Game Matrices Research Articles

Examining Competitive Strategies in Growing Coffee Industry with Game Theory

The coffee market in Turkey is growing rapidly, leading to fierce competition between international, national and local coffee shops. It has become extremely important for the local economy that local businesses can compete and survive with these more institutionalized structures that also have brand advantages in competition. In this study, game theory was used as a tool to analyse the competitive strategies of companies operating in the coffee market by designing both zero-sum and non-zero-sum games. Considering that the expectations, tastes, preferences and consumption habits of the generations may be different, 1600 people were surveyed in Erzurum province by grouping them as Generation X, Y and Z according to their year of birth. Different game matrices were created with the data obtained from the surveys. The games were designed to identify the prominent strategies among coffee shops for each generation. The results showed that, regardless of generation, local coffee shops stand out from international and national coffee chains in terms of space comfort. In addition, local shops also stand out in terms of meeting consumer expectations. In order to gain a competitive advantage, local coffee shops should focus on developing strategies related to the space comfort. Developing this strategy provides opportunities for local businesses to increase their market share. It is also observed that Generation Z has higher expectations in all strategies compared to other generations.

Study on the Evolutionary Game of Information Security Supervision in Smart Cities under Different Reward and Punishment Mechanisms

At present, the information security problems of smart city show a high incidence, and it is necessary to strengthen the information security supervision of smart city. In the process of supervision, there is a game relationship between local government and smart city enterprises. This paper firstly constructs the game matrices of local government and enterprises under the static and three dynamic reward and punishment mechanisms, then conducts numerical simulation with the help of MATLAB to arrive at the optimal reward and punishment mechanism through comparison, and finally explores the influence of the change of the upper limit value of each key variable on the directionality and sensitivity of the decision‐making behavior of game subjects under the optimal mechanism. The result shows that initial value is one of the decisive factors influencing the choice of management strategy by enterprise. Dynamic reward and dynamic punishment mechanism is the best reward and punishment mechanism for information security supervision in smart cities. In case the upper limit value of key parameters is increased, a larger punishment has a strong influence on the positive strategy choice of the enterprise, and a reasonable adjustment of the reward policy can likewise mobilize the probability that the enterprise actively chooses to strengthen information security management. Based on the simulation results, we propose a feasible strategy.

Using Visual Salience in Empirical Game Theory

“Bottom-up” visual perception is fast, universal, and does not depend on personal goals or experience. Where exactly bottom-up attention is directed in a visual image-- usually based on contrast, color, feature orientation, and centrality-- can be accurately predicted by machine-learned algorithms. We test whether this type of salience can help explain economic decisions in four experimental analyses (three are new). When people pick between sets of valued fruits, salience can lead to mistakes (lower-value sets are sometimes chosen because they are salient). Two studies show strong and weak influence on choices, respectively, of salience in stock price charts and salient payoffs in game matrices. The central analysis is evidence from games in which choices are locations in images. When players are trying to cooperatively match locations, concentration of salience is associated with the success of matching (r=.57). In competitive hider-seeker location games, all players choose salient locations more often. This fact creates a “seeker’s advantage” (9% wins compared to the 7% predicted by unique equilibrium). The location game data are consistent with cognitive hierarchy and level-k models in which predicted salience influences nonstrategic level-0 choices. Model estimates from hider-seeker games can predict behavior in matching games despite their opposite strategic structures. Looking forward, bottom-up salience could be included as a predictable type of costless involuntary “pre-attention” in expanded models of Bayesian-rational inattention. Novel applications involving prices and taxes, disability, ethnic markers, and visually-influenced beliefs are suggested.

A functional MRI study on how oxytocin affects decision making in social dilemmas: Cooperate as long as it pays off, aggress only when you think you can win

We investigate if the neuropeptide oxytocin (OT), known to moderate social behaviour, influences strategic decision making in social dilemmas by facilitating the integration of incentives and social cues. Participants (N=29) played two economic games with different incentive structures in the fMRI scanner after receiving OT or placebo (following a double blind, within-subject design). Pictures of angry or neutral faces (the social cues) were displayed alongside the game matrices. Consistent with a priori hypotheses based on the modulatory role of OT in mesolimbic dopaminergic brain regions, the results indicate that, compared to placebo, OT significantly increases the activation of the nucleus accumbens during an assurance (coordination) game that rewards mutual cooperation. This increases appetitive motivation so that cooperative behaviour is facilitated for risk averse individuals. OT also significantly attenuates the amygdala, thereby reducing the orienting response to social cues. The corresponding change in behaviour is only apparent in the chicken (or anti-coordination) game, where aggression is incentivized but fatal if the partner also aggresses. Because of this ambiguity, decision making can be improved by additional information, and OT steers decisions in the chicken game in accordance with the valence of the facial cue: aggress when face is neutral; retreat when it is angry. Through its combined influence on amygdala and nucleus accumbens, OT improves the selection of a cooperative or aggressive strategy in function of the best match between the incentives of the game and the social cues present in the decision environment.

