Interactive Epistemology and Reasoning: On the Foundations of Game Theory

Game theory describes and analyzes strategic interaction. It is usually distinguished between static games, which are strategic situations in which the players choose only once as well as simultaneously, and dynamic games, which are strategic situations involving sequential choices. In addition, dynamic games can be further classified according to perfect and imperfect information. Indeed, a dynamic game is said to exhibit perfect information, whenever at any point of the game every player has full informational access to all choices that have been conducted so far. However, in the case of imperfect information some players are not fully informed about some choices. Game-theoretic analysis proceeds in two steps. Firstly, games are modelled by so-called form structures which extract and formalize the significant parts of the underlying strategic interaction. The basic and most commonly used models of games are the normal form, which rather sparsely describes a game merely in terms of the players' strategy sets and utilities, and the extensive form, which models a game in a more detailed way as a tree. In fact, it is standard to formalize static games with the normal form and dynamic games with the extensive form. Secondly, solution concepts are developed to solve models of games in the sense of identifying the choices that should be taken by rational players. Indeed, the ultimate objective of the classical approach to game theory, which is of normative character, is the development of a solution concept that is capable of identifying a unique choice for every player in an arbitrary game. However, given the large variety of games, it is not at all certain whether it is possible to device a solution concept with such universal capability. Alternatively, interactive epistemology provides an epistemic approach to game theory of descriptive character. This rather recent discipline analyzes the relation between knowledge, belief and choice of game-playing agents in an epistemic framework. The description of the players' choices in a given game relative to various epistemic assumptions constitutes the fundamental problem addressed by an epistemic approach to game theory. In a general sense, the objective of interactive epistemology consists in characterizing existing game-theoretic solution concepts in terms of epistemic assumptions as well as in proposing novel solution concepts by studying the game-theoretic implications of refined or new epistemic hypotheses. Intuitively, an epistemic model of a game can be interpreted as representing the reasoning of the players. Indeed, before making a decision in a game, the players reason about the game and their respective opponents, given their knowledge and beliefs. Precisely these epistemic mental states on which players base their decisions are explicitly expressible in an epistemic framework. In this PhD thesis, we consider an epistemic approach to game theory from a foundational point of view. In Chapter 1, basic game-theoretic notions as well as Aumann's epistemic framework for games are expounded and illustrated. Also, Aumann's sufficient conditions for backward induction are presented and his conceptual views discussed. In Chapter 2, Aumann's interactive epistemology is conceptually analyzed. In Chapter 3, which is based on joint work with Conrad Heilmann, a three-stage account for dynamic games is introduced and a type-based epistemic model is extended with a notion of agent connectedness. Then, sufficient conditions for backward induction are derived. In Chapter 4, which is based on joint work with Jérémie Cabessa, a topological approach to interactive epistemology is initiated. In particular, the epistemic-topological operator limit knowledge is defined and some implications for games considered. In Chapter 5, which is based on joint work with Jérémie Cabessa and Andrés Perea, Aumann's impossibility theorem on agreeing to disagree is revisited and weakened in the sense that possible contexts are provided in which agents can indeed agree to disagree.

Human cooperation based on punishment reputation.

The threat of punishment usually promotes cooperation. However, punishing itself is costly, rare in nonhuman animals, and humans who punish often finish with low payoffs in economic experiments. The evolution of punishment has therefore been unclear. Recent theoretical developments suggest that punishment has evolved in the context of reputation games. We tested this idea in a simple helping game with observers and with punishment and punishment reputation (experimentally controlling for other possible reputational effects). We show that punishers fully compensate their costs as they receive help more often. The more likely defection is punished within a group, the higher the level of within-group cooperation. These beneficial effects perish if the punishment reputation is removed. We conclude that reputation is key to the evolution of punishment.

A cognitive hierarchy model of behavior in the action commitment game

We apply the cognitive hierarchy model of Camerer et al. (Q J Econ 119(3):861-898, 2004)-where players have different levels of reasoning-to Huck et al. (Games Econ Behav 38:240-264, 2002) discrete version of Hamilton and Slutsky (Games Econ Behav 2:29-46, 1990) action commitment game-a duopoly with endogenous timing of entry. We show that, for an empirically reasonable average number of thinking steps, the model rules out Stackelberg equilibria, generates Cournot outcomes including delay, and outcomes where the first mover commits to a quantity higher than Cournot but lower than Stackelberg leader. We show that a cognitive hierarchy model with quantal responses can explain the most important features of the experimental data on the action commitment game in (2002). In order to gauge the success of the model in fitting the data, we compare it to a noisy Nash model. We find that the cognitive hierarchy model with quantal responses fits the data better than the noisy Nash model.

