Power and welfare in bargaining for coalition structure formation

We investigate a noncooperative bargaining game for partitioning n agents into non-overlapping coalitions. The game has n time periods during which the players are called according to an exogenous agenda to propose offers. With probability \(\delta \), the game ends during any time period \(t<n\). If it does, the first t players on the agenda get a chance to propose but the others do not. Thus, \(\delta \) is a measure of the degree of democracy within the game (ranging from democracy for \(\delta =0\), through increasing levels of authoritarianism as \(\delta \) approaches 1, to dictatorship for \(\delta =1\)). We determine the subgame perfect equilibrium (SPE) and study how a player’s position on the agenda affects his bargaining power. We analyze the relation between the distribution of power of individual players, the level of democracy, and the welfare efficiency of the game. We find that purely democratic games are welfare inefficient and that introducing a degree of authoritarianism into the game makes the distribution of power more equitable and also maximizes welfare. These results remain invariant under two types of player preferences: one where each player’s preference is a total order on the space of possible coalition structures and the other where each player either likes or dislikes a coalition structure. Finally, we show that the SPE partition may or may not be core stable.     

A model of best choice under incomplete information

This paper suggests two approaches to the construction of a two-player game of best choice under incomplete information with the choice priority of one player and the equal weights of both players. We consider a sequence of independent identically distributed random variables (x _i , y _i ), i = 1..., n, which represent the quality of incoming objects. The first component is announced to the players and the second component is hidden. Each player chooses an object based on the information available. The winner is the player whose object has a greater sum of the quality components than the opponent’s object. We derive the optimal threshold strategies and compare them for both approaches.     

A game-theoretic model of TV show “The Voice”

This paper proposes a game-theoretic model of the two-player best-choice problem with incomplete information. The players (experts) choose between objects by observing their quality in the form of two components forming a sequence of random variables (x_i, y_i), i = 1,..., n. By assumption, the first quality component x_i is known to the players and the second one y_i is hidden. A player accepts or declines an object based on the first quality component only. A player with the maximal sum of the components becomes the winner in the game. The optimal strategies are derived in the cases of independent and correlated quality components.     

Majority bargaining for resource division

We address the problem of how a set of agents can decide to share a resource, represented as a unit-sized pie. The pie can be generated by the entire set but also by some of its subsets. We investigate a finite horizon non-cooperative bargaining game, in which the players take it in turns to make proposals on how the resource should for this purpose be allocated, and the other players vote on whether or not to accept the allocation. Voting is modelled as a Bayesian weighted voting game with uncertainty about the players’ weights. The agenda, (i.e., the order in which the players are called to make offers), is defined exogenously. We focus on impatient players with heterogeneous discount factors. In the case of a conflict, (i.e., no agreement by the deadline), no player receives anything. We provide a Bayesian subgame perfect equilibrium for the bargaining game and conduct an ex-ante analysis of the resulting outcome. We show that the equilibrium is unique, computable in polynomial time, results in an instant Pareto optimal outcome, and, under certain conditions provides a foundation for the core and also the nucleolus of the Bayesian voting game. In addition, our analysis leads to insights on how an individual’s bargained share is influenced by his position on the agenda. Finally, we show that, if the conflict point of the bargaining game changes, then the problem of determining the non-cooperative equilibrium becomes NP-hard even under the perfect information assumption. Our research also reveals how this change in conflict point impacts on the above mentioned results.     

Validating gameplay activity inventory (GAIN) for modeling player profiles

In the present study, we validated Gameplay Activity Inventory (GAIN), a short and psychometrically sound instrument for measuring players’ gameplay preferences and modeling player profiles. In Study 1, participants in Finland (\(N=879\)) responded to a 52-item version of GAIN. An exploratory factor analysis was used to identify five latent factors of gameplay activity appreciation: Aggression, Management, Exploration, Coordination, and Caretaking. In Study 2, respondents in Canada (\(N=1322\)) and Japan (\(N=1178\)) responded to GAIN, and the factor structure of a 15-item version was examined using a Confirmatory Factor Analysis. The results showed that the short version of GAIN has good construct validity, convergent validity, and discriminant validity in Japan and in Canada. We demonstrated the usefulness of GAIN by conducting a cluster analysis to identify player types that differ in both demographics and game choice. GAIN can be used in research as a tool for investigating player profiles. Game companies, publishers and analysts can utilize GAIN in player-centric game development and targeted marketing and in generating personalized game recommendations.     