Tightness of correlation inequalities with no quantum violation

We study the faces of the set of quantum correlations, i.e., the Bell and noncontextuality inequalities without any quantum violation. First, we investigate the question whether every proper (tight) Bell inequality for two parties, other than the trivial ones from positivity, normalization and no-signaling can be violated by quantum correlations, i.e., whether the classical Bell polytope or the smaller correlation polytope share any facets with their respective quantum sets. To do this, we develop a recently derived bound on the quantum value of linear games based on the norms of game matrices to give a simple sufficient condition to identify linear games with no quantum advantage. Additionally we show how this bound can be extended to the general class of unique games, illustrating it for the case of three outcomes. We then show as a main result that the paradigmatic examples of correlation Bell inequalities with no quantum violation, namely the non-local computation games do not constitute tight Bell inequalities, not even for the correlation polytope. We also extend this to an arbitrary prime number of outcomes for a specific class of these games. We then study the faces in the simplest CHSH Bell scenario of binary dichotomic measurements, and identify edges in the set of quantum correlations in this scenario. Finally, we relate the non-contextual polytope of single-party correlation inequalities with the cut polytope $CUT(\nabla G)$, where $G$ denotes the compatibility graph of observables in the contextuality scenario and $\nabla G$ denotes the suspension graph of $G$. We observe that there exist tight non-contextuality inequalities with no quantum violation, and furthermore that this set of inequalities is beyond those implied by the Consistent Exclusivity principle.

Stochastic Identification of Stability of Competitive Interactions in Ecosystems.

The problem of finding an optimum within a set of possibilities that represent the varying successfulness of numerous subjects competing with one another is highly relevant in the field of ecosystem interactions. We propose a method for solving this problem by the application of the Nash equilibrium concept, which is frequently used in ecology. The proposed model is based on the transformation of the initial payoff vectors of subjects that interact in different situations into a statistical set of symmetrical game matrices that consist of permutations of payoff values. The equilibrium solution is expressed as values of the probability of Nash equilibrium occurrence with uniform distribution over all possible permutations based on uncertainty of positions of payoff values in the matrix. We assume that this equilibrium solution provides information on the distribution of the degree of stability among individual situations and interacting subjects. In this paper, we validate this assumption and demonstrate its application to a dataset that represents interspecies interactions in plant ecology. We propose that the use of the Nash equilibrium in the analysis of datasets formalized according to the Pareto optimality scheme is applicable in numerous other contexts.

Modeling Learning and Cooperation in Iterative Games

In this paper, we describe the general framework of neural networks and how such frameworks can be adapted to model human game play and the learning that takes place during iterative games. We introduce a method of pre-processing game matrices in an effort to produce cooperative strategies in games with non-cooperative dominant strategies, such as the Nash Equilibrium solution to the Prisoner’s Dilemma. We find that the introduction of the pre-processed matrix increases the probability that the network plays a cooperative strategy significantly when compared to the network behavior without pre-processing.

Efficiency estimation technique of applying the model of mixed expansion of non-classical type matrix games for problems of controlling the technical condition of complex systems

This study presents a technique for estimating the efficiency of applying the model of mixed expansion of non-classical type matrix games for minimization of generalized costs required to organize and control the technical condition of a group of complex systems by the criterion of the reliability of substantiation of the decision made under conditions of uncertainty for a set of game matrices specified by the characteristic property.

Studying interval valued matrix games with fuzzy logic

Matrix games have been widely used in decision making systems. In practice, for the same strategies players take, the corresponding payoffs may be within certain ranges rather than exact values. To model such uncertainty in matrix games, we consider interval-valued game matrices in this paper; and extend the results of classical strictly determined matrix games to fuzzily determined interval matrix games. Finally, we give an initial investigation into mixed strategies for such games.

The economic basis of cooperation: tradeoffs between selfishness and generosity

The current study examined the economics of cooperation in controlled-payoff games by using captive blue jays, Cyanocitta cristata. This investigation used a special feeding apparatus to test for the stability of cooperative choice in a series of iterated games. The jays experienced experimentally determined game theoretical payoff matrices, which determined the distribution of food to themselves and their opponent, depending on their decision to cooperate or defect. The experiment tested four game matrices, called the cooperate only, defect only, prisoner’s dilemma, and opponent control treatments. This study found little cooperation in the defect only and prisoner’s dilemma treatments. Cooperation occurred significantly more often in the opponent control treatment. These findings suggest that the jays attend to short-term consequences; they do not cooperate in the absence of an immediate benefit (defect only), even if a long-term benefit may exist (prisoner’s dilemma). The opponent control treatment suggests that cooperation can occur when an individual’s benefits depend completely on the actions of others; therefore, generosity is cheap. This study, therefore, agrees with recent studies in proposing alternative models of cooperation.

