Marginality,differential marginality,and the Banzhaf value

We revisit the Nowak (Int J Game Theory 26:137–141, 1997) characterization of the Banzhaf value via 2-efficiency, the Dummy player axiom, symmetry, and marginality. In particular, we provide a brief proof that also works within the classes of superadditive games and of simple games. Within the intersection of these classes, one even can drop marginality. Further, we show that marginality and symmetry can be replaced by van den Brink fairness/differential marginality. For this axiomatization, 2-efficiency can be relaxed into superadditivity on the full domain of games.     

The <Emphasis Type="Italic">γ</Emphasis>-core in Cournot oligopoly TU-games with capacity constraints

In cooperative Cournot oligopoly games, it is known that the β-core is equal to the α-core, and both are non-empty if every individual profit function is continuous and concave (Zhao, Games Econ Behav 27:153–168, 1999b). Following Chander and Tulkens (Int J Game Theory 26:379–401, 1997), we assume that firms react to a deviating coalition by choosing individual best reply strategies. We deal with the problem of the non-emptiness of the induced core, the γ-core, by two different approaches. The first establishes that the associated Cournot oligopoly Transferable Utility (TU)-games are balanced if the inverse demand function is differentiable and every individual profit function is continuous and concave on the set of strategy profiles, which is a step forward beyond Zhao’s core existence result for this class of games. The second approach, restricted to the class of Cournot oligopoly TU-games with linear cost functions, provides a single-valued allocation rule in the γ-core called Nash Pro rata (NP)-value. This result generalizes Funaki and Yamato’s (Int J Game Theory 28:157–171, 1999) core existence result from no capacity constraint to asymmetric capacity constraints. Moreover, we provide an axiomatic characterization of this solution by means of four properties: efficiency, null firm, monotonicity, and non-cooperative fairness.     

Values for rooted-tree and sink-tree digraph games and sharing a river

We introduce values for rooted-tree and sink-tree digraph games axiomatically and provide their explicit formula representation. These values may be considered as natural extensions of the lower equivalent and upper equivalent solutions for line-graph games studied in van den Brink et al. (Econ Theory 33:349–349, 2007). We study the distribution of Harsanyi dividends. We show that the problem of sharing a river with a delta or with multiple sources among different agents located at different levels along the riverbed can be embedded into the framework of a rooted-tree or sink-tree digraph game correspondingly.     

NTU core,TU core and strong equilibria of coalitional population games with infinitely many pure strategies

Theory and Decision - Inspired by Scarf&nbsp;(J Econ Theory 3: 169–181, &nbsp;1971), Zhao (Int J Game Theory 28: 25–34, 1999), Sandholm&nbsp;(Population games and...     

On Loss Aversion in Bimatrix Games

In this article three different types of loss aversion equilibria in bimatrix games are studied. Loss aversion equilibria are Nash equilibria of games where players are loss averse and where the reference points—points below which they consider payoffs to be losses—are endogenous to the equilibrium calculation. The first type is the fixed point loss aversion equilibrium, introduced in Shalev (2000; Int. J. Game Theory 29(2):269) under the name of ‘myopic loss aversion equilibrium.’ There, the players’ reference points depend on the beliefs about their opponents’ strategies. The second type, the maximin loss aversion equilibrium, differs from the fixed point loss aversion equilibrium in that the reference points are only based on the carriers of the strategies, not on the exact probabilities. In the third type, the safety level loss aversion equilibrium, the reference points depend on the values of the own payoff matrices. Finally, a comparative statics analysis is carried out of all three equilibrium concepts in 2 × 2 bimatrix games. It is established when a player benefits from his opponent falsely believing that he is loss averse.     

Hierarchies achievable in simple games

A previous work by Friedman et al. (Theory and Decision, 61:305–318, 2006) introduces the concept of a hierarchy of a simple voting game and characterizes which hierarchies, induced by the desirability relation, are achievable in linear games. In this paper, we consider the problem of determining all hierarchies, conserving the ordinal equivalence between the Shapley–Shubik and the Penrose–Banzhaf–Coleman power indices, achievable in simple games. It is proved that only four hierarchies are non-achievable in simple games. Moreover, it is also proved that all achievable hierarchies are already obtainable in the class of weakly linear games. Our results prove that given an arbitrary complete pre-ordering defined on a finite set with more than five elements, it is possible to construct a simple game such that the pre-ordering induced by the Shapley–Shubik and the Penrose–Banzhaf–Coleman power indices coincides with the given pre-ordering.     

