scholarly journals The Refined Best Reply Correspondence and Backward Induction

2019 ◽  
Vol 20 (1) ◽  
pp. 52-66
Author(s):  
Dieter Balkenborg ◽  
Christoph Kuzmics ◽  
Josef Hofbauer
Keyword(s):  
Normal Form ◽  
Fixed Points ◽  
The Other ◽  
Backward Induction ◽  
Perfect Recall ◽  
Extensive Form ◽  
Remarkable Property ◽  
Extensive Form Games ◽  
Other Hand ◽  
Perfect Equilibria

Abstract Fixed points of the (most) refined best reply correspondence, introduced in Balkenborg et al. (2013), in the agent normal form of extensive form games with perfect recall have a remarkable property. They induce fixed points of the same correspondence in the agent normal form of every subgame. Furthermore, in a well-defined sense, fixed points of this correspondence refine even trembling hand perfect equilibria, while, on the other hand, reasonable equilibria that are not weak perfect Bayesian equilibria are fixed points of this correspondence.

scholarly journals Non-altruistic Equilibria

2019 ◽  
Vol 67 (3-4) ◽  
pp. 185-195
Author(s):  
Kazuhiro Ohnishi
Keyword(s):  
Normal Form ◽  
Cooperative Games ◽  
The Other ◽  
Extensive Form ◽  
Extensive Form Games ◽  
Normal Form Games ◽  
Subgame Perfect Equilibria ◽  
Perfect Equilibria ◽  
Non Cooperative Games

Which choice will a player make if he can make one of two choices in which his own payoffs are equal, but his rival’s payoffs are not equal, that is, one with a large payoff for his rival and the other with a small payoff for his rival? This paper introduces non-altruistic equilibria for normal-form games and extensive-form non-altruistic equilibria for extensive-form games as equilibrium concepts of non-cooperative games by discussing such a problem and examines the connections between their equilibrium concepts and Nash and subgame perfect equilibria that are important and frequently encountered equilibrium concepts.

A SOLUTION TO THE SURPRISE EXAM PARADOX IN CONSTRUCTIVE MATHEMATICS

2012 ◽  
Vol 5 (4) ◽  
pp. 679-686 ◽  
Author(s):  
MOHAMMAD ARDESHIR ◽  
RASOUL RAMEZANIAN
Keyword(s):  
The Other ◽  
Backward Induction ◽  
Constructive Mathematics ◽  
Constructive Logic ◽  
Extensive Form ◽  
Turing Computability ◽  
Extensive Form Games ◽  
Continuity Principle ◽  
Classical Mathematics

AbstractWe represent the well-known surprise exam paradox in constructive and computable mathematics and offer solutions. One solution is based on Brouwer’s continuity principle in constructive mathematics, and the other involves type 2 Turing computability in classical mathematics. We also discuss the backward induction paradox for extensive form games in constructive logic.

Extensive Form Rationalizability

2014 ◽  
Author(s):  
Herbert Gintis
Keyword(s):  
Normal Form ◽  
Nash Equilibria ◽  
Common Knowledge ◽  
Backward Induction ◽  
Forward Induction ◽  
Extensive Form ◽  
Extensive Form Games ◽  
Dominated Strategies ◽  
Extensive Form Game ◽  
Common Knowledge Of Rationality

The extensive form of a game is informationally richer than the normal form since players gather information that allows them to update their subjective priors as the game progresses. For this reason, the study of rationalizability in extensive form games is more complex than the corresponding study in normal form games. There are two ways to use the added information to eliminate strategies that would not be chosen by a rational agent: backward induction and forward induction. The latter is relatively exotic (although more defensible). Backward induction, by far the most popular technique, employs the iterated elimination of weakly dominated strategies, arriving at the subgame perfect Nash equilibria—the equilibria that remain Nash equilibria in all subgames. An extensive form game is considered generic if it has a unique subgame perfect Nash equilibrium. This chapter develops the tools of modal logic and presents Robert Aumann's famous proof that common knowledge of rationality (CKR) implies backward induction. It concludes that Aumann is perfectly correct, and the real culprit is CKR itself. CKR is in fact self-contradictory when applied to extensive form games.

Closing the GAP—A 5Å Scanning Microscope

1969 ◽  
Vol 27 ◽  
pp. 6-7
Author(s):  
A. V. Crewe
Keyword(s):  
Normal Form ◽  
The Other ◽  
Resolving Power ◽  
Scanning Microscope ◽  
Conventional Microscope ◽  
Other Hand ◽  
Closing The Gap ◽  
Thermal Sources ◽  
Better Than ◽  
Transmission Microscope

We have become accustomed to differentiating between the scanning microscope and the conventional transmission microscope according to the resolving power which the two instruments offer. The conventional microscope is capable of a point resolution of a few angstroms and line resolutions of periodic objects of about 1Å. On the other hand, the scanning microscope, in its normal form, is not ordinarily capable of a point resolution better than 100Å. Upon examining reasons for the 100Å limitation, it becomes clear that this is based more on tradition than reason, and in particular, it is a condition imposed upon the microscope by adherence to thermal sources of electrons.

scholarly journals Timed ATL: Forget Memory, Just Count

2019 ◽  
Vol 66 ◽  
pp. 197-223
Author(s):  
Michal Jozef Knapik ◽  
Etienne Andre ◽  
Laure Petrucci ◽  
Wojciech Jamroga ◽  
Wojciech Penczek
Keyword(s):  
Model Checking ◽  
Temporal Logic ◽  
Discrete Time ◽  
The Other ◽  
Simple Pattern ◽  
Perfect Recall ◽  
Other Hand ◽  
Compact Representations ◽  
Time Extension ◽  
Number Of Visits

