scholarly journals Computing Nash Equilibria of Unbounded Games

10.29007/1wpl ◽  
2018 ◽  
Author(s):  
Martin Escardo ◽  
Paulo Oliva
Keyword(s):  
Nash Equilibria ◽  
Proof Theory ◽  
A Priori ◽  
Subgame Perfect Equilibrium ◽  
Backward Induction ◽  
Closed Formula ◽  
Perfect Equilibrium ◽  
Finite Games ◽  
Game Theoretic ◽  
Theoretic Technique

Using techniques from higher-type computability theory and proof theory we extend the well-known game-theoretic technique of backward induction to finite games of unbounded length. The main application is a closed formula for calculating strategy profiles in Nash equilibrium and subgame perfect equilibrium even in the case of games where the length of play is not a-priori fixed.

scholarly journals BACKWARD INDUCTION: MERITS AND FLAWS

2017 ◽  
Vol 50 (1) ◽  
pp. 9-24
Author(s):  
Marek M. Kamiński
Keyword(s):  
Infinite Number ◽  
Nash Equilibria ◽  
Complex Structure ◽  
Subgame Perfect Equilibrium ◽  
Perfect Information ◽  
Backward Induction ◽  
Perfect Equilibrium ◽  
Sequential Games ◽  
Theoretical Predictions

Abstract Backward induction (BI) was one of the earliest methods developed for solving finite sequential games with perfect information. It proved to be especially useful in the context of Tom Schelling’s ideas of credible versus incredible threats. BI can be also extended to solve complex games that include an infinite number of actions or an infinite number of periods. However, some more complex empirical or experimental predictions remain dramatically at odds with theoretical predictions obtained by BI. The primary example of such a troublesome game is Centipede. The problems appear in other long games with sufficiently complex structure. BI also shares the problems of subgame perfect equilibrium and fails to eliminate certain unreasonable Nash equilibria.

scholarly journals Dinamicke igre ulaska na trziste

2005 ◽  
Vol 50 (165) ◽  
pp. 121-144
Author(s):  
Bozo Stojanovic
Keyword(s):  
Nash Equilibrium ◽  
Dynamic Games ◽  
Nash Equilibria ◽  
Dynamic Game ◽  
Backward Induction ◽  
Perfect Equilibrium ◽  
Equilibrium Concept ◽  
Reasonable Solution ◽  
Multiple Nash Equilibria ◽  
Entry Games

Market processes can be analyzed by means of dynamic games. In a number of dynamic games multiple Nash equilibria appear. These equilibria often involve no credible threats the implementation of which is not in the interests of the players making them. The concept of sub game perfect equilibrium rules out these situations by stating that a reasonable solution to a game cannot involve players believing and acting upon noncredible threats or promises. A simple way of finding the sub game perfect Nash equilibrium of a dynamic game is by using the principle of backward induction. To explain how this equilibrium concept is applied, we analyze the dynamic entry games.

Quantifying Commitment in Nash Equilibria

2019 ◽  
Vol 21 (02) ◽  
pp. 1940011
Author(s):  
Thomas A. Weber
Keyword(s):  
Nash Equilibrium ◽  
Normal Form ◽  
Nash Equilibria ◽  
Dynamic Game ◽  
Subgame Perfect Equilibrium ◽  
Canonical Extension ◽  
Perfect Equilibrium ◽  
Form Game ◽  
Normal Form Game ◽  
The Given

To quantify a player’s commitment in a given Nash equilibrium of a finite dynamic game, we map the corresponding normal-form game to a “canonical extension,” which allows each player to adjust his or her move with a certain probability. The commitment measure relates to the average overall adjustment probabilities for which the given Nash equilibrium can be implemented as a subgame-perfect equilibrium in the canonical extension.

A Formal Game-Theoretic Model for Rational Exchange Protocol

2011 ◽  
Vol 204-210 ◽  
pp. 2033-2040
Author(s):  
Xiu Ting Tao ◽  
Yong Gen Gu ◽  
Guo Qiang Li
Keyword(s):  
Subgame Perfect Equilibrium ◽  
Theoretic Model ◽  
Formal Definition ◽  
Perfect Equilibrium ◽  
Game Tree ◽  
Extensive Game ◽  
Formal Framework ◽  
Game Theoretic ◽  
Game Theoretic Model ◽  
The Relationship

A rational exchange protocol is a useful protocol in which two (or more) strange parties exchange their items successfully in a way that every rational party has enough reasons to follow the protocol faithfully rather than deviate from it. This means neither parties of the protocol can gain an advantage by deviating from the protocol, but he may bring a disadvantage to the other party even if who is correctly behaving party. We introduce game theory as a formal framework in this paper. We give a formal definition for the rational exchange related to the concept of the subgame perfect equilibrium of an extensive game. We use our model to analyze the Syverson protocol by the game tree and show the relationship with Buttyan's Model.

scholarly journals A generalization of Nash's theorem with higher-order functionals

2013 ◽  
Vol 469 (2154) ◽  
pp. 20130041 ◽  
Author(s):  
Julian Hedges
Keyword(s):  
Normal Form ◽  
Proof Theory ◽  
Mixed Strategy ◽  
Recent Theory ◽  
Sequential Games ◽  
New Class ◽  
Finite Games ◽  
Minimax Strategies ◽  
Game Theoretic ◽  
Axiom Of Countable Choice

