Game Theory

[Racoon]

Was wondering and reading about Game Theory.

everyone I assume has seen "A Beautiful Mind",

but John von Neumann [1903 - 1957] also had a few things to say:

 

Life is a game strategy as well.

Strategy as "that course of action, a series of plays, which guarantees minimum losses".

 

The Minimax Theorem : Using mathmatical matrices

..."Examining the outcomes of play at any stage, it's always possible to point out at least one value that is the best of the possible bad results, and at least one that is the least of all good outcomes"...

They are "Maxmin" & "Minmax"

 

Thoughts on Saddle Points?

'commuting' matrices?

Turtle has/knows a Game Theory called "Tit for Tat"

 

More later..., ;) :) :esmoking:

[Racoon]

According to von Neumann there are 2 classifications of Game;

Zero-sum or Nonzero-sum...

 

When 1 players gain's can only be made up from another players losses, they are competing for a larger or smaller slices of the same prize - a Zero-sum

 

On the other hand, if in playing the game everyone's stake is somehow made to grow, it's conceivable all players could finish ahead of the game - a Nonzero-sum.

 

This can be especially important and likened to Economics. :)

 

How can Game theory be applied to games like Chess?

To important economic considerations?

To games where Chance and Bluff come into play? (like in poker)

 

;) ;) :)

[Racoon]

__Very fascinated with Game theory now

 

__ Imagine Sherlock Holmes vs. Dr. Moriarty, each attempting to anticipate the moves of the other in a train pursuit! ;) :)

 

__How can you use Game theory to your advantage?

Economic advantage?

[CraigD]

I had a few semester hours of Game Theory as an undergrad, and have used it for a few practical applications since then. Here are a few comments and answers

…Turtle has/knows a Game Theory called "Tit for Tat"

”Tit for Tat” is the common name for a well known strategy for the itterated (the strategy is allowed to use information about previous plays of the game) version of the non-zero sum game “Prisoners Dilemma”. It’s a very simple strategy, consisting of the rule “begin by cooperating, then do whatever your opponent did in the previous turn” (in PD, one has only 2 choices, “cooperate” or “defect”). Despite its simplicity, it’s a very effective, typically besting much more complicated ones in various “tournament” situations.

 

PD is one of the most extensively studied games in existence – one could spend days just getting acquainted with its literature.

How can Game theory be applied to games like Chess?

It’s trivial to prove that a variant of Chess that always has a winner – that is, one with some rule to settle a stalemate – has an optimal strategy. Because Chess has so many possible board states, representing a strategy in the usual way – a table of state:next move pairs – is impractical. A straightforward minmax approach to calculating this vast strategy, while not difficult, is computationally impractical, requiring essentially that one play out every possible game of chess.

 

Perhaps some day, computation and/or data storage will become so cheap that the optimal strategy for Chess will be calculated. For now, chess strategies remain an art, more intuition and an ability to recognize and apply “book” knowledge than Mathematics.

To important economic considerations?

Along with a related field, linear programming, game theory can and has been applied to economic problems of many kinds. Notably, a logistics team headed by none other than John von Neumann are thought to have made critical contributions to the Berlin airlift relief efforts following WW2.

 

Large economic models are typically intractable to pure mathematical game theory approaches.

 

Most complicated games suffering from the lack of realism of a key assumption of the minmax theorem: that one’s opponent can be expected to play according to an optimal strategy. A more sophisticated approach that takes into account the ways that a realistic opponents strategy diverges from the optimal can often produce more effective strategies: for example, if I know that my opponent in a game of chess consistently makes the same tactical error, I can exploit that to have a winning strategy that would not win against another player who does not make the same error

To games where Chance and Bluff come into play? (like in poker)

In game theory, games of chance are considered to have imperfect information. This complicates their game theory approaches.

 

Games were the optimal strategy is not a simple “when A, always do B”, but rather a mixed strategy of “when A, do B PB percent of the time, C PC percent of the time, etc.” are common even for very simple games with perfect information.