Altruistic duality in evolutionary game theory

A game-theoretic dynamical model of social preference and enlightened self-interest is formulated. Existence of symmetry and duality in the game matrices with altruistic social preference is revealed. The model quantitatively describes the dynamical evolution of altruism in prisoner's dilemma and the regime change in prey–predator dynamics.

Learning in multilevel games with incomplete information. II

Multilevel games are abstractions of situations where decision makers are distributed in a network environment. In Part I of this paper, the authors present several of the challenging problems that arise in the analysis of multilevel games. In this paper a specific set up is considered where the two games being played are zero-sum games and where the decision makers use the linear reward-inaction algorithm of stochastic learning automata. It is shown that the effective game matrix is decided by the willingness and the ability to cooperate and is a convex combination of two zero-sum game matrices. Analysis of the properties of this effective game matrix and the convergence of the decision process shows that players tend toward noncooperation in these specific environments. Simulation results illustrate this noncooperative behavior.

GAMELPP: A PASCAL program for solving two-person zero-sum games on a microcomputer

The solution of a two-person zero-sum game provides optimal strategies for both players. In game-theoretic terms, the solution specifies a minimax strategy for one player and a maximin strategy for the other. These strategies can be pure strategies, that is, single strategies selected from the set ofstrategies available to each player, or they may be mixed strategies that are weighted com­ posites of the single strategies. The weights in these com­ posites consist of probabilities for playing the various strategies. The probabilities in a pair of mixed strategies are selected so that the expected value of the game is optimal for each player. The solution of a two-person zero-sum game, therefore, rests on fmding a vector of probabilities x that would allow one player to maximize the game value and another vector of probabilities y enabling the other player to minimize the game value. Solutions consisting of pure strategies can be readily obtained by examination of the game matrix, and the mixed strategies in a 2 by 2 game matrix can be deter­ mined by applying straightforward computational formulas (e.g., Coombs, Dawes, & Tversky, 1970). However, to obtain mixed-strategy solutions for game matrices with dimensionality greater than two, linear programming is required (Luce & Raiffa, 1957). Linear programming is a technique for maximizing and mini­ mizing linear functions that are subject to a number of linear constraints and nonnegativity conditions. Since the value of a game can easily be expressed as a linear function, and the rows and columns of a game matrix can serve as constraints, linear programming may be utilized to determine the vectors of probabilities x and y that, respectively, maximize and minimize the value of a two-person zero-sum game. Although linear programming computer programs have been available for some time, they are typically written for general linear programming applications. Many of the programs (such as MPOS,t for example) are large canned packages capable of solving a wide variety of linear programming problems. Since the primary application of linear programming in psychol­ ogy is for solving zero-sum games, the rationale for implementing a large general-purpose optimization package for solving a small class of problems is question­ able. These general programs require manual conversion of a game problem to a linear programming problem and manual conversion of a linear programming solution back to a game solution. Since linear programming is frequently used in business applications, many programs

Commitment to Future Interaction and Relative Power Under Conditions of Interdependence

A 2 (committed or uncommitted) × 2 (high power or low power) × 3 (structure: Prisoner's Dilemma or Chicken or No Conflict) factorial design was used to examine the effects of commitment to future interaction, relative power, and the structure of interdependence. The dependent variable was the proportion of cooperative choices on a series of single-play game matrices. Data were collected from 120 undergraduate students in a laboratory situation. The task consisted of responding either cooperatively or competitively to 18 game matrices. Each matrix was presented individually and without feedback concerning previous outcomes. As hypothesized, the results showed an interaction between commitment and power. Low reward power subjects were more cooperative when uncommitted than when committed in Chicken and the No Conflict game. High power subjects were more cooperative when committed than when uncommitted in the Prisoner's Dilemma. These results suggest caution when generalizing from the prevalent propositions that low power and commitment to future interaction have a cooperative effect.

Analysis of the Individualistic, Competitive and Cooperative Motives in Non-Zero-Sum Games

According to Messick and McClintock (1968), differences in choice behavior of strategy games of the non-zero-sum type may be explained mainly by three motives: the individualistic, the competitive, and the cooperative. The researchers' operational definitions of the motives are based on the payoffs in the game matrices. This article critically examines Messick and Mc-Clintock's expositions and demonstrates that the payoff consideration cannot be the sole criterion for the identification of motivational goals. Disregarding the opponent's choice may lead to mistaken conclusions concerning the participant's motive as inferred from his decision. In the wake of this oversight, the proposal for measuring the three motives, stated in this article, is based on the following principles: (1) A pre-programmed plan for one participant in the game in order to standardize the situation the subjects face. (2) A large number of trials in order to ensure the subject's awareness of the opponent's fixed strategy. (3) The combination of 1 and 2 with appropriate payoff values enables the construction of the conflict situation confronting the subject.

