A Game Theory-Based Heuristic for the Two-Dimensional VLSI Global Routing Problem

2015 ◽  
Vol 24 (06) ◽  
pp. 1550082 ◽  
Author(s):  
Umair F. Siddiqi ◽  
Sadiq M. Sait ◽  
Yoichi Shiraishi
Keyword(s):  
Game Theory ◽  
Spanning Tree ◽  
Pure Strategy ◽  
Main Task ◽  
Mixed Strategies ◽  
Routing Problem ◽  
Pure Strategies ◽  
R Process ◽  
Valid Solution ◽  
Number Of Iterations

In this work we propose a game theory (GT)-based global router. It works in two steps: (i) Initial routing of all nets using maze routing with framing (MRF) and (ii) GT-based rip-up and reroute (R&R) process. In initial routing, the nets are divided into several small subsets which are routed concurrently using multithreading (MT). The main task of the GT-based R&R process is to eliminate congestion. Nets are considered as players and each player employs two pure strategies: (attempt to improve its spanning tree, and, do not attempt to improve its spanning tree). The nets also have mixed strategies whose values act as probabilities for them to select any particular pure strategy. The nets which select their first strategy will go through the R&R operation. We also propose an algorithm which computes the mixed strategies of nets. The advantage of using GT to select nets is that it reduces the number of nets and the number of iterations in the R&R process. The performance of the proposed global router was evaluated on ISPD'98 benchmarks and compared with two recent global routers, namely, Box Router 2.0 (configured for speed) and Side-winder. The results show that the proposed global router with MT has a shorter runtime to converge to a valid solution than that of Box Router 2.0. It also outperforms Side-winder in terms of routability. The experimental results demonstrated that GT is a valuable technique in reducing the runtime of global routers.

On the Foundations of Nash Equilibrium

1996 ◽  
Vol 12 (1) ◽  
pp. 67-88 ◽  
Author(s):  
Hans Jørgen Jacobsen
Keyword(s):  
Game Theory ◽  
Nash Equilibrium ◽  
Cooperative Game Theory ◽  
Mixed Strategy ◽  
Pure Strategy ◽  
Mixed Strategies ◽  
Positive Theory ◽  
Equilibrium Concept ◽  
Strategy Equilibrium ◽  
Pure Strategies

The most important analytical tool in non-cooperative game theory is the concept of a Nash equilibrium, which is a collection of possibly mixed strategies, one for each player, with the property that each player's strategy is a best reply to the strategies of the other players. If we do not go into normative game theory, which concerns itself with the recommendation of strategies, and focus instead entirely on the positive theory of prediction, two alternative interpretations of the Nash equilibrium concept are predominantly available.In the more traditional one, a Nash equilibrium is a prediction of actual play. A game may not have a Nash equilibrium in pure strategies, and a mixed strategy equilibrium may be difficult to incorporate into this interpretation if it involves the idea of actual randomization over equally good pure strategies. In another interpretation originating from Harsanyi (1973a), see also Rubinstein (1991), and Aumann and Brandenburger (1991), a Nash equilibrium is a ‘consistent’ collection of probabilistic expectations, conjectures, on the players. It is consistent in the sense that for each player each pure strategy, which has positive probability according to the conjecture about that player, is indeed a best reply to the conjectures about others.

scholarly journals Cooperation in the Prisoner’s Dilemma: an experimental comparison between pure and mixed strategies

2019 ◽  
Vol 6 (7) ◽  
pp. 182142
Author(s):  
Leonie Heuer ◽  
Andreas Orland
Keyword(s):  
Treatment Group ◽  
Prisoner's Dilemma ◽  
Mixed Strategy ◽  
Pure Strategy ◽  
Prisoner’S Dilemma ◽  
Experimental Comparison ◽  
Control Group ◽  
Strategy Group ◽  
Mixed Strategies ◽  
Pure Strategies

