scholarly journals A symbolic shortest path algorithm for computing subgame-perfect Nash equilibria

2015 ◽  
Vol 25 (3) ◽  
pp. 577-596 ◽  
Author(s):  
Pedro A. Góngora ◽  
David A. Rosenblueth
Keyword(s):  
Nash Equilibrium ◽  
Model Checking ◽  
Critical Path ◽  
Shortest Paths ◽  
Subgame Perfect Nash Equilibrium ◽  
Graph Games ◽  
Optimal Paths ◽  
Minimum Number ◽  
The Cost ◽  
Extensive Games

AbstractConsider games where players wish to minimize the cost to reach some state. A subgame-perfect Nash equilibrium can be regarded as a collection of optimal paths on such games. Similarly, the well-known state-labeling algorithm used in model checking can be viewed as computing optimal paths on a Kripke structure, where each path has a minimum number of transitions. We exploit these similarities in a common generalization of extensive games and Kripke structures that we name “graph games”. By extending the Bellman-Ford algorithm for computing shortest paths, we obtain a model-checking algorithm for graph games with respect to formulas in an appropriate logic. Hence, when given a certain formula, our model-checking algorithm computes the subgame-perfect Nash equilibrium (as opposed to simply determining whether or not a given collection of paths is a Nash equilibrium). Next, we develop a symbolic version of our model checker allowing us to handle larger graph games. We illustrate our formalism on the critical-path method as well as games with perfect information. Finally, we report on the execution time of benchmarks of an implementation of our algorithms

A Generic Approach on How to Formally Specify and Model Check Path Finding Algorithms: Dijkstra, A* and LPA*

2020 ◽  
Vol 30 (10) ◽  
pp. 1481-1523
Author(s):  
Kazuhiro Ogata
Keyword(s):  
Model Checking ◽  
Shortest Path ◽  
Formal Specification ◽  
Model Check ◽  
Shortest Paths ◽  
Path Finding ◽  
Property A ◽  
Goal Node ◽  
The Cost ◽  
Start Node

The paper describes how to formally specify three path finding algorithms in Maude, a rewriting logic-based programming/specification language, and how to model check if they enjoy desired properties with the Maude LTL model checker. The three algorithms are Dijkstra Shortest Path Finding Algorithm (DA), A* Algorithm and LPA* Algorithm. One desired property is that the algorithms always find the shortest path. To this end, we use a path finding algorithm (BFS) based on breadth-first search. BFS finds all paths from a start node to a goal node and the set of all shortest paths is extracted. We check if the path found by each algorithm is included in the set of all shortest paths for the property. A* is an extension of DA in that for each node [Formula: see text] an estimation [Formula: see text] of the distance to the goal node from [Formula: see text] is used and LPA* is an incremental version of A*. It is known that if [Formula: see text] is admissible, A* always finds the shortest path. We have found a possible relaxed sufficient condition. The relaxed condition is that there exists the shortest path such that for each node [Formula: see text] except for the start node on the path [Formula: see text] plus the cost to [Formula: see text] from the start node is less than the cost of any non-shortest path to the goal from the start. We informally justify the relaxed condition. For LPA*, if the relaxed condition holds in each updated version of a graph concerned including the initial graph, the shortest path is constructed. Based on the three case studies for DA, A* and LPA*, we summarize the formal specification and model checking techniques used as a generic approach to formal specification and model checking of path finding algorithms.

scholarly journals Pure Nash Equilibria in Resource Graph Games

2021 ◽  
Vol 72 ◽  
Author(s):  
Tobias Harks ◽  
Max Klimm ◽  
Jannik Matuschke
Keyword(s):  
Nash Equilibrium ◽  
Arbitrary Function ◽  
Nash Equilibria ◽  
Mutual Influence ◽  
Pure Nash Equilibrium ◽  
Strategy Space ◽  
Graph Games ◽  
Finite Set ◽  
Load Vector ◽  
The Cost

