The Buck-Passing Game

2021 ◽  
Author(s):  
Roberto Cominetti ◽  
Matteo Quattropani ◽  
Marco Scarsini
Keyword(s):  
Nash Equilibrium ◽  
Directed Graph ◽  
Hamiltonian Cycle ◽  
Price Of Anarchy ◽  
Initial Distribution ◽  
The Other ◽  
Expected Frequency ◽  
Fixed Distribution ◽  
Pure Equilibria ◽  
Buck Passing

We consider two classes of games in which players are the vertices of a directed graph. Initially, nature chooses one player according to some fixed distribution and gives the player a buck. This player passes the buck to one of the player’s out-neighbors in the graph. The procedure is repeated indefinitely. In one class of games, each player wants to minimize the asymptotic expected frequency of times that the player receives the buck. In the other class of games, the player wants to maximize it. The PageRank game is a particular case of these maximizing games. We consider deterministic and stochastic versions of the game, depending on how players select the neighbor to which to pass the buck. In both cases, we prove the existence of pure equilibria that do not depend on the initial distribution; this is achieved by showing the existence of a generalized ordinal potential. If the graph on which the game is played admits a Hamiltonian cycle, then this is the outcome of prior-free Nash equilibrium in the minimizing game. For the minimizing game, we then use the price of anarchy and stability to measure fairness of these equilibria.

Hamiltonicity of a class of toroidal graphs

2020 ◽  
Vol 70 (2) ◽  
pp. 497-503
Author(s):  
Dipendu Maity ◽  
Ashish Kumar Upadhyay
Keyword(s):  
Connected Graph ◽  
Hamiltonian Cycle ◽  
Partial Solution ◽  
The Other ◽  
Hamiltonian Cycles ◽  
The Face ◽  
Toroidal Graphs

Abstract If the face-cycles at all the vertices in a map are of same type then the map is said to be a semi-equivelar map. There are eleven types of semi-equivelar maps on the torus. In 1972 Altshuler has presented a study of Hamiltonian cycles in semi-equivelar maps of three types {36}, {44} and {63} on the torus. In this article we study Hamiltonicity of semi-equivelar maps of the other eight types {33, 42}, {32, 41, 31, 41}, {31, 61, 31, 61}, {34, 61}, {41, 82}, {31, 122}, {41, 61, 121} and {31, 41, 61, 41} on the torus. This gives a partial solution to the well known Conjecture that every 4-connected graph on the torus has a Hamiltonian cycle.

scholarly journals Approximating Cayley Diagrams Versus Cayley Graphs

2012 ◽  
Vol 21 (4) ◽  
pp. 635-641
Author(s):  
ÁDÁM TIMÁR
Keyword(s):  
Cayley Graph ◽  
Hamiltonian Cycle ◽  
Cayley Graphs ◽  
The Other ◽  
Finite Graphs ◽  
Similar Construction ◽  
Graph Property

We construct a sequence of finite graphs that weakly converge to a Cayley graph, but there is no labelling of the edges that would converge to the corresponding Cayley diagram. A similar construction is used to give graph sequences that converge to the same limit, and such that a Hamiltonian cycle in one of them has a limit that is not approximable by any subgraph of the other. We give an example where this holds, but convergence is meant in a stronger sense. This is related to whether having a Hamiltonian cycle is a testable graph property.

scholarly journals ENDOGENOUS BORROWING CONSTRAINTS AND WEALTH INEQUALITY

2016 ◽  
Vol 20 (6) ◽  
pp. 1413-1431 ◽  
Author(s):  
Joydeep Bhattacharya ◽  
Xue Qiao ◽  
Min Wang
Keyword(s):  
Steady State ◽  
Credit Market ◽  
Wealth Inequality ◽  
Borrowing Constraints ◽  
Initial Distribution ◽  
The Other ◽  
Capital Investments ◽  
Loan Default ◽  
Long Run ◽  
Human Capital Investments

This paper studies the evolution of wealth inequality in an economy with endogenous borrowing constraints. In the model economy, young agents need to borrow to finance human capital investments but cannot commit to repaying their loans. Creditors can punish defaulters by banishing them permanently from the credit market. At equilibrium, loan default is prevented by imposing a borrowing limit tied to the borrower's inheritance. The heterogeneity in inheritances translates into heterogeneity in borrowing limits: endogenously, some borrowers face a zero borrowing limit, and some are partly constrained, whereas others are unconstrained. Depending on the initial distribution of inheritances, it is possible that all lineages are attracted either to the zero-borrowing-limit steady state or to the unconstrained-borrowing steady state—long-run equality. It is also possible that some lineages end up in one steady state and the rest in the other—complete polarization.

scholarly journals Friendly-rivalry solution to the iterated n-person public-goods game

2021 ◽  
Vol 17 (1) ◽  
pp. e1008217
Author(s):  
Yohsuke Murase ◽  
Seung Ki Baek
Keyword(s):  
Nash Equilibrium ◽  
Public Goods ◽  
The Other ◽  
Fixation Probability ◽  
Public Goods Game ◽  
Excellent Performance ◽  
Repeated Interaction ◽  
Evolutionary Simulation ◽  
Evolutionary Robustness ◽  
Full Cooperation