Strategic games with security and potential level players

This paper examines the existence of strategic solutions to finite normal form games under the assumption that strategy choices can be described as choices among lotteries where players have security- and potential level preferences over lotteries (e.g., Cohen, Theory and Decision, 33, 101–104, 1992, Gilboa, Journal of Mathematical Psychology, 32, 405–420, 1988, Jaffray, Theory and Decision, 24, 169–200, 1988). Since security- and potential level preferences require discontinuous utility representations, standard existence results for Nash equilibria in mixed strategies (Nash, Proceedings of the National Academy of Sciences, 36, 48–49, 1950a, Non-Cooperative Games, Ph.D. Dissertation, Princeton University Press, 1950b) or for equilibria in beliefs (Crawford, Journal of Economic Theory, 50, 127–154, 1990) do not apply. As a key insight this paper proves that non-existence of equilibria in beliefs, and therefore non-existence of Nash equilibria in mixed strategies, is possible in finite games with security- and potential level players. But, as this paper also shows, rationalizable strategies (Bernheim, Econometrica, 52, 1007–1028, 1984, Moulin, Mathematical Social Sciences, 7, 83–102, 1984, Pearce, Econometrica, 52, 1029–1050, 1984) exist for such games. Rationalizability rather than equilibrium in beliefs therefore appears to be a more favorable solution concept for games with security- and potential level players.      

Maximin play in completely mixed strategic games

Since the seminal paper of Nash (Proc Natl Acad Sci USA 36:48–49, 1950) game theoretic literature has focused mostly on equilibrium and not on maximin (minimax) strategies. In a recent paper of Pruzhansky (Int J Game Theory 40:351–365, 2011) it was shown that under fairy general conditions maximin strategies in completely mixed games can guarantee the same expected payoff as completely mixed Nash equilibrium strategies. Based on this finding, the current paper argues that maximin strategies have important properties. For instance, maximin strategies may refine Nash equilibria in subjective mixed strategies. Further, Bayesian rationality of the players may favor maximin strategies more often than Nash equilibrium strategies. The paper concludes with several suggestions for further experimental research that may shed more light on whether maximin behavior can explain reality better than Nash equilibrium.     

What are the minimal requirements of rational choice? Arguments from the sequential-decision setting

There are at least two plausible generalisations of subjective expected utility (SEU) theory: cumulative prospect theory (which relaxes the independence axiom) and Levi’s decision theory (which relaxes at least ordering). These theories call for a re-assessment of the minimal requirements of rational choice. Here, I consider how an analysis of sequential decision making contributes to this assessment. I criticise Hammond’s (Economica 44(176):337–350, 1977; Econ Philos 4:292–297, 1988a; Risk, decision and rationality, 1988b; Theory Decis 25:25–78, 1988c) ‘consequentialist’ argument for the SEU preference axioms, but go on to formulate a related diachronic-Dutch-book-style’ argument that better achieves Hammond’s aims. Some deny the importance of Dutch-book sure losses, however, in which case, Seidenfeld’s (Econ Philos 4:267–290, 1988a) argument that distinguishes between theories that relax independence and those that relax ordering is relevant. I unravel Seidenfeld’s argument in light of the various criticisms of it and show that the crux of the argument is somewhat different and much more persuasive than what others have taken it to be; the critical issue is the modelling of future choices between ‘indifferent’ decision-tree branches in the sequential setting. Finally, I consider how Seidenfeld’s conclusions might nonetheless be resisted.     

Conservative Stable Standards of Behavior and φ-Stable Sets

Within Luo’s (2001, J. Math. Econ. 36, 95–109) framework of a general system, I establish an equivalence theorem for the conservative stable standard of behavior (CSSB for short) developed by J. Greenberg (“The Theory of Social Situation: An Alternative Game-Theoretic Approach,” Cambridge University Press, 1990). It is shown that a standard of behavior for a situation is a CSSB if, and only if, its graph is a φ-stable set for the associated general system.     

Stability and Nash implementation in matching markets with couples

We consider two-sided matching markets with couples. First, we extend a result by Klaus and Klijn (J Econ Theory 21: 75–106, 2005, Theorem 3.3) and show that for any weakly responsive couples market, there always exists a “double stable” matching, i.e., a matching that is stable for the couples market and for any associated singles market. Second, we show that for weakly responsive couples markets, the associated stable correspondence is (Maskin) monotonic and Nash implementable. In contrast, the correspondence that assigns all double stable matchings is neither monotonic nor Nash implementable.     

Political influence in multi-choice institutions: cyclicity,anonymity, and transitivity

We study political influence in institutions where each member chooses a level of support for a collective goal. These individual choices determine the degree to which the goal is reached. Influence is assessed by newly defined binary relations, each of which ranks members on the basis of their relative performance at a corresponding level of participation. For institutions with three options (e.g., voting games in which each voter may vote “yes”, “abstain”, or vote “no”), we obtain three influence relations, and show that their strict components may be cyclic. This latter property describes a “paradox of power” which contrasts with the transitivity of the unique influence relation of binary voting games. Weak conditions of anonymity suffice for each of these relations to be transitive. We also obtain a necessary and sufficient condition for each of these relations to be complete. Further, we characterize institutions in which the rankings induced by these relations, and the Banzhaf–Coleman and Shapley–Shubik power indices coincide. We argue that extending the influence relations to firms would be useful in efficiently assigning workers to different units of production. Finally, we provide applications to various forms of political and economic organizations.     

