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 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.

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 Example of a Finite Game with No Berge Equilibria at All

Games ◽  
2019 ◽  
Vol 10 (1) ◽  
pp. 7 ◽  
Author(s):  
Jarosław Pykacz ◽  
Paweł Bytner ◽  
Piotr Frąckiewicz
Keyword(s):  
Normal Form ◽  
Mixed Strategies ◽  
Finite Games ◽  
Berge Equilibrium ◽  
Finite Game ◽  
Player Game

The problem of the existence of Berge equilibria in the sense of Zhukovskii in normal-form finite games in pure and in mixed strategies is studied. The example of a three-player game that has Berge equilibrium neither in pure, nor in mixed strategies is given.

scholarly journals An application of graphical enumeration to PA

2003 ◽  
Vol 68 (1) ◽  
pp. 5-16
Author(s):  
Andreas Weiermann
Keyword(s):  
Normal Form ◽  
Asymptotic Analysis ◽  
Natural Number ◽  
Proof Theory ◽  
Peano Arithmetic ◽  
First Order ◽  
Primitive Recursive Arithmetic ◽  
Independence Result ◽  
Primitive Recursive

AbstractFor α less than ε0 let Nα be the number of occurrences of ω in the Cantor normal form of α. Further let ∣n∣ denote the binary length of a natural number n, let ∣n∣h denote the h-times iterated binary length of n and let inv(n) be the least h such that ∣n∣h ≤ 2. We show that for any natural number h first order Peano arithmetic, PA, does not prove the following sentence: For all K there exists an M which bounds the lengths n of all strictly descending sequences 〈α0, …, αn〉 of ordinals less than ε0 which satisfy the condition that the Norm Nαi of the i-th term αi is bounded by K + ∣i∣ · ∣i∣i.As a supplement to this (refined Friedman style) independence result we further show that e.g., primitive recursive arithmetic, PRA, proves that for all K there is an M which bounds the length n of any strictly descending sequence 〈α0, …, αn〉 of ordinals less than ε0 which satisfies the condition that the Norm Nαi of the i-th term αi is bounded by K + ∣i∣ · inv(i). The proofs are based on results from proof theory and techniques from asymptotic analysis of Polya-style enumerations.Using results from Otter and from Matoušek and Loebl we obtain similar characterizations for finite bad sequences of finite trees in terms of Otter's tree constant 2.9557652856.…

scholarly journals The Price of Usability: Designing Operationalizable Strategies for Security Games

2018 ◽  
Author(s):  
Sara Marie Mc Carthy ◽  
Corine M. Laan ◽  
Kai Wang ◽  
Phebe Vayanos ◽  
Arunesh Sinha ◽  
...  
Keyword(s):  
Optimization Problem ◽  
Mixed Strategy ◽  
Static Solution ◽  
Mixed Integer ◽  
Integer Optimization ◽  
Security Personnel ◽  
Security Games ◽  
Game Theoretic ◽  
Pure Strategies ◽  
Security Game

We consider the problem of allocating scarce security resources among heterogeneous targets to thwart a possible attack. It is well known that deterministic solutions to this problem being highly predictable are severely suboptimal. To mitigate this predictability, the game-theoretic security game model was proposed which randomizes over pure (deterministic) strategies, causing confusion in the adversary. Unfortunately, such mixed strategies typically involve randomizing over a large number of strategies, requiring security personnel to be familiar with numerous protocols, making them hard to operationalize. Motivated by these practical considerations, we propose an easy to use approach for computing  strategies that are easy to operationalize and that bridge the gap between the static solution and the optimal mixed strategy. These strategies only randomize over an optimally chosen subset of pure strategies whose cardinality is selected by the defender, enabling them to conveniently tune the trade-off between ease of operationalization and efficiency using a single design parameter. We show that the problem of computing such operationalizable strategies is NP-hard, formulate it as a mixed-integer optimization problem, provide an algorithm for computing epsilon-optimal equilibria, and an efficient heuristic. We evaluate the performance of our approach on the problem of screening for threats at airport checkpoints and show that the Price of Usability, i.e., the loss in optimality to obtain a strategy that is easier to operationalize, is typically not high.