Repeated interaction promotes cooperation among rational individuals under the shadow of future, but it is hard to maintain cooperation when a large number of error-prone individuals are involved. One way to construct a cooperative Nash equilibrium is to find a ‘friendly-rivalry’ strategy, which aims at full cooperation but never allows the co-players to be better off. Recently it has been shown that for the iterated Prisoner’s Dilemma in the presence of error, a friendly rival can be designed with the following five rules: Cooperate if everyone did, accept punishment for your own mistake, punish defection, recover cooperation if you find a chance, and defect in all the other circumstances. In this work, we construct such a friendly-rivalry strategy for the iterated n-person public-goods game by generalizing those five rules. The resulting strategy makes a decision with referring to the previous m = 2n − 1 rounds. A friendly-rivalry strategy for n = 2 inherently has evolutionary robustness in the sense that no mutant strategy has higher fixation probability in this population than that of a neutral mutant. Our evolutionary simulation indeed shows excellent performance of the proposed strategy in a broad range of environmental conditions when n = 2 and 3.

scholarly journals Semi-Degree Threshold for Anti-Directed Hamiltonian Cycles

10.37236/3610 ◽  
2015 ◽  
Vol 22 (4) ◽  
Author(s):  
Louis DeBiasio ◽  
Theodore Molla
Keyword(s):  
Directed Graph ◽  
Hamiltonian Cycle ◽  
Minimum Degree ◽  
Directed Graphs ◽  
Hamiltonian Cycles ◽  
Alternate Direction ◽  
Degree Conditions

In 1960 Ghouila-Houri extended Dirac's theorem to directed graphs by proving that if $D$ is a directed graph on $n$ vertices with minimum out-degree and in-degree at least $n/2$, then $D$ contains a directed Hamiltonian cycle. For directed graphs one may ask for other orientations of a Hamiltonian cycle and in 1980 Grant initiated the problem of determining minimum degree conditions for a directed graph $D$ to contain an anti-directed Hamiltonian cycle (an orientation in which consecutive edges alternate direction). We prove that for sufficiently large even $n$, if $D$ is a directed graph on $n$ vertices with minimum out-degree and in-degree at least $\frac{n}{2}+1$, then $D$ contains an anti-directed Hamiltonian cycle. In fact, we prove the stronger result that $\frac{n}{2}$ is sufficient unless $D$ is one of two counterexamples. This result is sharp.

Complexity

1995 ◽  
Author(s):  
Raymond Greenlaw ◽  
H. James Hoover ◽  
Walter L. Ruzzo
Keyword(s):  
Directed Graph ◽  
Decision Problem ◽  
Flow Problem ◽  
Maximum Flow ◽  
The Other ◽  
Complexity Classes ◽  
Search Problem ◽  
Specific Question ◽  
Key Concepts ◽  
Formal Basis

The goal of this chapter is to provide the formal basis for many key concepts that are used throughout the book. These include the notions of problem, definitions of important complexity classes, reducibility, and completeness, among others. Thus far, we have used the term "problem" somewhat vaguely. In order to compare the difficulty of various problems we need to make this concept precise. Problems typically come in two flavors: search problems and decision problems. Consider the following search problem, to find the value of the maximum flow in a network. Example 3.1.1 Maximum Flow Value (MaxFlow-V) Given: A directed graph G = (V,E) with each edge e labeled by an integer capacity c(e) ≥ 0, and two distinguished vertices, s and t. Problem: Compute the value of the maximum flow from source s to sink t in G. The problem requires us to compute a number — the value of the maximum flow. Note, in this case we are actually computing a function. Now consider a variant of this problem. Example 3.1.2 Maximum Flow Bit (MaxFlow-B) Given: A directed graph G = (V, E) with each edge e labeled by an integer capacity c(e)≥ 0, and two distinguished vertices, s and t, and an integer i. Problem: Is the ith bit of the value of the maximum flow from source s to sink t in G a 1? This is a decision problem version of the flow problem. Rather than asking for the computation of some value, the problem is asking for a "yes" or "no" answer to a specific question. Yet the decision problem MaxFlow-B is equivalent to the search problem MaxFlow-V in the sense that if one can be solved efficiently in parallel, so can the other. Why is this? First consider how solving an instance of MaxFlow-B can be reduced to solving an instance of MaxFlow-V. Suppose that you are asked a question for MaxFlow-B, that is, "Is bit i of the maximum flow a 1?" It is easy to answer this question by solving MaxFlow-V and then looking at bit i of the flow.

