Characterization of Pareto Dominance

The Pareto dominance relation is shown to be the unique nontrivial partial order on the set of finite-dimensional real vectors satisfying a number of intuitive properties.Pareto dominance; characterization

Preparation and toolkit learning

A product set of pure strategies is a prep set ("prep" is short for "preparation") if it contains at least one best reply to any consistent belief that a player may have about the strategic behavior of his opponents. Minimal prep sets are shown to exists in a class of strategic games satisfying minor topological conditions. The concept of minimal prep sets is compared with (pure and mixed) Nash equilibria, minimal curb sets, and rationalizability. Additional dynamic motivation for the concept is provided by a model of adaptive play that is shown to settle down in minimal prep sets.noncooperative games; inertia; status quo bias; adaptive play; procedural rationality

From preferences to Cobb-Douglas utility

We provide characterizations of preferences representable by a Cobb-Douglas utility function.Preferences; Utility theory; Cobb-Douglas

The possibility of impossible stairways and greener grass

In classical game theory, players have finitely many actions and evaluate outcomes of mixed strategies using a von Neumann-Morgenstern utility function. Allowing a larger, but countable, player set introduces a host of phenomena that are impossible in finite games. Firstly, in coordination games, all players have the same preferences: switching to a weakly dominant action makes everyone at least as well off as before. Nevertheless, there are coordination games where the best outcome occurs if everyone chooses a weakly dominated action, while the worst outcome occurs if everyone chooses the weakly dominant action. Secondly, the location of payoff-dominant equilibria behaves capriciously: two coordination games that look so much alike that even the consequences of unilateral deviations are the same may nevertheless have disjoint sets of payoff-dominant equilibria. Thirdly, a large class of games has no (pure or mixed) Nash equilibria. Following the proverb ``the grass is always greener on the other side of the hedge'', greener-grass games model constant discontent: in one part of the strategy space, players would rather switch to its complement. Once there, they'd rather switch back.coordination games; dominant strategies; payoff-dominance; nonexistence of equilibrium; tail events

Numerical Representation of Incomplete and Nontransitive Preferences and Indifferences on a Countable Set

This note considers preference structures over countable sets which allow incomparable outcomes and nontransitive preferences and indifferences. Necessary and sufficient conditions are provided under which such a preference structure can be represented by means of utility function and a threshold function.Incomplete preferences; nontransitive preferences; threshold functions; utility theory

Probabilistic choice in games: properties of Rosenthal's t-solutions

In t-solutions, quantal response equilibria based on the linear probability model as introduced in R.W. Rosenthal (1989, Int. J. Game Theory 18, 273-292), choice probabilities are related to the determination of leveling taxes. The set of t-solutions coincides with the set of Nash equilibria of a game with quadratic control costs. Increasing the rationality of the players allows them to successively eliminate higher levels of strictly dominated actions. Moreover, there exists a path of t-solutions linking uniform randomization to Nash equilibrium.quantal response equilibrium; t-solutions; linear probability model

Congestion, equilibrium and learning: The minority game

The minority game is a simple congestion game in which the players' main goal is to choose among two options the one that is adopted by the smallest number of players. We characterize the set of Nash equilibria and the limiting behavior of several well-known learning processes in the minority game with an arbitrary odd number of players. Interestingly, different learning processes provide considerably different predictions

The cutting power of preparation

In a strategic game, a curb set [Basu and Weibull, Econ. Letters 36 (1991) 141] is a product set of pure strategies containing all best responses ro every possible belief restricted to this set. Prep sets [Voorneveld, Games Econ. Behav. 48 (2004) 403] relax this condition by only requiring the presence of at least one best response to such a belief. The purpose of this paper is to provide economically interesting classes of games in which minimal prep sets give sharp predictions, whereas in relevant subclasses of these games, minimal curb sets have no cutting power whatsoever and simply consist of the entire strategy space. These classes include potential games, congestion games with player-specific payoffs, and supermodular games.curb sets; prep sets; potential games; congestion games; supermodular games

Learning to be prepared

Behavioral economics provides several motivations for the common observation that agents appear somewhat unwilling to deviate from recent choices: salience, inertia, the formation of habits, the use of rules of thumb, or the locking in on certain modes of behavior due to learning by doing. This paper provides discrete-time adjustment processes for strategic games in which players display precisely such a bias towards recent choices. In addition, players choose best replies to beliefs supported by observed play in the recent past, in line with much of the literature on learning. These processes eventually settle down in the minimal prep sets of Voorneveld (2004, 2005).adjustment; learning; minimal prep sets; behavioral bias; salience