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.      

Probabilities and beliefs

Choice-theoretic definitions of subjective probabilities originated with the work of Ramsey and de Finetti and attained their definitive form in the work of Savage. These probabilities are intended to provide a numerical representation of a decision maker's beliefs regarding the likely realization of alternative events. In this article, I argue that the choice-theoretic definitions of subjective probabilities involve a tacit convention—namely, state-independent utility functions—that is not implied by the axioms, and, as a consequence, choice-theoretic subjective probabilities, even when they exist, do not necessarily represent the decision makers' beliefs.     

Self enforceable paths in extensive form games

This paper explores the idea of forward induction for extensive games. It interprets this idea as a general behavioral principle the technical details of which have to be worked out in each specific case. Because of its cooperative ingredient, this approach should be contrasted with the usual approaches of non-cooperative game theory which are rooted in individual rationality.     

Endogenous correlated equilibria in noncooperative games

Most of the results of modern game theory presuppose that the choices rational agents make in noncooperative games are probabilistically independent. In this paper I argue that there is noa priori reason for rational agents to assume probabilistic independence. I introduce a solution concept for noncooperative games called anendogenous correlated equilibrium, which generalizes the Nash equilibrium concept by dropping probabilistic independence. I contrast the endogenous correlated equilibrium with the correlated equilibrium defined by Aumann (1974, 1987). I conclude that in general the endogenous correlated equilibrium concept is a more appropriate solution concept for noncooperative game theory than the less general Nash equilibrium concept. I close by discussing the relationship between endogenous correlated equilibrium and the game solution concept calledrationalizability introduced by Bernheim (1984) and Pearce (1984).     

Markov interactions in a class of dynamic games

This paper contributes to the understanding of economic strategic behaviors in inter-temporal settings. Comparing the MPE and the OLNE of a widely used class of differential games it is shown: (i) what qualifications on behaviors a markov (dynamic) information structure brings about compared with an open-loop (static) information structure, (ii) what is the reason leading to intensified or reduced competition between the agents in the long run. It depends on whether agents’ interactions are characterized by markov substitutability or markov complementarity, which can be seen as dynamic translations of the ideas of strategic substitutability and strategic complementarity (Bulow et al. 1985, Journal of Political Economy 93:488–511). In addition, an important practical contribution of the paper for modelers is to show that these results can be directly deduced from the payoff structure, with no need to compute equilibria first. I dedicate this paper to Philippe Michel, who introduced me to the literature on differential games.     

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.     

A reconstruction of Jeffrey's notion of ratifiability in terms of counterfactual beliefs

We formalize Jeffrey's (1983) notion of ratifiability and show that the resulting formal structure can be obtained more directly by means of a theory of counterfactual beliefs. One implication is that, under the appropriate formalizations, together with certain restrictions on beliefs, Bayesian decision theory and causal decision theory coincide.     

Five legitimate definitions of correlated equilibrium in games with incomplete information

Aumann's (1987) theorem shows that correlated equilibrium is an expression of Bayesian rationality. We extend this result to games with incomplete information.First, we rely on Harsanyi's (1967) model and represent the underlying multiperson decision problem as a fixed game with imperfect information. We survey four definitions of correlated equilibrium which have appeared in the literature. We show that these definitions are not equivalent to each other. We prove that one of them fits Aumann's framework; the agents normal form correlated equilibrium is an expression of Bayesian rationality in games with incomplete information.We also follow a universal Bayesian approach based on Mertens and Zamir's (1985) construction of the universal beliefs space. Hierarchies of beliefs over independent variables (states of nature) and dependent variables (actions) are then constructed simultaneously. We establish that the universal set of Bayesian solutions satisfies another extension of Aumann's theorem.We get the following corollary: once the types of the players are not fixed by the model, the various definitions of correlated equilibrium previously considered are equivalent.     

