Bernard Walliser

A Behavioral Learning Process in Games

The paper studies a behavioral learning process where an agent plays, at each period, an action with a probability which is proportional to the cumulative utility he got in the past with that action. The so-called CPR learning rule and the dynamic process it induces are formally stated and compared to other reinforcement rules as well as to fictitious play or the replicator dynamics.

Ising model versus normal form game

The 2-spin Ising model in statistical mechanics and the 2×2 normal form game in game theory are compared. All configurations allowed by the second are recovered by the first when the only concern is about Nash equilibria. But it holds no longer when Pareto optimum considerations are introduced as in the prisoner’s dilemma. This gap can nevertheless be filled by adding a new coupling term to the Ising model, even if that term has up to now no physical meaning. An individual complete bilinear objective function is thus found to be sufficient to reproduce all possible configurations of a 2×2 game. Using this one-to-one mapping new perspectives for future research in both fields can be envisioned.

Infinite Regressions in the Optimizing Theory of Decision

It is a crucial assumption of economic theory that individuals make decisions by maximizing an objective (i.e. preference-representing) function under specified constraints. This decision-theoretic model has been subjected to strong empirical criticism on the grounds that it lacks psychological realism and does not fit well with known data on individual decision-making (e.g. Lester, 1945, Simon, 1959 and 1979). A related, methodological rather than empirical, strand of criticism emphasizes the barrenness of an approach that discards the cognitive and deliberative import of decisions as scientifically irrelevant and can more or less trivially be reconciled with any known observable behaviour (e.g. Simon, 1976, Latsis, 1976). Finally, there is a logical critique of optimization which is not as well developped in the technical literature. This paper aims at making it more precise.

A reinforcement learning process in extensive form games

The CPR (“cumulative proportional reinforcement”) learning rule stipulates that an agent chooses a move with a probability proportional to the cumulative payoff she obtained in the past with that move. Previously considered for strategies in normal form games (Laslier, Topol and Walliser, Games and Econ. Behav., 2001), the CPR rule is here adapted for actions in perfect information extensive form games. The paper shows that the action-based CPR process converges with probability one to the (unique) subgame perfect equilibrium.

Epistemic properties of knowledge hierarchies

Abstract After proving correspondence theorems between syntactic and semantic properties of a Kripke Structure, these properties are combined in order to generate two polar semantic structures, that is, partitions and nestings. Then, the relevant properties are extended to Probabilistic Kripke Structures and the concept of a Mixed Knowledge Hierarchy is introduced and associated with two dual measures of events. Finally, an application to Decision Theory is provided for the particular case of a 2-level hierarchy.

Change rules for hierarchical beliefs

The paper builds a belief hierarchy as a framework common to all uncertainty measures expressing that an actor is ambiguous about his uncertain beliefs. The belief hierarchy is further interpreted by distinguishing physical and psychical worlds, associated to objective and subjective probabilities. Various rules of transformation of a belief hierarchy are introduced, especially changing subjective beliefs into objective ones. These principles are applied in order to relate different contexts of belief change, revising, updating and even focusing. The numerous belief change rules already proposed in the literature receive epistemic justifications by associating them to specific belief hierarchies and change contexts. As a result, it is shown that the resiliency of probability judgments may have some limits and be reconciled with the possibility of learning from factual messages.

Topics of Cognitive Economics

This chapter presents the main topics which can be subsumed under the label “cognitive economics” and which correspond to current work. It first presents the cognitive processes carried out by an individual agent concerning the structure and revision of his beliefs and the mental deliberation preceding his choice. It then presents the mechanisms of interaction between several agents in the strategic and face-to-face form of game theory or in the more parametric and indirect form of economic theory. Each of the four successive themes is treated from two points of view, an epistemic one where hyper-intelligent agents act and coordinate instantaneously through sophisticated reasoning and an evolutionary one where boundedly rational agents adapt reciprocally through dynamic learning processes. Concluding remarks concern the ontological and epistemological positions of cognitive economics.

A simplified taxonomy of 2 × 2 games

All 2 × 2 games are classified into eight configurations, following three natural criteria, and prototypes given for each, especially as concerns the Newcomb and chain-store paradoxes. Two pseudo-dynamic properties, move priority and dynamic inconsistency, are examined in that framework, as well as more specifically, the problem of the origin of social institutions.

BELIEF REVISION AND DECISION UNDER COMPLEX UNCERTAINTY

ABSTRACT Uncertainty can be formalized in three increasingly stronger frameworks based on possible worlds: qualitative (Kripke structures), quantitative (Dempster - Shafer distributions) and hierarchical (two-level probabilities). It is possible then to compare and connect in a systematic way all the methods proposed in these frameworks for knowledge updating and decision-making. It is shown that the Gardenfors procedure for belief revision under qualitative uncertainty defines a specific qualitative possibility distribution and that several revision rules can be exhibited correspondingly for quantitative and hierarchical uncertainty. It is also shown that Gilboa-Schmeidlers non-additive probabilistic expected utility is identical with two-level probabilistic expected utility when the decision-maker has infinite aversion to second level uncertainty.

The Principles of Game Theory

This chapter recalls the main principles of non-cooperative game theory and especially its main equilibrium concepts. It proceeds by progressively introducing more and more complexity into the basic interaction model between rational players. The basic model is static with complete and perfect information and introduces the concept of Nash equilibrium as a fixed point of players’ best responses. Then, sequential action is explicitly introduced and leads to the concept of subgame perfect equilibrium, usually obtained by a backward induction procedure. Coming back to a static framework, structural uncertainty is embedded in the Bayesian equilibrium concept, where beliefs about others’ types are confirmed. Finally, combining temporality and factual uncertainty, one obtains the Bayesian Nash equilibrium concept, grounded on a fixed point between players’ beliefs about others’ past moves and their own future actions.