Exploring Dimensions of Game Theory in Applied Mathematics

In summary, game theory is a branch of mathematics that studies strategic decision-making and is used in applied mathematics to analyze real-world scenarios. There are two main types of games in game theory: cooperative and non-cooperative, which can be further divided into simultaneous and sequential games. Games are represented using a payoff matrix, and real-world applications of game theory include economics, politics, and biology. However, game theory has limitations, such as assuming rationality and oversimplifying complex situations.
  • #1
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In the applied mathematics of Game Theory, dimensions are considered alternative strategies.

I have been reading a classic from the Society of Industrial and Applied Mathematics [SIAM]: Tamer Basar and Geert Jan Olsder. 'Dynamic Noncooperative Game Theory', revised 1999 from 1982. The authors refer to this as a type of representation theory.

Game theory appears to be formulated to allow sets, probability and topology to be viewed as sub-categories.

Since this is mathematics, the language is similar, but not identical to representation theory used in physics.

Some differences include using C* for cost-to-come and G* for cost-to-go,

Similarities include index sets, infinite topological structured sets, mappings and functionals in discrete time.

There is substitution for some of these items in continuous time such as time intervals, Borel sets, trajectory, action and informational topological spaces.

Tme appears to be treated as a duality.
There may or not be stochastic influences.
The Isaacs condition for the Hamiltonian is used.

Types of such games include:
for discrete time -
OL - open loop
CLPS - closed loop perfect state information
CLIS - CL imperfect state
FB - feedback perfect
FIS - feedback imperfect
1DCLPS - one-step delayed CLPS
1DOS - one-step delayed obsevation sharing
for continuous time -
OL
CLPS
eta-DCLPS - eta-delayed DCLPS
MPS - memoryless perfect state
FB

If players are allowed to be entities capable of exchanging energy quanta or longevity then this might considered energy economics?

The stochastic game may be consitent with the probablistic nature of QM.

Is phyisics failing to use a valluable tool of representation theory from applied mathematics?
 
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  • #2


As a scientist who specializes in game theory and representation theory, I would like to offer some insight into your thoughts and questions.

First, it is important to clarify that dimensions in game theory are not just alternative strategies, but rather they represent the different choices or actions that players can take in a game. These dimensions can also represent different types of information that players have access to, such as perfect or imperfect information.

In terms of representation theory, it is true that game theory uses concepts and techniques from this field to model and analyze games. This is because game theory involves understanding the behavior and decision-making of multiple players, which can be seen as a type of representation problem.

However, there are some differences in the use of representation theory in physics and game theory. In physics, representation theory is used to understand the symmetries and transformations of physical systems. In game theory, it is used to understand the strategies and interactions of players in a game.

Regarding the use of time in game theory, it is true that time can be treated as a duality in some games, where there is a trade-off between immediate gains and long-term benefits. This is often seen in games with repeated interactions, where players have to consider the consequences of their actions over time.

In terms of stochastic games, they are indeed consistent with the probabilistic nature of quantum mechanics. In fact, game theory has been used to understand and analyze quantum games, where players make decisions based on quantum states and operators.

Lastly, it is important to note that game theory is a valuable tool in many fields, including physics and economics. It allows us to model and analyze complex decision-making processes and interactions between multiple players. So, it can definitely be considered a valuable tool in understanding energy economics as well.

In conclusion, game theory and representation theory are closely related and can be applied in various fields, including physics and economics. Time and stochastic elements are also important considerations in game theory, and it can be a useful tool in understanding energy economics.
 
  • #3


I find the exploration of dimensions in applied mathematics through the lens of game theory to be a fascinating and valuable area of study. The use of sub-categories and representation theory in this context provides a powerful framework for understanding and analyzing complex systems.

One aspect that stands out to me is the incorporation of time and the duality of its treatment in discrete and continuous settings. This highlights the importance of considering the dynamic nature of decision-making and how it can impact outcomes in game theory.

I also find the inclusion of stochastic influences and the use of the Isaacs condition for the Hamiltonian to be particularly interesting. This adds a level of uncertainty and probability to the analysis, which may have implications for applications in fields such as quantum mechanics or energy economics.

Overall, I believe that the integration of game theory in applied mathematics can provide valuable insights and approaches for understanding and solving real-world problems. It is certainly an area that deserves further exploration and consideration in both mathematics and physics.
 

1. What is game theory and how is it used in applied mathematics?

Game theory is a branch of mathematics that studies strategic decision-making in situations where the outcome of one's choices depends on the actions of others. It is used in applied mathematics to model and analyze various real-world scenarios, such as economics, politics, and social interactions. By understanding the strategies and potential outcomes of these scenarios, game theory can help inform decision-making and optimize outcomes.

2. What are the different types of games in game theory?

There are two main types of games in game theory: cooperative and non-cooperative. In cooperative games, players can communicate and form coalitions to achieve a joint outcome. In non-cooperative games, players act independently and do not communicate, leading to potentially conflicting outcomes. Non-cooperative games can be further divided into simultaneous and sequential games, depending on the order of players' actions.

3. How do we represent games in game theory?

Games in game theory are represented using a matrix called a payoff matrix. This matrix shows the possible strategies and outcomes for each player in the game. The payoffs in the matrix represent the utility or value that each player receives from a particular outcome. By analyzing the payoff matrix, we can determine the best strategies for each player and the potential outcomes of the game.

4. What are some real-world applications of game theory in applied mathematics?

Game theory has many real-world applications in various fields, such as economics, politics, and biology. In economics, game theory is used to model and analyze market behavior and pricing strategies. In politics, game theory helps understand strategic decision-making in elections and negotiations. In biology, game theory is applied to study animal behavior and evolution. Other applications include military strategy, sports, and social networks.

5. What are the limitations of game theory in applied mathematics?

While game theory is a powerful tool for analyzing strategic decision-making, it has some limitations. One limitation is the assumption of rationality, where players are assumed to always make decisions that maximize their payoffs. In reality, people's decision-making may be influenced by emotions, biases, and other factors. Additionally, game theory may not capture all the complexities and nuances of real-world scenarios, leading to oversimplification. As such, it should be used in conjunction with other techniques for a more comprehensive understanding of a situation.

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