Parents' beliefs about children's emotions,children's emotion understanding,and classroom adjustment in middle childhood

To explore how parental socialization of emotion may influence children's emotion understanding, which then guides children's interpretations of emotion‐related situations across contexts, we examined the pathways between socialization of emotion and children's adjustment in the classroom, with children's emotion understanding as an intervening variable. Specifically, children's emotion understanding was examined as a mediator of associations between mothers' beliefs about the value and danger of children's emotions and children's adjustment in the classroom within an SEM framework. Classroom adjustment was estimated as a latent variable and included social, emotional, and behavioral indices. Covariates included maternal education, and child gender and ethnicity. Participants were a diverse group of 201 third‐graders (116 African American, 81 European American, 4 Biracial; 48.8% female), their mothers, and teachers. Results revealed that emotion‐related beliefs (value and danger) had no direct influence on classroom adjustment. However, children whose mothers endorsed the belief that emotions are dangerous demonstrated less emotion understanding and were less well‐adjusted in the classroom. Mothers' belief that emotions are valuable was not independently associated with emotion understanding. Findings point to the important role of emotion understanding in children's development across contexts (family, classroom) and developmental domains (social, emotional, behavioral) during the middle childhood years.     

Predicting the development of dependency on the teacher in late childhood: The role of peer victimization and peer beliefs

Evidence indicates that being overly dependent on the teacher places children's academic and socioemotional development at risk. However, little is known about what predicts dependency on the teacher or how the quality of interactions occurring within the classroom peer system may impact how children relate to their teacher the following school year. The current study tested the proposition that peer victimization may result in negative perceptions of classmates (i.e., peer beliefs), leading to overreliance on teachers. Data were collected from 365 children in the fall and spring of their third/fourth grade year and the fall of their fourth/fifth grade year (195 girls; M_age = 8.92 years; 86.8% white). Peer-reports of peer victimization and self-reports of peer beliefs were collected at each wave of the study. Teacher-reports of dependency were collected in the fall of the first and second years of the study. Path analyses showed that for boys peer victimization directly predicted higher levels of dependency on the next year's teacher, and, for boys and girls, peer victimization indirectly predicted dependency through lower levels of prosocial peer beliefs. Supplemental analyses assessing teacher-child conflict and closeness confirmed that findings were unique to dependency. These results underscore how children's perceptions of their classmates may contribute to dependency in their relationships with teachers and highlight the need for further research into the transactional and cumulative impact of difficulties within peer and teacher relationships.     

Another impossibility result for normal form games

It is shown that the axioms Cubitt and Sugden (1994; Economic J. 104: 798) impose on a theory of rationally justifiable play (TRJP) do not prevent the possibility that two players necessarily disagree concerning the probability they ascribe to the choice of a third player. This appears to indicate that those axioms are not sufficient for defining a `reasonable' TRJP. In addition, for the case in which a player's beliefs are statistically independent, conditions for a TRJP are suggested under which the existence of a quasi-strict equilibrium is sufficient, but the existence of a consistent n-pair is not, for defining a TRJP meeting those requirements.     

Cultural family beliefs,maternal sacrifice,and adolescent psychological competence in Chinese poor single‐mother families

Research on cultural family beliefs and family processes as protective factors of adolescent development is severely lacking in the Chinese culture. Based on 432 Chinese single‐mother families living in poverty in Hong Kong, the relationships among Chinese cultural beliefs of familism, adolescent perceived maternal sacrifice, and psychological competence (indexed by a clear and healthy identity, cognitive competence, and a positive future outlook) were examined. Results showed that adolescents' perceived maternal sacrifice mediated the influence between maternal Chinese cultural beliefs of familism and the psychological competence of adolescents raised in poor single‐mother families in Hong Kong. The present study underscores the importance of cultural family beliefs and parental sacrifice on nurturing adolescent psychological competence in Chinese single‐mother families living in poverty, which contributes to the construction of a family resilience model applicable to Chinese communities.     

On social utility payoffs in games: a methodological comparison between Behavioural and Rational Game Theory

Are the recent findings of Behavioural Game Theory (BGT) on unselfish behaviours relevant for the progress of game theory? Is the methodology of BGT, centred around the attempt to study theoretically players’ utility functions in the light of the feedback that experimental evidence can produce on the theory, a satisfactory one? Or is the creation of various types of ‘social preferences’ just wasteful tinkering? This article compares BGT with the methodology of Rational Game Theory (RGT). BGT is viewed as a more promising and constructive approach, with regard to the relationship between experimental data and theoretical modelling. However, I also argue that today RGT and BGT are closer to one another than often thought.     

