Relaxed Game Research Articles

On the convexity of step out\u2013step in sequencing games

The main result of this paper is the convexity of step out–step in (SoSi) sequencing games, a class of relaxed sequencing games first analyzed by Musegaas et al. (Eur J Oper Res 246:894–906, 2015). The proof makes use of a polynomial time algorithm determining the value and an optimal processing order for an arbitrary coalition in a SoSi sequencing game. In particular, we use that in determining an optimal processing order of a coalition, the algorithm can start from the optimal processing order found for any subcoalition of smaller size and thus all information on such an optimal processing order can be used.

Differential Games in $$L^{\infty }$$

The Isaacs equations for differential games in \({L^{\infty }}\) first derived in Barron (Nonlinear Anal 14:971–989, 1990) are reformulated so that the Hamiltonians are continuous and result in a simpler problem to analyze numerically. Relaxed differential games in \({L^{\infty }}\) are considered. \(L^{\infty }\) differential games with time and state independent dynamics and convex or quasiconvex terminal data are solved explicitly using a type of Hopf–Lax formula. The stochastic differential game in \({L^{\infty }}\) connected to stochastic target problems is also discussed.

THE CORES OF PAIRED-DOMINATION GAMES

Velzen introduced the rigid and relaxed dominating set games and showed that the rigid game being balanced is equivalent to the relaxed game being balanced in 2004. After then various variants of dominating set games were introduced and it was shown that for each variant, a rigid game being balanced is equivalent to a relaxed game being balanced. It is natural to ask if for any other variant of dominating set game, the balancedness of a rigid game and the balancedness of a relaxed game are equivalent. In this paper, we show that the answer for the question is negative by considering the rigid and relaxed paired-domination games, which is considered as a variant of dominating set games. We characterize the cores of both games and show that the rigid game being balanced is not equivalent to the relaxed game being balanced. In addition, we study the cores of paired-dominations games on paths and cycles.

Assessment of crowdsourcing and gamification loss in user-assisted object segmentation

There has been a growing interest in applying human computation --- particularly crowdsourcing techniques --- to assist in the solution of multimedia, image processing, and computer vision problems which are still too difficult to solve using fully automatic algorithms, and yet relatively easy for humans. In this paper we focus on a specific problem --- object segmentation within color images --- and compare different solutions which combine color image segmentation algorithms with human efforts, either in the form of an explicit interactive segmentation task or through an implicit collection of valuable human traces with a game. We use Click'n'Cut, a friendly, web-based, interactive segmentation tool that allows segmentation tasks to be assigned to many users, and Ask'nSeek, a game with a purpose designed for object detection and segmentation. The two main contributions of this paper are: (i) We use the results of Click'n'Cut campaigns with different groups of users to examine and quantify the crowdsourcing loss incurred when an interactive segmentation task is assigned to paid crowd-workers, comparing their results to the ones obtained when computer vision experts are asked to perform the same tasks. (ii) Since interactive segmentation tasks are inherently tedious and prone to fatigue, we compare the quality of the results obtained with Click'n'Cut with the ones obtained using a (fun, interactive, and potentially less tedious) game designed for the same purpose. We call this contribution the assessment of the gamification loss, since it refers to how much quality of segmentation results may be lost when we switch to a game-based approach to the same task. We demonstrate that the crowdsourcing loss is significant when using all the data points from workers, but decreases substantially (and becomes comparable to the quality of expert users performing similar tasks) after performing a modest amount of data analysis and filtering out of users whose data are clearly not useful. We also show that --- on the other hand --- the gamification loss is significantly more severe: the quality of the results drops roughly by half when switching from a focused (yet tedious) task to a more fun and relaxed game environment.

