equilibrium selection problem. While cautious (resp. dangerous) play is decreasing (resp. increasing) in the probability that the threshold is high, coordination failures are salient for intermediate probabilities where the likelihood of coexistence of both type of equilibria is high. We find that there is a U-shaped relationship between overall coordination and the probability that the threshold is high.

Stakeholder Fairness Under an Induced ‘Veil of Ignorance'

John Rawls introduced the ‘veil of ignorance' in social contract theory to bring about a common conception of justice, and hypothesized that it will enable rational individuals to choose distributive shares on basis of ‘maximin' principle. R. E. Freeman conceptualised stakeholder fairness using the Rawlsian ‘veil of ignorance'. In contrast to Rawls' theory, John Harsanyi postulated that rational individuals behind the ‘veil of ignorance' will choose allocation to maximise expected utility. This article investigates how subjects choose allocations behind the ‘veil of ignorance,' in a laboratory experiment, and interprets the findings in light of stakeholder fairness. The ‘veil of ignorance' was induced on randomly paired and mutually anonymous subjects, who were asked to choose allocations in a simultaneous move discrete choice Nash demand game. Both ‘maximin' principle and expected utility maximisation was found to be used by the subjects. Choice of allocations where no one is worse off vis-à-vis status quo was salient. This is consistent with Freeman's Principle of Governance.

Bargaining by Heterogeneously Responsive Individuals

I study the process of bargaining over a pie of fixed size under the Nash demand game protocol. Motivated by the argument that the manner in which bargaining unfolds often depends on antecedents, I embed the bargaining process in a recurrent framework where individuals obtain random samples from the demands made in the past and respond to this information in a manner that is dictated by their respective behavioural traits. I pursue two specific issues: firstly, is there a behavioural trait (i.e. manner of responding to past information) that outperforms any other behavioural trait in the bargaining game, and secondly, to assess the evolutionary stability of behavioural traits in the bargaining game. In context of the first question, using a bargaining framework where individuals described by a particular behavioural trait bargain with individuals described by another behavioural trait, I find that a population of 'wildly optimistic' individuals obtains almost the entire pie against a population comprised of 'almost any other' behavioural type. Secondly, using a playing-the-field model of bargaining I find that all behavioural traits are unstable, i.e. there exists no state where all the incumbent behavioural traits always obtain a higher fitness than any mutant behavioural trait.The only state where any behavioural trait can co-exist with any other behavioural trait is when each individual demands exactly half of the pie. This demonstrates the importance of the equal-splitting norm for co-existence of various behavioural traits. I also prove a hardwired behaviour-responsive behaviour equivalence theorem, which shows that the analysis of evolutionary stability of a population comprising of various behavioural traits is equivalent to the analysis of evolutionary stability of an invariant population level strategy profiles.

Bargaining by Heterogeneously Responsive Populations

I study the process of bargaining over a pie of fixed size. Motivated by the argument that the manner in which bargaining unfolds often depends on antecedents, I embed the bargaining process in an evolutionary framework. First, I posit that there exist two separate populations, and in each period, one individual is randomly drawn from each population to bargain. The bargaining protocol used is the Nash demand game, and the demand that each individual makes is guided by the demands made by the other population in the recent past. I show that under very general conditions, the bargaining process reaches a convention, where each population settles on demanding a fixed share of the pie, and the demands of each population are both compatible with each other, and cumulatively exhaust the pie. Next, I identify the most advantageous behavioural trait in the long-run: in the convention that is stable in the long-run, a population of 'wildly optimistic' individuals obtains almost the entire pie against a population comprised of 'almost any other' behavioural type. Secondly, since this two-population bargaining game does not allow for evolutionary selection, I analyse the stability of a behavioural trait in a playing-the-field model of bargaining by examining the relative performance of an incumbent population described by a particular behavioural trait against a mutant of another behavioural trait. I show that all behavioural traits are unstable as they are susceptible to invasion by any mutant trait. However, the only state where any behavioural trait can co-exist with any other mutant trait is when the pie is shared equally. This demonstrates the importance of the equal-splitting norm for co-existence of various behavioural traits, and hence, for sustainable population diversity.

How fully do people exploit their bargaining position? The effects of bargaining institution and the 50–50 norm

A recurring puzzle in bargaining experiments is that individuals under-exploit their bargaining position, compared to theoretical predictions. We conduct an experiment using two institutions: Nash demand game (NDG) and unstructured bargaining game (UBG). Unlike most previous experiments, disagreement payoffs are earned rather than assigned, and about one-fourth of the time, one bargainer's disagreement payoff is more than half the cake size (“dominant bargaining power”), so that equal splits are not individually rational.Subjects under-respond to their bargaining position most severely in the NDG without dominant bargaining power. Responsiveness increases in the UBG, but is still lower than predicted; the same is true for the NDG with dominant bargaining power. Only in the UBG with dominant bargaining power – the combination of a bargaining institution with low strategic uncertainty and elimination of the 50–50 “security blanket” – do subjects approximately fully exploit their bargaining position.