Differential games of <Emphasis Type="Italic">N</Emphasis> players in stochastic systems with controlled diffusion terms

Differential Games of N players defined by stochastic systems with controlled diffusion terms are considered. Necessary conditions of equilibrium strategy are obtained. These conditions are specified for the linear quadratic differential game of N players in terms of differential Riccati equation for program and positional equilibrium situations.     

When Can Limited Randomness Be Used in Repeated Games?

The central result of classical game theory states that every finite normal form game has a Nash equilibrium, provided that players are allowed to use randomized (mixed) strategies. However, in practice, humans are known to be bad at generating random-like sequences, and true random bits may be unavailable. Even if the players have access to enough random bits for a single instance of the game their randomness might be insufficient if the game is played many times. In this work, we ask whether randomness is necessary for equilibria to exist in finitely repeated games. We show that for a large class of games containing arbitrary two-player zero-sum games, approximate Nash equilibria of the n-stage repeated version of the game exist if and only if both players have Ω(n) random bits. In contrast, we show that there exists a class of games for which no equilibrium exists in pure strategies, yet the n-stage repeated version of the game has an exact Nash equilibrium in which each player uses only a constant number of random bits. When the players are assumed to be computationally bounded, if cryptographic pseudorandom generators (or, equivalently, one-way functions) exist, then the players can base their strategies on “random-like” sequences derived from only a small number of truly random bits. We show that, in contrast, in repeated two-player zero-sum games, if pseudorandom generators do not exist, then Ω(n) random bits remain necessary for equilibria to exist.     

Chain equilibria in secure strategies

In this paper we introduce a modification of the concept of Equilibrium in Secure Strategies (EinSS), which takes into account the non-uniform attitudes of players to security in non-cooperative games. In particular, we examine an asymmetric attitude of players to mutual threats in the simplest case, when all players are strictly ordered by their relation to security. Namely, we assume that the players can be reindexed so that each player i in his behavior takes into account the threats posed by players j > i but ignores the threats of players j < i provided that these threats are effectively contained by some counterthreats. A corresponding equilibrium will be called a Chain EinSS. The conceptual meaning of this equilibrium is illustrated by two continuous games that have no pure Nash equilibrium or (conventional) EinSS. The Colonel Blotto two-player game (Borel 1953; Owen 1968) for two battlefields with different price always admits a Chain EinSS with intuitive interpretation. The product competition of many players on a segment (Eaton, Lipsey 1975; Shaked 1975) with the linear distribution of consumer preferences always admits a unique Chain EinSS solution (up to a permutation of players). Finally, we compare Chain EinSS with Stackelberg equilibrium.     

Network Formation for Asymmetric Players and Bilateral Contracting

We study a network formation game where players wish to send traffic to other players. Players can be seen as nodes of an undirected graph whose edges are defined by contracts between the corresponding players. Each player can contract bilaterally with others to form bidirectional links or break unilaterally contracts to eliminate the corresponding links. Our model is an extension of the traffic routing model considered in Arcaute, E., Johari, R., Mannor, S., (IEEE Trans. Automat. Contr. 54(8), 1765–1778 2009) in which we do not require the traffic to be uniform and all-to-all. Player i specifies the amount of traffic t_ij ≥ 0 that wants to send to player j. Our notion of stability is the network pairwise Nash stability, when no node wishes to deviate unilaterally and no pair of nodes can obtain benefit from deviating bilaterally. We show a characterization of the topologies that are pairwise Nash stable for a given traffic matrix. We prove that the best response problem is NP-hard and devise a myopic dynamics so that the deviation of the active node can be computed in polynomial time. We show the convergence of the dynamics to pairwise Nash configurations, when the contracting functions are anti-symmetric and affine, and that the expected convergence time is polynomial in the number of nodes when the node activation process is uniform.     

On an approach to constructing a characteristic function in cooperative differential games

We propose a novel approach to constructing characteristic functions in cooperative differential games. A characteristic function of a coalition S is computed in two stages: first, optimal control strategies maximizing the total payoff of the players are found, and next, these strategies are used by the players from the coalition S, while the other players, those from N S, use strategies minimizing the total payoff of the players from S. The characteristic function obtained in this way is superadditive. In addition, it possesses a number of other useful properties. As an example, we compute values of a characteristic function for a specific differential game of pollution control.     

