Game Theory - A graphical solution to all 2-person n x n strategy games?

In summary, the conversation discusses the use of the prism method in game theory for representing 2-person non-zero-sum games with 2n and 3n strategies. While the method is applicable and useful for 2n strategy games, it becomes more complex and less practical for higher n strategy games. Other methods may be more effective in these cases.
  • #1
hyrodi
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I was not sure where to post this. It didn't seem to fit in the homework/self-study category, and I believe it is thought-provoking enough to warrant posting in the general forums. I am sorry if I am wrong.

In game theory, 2-person non-zero-sum games with 2n strategies can be graphically solved by by placing the two strategies of player 1 on two endpoints of a line segment that has a length of one, then graphing the expected values of player 2s strategies as lines from a corresponding y-value above one end of the line segment to another. Where these lines intersect and form either an upper or lower envelope, the corresponding x-value denotes the optimal mixed strategy.
My question is, can 2-person 3n strategy games also be represented in this manner by forming a triangular prism in which the vertices of a base are the three strategies for player 1, expected values are graphed as lines between any two of the three player 1 strategies, and various triangular planes are formed, the corresponding xy coordinate value of the intersection of these planes and their respective distances to the vertices denoting the optimal mixed strategies?
Further developing this intuitive line of thought, could any 2-person non-zero-sum game be theoretically represented by an n-polygonal prism, with the intersections of the various planes formed denoting the solution? Though I lack the knowledge in topology to pursue the topic further, I imagine that the shape of any of the polygons past a triangle would assume a "soap bubble" - like behavior once the vertices of the polygon would not form a flat plane?
Help would be appreciated.
 
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  • #2


Thank you for your thought-provoking post. I find game theory to be a fascinating and useful tool for understanding decision-making and strategic interactions. To answer your question, yes, it is possible to represent 2-person 3n strategy games in the manner you have described. In fact, this method is known as the "prism method" in game theory literature.

The prism method is an extension of the "line segment method" you described for 2n strategy games. It allows for a more visual and intuitive representation of the game, making it easier to identify optimal strategies. The idea of using a triangular prism to represent 3n strategy games is a valid approach and has been used in some game theory studies.

However, as you have correctly pointed out, the complexity of representing higher n strategy games increases significantly. While it may be possible to theoretically represent any 2-person non-zero-sum game with an n-polygonal prism, the practicality and usefulness of such a representation may be limited. This is because the number of planes and intersections would increase exponentially with the number of strategies, making it difficult to visualize and analyze the game.

In addition, as you mentioned, the shape of the prism may not always be a flat plane, especially for higher n strategy games. This can make it challenging to accurately determine the optimal mixed strategies.

Overall, the prism method is a useful tool for understanding and analyzing 2-person non-zero-sum games. However, for higher n strategy games, other methods such as the linear programming approach may be more effective. I hope this helps answer your question and provides some insight into the representation of games with a larger number of strategies. Thank you for your contribution to the forum.
 

What is game theory?

Game theory is a mathematical framework used to analyze decision-making and strategic interactions between rational individuals.

What is a graphical solution to a game?

A graphical solution to a game is a visual representation of all possible outcomes and strategies in a game, using graphs or diagrams.

What is a 2-person n x n strategy game?

A 2-person n x n strategy game is a game in which there are two players who have n strategies each, and the outcome of the game depends on the strategies chosen by both players.

How is a game solved using a graphical solution?

A game is solved using a graphical solution by identifying the dominant strategies of each player and finding the Nash equilibrium, which is the best possible outcome for both players.

What is the significance of game theory in real-world applications?

Game theory has various applications in fields such as economics, political science, and biology, as it provides a framework for understanding and predicting the behavior of individuals and groups in strategic situations.

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