Cooperation is—despite not being predicted by game theory—a widely documented aspect of human behaviour in Prisoner’s Dilemma (PD) situations. This article presents a comparison between subjects restricted to playing pure strategies and subjects allowed to play mixed strategies in a one-shot symmetric PD laboratory experiment. Subjects interact with 10 other subjects and take their decisions all at once. Because subjects in the mixed-strategy treatment group are allowed to condition their level of cooperation more precisely on their beliefs about their counterparts’ level of cooperation, we predicted the cooperation rate in the mixed-strategy treatment group to be higher than in the pure-strategy control group. The results of our experiment reject our prediction: even after controlling for beliefs about the other subjects’ level of cooperation, we find that cooperation in the mixed-strategy group is lower than in the pure-strategy group. We also find, however, that subjects in the mixed-strategy group condition their cooperative behaviour more closely on their beliefs than in the pure-strategy group. In the mixed-strategy group, most subjects choose intermediate levels of cooperation.

scholarly journals Converting MST to TSP Path by Branch Elimination

2020 ◽  
Vol 11 (1) ◽  
pp. 177
Author(s):  
Pasi Fränti ◽  
Teemu Nenonen ◽  
Mingchuan Yuan
Keyword(s):  
Spanning Tree ◽  
Closed Loop ◽  
Minimum Spanning Tree ◽  
Travelling Salesman Problem ◽  
Open Loop ◽  
Travelling Salesman ◽  
Elimination Algorithm ◽  
Number Of Iterations ◽  
Number Of Branches

Travelling salesman problem (TSP) has been widely studied for the classical closed loop variant but less attention has been paid to the open loop variant. Open loop solution has property of being also a spanning tree, although not necessarily the minimum spanning tree (MST). In this paper, we present a simple branch elimination algorithm that removes the branches from MST by cutting one link and then reconnecting the resulting subtrees via selected leaf nodes. The number of iterations equals to the number of branches (b) in the MST. Typically, b << n where n is the number of nodes. With O-Mopsi and Dots datasets, the algorithm reaches gap of 1.69% and 0.61 %, respectively. The algorithm is suitable especially for educational purposes by showing the connection between MST and TSP, but it can also serve as a quick approximation for more complex metaheuristics whose efficiency relies on quality of the initial solution.

MONTE CARLO METHODS IN FUZZY GAME THEORY

2007 ◽  
Vol 03 (02) ◽  
pp. 259-269 ◽  
Author(s):  
AREEG ABDALLA ◽  
JAMES BUCKLEY
Keyword(s):  
Game Theory ◽  
Monte Carlo ◽  
Monte Carlo Method ◽  
Monte Carlo Methods ◽  
Mixed Strategy ◽  
Minimax Theorem ◽  
Approximate Solutions ◽  
Mixed Strategies ◽  
Fuzzy Game ◽  
Zero Sum

In this paper, we consider a two-person zero-sum game with fuzzy payoffs and fuzzy mixed strategies for both players. We define the fuzzy value of the game for both players [Formula: see text] and also define an optimal fuzzy mixed strategy for both players. We then employ our fuzzy Monte Carlo method to produce approximate solutions, to an example fuzzy game, for the fuzzy values [Formula: see text] for Player I and [Formula: see text] for Player II; and also approximate solutions for the optimal fuzzy mixed strategies for both players. We then look at [Formula: see text] and [Formula: see text] to see if there is a Minimax theorem [Formula: see text] for this fuzzy game.

Why Students Don't Study and Employers Don't Pay High Salaries?

2021 ◽  
Vol 29 (6) ◽  
pp. 557-567
Author(s):  
Georgi Kiranchev
Keyword(s):  
Higher Education ◽  
Game Theory ◽  
Optimal Strategy ◽  
Bimatrix Game ◽  
Pure Strategies ◽  
Made In ◽  
Low Wages

The article examines the behavior of students and employers as a bimatrix game. With the tools of game theory, it is generally proven that the optimal strategy for employers is to pay low wages, and for students – not to study or to study too little. These two strategies form the Nash’s equilibrium in pure strategies. No specific numbers were used in the evidence, but only plausible assumptions about the relationships between the used parameters. This generalizes the conclusions made in the general case of higher education. Such a study of the question using game theory has not been done yet.