Dynamic consistency in extensive form decision problems

In a stimulating paper, Piccione and Rubinstein (1997) argued how a decision maker could undertake dynamically inconsistent choices when, in an extensive form decision problem, she has a particular type of imperfect recall named absentmindedness. Such memory limitation obtains whenever information sets include decision histories along the same decision path. Starting from work focusing on the absentminded driver example, and independently developed by Segal (2000) and Dimitri (1999), the main theorem of this article provides a general result of dynamically consistent choices, valid for a large class of finite extensive form decision problems without nature.      

Dynamic focal points in N-person coordination games

To understand how groups coordinate, we study infinitely repeated N-player coordination games in the context of strategic uncertainty. In a situation where players share no common language or culture, ambiguity is always present. However, finding an adequate principle for a common language is not easy: a tradeoff between simplicity and efficiency has to be made. All these points are illustrated on repeated N-player coordination games on m loci. In particular, we demonstrate how a common principle can accelerate coordination. We present very simple rules that are optimal in the space of all languages for m (number of coordination loci) from 2 to 5 and for all N, the number of players. We also show that when more memory is used in the language (strategies), players may not coordinate, whereas this is never the case when players remember only the previous period.     

Communication and its cost in graph-restricted games

We consider cooperative games which are modified by subordination to communication networks. Two players i and j will be able to cooperate only if they pay for the cost w _ij of their communication link (i,j). Coalitions of players are treated similarly and a new characteristic function form game is developed thus. We also examine incentive for players to cooperate in such situations as well as a related index of a player's communicative strength.     

Joint Beliefs in Conflictual Coordination Games

The traditional solution concept for noncooperative game theory is the Nash equilibrium, which contains an implicit assumption that players probability distributions satisfy t probabilistic independence. However, in games with more than two players, relaxing this assumption results in a more general equilibrium concept based on joint beliefs (Vanderschraaf, 1995). This article explores the implications of this joint-beliefs equilibrium concept for two kinds of conflictual coordination games: crisis bargaining and public goods provision. We find that, using updating consistent with Bayes rule, players beliefs converge to equilibria in joint beliefs which do not satisfy probabilistic independence. In addition, joint beliefs greatly expand the set of mixed equilibria. On the face of it, allowing for joint beliefs might be expected to increase the prospects for coordination. However, we show that if players use joint beliefs, which may be more likely as the number of players increases, then the prospects for coordination in these games declines vis-à-vis independent beliefs.     

More nonconcavities in information processing functions

The productivity of (human) information processing as an economic activity is a question that is raising some interest. Using Marschak's evaluation framework, Radner and Stiglitz have shown that, under certain conditions, the production function of this activity has increasing marginal returns in its initial stage. This paper shows that, under slightly different conditions, this information processing function has repeated convexities with ongoing processing activity. Even for smooth changes in the signals' likelihoods, the function is only piecewise smooth with non-differentiable convexities at points of conditional changes of action. For linear likelihood functions the processing value proves to be piecewise linear with convexities at these levels. This revised version was published online in June 2006 with corrections to the Cover Date.     

Ordinal equivalence of power notions in voting games

In this paper, we are concerned with the preorderings (SS) and (BC) induced in the set of players of a simple game by the Shapley–Shubik and the Banzhaf–Coleman's indices, respectively. Our main result is a generalization of Tomiyama's 1987 result on ordinal power equivalence in simple games; more precisely, we obtain a characterization of the simple games for which the (SS) and the (BC) preorderings coincide with the desirability preordering (T), a concept introduced by Isbell (1958), and recently reconsidered by Taylor (1995): this happens if and only if the game is swap robust, a concept introduced by Taylor and Zwicker (1993). Since any weighted majority game is swap robust, our result is therefore a generalization of Tomiyama's. Other results obtained in this paper say that the desirability relation keeps itself in all the veto-holder extensions of any simple game, and so does the (SS) preordering in all the veto-holder extensions of any swap robust simple game.