An experimental study of interethnic competition in segregated schools

Decomposed matrix games were used to study interethnic competition in the triethnic school system of a southwestern city. The sample consisted of sixth-grade males from segragated schools. In addition to the data from the matrix games, brief questionnaire measures of internal vs. external locus of control, self-esteem, and authoritarianism were also obtained. The results on the game matrices indicated that the anglos competed more and were less trusting than the blacks or chicanos. The questionnaire results indicated that the blacks were highest on feelings of external control and on authoritarianism, while the chicanos were lowest in self-esteem. Cooperative interethnic work groups were suggested as a means of coping with the problems that these differences might be expected to cause in integrated schools.

Responsiveness in two-person zero-sum games

Thirty-six subjects participated in two-person zero-sum games played against an experimenter in an examination of the effect of different types of game matrices and different mixed strategies on behavior. Results showed that when the opponent played nonoptimally, subjects were able to detect this nonoptimality and to exploit it to their own benefit. When the opponent played according to the minimax prescription, subjects' performance was not optimal but was sufficiently and consistently close to it that arguments in terms of differences between perceived and objective probabilities provide an attractive explanation for the differences.

On Decision Theory Under Competition

Decision theory treats models in which a decision maker must choose an information function and a decision function, where the former transforms states of nature into messages, the latter transforms messages into actions, and the ultimate payoff depends upon the state of nature, the message, and the action. This paper treats similar models in which nature's role is played by a selfseeking rational opponent. These models fall into the purview of the theory of two-person games, and the main results concern the maximin and minimax theories of these games. The various games treated differ in the assumptions made about the decision maker's ability to probability mix his choices and about his opponent's information concerning these choices. Twenty different sets of assumptions are considered, but it is shown that the maximin and minimax theories of the resulting games are all reducible in substance to those of just three games: (i) in which the decision maker cannot probability mix his choices at all, and his opponent observes both choices; (ii) in which the decision maker can mix arbitrarily, and his opponent observes neither of his choices; or (iii) in which the decision maker can mix decision functions only, and his opponent observes only the information function. These games are represented in terms of matrix games, and the matrices are compressed by the deletion of dominated strategies for the decision maker. Further compressions are obtained for the case in which the cost of information is separable from the payoff resulting from the actions of the decision maker and his opponent. The game matrices are finally reduced to manageable proportions for the further specialized cases in which either the capacity of the decision maker's information channel is significantly limited and information cost is constant, or the capacity of the information channel is large and there is one low—cost “non-message” available. A numerical example is given, and the concluding comments point to directions for further research.

The Probability that a Random Game is Unfair

To every matrix $A$ of real valued payoffs there corresponds a unique quantity $V(A)$, the value of a game with payoff matrix $A$, defined by $V(A) = \min_y \max_x x^t Ay = \max_x \min_y x^t Ay,$ where the maximum and minimum are taken over all $x$ and $y$ having nonnegative components summing to one. We wish to show that, for a relatively wide class of probability distributions on the class of all $m \times n$ game matrices, the probability $P(m, n)$ that the value of a random game is greater than zero is given by the cumulative binomial distribution $P(m, n) = (\frac{1}{2})^{m + n - 1} \sum^{m - 1}_{k = 0} \binom{m + n -1}{k}.$ The observation that, for any $\epsilon > 0$, $\lim_{m \rightarrow \infty} P(m, (1 + \epsilon)m) = 0, P(m, m) = \frac{1}{2},\quad \text{for all m}, \lim_{m \rightarrow \infty} P(m, (1 - \epsilon)m) = 1,$ suggests that large rectangular games tend to be strongly biased in favor of the player having the greater number of alternatives. Thus we characterize the extent to which the nonsquareness of a game is reflected in a bias for one of the players. Our method involves an application of a theorem in combinatorial geometry due to Schlafli [5]. A discussion of the consequences of Schlafli's theorem in geometrical probability is given in [2], [3], [9].

Some reflections on the design of game theoretic models for the study of negotiation and threats

One of the greatest sources of misunderstanding in the social sciences is the inability of many political scientists, sociologists, generalists, and a great many economists and psychologists, to understand what is meant by a formal model, and to comprehend what constitutes a legitimate or an unsound use of that model. On certain occasions, more mathematically inclined social scientists may use simple examples to illustrate methodology. Unfortunately, these examples can easily be misconstrued and given significance and content matter far beyond the initial intent of the expositor. Thus, in the past few years, there has been a rash of writing using two by two game matrices as a basis for discussions of threat and deterrence policies. It is unlikely that any international power situation can be adequately modelled for any purpose other than very simplified teaching by a two by two matrix.