In this paper we investigate the Timed Alternating-Time Temporal Logic (TATL), a discrete-time extension of ATL. In particular, we propose, systematize, and further study semantic variants of TATL, based on different notions of a strategy. The notions are derived from different assumptions about the agents’ memory and observational capabilities, and range from timed perfect recall to untimed memoryless plans. We also introduce a new semantics based on counting the number of visits to locations during the play. We show that all the semantics, except for the untimed memoryless one, are equivalent when punctuality constraints are not allowed in the formulae. In fact, abilities in all those notions of a strategy collapse to the “counting” semantics with only two actions allowed per location. On the other hand, this simple pattern does not extend to the full TATL. As a consequence, we establish a hierarchy of TATL semantics, based on the expressivity of the underlying strategies, and we show when some of the semantics coincide. In particular, we prove that more compact representations are possible for a reasonable subset of TATL specifications, which should improve the efficiency of model checking and strategy synthesis.

scholarly journals The Murky Origin of Snow White and Her T-Even Dwarfs

Genetics ◽  
2000 ◽  
Vol 155 (2) ◽  
pp. 481-486 ◽  
Author(s):  
Stephen T Abedon
Keyword(s):  
The Other ◽  
Heritable Variation ◽  
Remarkable Property ◽  
Snow White ◽  
Other Hand

Abstract That two distinct kinds of substances—the d'Hérelle substances and the genes—should both possess this most remarkable property of heritable variation or “mutability,” each working by a totally different mechanism, is quite conceivable, considering the complexity of protoplasm, yet it would seem a curious coincidence indeed. It would open up the possibility of two totally different kinds of life, working by different mechanisms. On the other hand, if these d'Hérelle bodies were really genes, fundamentally like our chromosome genes, they would give us an utterly new angle from which to attack the gene problem. They are filterable, to some extent isolable, can be handled in test-tubes, and their properties, as shown by their effects on the bacteria, can then be studied after treatment. It would be very rash to call these bodies genes, and yet at present we must confess that there is no distinction known between the genes and them. Hence we cannot categorically deny that perhaps we may be able to grind genes in a mortar and cook them in a beaker after all. Must we geneticists become bacteriologists, physiological chemists, and physicists, simultaneously with being zoologists and botanists? Let us hope so. H. J. Muller (1922, pp. 48–49)

scholarly journals An Algorithm for Constructing and Solving Imperfect Recall Abstractions of Large Extensive-Form Games

2017 ◽  
Author(s):  
Jiri Cermak ◽  
Branislav Bošanský ◽  
Viliam Lisý
Keyword(s):  
Relative Size ◽  
Fictitious Play ◽  
Perfect Recall ◽  
Extensive Form ◽  
Extensive Form Games ◽  
Imperfect Recall ◽  
Information Sets ◽  
Information Set ◽  
Zero Sum ◽  
Large Extensive Form Games

We solve large two-player zero-sum extensive-form games with perfect recall. We propose a new algorithm based on fictitious play that significantly reduces memory requirements for storing average strategies. The key feature is exploiting imperfect recall abstractions while preserving the convergence rate and guarantees of fictitious play applied directly to the perfect recall game. The algorithm creates a coarse imperfect recall abstraction of the perfect recall game and automatically refines its information set structure only where the imperfect recall might cause problems. Experimental evaluation shows that our novel algorithm is able to solve a simplified poker game with 7.10^5 information sets using an abstracted game with only 1.8% of information sets of the original game. Additional experiments on poker and randomly generated games suggest that the relative size of the abstraction decreases as the size of the solved games increases.

A result on implications of Σ1-sentences and its application to normal form theorems

1981 ◽  
Vol 46 (3) ◽  
pp. 634-642
Author(s):  
Jean-Yves Girard ◽  
Peter Päppinghaus
Keyword(s):  
Normal Form ◽  
Reflection Principle ◽  
The Other ◽  
Other Hand ◽  
Primitive Recursive Arithmetic ◽  
Incompleteness Theorems ◽  
False Premise ◽  
Primitive Recursive ◽  
Gödel’S Incompleteness Theorems

In this paper we are concerned with the formal provability of certain true implications of Σ1-sentences. Old completeness and incompleteness results already give some information about this. For example by Σ1-completeness of PRA (primitive recursive arithmetic) every true implication of the form D → E, where D is a Σ0-sentence and E a Σ1-sentence, or D a Σ1-sentence and E a true Σ1-sentence, is provable in PRA. On the other hand, by Gödel's incompleteness theorems one can define for every suitable theory S a false Σ1-sentence Ds such that for every false Σ0-sentence E the true implication Ds → E is not provable in S, but is provable in PRA + Cons. So one sees that for a suitable fixed conclusion the provability of true implications of Σ1-sentences depends on the content of the premise.Now we ask, if and how for a suitable fixed premise the provability of true implications of Σ1-sentences depends on the conclusion. As remarked above, by Σ1-completeness of PRA this question is settled, if the premise is true. For a false premise it is answered in § 1 as follows: Let D be a false Σ1-sentence, S an extension of PRA, and S+ ≔ PRA + IAΣ1 + RFNΣ1S). Depending on D and S one can define a Σ1-sentence Es such that S+ ⊢ D → Es, but S ⊬ D → Es provided that S+ is not strong enough to refute D. (IAΣ1 denotes the scheme of the induction axiom for Σ1-formulas, and RFNΣ1(S) the uniform Σ1-reflection principle for S.)

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