This paper studies the existence of pure Nash equilibria in resource graph games, a general class of strategic games succinctly representing the players’ private costs. These games are defined relative to a finite set of resources and the strategy set of each player corresponds to a set of subsets of resources. The cost of a resource is an arbitrary function of the load vector of a certain subset of resources. As our main result, we give complete characterizations of the cost functions guaranteeing the existence of pure Nash equilibria for weighted and unweighted players, respectively. For unweighted players, pure Nash equilibria are guaranteed to exist for any choice of the players’ strategy space if and only if the cost of each resource is an arbitrary function of the load of the resource itself and linear in the load of all other resources where the linear coefficients of mutual influence of different resources are symmetric. This implies in particular that for any other cost structure there is a resource graph game that does not have a pure Nash equilibrium. For weighted games where players have intrinsic weights and the cost of each resource depends on the aggregated weight of its users, pure Nash equilibria are guaranteed to exist if and only if the cost of a resource is linear in all resource loads, and the linear factors of mutual influence are symmetric, or there is no interaction among resources and the cost is an exponential function of the local resource load. We further discuss the computational complexity of pure Nash equilibria in resource graph games showing that for unweighted games where pure Nash equilibria are guaranteed to exist, it is coNP-complete to decide for a given strategy profile whether it is a pure Nash equilibrium. For general resource graph games, we prove that the decision whether a pure Nash equilibrium exists is Σ p 2 -complete.

Determination of the minimum number of technological bases of a part

2020 ◽  
pp. 73-75
Author(s):  
B.M. Bazrov ◽  
T.M. Gaynutdinov
Keyword(s):  
Cost Price ◽  
Labor Input ◽  
Part Surface ◽  
Technological Base ◽  
Size Accuracy ◽  
Minimum Number ◽  
The Cost ◽  
Input Cost ◽  
Selection Of

The selection of technological bases is considered before the choice of the type of billet and the development of the route of the technological process. A technique is proposed for selecting the minimum number of sets of technological bases according to the criterion of equality in the cost price of manufacturing the part according to the principle of unity and combination of bases at this stage. Keywords: part, surface, coordinating size, accuracy, design and technological base, labor input, cost price. [email protected]

A Harmonic Source Location Method Based on PMU Measurements in Distribution Network

2015 ◽  
Vol 775 ◽  
pp. 409-414
Author(s):  
Bing Jun Li ◽  
Su Quan Zhou ◽  
Xiao Xiang Lun
Keyword(s):  
Power System ◽  
State Estimation ◽  
Distribution Network ◽  
Measured Data ◽  
Optimal Measurement ◽  
Location Method ◽  
Measurement Placement ◽  
Minimum Number ◽  
Harmonic Source ◽  
The Cost

It is of great importance to identify the location of the harmonic sources for the harmonic governance in the power system. Applied with optimal measurement placement (OMP) and harmonic state estimation (HSE), this paper presents a novel process based on PMU measurements to locate the harmonic sources in the distribution network. Considering the cost and the observability, the OMP can provide a scheme of the measurement placement with the minimum number of PMU measurements. In order to simplify the HSE equation, the measured data are converted to the form of voltage by the method proposed in this paper.By solving the HSE equation, the location and magnitude of the harmonic source are evaluated. The methodology is applied to the IEEE 33-bus system, and the obtained results are properly analyzed.

scholarly journals AN EXACT SOLUTION FOR THE RESOURCE-CONSTRAINED CONSTRUCTION SCHEDULING PROBLEM

2016 ◽  
Vol 8 (2) ◽  
pp. 71-78
Author(s):  
Bartłomiej Sroka ◽  
Elżbieta Radziszewska-Zielina
Keyword(s):  
Execution Time ◽  
Critical Path ◽  
Scheduling Problem ◽  
Resource Constrained ◽  
Building Projects ◽  
Construction Scheduling ◽  
Program Sample ◽  
Resource Constrained Scheduling Problem ◽  
The Cost ◽  
Exponential Complexity

Reduced time and, by the same token, the cost of the project is a crucial factor in contemporary construction. This article presents a method for the exact optimisation of a resource-constrained scheduling problem. Based on the Critical Path Method, graph theory and linear programming, an algorithm was developed and the FROPT program was written in Matlab to minimise the execution time of the task. By using the newly-created program, sample networks were calculated and the results were compared with results obtained by using the MS Project scheduling program (using approximation algorithm). The execution time obtained by using FROPT were on average 10% shorter than those obtained using MS Project. In selected cases the improvement in execution time reached 25%. A deterministic approach to the problem may shorten planned project times and bring financial benefits. Due to the exponential complexity of the algorithm, it is most useful in solving small or highly coherent networks. The algorithm and program may result in benefits not offered by commercial software for planners of building projects.

