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# John F. Nash

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Have you guys seen the movie "A beatiful mind"? It is fantastic. john nash won the nobel prize because of his "game theory" applied to economics. i wonder what can we use for game design. more: http://cepa.newschool.edu/het/schools/game.htm http://www.amiganr1.com

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The 'game theory' being that the most beneficial thing for an individual to do within a group is that which brings himself AND the group the best result?

EDIT: That was what I'd gathered from the movie, but I'll make sure to read more info on the site you listed.

[edited by - Silvermyst on March 19, 2002 1:11:37 PM]

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While watching the film, I was thinking about how it could apply to unit AI in RTS''s.....

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yes, that´s what i mean. if it can we used in AI for games or to design game rules etc.

http://www.princeton.edu/~mdaniels/PD/PD.html

it is called The Prisoner''s Dilemma and it is a very interesting and strategic game.

what do you think?

http://www.amiganr1.com

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Disclaimer: I can''t program and know nothing about Math or Game Theory, so please take nothing I say seriously, as I am the dumbest person ever to walk the earth.

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You can visit http://www.gametheory.net/ , it looks pretty informative.

Game Theory is generally attributed to Von Neumann I believe. What Nash invented was the Nash Equilibrium. The Nash Equilibrium is (I forget the exact definition) all the cominations of strategies that make sense to be rationally played.

For example, consider the following matrix. a b c are stratgies for player 1, xyz strategies for player 2. The numbers represent payoff to player 1, which player 2 is trying to minimize :

a b c
x 0 3 6
y 1 9 7
z 9 2 8

Now, remember player 2 is trying to minimize these numbers. Player 2 should NEVER play strategy y. Because y is strictly worse than x. No matter what strategy player 1 chooses, strategy x is better for player 2 that strategy y. So we ignore strategy y, and get a new matrix of rational choices:

a b c
x 0 3 6
z 9 2 8

Now, note that for p1, strategy c is ALWAYS better than b. (p1 is trying to maximize values). Wherher or not p2 chooses x or z, player2 is better with c than b.

So:

a c
x 0 6
z 9 8

At this point, strategy x is better for player 2 than z, in all cases. (p2 is tring to minimize) So p2 should play x.

a c
x 0 6

Which means that c should play c. So, if both players play rationally, there is a single Nash Equilibrium at (c,x)=6. it is possible to have more than one Nash Equilibrium, there just happened to be one here.

Now, about Prisoner''s Dilemma. The Prisoners Dilemma is this: 2 guys are arrested. If both stay quiet, they both get 3 years in jail. If one rats out his friend, the talker gets only 1 year, but the guy who kept quiet gets 10. If they BOTH rat out each other, they each get 7 years.

Prisoners Dilemma Matrix: T stand for Talk, DT for Don''t Talk. First values are years player 1 gets, second are years player 2 gets. So, for DT/T, p1 gets 10 years, p2 gets 1.

T DT
T 7/7 10/1
DT 1/10 3

To simplify, just look at the values for player one:
T DT
T 7 10
DT 1 3

Player 1 wants to minimize his years in jail. The strategy talk gives him 7 or 1 years, the strategy DT gives him 10 or 3. So, T is the better strategy in all cases.

The same logic applies to player 2, his best strategy is also to talk. (Situation is symmetric) So they both talk, both get 7 years in jail.

But wait! If they had both kept quiet, they would have gotten only 3 years in jail each! That is what makes prisoner''s dilemma interesting. By both playing the "best" strategies, they did worse than if they had both played other strategies.

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Game theory is pretty interesting in general. There are a lot of neat things. For example, most people know you can cut a cake with 2 people so that both are happy. Well, you can actually do it with any number of people. (Not very realistically though) Or, if you are looking to split an apartment with roomates, you can assure that when splitting rooms each person thinks the deal they are getting is equal to or better than any other deal.

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Game Theory sounds like it would be very useful for AI, and it probably could be. But also it is useful for the game design as well. For example, if the original matrix in the first example represented possible strategies, we see that even though each side has 3 strategies realistically the game devolves into utter predictability. So while the game has the appearance of depth, most of the strategies (or units, or whatever) end up being worthless.

Studying game theory may help you think more analytically about those types of issues, and help with rule formation, evaluating and building strategies into games, balancing units, etc. I would guess that in many games if you made a matrix of say RTS units you would find you can do the same sort of reduction and arrive at the conclusion that certain units add no value to the game.

Obviously you can get very technical, but exact quantifications are never that easy. But at least it will set you on the right direction of things to look out for. A good way of putting it is that the percieved complexity of a game may be very different from the actual complexity. An example is tic-tac-toe. Although you have the illusion of choice, there is a best way to play that always ensures a tie, making for a very boring game.

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Of course, I''m just a caveman, so take the above with the appropriate grain(s) of salt.

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quote:
Original post by Silvermyst
The ''game theory'' being that the most beneficial thing for an individual to do within a group is that which brings himself AND the group the best result?

Game theory is just generally the study of situations where different entities are engaged in some sort of activity and they can choose from a set of strategies to play. These situations are usually competitive, although they don''t have to be.

What game theory is is a collection of techniques and observations for giving a good answer to the question "given this game and these goals, what should the players do?" The answer will change from game to game, game theory is just the study of ways of finding those answers.

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a b c are stratgies for player 1, xyz strategies for player 2. The numbers represent payoff to player 1, which player 2 is trying to minimize :

a b c
x 0 3 6
y 1 9 7
z 9 2 8

Now, remember player 2 is trying to minimize these numbers. Player 2 should NEVER play strategy y. Because y is strictly worse than x. No matter what strategy player 1 chooses, strategy x is better for player 2 that strategy y. So we ignore strategy y, and get a new matrix of rational choices:

wait, wait. what about z? if p2 uses z against p1, then p1 always gets results equal to or better than those from p2 using y. why wouldn''t p2 want to disregard this strategy and keep the y?

the rest of your examples made sense to me, but this was confusing

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It wans''t game theory, that was john von neumon

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A Beautiful Mind = revisionist history, the only thing Hollywood is good for. Plus Jennifer Connoley is hot.

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quote:
Original post by Retro_Joe

a b c are stratgies for player 1, xyz strategies for player 2. The numbers represent payoff to player 1, which player 2 is trying to minimize :

a b c
x 0 3 6
y 1 9 7
z 9 2 8

...

wait, wait. what about z? if p2 uses z against p1, then p1 always gets results equal to or better than those from p2 using y. why wouldn''t p2 want to disregard this strategy and keep the y?

Player 2 wants to minimize the numbers. X is always less than Y, no matter what player 1 chooses. However, while z looks pretty bad overall, take a look a column b. The z value for column b is 2, which is less than the x or y value. So, if player 1 chose b, z is the best option for player 2. So z is a valid choice under some circumstances, at least as long as b is a valid choice for player 1.

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