Coevolution of adaptive technology, maladaptive culture and population size in a producer-scrounger game.

Technology (i.e. tools, methods of cultivation and domestication, systems of construction and appropriation, machines) has increased the vital rates of humans, and is one of the defining features of the transition from Malthusian ecological stagnation to a potentially perpetual rising population growth. Maladaptations, on the other hand, encompass behaviours, customs and practices that decrease the vital rates of individuals. Technology and maladaptations are part of the total stock of culture carried by the individuals in a population. Here, we develop a quantitative model for the coevolution of cumulative adaptive technology and maladaptive culture in a 'producer-scrounger' game, which can also usefully be interpreted as an 'individual-social' learner interaction. Producers (individual learners) are assumed to invent new adaptations and maladaptations by trial-and-error learning, insight or deduction, and they pay the cost of innovation. Scroungers (social learners) are assumed to copy or imitate (cultural transmission) both the adaptations and maladaptations generated by producers. We show that the coevolutionary dynamics of producers and scroungers in the presence of cultural transmission can have a variety of effects on population carrying capacity. From stable polymorphism, where scroungers bring an advantage to the population (increase in carrying capacity), to periodic cycling, where scroungers decrease carrying capacity, we find that selection-driven cultural innovation and transmission may send a population on the path of indefinite growth or to extinction.

The genetical theory of social behaviour.

We survey the population genetic basis of social evolution, using a logically consistent set of arguments to cover a wide range of biological scenarios. We start by reconsidering Hamilton's (Hamilton 1964 J. Theoret. Biol. 7, 1-16 (doi:10.1016/0022-5193(64)90038-4)) results for selection on a social trait under the assumptions of additive gene action, weak selection and constant environment and demography. This yields a prediction for the direction of allele frequency change in terms of phenotypic costs and benefits and genealogical concepts of relatedness, which holds for any frequency of the trait in the population, and provides the foundation for further developments and extensions. We then allow for any type of gene interaction within and between individuals, strong selection and fluctuating environments and demography, which may depend on the evolving trait itself. We reach three conclusions pertaining to selection on social behaviours under broad conditions. (i) Selection can be understood by focusing on a one-generation change in mean allele frequency, a computation which underpins the utility of reproductive value weights; (ii) in large populations under the assumptions of additive gene action and weak selection, this change is of constant sign for any allele frequency and is predicted by a phenotypic selection gradient; (iii) under the assumptions of trait substitution sequences, such phenotypic selection gradients suffice to characterize long-term multi-dimensional stochastic evolution, with almost no knowledge about the genetic details underlying the coevolving traits. Having such simple results about the effect of selection regardless of population structure and type of social interactions can help to delineate the common features of distinct biological processes. Finally, we clarify some persistent divergences within social evolution theory, with respect to exactness, synergies, maximization, dynamic sufficiency and the role of genetic arguments.

The evolution of punishment through reputation.

Punishment of non-cooperators has been observed to promote cooperation. Such punishment is an evolutionary puzzle because it is costly to the punisher while beneficial to others, for example, through increased social cohesion. Recent studies have concluded that punishing strategies usually pay less than some non-punishing strategies. These findings suggest that punishment could not have directly evolved to promote cooperation. However, while it is well established that reputation plays a key role in human cooperation, the simple threat from a reputation of being a punisher may not have been sufficiently explored yet in order to explain the evolution of costly punishment. Here, we first show analytically that punishment can lead to long-term benefits if it influences one's reputation and thereby makes the punisher more likely to receive help in future interactions. Then, in computer simulations, we incorporate up to 40 more complex strategies that use different kinds of reputations (e.g. from generous actions), or strategies that not only include punitive behaviours directed towards defectors but also towards cooperators for example. Our findings demonstrate that punishment can directly evolve through a simple reputation system. We conclude that reputation is crucial for the evolution of punishment by making a punisher more likely to receive help in future interactions, and that experiments investigating the beneficial effects of punishment in humans should include reputation as an explicit feature.