On Cycles and Connectivity in Planar Graphs

1973 ◽  
Vol 16 (2) ◽  
pp. 283-288 ◽  
Author(s):  
M. D. Plummer ◽  
E. L. Wilson
Keyword(s):  
Planar Graphs ◽  
Hamiltonian Cycle ◽  
The Other ◽  
Other Hand

Let G be a graph and ζ(G) be the greatest integer n such that every set of n points in G lies on a cycle [8]. It is clear that ζ(G)≥2 for 2-connected planar graphs. Moreover, it is easy to construct arbitrarily large 2-connected planar graphs for which ζ=2. On the other hand, by a well-known theorem of Tutte [5], [6], if G is planar and 4-connected, it has a Hamiltonian cycle, i.e., ζ(G)=|V(G)| for all 4-connected (and hence for all 5-connected) planar graphs.

THE PRICE OF NASH EQUILIBRIA IN MULTICAST TRANSMISSION GAMES

2010 ◽  
Vol 11 (03n04) ◽  
pp. 97-120 ◽  
Author(s):  
VITTORIO BILÒ
Keyword(s):  
Nash Equilibrium ◽  
Cost Sharing ◽  
Price Of Anarchy ◽  
Optimal Solution ◽  
Minimum Cost ◽  
Price Of Stability ◽  
Common Source ◽  
Cost Share ◽  
Worst Case ◽  
The Cost

We consider the problem of sharing the cost of multicast transmissions in non-cooperative undirected networks where a set of receivers R wants to be connected to a common source s. The set of choices available to each receiver r ∈ R is represented by the set of all (s, r)-paths in the network. Given the choices performed by all the receivers, a public known cost sharing method determines the cost share to be charged to each of them. Receivers are selfish agents aiming to obtain the transmission at the minimum cost share and their interactions create a non-cooperative game. Devising cost sharing methods yielding games whose price of anarchy (price of stability), defined as the worst-case (best-case) ratio between the cost of a Nash equilibrium and that of an optimal solution, is not too high is thus of fundamental importance in non-cooperative network design. Moreover, since cost sharing games naturally arise in socio-economical contests, it is convenient for a cost sharing method to meet some constraining properties. In this paper, we first define several such properties and analyze their impact on the prices of anarchy and stability. We also reconsider all the methods known so far by classifying them according to which properties they satisfy and giving the first non-trivial lower bounds on their price of stability. Finally, we propose a new method, namely the free-riders method, which admits a polynomial time algorithm for computing a pure Nash equilibrium whose cost is at most twice the optimal one. Some of the ideas characterizing our approach have been independently proposed in Ref. 10.

SYSTEMIC RISK: THE EFFECT OF MARKET CONFIDENCE

2020 ◽  
Vol 23 (07) ◽  
pp. 2050043
Author(s):  
MAXIM BICHUCH ◽  
KE CHEN
Keyword(s):  
Nash Equilibrium ◽  
Simple Model ◽  
Empirical Study ◽  
Systemic Risk ◽  
Existence And Uniqueness ◽  
Maximum Amount ◽  
The Other ◽  
Nash Equilibrium Point ◽  
Optimal Mix ◽  
Uniqueness Of Nash Equilibrium

In a crisis, when faced with insolvency, banks can sell stock in a dilutive offering in the stock market and borrow money in order to raise funds. We propose a simple model to find the maximum amount of new funds the banks can raise in these ways. To do this, we incorporate market confidence of the bank together with market confidence of all the other banks in the system into the overnight borrowing rate. Additionally, for a given cash shortfall, we find the optimal mix of borrowing and stock selling strategy. We show the existence and uniqueness of Nash equilibrium point for all these problems. Finally, using this model we investigate if banks have become safer since the crisis. We calibrate this model with market data and conduct an empirical study to assess safety of the financial system before, during after the last financial crisis.

scholarly journals Maximin and Minimax Strategies in Two-Players Game with Two Strategic Variables

2018 ◽  
Vol 20 (01) ◽  
pp. 1750030 ◽  
Author(s):  
Atsuhiro Satoh ◽  
Yasuhito Tanaka
Keyword(s):  
Nash Equilibrium ◽  
Objective Function ◽  
The Other ◽  
Minimax Strategy ◽  
Equilibrium Strategies ◽  
Minimax Strategies ◽  
Nash Equilibrium Strategies ◽  
Special Case ◽  
Maximin Strategy

We examine maximin and minimax strategies for players in a two-players game with two strategic variables, [Formula: see text] and [Formula: see text]. We consider two patterns of game; one is the [Formula: see text]-game in which the strategic variables of players are [Formula: see text]’s, and the other is the [Formula: see text]-game in which the strategic variables of players are [Formula: see text]’s. We call two players Players A and B, and will show that the maximin strategy and the minimax strategy in the [Formula: see text]-game, and the maximin strategy and the minimax strategy in the [Formula: see text]-game are all equivalent for each player. However, the maximin strategy for Player A and that for Player B are not necessarily equivalent, and they are not necessarily equivalent to their Nash equilibrium strategies in the [Formula: see text]-game nor the [Formula: see text]-game. But, in a special case, where the objective function of Player B is the opposite of the objective function of Player A, the maximin strategy for Player A and that for Player B are equivalent, and they constitute the Nash equilibrium both in the [Formula: see text]-game and the [Formula: see text]-game.

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