A cellular automaton for blocking queen games

We show that the winning positions of a certain type of two-player game form interesting patterns which often defy analysis, yet can be computed by a cellular automaton. The game, known as Blocking Wythoff Nim, consists of moving a queen as in chess, but always towards (0, 0), and it may not be moved to any of \(k-1\) temporarily “blocked” positions specified on the previous turn by the other player. The game ends when a player wins by blocking all possible moves of the other player. The value of k is a parameter that defines the game, and the pattern of winning positions can be very sensitive to k. As k becomes large, parts of the pattern of winning positions converge to recurring chaotic patterns that are independent of k. The patterns for large k display an unprecedented amount of self-organization at many scales, and here we attempt to describe the self-organized structure that appears. This paper extends a previous study (Cook et al. in Cellular automata and discrete complex systems, AUTOMATA 2015, Lecture Notes in Computer Science, vol 9099, pp 71–84, 2015), containing further analysis and new insights into the long term behaviour and structures generated by our blocking queen cellular automaton.     

Procedural content generation for platformers: designing and testing FUN PLEdGE

Video games are a peculiar medium, standing at the crossing point between art and software application, and characterized by an active involvement of its audience. The complexity of the product generates a huge challenge for the companies that develop video games. In the development process, level designers play a crucial role: they are in charge of declining the theoretical framework developed by the game designer into game levels, which contain the actual gameplay scenarios. Hence, the final goal of any level designer is to valorize the game design by creating enjoyable gaming experiences while, at the same time, respecting several constraints. To lighten the burden on level designers, several semi-automated approaches to level generation have appeared in time, but the majority of these tools suffer from several drawbacks. In the present work, we tackle the issue of designing, prototyping and testing FUN PLEdGE, a general-purpose automated levels generator and editor for platform video games. Its main goal is to shrink development time while producing – unassisted – levels that are both playable and fun. Moreover, our tool provides the maximum freedom to the level designer, by avoiding to impose unnecessary constraints on the structure of the levels and by guaranteeing the possibility to modify and personalize by hand the generated levels. During this process, the generator assists the designer by suggesting corrections functional to the quality of the player experience. To prove the effectiveness of our prototypal application we have also developed and tested with players a platform game. In the same vein, we asked to a group of game developers to test FUN PLEdGE.     

Interactive Restless Multi-armed Bandit Game and Swarm Intelligence Effect

We obtain the conditions for the emergence of the swarm intelligence effect in an interactive game of restless multi-armed bandit (rMAB). A player competes with multiple agents. Each bandit has a payoff that changes with a probability p _c per round. The agents and player choose one of three options: (1) Exploit (a good bandit), (2) Innovate (asocial learning for a good bandit among n _I randomly chosen bandits), and (3) Observe (social learning for a good bandit). Each agent has two parameters (c, p _obs ) to specify the decision: (i) c, the threshold value for Exploit, and (ii) p _obs , the probability for Observe in learning. The parameters (c, p _obs ) are uniformly distributed. We determine the optimal strategies for the player using complete knowledge about the rMAB. We show whether or not social or asocial learning is more optimal in the (p _c , n _I ) space and define the swarm intelligence effect. We conduct a laboratory experiment (67 subjects) and observe the swarm intelligence effect only if (p _c , n _I ) are chosen so that social learning is far more optimal than asocial learning.     

The bounded core for games with restricted cooperation

A game with restricted (incomplete) cooperation is a triple (N, v, Ω), where N represents a finite set of players, Ω ? 2^N is a set of feasible coalitions such that N ∈ Ω, and v: Ω → R denotes a characteristic function. Unlike the classical TU games, the core of a game with restricted cooperation can be unbounded. Recently Grabisch and Sudhölter [9] proposed a new solution concept—the bounded core—that associates a game (N, v,Ω) with the union of all bounded faces of the core. The bounded core can be empty even if the core is nonempty. This paper gives two axiomatizations of the bounded core. The first axiomatization characterizes the bounded core for the class G^r of all games with restricted cooperation, whereas the second one for the subclass G_bc^r ? G^r of the games with nonempty bounded cores.     

Lion and Man Game and Fixed Point-Free Maps

This paper is dedicated to the pursuit-evasion game in which both players (Lion and Man) move in a metric space, have equal maximum speeds and complete information about the location of each other. We assume that evasion is successful if, for some initial positions of players, there exists a positive number p and an evader’s non-anticipative strategy guaranteeing that the distance between the players is always greater than p. We consider connection between successful evasion and such properties of the phase space as geodesics behavior and the existence of non-expanding fixed point-free self-maps.     