Game theoretic modeling of economic denial of sustainability (EDoS) attack in cloud computing

2021 ◽  
pp. 1-25
Author(s):  
KC Lalropuia ◽  
Vandana Khaitan (nee Gupta)
Keyword(s):  
Mixed Strategy ◽  
Dynamic Game ◽  
Theoretic Model ◽  
Numerical Illustration ◽  
Defense Strategies ◽  
Bayesian Nash Equilibrium ◽  
Proposed Model ◽  
Edos Attacks ◽  
Game Theoretic ◽  
Pooling Equilibrium

Abstract In this paper, we develop a novel game theoretic model of the interactions between an EDoS attacker and the defender based on a signaling game that is a dynamic game of incomplete information. We then derive the best defense strategies for the network defender to respond to the EDoS attacks. That is, we compute the perfect Bayesian Nash Equilibrium (PBE) of the proposed game model such as the pooling PBE, separating PBE and mixed strategy PBE. In the pooling equilibrium, each type of the attacker takes the same action and the attacker's type is not revealed to the defender, whereas in the separating equilibrium, each type of the attacker uses different actions and hence the attacker's type is completely revealed to the defender. On the other hand, in the mixed strategy PBE, both the attacker and the defender randomize their strategies to optimize their payoffs. Numerical illustration is also presented to show the efficacy of the proposed model.

Paradox of crosses in association football (soccer) – a game-theoretic explanation

2018 ◽  
Vol 14 (1) ◽  
pp. 25-36 ◽  
Author(s):  
Sumit Sarkar
Keyword(s):  
Nash Equilibrium ◽  
Theoretical Model ◽  
Mixed Strategy ◽  
Inverse Relation ◽  
Association Football ◽  
Game Theoretical Model ◽  
Mixed Strategy Nash Equilibrium ◽  
Game Theoretic ◽  
Strategy Nash Equilibrium

Abstract In association football, crosses from the wide areas of the pitch in the attacking third is a standard tactic for creating goal-scoring opportunities. But recent studies show that crosses adversely impact goals. Regression run in this paper on data from the premier soccer leagues of England, Spain, Germany, France and Italy for 2016–2017 season also found this inverse relation. However, there is no research that explains the reason for this inverse relation between crosses and goals. A game-theoretical model developed in this paper explains why crosses adversely affect goal-scoring. The model identifies a mixed strategy Nash equilibrium (MSNE), wherein the attacking team’s probability of playing a cross decreases with increase in their crossing accuracy, heading accuracy and probability of winning aerial balls. If the attacking team is good in terms of these parameters, the defending team’s probability of using an offside trap increases and that forces the attacking team to use crosses less frequently. In the MSNE, teams with a greater chance of scoring from crosses use the crosses less frequently than teams having a smaller chance of scoring from crosses. The theory was subsequently validated using the data of the 2016–2017 football season.

The Economics of Crime Reconsidered

2012 ◽  
pp. 352-379 ◽  
Author(s):  
Joseph P. McGarrity
Keyword(s):  
Mixed Strategy ◽  
Major League Baseball ◽  
Total Effect ◽  
Theoretic Model ◽  
Theoretic Approach ◽  
Economics Of Crime ◽  
Game Theoretic ◽  
Strategy Game ◽  
Game Theoretic Model ◽  
Game Theoretic Approach

This article uses data on hit batsmen from Major League Baseball to illustrate a mixed-strategy, game theoretic approach to the decisions of the pitcher and the batter. The pitcher would like to throw to a batter who stands in the middle of the batter's box. The game theoretic model predicts that the pitcher will throw at fewer batters as velocity increases, while the standard crime model would assume that the pitcher's throw-ats would remain unchanged and the batter would respond by leaning in less often. The Total Effect curves suggest that there will be more throw-ats in the American League for any level of velocity. The number of purposeful inside pitches will decrease at an increasing rate as velocity increases. The game theoretic model predicts that a pitcher who can throw with greater velocity will have to waste fewer inside pitches to keep a batter from leaning into a pitch.

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