Vocal sib-sib interactions: how siblings adjust signaling level to each other

Game theory states that iterative interactions between individuals are necessary to adjust behaviour optimally to one another. Although our understanding of the role of begging signals in the resolution of parent-offspring conflict over parental investment rests on game theory implying repeated interactions between family members, empiricists usually consider interactions at the exact moment when parents allocate food among the brood. Therefore, the mechanisms by which siblings adjust signalling level to one another remain unclear. We tackled this issue in the barn owl, Tyto alba. In the absence of parents, hungry nestlings signal vocally to siblings their intention to contest vigorously the next, indivisible, food item. Such behaviour deters siblings from competing and begging when parents return to the nest. In experimental two-chick broods, nestlings producing the longest calls in the absence of parents, a signal of hunger level, were more successful at monopolizing the food item at the first parental feeding visit of the night. Moreover, nestlings increased (versus decreased) call duration when their sibling produced longer (versus shorter) calls, and an individual was more likely to call again if its sibling began to vocalize before or just after it had ended its previous call. These results are in agreement with the hypothesis that siblings challenge each other vocally to reinforce the honesty of sib-sib communication and to resolve conflicts over which individual will have priority of access to the next delivered food item. Siblings challenge each other vocally to confirm that the level of signalling accurately reflects motivation.

Exposure to superfluous information reduces cooperation and increases antisocial punishment in reputation-based interactions

Human cooperation is often based on reputation gained from previous interactions with third parties. Such reputation can be built on generous or punitive actions, and both, one's own reputation and the reputation of others have been shown to influence decision making in experimental games that control for confounding variables. Here we test how reputation-based cooperation and punishment react to disruption of the cognitive processing in different kinds of helping games with observers. Saying a few superfluous words before each interaction was used to possibly interfere with working memory. In a first set of experiments, where reputation could only be based on generosity, the disruption reduced the frequency of cooperation and lowered mean final payoffs. In a second set of experiments where reputation could only be based on punishment, the disruption increased the frequency of antisocial punishment (i.e. of punishing those who helped) and reduced the frequency of punishing defectors. Our findings suggest that working memory can easily be constraining in reputation-based interactions within experimental games, even if these games are based on a few simple rules with a visual display that provides all the information the subjects need to play the strategies predicted from current theory. Our findings also highlight a weakness of experimental games, namely that they can be very sensitive to environmental variation and that quantitative conclusions about antisocial punishment or other behavioral strategies can easily be misleading.

Resistance to extreme strategies, rather than prosocial preferences, can explain human cooperation in public goods games.

The results of numerous economic games suggest that humans behave more cooperatively than would be expected if they were maximizing selfish interests. It has been argued that this is because individuals gain satisfaction from the success of others, and that such prosocial preferences require a novel evolutionary explanation. However, in previous games, imperfect behavior would automatically lead to an increase in cooperation, making it impossible to decouple any form of mistake or error from prosocial cooperative decisions. Here we empirically test between these alternatives by decoupling imperfect behavior from prosocial preferences in modified versions of the public goods game, in which individuals would maximize their selfish gain by completely (100%) cooperating. We found that, although this led to higher levels of cooperation, it did not lead to full cooperation, and individuals still perceived their group mates as competitors. This is inconsistent with either selfish or prosocial preferences, suggesting that the most parsimonious explanation is imperfect behavior triggered by psychological drives that can prevent both complete defection and complete cooperation. More generally, our results illustrate the caution that must be exercised when interpreting the evolutionary implications of economic experiments, especially the absolute level of cooperation in a particular treatment.

Is sociality driven by the costs of dispersal or the benefits of philopatry? A role for kin-discrimination mechanisms.

The role of ecological constraints in promoting sociality is currently much debated. Using a direct-fitness approach, we show this role to depend on the kin-discrimination mechanisms underlying social interactions. Altruism cannot evolve under spatially based discrimination, unless ecological constraints prevent complete dispersal. Increasing constraints enhances both the proportion of philopatric (and thereby altruistic) individuals and the level of altruistic investments conceded in pairwise interactions. Familiarity-based discrimination, by contrast, allows philopatry and altruism to evolve at significant levels even in the absence of ecological constraints. Increasing constraints further enhances the proportion of philopatric (and thereby altruistic) individuals but not the level of altruism conceded. Ecological constraints are thus more likely to affect social evolution in species in which restricted cognitive abilities, large group size, and/or limited period of associative learning force investments to be made on the basis of spatial cues.