Cycles versus equilibrium in evolutionary games

Mixed-strategy equilibria are typically rather unstable in evolutionary game theory. “Monocyclic” games, such as Rock–Paper–Scissors, have only mixed equilibria, some of which are “stable” in the sense that sequential best replies lead to them; yet, even these games are prone to stable cycles under discrete-time simultaneous best replies, giving an unusual equilibrium-selection problem. This article analyzes such games in a random-utility setting where changing strategies is costly, and the speed of the dynamic is, thus, endogenous. The stochastically stable outcome is determined by the cost of switching strategies; when switching costs are high, mixed equilibria are selected, whereas when switching costs are low, cycles are selected.     

Strategic ambiguity and decision-making: an experimental study

We conducted a set of experiments to compare the effect of ambiguity in single-person decisions and games. Our results suggest that ambiguity has a bigger impact in games than in ball and urn problems. We find that ambiguity has the opposite effect in games of strategic substitutes and complements. This confirms a theoretical prediction made by Eichberger and Kelsey (J Econ Theory 106:436–466, 2002). In addition, we note that subjects’ ambiguity attitudes appear to be context dependent: ambiguity loving in single-person decisions and ambiguity averse in games. This is consistent with the findings of Kelsey and le Roux (Theory Decis 79:667–688, 2015).     

Characterization of dominance relations in finite coalitional games

McGarvey (Econometrica, 21(4), 608–610, 1953) has shown that any irreflexive and anti-symmetric relation can be obtained as a relation induced by majority rule. We address the analogous issue for dominance relations of finite cooperative games with non-transferable utility (coalitional NTU games). We find any irreflexive relation over a finite set can be obtained as the dominance relation of some finite coalitional NTU game. We also show that any such dominance relation is induced by a non-cooperative game through β-effectivity. Dominance relations obtainable through α-effectivity, however, have to comply with a more restrictive condition, which we refer to as the edge-mapping property.     

Equivalent comparisons of information channels

This article constructs a static model of information acquisition when the agent does not know exactly what pieces of information he is missing. A representation of preferences over information channels and menus of lotteries is shown by adapting the model of unforeseen contingencies by Dekel et al. (Econometrica 69:891–934, 2001; Econometrica 75:591–600, 2007), which is an extension of Kreps (Econometrica 47:565–576, 1979; Economic analysis of markets and games: essays in honor of Frank Hahn, 1992). Also, characterisation of informativeness of an information channel analogous to the one by Blackwell (Ann Math Stud 24:265–272, 1953) is examined in conjunction with the preference for flexibility by applying the structure of Shapley value.     

Coalitional desirability and the equal division value

Theory and Decision - We introduce three natural collective variants of the well-known axiom of desirability (Maschler and Peleg in Pac J Math 18:289–328, 1966), which require that if the...     

Elementary proof that mean–variance implies quadratic utility

An extensive literature overlapping economics, statistical decision theory and finance, contrasts expected utility [EU] with the more recent framework of mean–variance (MV). A basic proposition is that MV follows from EU under the assumption of quadratic utility. A less recognized proposition, first raised by Markowitz, is that MV is fully justified under EU, if and only if utility is quadratic. The existing proof of this proposition relies on an assumption from EU, described here as “Buridan’s axiom” after the French philosopher’s fable of the ass that starved out of indifference between two bales of hay. To satisfy this axiom, MV must represent not only “pure” strategies, but also their probability mixtures, as points in the (σ, μ) plane. Markowitz and others have argued that probability mixtures are represented sufficiently by (σ, μ) only under quadratic utility, and hence that MV, interpreted as a mathematical re-expression of EU, implies quadratic utility. We prove a stronger form of this theorem, not involving or contradicting Buridan’s axiom, nor any more fundamental axiom of utility theory.     

Solving the St. Petersburg Paradox in cumulative prospect theory: the right amount of probability weighting

Cumulative Prospect Theory (CPT) does not explain the St. Petersburg Paradox. We show that the solutions related to probability weighting proposed to solve this paradox, (Blavatskyy, Management Science 51:677–678, 2005; Rieger and Wang, Economic Theory 28:665–679, 2006) have to cope with limitations. In that framework, CPT fails to accommodate both gambling and insurance behavior. We suggest replacing the weighting functions generally proposed in the literature by another specification which respects the following properties: (1) to solve the paradox, the slope at zero has to be finite. (2) to account for the fourfold pattern of risk attitudes, the probability weighting has to be strong enough.