The recent theory of sequential games and selection functions by Escardó & Oliva is extended to games in which players move simultaneously. The Nash existence theorem for mixed-strategy equilibria of finite games is generalized to games defined by selection functions. A normal form construction is given, which generalizes the game-theoretic normal form, and its soundness is proved. Minimax strategies also generalize to the new class of games, and are computed by the Berardi–Bezem–Coquand functional, studied in proof theory as an interpretation of the axiom of countable choice.

scholarly journals Non-Determinism and Nash Equilibria for Sequential Game over Partial Order

2005 ◽  
Vol DMTCS Proceedings vol. AF,... (Proceedings) ◽  
Author(s):  
Stéphane Le Roux
Keyword(s):  
Nash Equilibrium ◽  
Nash Equilibria ◽  
Ordered Set ◽  
Backward Induction ◽  
Proof Assistant ◽  
Perfect Equilibrium ◽  
Sequential Games ◽  
International Audience ◽  
Traditional Game ◽  
Deterministic Strategy

International audience In sequential games of traditional game theory, backward induction guarantees existence of Nash equilibrium by yielding a sub-game perfect equilibrium. But if payoffs range over a partially ordered set instead of the reals, then the backward induction predicate does no longer imply the Nash equilibrium predicate. Non-determinism is a solution: a suitable non-deterministic backward induction function returns a non-deterministic strategy profile which is a non-deterministic Nash equilibrium. The main notions and results in this article are constructive, conceptually simple and formalised in the proof assistant Coq.

From Jungle to Civilized Economy: The Power Foundation of Exchange Economy Equilibrium

2019 ◽  
Vol 19 (2) ◽  
Author(s):  
Mordechai E. Schwarz
Keyword(s):  
Subgame Perfect Equilibrium ◽  
Exchange Economy ◽  
Perfect Equilibrium ◽  
Initial Endowment ◽  
Theoretic Analysis ◽  
Bankruptcy Problem ◽  
Game Theoretic Analysis ◽  
Talmud Rule ◽  
Game Theoretic ◽  
Walrasian Equilibria

Abstract This article explores the evolution of a civilized exchange economy from an anarchistic environment. I analyze a model of stochastic jungle bargaining mechanism and show that it implements the Talmud Rule allocation (Aumann, R. J., and M. Maschler. 1985. “Game Theoretic Analysis of a Bankruptcy Problem from the Talmud.” Journal of Economic Theory 36 (2): 195–213.) in subgame perfect equilibrium. This Pareto-inefficient allocation constitutes the initial endowment of a stable exchange economy and supports stable Walrasian equilibria, implying that civilized economies could evolve from a Hobbesian state of nature without social contract or regulator. The moral implications of these results are also briefly discussed.

scholarly journals Selection functions, bar recursion and backward induction

2010 ◽  
Vol 20 (2) ◽  
pp. 127-168 ◽  
Author(s):  
MARTÍN ESCARDÓ ◽  
PAULO OLIVA
Keyword(s):  
Proof Theory ◽  
Optimal Strategies ◽  
Backward Induction ◽  
Constructive Mathematics ◽  
Infinite Games ◽  
Sequential Games ◽  
Finite Games ◽  
Conceptual Explanation ◽  
Cartesian Closed ◽  
Countably Infinite

Bar recursion arises in constructive mathematics, logic, proof theory and higher-type computability theory. We explain bar recursion in terms of sequential games, and show how it can be naturally understood as a generalisation of the principle of backward induction that arises in game theory. In summary, bar recursion calculates optimal plays and optimal strategies, which, for particular games of interest, amount to equilibria. We consider finite games and continuous countably infinite games, and relate the two. The above development is followed by a conceptual explanation of how the finite version of the main form of bar recursion considered here arises from a strong monad of selections functions that can be defined in any cartesian closed category. Finite bar recursion turns out to be a well-known morphism available in any strong monad, specialised to the selection monad.

REFINEMENTS OF NASH EQUILIBRIA IN VIEW OF JEALOUS OR FRIENDLY BEHAVIOR OF PLAYERS

2002 ◽  
Vol 04 (03) ◽  
pp. 281-299 ◽  
Author(s):  
AGNIESZKA RUSINOWSKA
Keyword(s):  
Nash Equilibria ◽  
Subgame Perfect Equilibrium ◽  
Perfect Equilibrium ◽  
Infinitely Divisible ◽  
Common Assumption ◽  
Bargaining Models ◽  
The Common ◽  
Simultaneous Moves ◽  
The One ◽  
Alternating Offers

In this paper, several bargaining models, differing in some assumptions from each other, are analyzed. We consider a discrete case and a continuous case. In the former model, players bargain over a division of n objects. In the latter, parties divide one unit of infinitely divisible good. We start with an analysis of the one-round model, and then we consider a model in which players can continue to bargain. For each model, simultaneous moves as well as alternating offers of players are considered. The assumption that each player receives no more than his/her opponent proposes giving to him/her is the common assumption for all cases analyzed. Moreover, we adopt some assumptions concerning players' attitudes towards their opponents' payments, assuming that players can be either jealous or friendly. In view of the jealousy or friendliness of players, Nash equilibrium and subgame perfect equilibrium are described.

Sign in / Sign up

Export Citation Format

Share Document