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Using analytical tools from game theory, we investigate the relevance of a series of hypotheses concerning natal dispersal, focusing in particular on the interaction between inbreeding and kin competition, as well as on the components of mating and social systems that are likely to interfere with these phenomena. A null model of pure kin competition avoidance predicts a balanced equilibrium in wich both sexes disperse equally. Inbreeding costs have the potential to destabilize the equilibrium, resulting in strongly sex-biased dispersal. This effect is mostly evident when the peculiarities of the mating system induce asymmetries in dispersal and/or inbreeding costs, or when kin cooperation counteracts kin competition. Inbreeding depression, however, is not the only possible cause for sex biases. The relevance of our results to empirical findings is dicussed and suggestions are made for further empirical or modelling work.

Fitness, inclusive fitness, and optimization

Individual-as-maximizing agent analogies result in a simple understanding of the functioning of the biological world. Identifying the conditions under which individuals can be regarded as fitness maximizing agents is thus of considerable interest to biologists. Here, we compare different concepts of fitness maximization, and discuss within a single framework the relationship between Hamilton's (J Theor Biol 7: 1-16, 1964) model of social interactions, Grafen's (J Evol Biol 20: 1243-1254, 2007a) formal Darwinism project, and the idea of evolutionary stable strategies. We distinguish cases where phenotypic effects are additive separable or not, the latter not being covered by Grafen's analysis. In both cases it is possible to define a maximand, in the form of an objective function phi(z), whose argument is the phenotype of an individual and whose derivative is proportional to Hamilton's inclusive fitness effect. However, this maximand can be identified with the expression for fecundity or fitness only in the case of additive separable phenotypic effects, making individual-as-maximizing agent analogies unattractive (although formally correct) under general situations of social interactions. We also feel that there is an inconsistency in Grafen's characterization of the solution of his maximization program by use of inclusive fitness arguments. His results are in conflict with those on evolutionary stable strategies obtained by applying inclusive fitness theory, and can be repaired only by changing the definition of the problem.

Local Competition, Inbreeding, and the Evolution of Sex-Biased Dispersal.

Using game theory, we developed a kin-selection model to investigate the consequences of local competition and inbreeding depression on the evolution of natal dispersal. Mating systems have the potential to favor strong sex biases in dispersal because sex differences in potential reproductive success affect the balance between local resource competition and local mate competition. No bias is expected when local competition equally affects males and females, as happens in monogamous systems and also in polygynous or promiscuous ones as long as female fitness is limited by extrinsic factors (breeding resources). In contrast, a male-biased dispersal is predicted when local mate competition exceeds local resource competition, as happens under polygyny/promiscuity when female fitness is limited by intrinsic factors (maximal rate of processing resources rather than resources themselves). This bias is reinforced by among-sex interactions: female philopatry enhances breeding opportunities for related males, while male dispersal decreases the chances that related females will inbreed. These results meet empirical patterns in mammals: polygynous/promiscuous species usually display a male-biased dispersal, while both sexes disperse in monogamous species. A parallel is drawn with sex-ratio theory, which also predicts biases toward the sex that suffers less from local competition. Optimal sex ratios and optimal sex-specific dispersal show mutual dependence, which argues for the development of coevolution models.

Reputation based on punishment rather than generosity allows for evolution of cooperation in sizable groups

Cooperation among unrelated individuals can arise if decisions to help others can be based on reputation. While working for dyadic interactions, reputation-use in social dilemmas involving many individuals (e.g. public goods games) becomes increasingly difficult as groups become larger and errors more frequent. Reputation is therefore believed to have played a minor role for the evolution of cooperation in collective action dilemmas such as those faced by early humans. Here, we show in computer simulations that a reputation system based on punitive actions can overcome these problems and, compared to reputation system based on generous actions, (i) is more likely to lead to the evolution of cooperation in sizable groups, (ii) more effectively sustains cooperation within larger groups, and (iii) is more robust to errors in reputation assessment. Punishment and punishment reputation could therefore have played crucial roles in the evolution of cooperation within larger groups of humans.