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on Microeconomics |
| By: | Dirk Bergemann (Cowles Foundation, Yale University); Stephen Morris (Dept. of Economics, Princeton University) |
| Abstract: | We define a notion of correlated equilibrium for games with incomplete information in a general setting with finite players, finite actions, and finite states, which we call Bayes correlated equilibrium. The set of Bayes correlated equilibria of a fixed incomplete information game equals the set of probability distributions over actions, states and types that might arise in any Bayes Nash equilibrium where players observed additional information. We show that more information always shrinks the set of Bayes correlated equilibria. |
| Keywords: | Correlated equilibrium, Incomplete information, Robust predictions, Information structure |
| JEL: | C72 D82 D83 |
| Date: | 2011–10 |
| URL: | http://d.repec.org/n?u=RePEc:cwl:cwldpp:1822&r=mic |
| By: | Chiu Yu Ko (Boston College) |
| Abstract: | This paper extends Bernheim and Whinston's (1986) menu auction model under transferable utilities to a framework with non-transferable utilities and budget constraints. Under appropriate definitions of equilibria, it is shown that every truthful Nash equilibrium (TNE) is a coalition-proof Nash equilibrium (CPNE) and that the set of TNE payoffs and the set of CPNE payoffs are equivalent, as in a transferable utility framework. The existence of a CPNE is assured in contrast with the possible non-existence of Nash equilibrium under the definition by Dixit, Grossman, and Helpman (1997). Moreover, the set of CPNE payoffs is equivalent to the bidder-optimal weak core. |
| Keywords: | non-transferable utility, menu auction, coalition-proof Nash equilibrium, truthful Nash equilibrium |
| JEL: | C72 D79 |
| Date: | 2011–10–20 |
| URL: | http://d.repec.org/n?u=RePEc:boc:bocoec:787&r=mic |
| By: | Kieron Meagher |
| Abstract: | In Hotelling style duopoly location games the product variety (or firm locations) is typically not socially optimal. This occurs because the competitive outcome is driven by the density of consumers at the margin while the socially optimal outcome depends on the whole distribution of consumer locations/tastes. We consider a natural extension of the standard model in which firms are imperfectly informed about the distribution of consumers, in particular firms are uncertain about the consumer mean. In the uniform case, as the aggregate uncertainty about the mean becomes large relative to the dispersion of consumers about the mean, competitive locations become socially optimal. A limit result on prices for discontinuous, log-concave densities shows the result will hold in a range of cases. |
| JEL: | C72 D43 D81 L10 L13 R30 R39 |
| Date: | 2011–10 |
| URL: | http://d.repec.org/n?u=RePEc:acb:cbeeco:2011-555&r=mic |
| By: | Pintér, Miklós; Udvari, Zsolt |
| Abstract: | Ordinary type spaces (Heifetz and Samet, 1998) are essential ingredients of incomplete information games. With ordinary type spaces one can grab the notions of beliefs, belief hierarchies and common prior etc. However, ordinary type spaces cannot handle the notions of finite belief hierarchy and unawareness among others. In this paper we consider a generalization of ordinary type spaces, and introduce the so called generalized type spaces which can grab all notions ordinary type spaces can and more, finite belief hierarchies and unawareness among others. We also demonstrate that the universal generalized type space exists. |
| Keywords: | type space; unawareness; finite belief hierarchy; generalized type space; generalized belief hierarchy; incomplete information games |
| JEL: | D83 C72 |
| Date: | 2011 |
| URL: | http://d.repec.org/n?u=RePEc:pra:mprapa:34107&r=mic |
| By: | Pintér, Miklós |
| Abstract: | The notion of common prior is well-understood and widely-used in the incomplete information games literature. For ordinary type spaces the common prior is defined. Pinter and Udvari (2011) introduce the notion of generalized type space. Generalized type spaces are models for various bonded rationality issues, for finite belief hierarchies, unawareness among others. In this paper we define the notion of common prior for generalized types spaces. Our results are as follows: the generalization (1) suggests a new form of common prior for ordinary type spaces, (2) shows some quantum game theoretic results (Brandenburger and La Mura, 2011) in new light. |
| Keywords: | Type spaces; Generalized type spaces; Common prior; Harsányi Doctrine; Quantum games |
