Common Knowledge and Game Theory

Perhaps the most important area in which common knowledge problems arise is in the study of rational expectations equilibria in the trading of risky securities. How can there be trade if everybody's willingness to trade means that everybody knows that everybody expects to be a winner? (see Milgrom/Stokey [1982] and Geanakoplos [1988].) Since risky securities are traded on the basis of private information, there must presumably be some "agreeing to disagree" in the real world. But to assess its extent and its implications, one needs to have a precise theory of the norm from wich "agreeing to disagree" is seen as a deviation. The beginnings of such a theory are presented here. Some formalism is necessary in such a presentation because the English language is not geared up to express the appropriate ideas compactly. Without some formalism, it is therefore very easy to get confused. However, nothing requiring any mathematical expertise is to be described.Center for Research on Economic and Social Theory, Department of Economics, University of Michiganhttp://deepblue.lib.umich.edu/bitstream/2027.42/100630/1/ECON107.pd

Admissibility and Common Knowledge

The implications of assuming that it is commonly known that players consider only admissible best responses are investigated.Within a states-of-the-world model where a state, for each player, determines a strategy set rather than a strategy the concept of fully permissible sets is defined.General existence is established, and a finite algorithm (eliminating strategy sets instead of strategies) is provided.The concept refines rationalizability as well as the Dekel-Fudenberg procedure, and captures a notion of forward induction.When players consider all best responses, the same framework can be used to define the concept of rationalizable sets, which characterizes rationalizability.game theory

Market uncertainty: correlated and sunspot equilibria in imperfectly competitive economies

An imperfectly competitive economy is very prone to market uncertainty, including uncertainty about the liquidity (or "thickness") of markets. We show, in particular, that there exist stochastic equilibrium outcomes in nonstochastic market games if (and only if) the endowments are not Pareto optimal. We also provide a link between extrinsic uncertainty arising in games (e.g. correlated equilibria) and extrinsic uncertainty in market economies (e.g. sunspot equilibria). A correlated equilibria to the market game is either a sunspot equilibrium or a non-sunspot equilibrium to the related securities games, but the converse is not true in general. 1

Admissibility and Common Knowledge

The implications of assuming that it is commonly known that players consider only admissible best responses are investigated.Within a states-of-the-world model where a state, for each player, determines a strategy set rather than a strategy the concept of fully permissible sets is defined.General existence is established, and a finite algorithm (eliminating strategy sets instead of strategies) is provided.The concept refines rationalizability as well as the Dekel-Fudenberg procedure, and captures a notion of forward induction.When players consider all best responses, the same framework can be used to define the concept of rationalizable sets, which characterizes rationalizability.

The Impartial Observer Theorem of Social Ethics

Following a long-standing philosophical tradition, impartiality is a distinctive and determining feature of moral judgments, especially in matters of distributive justice. This broad ethical tradition was revived in welfare economics by Vickrey, and above all, Harsanyi, under the form of the so-called Impartial Observer Theorem. The paper offers an analytical reconstruction of this argument, using a simple mathematical formalism, as well as a conceptual critique of each its premisses.Utilitarianism, Impartiality, Sympathy, Von Neumann- Morgenstern Utility Theory, Subjective Probability.

On Large Games with a Bio-Social Typology

We present a comprehensive theory of large non-anonymous games in which agents have a name and a determinate social-type and/or biological trait to resolve the dissonance of a (matching-pennies type) game with an exact pure-strategy Nash equilibrium with finite agents, but without one when modeled on the Lebesgue unit interval. We (i) establish saturated player spaces as both necessary and sufficient for an existence result for Nash equilibrium in pure strategies, (ii) clarify the relationship between pure, mixed and behavioral strategies via the exact law of large numbers in a framework of Fubini extension, (iii) illustrate corresponding asymptotic results.

Communication Equilibria and Bounded Rationality

In this paper, we generalize the notion of a communication equilibrium (Forges 1986, Myerson 1986) of a game with incomplete information by introducing two new types of correlation device, namely extended and Bayesian devices. These new devices explicitly model the `thinking process' of the device, i.e. the manner in which it generates outputs conditional on inputs. We proceed to endow these devices with both information processing errors, in the form of non-partitional information, and multiple transition and prior distributions, and prove that these two properties are equivalent in this context, thereby generalizing the result of Brandenburger, Dekel and Geanakoplos (1988). We proceed to discuss the Revelation Principle for each device, and conclude by nesting a certain class of `cheap-talk' equilibria of the underlying game within Bayesian communication equilibria. These so-called fallible talk equilibria cannot be generated by standard communication equilibria.

Bayesian models and repeated games

A game is a theoretical model of a social situation where the people involved have individually only partial control over the outcomes. Game theory is then the method used to analyse these models. As a player's outcome from a game depends upon the actions of his opponents, there is some uncertainty in these models. This uncertainty is described probabilistically, in terms of a player's subjective beliefs about the future play of his opponent. Any additional information that is acquired by the player can be incorporated into the analysis and these subjective beliefs are revised. Hence, the approach taken is `Bayesian'. Each outcome from the game has a value to each of the players, and the measure of merit from an outcome is referred to as a player's utility. This concept of utility is combined with a player's subjective probabilities to form an expected utility, and it is assumed that each player is trying to maximise his expected utility. Bayesian models for games are constructed in order to determine strategies for the players that are expected utility maximising. These models are guided by the belief that the other players are also trying to maximise their own expected utilities. It is shown that a player's beliefs about the other players form an infinite regress. This regress can be truncated to a finite number of levels of beliefs, under some assumptions about the utility functions and beliefs of the other players. It is shown how the dichotomy between prescribed good play and observed good play exists because of the lack of assumptions about the rationality of the opponents (i. e. the ability of the opponents to be utility maximising). It is shown how a model for a game can be built which is both faithful to the observed common sense behaviour of the subjects of an experimental game, and is also rational (in a Bayesian sense). It is illustrated how the mathematical form of an optimal solution to a game can be found, and then used with an inductive algorithm to determine an explicit optimal strategy. It is argued that the derived form of the optimal solution can be used to gain more insight into the game, and to determine whether an assumed model is realistic. It is also shown that under weak regularity conditions, and assuming that an opponent is playing a strategy from a given class of strategies, S, it is not optimal for the player to adopt any strategy from S, thus compromising the chosen model

The Framing of Games and the Psychology of Strategic Choice

Psychological game theory can provide a rational choice explanation of framing effects; frames influence beliefs, and beliefs influence motivations. We explain this point theoretically, and explore its empirical relevance experimentally. In a 2×2-factorial framing design of one-shot public good experiments we show that frames affect subject’s first- and second-order beliefs, and contributions. From a psychological game-theoretic framework we derive two mutually compatible hypotheses about guilt aversion and reciprocity under which contributions are related to second- and first-order beliefs, respectively. Our results are consistent with either.Framing; psychological games; guilt aversion; reciprocity; public good games; voluntary cooperation