scholarly journals A MODIFIED GENETIC ALGORITHM FOR FINDING FUZZY SHORTEST PATHS IN UNCERTAIN NETWORKS

2016 ◽  
Vol XLI-B2 ◽  
pp. 299-304 ◽  
Author(s):  
A. A. Heidari ◽  
M. R. Delavar
Keyword(s):  
Genetic Algorithm ◽  
Shortest Path ◽  
Shortest Paths ◽  
The Body ◽  
Shortest Path Problems ◽  
Conventional Procedure ◽  
Path Lengths ◽  
The Cost ◽  
Uncertain Networks ◽  
Exploration Exploitation

In realistic network analysis, there are several uncertainties in the measurements and computation of the arcs and vertices. These uncertainties should also be considered in realizing the shortest path problem (SPP) due to the inherent fuzziness in the body of expert's knowledge. In this paper, we investigated the SPP under uncertainty to evaluate our modified genetic strategy. We improved the performance of genetic algorithm (GA) to investigate a class of shortest path problems on networks with vague arc weights. The solutions of the uncertain SPP with considering fuzzy path lengths are examined and compared in detail. As a robust metaheuristic, GA algorithm is modified and evaluated to tackle the fuzzy SPP (FSPP) with uncertain arcs. For this purpose, first, a dynamic operation is implemented to enrich the exploration/exploitation patterns of the conventional procedure and mitigate the premature convergence of GA technique. Then, the modified GA (MGA) strategy is used to resolve the FSPP. The attained results of the proposed strategy are compared to those of GA with regard to the cost, quality of paths and CPU times. Numerical instances are provided to demonstrate the success of the proposed MGA-FSPP strategy in comparison with GA. The simulations affirm that not only the proposed technique can outperform GA, but also the qualities of the paths are effectively improved. The results clarify that the competence of the proposed GA is preferred in view of quality quantities. The results also demonstrate that the proposed method can efficiently be utilized to handle FSPP in uncertain networks.

scholarly journals The Number of Moves of the Largest Disc in Shortest Paths on Hanoi Graphs

10.37236/4252 ◽  
2014 ◽  
Vol 21 (4) ◽  
Author(s):  
Simon Aumann ◽  
Katharina A.M. Götz ◽  
Andreas M. Hinz ◽  
Ciril Petr
Keyword(s):  
Shortest Paths ◽  
General Setting ◽  
Optimal Number ◽  
Numerical Results ◽  
Breadth First Search ◽  
Tower Of Hanoi ◽  
Optimal Paths ◽  
Widespread Interest ◽  
Binary Strings

In contrast to the widespread interest in the Frame-Stewart conjecture (FSC) about the optimal number of moves in the classical Tower of Hanoi task with more than three pegs, this is the first study of the question of investigating shortest paths in Hanoi graphs $H_p^n$ in a more general setting. Here $p$ stands for the number of pegs and $n$ for the number of discs in the Tower of Hanoi interpretation of these graphs. The analysis depends crucially on the number of largest disc moves (LDMs). The patterns of these LDMs will be coded as binary strings of length $p-1$ assigned to each pair of starting and goal states individually. This will be approached both analytically and numerically. The main theoretical achievement is the existence, at least for all $n\geqslant p(p-2)$, of optimal paths where $p-1$ LDMs are necessary. Numerical results, obtained by an algorithm based on a modified breadth-first search making use of symmetries of the graphs, lead to a couple of conjectures about some cases not covered by our ascertained results. These, in turn, may shed some light on the notoriously open FSC.

Applications to Economics

2016 ◽  
Author(s):  
Jacob K. Goeree ◽  
Charles A. Holt ◽  
Thomas R. Palfrey
Keyword(s):  
Nash Equilibrium ◽  
Private Information ◽  
Price Competition ◽  
Quantal Response ◽  
Unique Nash Equilibrium ◽  
Common Effort ◽  
The Cost ◽  
All Pay Auction ◽  
Individual Effort ◽  
Applications To Economics

This chapter explores whether the equilibrium effects of noisy behavior can cause large deviations from standard predictions in economically relevant situations. It considers a simple price-competition game, which is also partly motivated by the possibility of changing a payoff parameter that has no effect on the unique Nash equilibrium, but which may be expected to affect quantal response equilibrium. In the minimum-effort coordination game studied, any common effort in the range of feasible effort levels is a Nash equilibrium, but one would expect that an increase in the cost of individual effort or an increase in the number of players who are trying to coordinate would reduce the effort levels observed in an experiment. The chapter presents an analysis of the logit equilibrium and rent dissipation for a rent-seeking contest that is modeled as an “all-pay auction.” The final two applications in this chapter deal with auctions with private information.

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