Exponential Families and Game Dynamics

1982 ◽  
Vol 34 (2) ◽  
pp. 374-405 ◽  
Author(s):  
Ethan Akin
Keyword(s):  
Game Theory ◽  
Evolutionary Game ◽  
Payoff Matrix ◽  
Exponential Families ◽  
Von Neumann ◽  
Rational Player ◽  
Large Populations ◽  
Pure Strategies ◽  
Classical Game ◽  
Classical Game Theory

A symmetric game consists of a set of pure strategies indexed by {0, …, n} and a real payoff matrix (aij). When two players choose strategies i and j the payoffs are aij and aji to the i-player and j-player respectively. In classical game theory of Von Neumann and Morgenstern [16] the payoffs are measured in units of utility, i.e., desirability, or in units of some desirable good, e.g. money. The problem of game theory is that of a rational player who seeks to choose a strategy or mixture of strategies which will maximize his return. In evolutionary game theory of Maynard Smith and Price [13] we look at large populations of game players. Each player's opponents are selected randomly from the population, and no information about the opponent is available to the player. For each one the choice of strategy is a fixed inherited characteristic.

The behaviour of strategy-frequencies in Parker's model

1980 ◽  
Vol 12 (01) ◽  
pp. 5-7
Author(s):  
D. Gardiner
Keyword(s):  
Finite Number ◽  
Evolutionarily Stable Strategy ◽  
Pure Strategy ◽  
Payoff Matrix ◽  
Stable Strategy ◽  
Expected Payoff ◽  
Pure Strategies ◽  
Image Position ◽  
Evolutionarily Stable

Parker's model (or the Scotch Auction) for a contest between two competitors has been studied by Rose (1978). He considers a form of the model in which every pure strategy is playable, and shows that there is no evolutionarily stable strategy (ess). In this paper, in order to discover more about the behaviour of strategies under the model, we shall assume that there are only a finite number of playable pure strategies I 1, I 2, ···, I n where I j is the strategy ‘play value m j ′ and m 1 &lt; m 2 &lt; ··· &lt; m n . The payoff matrix A for the contest is then given by where V is the reward for winning the contest, C is a constant added to ensure that each entry in A is non-negative (see Bishop and Cannings (1978)), and E[I i , I j ] is the expected payoff for playing I i against I j . We also assume that A is regular (Taylor and Jonker (1978)) i.e. that all its rows are independent.

BENEFIT FUNCTION AND DUALITY IN FINITE NORMAL FORM GAMES

2007 ◽  
Vol 09 (03) ◽  
pp. 495-513
Author(s):  
WALTER BRIEC
Keyword(s):  
Game Theory ◽  
Normal Form ◽  
Mixed Strategies ◽  
Numerical Function ◽  
Normal Form Games ◽  
Benefit Function ◽  
Pareto Efficient

Luenberger (1992, 1994) introduced a function he terms the benefit function, that converts preferences into a numerical function and has some cardinal meaning. In this paper, we show that the benefit function enjoys many interesting properties in a game theory context. We point out that the benefit function can be adapted to compare the mixed profiles of a game. Along this line, inspired from the Luenberger's approach, we propose a dual framework and establish a characterization of Nash equilibriums in terms of the benefit function. Moreover, some criterions are provided to identify the efficient mixed strategies of a game (which differ from the Pareto efficient strategies). Finally, we go a bit further proposing some issue in comparing profiles and equilibriums of a game. This we do using the so-called Σ-subdifferential of the benefit function.

Game theory and the evolution of behaviour

1979 ◽  
Vol 205 (1161) ◽  
pp. 475-488 ◽  
Keyword(s):  
Game Theory ◽  
Evolutionarily Stable Strategy ◽  
Field Studies ◽  
Mixed Strategies ◽  
Conditional Strategy ◽  
Stable Strategy ◽  
Frequency Dependent Selection ◽  
Contest Behaviour ◽  
Assessment Strategies ◽  
Evolutionarily Stable

How far can game theory account for the evolution of contest behaviour in animals? The first qualitative prediction of the theory was that symmetric contests in which escalation is expensive should lead to mixed strategies. As yet it is hard to say how far this is borne out, because of the difficulty of distinguishing a ‘mixed evolutionarily stable strategy’ maintained by frequency-dependent selection from a ‘pure conditional strategy'; the distinction is discussed in relation to several field studies. The second prediction was that if a contest is asymmetric (e. g. in ownership) then the asymmetry will be used as a conventional cue to settle it. This prediction has been well supported by observation. A third important issue is whether or not information about intentions is exchanged during contests. The significance of ‘assessment’ strategies is discussed.

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