Quantum secure two-party computation for set intersection with rational players

Recently, Shi et al. (Phys Rev A 92:022309, 2015) proposed quantum oblivious set member decision protocol where two legitimate parties, namely Alice and Bob, play a game. Alice has a secret k, and Bob has a set \(\{k_1,k_2,\ldots k_n\}\). The game is designed towards testing if the secret k is a member of the set possessed by Bob without revealing the identity of k. The output of the game will be either “Yes” (bit 1) or “No” (bit 0) and is generated at Bob’s place. Bob does not know the identity of k, and Alice does not know any element of the set. In a subsequent work (Shi et al in Quant Inf Process 15:363–371, 2016), the authors proposed a quantum scheme for private set intersection (PSI) where the client (Alice) gets the intersected elements with the help of a server (Bob) and the server knows nothing. In the present draft, we extended the game to compute the intersection of two computationally indistinguishable sets X and Y possessed by Alice and Bob, respectively. We consider Alice and Bob as rational players, i.e. they are neither “good” nor “bad”. They participate in the game towards maximizing their utilities. We prove that in this rational setting, the strategy profile ((cooperate, abort), (cooperate, abort)) is a strict Nash equilibrium. If ((cooperate, abort), (cooperate, abort)) is strict Nash, then fairness and correctness of the protocol are guaranteed.     

ACE-RL-Checkers: decision-making adaptability through integration of automatic case elicitation,reinforcement learning,and sequential pattern mining

In agents that operate in environments where decision-making needs to take into account, not only the environment, but also the minimizing actions of an opponent (as in games), it is fundamental that the agent is endowed with the ability of progressively tracing the profile of its adversaries, in such a manner that this profile aids in the process of selecting appropriate actions. However, it would be unsuitable to construct an agent with a decision-making system based only on the elaboration of such a profile, as this would prevent the agent from having its “own identity,” which would leave the agent at the mercy of its opponent. Following this direction, this study proposes an automatic Checkers player, called ACE-RL-Checkers, equipped with a dynamic decision-making module, which adapts to the profile of the opponent over the course of the game. In such a system, the action selection process is conducted through a composition of multilayer perceptron neural network and case library. In this case, the neural network represents the “identity” of the agent, i.e., it is an already trained static decision-making module. On the other hand, the case library represents the dynamic decision-making module of the agent, which is generated by the Automatic Case Elicitation technique. This technique has a pseudo-random exploratory behavior, which allows the dynamic decision-making of the agent to be directed either by the opponent’s game profile or randomly. In order to avoid a high occurrence of pseudo-random decision-making in the game initial phases—in which the agent counts on very little information about its opponent—this work proposes a new module based on sequential pattern mining for generating a base of experience rules extracted from human expert’s game records. This module will improve the agent’s move selection in the game initial phases. Experiments carried out in tournaments involving ACE-RL-Checkers and other agents correlated to this work, confirm the superiority of the dynamic architecture proposed herein.     

Adaptive user modelling in car racing games using behavioural and physiological data

Personalised content adaptation has great potential to increase user engagement in video games. Procedural generation of user-tailored content increases the self-motivation of players as they immerse themselves in the virtual world. An adaptive user model is needed to capture the skills of the player and enable automatic game content altering algorithms to fit the individual user. We propose an adaptive user modelling approach using a combination of unobtrusive physiological data to identify strengths and weaknesses in user performance in car racing games. Our system creates user-tailored tracks to improve driving habits and user experience, and to keep engagement at high levels. The user modelling approach adopts concepts from the Trace Theory framework; it uses machine learning to extract features from the user’s physiological data and game-related actions, and cluster them into low level primitives. These primitives are transformed and evaluated into higher level abstractions such as experience, exploration and attention. These abstractions are subsequently used to provide track alteration decisions for the player. Collection of data and feedback from 52 users allowed us to associate key model variables and outcomes to user responses, and to verify that the model provides statistically significant decisions personalised to the individual player. Tailored game content variations between users in our experiments, as well as the correlations with user satisfaction demonstrate that our algorithm is able to automatically incorporate user feedback in subsequent procedural content generation.     

Guaranteed approach with the farthest of the runaways

Games of the family {Λ _N } _N?2 are formulated and studied with the application of generalized Isaacs’s approach. The game Λ _N is a simplest model of the counteraction of one persecutor P and coalition N of E ^N runaways for the case when the payoff is the distance up to the coalition of E ^N equal to the Euclidean distance between P and the farthest from the runaways; P is in command of the termination moment. Moreover, an approach within the limits of which in games with a smooth terminal payoff are generated strategies prescribing players’ motions in the directions of local gradients of the payoff is described. The approach is used for constructing pursuit strategies in games in which smooth approximations of the maximum of Euclidean distances up to the runaways are in place of payoffs. Pursuit strategies prescribing the motion in the direction of the farthest of the runaways are studied. A numerical simulation of the development of the games Λ₂ and Λ₃ is conducted in using different strategies by the players.