 

Having more perfect information than one’s opponent usually give one an advantage. The ability to bluff, and tell when an opponent is or isn’t bluffing, work toward this advantage.

 

There is, of course, much more to game theory than a short post can begin to outline. It’s a fascinating field, requiring only relatively simple mathematics. I’d recommend reading a textbook on it – the one I studied was McKinsey’s “Introduction to the Theory of Games” http://www.amazon.com/gp/product/0486428117, though, not having read any other, I can’t judge it relative goodness. If you’ve not studied basic linear algebra, you’ll likely need to, before or at the same time you study game theory.

[Racoon]

I had a few semester hours of Game Theory as an undergrad, and have used it for a few practical applications since then. Here are a few comments and answers

 

There is, of course, much more to game theory than a short post can begin to outline. It’s a fascinating field, requiring only relatively simple mathematics. I’d recommend reading a textbook on it – the one I studied was McKinsey’s “Introduction to the Theory of Games”, though, not having read any other, I can’t judge it relative goodness. If you’ve not studied basic linear algebra, you’ll likely need to, before or at the same time you study game theory.

 

Nice CraigD!

you indeed are a bright and shining star. :cup: :cup:

(read above post)

Have had Math up to College Algebra 111.

But I am no John von Neumann! :cup:

 

Chess would need a staggering size of computations to materialize an actual strategy. And if/when it does, Humans i think would toss chess into the heap of things not worth doing anymore. :hihi:

 

Game theory cannot prescribe a theory when none exist.:(

In a game of pure chance with identical odds on all outcomes- like flipping a coin- game theory can't improve anyones expectations by some mathmatical sleight of hand.

 

The original book published by von Neumann and Morgenstern back in 1943 is "Theory of Games and Economic Behavior"

 

An Oldie but Goodie,

[InfiniteNow]

Heck with chess (although, it is fun in it's own right)... play Go. :hihi:

[CraigD]

Game theory cannot prescribe a theory when none exist.:cup:

In a game of pure chance with identical odds on all outcomes- like flipping a coin- game theory can't improve anyones expectations by some mathmatical sleight of hand.

For a slight variation of the usual game of coin flipping, where one player chooses heads or tails before a coin that the other player has previously placed heads up or down, is revealed – the minmax theorem does prescribe an optimal strategy: both players should play randomly, equally choosing heads or tails. If either player deviates from this strategy, the other can choose a strategy that will result in him tending to win more than half of the tosses.

 

For this coin flip game, which has payoff matrix

|1 -1|

|-1 1|

, the optimal mixed strategy for both players,

(.5,.5)

, is trivial, and just confirms common sense – since there’s no “pattern” to the game, both should play it “purely randomly”. However, a slight variation such as

|2 -1|

|-4 3|

(that is, player 1 wins $2 if he picks head and is correct, loses $1 if he picks head and is incorrect, wins $3 if he picks tails and is correct, and loses $4 if he picks tails and is incorrect), has an optimal strategy for player 1 of

(.7,.3)

and for player 2,

(.4,.6)

.

 

It’s easy, using ordinary rather than matrix algebra, to see why this is so. Just write the expression for the expected payoff of the game in terms of the probability of player 1 choosing heads (A) and player 2 placing the coin heads up (:hihi::

2AB –A(1-:cup: -4(1-A)B +3(1-A)(1-:cup:

Then get it into a form x(A -p)(B -q) +y:

2AB +AB –A -4B +4AB +3 -3B -3A +3AB

10AB -4A -7B +3

10(A -7/10)(B -4/10) +(3 -28/10)

10(A -7/10)(B -4/10) +2/10

 

If either player uses their optimal strategy, the game will have an average payoff of .2 (player 1 wins $0.20). If either player uses something other than their optimal strategy, the other can chose a strategy so that the game’s average payoff is anywhere from -4 (player 1 loses $4) to 3 (player 1 wins $3).

[ughaibu]

Does this suggest anything other than level stakes as an optimal staking plan for backers of horses?