Robustness to strategic uncertainty in the Nash demand game

This paper studies the role of strategic uncertainty in the Nash demand game. A player’s uncertainty about another player’s strategy is modeled as an atomless probability distribution over that player’s strategy set. A strategy profile is robust to strategic uncertainty if it is the limit, as uncertainty vanishes, of some sequence of strategy profiles in which every player’s strategy is optimal under his or her uncertainty about the others (Andersson et al., 2014). In the context of the Nash demand game, we show that robustness to symmetric (asymmetric) strategic uncertainty singles out the (generalized) Nash bargaining solution. The least uncertain party obtains the bigger share.

On the Coevolution of Social Norms in Primitive Societies

Two parties bargaining over a pie, the size of which is determined by their previous investment decisions. The bargaining rule is sensitive to investment behavior. Two games are considered. In both, bargaining proceeds according to the Nash Demand Game when a symmetric investments profi le is observed. When, on the other hand, an asymmetric investments profi le is observed, we assume that bargaining proceeds according to the Ultimatum Game in one case and according to a Dictator Game in the other. We hereby show that in both games when a unique stochastically stable outcome exists it supports an homogeneous behavior in the whole population both at the investment stage and at the distribution stage. A norm of investment and a norm of division must therefore coevolve in the two games, supporting both the efficient investment pro le and the egalitarian distribution of the surplus, respectively. The two games differ depending on the conditions needed for the two norms to coevolve. The games are proposed to explain the social norms used in modern hunter-gatherer societies.

Does Rawls' 'Original Position' Induce Fairness? Experimental Findings on Selection Criteria in a Discrete Nash Demand Game Played from Behind the 'Veil of Ignorance'

Rawls (1958) suggested that it is possible to arrive at a fair allocation in a 2-player Nash demand game by granting equal gains to both players. Rawls theorized that players themselves would select this allocation if they bargain from the 'original position'. Harsanyi (1958) suggested the utilitarian solution, wherein rational players playing the Nash demand game from behind the 'veil of ignorance' should maximize the aggregate payoff instead of equalizing the gains vis-a-vis endowment. Harsanyi (1975) shows that the utilitarian solution generates the fairest allocation. In this paper we experimentally examined whether subjects select the utilitarian solution while playing a Nash demand game from behind the 'veil of ignorance'. For the purpose of the experiment we used a Nash demand game with discrete strategies. We found that subjects without any exposure to game theory found it difficult to identify the utilitarian solution. Subjects with exposure to basic game theory could identify the utilitarian solution when it was one of the Nash equilibria of the Nash demand game. However, only a few could identify the utilitarian solution when it was not a Nash equilibrium.

The Coalitional Nash Bargaining Solution with Simultaneous Payoff Demands

We consider a standard coalitional bargaining game where once a coalition forms it exits as in Okada (2011), however, instead of alternating offers, we have simultaneous payoff demands. We focus in the producer game he studies. Each player is chosen with equal probability. If that is the case, she can choose any coalition she belongs to. However, a coalition can form if an only if payoff demands are feasible as in the Nash (1953) demand game. After smoothing the game (as in Van Damme (1991)), when the noise vanishes, when the discount factor is close to 1, and as in Okada's (2011), the coalitional Nash bargaining solution is the unique stationary subgameperfect equilibrium.

Evolution of Three Norms of Distributive Justice in an Extended Nash Demand Game

The Nash demand game (NDG) has been at the center of attention when explaining moral norms of distributive justice on the basis of the game theory. This paper describes the demand-intensity game (D-I game), which adds an “intensity” dimension to NDG in order to discuss various scenarios for the evolution of norms concerning distributive justice, while keeping such simplicity that it can be analyzed by the concepts and tools of the game theory. We perform an ESS analysis and evolutionary simulations, followed by the analysis of replicator dynamics. It is shown that the three norms emerge: the one claiming an equal distribution (Egalitarianism), the one claiming the full amount (Libertarianism), and, as the special case of Libertarianism, the one claiming the full amount but conceding the resource in conflict (Wimpy libertarianism). The evolution of these norms strongly depends on the conflict cost parameter. Egalitarianism emerges with a larger conflict cost while Libertarianism with a smaller cost. Wimpy libertarianism emerges with a relatively larger conflict cost in libertarianism. The simulation results show that there are three types of evolutionary scenarios in general. We see in most of the trials the population straightforwardly converges to Libertarianism or Egalitarianism. It is also shown that, in some range of the conflict cost, the population nearly converges to Egalitarianism, which is followed by the convergence to Libertarianism. It is shown that this evolutionary transition depends on the quasi stability of Egalitarianism.