| JEL: | D83 C72 |
| Date: | 2011 |
| URL: | http://d.repec.org/n?u=RePEc:pra:mprapa:34118&r=mic |
| By: | Thomas, C.D. |
| Abstract: | In this thesis we present a new take on two classic problems of game theory: the "multiarmed bandit" problem of dynamic learning, and the "Colonel Blotto" game, a multidi- mensional contest. In Chapters 2-4 we treat the questions of experimentation with congestion: how do players search and learn about options when they are competing for access with other players? We consider a bandit model in which two players choose between learning about the quality of a risky option (modelled as a Poisson process with unknown arrival rate), and competing for the use of a single shared safe option that can only be used by one agent at the time. We present the equilibria of the game when switching to the safe option is irrevocable, and when it is not. We show that the equilibrium is always inefficient: it involves too little experimentation when compared to the planner solution. The striking equilibrium dynamics of the game with revocable exit are driven by a strategic option-value arising purely from competition between the players. This constitutes a new result in the bandit literature. Finally we present extensions to the model. In particular we assume that players do not observe the result of their opponent's experimentation. In Chapter 5 we turn to the n-dimensional Blotto game and allow battlefields to have different values. We describe a geometrical method for constructing equilibrium distribution in the Colonel Blotto game with asymmetric battlfield values. It generalises the 3-dimensional construction method first described by Gross and Wagner (1950). The proposed method does particularly well in instances of the Colonel Blotto game in which the battlefield weights satisfy some clearly defined regularity conditions. The chapter also explores the parallel between these conditions and the integer partitioning problem in combinatorial optimisation. |
| Date: | 2011–08–28 |
| URL: | http://d.repec.org/n?u=RePEc:ner:ucllon:http://discovery.ucl.ac.uk/1325637/&r=mic |
| By: | Bosi, Gianni; Zuanon, Magalì |
| Abstract: | We characterize weak continuity of an interval order on a topological space by using the concept of a scale in a topological space. |
| Keywords: | Weakly continuous interval order; continuous numerical representation |
| JEL: | D00 C60 |
| Date: | 2011–10–17 |
| URL: | http://d.repec.org/n?u=RePEc:pra:mprapa:34182&r=mic |
| By: | Barbanel, Julius B.; Brams, Steven J. |
| Abstract: | A cake is a metaphor for a heterogeneous, divisible good. When two players divide such a good, there is always a perfect division—one that is efficient (Pareto-optimal), envy-free, and equitable—which can be effected with a finite number of cuts under certain mild conditions; this is not always the case when there are more than two players (Brams, Jones, and Klamler, 2011b). We not only establish the existence of such a division but also provide an algorithm for determining where and how many cuts must be made, relating it to an algorithm, “Adjusted Winner” (Brams and Taylor, 1996, 1999), that yields a perfect division of multiple homogenous goods. |
| Keywords: | Cake-cutting; fair division; envy-freeness; adjusted winner; heterogeneous good |
| JEL: | D63 D30 D74 C61 D61 C72 |
| Date: | 2011–10–22 |
| URL: | http://d.repec.org/n?u=RePEc:pra:mprapa:34263&r=mic |
| By: | Brams, Steven J.; Jones, Michael A.; Klamler, Christian |
| Abstract: | A cake is a metaphor for a heterogeneous, divisible good, such as land. A perfect division of cake is efficient (also called Pareto-optimal), envy-free, and equitable. We give an example of a cake in which it is impossible to divide it among three players such that these three properties are satisfied, however many cuts are made. It turns out that two of the three properties can be satisfied by a 3-cut and a 4-cut division, which raises the question of whether the 3-cut division, which is not efficient, or the 4-cut division, which is not envy-free, is more desirable (a 2-cut division can at best satisfy either envy-freeness or equitability but not both). We prove that no perfect division exists for an extension of the example for three or more players. |
| Keywords: | Cake-cutting; fair division; efficiency; envy-freeness; equitability; heterogeneous good |
| JEL: | D71 D63 D30 D74 D61 C61 C72 |
| Date: | 2011–10–22 |
| URL: | http://d.repec.org/n?u=RePEc:pra:mprapa:34264&r=mic |