The Relaxed Game Chromatic Number of Graphs with Cut-Vertices

In the $$(r,d)$$(r,d)-relaxed colouring game, two players, Alice and Bob alternately colour the vertices of $$G$$G, using colours from a set $$\mathcal{C}$$C, with $$|\mathcal{C}|=r$$|C|=r. A vertex $$v$$v can be coloured with $$c,c\in \mathcal{C}$$c,c?C if after colouring $$v$$v, the subgraph induced by all vertices with $$c$$c has maximum degree at most $$d$$d. Alice wins the game if all vertices of the graph are coloured. The $$d$$d-relaxed game chromatic number of $$G$$G, denoted by $$\chi _g^{(d)}(G)$$?g(d)(G), is the least number $$r$$r for which Alice has a winning strategy for the $$(r,d)$$(r,d)-relaxed colouring game on $$G$$G. A $$(r,d)$$(r,d)-relaxed edge-colouring game is the version of the $$(r,d)$$(r,d)-relaxed colouring game which is played on edges a graph. The parameter associated with the $$(r,d)$$(r,d)-relaxed edge-colouring game is called the $$d$$d-relaxed game chromatic index and denoted by $$^{(d)}\chi '_g(G)$$(d)?g?(G). We consider the game on graphs containing cut-vertices. For $$k\ge 2$$k?2 we define a class of graphs $$\mathcal{H}_k =\{G|\mathrm{\;every \;block \;of\;} G \; \mathrm{has \;at \;most}\; k \;\mathrm{vertices}\}$$Hk={G|everyblockofGhasatmostkvertices}. We find upper bounds on the $$d$$d-relaxed game chromatic number of graphs from $$\mathcal{H}_k\;(k\ge 5)$$Hk(k?5). Since the line graph of the forest with maximum degree $$k$$k belongs to $$\mathcal{H}_k$$Hk, from results for $$\mathcal{H}_k$$Hk we obtain some new results for line graphs of forests, i.e., for the relaxed game chromatic index of forests. We prove that $$^{(d)}\chi '_g(T)\le \Delta (T)+2-d$$(d)?g?(T)≤Δ(T)+2-d for any forest $$T$$T. Moreover, we show that for forests with large maximum degree we can derive better bounds. These results improve the upper bound on the relaxed game chromatic index of forests obtained in Dunn (Discrete Math 307:1767---1775, 2007). Furthermore, we determine minimum $$d$$d that guarantee Alice to win the $$(2,d)$$(2,d)-relaxed edge-colouring game on forests.

On the Relaxed Colouring Game and the Unilateral Colouring Game

In the paper we introduce the new game--the unilateral $${\mathcal{P}}$$ -colouring game which can be used as a tool to study the r-colouring game and the (r, d)-relaxed colouring game. Let be given a graph G, an additive hereditary property $${\mathcal {P}}$$ and a set C of r colours. In the unilateral $${\mathcal {P}}$$ -colouring game similarly as in the r-colouring game, two players, Alice and Bob, colour the uncoloured vertices of the graph G, but in the unilateral $${\mathcal {P}}$$ -colouring game Bob is more powerful than Alice. Alice starts the game, the players play alternately, but Bob can miss his move. Bob can colour the vertex with an arbitrary colour from C, while Alice must colour the vertex with a colour from C in such a way that she cannot create a monochromatic minimal forbidden subgraph for the property $${\mathcal {P}}$$ . If after |V(G)| moves the graph G is coloured, then Alice wins the game, otherwise Bob wins. The $${\mathcal {P}}$$ -unilateral game chromatic number, denoted by $${\chi_{ug}^\mathcal {P}(G)}$$ , is the least number r for which Alice has a winning strategy for the unilateral $${\mathcal {P}}$$ -colouring game with r colours on G. We prove that the $${\mathcal {P}}$$ -unilateral game chromatic number is monotone and is the upper bound for the game chromatic number and the relaxed game chromatic number. We give the winning strategy for Alice to play the unilateral $${\mathcal {P}}$$ -colouring game. Moreover, for k ≥ 2 we define a class of graphs $${\mathcal {H}_k =\{G|{\rm every \;block \;of\;}G \; {\rm has \;at \;most}\; k \;{\rm vertices}\}}$$ . The class $${\mathcal {H}_k }$$ contains, e.g., forests, Husimi trees, line graphs of forests, cactus graphs. Let $${\mathcal {S}_d}$$ be the class of graphs with maximum degree at most d. We find the upper bound for the $${\mathcal {S}_2}$$ -unilateral game chromatic number for graphs from $${\mathcal {H}_3}$$ and we study the $${\mathcal {S}_d}$$ -unilateral game chromatic number for graphs from $${\mathcal {H}_4}$$ for $${d \in \{2,3\}}$$ . As the conclusion from these results we obtain the result for the d-relaxed game chromatic number: if $${G \in \mathcal {H}_k}$$ , then $${\chi_g^{(d)}(G) \leq k + 2-d}$$ , for $${k \in \{3, 4\}}$$ and $${d \in \{0, \ldots, k-1\}}$$ . This generalizes a known result for trees.

On the cores of games arising from integer edge covering functions of graphs

In this paper, we study cooperative games arising from integer edge covering problems on graphs. We introduce two games, a rigid k-edge covering game and its relaxed game, as generalizations of a rigid edge covering game and its relaxed game studied by Liu and Fang (2007). Then we give a characterization of the cores of both games, find relationships between them, and give necessary and sufficient conditions for the balancedness of a rigid k-edge covering game and its relaxed game.

