MQE Mixed strategy QoE equilibrium

MQE, or Mixed-Strategy Quality-of-Experience (QoE) Equilibrium, is a concept used in game theory to analyze how players make decisions and allocate resources in situations where the outcome is influenced by the players' subjective experiences.

In a game, players are typically assumed to be rational and seek to maximize their expected utility, which is a measure of how much they value the potential outcomes of the game. However, in many situations, players' utility is not only determined by the objective outcomes of the game, but also by their subjective experiences, such as their satisfaction with the game, their level of stress, or their level of frustration.

MQE takes into account the subjective experiences of players and their impact on decision-making in games. It assumes that players have multiple strategies available to them, and each strategy leads to a different QoE. QoE can be defined as the subjective evaluation of the quality of the experience that the player has while playing the game, which can include factors such as the level of excitement, the level of frustration, the level of challenge, or the level of engagement.

In a game with two players, player A and player B, each player has a set of strategies to choose from. Player A has a set of strategies {a1, a2, ..., am}, and player B has a set of strategies {b1, b2, ..., bn}. The QoE of player A when playing strategy ai against player B's strategy bj is denoted by Qi,j(a,b), and the QoE of player B is denoted by Qj,i(b,a).

In a mixed-strategy equilibrium, each player chooses a probability distribution over their set of strategies, rather than selecting a single strategy. The probability that player A selects strategy ai is denoted by x_i, and the probability that player B selects strategy bj is denoted by y_j. The mixed-strategy equilibrium is defined as the pair of probability distributions (x*, y*) such that:

Player A's expected utility is maximized for any response by player B:scssCopy codemax_j [Qi,j(ai, bj) * y_j] = u_A(x*, y*)

Player B's expected utility is maximized for any response by player A:scssCopy codemax_i [Qj,i(bj, ai) * x_i] = u_B(x*, y*)

The probabilities x* and y* are themselves probabilities, meaning that they must sum to 1:javaCopy codesum_i x_i = 1sum_j y_j = 1

MQE is a generalization of Nash equilibrium, which is a concept that describes a situation where each player's strategy is a best response to the strategies chosen by the other players. In MQE, each player's mixed-strategy is a best response to the mixed-strategy chosen by the other player, considering the QoE associated with each possible combination of strategies.

MQE can be applied to a wide range of games and situations, such as online advertising, video games, or social networks. For example, in an online advertising scenario, advertisers may compete for the attention of users by choosing different ads to display. Each ad may have a different QoE for the user, depending on factors such as relevance, attractiveness, or annoyance. MQE can be used to model how advertisers should allocate their budgets across different ads to maximize their expected utility, taking into account the QoE of the user for each ad.

Similarly, in a video game, players may have different strategies available to them, such as attacking, defending, or avoiding. Each strategy may have a different QoE for the player, depending on factors such as the level of challenge, the level of reward, or the level of frustration Certainly! In the context of a video game, MQE can be used to analyze how players should allocate their resources, such as time, energy, or in-game currency, across different strategies to maximize their expected utility, considering the subjective experiences associated with each strategy.

To determine the mixed-strategy equilibrium, the players' utilities need to be quantified. This can be done through various methods, such as surveys, user feedback, or experimental studies. By collecting data on players' subjective experiences and preferences, researchers can estimate the QoE functions (Qi,j and Qj,i) that capture the relationship between strategies and subjective experiences.

Once the QoE functions are estimated, the next step is to find the mixed-strategy equilibrium. This can be achieved through mathematical optimization techniques, such as linear programming or numerical algorithms. The goal is to find the probability distributions (x*, y*) that satisfy the three conditions mentioned earlier: maximizing the expected utility for each player and summing to 1.

Finding the mixed-strategy equilibrium requires solving a system of equations, which can be computationally intensive, especially for complex games with a large number of strategies. Researchers often use iterative algorithms, such as the Lemke-Howson algorithm or the support enumeration algorithm, to find the equilibrium efficiently.

Once the mixed-strategy equilibrium is obtained, it provides valuable insights into the decision-making process of the players. It reveals the optimal allocation of resources and the probabilities associated with each strategy, taking into account the subjective experiences of the players. This information can be used to design better game mechanics, improve user satisfaction, or optimize resource allocation in various applications.

It is important to note that the concept of MQE relies on the assumption that players are rational decision-makers who seek to maximize their expected utility. However, in practice, players' behaviors may deviate from this assumption due to factors like bounded rationality, emotions, or social influences. Therefore, MQE serves as a useful theoretical framework but may not fully capture the complexity of real-world decision-making.

Furthermore, QoE is a subjective concept, and different players may have different preferences and experiences. Accounting for individual differences and personalization is an ongoing challenge in the application of MQE. Advanced techniques, such as machine learning algorithms or user profiling, can be employed to address these challenges and tailor the analysis to individual players.

In conclusion, MQE is a concept in game theory that extends the notion of equilibrium to consider players' subjective experiences, quantified as Quality-of-Experience (QoE). By analyzing mixed strategies and their associated QoE, MQE provides insights into how players make decisions and allocate resources in games. It has applications in various domains, such as online advertising, video games, and social networks, where subjective experiences play a crucial role in user engagement and satisfaction.