Solving Cournot equilibriums with variational inequalities algorithms

Over the past two decades, the fact that many games have been formulated as variational inequalities problems has led to relevant developments related with the existence and uniqueness of equilibriums, and with their calculation methodologies. Based on this approach, this study applies sufficient conditions for Cournot equilibriums existence and uniqueness, proving that while existence holds, uniqueness cannot be proved in general. To find one of the existent equilibriums, this study also proposes a novel variational inequalities algorithm which is globally convergent and easy to implement. It iteratively computes searching directions of the equilibrium by generating hyper-planes that separate the equilibrium from the intermediate solutions obtained relaxing the original Cournot game. Unlike other related algorithms, the proposed algorithm does not require Jacobian matrixes evaluation, and the iterative relaxed games can easily be solved using convex and quadratic optimisation models. Numerical results show the operation and convergence of the algorithm.

The 6-relaxed game chromatic number of outerplanar graphs

Suppose G is a graph and k , d are integers. The ( k , d ) -relaxed colouring game on G is a game played by two players, Alice and Bob, who take turns colouring the vertices of G with legal colours from a set X of k colours. Here a colour i is legal for an uncoloured vertex x if after colouring x with colour i , the subgraph induced by vertices of colour i has maximum degree at most d . Alice’s goal is to have all the vertices coloured, and Bob’s goal is the opposite: to have an uncoloured vertex without a legal colour. The d -relaxed game chromatic number of G , denoted by χ g ( d ) ( G ) , is the least number k so that when playing the ( k , d ) -relaxed colouring game on G , Alice has a winning strategy. This paper proves that if G is an outerplanar graph, then χ g ( d ) ( G ) ≤ 2 for d ≥ 6 .

The relaxed game chromatic index of k-degenerate graphs

The (r,d)-relaxed coloring game is a two-player game played on the vertex set of a graph G. We consider a natural analogue to this game on the edge set of G called the (r,d)-relaxed edge-coloring game. We consider this game on trees and more generally, on k-degenerate graphs. We show that if G is k-degenerate with Δ(G)=Δ, then the first player, Alice, has a winning strategy for this game with r=Δ+k-1 and d⩾2k2+4k.

BALANCEDNESS OF INTEGER DOMINATION GAMES

In this paper, we consider cooperative games arising from integer domination problem on graphs. We introduce two games, <TEX>${\kappa}-domination$</TEX> game and its monotonic relaxed game, and focus on their cores. We first give characterizations of the cores and the relationship between them. Furthermore, a common necessary and sufficient condition for the balancedness of both games is obtained by making use of the technique of linear programming and its duality.

The relaxed game chromatic number of outerplanar graphs

AbstractThe (r,d)‐relaxed coloring game is played by two players, Alice and Bob, on a graph G with a set of r colors. The players take turns coloring uncolored vertices with legal colors. A color α is legal for an uncolored vertex u if u is adjacent to at most d vertices that have already been colored with α, and every neighbor of u that has already been colored with α is adjacent to at most d – 1 vertices that have already been colored with α. Alice wins the game if eventually all the vertices are legally colored; otherwise, Bob wins the game when there comes a time when there is no legal move left. We show that if G is outerplanar then Alice can win the (2,8)‐relaxed coloring game on G. It is known that there exists an outerplanar graph G such that Bob can win the (2,4)‐relaxed coloring game on G. © 2004 Wiley Periodicals, Inc. J Graph Theory 46:69–78, 2004

Relaxed game chromatic number of trees and outerplanar graphs

This paper studies the relaxed game chromatic number of trees and outerplanar graphs. It is proved that if G is a tree then χ g d ( G)⩽2 for d⩾2. If G is an outerplanar graph, then χ g d ( G)⩽5 for d⩾2, and χ g d ( G)⩾3 for d⩽4.

On the balancedness of relaxed sequencing games

This paper shows that some classes of relaxed sequencing games, which arise from the class of sequencing games as introduced in Curiel, Pederzoli, Tijs (1989), are balanced.

Relaxed game chromatic number of graphs

This paper discusses a coloring game on graphs. Let k,d be non-negative integers and C a set of k colors. Two persons, Alice and Bob, alternately color the vertices of G with colors from C, with Alice having the first move. A color i is legal for an uncolored vertex x if by coloring x with color i, the subgraph of G induced by those vertices of color i has maximum degree at most d. Each move of Alice or Bob colors an uncolored vertex with a legal color. The game is over if either all vertices are colored, or no more vertices can be colored with a legal color. Alice's goal is to produce a legal coloring which colors all the vertices of G, and Bob's goal is to prevent this from happening. We shall prove that if G is a forest, then for k=3,d⩾1, Alice has a winning strategy. If G is an outerplanar graph, then for k=6 and d⩾1, Alice has a winning strategy.